POLITECNICO DI TORINO

Master of Science in Mechanical Engineering

Analysis and Structural Optimization of a

Mechanical Component for the Heavy Industry

Supervisors:

Illustrious Prof. Dr. Eugenio Brusa

Illustrious V. P. Andrea De Luca (Company Tutor)

Speaker:

Alessandro Musu

Academic Year 2015-2016

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Dedico questo mio modesto lavoro che si pone a conclusione di un lungo e faticoso cammino

alle persone che hanno reso possibile che tutto ciò si avverasse.

Ai miei genitori, alle mie sorelle Giulia e Valentina, per avermi aiutato e incoraggiato

nelle difficoltà incontrate in questi anni.

Ai miei amici più cari, per non aver mai smesso di credere in me.

Al mio relatore, Prof. Eugenio Brusa,

per la disponibilità, la pazienza, l’incoraggiamento e per avermi aiutato nella stesura di questa tesi.

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Table of Contents

Preface…………………………………………………………………………...…………………………………..7

1. Introduction to Structural Optimization…..….……………………………………………11

1.1. Structural Optimization in the Design Process..….………………………..…………15

2. Shape Optimization………….....…………………………………………………………………...17

2.1. Shape Optimization Methods……………………………...……………..…………………..19

3. Topology Optimization.…………..………………………………………………………………..21

3.1. Minimum Compliance Design Formulation……………………..…………...…………..21

3.2. Conditions of Optimality……………………………………………..……………..……………26

3.3. Computational Procedure………………………………………………………….……………29

3.4. Analysis Refinement and Issues……………………………………..………………………34

3.5. Bi-Directional Evolutionary Structural Optimization Method…………..…….…36

3.6. BESO with Material Interpolation Scheme and Penalization……………..……....45

3.7. BESO for Extended Topology Optimization Problems……………........................47

3.7.1. Minimizing Structural Volume with a Displacement Constraint...........48

3.7.2. Topology Optimization for Natural Frequency...………………...................51

3.7.3. Topology Optimization for Multiple Load Cases...………………..……….…54

3.7.4. BESO based on von Mises Stress…………………………………………………..54

4. Structural Optimization with Inspire………………………………………………………57

4.1. Optimization Terminology and Definitions…………………………………………...63

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5. Structural Optimization: a Practical Example…………………………………………...65

5.1. DANIELI’s SRW 18 Guide System Description…………………….…………………….67

5.2. Loads and Constraints Analysis………………...…………………………………………….72

5.3. Finite Element Analysis of the Current Roller Holder……………………………….75

5.4. Structural Optimization Setting………………...…………………………………………….79

5.5. Maximize Stiffness - Results………………...…………………………………..………..…….87

5.5.1. Free Shape Optimization, Stainless Steel 316L.……….…….......……………87

5.5.2. Extrusion along Z-axis Optimization, Stainless Steel 316L.…..…...……..89

5.5.3. Extrusion along Y-axis Optimization, Stainless Steel 316L.…….………..90

5.5.4. Free Shape Optimization, CoCrMo Alloy.…….....………..………...……………92

5.5.5. Extrusion along Z-axis Optimization, CoCrMo Alloy….…..…..………..…...93

5.5.6. Extrusion along Y-axis Optimization, CoCrMo Alloy……..…..….………..94

5.5.7. Free Shape Optimization, Ti6Al4V-ELI Alloy………..................……………96

5.5.8. Extrusion along Z-axis Optimization, Ti6Al4V-ELI Alloy..………………..97

5.5.9. Extrusion along Y-axis Optimization, Ti6Al4V-ELI Alloy...….…..………..99

5.6. Minimize Mass - Results.………………………………………………….……………………101

5.6.1. Free Shape Optimization, Stainless Steel 316L…....………………………..101

5.6.2. Extrusion along Z- and Y-axis Optimization, Stainless Steel 316L..…103

5.6.3. Free Shape Optimization, CoCrMo Alloy…………..…...……………………...105

5.6.4. Extrusion along Z- and Y-axis Optimization, CoCrMo Alloy….….....….106

5.6.5. Free Shape Optimization, Ti6Al4V-ELI Alloy..….…...……………………….108

5.6.6. Extrusion along Z- and Y-axis Optimization, Ti6Al4V-ELI Alloy.........110

6. Conclusion..………..………………………………………………………………………..……………113

7. Bibliography and References……….…….…………………………………………………….117

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Preface

This innovative work studies a practical application that a cutting edge technology such as Additive Manufacturing offers to the heavy industry sector. Additive Manufacturing (from now on referred as AM) is that set of new technologies which includes three-dimensional components production by adding layer upon layer of material, whether this material may be plastic, metal or concrete. AM application is limitless. Although its main early application was rapid prototyping in the form of pre-production visualization models, nowadays is being developed to produce high performance lightweight components for the aircraft industry but also shows growing applications in medical and automotive sectors. The main common characteristics of AM technology are the possibility to obtain complex shapes, with internal canals and ducts, to regulate density properties of the material if a sponge-like structure is needed or to re-design a component to improve its performance in critical areas or even to reduce its weight. Another major advantage is the possibility to accelerate design and production processes, it is in fact possible to design a particularly complex component with a computer aided design software and then, once design process is completed, proceed to produce the component on a suitable AM machine by inserting into the machine software the design file with all the specification of the part. With this technology, design and production processes may take some weeks whereas with conventional production processes may require some months, due to several different production steps. Apparently, once the production process is completed, the new component is ready to be employed and does not require further tooling processes; there is however a limit, which is worth being mentioned here, that is given by the often poor surface finish that is currently achievable with the current technology and that usually require further surface treatment to make the component suitable for its application. The other main disadvantage is the particularly high production costs caused by AM technology and that ranges from metallic powder production to AM machine development; however it is commonly believed that the actual development and improvement trend will make the technology easily accessible and less expensive in the near future. All in all, AM is a technology which development is finally experiencing an exponential growth in recent years and with this process, new applications for the technology are being discovered every day. The aim of this work is to define a suitable design process to rethink current components for heavy industry in order to take full advantage of all the possibility AM technology currently offers and which are already being discovered by other main industrial actors. Even though a free forming technology can provide huge benefit for any industrial application, heavy industry requirements are generally different from aerospace, medical and automotive ones, in fact the latter consider lightweight characteristic a key factor. Some example may clarify this point: lightweight components make aircrafts reduce their fuel consumption; lightweight components in cars wheel struts make cars more stable since their unsuspended mass is reduced; a lightweight internal prosthesis will certain make injured people recover faster and

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prosecute their life with less ailment; thousands of other examples may be done following this reasoning line. On the other side, heavy industry applications must withstand heavy loads for long periods of time in generally hostile environments, from both chemical and thermal points of view. So lightweight components may be interesting when they allow reducing metallic material consumption and waste, but they still have to fully withstand the heavy applications that the components they are replacing were thought for. Heavy industry requirements can easily exclude any application of plastic materials, except maybe for insulating components or to build visualization models; but AM technology is now being applied to a broader range of metallic materials and alloys than ever. AM technology may be a winning choice when it is used to reduce the number of secondary machining operations, to build single-piece components out of components that previously were made out of several different pieces welded together, to reduce the quantity of metal waste caused by subtractive manufacturing, to produce complex mould components with inner channels to improve heat dissipation and solidification processes and so on. It is thus clear that Additive Manufacturing offers different applications also to the heavy industry sector and from this acknowledgement is born the interest of Danieli Group for the technology. Danieli Group is an Italian multinational company that is specialized in building and development of turnkey plants for heavy industry. The company is also a worldwide benchmark for ironmaking and steelmaking production, flat products, long products and non-ferrous metal production; within this broad production range, it has included a special attention for sustainable production, material recycling and environment pollution. Danieli Group’s Head Quarters are located in Buttrio (Udine, Italy) where the company’s main technical and administrative offices are located. The Group has seven different workshops across Italy and many others in several foreign countries across Europe but also in Russia, Thailand, China, India and the U.S. Danieli Group is proud of an asset of more than ten thousand employees and an overall production area of more than 2M m2; this remarkably broad extension is the result of more than a hundred years of growth and development. During its history, Danieli Group has become well aware that their know-how reflects not only technological process and design but also manufacturing capability: this awareness has pushed the company towards the decision to develop an in-house ability to fully control any detail of their design and production processes, in order to avoid any compromise on quality and reliability levels of the services and equipment supplied. In this way, any workshop area across the world is owned and managed directly by Danieli and operates according to Danieli’s manufacturing know-how to guarantee the same quality level worldwide. Danieli Group manufactures most of its equipment in its own workshops in Italy, China and Thailand obtaining the competitive production costs and the desired quality standard. Over the years, Danieli Group has strengthen its technical know-how regarding any step of metal production, from iron ores to electric arc furnaces, from hot and cold rolling mills to recycling plants, with an exceptional specialization on flat and long

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steel products production and a further specialization on non-ferrous metallic materials products to broaden its production range. During the last eight years, Danieli Group introduced a large number of innovations, thanks to the investment of an average of 140M Euro/Year in research and development over the last eight years [13]. In conclusion it is presented a general description about this work that has been divided into five chapters. The first chapter is an introduction to the reasoning methodology behind structural optimization, a simplified structural optimization process is described together with a comparison between the classic design process and the structural optimization process: the importance of defining the optimization objective and the influence of design variables on the design process are presented in this chapter. The second chapter describes a simplified optimization process called shape optimization, this optimization process can lead to minimization of mass by changing or determining boundary shape while satisfying all design requirements; shape optimization comes from the need to achieve best results with limited resources. The third chapter presents two of the main topology optimization methods: Minimum Compliance Optimization method and Bi-Directional Evolutionary Structural Optimization (BESO) method; the two methods are presented from the point of view of their mathematical implementation. Topology optimization is a phrase used to describe design optimization formulation that allows the prediction of the layout of a structural mechanical system: the topology is an outcome of the optimization procedure. The fourth chapter presents an introduction to the software used to carry out a practical example of structural optimization: Altair’s solidThinking Inspire. In the chapter it is described the way the software works to compute an optimal solution out of a problem together with the most important methods used to define the optimal solution itself. The fifth and last chapter presents a practical example of structural optimization applied to a mechanical component for the heavy industry sector: here a methodology to carry out the structural optimization process is presented along with some possible optimal results; however the limited amount of data on the component working conditions has prevented the research to come to an univocal and reliable design to be ready for production with Additive Manufacturing technology. Finally, conclusions are drawn on the basis of the findings of the present study.

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1. Introduction to Structural Optimization

The aim of this research is to define a procedure to improve design process in order to produce better components or parts than the ones that are currently employed so to make parts that are lighter, stiffer and perform in a better way. Improvements in terms of structural aspects of parts design consist in defining better design procedure, better materials and better ways to manufacture the final products: this process of improving a component, considered as an evolution of its properties and capacity is called optimization. Optimal structural design is becoming increasingly important due to the limited material resources, environmental impact and technological competition, all of which demand lightweight low-cost and high-performance structures. Optimization is defined as the process of selecting the best variables from a wide range of feasible solutions and can be applied to many different fields, such as aircraft, automotive and medical industry. The following are some practical examples:

Design of a bicycle frame for minimum weight; Design of a beam for maximum stiffness; Design of a bridge for lowest natural frequency; Design of thermal conduction systems to maximize heat transfer.

In this research will be presented an example of structural optimization problem. Structural optimization has the objective to make assemblage of materials that can best sustain the applied load; the objective can be to minimize the component weight or to maximize its stiffness (which is the same of minimize its compliance). To achieve this goal certain constraints must be applied to the problem, to name a few, these constraints may be on the volume of material, on maximum displacements or on maximum allowed stress. Figure 1 below explains this concept.

Figure 1: Structural optimization problem [5].

The optimization process consists in finding the best possible way to minimize or maximize the objective function inside the design domain, so the applied load is

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transmitted to the supports in the most efficient and safe way, keeping all the defined constraints in check. The following task formulation is typical:

Minimize weight of a carrying structure so that the allowed tensions and a certain distortion are not crossed;

Maximize the first natural frequency so that the weight is identical to the initial design and higher natural frequencies are not reduced.

In order to carry out correctly a structural optimization process, it is often needed a trusted model of the mechanical behaviour, called analysis model, which is a key component in the process. The analysis model can be based on different methods: for example for simple problem definitions analytic approaches are enough; commonly numerical methods such as finite elements methods are employed and sometime this model can also be built up from available data points. In the broadest general form, the aim of structural optimization it to improve a component behaviour concerning all the given requirements; the definition of these requirements is the first step in a structural optimization process. The questions are:

Definition of the optimization objectives; Definition of design variables, also called influence parameters, which allow

the process to meet the set objectives. The second step is the definition of the design variables; in this case the questions are:

Definition of the dimensions in the analysis model that are possible to change;

Expected dimension influence on the component behaviour. The core passage of the optimization procedure consists is coupling the analysis model with an optimization algorithm able to modify the design variables so that the component behaviour and performance are improved. Once all design variables have been defined from the initial design, the assignment is analysed and evaluated; the optimization algorithm improves the component in an iterative loop that has to be gone through again and again until the optimum design is reached. Topology optimization of continuum structures is by far the most challenging technically and at the same time the most rewarding economically. Rather than limiting the changes to the sizes of structural components, topology optimization provides much more freedom and allows the designer to create totally new and highly efficient conceptual designs for continuum structures; topology optimization can be applied to large-scale structures such as buildings and bridges but also to design materials at micro- and nano-levels. Most of the methods developed for topology optimization are based on finite element analysis where the design domain is discretized into a fine mesh of elements: in such a setting, the aim is to find the topology of a structure by determining for every point in the design domain whether there should be material (solid element) or not (void element).

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In conclusion of this introductive chapter, a simplified scheme of the structural topology optimization process is reported below in Fig. 2.

Figure 2: Scheme of the topology optimization process [4].

In a general way, structural optimization tasks are classified based on the kind of design variables because afterwards the applied solution strategies are selected according to different considerations; there are then three main design methods, called size optimization, shape optimization and topology optimization, they are represented in Figure 3.

Size Optimization – design variables are the parameters that dictate the size of the structure: this often consists in computing the optimal cross-sectional area of each strut in a truss structure or the optimal wall thickness.

Shape Optimization – design variables describe the shape of the component boundary: the optimal form or shape that defines the boundary curves and surfaces of the body is computed and the bringing in of new structural elements like cavities and braces is excluded.

Topology Optimization – this process has the aim to determine the areas of a component where material has to be added to improve overall performance of the component itself and the areas where material can be removed to reduce its weight without compromising its structural performance.

Size and shape optimizations allow the material distribution in the structure to satisfy certain loading conditions without modifying the topology of the component; on the other hand, initial and optimized structures are completely different after a process of topology optimization. The optimization task has to be defined exactly through the definition of a specification list that should contain the available possibilities for the change of the structure (design variables), the requirements for the component (objective and restriction functions) and the load cases to be considered. The specification list

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should contain all requirements, constraint functions and load cases otherwise the solver will define an optimum design that is not able to fulfil completely the functions to which is destined.

Figure 3: Classification of structural optimization tasks for a bridge [4].

Each analysis model is referred to single design conditions: to adjust to the variation in the design domain, the analysis model must be able to automatically update, this variation is decided on the input side with the design variables. On the output side the numerical values with which the objective and constraint functions have to be evaluated are then selected. Input and output of the analysis model occur with variable parameters; if this configuration cannot be used directly, the analysis model must become parameterized so must be constructed is such a way it can be modified by modifying its parameters. In this way a universally working algorithm can be applied to different tasks. All requirements that a component must fulfil have to be considered in the optimization process; requirements have to be parameterized in order to compare them to the parametric structural responses. As an example, for the requirement “the maximum stress in the component have to be up to 100 N/mm2”, all stresses have to be evaluated in the component and the highest stress value have to be compared to the defined stress limit. In this example, parameterization of the requirements appears to be very simple, however it becomes difficult with requirements like appearance, haptic or acoustic properties; these requirements must be described by physical dimensions and sometimes they require specific field studies. In the simplest case, a goal or a constraint function correspond directly to a certain parameters in the output file of the simulation program, although this condition is rather rare; generally the output has to be processed with additional routines. Whenever requirements for the structure can be described as a function value of a domain integral, even this function value must be considered in the optimization process for these requirements; this a beneficial effect on the optimization process. Particular importance comes up to these domain integrals: for example, in a lot of sectors, mass has a very important meaning; with constant density, mass correlates with the simplest domain integral, the volume of the component.

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1.1. Structural Optimization in the Design Process One thing that is in common to all engineering sectors is the fact that designers are constantly under pressure to create better products in less time and at a lower price: this is why optimization plays such an important role in product design. By considering the “typical” design cycle (Fig. 4), this almost always originates with a drawing – a sketch to illustrate a concept – and almost always ends with a drawing – the manufacturing drawing.

Figure 4: Typical design cycle [4].

The biggest problem consists is translating the sketch into an acceptable and manufacturable design. Common trade-offs of the typical design cycle are often appearance vs. function, cost vs. ease of manufacture, etc. Each single trade-off affects the design in different ways. In the conventional design process, the designer would have to rely on experience or insight to come up with acceptable proposals. An analysis tool is then used to evaluate each proposal so the designer can use these results to choose the “best proposal” among the available ones.

Figure 5: Typical design cycle vs. optimization driven design cycle [4].

In this process, design optimization becomes a part of Computer Aided Engineering (CAE) and allows to think the new design and to analyse it at the same time: the designer outlines the constraints and leaves them to the optimization tool to produce compliant proposals, the optimizer tool itself uses the analysis tool to decide how to modify the initial design to produce a better one (Fig. 5). Often the shapes and sizes proposed by the optimization tool are the ones that are most likely to pass the subsequent analyst’s verification.

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This process is called Optimization Driven Design; the designer should always look for an optimum design. At this point it is fundamental to understand what an “optimum design” is and how to recognise it. A good point to start is the dictionary definition of the word “optimum”: optimum is the greatest degree or best result obtained or obtainable under specific conditions where “specific conditions” refers to the allowed design freedom and differs for each application. The designer defines the conditions to evaluate all the design alternatives: in engineering terms this means drawing up mathematical equations that quantify the performance of a certain design. The quantitative parameter used to evaluate a design result is called “objective”; unfortunately in many cases different objectives are required and those are often contradictory, making difficult for the designer to define the best compromise. A working design almost always involves a compromise of some sort, especially because very few designers have the luxury of “infinite” resources to pursuit their objectives. These limits then give rise to the concept of constrained optimization and a design that satisfies the constraints is called a feasible solution. It is important to note that not all design are done from scratch and the optimization philosophy can be applied also to existing designs in order to improve them to the best extent possible. In this case, things are a little harder since the flexibility to modify things is often much lower. One further requirement consists in the necessity of a component to fit within an assembly of other components: this implies to work with a package space within which the component needs to fit and with assembly points that cannot be varied. From a mathematics point of view, the package space is considered as a design space or optimization domain. Finally, often is not possible to change every possible parameter: those parameters that it is possible to vary are called design variables. The dependence of the objective on the design variables is expressed as an equation, called objective function. The statement of design optimization problem consists of:

Package space, Design variables, Constraints, Objectives.

All of the above mentioned requirements must be satisfied so that the new design proposals are useful in any way.

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2. Shape Optimization

Shape optimization has been implemented into several commercial finite element programs to meet industrial need to lower cost and improve performance. In this chapter the geometric boundary method of optimization is presented: geometric boundary method defines design variables as Computer Aided Design (CAD) based curves: surfaces and solids are created and meshes are generated within finite element analysis whereas shape optimization is performed outside of finite element program. Shape optimization can lead to minimization of mass by changing or determining boundary shape while satisfying all design requirements; shape optimization comes from the need to achieve best results with limited resources. The simplest shape optimization problem is the isoperimetric problem that regards the determination of the shape of a closed curve of given length and enclosing the maximum area on a plane. The result of this simple problem is that the maximum area is enclosed by a semicircle, as shown in Figure 6 below.

Figure 6: Isoperimetric problem [1].

In engineering field, Galileo was the first to define a shape optimization problem in 1638 at his book titled “Dialogues Concerning Two New Sciences” where he presented a logical definition and solution for the shape of a cantilever beam for uniform strength (Fig. 7).

Figure 7: Galileo’s shape optimization problem and solution for a cantilever beam [1].

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Shape optimization has been a topic of in-depth research for the last three decades and structural optimization methodologies have broadly been implemented into commercial Finite Element (FE) software: it is nowadays possible to treat large shape changes without mesh distortion during shape design process. The mesh distortion problem regards the areas of the domain where the mesh elements cannot keep the pre-imposed shape and size (as for example rectangular, triangular, etc.) and consequently the mesh in these areas is not regular; this problem heavy affects the results and the computation time of the process. A complete presentation of the shape optimization problem and its solution schemes, together with some limits of the method, is described here. Structural problem can be governed by means of the principle of virtual works for a deformable continuum body in static equilibrium under the action of the external force fi and surface traction ti0 as follows:

∫ 𝑓𝑖𝛿𝑢𝑖dV

𝑉+ ∫ 𝑡𝑖

0𝛿𝑢𝑖𝑑Γ𝑡

Γ𝑡= ∫ 𝜎𝑖𝑗𝛿𝑢𝑖,𝑗dV

𝑉 (2.1)

𝑢𝑖 = 𝑢𝑖0 𝑜𝑛 Γ𝑢 𝑎𝑛𝑑 𝑡𝑖

= 𝑡𝑖0 𝑜𝑛 Γ𝑡

(2.1)

Where 𝛿𝑢𝑖 is the admissible virtual displacement. V denotes the domain during the analysis phase while 𝛤𝑢 and 𝛤𝑡

are displacement and traction boundaries respectively; Fig. 8 represents the described domain. The shape optimization problem can be defined as follows: find the boundary of V(𝛤) to minimize a cost function m(V, 𝑢𝑖) subject to:

𝑔𝑗(V, 𝑢𝑖) ≤ 𝑔𝑖

0𝑎𝑛𝑑 𝑡𝑜 ℎ𝑘(V, 𝑢𝑖) ≤ ℎ𝑘0

(2.2)

𝑢𝑖 satisfies the governing equations; 𝑔𝑗(V, 𝑢𝑖) and ℎ𝑘(V, 𝑢𝑖) denote inequality and

equality constraints respectively. Each constraint describes a design requirement.

Figure 8: Deformable body with applied external loads [1].

It is important to note that shape optimization problems may have multiple solutions and a unique solution is not guaranteed, mostly because design problems are often ill posed; moreover the final design domain cannot be known at priori. The goal in this case is not to produce an absolute optimum design but to improve the design within a neighborhood of small changes. Shape optimization based on Finite Element Analysis (from now on FEA) has received continuously growing interest in the practical design since FEA can replace

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physical experiments in many engineering fields. On the opposite side, it is almost impossible to provide continuous shape changes during shape optimization without distorting mesh elements of FEA. Mathematical representation of geometric boundary, mesh generation and manipulation affects extensively the result of optimization process; currently design boundaries can be parameterized using parametric language of FE solver. Techniques for representation of geometric boundary and geometric boundary method of formulation are discussed below.

2.1. Shape Optimization Methods Element Nodal Coordinate Method – It is an early method for shape optimization using finite element nodal coordinates as design variables; it is commonly affected by deterioration of mesh quality caused by relocation of nodal boundary points: this effect easily leads to unacceptable results; an example of unacceptable result is shown in Fig. 9.

Figure 9: Shape optimization problem of a square plate [1].

To limit the mesh distortion, additional constraints must be added to control the movement of each nodal coordinate; it is a process of trial and error. The general configuration to implement this method requires integrating with a CAD system to define suitable design boundary and with a good mesh generator to update the finite element model while changing design variables. Boundary shape can also be obtained as a linear combination of several different basis shapes represented by boundary elements or fictitious loads (to control nodal points movement). In order to characterize shape changes with a reasonable (finite) number of design variables, the reduced-basis method uses few design vectors to exhaustively describe shape changes in finite element analysis. Geometric Boundary Method – Geometric boundary can be defined by CAD-based curves, which are referred to as geometric boundary method. For shell type structures appropriate curves are predefined, surface is generated from those predefined curves and an automatic mesh generator creates the meshes for the surface. Once design in changed, also CAD-based curves are changed; then surface

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modification and new mesh generations are sequentially followed during shape optimization procedure. For each shape optimization study, it is possible to define one or more design variables for each axes, more design variables on a single axis allow obtaining a better shape. For the simplest case of one design variable on each axes, the shape optimization problem can be presented as follows:

Minimize σmax(wi) (2.3)

Subject to m(wi)≤m0 (2.4)

The solution of the shape optimization problem using finite element method for the analysis procedure has to be able to handle the shape variation introduced after each optimization iteration, these changes often require the construction of a new discrete model of the structure after each optimization step. The mesh model should be updated automatically and directly from the design variables used to parameterize the shape. The complication in the boundary method lies almost entirely in the analysis and design sensitivity whereas the optimization process is simplified by the small number of design variables that are typically used in this kind of problems. For highly time-consuming simulations, high-fidelity simulation models can replace approximated models, to efficiently predict performances during shape optimization process; approximate models need to be employed to predict the performance of the actual component. Shape optimization can be employed for daily computer aided design tool because the manual efforts to integrate CAD systems, finite element programs and optimizer at high fidelity levels have been considerably reduced over the recent years. Design variables for shape optimization problems have been implemented into commercial finite element codes by using geometric boundary method: curve, surface and mesh generations are performed in the finite element software by using the parametric language. Often shape optimization software is integrated with topology optimization software in order to convert the optimum topology into an initial shape for the shape optimization process.

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3. Topology Optimization

Topology optimization is a phrase used to describe design optimization formulation that allows the prediction of the lay-out of a structural mechanical system: the topology is an outcome of the optimization procedure. Topology optimization is often restricted to design situations with a moderate number of constraints, thus offering new types of designs that may be further processed directly or refined through shape and size optimization procedures. In the early 1900 the classical concept of lay-out was developed: the design of “thin” frame structures was characterized by setting the plastic limit of material as design limit for the design; nowadays thanks to linear programming techniques allows to design minimum weight components characterized by truss structures. By setting as design variable the cross-sectional area with a lower bound equal to zero, minimum compliance optimal topology of a truss can be defined solving a linear set of equations. Topology in this case defines which nodes are connected, starting from the so-called ground structure consisting of a given, fixed set of nodal points with an associated set of potential bars. The idea of working with a fixed reference domain, the ground structure, was carried over to the case of continuum structures; homogenization techniques were developed to define the criteria for well-posed problems. This constituted the foundation for the new computational method called material distribution technique, which allows to define a design parameterization to predict the optimal material distribution in a given reference domain. Many of the fundamental solution techniques of the material distribution methods are based on mathematical programming and FEA, thus being very similar to the methods developed for sizing optimization problems. A recent development consists in the application of level-set methods for the description of the design, which is carried out though the level-set curves obtained form level-set functions. This methodology relies on sensitivity analysis results from shape design, but contrary to standard shape design techniques, the level-set system allows for topology variation. In the following the material distribution method for structural problems is presented along with some examples of the use of the methodology in an industrial setting.

3.1. Minimum Compliance Design Formulation The general set-up for the optimal design formulated as material distribution is presented in the following. The problem type presented here is, from a computational point of view, a large-scale problem for both in state and in design variables. Consider a mechanical element as a body occupying a domain Ω𝑚 which is a part of a larger domain Ω in 𝑹2; the chosen domain allows defining loads and boundary conditions: it is the ground structure (Fig. 9). Referring to the reference domain Ω it is possible to define the optimal shape design problem as a problem of finding the optimal choice of elasticity tensor 𝐸𝑖𝑗𝑘𝑙(𝑥) that

is variable over the domain.

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Figure 9: Generalized problem of finding the optimal material distribution [11].

Introducing the energy bilinear form (internal virtual work of an elastic body at equilibrium u and for an arbitrary virtual displacement v):

𝑎(𝑢, 𝑣) = ∫ 𝐸𝑖𝑗𝑘𝑙(𝑥)휀𝑖𝑗(𝑢)

Ω휀𝑘𝑙(𝑣)𝑑Ω (3.1)

Where 휀𝑖𝑗(𝑢) is the linearized strain energy. The load linear form is:

𝑙(𝑢) = ∫ 𝑝𝑢

Ω𝑑Ω + ∫ 𝑡𝑢

Γ𝑇𝑑s (3.2)

The minimum compliance problem (or maximum global stiffness problem) takes then the form:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝑢∈𝑈,𝐸 𝑙(𝑢) (3.3)

Subject to: 𝑎𝐸(𝑢, 𝑣) = 𝑙(𝑣) (3.4)

𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣 ∈ 𝑈 𝑎𝑛𝑑 𝐸 ∈ 𝐸𝑎𝑑 (3.5)

In this formulation the equilibrium equation is written in its weak variation form, U denotes the space of kinematically admissible displacement fields, p are the body forces and t the boundary tractions on the tractions part 𝛤𝑡 ⊂ 𝛤

≡ 𝛿𝛺 of the boundary. The term 𝐸𝑎𝑑 denotes the set of admissible rigidity tensors for the design problem. It is possible to consider the case of multiple load conditions, to simplify computations only the case of minimizing weighted average of the compliances for each load cases is reported here. The simple multiple load formulation is:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝑢∈𝑈,𝐸 ∑ 𝑤𝑘𝑙𝑘(𝑢)𝑘𝑀𝑘=1 (3.6)

Subject to:

𝑎𝐸(𝑢𝑘, 𝑣) = 𝑙𝑘(𝑣) (3.7)

𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣 ∈ 𝑈; 𝑘 = 1,2, …𝑀 𝑎𝑛𝑑 𝐸 ∈ 𝐸𝑎𝑑 (3.8)

23

For a set 𝑤𝑘, 𝑝𝑘, 𝑡𝑘 , 𝑘 = 1,2, …𝑀 of weighting factors, loads and tractions and corresponding load linear forms given as:

𝑙𝑘(𝑢) = ∫ 𝑝𝑘𝑢

𝛺𝑑𝛺 + ∫ 𝑡𝑘𝑢

𝛤𝑇𝑘

𝑑𝑠 (3.9)

The topology optimization problem presented here has been cast as a problem in a fixed reference domain. In the design of the topology of a structure, it is fundamental to define the optimal placement of a given isotropic material in space, thus determining which points of space should be material points and which points should remain void (no material). From the reference domain Ω will be defined the optimal design domain Ω𝑚 of material points. The set 𝐸𝑎𝑑 of admissible rigidity tensors consists of those sensor for which:

𝐸𝑖𝑗𝑘𝑙 ∈ 𝐿∞(Ω) (3.10)

𝐸𝑖𝑗𝑘𝑙 = 1Ω𝑚𝐸𝑖𝑗𝑘𝑙0 𝑤𝑖𝑡ℎ 1Ω𝑚 = {

1 𝑖𝑓 𝑥 ∈ Ω𝑚

0 𝑖𝑓 𝑥 ∈ Ω

Ω𝑚 (3.11)

∫ 1Ω𝑚𝑑Ω = Vol(

ΩΩ𝑚) ≤ 𝑉 (3.12)

The last inequality expresses a limit on the amount of material at our disposal, so that the minimum compliance design is for limited (fixed) volume; the tensor 𝐸𝑖𝑗𝑘𝑙

0 is

the rigidity tensor for the given isotropic material. This given definition of 𝐸𝑎𝑑 means that we have formulated a distributed, discrete value design problem, defined as 0-1 problem: the fundamental idea of the material distribution technique is to associate to each point a discretization of material distribution in a continuum formulation. For the topology design problem of determining which elements of the ground structure should be part of the final structure, the design variables 𝜌𝑒 are discrete variables and 𝜌𝑒 ∈ {0,1} : 𝜌𝑒=0 indicates a void, 𝜌𝑒=1 indicates material and 0 < 𝜌𝑒 < 1 indicates the porous areas with voids at micro level. It is now possible to formulate the optimization problem for the reference domain Ω in the following form:

𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑎𝑟𝑒: 𝜇, 𝛾 … ∈ 𝐿∞(Ω) (3.13)

𝐴𝑛𝑔𝑙𝑒: 𝜃 ∈ 𝐿∞(Ω) (3.14)

𝐸𝑖𝑗𝑘𝑙(𝑥) = Ẽ𝑖𝑗𝑘𝑙(𝜇(𝑥), 𝛾(𝑥)…𝜃(𝑥)) (3.15)

Density of material: 𝜌(𝑥) = 𝜌(𝜇(𝑥), 𝛾(𝑥)…𝜃(𝑥)) (3.16)

∫ 𝜌(𝑥)𝑑Ω ≤ V;

Ω0 ≤ 𝜌(𝑥) ≤ 1; 𝑥 ∈ Ω (3.17)

Ẽ𝑖𝑗𝑘𝑙 represents the effective material parameters over the domain.

These quantities can be obtained analytically or numerically through a suitable micro-mechanical modelling; the final material will be anisotropic (or orthotropic) so the angle of rotation of the directions of orthotropy becomes an additional variable. The density of material is a function of a number of design variables that

24

describe the geometry of holes at micro-level, it is on these variables that the optimization process should be applied. For any material consisting of a given elastic material with microscopic inclusions of void and intermediate values of base material density, will provide the structure with strictly less than proportional rigidity. In an optimal structure there should be density values of 0 or 1 in most elements; this depends directly on the choice of the microstructure since the use of an optimal microstructure results in a very efficient use of intermediate densities of material. A body with an optimal distribution of material in considered as formed out of cells of infinitely small dimension and infinitely big number. It is however possible to regularize the minimum compliance problem formulated as 0-1 problem by restricting the possible range of material sets to measurable sets of bounded perimeter, as for example constraining the total length of the boundaries of the structure. The imposition of a constraint does not change the discrete valued nature of the problem and the perimeter constraint is particularly valuable because prevents the formation of microstructures with rapid variation of material density or rapid variation of material thickness. Material distribution approach to topology design of continuum structures allows describing structure by density of material. From now on, the porous material with microstructure is constructed from a basic unit cell consisting at macroscopic level of material and void; the body is then composed of infinitely many of such cells, of infinitely small dimension and repeated periodically through the medium. It is possible to have also continuously varying density of material through the medium as required by topology optimization problems. The resulting medium is exhaustively described by effective macroscopic material properties that depend on the geometry of the basic cell, these properties are computed through homogenization theory formulation. Computation of these effective properties plays a key role for the topology optimization; the homogenization formulation is thus presented here for a two dimensional case. Suppose that a periodic microstructure, which is a structure where the basic unit cell is repeated throughout the whole volume of the domain, is assumed in the neighbourhood of an arbitrary point x of a given linearly elastic structure, periodicity is represented by the parameter 𝛿 (of very small value) and the elasticity

tensor 𝐸𝑖𝑗𝑘𝑙𝛿 has the form:

𝐸𝑖𝑗𝑘𝑙𝛿 (𝑥) = 𝐸𝑖𝑗𝑘𝑙

(𝑥,𝑥

𝛿) (3.18)

𝐸𝑖𝑗𝑘𝑙𝛿 (𝑦) = 𝐸𝑖𝑗𝑘𝑙

(𝑦,𝑦

𝛿) (3.19)

𝑥, 𝑦 → 𝐸𝑖𝑗𝑘𝑙 (𝑥, 𝑦) (3.20)

Where x represents the macroscopic variation of material parameters and 𝑥/𝛿 the microscopic periodic variation. Suppose now that the structure is subjected to a macroscopic body force and a macroscopic surface traction; these external loads cause the displacement 𝑢𝛿(𝑥):

𝑢𝛿(𝑥) = 𝑢0(𝑥) + 𝛿𝑢1 (𝑥,𝑥

𝛿) + ⋯ (3.21)

25

Where 𝑢0 is the macroscopic deformation field that is independent of the microscopic variable y. The effective displacement field is the macroscopic deformation field that arises due to the applied force when the rigidity of the structure is assumed as given by the effective rigidity tensor:

𝐸𝑖𝑗𝑘𝑙𝐻 (𝑥) =

1

|𝑌|∫ [𝐸𝑖𝑗𝑘𝑙

(𝑥, 𝑦) − 𝐸𝑖𝑗𝑘𝑙 (𝑥, 𝑦)

𝜕𝜒𝑝𝑘𝑙

𝜕𝑦𝑞] 𝑑𝑦

𝑌 (3.22)

With 𝜒

𝑘𝑙 macroscopic displacement field that is given as the Y-periodic solution of the cell-problem. The variational form of the previous equation is:

𝐸𝑖𝑗𝑘𝑙𝐻 (𝑥) = 𝑚𝑖𝑛𝜑∈𝑈𝑌

1

|𝑌|𝑎𝑌(𝑦

𝑖𝑗 − 𝜒𝑖𝑗 , 𝑦𝑘𝑙 − 𝜒𝑘𝑙) (3.23)

𝑎𝑌(𝑦𝑘𝑙 − 𝜒𝑘𝑙, 𝜑) = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜑 ∈ 𝑈𝑌 (3.24)

𝑈𝑌 denotes the set of all Y-periodic virtual displacement fields. The effective elastic moduli for plane problems can be computed by solving three different analysis problems for the unit cell Y; for most geometry this process has to be carried out using finite element methods. To use the homogenization method in an actual design process, it should be implemented in an easy to use pre-processor and should hold for mixtures of linearity, elastic materials and for materials with voids. Consider now a layered material, with layers directed along the y2-direction and repeated periodically along the y1-axis: the consequent unit cell is [0,1] × 𝑹 and the unit fields 𝜒𝑘𝑙 are independent of the variable y2. Using periodicity and appropriate test functions and assuming that the direction of layering coalesces with direction of orthotropy of material, the only non-zero elements of the tensor 𝐸𝑖𝑗𝑘𝑙

are listed below:

𝐸1111 , 𝐸2222 , 𝐸1212 (= 𝐸1221

= 𝐸2121 = 𝐸2112

), (3.25)

𝐸1122 (= 𝐸2211

).

For a layering of two isotropic materials, with same Poisson ratio v but different density, elasticity modulus E+ and E- and with layer thickness 𝛾 𝑎𝑛𝑑 (1 − 𝛾) respectively, the layering formulas are:

𝐸1111𝐻 = 𝐼1, 𝐸2222𝐻 = 𝐼2 + 𝑣

2𝐼1,

𝐸1212𝐻 =

1−𝑣

2𝐼1, (3.26)

𝐸1122𝐻 = 𝑣𝐼1,

𝐼1 =1

1−𝑣2𝐸+𝐸−

𝛾𝐸−+(1−𝛾)𝐸+, (3.27)

𝐼2 = 𝛾𝐸+ + (1 − 𝛾)𝐸− (3.28)

26

The above mentioned set of equations allow to define the effective material properties of the resulting material; the elasticity moduli are computed as the material is constructed, the resulting material properties are:

𝐸1111 =

𝛾𝐸

𝜌𝛾(1 − 𝑣2) + (1 − 𝜌),

𝐸1122 = 𝜌𝑣𝐸1111

, 𝐸2222 = 𝜌𝐸 + 𝜌2𝑣2𝐸1111

, (3.29)

𝐸1212 = 0.

From the results listed above, for layered materials consisting of material and void, the dimension two does not possess any shear stiffness. For a computational topology design scheme that is based on equilibrium analyses with these materials, voids should be represented by a very weak material (with a very low stiffness, but non-zero) in order to avoid singular stiffness matrix. On the other hand, layered materials have analytical expression for the effective elasticity moduli that have distinct advantages for optimization. It is important to underline that the use of homogenized material coefficients is consistent with the basic properties of the minimum compliance problem. Consider a minimizing sequence of designs in the set of 0-1 designs and assume this sequence to be composed by a sequence of microcells given by a scaling factor 𝛿 > 0. In the limit 𝛿 → 0, the design sequence has a response governed by the homogenized parameters. It is a fundamental property of the homogenization process that the displacement 𝑢𝛿(𝑥) will converge weakly to the displacement 𝑢0(𝑥) of the homogenized design. As the compliance functional is a weakly continuous functional of the displacements, this implies the convergence of the compliance values.

3.2. Conditions of Optimality In this chapter the necessary conditions of optimality for the minimum compliance design problem that employs composite materials in the parameterization of design. There are two different types of design variables: the composite material is an anisotropic (orthotropic) material for which the angle of rotation of the unit cell is a fundamental design variable; the other variable regards the size of the unit cell and thus material density across the volume. The formulation of the material distribution method for optimal continuum structures involves working with a composite material consisting of base material and periodically repeated micro voids. Composite materials with cell symmetry are orthotropic, the angle of rotation of the material axes will influence the effective compliance of the structure: it is possible to compute analytically the optimal rotation of the cell. Here will be derived the optimal conditions for material rotations in plane stress/strain problems. Assume an orthotropic material according to properties described before; in the frame of reference given by material axes of the chosen material there is the following stress/strain relation:

𝜎𝑖𝑗 = 𝐸𝑖𝑗𝑘𝑙휀𝑘𝑙 (3.30)

27

With 𝐸1111 , 𝐸2222

, 𝐸1122 , 𝐸1212

being the only non-zero components of the rigidity tensor 𝐸𝑖𝑗𝑘𝑙; 𝐸1111

is assumed greater or equal to 𝐸2222 and a given set 휀𝑖𝑗

𝑘 (with k=1,

2, … M) of strain field for a number of load cases. With minimum compliance design in mind, the problem is to maximize the weighted sum of a number of strain energy densities:

𝑊 = ∑ 𝑤𝑘 [

1

2𝐸1111휀11

𝑘2 +1

2𝐸2222휀22

𝑘2 + 𝐸1122휀11𝑘 휀22

𝑘 + 2𝐸1212휀12𝑘2]𝑀

𝑘=1 (3.31)

If the strains are expressed in terms of principal strains 휀𝐼

𝑘𝑎𝑛𝑑 휀𝐼𝐼𝑘 with the first

greater than the second:

휀11𝑘 =

1

2[(휀𝐼

𝑘 + 휀𝐼𝐼𝑘) + (휀𝐼

𝑘 − 휀𝐼𝐼𝑘)𝑐𝑜𝑠2𝜓𝑘]

휀22𝑘 =

1

2[(휀𝐼

𝑘 + 휀𝐼𝐼𝑘) − (휀𝐼

𝑘 − 휀𝐼𝐼𝑘)𝑐𝑜𝑠2𝜓𝑘] (3.32)

휀12𝑘 = −

1

2[(휀𝐼

𝑘 − 휀𝐼𝐼𝑘)𝑠𝑖𝑛2𝜓𝑘]

Here 𝜓𝑘 is the angle of rotation of the material frame relative to the frame of the k-th principal strains. In this analysis the aim is to determine the angle Θ of rotation of the material relative to a chosen frame of reference that maximizes function W. Each angle 𝜓𝑘 is thus written as Θ = 𝜓𝑘 − 𝛼𝑘 where 𝛼𝑘 is the angle of rotation of the k-th strain field. Once the new expression for the strains is inserted in the equation for W and the latter is differentiated, stationary condition is found:

∑ 𝑤𝑘[𝐴𝑘𝑠𝑖𝑛2(Θ − 𝛼𝑘) + 𝐵𝑘𝑠𝑖𝑛2(Θ − 𝛼𝑘)𝑐𝑜𝑠2(Θ − 𝛼𝑘)]𝑀𝑘=1 = 0 (3.33)

𝐴𝑘 = (휀𝐼𝑘2 + 휀𝐼𝐼

𝑘2)(𝐸1111 − 𝐸2222) (3.34)

𝐵𝑘 = (휀𝐼𝑘 + 휀𝐼𝐼

𝑘)2(𝐸1111 + 𝐸2222 − 2𝐸1122 − 4𝐸1212) (3.35)

Stationary condition is then achieved if the following 4-th order polynomial in 𝑠𝑖𝑛2Θ is zero:

𝑃(𝑠𝑖𝑛2Θ) = 𝑎4𝑠𝑖𝑛42Θ + 𝑎3𝑠𝑖𝑛

32Θ + 𝑎2𝑠𝑖𝑛22Θ + 𝑎1𝑠𝑖𝑛

2Θ + 𝑎0 (3.36)

W is periodic so there exist at least two real roots of P; since the last equation is of the 4-th order, it can be solved analytically. The actual minimizer of the compliance is finally found evaluating W for four or eight stationary rotations. For the single load case, the stationary angle 𝜓 can be expressed as:

𝑠𝑖𝑛2𝜓 = 0 or 𝑐𝑜𝑠2𝜓 = −𝛾

𝛾 =𝛼

𝛽

𝐼 + 𝐼𝐼

𝐼 − 𝐼𝐼

(3.37)

𝛼 = (𝐸1111 + 𝐸2222) ≥ 0 𝑎𝑛𝑑 𝛽 = (𝐸1111 + 𝐸2222 − 2𝐸1122 − 4𝐸1212)

With the values defined above, it is possible to maximize 𝜓 (it depends on the sign of the parameter 𝛽, this parameter is a measure of shear stiffness of the material). For low shear stiffness values (𝛽 ≥ 0) the globally minimal compliance is achieved with 𝜓 = 0; in this case the largest principal strain is aligned with the strongest material

28

axis and analysing the stress-strain relation it results that these axes are aligned with the principal stresses too. Normally for a single field of stress-strain, principal stresses, principal strains and material axes should all be aligned at the optimum for materials that are weak in shear. For the used material it is possible to show stationary of the alignment of material axes, principal strain axes and principal stress axes. For the problem at hand, the tensor 𝐸𝑖𝑗𝑘𝑙 depends on geometric quantities that

define the microstructure: for a square 1 by 1, micro cell with a rectangular hole of dimension (1 − 𝜇) × (1 − 𝛾), the density of material is given as 𝜌 = 𝜇 + 𝛾 − 𝜇𝛾 and the constraints of the design variables 𝜇, 𝛾 are:

∫ (𝜇(𝑥) + 𝛾(𝑥) − 𝜇(𝑥)𝛾(𝑥)𝑑Ω

Ω= 𝑉 (3.38)

0 ≤ 𝜇(𝑥) ≤ 1; 0 ≤ 𝛾(𝑥) ≤ 1; 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ Ω (3.39)

The necessary conditions for optimality for the sizing variables 𝜇, 𝛾 are a subset of the stationary conditions for the Lagrange function:

𝐿 = ∑(𝑤𝑘𝑙𝑘𝑢𝑘) − {∑𝑎𝐸(𝑢𝑘, 𝑣𝑘) − 𝑙𝑘𝑢𝑘

𝑀

𝑘=1

}

𝑀

𝑘=1

+

+Λ(∫ [𝜇(𝑥) + 𝛾(𝑥) − 𝜇(𝑥)𝛾(𝑥)]

Ω

𝑑Ω − V) +

+∫𝑧𝜇+

𝛺

(𝑥)(𝜇(𝑥) − 1)𝑑𝛺 + ∫𝑧𝜇−

𝛺

(𝑥)𝜇(𝑥)𝑑𝛺 +

+∫ 𝑧𝛾+

𝛺(𝑥)(𝛾(𝑥) − 1)𝑑𝛺 + ∫ 𝑧𝛾

−

𝛺(𝑥)𝛾(𝑥)𝑑𝛺 (3.40)

In this equation 𝑢𝑘, 𝑣𝑘 (𝑘 = 1,2…𝑀) are Lagrange multipliers for the equilibrium constraints; 𝑣𝑘 belongs to the set U of kinematically admissible displacement fields. For intermediate densities (0 < 𝜇 < 1; 0 < 𝛾 < 1) the conditions of optimality can be stated as:

1

𝛬(1−𝛾)∑ 𝑤𝑘𝑀𝑘=1

𝜕𝐸𝑖𝑗𝑝𝑞

𝜕𝜇휀𝑖𝑗(𝑢

𝑘)휀𝑝𝑞(𝑢𝑘) = 1 (3.41)

1

𝛬(1−𝜇)∑ 𝑤𝑘𝑀𝑘=1

𝜕𝐸𝑖𝑗𝑝𝑞

𝜕𝛾휀𝑖𝑗(𝑢

𝑘)휀𝑝𝑞(𝑢𝑘) = 1 (3.42)

The above equation can be interpreted as a statement that expresses that the energy-like left hand side terms are constant equal to 1 for all intermediate densities: this is thus a condition that reminds us of the fully stressed design condition in plastic design. Normally areas with high energy are often too low on rigidity [2]; the expressions for energy are:

𝐵𝑘 = [Λ𝑘(1 − 𝛾𝑘)]−1∑ 𝑤𝑘𝑀

𝑘=1

𝜕𝐸𝑖𝑗𝑝𝑞

𝜕𝜇(𝜇𝑘,𝛾𝑘)휀𝑖𝑗(𝑢

𝑘)휀𝑝𝑞(𝑢𝑘) (3.43)

𝐸𝑘 = [Λ𝑘(1 − 𝜇𝑘)]−1∑ 𝑤𝑘𝑀

𝑘=1

𝜕𝐸𝑖𝑗𝑝𝑞

𝜕𝛾(𝜇𝑘,𝛾𝑘)휀𝑖𝑗(𝑢

𝑘)휀𝑝𝑞(𝑢𝑘) (3.44)

It is possible to define two adjustable parameters 𝜂 𝑎𝑛𝑑 𝜉 that represent a tuning parameter and a move limit respectively, in order to improve the efficiency of the

29

method. Values of 𝜇𝑘 𝑎𝑛𝑑 𝛾𝑘 depend on present values of the Lagrange multiplier Λ, the multiplier should be adjusted in an inner iteration loop in order to satisfy the active volume constraint: the volume of the updated values of density is continuous and decreasing function of the multiplier. The volume is strictly decreasing in the interesting intervals, where the bounds on density are not active in all points, in this way it is possible to determine a unique value of multiplier. The values of the parameters 𝜂 𝑎𝑛𝑑 𝜉 are chosen conveniently to achieve a rapid and stable convergence on the iteration scheme, typical values of 𝜂 𝑎𝑛𝑑 𝜉 are 0.8 and 0.5 respectively. If the density is given through a number of other design variables describing the micro geometry of voids, it is necessary to define updated schemes for those variables too. The angle of rotation of the material with voids should also be updated using the axes of principal strains or of principal stresses as axes of orthotropy.

3.3. Computational Procedure Homogenization modelling is based on the numerical calculation of the globally optimal distribution of the design variables that define the microstructure being used, this in turn determines the density distribution of material, which is the primary target of the process. In the following the main steps to define the optimal topology of a structure will be described, starting from an initial layout to obtain a final optimal solution. Step 1 – Pre-processing of geometry, of loading and of material properties, Start the analysis by choosing a suitable reference domain (ground structure) that allows defining surface tractions, fixed boundaries, etc. It is now possible to define those areas of the ground structure that have to be left untouched as solid material or as voids and the rest which represent the design area and can be modified during the optimization process. Once this setting procedure is completed, it is possible to construct a FEM mesh for the ground structure, the mesh should be fine enough to describe accurately all the areas of the structure, the mesh remains unchanged during the whole design process. Construct now finite element spaces for the independent fields of displacements and design variables. The nature of the problem easily causes that the finite element models involved in the material distribution method become large scale, especially in 3D; since the process works on a fixed grid, no re-meshing of the design in necessary. The FEM analysis can be further optimized if rectangular (box-like) domains are used and are discretized with the same element throughout: in this case only one element matrix needs to be calculated. For large-scale computations iterative solvers and parallel implementation are able to sensibly reduce computation time, normally solving equilibrium equations is the most time consuming process of topology optimization problems. Topology design problems require working with a huge number of design variables, even though it is often possible to reduce the number of constraints in the problem

30

statements. The application of an adjoint method for the computation is often required, so here it will be presented briefly. For the functional 𝑢𝑜𝑢𝑡 = 𝑙

𝑇𝑢, the equilibrium equation is satisfied by u so that 𝑲𝑢 − 𝑓 = 0. For any vector 𝜆 the relation becomes:

𝑢𝑜𝑢𝑡 = 𝑙𝑇𝑢 − 𝜆𝑇(𝑲𝑢 − 𝑓) (3.45)

After differentiating the previous relation, this can be written as:

𝜕𝑢𝑜𝑢𝑡

𝜕𝜌𝑘= (𝑙𝑇 − 𝜆𝑇𝑲)

𝜕𝑢

𝜕𝜌𝑘− 𝜆𝑇

𝜕𝑲

𝜕𝜌𝑘𝑢 (3.46)

If the adjoint variable now satisfies the adjoint equation 𝑙𝑇 − 𝜆𝑇𝑲 = 0, then the simple expression for the derivative of the output displacement is:

𝜕𝑢𝑜𝑢𝑡

𝜕𝜌𝑘= −𝑝𝜌𝑘

𝑝−1 − 𝜆𝑇𝑲𝑢 (3.47)

It is important to not neglect material characteristics. Choose a composite, constructed by periodic repetition of a unit cell consisting of the given material with one or more holes or a layered layout. Compute then the effective material properties of the composite according to the homogenization theory, this allows obtaining a functional relationship between density in the composite material and the effective material properties of the resulting orthotropic material. At this point it is possible to generate a database of material properties as function of design variables with a specific set of data for each value of Poisson ratio. Step 2 – Optimization Process At this point it is possible to compute the optimal distribution over the reference domain of the design variables that describe the properties of the composite material. The process uses a displacement-based analysis together with the optimality update schemes for density and for optimal angle of rotation for the cell-related computation. The structure of the algorithm is the following:

Analysis of the current design together with objective evaluation, the starting design is often characterized by a homogeneous distribution of material;

The iterative part of the algorithm comprehends:

For the present design defined by density and angle of rotation of cell (as

variables), compute the rigidity tensor throughout the whole structure;

For the defined distribution of rigidity, compute using FEM the resulting displacements and associated stress and strains, for each load case;

Compute the compliance of this design, if there is only a marginal improvement over the previous design, stop the iteration process otherwise continue;

31

Update angle of cell rotations based on the optimality criteria described before and base this calculation on principal stresses;

Update density variables according to the criteria shown before, compute energy equations B and E from principal strains; at this step it is also possible to compute the effective value of the Lagrange multiplier for the volume constraint.

Repeat the iteration loop until the difference between two consecutive iterations is lower than the chosen tolerance.

Normally the design process regards a structure composed of fixed areas (such as solids and voids) and the updating of the design variables should regard only the areas of the ground structure that are set to be re-designed. For the problem above is thus necessary to decide at an early stage on a choice of basic unit cells as the basis for computation of the effective elasticity moduli; the most important quantity in this analysis is the density of material and the underlying geometric quantities that define density are less interesting. Continuing the dissertation on a hypothetical two-dimensional problem, micro voids are made out of square holes in square unit cells, in this case density is described by just one geometric variable (the length of the square sides) and can take all values between 0 and 1; if voids are made out of circular holes in square cells, the option of 0 density is not admissible. On the other hand, using rectangular micro voids in square cells gives a more complicated microstructure with a doubled number of geometric variables but at the same time, experiments have proven that this choice results in a more stable iteration history and in slightly better compliance values. For three-dimensional problems, box-like holes in cubic cells are a simple choice of microstructure. The use of single inclusion cells as outlined above is not justified from a mathematical point of view since these composites do not assure existence of solutions to the optimization problem. These composites should be seen only as simple type of composites useful to remove the 0-1 nature of the generalized shape problem. It is possible to demonstrate that layered materials are able to generate the strongest microstructure constructed from a given material, moreover the rigidity tensors are given analytically thus simplifying the pre-processing step to define the effective material properties. For layered material however it is necessary to use different refinements of unit cell, this condition depends on spatial dimension and on whether the problem is a single or multiple load problem. From an engineering point of view, it is interesting to note that topology optimization using square holes in square cells gives rise to very well defined designs consisting almost of areas with material or void and very little areas with intermediate density so with composite material; the use of layered materials on the other hand gives less well defined shapes with larger areas of intermediate density but this design tends to be more efficient than the design that uses single inclusion cells. The use of square or rectangular holes in square cells at a single level of micro-geometry is considered as sub-optimal. However, the well-defined shapes obtained using square holes in square cells as well as their simplicity tend to favour the use of this micro-geometry.

32

Optimal Microstructures

Dimension of space is 2

Dimension of space is 3

Single Load

Rank 2 materials with orthogonal layers along directions of principal

stress/strains

Rank 3 materials with orthogonal layers along directions of principal

stress/strains

Multiple Loads

Rank 3 materials with non-orthogonal layers

Rank 6 material with non- orthogonal layers

Table 1: possible optimal microstructures [2].

Notice also that the way the angle of rotation of cells is updated in the optimality criteria influences directly the resulting design: the layered material tend to give sub-optimal designs if the rotation angle is updated in single steps, instead of using alignment of material and stress/strain axes at each iteration step. A final consideration about the optimality criteria method described above regards that the assembled stiffness matrix is positive definite, densities are thus bounded by the following inequalities:

𝜇 ≤ 𝜇𝑚𝑖𝑛 < 0 𝑎𝑛𝑑 𝛾 ≤ 𝛾𝑚𝑖𝑛 < 0 (3.48)

𝜇𝑚𝑖𝑛, 𝛾𝑚𝑖𝑛 are the suitable lower bounds, they can never be equal to zero. The type of algorithm such the one described above have been used to great effect in a large number of structural topology design studies and is established for solving large-scale problems. The effectiveness of the algorithm comes from the fact that each design variable is updated independently of the others, except from rescaling that has to take place to satisfy the volume constraint. A major challenge for the computational implementation of topology design is to cope with the high number of design variables. Optimality criteria methods were first applied, however the use of mathematical programming algorithms typically implies greater flexibility. The high number of design variables is combined with a moderate number of constraints: an algorithm that is suitable to work for large-scale topology optimization problems is the MMA algorithm (Method of Moving Asymptotes). This method works with a sequence of approximate sub-problems that are constructed from sensitivity information at the current iteration point as well as some iteration history. As conclusion of the paragraph, Figure 10 represents the flowchart of the homogenization method for topology optimization.

33

Figure 10: Flowchart of the homogenization method for topology optimization

[2],[12].

34

3.4. Analysis Refinement and Issues After implementing a computational scheme for topology design along the lines for standard sizing design problems, several additional issues have to be addressed: if low order elements are used for the analysis the results will be affected by a lot of checkerboard patterns of black and white elements. At the same time, refining the mesh can have dramatic effects on the results since this allows adding finer and finer details in the design. Checkerboard pattern correspond to areas of a structure where the density of material assigned to continuous finite element varies in a periodic fashion similar to a checkerboard consisting of alternating voids and solid elements. This effect is un-physical and results from bad FEM modelling being exploited by the optimization procedure; at the same time the affected area seems to have the best performances, for example a checkerboard of material in a uniform grid of square elements has a greater stiffness than any other possible material distribution. The occurrence of checkerboards patterns can be easily be prevented and any method to achieve mesh-independency is also able to solve the problem when the mesh becomes fine enough; geometry control measures also help avoiding checkerboard formation. A fixed scale geometric restriction on the design could be a counter-productive solution when using a numerical method to obtain a overview on the behaviour of optimal topology at a fair fine scale, when designing low volume fraction structures or when composite materials are used as basis for optimization. The most general approach is to use FEM discretization where checkerboards are not present, this involves using high order elements for displacement and thus higher computational cost. Another serious problem associated with the 0-1 statement is that normally there is not a solution to the continuous problem, since it has been presented in a discretized form: this is drawback of making the problem sensitive to the mesh element dimension. The physical explanation for the mesh dependent results is that by introducing finer and finer scales, the design space is expanded and the optimal design is not a classical solution with finite size features but a composite material with different density throughout the volume. For production reason, a design with fine scale variation should be avoided and a design tool able to give a mesh-independent result is generally preferred. Several different techniques have been proposed to limit geometric variation of the design field by imposing additional constraints on the problem, by restricting the size of the gradient of density distribution. This constraint can be set either on the perimeter or on some 𝐿𝑞-norm of the gradient, in both case experimentation is needed to define the suitable constraint value. An alternative consists in imposing a point-wise limitation on the gradient of density field, in this case the constraint has an immediate geometric meaning, it may be for example the thinnest possible features in a design. Implementation may be problematic but can be handed easily via a move limit technique. Another way to limit geometric variations of the design can be achieved applying filters, in a similar way to image processing: it is possible then to work with filtered densities in the stiffness matrix so that the equilibrium constraint is modified to the format:

𝑲𝒔𝒊𝒎𝒑(𝐻(𝜌))𝑢 = 𝑓 (3.49)

35

From the original format: 𝑲𝒔𝒊𝒎𝒑(𝜌)𝑢 = 𝑓 (3.50)

H denotes a filtering applied on density 𝜌. The filter may be a linear filter with a defined minimum radius 𝑟𝑚𝑖𝑛 that gives the modified density 𝐻(𝜌)𝑘 in the k-th element as:

𝐻(𝜌)𝑘 = ∑ 𝐻𝑗𝑘𝜌𝑗

𝑁𝑗=1 (3.51)

𝐻𝑗𝑘 =

𝐴𝑗𝑘

∑ 𝐴𝑗𝑘𝑁

𝑗=1

𝑤𝑖𝑡ℎ 𝐴𝑗𝑘 = 𝑟𝑚𝑖𝑛

– 𝑑𝑖𝑠𝑡(𝑘, 𝑗) (3.52)

{𝑗 ∈ 𝑁 | 𝑑𝑖𝑠𝑡(𝑘, 𝑗) ≤ 𝑟𝑚𝑖𝑛 } 𝑤𝑖𝑡ℎ 𝑘 = 1, 2, …𝑁

𝐻𝑗𝑘 is the normalized weight factor and 𝑑𝑖𝑠𝑡(𝑘, 𝑗) denotes the distance between the

centre of the element k and the centre of element j. the convolution weight 𝐻𝑗𝑘 is

zero outside the filter area and the weight for the element j decay linearly with distance from element k. The filtering means that stiffness in an element depends on the density 𝜌 in all elements in a neighbourhood of the element itself, causing a smoothing of the density. The filter radius 𝑟𝑚𝑖𝑛

is fixed in the formulation and implies the enforcement of a fixed length-scale in the designs and convergence with mesh refinement. Generally filtering results in density fields that are bi-valued, the stiffness distribution is then more “blurred” [2] with grey boundaries. To implement the filter in the procedure described before for the optimization problem is necessary to modify sensitivity information so that the stiffness matrix is refined in such a way the sensitivity of the output displacement with respect to 𝜌k will be affected by adjacent elements. Compared to the other constraining approaches, the application of a filter does not require any additional constraint to be added to the problem. Another alternative to the direct filtering of the densities consists in filtering the sensitivity information of the optimization problem, experience has proven that this is the best way to ensure mesh-independency. The method works modifying element sensitivities as follows:

𝜕𝑢𝑜𝑢�̂�

𝜕𝜌𝑘=

1

𝜌𝑘∑ 𝐻𝑗

𝑘𝜌𝑗𝜕𝑢𝑜𝑢𝑡

𝜕𝜌𝑗

𝑁𝑗=1 (3.53)

The standard expression of element sensitivities was:

𝜕𝑢𝑜𝑢𝑡

𝜕𝜌𝑘= −𝑝𝜌𝑘

𝑝−1 − 𝜆𝑇𝑲𝑢 (3.54)

Filtering on sensitivities is not the same of applying a filter H to sensitivities as the densities in this case influence the result; however with a little extra CPU-time this procedure comes with ease of implementation and very similar results to those from the local gradient constraint. Sensitivity converges to the original sensitivity when the filter radius 𝑟𝑚𝑖𝑛

approaches zero and all sensitivities will be equal when 𝑟𝑚𝑖𝑛 → ∞. An interesting side effect of this technique is that it improves the

computational behaviour of the topology design procedure and allows for greater design variation before defining an optimal solution and this is obtained thanks to the inclusion of 𝜌𝑘 in the filtering expression.

36

The filtering techniques described above can also be used to control checkerboard formation without imposing mesh-independent length scale, in this case it is necessary to that the filtering is adjusted to be mesh-dependent (𝑟𝑚𝑖𝑛

has to vary with mesh size).

3.5. Bi-Directional Evolutionary Structural Optimization Method (BESO) Evolutionary structural optimization (ESO) method is based on the simple concept of gradually removing inefficient material from a structure, in this way the structure will evolve towards its optimal shape and topology. Even though the is it not possible to guarantee that the method will produce the best solution, which means that the found solution is the best among all the possible solutions, the method is a useful tool for engineers to explore structurally efficient forms and shapes during the conceptual design stage of a project. Through a finite element analysis on a component, it is possible to define the stress level in any part of its structure: a reliable indicator of inefficient use of material is the low value of stress or strain in some part of the structure. Ideally in a structure the stress level should be close to the same safe level. This concept leads to a rejection criterion based on the local stress level where low-stressed areas of material are assumed to be under-utilized and can be removed subsequently; the removal process can be undertaken by deleting elements from the finite element model. A simple way to define the stress level consists in comparing the effective von Mises stress of the element 𝜎𝑘

𝑣𝑀 with the maximum von Mises stress level of the whole structure 𝜎𝑚𝑎𝑥

𝑣𝑀 . After the finite element analysis, all the elements that satisfy the following rule can be removed from the model:

𝜎𝑘𝑣𝑀

𝜎𝑚𝑎𝑥𝑣𝑀 < 𝑅𝑅𝑖 (3.55)

Where 𝑅𝑅𝑖 is the current rejection ratio. The von Mises theory proposes that the total strain energy can be separated into two components: the volumetric (hydrostatic) strain energy and the shape (distortion or shear) strain energy. It is proposed that yield occurs when the distortion component exceeds that at the yield point for a simple tensile test. This theory is approximately acceptable for ductile materials but not for brittle ones; for brittle materials the maximum principle stress theory in considered more correct: according to this theory failure will occur when the maximum principal stress in a system reaches the value of the maximum stress at elastic limit in simple tension. Another consideration on the von Mises method regards the way it considers the tension layout: it does not consider correctly a strong anisotropy load layout that produces a non-homogeneous tensions layout with a prevalent tension in one direction and moreover cannot allows to analyze each tension separately. It is important to underline here that the optimization algorithm is not developed to work with any condition or any material but often is improved to work only with certain specific cases, for all the other cases a different algorithm, that uses different tension analysis methods, may be necessary to produce trustworthy results.

37

Such a cycle of FEA and element removal is repeated using the same value of 𝑅𝑅𝑖 until a steady state is reached, at this point there are no more elements being deleted with the current rejection ratio. At this stage an evolutionary rate 𝐸𝑅 is added to the rejection ratio that becomes:

𝑅𝑅𝑖+1 = 𝑅𝑅𝑖 + 𝐸𝑅 (3.56)

With the increased ratio the iteration process takes place again until a new steady state is reached. The process continues until a desired optimum is obtained, for example when there is no material in the final structure that has a stress level lower than 25% of the maximum allowable. The optimization procedure is summarized in five steps, presented below:

Step 1: discretize the structure using a fine mesh of finite elements; Step 2: carry out a finite element analysis for the structure; Step 3: remove all the elements that satisfy the defined rule for the stress

ratio; Step 4: increase the rejection ratio once steady state is reached; Step 5: repeat steps 3 and 4 until a satisfying optimum solution is reached.

Stiffness is one of the key factors that must me taken into account when designing a mechanical structure, commonly the mean compliance C (inverse measure of the overall stiffness) of a structure is considered. The mean compliance is considered as the total strain energy of the structure (external work done by applied loads):

𝐶 =1

2𝑓𝑇𝑢 (3.57)

Where f and u are the external force and the displacement vectors respectively. Even if this equation has been presented before, it is better to remember that in FEA, the static equilibrium of a structure is expressed as:

𝑲𝑢 = 𝑓 (3.58)

K is the global stiffness matrix. When the i-th element is removed from the structure, the stiffness matrix will change by:

∆𝑲 = 𝑲∗ −𝑲 = −𝑲𝒊 (3.59)

In the previous equation, 𝑲∗ is the stiffness matrix of the resulting structure and 𝑲𝒊 is the element i-th stiffness. The most general assumption is that vector f, the vector of the applied external loads, does not get modified by the element removal process. Consequently to the elements removal, the displacement and the displacement vector may experience a variation too, expressed as:

∆𝑢 = −𝑲−1∆𝑲𝑢 (3.60)

38

In conclusion, the mean compliance is expressed as:

∆𝐶 =1

2𝑓𝑇∆𝑢 = −

1

2𝑓𝑇𝑲−1∆𝐾𝑢 =

1

2𝑢𝑖𝑇𝑲𝒊𝑢𝑖 (3.61)

The sensitivity is:

𝛼𝑖𝑘 =

1

2𝑢𝑖𝑇𝑲𝑖𝑢𝑖 (3.62)

The previous equation indicates that the increase in the mean compliance as a consequence of the removal of the i-th element is equal to its elemental strain energy. Minimizing the mean compliance (that is the equivalent to maximizing the stiffness) through the element removal process is often the main objective of optimization process, it can be obtained in an effective way by removing elements with the lowest values of 𝛼𝑖 so that the increase in C will be minimal. The number of elements to be removed is determined by the element removal ratio, which is defined as the ratio of the number of elements removed at each iteration to the total number of elements of the initial FEA model. The stiffness optimization procedure can be summarized into the following steps:

Step 1: discretize the structure using a fine mesh of finite elements; Step 2: carry out a finite element analysis for the structure; Step 3: calculate the sensitivity number for each element; Step 4: remove a number of elements with the lowest sensitivity value

according to the predefined element removal ratio ERR; Step 5: repeat steps 2 to 4 until the mean compliance (or maximum

displacement) of the analysed structure reaches the defined limit. Contrarily to the optimization procedure based on the stress level, the optimization procedure for stiffness does not require a specified steady state: in this case it is possible to improve the computational efficiency with fair less iterations but sometimes it may result in numerical problems such as the production of unstable structures. A sensitivity number can be derived for a displacement constraint where the maximum displacement of the structure or the displacement at a specific location of the structure has to be within a predefined limit. The ESO method starts from the full structure design and removes inefficient material from the structure according to stress and strain energy levels of the elements: it is a simple concept, does not require sophisticated mathematical programming techniques, it can be implemented on available FEA software packages. One of the main advantages of the method is that it does not require regenerating new finite element meshes even when the final structure has departed substantially from the initial design; element removal is done by simply assigning the material property number of the rejected elements to zero and then ignoring those elements when the global stiffness matrix is assembled in the subsequent finite element analysis. As more and more elements are removed, the number of equations to be solved diminishes consequently obtaining a substantial reduction of computation time, also for large three-dimensional structures. To minimize the material usage under a given performance constraint, the ESO method acts on the structure reducing its weight (or volume) by gradually removing

39

material until the constraint cannot be satisfied anymore. However there is a limit in this procedure, sometimes it is possible that part of material removed in an initial iteration might be required later to be part of the optimal design and the ESO method is not able to recover that material once it has been deleted from the design. The ESO method may be able to provide an improved solution over an initial design, but it often cannot be considered as an absolute optimum result. The bi-directional evolutionary structural optimization (BESO) method goes past the limits of the ESO method and allows material to be removed and added simultaneously. In the BESO method applied to stiffness optimization, the sensitivity numbers of void elements are estimated through a linear extrapolation of the displacement field after the finite element analysis. At this point, the solid elements with the lowest sensitivity number are removed from the structure and the void elements with the highest sensitivity are changed back into solid elements. The numbers of removed and added elements in each iteration are determined by two unrelated parameters, RR and IR that are the rejection ratio and the inclusion ratio respectively. The BESO concept has also been applied to “full stress design” by using the von Mises criterion where elements with the lowest stresses are removed and void elements near the regions with the highest stressed are switched back to solid. In a comparable way to the stiffness optimization problem, the numbers of elements to be rejected and added are defined by a rejection ratio and an inclusion ratio. Topology optimization main aim is searching for the stiffest structure with a given volume of material, in BESO method a structure is optimized by removing and adding elements: in this case the element itself is treated a design variable. The optimization problem with the volume constraint is defined as:

𝐶 =1

2𝑓𝑇𝑢 (3.63)

Subjected to:

𝑉∗ − ∑ 𝑉𝑖𝑁𝑖=1 𝑥𝑖 = 0 (3.64)

This problem formulation is the most used currently for topology optimization problems. 𝑥𝑖 is a binary design variable that expresses the absence (0) or presence (1) of an element. f and u are the external force and the displacement vectors respectively, C is the mean compliance, 𝑉∗ and 𝑉𝑖 are the total volume of the structure and the individual volume of an element respectively. When a solid element is removed from the structure, the change of the mean compliance or total strain energy is equal to the element strain energy and it is defined as the elemental sensitivity number:

𝛼𝑖𝑘 =

1

2𝑢𝑖𝑇𝑲𝑖𝑢𝑖 = Δ𝐶𝑖 (3.65)

𝑲𝑖 is the elemental stiffness matrix and 𝑢𝑖 is the nodal displacement vector. When a non-uniform mesh is assigned, the sensitivity number has to take into account the effect of the element volume, in this case the sensitivity number is replaced with the elemental strain energy:

𝛼𝑖𝑒 = 𝑒𝑖

=1

2𝑢𝑖𝑇𝑲𝑖𝑢𝑖

𝑉𝑖 (3.66)

40

The procedure for stiffness optimization is driven by removing the elements with the lowest sensitivity numbers, computed using one of the two equation presented before; for void elements the sensitivity number is initially set to zero. To add material into the design domain, a filter scheme is used to obtain the sensitivity number for void elements and to smooth the sensitivity number in the whole design domain. Thanks to the filter scheme, the checkerboard patterns problem and the mesh-dependency are resolved at once. When a continuum structure is discretized using low order bilinear (2D) or trilinear (3D) finite elements, the sensitivity number can become 𝐶0 discontinuous across element boundaries: this leads to checkerboard patterns in the resulting topologies, these patterns cause heavy difficulties in interpreting and manufacturing the optimal structure. To supress the formation of the checkerboard patterns a smoothing scheme to average the sensitivity numbers of neighbouring elements can be employed, but this solution does not overcome the mesh-dependency problem. The mesh-dependency problem refers to the problem of obtaining different topologies from using finite element meshes: when a finer mesh is used, the numerical process of structural optimization will produce a final topology with more elements of smaller size. Ideally mesh-refinement should result in a better finite element modelling of the same optimal structure (with a better description of boundaries) and not in a more detailed or qualitatively different structure. Commonly employed methods to overcome the mesh-dependency problem are the perimeter control method and the sensitivity filter scheme: the perimeter control method applied to BESO method allows obtaining mesh-independent solutions thanks to the extra constraint (the perimeter length) imposed to the optimization problem. The biggest challenge given by the perimeter control method consists in predicting the adequate value of perimeter length for a new design, for this reason the sensitivity filter scheme is the most used solution for BESO method implementation. In order to apply the filter scheme it is necessary to define the nodal sensitivity numbers by averaging the elemental sensitivity numbers:

𝛼𝑖𝑛 = ∑ 𝑤𝑖𝛼𝑖

𝑒𝑀𝑖=1 (3.67)

M denotes the total number of elements connected to the j-th node, 𝑤𝑖 is the weight factor of the i-th element, it can be defined as (𝑟𝑖𝑗 is the distance between the centre

of the i-th node and the j-th node):

𝑤𝑖 =1

𝑀−1(1 −

𝑟𝑖𝑗

∑ 𝑟𝑖𝑗𝑀𝑖=1

) (3.68)

The weight factor indicates that the elemental sensitivity number has larger effect on the nodal sensitivity number when it is closer to the node. The nodal sensitivity number will be converted into smoothed elemental sensitivity number: nodal sensitivity numbers are projected on the design domain and the filter scheme is used to carry out this process. The filter posses a fixed scale length 𝑟𝑚𝑖𝑛 (does not change with mesh refinement), its primary role is to identify the nodes that influence the sensitivity of the i-th element. The filter scheme generates a circle of radius 𝑟𝑚𝑖𝑛 centred at the centroid of the i-th element, thus generating a circular sub-domain Ω𝑖 (Fig. 11). The value of

41

𝑟𝑚𝑖𝑛 should be big enough so that the sub-domain covers more than one element and as 𝑟𝑚𝑖𝑛, also the sub-domain does not change with mesh refinement. All the nodes inside the sub-domain Ω𝑖 contribute to the computation of the improved sensitivity number of the i-th element as:

𝛼𝑖 =

∑ 𝑤(𝑟𝑖𝑗)𝛼𝑖𝑛𝑁

𝑗=1

∑ 𝑤(𝑟𝑖𝑗)𝑁𝑗=1

(3.69)

Where N is the total number of nodes inside the sub-domain and 𝑤(𝑟𝑖𝑗) is the linear

weight factor, defined as: 𝑤(𝑟𝑖𝑗) = 𝑟𝑚𝑖𝑛 − 𝑟𝑖𝑗 .

Figure 11: Sub-domain defined by the filter scheme for the i-th element [3].

The filter scheme is able to smooth the sensitivity number in the whole design domain, in this way the sensitivity number for void elements is obtained too. It is possible that the sensitivity numbers of some void elements have high values, in this case those void elements can be changed into solid elements in the successive iterations. The filter scheme does not consider the element status (void or solid) and the initial sensitivity for voids is set to zero; as a result of the filtering technique non-zero sensitivity numbers for void elements are found and then it is possible to rank void elements alongside of solid elements in terms of structural importance. The filtering scheme addresses many numerical problems in topology optimization, such as checkerboard patterns and mesh dependency, it requires little extra computational time and it is easy to implement in the optimization algorithm; however the objective function and the corresponding topology may not be convergent. With BESO methods large oscillations are often observed in the evolution history of the objective function: the reason of such chaotic behaviour is that the sensitivity numbers of solids (1) and voids (0) are based on discrete design variables of element presence (1) and absence (0); this condition entails that the convergence between topology and the objective function is difficult to be achieved.

42

An efficient solution to this problem consists in averaging the sensitivity number with its historical information, the simple averaging scheme is:

𝛼𝑖 =𝛼𝑖𝑘+𝛼𝑖

𝑘+1

2 (3.70)

k is the current iteration number. After each iteration is solved, let 𝛼𝑖

𝑘 = 𝛼𝑖 that then will be used in the next iteration: in this way the updated sensitivity number includes the whole history of the sensitivity information in the previous iterations. Whilst the averaging scheme affects the searching path of the BESO algorithm, its effect on the final solution is very small when it becomes convergent and thanks to the evolution history, the final result is highly stable in both the topology and the objective function. Before adding or removing elements from the current design, the target volume for the next iteration 𝑉𝑘+1 needs to be given first: since the volume constraint 𝑉∗ can be greater or smaller than the volume of the initial guess design, the target volume in each iteration may decrease or increase step by step until the target value is achieved. Volume evolution is expressed as:

𝑉𝑘+1 = 𝑉𝑘(1 ± 𝐸𝑅) 𝑤𝑖𝑡ℎ 𝑘 = 1,2, … (3.71)

ER is the evolutionary volume ratio. Once the volume constraint is satisfied, the volume of the structure is kept constant for all the following iterations as 𝑉𝑘+1 = 𝑉

∗. At this point the sensitivity numbers of all elements (both solid and void) are calculated as described before; elements are then sorted according to the values of their sensitivity numbers. A solid (1) element will be removed (switched to 0) if 𝛼𝑖 ≤ 𝛼𝑑𝑒𝑙

𝑡ℎ whereas a void (0) element will be added if 𝛼𝑖 > 𝛼𝑎𝑑𝑑𝑡ℎ ; 𝛼𝑑𝑒𝑙

𝑡ℎ and 𝛼𝑎𝑑𝑑𝑡ℎ are

the thresholds sensitivity numbers for removing and adding elements respectively. To define the thresholds sensitivity numbers there are two different procedures that will be explained from now on. 𝛼𝑎𝑑𝑑𝑡ℎ is defined according to the following procedure:

1. Let 𝛼𝑎𝑑𝑑

𝑡ℎ = 𝛼𝑑𝑒𝑙𝑡ℎ = 𝛼𝑡ℎ

, 𝛼𝑡ℎ can be determined by 𝑉𝑘+1

. An example will explain this concept easily, if there are 1000 elements in the design domain with the sensitivity numbers listed as 𝛼1 > 𝛼2 > ⋯ >𝛼1000 and if 𝑉𝑘+1

corresponds to a design with 725 elements, then 𝛼𝑡ℎ = 𝛼725.

2. Calculate the volume addition ratio AR which is defined as the number of added elements divided by the total number of elements in the design domain; if 𝐴𝑅 < 𝐴𝑅𝑚𝑎𝑥 (𝐴𝑅𝑚𝑎𝑥 is the prescribed maximum volume addition ratio) then step 3 can be skipped, otherwise recalculate 𝛼𝑑𝑒𝑙

𝑡ℎ and 𝛼𝑎𝑑𝑑𝑡ℎ as prescribed is the third step.

3. Calculate 𝛼𝑎𝑑𝑑𝑡ℎ by first sorting the sensitivity number of void elements

(0), the number of elements to be switched from 0 to 1 will be equal to 𝐴𝑅𝑚𝑎𝑥 multiplied by the total number of element ranked just below the last added element. 𝛼𝑎𝑑𝑑

𝑡ℎ is the sensitivity number of the element ranked just below the last added element, 𝛼𝑑𝑒𝑙

𝑡ℎ is then determined as 𝛼𝑑𝑒𝑙𝑡ℎ ≤ 𝛼𝑎𝑑𝑑

𝑡ℎ ∗ 𝛼𝑑𝑒𝑙𝑡ℎ so that the removed volume is equal to

(𝑉𝑘 − 𝑉𝑘+1 + volume of added elements).

43

ARmax is introduced to ensure that not too many elements are added in a single iteration, otherwise the structure may lose its integrity when the BESO method starts from an initial design guess; generally ARmax is greater than 1% so that it does not supress the capability of adding elements. The cycle of finite element analysis and element removal/addition continues until the objective volume V* is reached and the following convergence criterion, defined in terms of variation of the objective function, is satisfied:

𝑒𝑟𝑟𝑜𝑟 =| ∑ 𝐶𝑘−𝑖+1

𝑁𝑖=1 −∑ 𝐶𝑘−𝑁−𝑖+1

𝑁𝑖=1 |

∑ 𝐶𝑘−𝑖+1𝑁𝑖=1

≤ 𝜏 (3.72)

𝜏 is the allowable convergence tolerance, k is the current iteration number and N is an integer number that normally is selected to be 5, this implies that the change in the mean compliance over the last 10 iterations is relatively small. The BESO method presented in this chapter is considered a “hard-kill” method due to the complete removal of an element instead of changing it into a very soft material; in this way the computational time is significantly reduced, especially for large 3D structures, since the removed elements are not involved in the finite element analysis.

44

The evolutionary iteration procedure of the presented BESO method is given as follows and is represented in Figure 12:

1. Discretize the design domain using a finite element mesh and assign initial property values (0 or 1) for elements to construct an initial design;

2. Perform finite element analysis and then calculate the elemental sensitivity number;

3. Average the sensitivity number with its history information and the save the resulting sensitivity number for the next iteration;

4. Determine the target volume for the next iteration;

5. Add and delete elements;

6. Repeat steps 2-5 until the constraint volume (V*) is achieved and the convergence criterion is satisfied. Figure 12: Flowchart of the BESO method [3].

45

3.6. BESO with Material Interpolation Scheme and Penalization The BESO method in characterized by the definition of the optimal topology according to the relative ranking of sensitivity numbers, these number are estimated by the approximate variation of the objective function due to the removal of individual elements. At the same time it is not so simple to estimate the sensitivity numbers of the void elements because normally information on void elements are not available since they are not included in the finite element analysis. The complete removal of an element from the design domain is an irrational choice and may result in theoretical difficulties in the topology optimization problem. An alternative way to effective remove an element is to reduce the elasticity modulus or one of the element characteristic dimensions (such as thickness) to a very small value. In the BESO method, a solution consists in replacing void elements with orthotropic cellular microstructure to obtain resulting topology designs close to the ones obtained with the original ESO methods (in the original ESO method applied to stiffness optimization, the sensitivity numbers are defined by the approximate variation of the objective function due to element removal). Al alternative to the hard-kill BESO method, here is presented a soft-kill BESO method that uses the material interpolation scheme with penalization. Material interpolation schemes with penalization have been widely used in the SIMP method (SIMP method is not discussed in this work) to steer the solution to nearly solid-void designs and now are used with the same goal in the BESO method. During the years different material interpolation schemes have been compared to various bounds for effective material properties in composite and showed that composite materials (composite materials made out of solid and void elements) from intermediate densities are physically realizable. To obtain a solid-void design with maximum stiffness, the mean compliance is minimized for a fixed volume of material; the resulting topology problem can be stated as:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐶 =1

2𝑓𝑇𝑢 (3.73)

Subjected to: 𝑉∗ − ∑ 𝑉𝑖

𝑁𝑖=1 𝑥𝑖 = 0 (3.74)

With 𝑥𝑖 = 𝑥𝑚𝑖𝑛 𝑜𝑟 1, it is a binary design variable that denotes the element i-th density; 𝑉∗and 𝑉𝑖 are the prescribed volume of the final structure and the volume of the i-th element respectively. The value of 𝑥𝑚𝑖𝑛 is usually very small, as for example 𝑥𝑚𝑖𝑛 = 0.001, this value is used to define void elements and implies that no element can be removed completely from the design domain: this is the biggest difference between the soft-kill approach and the hard-kill method where void elements are completely removed. To achieve a nearly solid-void design, the elasticity modulus of the intermediate material is interpolated as a function of the element density:

𝐸(𝑥𝑖) = 𝐸1𝑥𝑖𝑝

(3.75)

𝐸1 denotes the elasticity modulus of the solid material and p the penalty exponent. The Poisson ratio is assumed as independent of the design variables and the global

46

stiffness matrix K can be expressed by the elemental stiffness and the design variable 𝑥𝑖:

𝑲 = ∑ 𝑥𝑖𝑝𝑲𝑖

0𝑖 (3.76)

𝑲𝑖0 is the elemental stiffness matrix of the solid element.

In FEA the equilibrium equation of a static structure can be expressed as Ku=f. If the mean compliance is considered as the objective function and if the design variable 𝑥𝑖 continuously changes from 1 to 𝑥𝑚𝑖𝑛, the sensitivity of the objective function with respect to the change in the design variable is:

𝑑𝐶

𝑑𝑥𝑖=

1

2

𝑑𝑓𝑇

𝑑𝑥𝑖𝑢 +

1

2𝑓𝑇

𝑑𝑢

𝑑𝑥𝑖 (3.77)

The adjoint method is now used to determine the sensitivity of the displacement vector. The objective function is modified as consequence of the introduction of a Lagrangian multiplier vector 𝜆 but the equilibrium does not change:

𝐶 =1

2𝑓𝑇𝑢 + 𝜆𝑇(𝑓 − 𝑲𝑢) (3.78)

The sensitivity then changes too (elements variation has no effect on the load vector):

𝑑𝐶

𝑑𝑥𝑖= (

1

2𝑓𝑇 − 𝜆𝑇𝑲)

𝑑𝑢

𝑑𝑥𝑖− 𝜆𝑇

𝑑𝑲

𝑑𝑥𝑖𝑢 (3.79)

Since the term (𝑓 − 𝑲𝑢) is equal to zero, the Lagrangian multiplier 𝜆 can be chosen

freely, its correct value is then 𝜆 =1

2𝑢 and substituting in the sensitivity equation,

the latter becomes:

𝑑𝐶

𝑑𝑥𝑖= −

1

2𝑢𝑇

𝑑𝑲

𝑑𝑥𝑖𝑢 (3.80)

Finally the sensitivity of the objective function with regard to the change in the i-th element is found as:

𝜕𝐶

𝜕𝑥𝑖= −

1

2𝑝𝑥𝑖

𝑝−1𝑢𝑖𝑇𝑲𝒊

𝟎𝑢𝑖 (3.81)

In the BESO method the structure is optimized using discrete design variable, this means that only two bound materials are allowed in the design; the sensitivity number used in the method is defined by the relative ranking of the sensitivity of an individual element, as:

𝛼𝑖 = −1

𝑝

𝜕𝐶

𝜕𝑥𝑖{

1

2𝑢𝑖𝑇𝑲𝒊

𝟎𝑢𝑖 , 𝑤ℎ𝑒𝑛 𝑥𝑖 = 1

𝑥𝑚𝑖𝑛𝑝−1

2𝑢𝑖𝑇𝑲𝒊

𝟎𝑢𝑖 , 𝑤ℎ𝑒𝑛 𝑥𝑖 = 𝑥𝑚𝑖𝑛

(3.82)

47

The sensitivity number of the soft elements depends on the selection of the penalty exponent p, since the penalty exponent tends to infinity, the sensitivity number becomes:

𝛼𝑖 = {

1

2𝑢𝑖𝑇𝑲𝒊

𝟎𝑢𝑖 , 𝑤ℎ𝑒𝑛 𝑥𝑖 = 1

0, 𝑤ℎ𝑒𝑛 𝑥𝑖 = 𝑥𝑚𝑖𝑛

(3.83)

The above equation states that the sensitivity numbers of solid and soft elements are equal to the elemental strain energy and zero, respectively. The stiffness 𝑲 of soft elements also becomes zero as the penalty exponent approaches infinity. When the penalty exponent tends to infinity the soft elements are equivalent to void elements and can be completely removed from the design domain as in the hard-kill BESO: in conclusion the hard-kill BESO is a special case of the soft-kill BESO. The optimality criterion in this case can be derived if no restriction is imposed on the design variables 𝑥𝑖 , for example the strain energy densities of all elements should be equal); the elements with higher strain energy density should have 𝑥𝑖 increased and the elements with lower strain energy density should have 𝑥𝑖 decreased. In the soft-kill BESO method the design variables are restricted to be either 1 or 𝑥𝑚𝑖𝑛 so the optimality criterion can be described as strain energy densities of solid elements are always higher than those of soft elements. To improve the convergence of the soft-kill BESO method, the averaging scheme defined for the hard-kill BESO method should be applied to the sensitivity numbers to take into account the historical information of the element density from previous iterations. The averaging scheme will supress unwarranted changes of design variables for solid elements with high historical sensitivity numbers and void elements with low historical sensitivity numbers. Differently from the hard-kill method, the soft-kill BESO method can add and recover elements without any help of a filter scheme, however in this case the optimized design will be affected by checkerboard patterns so has little practical value and the filter technique is necessary to supress checkerboard patterns and to overcome mesh-dependency problems.

3.7. BESO for Extended Topology Optimization Problems Topology optimization problems may consider objective functions different from maximum stiffness and constraints from structural volume. For topology optimization techniques based on FEA, elements acting as the design variables cannot be directly eliminated from the design domain unless soft elements are fully equivalent to void elements: this means that for a new topology optimization problem is more reliable to develop a soft-kill BESO and then to explore the possibility of a hard-kill approach. Based on the given problem statement on the optimization objectives and constraints, the sensitivity of the objective function with respect to the design variables should be determined according to the assumed material interpolation scheme. Sensitivity numbers that provide the relative ranking of elemental sensitivities are established for discrete design variables (discrete design variables are updated according to sensitivity numbers) so the structure evolves to an optimum automatically.

48

In this paragraph a series of possible applications of the BESO method are presented.

3.7.1. Minimizing Structural Volume with a Displacement Constraint In this optimization problem, the objective is to save material in a structure and the constraint is imposed on the mean compliance or on the displacement; this problem has a significant application in practice, for example some structures require that the maximum displacement is below a defined value. The BESO method evolves a structure by removing and adding material so a volume constraint can be implemented easier than a displacement or a mean compliance constraint; the BESO procedure cannot be used directly for applications that require different constraints than structural volume. In this case, the optimization problem based on finite element analysis is stated as:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑉 = ∑ 𝑉𝑖𝑥𝑖

𝑁𝑖=1 (3.84)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑢𝑗 = 𝑢𝑗∗ 𝑎𝑛𝑑 𝑥𝑖 = 𝑥𝑚𝑖𝑛 𝑜𝑟 1 (3.85)

𝑢𝑗 and 𝑢𝑗

∗ denote the j-th displacement and its constraint, respectively.

In order to solve this problem using the BESO method, the displacement constraint is added to the objective function by introducing the Lagrange multiplier 𝜆.

𝑓1(𝑥) = ∑ 𝑉𝑖𝑥𝑖𝑁𝑖=1 + 𝜆(𝑢𝑗 − 𝑢𝑗

∗) (3.86)

The modified objective function is equivalent to the original one and the Lagrangian multiplier can be any constant if the displacement constraint is respected. The derivative of the modified objective function 𝑓1(𝑥) is:

𝑑𝑓1

𝑑𝑥𝑖= 𝑉𝑖 + 𝜆

𝑑𝑢𝑗

𝑑𝑥𝑖 (3.87)

In order to calculate 𝑑𝑢𝑗

𝑑𝑥𝑖, a virtual unit load 𝑓𝑗 is introduced where only the

corresponding j-th component is equal to unity and all other components are equal to zero.

𝑢𝑗 = 𝑓𝑗𝑇𝑢 (3.88)

𝑑𝑢𝑗

𝑑𝑥𝑖= −𝑝𝑥𝑖

𝑝−1𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖 (3.89)

𝑢𝑖𝑗 is found from the adjoint equation 𝑓𝑗 −𝑲𝑢𝑖𝑗 = 0.

𝑢𝑖𝑗 is the virtual displacement vector oth the i-th element resulted from a unit

virtual load 𝑓𝑗 . The dimension of 𝑢𝑖 and 𝑢𝑖𝑗 is the same of vector 𝑢 , all the

components in 𝑢𝑖 and 𝑢𝑖𝑗 that are unrelated to the element i-th are zero. Similarly

the dimension of 𝑲𝑖0 is the same of those of 𝑲

but all components in the first, that are unrelated to the i-th element are zero.

49

The derivative of the objective function becomes:

𝑑𝑓1

𝑑𝑥𝑖= 𝑉𝑖 − 𝜆𝑝𝑥𝑖

𝑝−1𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖 (3.90)

When a uniform mesh is used (all the elements of the mesh have the same volume), the relative ranking of sensitivity of each element is defined as:

𝛼𝑖 = −1

𝜆𝑝(𝑑𝑓1

𝑑𝑥𝑖− 𝑉𝑖) = 𝑥𝑖

𝑝−1𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖 (3.91)

In this application of the BESO method, only two discrete values are used and the sensitivity numbers for soft and solid elements are the following:

𝛼𝑖 = {𝑢𝑖𝑇𝑲𝒊

𝟎𝑢𝑖 , 𝑤ℎ𝑒𝑛 𝑥𝑖 = 1

𝑥𝑖𝑝−1𝑢𝑖

𝑇𝑲𝒊𝟎𝑢𝑖 , 𝑤ℎ𝑒𝑛 𝑥𝑖 = 𝑥𝑚𝑖𝑛

(3.92)

For the present optimization problem, the structural volume has to be minimized and defined according to the displacement constraint. Once the sensitivity of the displacement 𝑢𝑗 is known, it is possible to define the variation of displacement due

to change in the design variables, its estimation is quantified as:

𝑢𝑗𝑘+1 ≈ 𝑢𝑗

𝑘 + ∑𝑑𝑢𝑗

𝑘

𝑑𝑥𝑖

𝑖 ∆𝑥𝑖 (3.93)

𝑢𝑗𝑘 and 𝑢𝑗

𝑘+1 denote the j-th displacement in the current and in the successive

iterations respectively. From the above equation it is possible to compute the threshold of the sensitivity number as well as the corresponding volume 𝑉𝑐, these results are obtained setting 𝑢𝑗

𝑘+1 = 𝑢∗. A possible drawback consists in the

eventuality that the resulting volume 𝑉𝑐 is much larger or far smaller than the current design, so in order to have a gradual evolution of the topology, the following equation is adopted to determine the structural volume for the next iteration:

𝑉 𝑘+1 = {

max (𝑉𝑘(1 − 𝐸𝑅), 𝑉𝑐) , 𝑤ℎ𝑒𝑛 𝑉𝑘 > 𝑉𝑐

max (𝑉𝑘(1 + 𝐸𝑅), 𝑉𝑐) , 𝑤ℎ𝑒𝑛 𝑉𝑘 ≤ 𝑉𝑐 (3.94)

In this way the volume change in each iteration is less than the prescribed evolutionary volume ratio ER, which defines the maximum variation of structural volume in a single iteration. This procedure applies correctly only to soft-kill BESO method. In the hard-kill BESO method, the penalty exponent p is infinite and therefore the derivative of the displacement is infinite too for the solid elements: it is possible to avoid this problem and determine the structural volume using the following algorithm, where the current displacement is compared with its constraint value.

50

𝑉 𝑘+1 =

{

max [(𝑉𝑘(1 − 𝐸𝑅)), (𝑉𝑘 (1 −

𝑢𝑗𝑘−𝑢𝑗

∗

𝑢𝑗∗ ))]; 𝑤ℎ𝑒𝑛 𝑢𝑗

𝑘 > 𝑢𝑗∗

min [(𝑉𝑘(1 + 𝐸𝑅)) , (𝑉𝑘 (1 +𝑢𝑗∗−𝑢𝑗

𝑘

𝑢𝑗∗ ))]; 𝑤ℎ𝑒𝑛 𝑢𝑗

𝑘 ≤ 𝑢𝑗∗

(3.95)

The biggest challenge in this optimization problem is to properly select the displacement constraint, usually the chosen displacement relates directly to the overall structural performance: for example it is possible to choose the maximum deflection at the loading point as maximum displacement constraint. It is now possible to reconsider the previous problem taking into account an additional displacement constraint, for example a local displacement constraint imposed on the horizontal movement of a support, necessary when the displacement at a specific location is required to be within a prescribed limit. In topology optimization for normal structures (non-compliant mechanisms), a certain measure of the overall structural performance should be included in the objective function whereas a local displacement should be considered as a constraint. The topology optimization problem of maximizing stiffness with a volume constraint and an additional displacement constraint can be as:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐶 =1

2𝑓𝑇𝑢 (3.96)

Subjected to: 𝑢𝑗𝑘 ≤ 𝑢𝑗

∗ (3.97)

𝑉∗ − ∑ 𝑉𝑖𝑁𝑖=1 𝑥𝑖 = 0 (3.98)

And with: 𝑥𝑖 = 𝑥𝑚𝑖𝑛 𝑜𝑟 1 The above optimization problem has multiple constraints and the BESO method developed bellow displays the general principle for solving this type of problems. The local displacement constraint is added to the objective function by introducing a Lagrangian multiplier 𝜆, the modified objective function becomes:

𝑓1(𝑥) =1

2𝑓𝑇𝑢 + 𝜆(𝑢𝑗 − 𝑢𝑗

∗) =1

2𝑢𝑖𝑇𝑲𝒊𝑢𝑖 + 𝜆(𝑢𝑗 − 𝑢𝑗

∗) (3.99)

The modified objective function is equivalent to the original one when the displacement is equal to its constraint value, otherwise 𝜆 = 0 if 𝑢𝑗 < 𝑢𝑗

∗ which

means that the displacement constraint is already satisfied, 𝜆 → ∞ if 𝑢𝑗 > 𝑢𝑗∗ which

require to minimize 𝑢𝑗 in order to satisfy the constraint in later iterations. The main

function of the Lagrangian multiplier is to act as compromise between the objective function and the displacement constraint. Applying the adjoint method, the sensitivity of the modified objective function is:

𝑑𝑓1

𝑑𝑥𝑖= 𝑝𝑥𝑖

𝑝−1(−1

2𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖 − 𝜆𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖) (3.100)

Accordingly the elemental sensitivity number is:

𝛼𝑖 = −1

𝑝(𝑑𝑓1(𝑥)

𝑑𝑥𝑖) = 𝑥𝑖

𝑝−1(1

2𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖 + 𝜆𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖) (3.101)

51

And as a result, the sensitivity numbers for solid and soft elements are expressed as:

𝛼𝑖 = {

1

2𝑢𝑖𝑇𝑲𝒊

𝟎𝑢𝑖 + 𝜆𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖, 𝑤ℎ𝑒𝑛 𝑥𝑖 = 1

𝑥𝑖𝑝−1(

1

2𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖 + 𝜆𝑢𝑖𝑗𝑇𝑲𝒊

𝟎𝑢𝑖), 𝑤ℎ𝑒𝑛 𝑥𝑖 = 𝑥𝑚𝑖𝑛

(3.102)

The sensitivity numbers are dependent on the Lagrangian multiplier 𝜆, so it has to be defined first: an appropriate value of 𝜆 can be determined when both volume and displacement constraint are satisfied. 𝜆 is defines as:

𝜆 =1−𝑤

𝑤 (3.103)

w is ranging from a minimum value 𝑤𝑚𝑖𝑛 to 1. To define the appropriate value of w, two initial bound values wlower and wupper are set: wlower = 𝑤𝑚𝑖𝑛 and wupper = 1. The program starts from an initial guess of w = 1 and then the sensitivity number is defined: the threshold of sensitivity numbers can be determined when the volume constraint is satisfied by assuming that the elemental density is 𝑥𝑚𝑖𝑛 or 1if the sensitivity number is smaller or larger than the threshold accordingly. To define the displacement 𝑢𝑗

𝑘+1 of the successive iteration, the following equation

is employed:

𝑢𝑗𝑘+1 ≈ 𝑢𝑗

𝑘 + ∑𝑑𝑢𝑗

𝑘

𝑑𝑥𝑖

𝑖 ∆𝑥𝑖 (3.104)

If 𝑢𝑗𝑘+1 > 𝑢𝑗

∗, w is updated with a smaller value: �̂� =𝑤+𝑤𝑙𝑜𝑤𝑒𝑟

2, the upper bound

becomes wupper = w.

If 𝑢𝑗𝑘+1 < 𝑢𝑗

∗, w is updated with a bigger value: �̂� =𝑤+𝑤𝑢𝑝𝑝𝑒𝑟

2, the lower bound

becomes wlower = w. With the updated 𝑤 = �̂�, the above procedure is repeated until wupper – wlower is lower than the fixed minimum value. After a certain number of iterations, an accurate value for the Lagrangian multiplier is found. This procedure is typically used for any additional constraint except for the volume constraint.

3.7.2. Topology Optimization for Natural Frequency

Frequency optimization is of great importance in many engineering fiels, but compared with the extensive research on stiffness optimization, there has been much less work concerned with topology optimization for natural frequency. The standard BESO method uses only two discrete design variables, xmin and 1, so it is not suitable to apply a discontinuous material interpolation scheme, a modified soft-kill BESO method will be presented here for frequency optimization problems. In finite element analysis, the dynamic behaviour of a structure can be represented by the following general eigenvalue problem:

(𝑲 − 𝜔𝑗2𝑴)u𝑗 = 0 (3.105)

52

𝑲 is the global stiffness matrix, M is the global mass matrix, 𝜔𝑗 is the j-th natural

frequency and u𝑗 is the eigenvector corresponding to 𝜔𝑗 . u𝑗 and 𝜔𝑗

are related to

each other by the Rayleigh quotient:

𝜔𝑗2 =

𝑢𝑗𝑇𝑲u𝑗

𝑢𝑗𝑇𝑴u𝑗

(3.106 )

The topology optimization problem in this case consists in maximizing the natural frequency 𝜔𝑗

; for a solid void design, the problem can be stated as:

𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝜔𝑗

(3.107)

Subject to: 𝑉∗ − ∑ 𝑉𝑖

𝑁𝑖=1 𝑥𝑖 = 0 (3.108)

And with: 𝑥𝑖 = 𝑥𝑚𝑖𝑛 𝑜𝑟 1. 𝑉∗ and 𝑉𝑖 are the prescribed total structural volume and the volume of the i-th element respectively. To obtain the gradient information of the design variable, it is necessary to interpolate the material properties between 𝑥𝑚𝑖𝑛 and 1 : a popular material interpolation scheme is to apply the power law penalization model to the stiffness (𝐸(𝑥𝑖) = 𝐸1𝑥𝑖

𝑝) and a linear interpolation model to the density (𝜌(𝑥𝑖) = 𝑥𝑖𝜌0). This

scheme results in numerical difficulties, due to the extreme high ratio between mass and stiffness for small value of 𝑥𝑖 that causes artificial localized vibration modes in the low-density areas. A simple way to avoid this problem consists in keeping the ratio between mass and stiffness constant when 𝑥𝑖 = 𝑥𝑚𝑖𝑛, this requires that:

𝜌(𝑥min) = 𝑥𝑚𝑖𝑛𝜌0

𝐸(𝑥min ) = 𝑥𝑚𝑖𝑛𝐸0 (3.109)

𝜌0 and 𝐸0 denote the density and the elasticity modulus of the solid material. The above model is represented in Figure 13 below.

Figure 13: Alternative material interpolation scheme 𝑥𝑚𝑖𝑛 = 0.01 [3].

53

An alternative material interpolation scheme is expressed as:

𝜌(𝑥𝑖) = 𝑥𝑖𝜌0

𝐸(𝑥𝑖) = [ 𝑥𝑚𝑖𝑛−𝑥𝑚𝑖𝑛

𝑝

1−𝑥𝑚𝑖𝑛𝑝 (1 − 𝑥𝑖

𝑝) + 𝑥𝑖𝑝]𝐸0 (3.110)

0 < 𝑥𝑚𝑖𝑛 < 𝑥𝑖 < 1 At this point the derivatives of the global mass matrix M and the stiffness matrix K are obtained:

𝜕𝑴

𝜕𝑥𝑖= 𝑴𝒊

𝟎 (3.111)

𝜕𝑲

𝜕𝑥𝑖=

1−𝑥𝑚𝑖𝑛

1−𝑥𝑚𝑖𝑛𝑝 𝑝𝑥𝑖

𝑝−1𝑲𝑖0

(3.112)

𝑴𝑖0 and 𝑲𝑖

0 are the mass and stiffness matrices of the i-th element when it is solid. The sensitivity of the objective function 𝜔𝑗

is expressed as:

𝑑𝜔𝑗

𝑑𝑥𝑖 =

1

2𝜔𝑗 𝑢𝑗𝑇𝑴u𝑗

[𝑢𝑗𝑇 (

𝜕𝑲

𝜕𝑥𝑖− 𝜔𝑗

2 𝜕𝑴

𝜕𝑥𝑖) u𝑗] (3.113)

Substituting the derivatives of the global mass matrix M and the stiffness matrix K into the previous equation and assuming that the eigenvector u𝑗 is normalized with

respect to the global mass matrix M, the sensitivity of the j-th natural frequency can be found as:

𝑑𝜔𝑗

𝑑𝑥𝑖 =

1

2𝜔𝑗 𝑢𝑗

𝑇 (1−𝑥𝑚𝑖𝑛

1−𝑥𝑚𝑖𝑛𝑝 𝑝𝑥𝑖

𝑝−1𝑲𝑖0 − 𝜔𝑗

2𝑴𝑖0) u𝑗 (3.114)

The sensitivity numbers for solid and soft elements can be expressed explicitly as:

𝛼𝑖 =1

𝑝

𝑑𝜔𝑗

𝑑𝑥𝑖 =

{

1

2𝜔𝑗 𝑢𝑗

𝑇 (1−𝑥𝑚𝑖𝑛

1−𝑥𝑚𝑖𝑛𝑝 𝐾𝑖

0 −𝜔𝑗2

𝑝𝑀𝑖0) u𝑗 , 𝑥𝑖

= 1

1

2𝜔𝑗 𝑢𝑗

𝑇 (𝑥𝑚𝑖𝑛𝑝−1

−𝑥𝑚𝑖𝑛

1−𝑥𝑚𝑖𝑛𝑝 𝐾𝑖

0 −𝜔𝑗2

𝑝𝑀𝑖0) u𝑗 , 𝑥𝑖

= 𝑥𝑚𝑖𝑛

(3.115)

When 𝑥𝑚𝑖𝑛

tends to 0 (p>1), the sensitivity numbers can be simplified as:

𝛼𝑖 =1

𝑝

𝑑𝜔𝑗

𝑑𝑥𝑖 = {

1

2𝜔𝑗 𝑢𝑗

𝑇 (𝑲𝒊𝟎 −

𝜔𝑗2

𝑝𝑴𝒊𝟎) u𝑗 , 𝑥𝑖

= 1

−𝜔𝑗

2𝑝𝑢𝑗𝑇𝑴𝒊

𝟎u𝑗 , 𝑥𝑖 = 𝑥𝑚𝑖𝑛

(3.116)

It is important to note that 𝑥𝑖

= 𝑥𝑚𝑖𝑛 is still being used for void elements (rather

than 𝑥𝑖 = 0) because the components of the eigenvector u𝑗 related to void elements

would not be found if these elements were totally eliminated from the design domain; for this reason it is difficult to define a rigorous hard-kill BESO method using the present material interpolation scheme.

54

3.7.3. Topology Optimization for Multiple Load Cases

Most real structures are subjected to different loads at different times: these working conditions are referred to as multiple load cases. For example when a structure as a bridge is subjected to a moving load, the force acting on the structure changes from one location to another. A moving load such the one of the example can be approximated to multiple load cases, by applying the load sequentially to a finite number of locations along the path of the moving load, in this condition, the structure has to be designed to take into account all load cases. The application of the BESO method to structures with multiple load cases is relatively simple, the optimization problem can be stated as one of minimizing a weighted average of the mean compliance of all load cases. The topology optimization problem can be stated as:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓(𝑥) = ∑ 𝑤𝑘𝐶𝑘𝑀𝑘=1 (3.117)

Subject to: 𝑉∗ − ∑ 𝑉𝑖

𝑁𝑖=1 𝑥𝑖 = 0 (3.118)

And with: 𝑥𝑖 = 𝑥𝑚𝑖𝑛 𝑜𝑟 1. M is the total number of load cases, 𝑤𝑘 and 𝐶𝑘 are the weighting factor and the mean compliance for the k-th load case respectively. The displacement field of each load case is independent of the others, so the sensitivity of the weighted objective function is found as:

𝑑𝑓

𝑑𝑥𝑖= −

1

2𝑝𝑥𝑖

𝑝−1∑ 𝑤𝑘(𝑢𝑖𝑇𝑲𝒊

𝟎u𝑖)𝑘𝑀𝑘=1 (3.119)

Thus the sensitivity numbers used in BESO can be defined as:

𝛼𝑖 = −1

𝑝

𝑑𝑓

𝑑𝑥𝑖 = {

1

2∑ 𝑤𝑘(𝑢𝑖

𝑇𝑲𝒊𝟎u𝑖)𝑘

𝑀𝑘=1 , 𝑥𝑖

= 1

1

2𝑥𝑖𝑝−1∑ 𝑤𝑘(𝑢𝑖

𝑇𝑲𝒊𝟎u𝑖)𝑘

𝑀𝑘=1 , 𝑥𝑖

= 𝑥𝑚𝑖𝑛

(3.120)

When p tends to infinity, the sensitivity number for the hard-kill BESO method are simplified as:

𝛼𝑖 = {1

2∑ 𝑤𝑘(𝑢𝑖

𝑇𝑲𝒊𝟎u𝑖)𝑘

𝑀𝑘=1 , 𝑥𝑖

= 1

0, 𝑥𝑖 = 0

(3.121)

3.7.4. BESO based on von Mises Stress

The original ESO method was based on von Mises stress, lowly stressed material was assumed as underutilized and removed indeed. In a similar way, the removal and addition of material in the BESO method can be decided by the stress level. The stress criterion is, to a large extent, equivalent to the stiffness criterion but the problem statement for the topology optimization based on von Mises stress differs significantly from the stiffness optimization one.

55

Take a 2D isotropic elastic structure as an example, the von Mises stress of the i-th element is defined as:

𝜎𝑖𝑣𝑚 = √𝜎𝑥𝑥2 + 𝜎𝑦𝑦2 − 𝜎𝑥𝑥 𝜎𝑦𝑦 + 3𝜏𝑥𝑦2 = (𝜎𝑖

𝑇𝑻𝜎𝑖 )1

2 (3.122)

𝜎𝑖 = {𝜎𝑥𝑥

, 𝜎𝑦𝑦 , 𝜎𝑥𝑦

}𝑇

is the elemental stress vector; 𝜎𝑥𝑥 𝑎𝑛𝑑 𝜎𝑦𝑦

are the normal

stresses in x and y directions; 𝜏𝑥𝑦 is the shear stress; matrix 𝑻 is the coefficient

matrix, reported below:

𝑻 = [1 −0.5 0

−0.5 1 00 0 3

] (3.123)

In FEA, 𝜎𝑖

can be calculated from the nodal displacement vector 𝑢𝑖 of the i-th

element as follows: 𝜎𝑖 = 𝑫𝑩𝑢𝑖

(3.124)

D and B are the conventional elasticity and strain matrices, respectively. When the power law material interpolation scheme is applied, the stress vector 𝜎𝑖

can be written as:

𝜎𝑖 = 𝑥𝑚𝑖𝑛

𝑝 𝑫𝟎 𝑩𝑢𝑖 (3.125)

𝑫0 is the elasticity matrix of the solid element. Once the above equation is substituted into the von Mises equation, it yields:

𝜎𝑖𝑣𝑚 = 𝑥𝑚𝑖𝑛

𝑝 √𝑢𝑖𝑇𝑩𝑻𝑫𝟎

𝑻𝑻𝑫𝟎𝑩𝑢𝑖 = 𝑥𝑚𝑖𝑛

𝑝(𝜎𝑖0

𝑇𝑻𝜎𝑖0 )

1

2 = 𝑥𝑚𝑖𝑛𝑝

𝜎𝑖0𝑣𝑚

(3.126)

Where 𝜎𝑖0

𝑣𝑚 is the von Mises stress of the solid element. When the design variable 𝑥𝑖

change, also the correspondent von Mises stress changes. It is now possibile to define the sensitivity number as:

𝛼𝑖 =𝜎𝑖𝑣𝑚

𝑥𝑖 = 𝑥𝑖

𝑝−1𝜎𝑖0𝑣𝑚

(3.127)

For solid and soft elements, the sensitivity numbers are:

𝛼𝑖 = { 𝜎𝑖0

𝑣𝑚, 𝑥𝑖 = 1

𝑥𝑖𝑝−1

𝜎𝑖0𝑣𝑚, 𝑥𝑖

= 𝑥𝑚𝑖𝑛 (3.128)

And when the penalty exponent p tends to infinity, the sensitivity numbers reduce to:

𝛼𝑖 = {𝜎𝑖0𝑣𝑚, 𝑥𝑖

= 1 0, 𝑥𝑖

= 0 (3.129)

In this last case, 𝑥𝑚𝑖𝑛

is replaced with 0 because a soft element with extremely small stiffness is equivalent to a void element.

56

57

4. Structural Optimization with Inspire In this chapter a brief formulation of an optimization problem, applied to a linear case is presented with some key characteristics of the process itself. The optimization work has been conducted on Altair’s optimization package, called solidThinking Inspire which uses the proprietary solver OptiStruct. OptiStruct performs an iterative procedure known as the local approximation method to solve the optimization problem; the method determines the solution using the following steps:

1. Analysis of the physical problem using finite elements; 2. Convergence test to asses if the convergence can be achieved or not; 3. Response screening to retain potentially active responses for the current

iteration; 4. Design sensitivity analysis for retained responses – calculates derivatives of

structural responses with respect to design variables, this function takes FEA from a simple design validation tool to an automated design optimization framework;

5. Optimization of an explicit approximate problem formulated using the sensitivity information.

6. Back to the first step. To achieve a stable convergence, design variable changes during each iteration are limited to a narrow range within their bounds, called move limits. The biggest design variable changes occur within the first few iterations and convergence for practical applications is achieved with only a small number of FE analyses. Design updates are generated solving the explicit approximate optimization problem based on sensitivity information. OptiStruct comes implemented with two classes of optimization methods: Dual Method and Primal Method. The Dual Method solves the optimization problem in the dual space of Lagrange multipliers associated with active constraints; this method is highly effective for design problems involving a large number of design variables but with much fewer constraints: it is well suited for topology and topography optimization processes. The Primal Method searches the optimum solution in the original design variable space; it is used for problems that involve equally as many design variables as design constraints: it is thus suited for size and shape optimization processes. OptiStruct chooses automatically the optimizer method according on the characteristics of the optimization problem. In the previous chapter were mentioned the main elements to define an optimization problem, to recap they are the design space, the design variables, the constraints and the objective. Here, it is possible to express this set of elements as a mathematical statement:

Minimize f(x)=f(x1, x2, x3, x4, … xn) g(x)≤0, with j=1,2,….m (4.1)

Subject to xiL ≤xi≤xiU (4.2)

58

Where f(x) is the objective function, g(x) is the constraint function and x is the vector of design variables. A key part of the optimization process consists in evaluating the sensitivity of the responses to changes in design variables. The response quantity g is calculated from displacement as:

g=uTq (4.3)

The sensitivity of this response with respect to the design variable x, or the gradient of the response is:

𝝏𝒈

𝝏𝒙=

𝝏𝒒𝑻

𝝏𝒙𝒖 + 𝒒𝑻

𝝏𝒖

𝝏𝒙 (4.4)

As is common in the practice, some design problems have more constraints than design variables whereas for some other problems it is the opposite way; this broad range of working conditions requires the software to use different algorithms in each case in order to efficiently arrive to the optimum solution. The two main algorithms used by OptiStruct are Direct Variable method and Adjoin Variable method. Direct Variable method is used to solve problems of size and shape optimization with a low number of DVs and a high number of constraints:

𝑲𝝏𝒖

𝝏𝒙=

𝝏𝒇

𝝏𝒙−𝝏𝑲

𝝏𝒙𝒖 (4.5)

Adjoin Variable method is used to solve problems of topology optimization with a high number of DVs and a low number of constraints:

𝝏𝒈

𝝏𝒙=

𝝏𝒒𝑻

𝝏𝒙𝒖 + 𝒂𝑻[

𝝏𝒇

𝝏𝒙−𝝏𝑲

𝝏𝒙𝒖] (4.6)

The analysis may be carried evaluating the responses each time a variable is modified or by building an approximate model, turning back to the analysis software only when essential; in this way the computer time is strongly reduced and the optimization process is much faster. The model itself is an approximation of the physical behaviour of the product and the optimization model is an approximation too, the responses evaluated by the software are unlikely to be very precise so every final proposal of optimized model must be submitted to a verification analysis. As the optimizer searches through the design space, it needs to check whether it comes up with optimal proposals or not. A common problem consists in understanding if the obtained optimized problem is a local o a global optimum. Local approximation based methods (gradient based optimization) are more susceptible to find a local optimum whereas global approximation methods (response surface methods) are less susceptible to find a local optimum. As a base rule, the designer should remember that no algorithm can guarantee that the found optimum is actually a global one, an optimum is trusted to be global only if the optimization problem is convex, so the objective function and feasible domain have to be convex too (Fig. 14)

59

Figure 14: Convex function f(x), point A is the global minimum [4].

In reality, most engineering problems do not show to be convex, so for practical problems a global optimum is an elusive goal. Using different algorithms alters the chance to find a more global optimum but this often comes at a greater computational cost. In the case of non-convex problems solved using gradient-based techniques, the inherent behaviour is that the obtained optimized result is dependent on the starting point of the initial design. For this work, the employed software comes with a built in global search algorithm called Multiple Starting Points Optimization. This algorithm performs an extensive search of the design space for multiple starting points to improve the chances of finding a more global optimum. N different design starting points could potentially result in N different optimum solutions, it is also likely that different starting points may lead to the same optimum solution and this does not assure that the defined result is a global optimum anyway. This concept of multiple starting points optimization can be well explained considering a non-convex function, f(x), bounded by –a < x < b. The non-convex function f(x) is portrayed in Figure 15. Optimizing the design from the starting point A will result in the optimum solution P and optimizing the design from starting point B will result in the same optimum solution P. On the other hand, optimizing the design from a radically different starting point, C, the optimum solution Q will be found. Comparing the two solution points, P and Q, it is clear that a global optimum cannot be guaranteed but at the same time, solution Q is a more global solution than solution P, so the chances have clearly been improved to find a better solution. This process is an iterative procedure, it is possible to set the optimizer to compute a pre-defined number of iterations; it is also possible to set the “detail grade” for the search by defining the move limit. If the difference between two successive iterations is lower than the convergence tolerance, it is often recommended to stop the procedure and its result can be considered acceptable from a design perspective. It is now worth mentioning the ways the software achieves convergence at the end of the design cycle, there are two cases called Regular convergence and Soft convergence respectively.

60

Figure 15: Non-convex function f(x) [4].

Regular convergence is achieved when the convergence criteria are satisfied for two consecutive iterations, so the change in the objective function is less than the objective tolerance and constraint violations are lower than 1%. At least three analyses are necessary to achieve regular convergence since the convergence is based on the comparison of true objective values; in this case the design is feasible. There is an exception that verifies when the constraints are violated by more than 1%, for the three analyses the change in the objective function is lower than the tolerance and the constraints violation is lower than 0.2%, in this case the process produces a non-feasible design. Soft convergence is achieved when there is little or no change in the design variables for two consecutive iterations; in this case it is not necessary to evaluate the objective or the constraints for the final design point as the model is unchanged from the previous step. The main difference between Regular and Soft convergence is that the latter requires one less passage than the first convergence rule. The last aspect is going to be described in this chapter regards the methods used by the optimization software during its work.

Gradient Search Methods – this method uses the slope of the curve to guess at which direction the initial guess should be adjusted in, to increase or decrease it. The gradient is often computed using a finite difference method. The optimizer used this method to move from the initial configuration to the final solution. The process starts with an estimate of the optimum design, from this point the direction in which the objective function decreases more rapidly is computed, based on the gradient of the objective function. It is mandatory to move as much as possible in this direction before repeating the process again; after different iterations the process reaches convergence when the objective of the function gradient is zero.

61

Gradient method is used to find a minimum of a function using its gradient value; its algorithm is described as:

1. Start from a x0 point; 2. Evaluate the function f(xi) and its gradient nf(xi) at the point xi; 3. Determine the next point using a negative gradient direction:

xi + 1= xi – g(nf(xi)); 4. Repeat steps 2 and 3 until the function converges to the minimum.

Figure 16 summarizes the gradient method process.

Figure 16: Gradient search method [4].

Gradient-based methods are effective when the sensitivities (derivatives) of the system responses can be computed easily and inexpensively. The local approximation method is best suited to situations where Design Sensitivity Analysis is available and if the method is applied to linear static and dynamic problems (integrated in mostly FEA solvers). Gradient-based methods depend on the sensitivity of the system responses with respect to changes in design variables in order to understand the effect of the design changes and to optimize the system.

Constraint Screening, Constraint Linking and Constraint Deletion – the optimizer uses these methods to speed up the optimization process. Constraint screening is a technique used to identify the critical constraints to the current iteration; in an effort to reduce the number of variables, the optimizer uses one or more criteria to choose a subset of all variables during each iteration. Constraint screening is the process by which the number of responses in an optimization problem is trimmed to a representative set; this set contains the essence of the original design problem while keeping the size at an acceptable level. Screening technique considers all constraints that are far from their bounding values or which are less critical as not capable to affect the direction of the optimization problem and therefore can be removed from the problem for the current iteration. A practical example regards an optimization problem where the objective is to minimize mass of a finite element model composed of 100000 elements

62

while keeping their elemental stress below the material’s yield stress. This problem is composed of 100000 constraints and for every design variable, 100000 sensitivity calculations have to be performed at every iteration. Design variables are restricted by move limits so stresses are not expected to change drastically from one iteration to the next: it results to be a waste of time to calculate sensitivities for all those elements which stress is considerably below material’s yield stress. The highest elemental stresses drive the optimization direction so the number of required calculations can be further reduced by only considering a set of the highest elemental stresses. Screening technique implies an important trade-off, if too many constrained responses are removed, the solver may take considerably longer time to reach convergence or may not be able to compute a solution at all, especially when the number of retained responses is lower than the number of active constraints for the problem; on the other hand this technique allows to save a lot of time and computational effort. Constraint linking is able to use factors such as symmetry to reduce the number of constraints that need to be considered during each computation step. Constraint deletion is useful when constraints are violated during iteration, in this case it may be possible that such constraints are not important for the process, so they can be deleted (ignored).

Move Limit Adjustments – As the design moves away from its initial point in

the approximate optimization problem, the approximate values become less accurate and this effect often leads to slow overall convergence since the optimum design and actual design are very different. Move limits on the design variables and on intermediate steps are used to protect the accuracy of the approximations, they can be represented as:

x < xm < x < Xm < X (4.7)

Small move limits lead to smoother convergence even if in this case several iterations may be required due to the small design changes at each step. Large move limits on the other hand may lead to infeasible designs as critical constraints are computed inaccurately. Only if the approximations themselves are accurate, then large move limits can be used safely. Typical move limits in the optimization problem are 20% of the current design variable values but if advanced approximation concepts are used, move limits up to 50% are allowed. It is important to look at the approximations of the responses that are driving the design: these are the objective function and most critical constraints. If the objective function moves in the wrong direction or critical constrains are severely violated, it is a clear sign that approximation are not accurate and move limits have to be reduced consequently; if however move limits are too low, convergence is slowed since design variables that are far from the optimum design are forced to change slowly. If move limits are continuously reached during the optimization process, then their bound have to be adjusted consequently.

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4.1. Optimization Terminology and Definitions In this chapter the main optimization terms are listed with their definition. Design Variables – The design variables are the structural parameters that are free to be altered during an optimization process. Typical examples include material properties, topology and geometry of a structure and member sizes. Design variables may be either continuous or discrete, depending on the type of optimization being performed.

Design Space – Some selected parts that are designable during optimization process, such as material in the design space of a topology optimization. Non-design parameters are those that are pre-specified and may not be altered during the optimization. Any element where a force or a restraint is applied must be specified as non-design part. Response – Measurement of system performance. It may be mass, volume, maximum displacement, safety factor, etc. Objective function – It is the goal of the optimization process. The objective function is the function which least vale is looked for during an optimization process. It represents the single most important property of a design and its associated response is a function of the design variables mentioned above. Design constraint functions – They are restrictions placed on a problem by limiting the values that selected response functions of the system can take and must be satisfied for the design to be acceptable. Those constraints are, for example, the maximum permissible stress or a critical range of frequencies to avoid. Constraints are usually expressed as inequalities. Feasible design – Design that satisfies all the imposed constraints. The opposite case is an unfeasible design, so that violates one or more constraint functions. Optimum design – It is the optimization result (or design variable values) that satisfies all the constraints and gives the minimum (or maximum) possible value according to the objective function. Response surface – Typically there is not a continuous function that is able to relate the objective to the design variables. Consequently it is possible to use numerical experiments to generate a table of objective-function values vs. design-variable values. By fitting a surface to thus defined points, a response surface is created and it is used to find optimal locations.

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5. Structural Optimization: a Practical Example

In conclusion, here is presented this work objective: from the actual design of the Roller Holder of Danieli’s SRW 18 Guide System, a guideline for a new design will be defined looking forward for a lightweight and efficiency oriented component to replace the existing one. The optimized design process will produce a component that is able to satisfy all the work requirements for the existing one and possibly will be more efficient on the material use and to bear the applied working stress. Hover due to the limited amount of information on the actual working condition of the roller holder, a final proposal of design will not be presented. The guideline for structural optimization of a component are composed of the following steps:

Detailed analysis of the actual component to understand all its fundamental

characteristics and functions, the way it works and the limit of the actual design; Analysis of the applied loads and constraints; FEM analysis of the actual component to define its critical areas, such as the most

stressed areas, the way it deforms when a load is applied; in this phase also a modal analysis is carried out to define the first ten critical frequencies of the component;

The topology optimization process will comprehend the following steps:

o Structural optimization of the component which objective is minimizing the structure stiffness;

o Structural optimization of the component which objective is maximizing the structure mass;

o For each optimization process, different hypothesis of production directions (free versus extrusion) will be evaluated;

o Hypothesis on different materials according to what the market offers currently and to the component working conditions.

FEM analysis of the new design and comparison with the actual design, flaws and

improvements will be evidenced during this phase, multiple different solutions will be presented and compared;

Consideration on the results; Hypothesis and consideration on the production methods with some hints on

potential production costs.

For each material, a total of six different analyses will be reported in this work in order to produce a sufficient amount of material to make a meaningful comparison between the results. The results will be divided according to the objective of the structural optimization (maximize stiffness and minimize mass): each of the two resulting groups of analyses will comprehend an analysis for each material and for each building direction (free, extrusion along Z-axis, extrusion along Y-axis) for a total of eighteen different solutions. In Figure 17 is presented the flowchart of the structural optimization method that is applied in this work.

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Figure 17: Flowchart of the structural optimization process [2],[12].

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5.1 DANIELI’s SRW18 Guide System Description

The very first step of this work consists in fully understanding the SRW18 guide’s roller holder layout, working conditions and applied loads in order to define accurately all the constraints the new model will have to respect. A good starting point is a description of the guide’s components: taking as reference Figures 1 and 2, the guide is composed by a frame (d) on which are mounted an inlet cone (a), a position adjustment mechanism (b), the fork (e) and a couple of roller holders (g) that sustain a roller (f) each. The roller holder (f) that is visible in Figure 18 and 19 is linked to the position adjustment mechanism (b) on its left end by a screw whereas the main link to the frame is the pivot shaft (c), on the base of the shaft a set of disk springs (h) is placed to adjust the vertical position of the roller holder and on the side, the stability control spring is linked to the roller holder through a plate (i). The roller holder is a component made out of a particular kind of stainless steel which allows obtaining a high rigidity structure, its weight is about 1.8 kilograms and its dimensions are 210x90x45 millimetres circa. In Figure 18 is represented in its working position on the SRW18 Guide System.

Figure 18: Lateral view of the Danieli’s SRW 18 guide system.

Courtesy of Danieli & C. Officine Meccaniche.

It is important to fully understand the way each roller holder behaves; the SRW18 guide’s roller holder has been designed to keep the product, in this case a wire rod with a diameter between 5 and 10 millimetres, aligned with the inlet of the wire drawing machine. The wire rod (not represented), arriving from the previous drawing stand, enters the SRW18 guide through the inlet cone (a), travels through it, passes through the couple of rollers that impose to it the outlet direction and finally is directed to the next drawing stand. Thanks to the actions of two roller holders (g), which sole aim is to keep the wire rod aligned with the inlet of the next drawing stand, the wire can correctly enter the mill and be further processed.

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Figure 19: Isometric view of the Danieli’s SRW 18 guide system.

Courtesy of Danieli & C. Officine Meccaniche. In order to make the guide able to work with different size wire rods, the position adjustment mechanism is necessary to regulate the relative distance between the two rollers according to the diameter of the wire rod; to avoid any clearance during regulation process and working operations, a spring placed between the frame and the roller holder pushes the latter outward. A noticeable characteristic of the SRW 18 Guide System is that it can be easily separated from the drawing machine for maintenance and position adjusting regulation operations, the regulation operations comprehend guide setting and alignment on the mill and are carried out with a computerized optic device, called Danieli HiLINE. [16]

Figure 20: Detail of the two roller holders with their main components.

Courtesy of Danieli & C. Officine Meccaniche.

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Once this brief description of the guide has been finished, it is possible to focus on the component on which the structural optimization process will be applied: the roller holder. The system is a symmetric one, in fact each side of the guide is a mirrored copy of the other and this is particularly important when studying the two roller holders, they are equal but mirrored. This characteristic is useful to reduce the amount of work, each consideration that will be developed through this work for the right side roller holder can be applied to the other side without any limitation. At this point it is possible to revise in detail all the characteristics of the roller holder, for this purpose Figures 21, 22, 23 and 24 will come to help.

Figure 21: Right roller holder with all its components.

Courtesy of Danieli & C. Officine Meccaniche. The roller holder system is composed by the following set of components:

1. Roller holder: can be considered as a cantilever beam, it has to sustain the loads applied by the product on the roller, it is made of stainless steel (X20Cr13 EN 10088-3) and has a weight of approximately 1.8 kg, this is the component that will be the subject of the optimization process;

2. Roller: it is the component that is in direct contact with the wire rod during working operation. The roller is mounted on the roller holder through a shaft and a couple of SKF bearings model 6000-ZT-N9 (10x26x8 mm);

3. Pivot shaft: this shaft is the main link between the guide frame and the roller holder, it is assembled on the roller holder in a way that allows the roller holder to rotate but not to move along its axis;

4. Register screw: it is mounted on the left end of the roller holder, its end is all the time in direct contact with the cam disk of the position adjustment

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mechanism, when the cam disk is rotated, through this screw the position of the roller holder is modified;

5. Plate: this plate allows the assembly of the spring (not represented) to prevent any clearance formation during position adjusting process;

6. Locking screw: this screw has the unique function of locking in position the shaft of the roller, once it has been assembled on the roller holder;

7. Adjustment screw: to the end of this screw is linked another spring to limit clearance during adjustment process and working operation, the spring links the two roller holder together;

8. Gas stopper: this screw has the function to prevent the lubricating medium (a mixture of air and oil) to leak from its circuit.

The roller holder requires a more accurate analysis to define the building characteristics that cannot be changed during the optimization process: it is fundamental that the new design is perfectly interchangeable with the original component, in this way no changes in the layout of the guide system are required.

Figure 22: Roller holder detail view (external side).

Courtesy of Danieli & C. Officine Meccaniche. Taking pictures 22, 23 and 24 as reference, all the roller holder areas of interest will be listed in the following list:

1. Two holes are placed on the right end of the component, they are the roller shaft seat; the shaft is then locked in position thanks to an apposite skrew pin (see point 7), this is the most displaced area since on the roller the load is

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applied and then is transmitted to the roller holder, the forces applied in this case are the main components of the load condition to which the roller holder is subject;

2. This cylindrical hole runs through the whole height of the roller holder, the upper half has a greater diameter than the lower one, allowing to obtain an inner shoulder where the pivot shaft can be axially locked in position;

3. Threaded hole on which a particular bolt is placed, this is the link with the position adjusting mechanism, the bolt is fixed in its position whereas the mechanism can be moved;

4. The excavation on the bottom of the component is necessary to accommodate the set of disk springs to regulate the roller holder vertical position, the force applied by the disk springs set is relatively low and will not be taken into account during the definition of the load condition;

5. The housing on the side of the roller holder is necessary to accommodate the stability control mechanism, this is composed by a plate (5) of Figure 4 and an inner spring, the spring applies a certain force on the component and its action will be considered during the definition of the load condition;

Figure 23: Roller holder detail view (internal side).

Courtesy of Danieli & C. Officine Meccaniche.

6. Threaded hole for the pin screw, through this pin and another one on the other roller holder, a little spring links the two together;

7. Threaded hole for the screw that has the function to lock the roller shaft in position;

8. Two drilled holes form the inner lubricating circuit of the roller holder, they are necessary to bring the lubricant from the pivot shaft to the roller shaft to keep the bearings in optimal working conditions constantly, considering that with conventional production techniques (the roller holder is built by press

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moulding and refined with successive tooling operations) is not possible to obtain inner channels, two long drilled holes are necessary, their extremities are then sealed with a screw seal and some kind of sealing glue. The holes necessary for the lubricating circuit is evidenced in red in Figure 24, they have been obtained through two different drilling operations.

Figure 24: Roller holder X-Ray view, the duct in red represent the inner lubricant

circuit. Courtesy of Danieli & C. Officine Meccaniche.

5.2 Loads and Constraints Analysis

The roller holder has to keep in an exact position the wire rod before its entry into the drawing machine: considering that the drawing process is carried out with a particularly high velocity (up to 140 m/s [16]), the dynamic load on the roller holder is significant. Unfortunately at the moment the exact dynamic load applied on the component is not known: it is however known that the component experiences different working conditions during its life; each one of them should require a specific analysis to fully determine the loads and constraints applied to the component in each case. Only after a series of tests to record the most critical working conditions has been carried out, it is possible to build a complete load history for the roller holder; at this point it will be possible to apply a full topology optimization process to produce a trustworthy and improved component. Consequently to the condition stated before, the following load and constraint conditions will be taken into account, allowing to carry out only a static analysis of the roller holder. The critical components of the SRW 18 Guide System are the two bearing placed into each roller, they cope efficiently with the average working load but may

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experience breakage during the initial working phase, when the high speed wire rod enters for the first time into the guide: in this case the wire rod may impact on the side of one of the rollers before being directed correctly towards the mill. Consequently the bearings are subject to an impact load due to the wire rod high velocity and may collapse. For this reason the load applied on each side of the roller shaft seat (upper and lower sides) will be equal to the basic static load rating C0 of the bearing. For sake of completeness, the technical sheet of the SKF bearings, model 6000-ZT-N9 is reported below in Figure 25.

Figure 25: SKF bearing, series 6000, model ZT-N9 datasheet [14].

The other load that will be considered is the one applied by the spring of the position adjusting mechanism on the side of the roller holder. The spring is a Special Spring series B13, model 032. The load applied by the spring is directly dependent on its compression rate; in this case an average value, corresponding to the maximum compression experienced by the spring, has been computed and will be used in the analysis. Both loads are directed along the Y-axis. It is important to underline that the roller holder is subjected to several other small loads, however thanks to their limited entity, they will not be considered in the analysis process. It is worth mentioning that the neglected loads regard the force

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applied by the disk springs on the roller holder base, the lubricant pressure inside the lubricating circuit and the force applied by the linking spring on the pin screw placed on the upper left side on the roller holder (reference to Figure 22, point 7). All the other minor loads have been neglected thanks to their limited influence. On the constraint side, the roller holder is subjected to a “pin” type constraint applied on the pivot shaft seat that allows the roller holder to rotate, to an axial constraint applied on the inner shoulder of the pivot shaft seat that prevents axial motion alogn Z-axis and to an axial constraint applied on the hole for the position adjusting mechanism, where axial motion is permited only durign position tuning operations. The load and constraints layout is evidenced in Figure 26 below.

Figure 26: Scheme of the loads and constraints applied on the roller holder.

An important consideration on the described load condition has to be made: consequently to the limited amount of information about the working condition, the analysis carried out on the component and the relative results strictly refers to a static load case. In order to define a reliable optimal design, the mimimum amount of information necessary to carry out the analysis and optimization process would necessarily require to know the load history of the system, along with information about vibration entity that affects the system and its expected fatigue life. Only then it is possible to carriy out a full dynamic analysis to develop a fully functional optimial component out of the current design; so for all the reasons mentioned before, in this work a final proposal of optimized design is not presented but only a guideline to show the potential of the optimization process.

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5.3 Finite Element Analysis of the Current Roller Holder

The Finite Element Analysis on the current layout of the roller holder has been made to assess the static performance of the actual component in order to define a comparison term for the new layout. The FE analysis has been carried out considering the load and constraint conditions described in the previous paragraph; the mesh employed for the analysis uses elements which side is 2 mm long. The complete FE model is represented in Figure 27, it displays the meshed domain of the component and the applied loads and constraints. Taking as reference Figures 28 and 29 that display the results of the analysis, the roller holder is characterized by high rigidity, the axis of the roller shaft seat is considered as the area that experiences the greatest displacement, which is about 0.3 mm along the Y-axis direction (which is the load direction); on the other hand, the areas that goes from the pivot pin to the screw that is linked to the position adjusting mechanism undergoes a relatively small displacement.

Figure 27: Roller holder FE model.

In Figures 28a and 28b is represented a magnification of the deformation experienced by the component, along with the un-deformed shape of the component which is represented by the transparent, light grey shape on the right side. Figures 29a and 29b represent the current von Mises stress layout to which the component is subjected when the load and constraint conditions presented above are applied. It is possible to define the most stressed areas as the ones that are subjected to the highest stress levels, they are listed below:

1. This area is characterized by a sharp edge that behaves as a area of stress concentration, the actual component in this area is characterized by a casting

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fitting which radius is non-exactly known, so in this analysis a small radius of 3 mm has been used;

2. The pin screw connected to the position adjusting mechanism is affected by a flexural tension as consequence of the applied load, the external load is distributed on the pivot pin, on the roller holder side and on the screw pin; the screw pin hole, consequently to its shape and function, is heavily affected by the load;

3. The internal side of the roller holder transmits the load towards the pivot pin, it is affected by a traction load and the area evidenced in Figure 29a is the most stressed;

4. The external side of the roller holder transmits the load towards the pivot pin and the screw pin, it is affected by a compression load and the area evidenced in Figure 29b is the most stressed.

Apparently the area between the inner and outer surfaces is much less loaded than the surfaces, this condition will affect significantly the optimized design, as will be shown later in this chapter. An interesting consideration regards the peak stress values evidenced in the FE analysis, especially in the areas evidenced by Point 1 in Figures 29a and 29b. Consequently to the limited fitting radius, the sharp edge areas on the borders behave as notch points: the peak stress level reached in these areas is consequent to this apparent design flaw.

Figure 28a: Resulting displacement layout from the FE analysis (internal side view).

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Figure 28b: Resulting displacement layout from the FE analysis (external side view).

Figure 29a: Resulting von Mises stress layout from the FE analysis (internal side view).

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Figure 29b: Resulting von Mises stress layout from the FE analysis (external side view). Compared to the peak stress level, across the whole structure the average stress level is much lower; the peak stress at a first sight may give a false impression on the overall stress to which the component is subjected. Images from Figure 27, 28 and 29 have all been taken from the Finite Element Analysis carried out with Siemens NX 10. The same FE analysis (FEA set with the same parameters) has been performed on two different software: Siemens NX 10 and Altair HyperWorks 13; the obtained results are congruent but differ slightly on the maximum values of displacement and von Mises Stresses. Table 2 below recapitulates all the parameters used for the analysis and the results of the analysis itself, all the values are those produced by the HyperWorks package since this software will be used for the structural optimization process of the component.

Component

Right Roller Holder – Danieli’s SRW18 Guide System

Solvers

Siemens NX 10 with NASTRAN (solver)

Altair HyperWorks 13 with OptiStruct (solver)

Mesh

Element kind = CTETRA 10

Element size = 2 mm

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Material

Stainless Steel X20Cr13 EN 10088-3

Density 𝜌 = 7700 kg/m3

Elasticity modulus = 205 GPa Poisson Coefficient = 0.235 Yield strength = 440 MPa

Ultimate tensile strength = 640 MPa

Loads

Axial load on roller shaft seat (2X) = 2000N along Y-axis

Axial load on spring seat = 275N along Y-axis

Constraints

Pin constraint on pivot shaft seat, allows rotation only

Axial constraint on pivot shaft seat shoulder, prevents motion along Z- axis Pin constraint on screw seat, allows rotation only

Results

Maximum Displacement [mm]

X-axis = 0.33

Y-axis = 0.015

Z-axis = 0.11

Maximum von Mises Stress

220 MPa

First 10 Normal Modes [Hz]

F1=1198 F2=2774 F3=3815 F4=5003 F5=5833 F6=7445 F7=8762

F8=10354 F9=10977 F10=12174

Table 2: Parameters and results of the FE analysis.

5.4 Structural Optimization Setting

The structural optimization process has been carried out on Altair’s Inspire with the aid of OptiStruct, which is Altair’s solver that will take the computational effort toward the research of an optimal solution. The process will be divided in two main steps, one which aim is to produce a lightweight structure and another one which aim is to produce the stiffest structure possible, compatibly with a reduction of overall mass of the component. As a result of this work, a guideline for structural optimization of a component will be produced, together with some practical examples applied to the roller holder; however, consequently to a policy of non-disclosure of the Company that limits the amount on information on the actual working condition at disposal of this work, a definitive optimal design, ready for production, will not be presented.

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The key points of the guideline for structural optimization of a component are listed below:

Definition of the load cases which contain the working conditions that the component experiences during its life span;

Definition of non-design space to which loads, supports and constraints are applied;

Definition of design space that will be directly affected by the optimization process;

Choice of a material compatible with the component application and the Additive Manufacturing Technology;

Structural optimization process applied to the design space, two possible objectives are available, they are stiffness maximization and mass minimization;

Finite Element Analysis of the resulting optimal designs and comparison with the actual component performance;

Consideration on the potential possible optimal designs from the point of view of production with Additive Manufacturing Technology.

In order to run the optimization analysis, at least one load case must be created, each load case contains a combination of loads, supports and constraints; during the setting procedure, it is possible to choose which load case have to be considered for the current structural optimization procedure or to make the software consider multiple load cases in sequence in order to obtain directly a potential result that can satisfy all the considered load cases. The first thing to define, before proceeding to compute an optimal solution, consists in defining what portion of the design domain, which in this analysis is the actual roller holder domain, cannot be modified by the optimization process and what portion is the design space. The non-design space of the domain is composed by all the areas to which loads, supports and constraints are applied to and it is made out of little portions of the roller holder domain; the remaining domain is the design space. Taking Figure 30 as reference, the design space is the brownish area whereas the non-design space is the bluish area. The non-design space is defined manually through the set of modelling tools available in Inspire; it is worth mentioning here that even if design space and non-design space at this point appear to be two different, separated bodies, the software considers all the contact boundary surfaces between the two as perfectly glued, in order to consider the component as the single body which indeed it is. A design space is the initial part from which material is removed until a final shape is reached during optimization. It is possible to create a design space by using the modeling tools in Inspire, or by importing parts from other solid modelers. Any part that is a design space will be reshaped during optimization, while any part that is not a design space will remain as it is. Is it possible to have any number of design spaces in the model; a design space can have any shape or topology as long as it is a single solid volume. A part that is used as a design space should not be very detailed; to ensure the most freedom to generate a shape, the software should work with the simplest design space possible. The more fine details are present in the design space, the longer it will take to run the optimization.

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Figure 30: Design space (brown) and non-design space (blue).

In this analysis, thanks to the limited dimension of the component that will be analyzed, it has been preferred to limit the simplification process to the essential changes to improve the overall result. These little changes consisted in removing the portions of the lubricating circuit that have been excluded from the non-design space because they are not strictly necessary; the reason of these variation is due mainly to improve the aesthetics of the optimized shapes but also to simplify calculation operations. Any shape generated by optimization is contained entirely within the volume of the original design space, since material is only removed and not added. It is possible to specify the amount of material to keep, either as a percentage of the design space or by entering a target weight. In Figure 31 the detailed view of the non-design space is portrayed, it comprehends all the vital parts (pivot shaft seat, screw pin seat, stability controller mechanism seat, inner lubricant circuit and roller shaft seat) of the initial roller holder design so that the new component will be perfectly interchangeable with the old one and the SRW18 Guide System will not require any modification to accommodate the new component. Consequently to this last consideration, it is particularly important to underline that the software considers as an additional constraint the volume of the design space, this means that the new structure cannot exceed the boundary volume of the original structure.

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Figure 31: Detail of the non-design space.

At this point the interest can be focused on how the software computes the optimal design and on what settings it is possible to regulate in order to influence the analysis. Inspire can be set to produce two different optimal designs, each one with a different objective: maximizing stiffness and minimizing mass. If the objective is to maximize the stiffness of a design space, the resulting shape will resist deflection, but may be heavier as a consequence. If the objective is to minimize mass, the resulting shape will be lighter, but may deflect more. Once a particular design space and the set of loads, supports, and shape controls applied to the model have been defined, different results will be produced depending on which objective is selected in the Run Optimization window (Figure 32). Minimizing the mass of a design space will result in a shape that has the lightest weight possible which can still support the applied loads; once minimize mass is set as optimization objective, it is necessary to specify one or more constraints among the following:

Stress constraints – it is specified in terms of safety factors; a global stress constraint can be applied to limit the maximum stress in the model: Inspire analyzes the materials used in the model to determine the areas with the lowest yield stress and the divides them by its minimum safety factor. The higher the safety factor, the more material will be added to the optimized shape to reduce the overall stress in the model: more material is required to bear the applied external loads.

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Figure 32: Run Optimization windows, the objectives are Minimize Mass (left) and Maximize Stiffness (right).

Frequency constraint – in some cases it may be necessary to change the

natural frequency of the model in order to avoid resonance with other parts of a design: the frequency constraint setting is made to control the frequency at which an optimized part vibrates. When minimizing mass it is possible to define a minimum frequency and specify how many of the lowest mode can exceed that frequency: if Inspire is not able to achieve the specified frequency for the lowest modes, the optimization run may fail and it may be necessary to assign a stiffer material (or to increase the minimum frequency).

Displacement constraint – can be applied to a model to limit deflections in desired locations and directions, this kind of constraint can be applied only to non-design spaces, only if the objective is minimizing mass and alongside a stress constraint. The displacement constraint can have an upper bound, a lower bound, or both. The displacement constraint does not figure inside the Run Optimization window but has to be applied independently; it is then included in the current load case to which it has been applied. As a consequence of this constraint, the optimized model will result in a stiffer and heavier structure, due to the imposed limit on deflection.

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Maximizing the stiffness of a design space will result in a shape that is the stiffest possible for a given mass; once maximize stiffness is set as optimization objective, it is necessary to specify one or more constraints among the following:

Mass target – it is a constraint used to specify the amount of material to keep; mass target can be defined either as a percentage of the total volume of the design space or as the total mass of the entire model. If there are more than one design spaces, it is possible to set different targets for each one of them; it is also possible to set the mass target to none. One option for specifying a mass target is as percentage of the total volume of the design space: if a 30% of mass target is set, the remaining 70% of the material is carved away from the design space leaving only the 30% of the original material. The optimization may not always produce an answer for every percentage is specified, as the lower the percentage, the more challenging is to produce an answer; typical percentages range from 20% to 50%. The second option for specifying a mass target is to enter the total mass for the entire model: this is the total mass of the model after the optimization, including both design and non-design spaces. The lower is the mass target, the more challenging is for the solver to produce an answer.

Frequency constraint – in some cases it may be necessary to change the natural frequency of the model in order to avoid resonance with other parts of a design: the frequency constraint setting is made to control the frequency at which an optimized part vibrates. When maximizing stiffness it is possible to either maximize frequencies or to set a specific minimum frequency. If the option to maximize stiffness is chosen, Inspire will automatically maximize both stiffness and frequency of the model (the lowest natural frequencies will be displayed at the end of the process); if the resulting frequency is not high enough to cope with the design requirements, it is necessary to assign a stiffer material or to modify the mass target for the optimized shape.

In both cases of minimizing mass and maximizing stiffness it is possible to set a constraint for the minimum and maximum wall thickness of material, this constrain limit the overall dimensions of walls of the optimized shape. Inspire can also take into account the gravity direction and the eventuality of different load cases for which it automatically considers both effect to produce an optimized component that can cope with all the prescribed working conditions. The last consideration before running the optimization process regards the materials that can be employed: in this case the purpose is to showcase an example of component that could be produced with additive manufacturing technology so the choice of materials at disposal is limited to those currently available for production with AM technology. The original roller holder is a highly rigid structure that is subjected to work in a highly corrosive environment at high temperature, so to withstand the environmental working conditions and the external loads, stainless steel is a forced choice. Among the materials available for AM technology production, three materials that are suitable for the purpose of this work have been selected, a little summary of their properties is presented below:

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Stainless Steel 316L – corrosion resistant iron based alloy, it is an austenitic stainless steel which comprises iron alloyed with chromium of mass fraction up to 18%, nickel up to 14% and molybdenum up to 3%, along with other minor elements. The alloy is an extra-low carbon variation on the standard 316 alloy. The material properties are reported in the table below.

SS 316 L Density 𝜌 [kg/m3] 7700 Elasticity modulus E [GPa] 180 Ultimate Tensile Strength Rm [MPa] 600 Yield Strength Rp0,2 [MPa] 500 Poisson Coefficient 𝑣 0.250 General alloy characteristics High hardness and toughness

High corrosion resistance High machine-ability Can be highly polished

Table 3: Stainless Steel 316 L properties [9], [10].

CoCrMo Alloy – alloy that comprises cobalt alloyed with chromium of mass fraction up to 30% and molybdenum up to 7%, along with other minor elements. The alloy has a high melting point making it stable at high temperatures, along with its high level of corrosion resistance. The material properties are reported in the table below. CoCrMo Alloy Density 𝜌 [kg/m3] 8300 Elasticity modulus E [GPa] 220 Ultimate Tensile Strength Rm [MPa] 1100 Yield Strength Rp0,2 [MPa] 700 Poisson Coefficient 𝑣 0.3 General alloy characteristics High strength and hardness

Excellent biocompatibility High corrosion resistance High temperature resistance

Table 4: CoCrMo alloy properties [9].

Titanium Ti6Al4V-ELI – this alloy comprises titanium mass fraction up to 90% alloyed with aluminum up to 6.75% and vanadium up to 4.5%, along with other minor elements. Ti6Al4V grade 23 is otherwise referred to as ELI (Extra Low Interstitial) with regards to the interstitial impurities of oxygen, carbon, and nitrogen. It is a higher purity version of the most commonly used titanium alloy Ti6Al4V grade 5. The material properties are reported in the table below. Ti6Al4V-ELI Alloy Density 𝜌 [kg/m3] 4420 Elasticity modulus E [GPa] 114

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Ultimate Tensile Strength Rm [MPa] 1100 Yield Strength Rp0,2 [MPa] 1000 Poisson Coefficient 𝑣 0.31 General alloy characteristics High specific strength

High corrosion resistance Low thermal expansion Low thermal conductivity

Table 5: Ti6Al4V-ELI properties [9], [10].

Stainless Steel X20Cr13 EN 10088-3 – The steel grade 1.4021 (also called ASTM 420 and SS2303) is a high tensile strength martensitic stainless steel that comprises iron alloyed with chromium of mass fraction up to 13% and molybdenum up to 1% along with other minor elements. In order to be able to make a comparison with the materials that will be employed in the structural optimization analysis, the material properties are reported in the table below.

SS X20Cr13 EN 10088-3 Density 𝜌 [kg/m3] 7700 Elasticity modulus E [GPa] 205 Ultimate Tensile Strength Rm [MPa] 640 Yield Strength Rp0,2 [MPa] 440 Poisson Coefficient 𝑣 0.235 General alloy characteristics Good corrosion resistance

High tensile strength High machine-ability Can be highly polished

Table 6: Stainless Steel X20Cr13 EN 10088-3 properties [8].

The values represented in the tables above are average values since AM production is only able to produce components affected by a slight anisotropy: normally the produced object show lower performance on the building direction (Z-axis) than on the building plane (XY plane). In order to simplify the analysis and thus to neglect this effect, the averaged values listed above have been used. A different approach could have used the lower values referred to the building direction, but considering the difference is limited, probably the results would variate of a relatively small amount.

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5.5 Maximize Stiffness - Results

5.5.1 Free Shape Optimization, Stainless Steel 316L The aim of this analysis is to produce the stiffest structure possible; the free shape optimization has the objective to produce an optimal structure without following any particular direction of production. Three different results are reported, they correspond to a reduction of volume of 60%, 50% and 40% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 7 below whereas the images in Figure 33 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Free Direction of Optimization

SS 316L (Maximum Stiffness)

remaining 40% of total design space

volume

remaining 50% of total design space

volume

remaining 60% of total design space

volume

Max Displacement [mm] 0.491 0.412 0.368

Max von Mises Stress [MPa] 318 207 166

Min Safety Factor Area 1.5 2.5 3

First 6 Normal Modes [Hz] F1=1604 F2=3532 F3=4679 F4=4890 F5=5672 F6=7593

F1=1623 F2=3733 F3=4828 F4=5619 F5=5866 F6=7993

F1=1594 F2=3795 F3=4808 F4=5842 F5=6710 F6=8485

Total Weight [Kg] 0.876 1.05 1.23

Table 7: Structural optimization results, free shape direction, SS 316L. The choice criteria for the best result among the listed is aimed toward defining an optimal design which performance are compatible or better than those of the initial design: with a reduction of the 40% of the design space volume, a total reduction of weight of 32% is obtained, together with an overall performance close to the original one. An important observation for the best case regards the maximum von Mises stress: consequently to the topology optimization process, the notch points evidenced in the original design are eliminated (or their influence is reduced) so the maximum von Mises stress is much lower even if the mechanical performance of stainless steel 316L are slightly lower than those of stainless steel X20Cr13. It is worth to remember the reader that all the results are referred to a static load condition. An important observation regards the modal nodes that have been obtained out of each analysis by setting the software to maximize the frequency at the same time of the stiffness of the structure; in this way the first natural frequency of the structure

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are the highest possible. However there is no correlation between this result and the actual vibrations that affect the roller holder during its working conditions, to obtain a useful result, the vibration history of the component working conditions has to be recorded and analysed.

Figure 33: Optimal design for a free shape roller holder, SS316L. Another consideration regards the materials, stainless steel 316L offers lower performance than stainless steel X20Cr13; consequently the maximum weight reduction that can be obtained in this case is limited, unless a poorer performance is accepted or the design volume can be augmented on the external side of the component (at the state of the current analysis, a volume increase is not accepted). A particular consideration for this optimal design regards the way the original shape has been modified, it is possible to observe in Figure 33 that several internal cavities are present in the new design: although it is possible to obtain those cavities with AM technology, it may be necessary to adjust the design to remove the powder from the cavities with a suitable drain; another aspect regards the necessity to build supports structure during production for the internal cavities, in some cases it might be difficult to remove the supports from inside the cavities.

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5.5.2 Extrusion along Z-axis Optimization, Stainless Steel 316L The extrusion along Z-axis optimization has the objective to produce an optimal structure having a production direction that is directed along Z-axis; the aim of this analysis is to produce the stiffest structure possible. Three different results are reported, they correspond to a reduction of volume of 60%, 50% and 40% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 8 below whereas the images in Figure 34 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Extrusion along Z-axis Optimization

SS 316L (Maximum Stiffness)

remaining 40% of total design space

volume

remaining 50% of total design space

volume

remaining 60% of total design space

volume

Max Displacement [mm] 0.510 0.474 0.404

Max von Mises Stress [MPa] 208 244 315

Min Safety Factor Area 2.5 2.5 3

First 6 Normal Modes [Hz] F1=1358 F2=2946 F3=3814 F4=5018 F5=5593 F6=6552

F1=1442 F2=3191 F3=4102 F4=5067 F5=5611 F6=7002

F1=1468 F2=3414 F3=4379 F4=5459 F5=6459 F6=7803

Total Weight [Kg] 1 1.07 1.25

Table 8: Structural optimization results, extrusion along Z-axis, SS 316L. In this case a reduction of the 40% of the design space volume allows obtaining a total reduction of weight of 30% together with an overall performance poorer than the original one. Even if the hypothesis to produce the component by building it layer by layer along one of the structure axis is interesting since it does not require powder drains for internal cavities and shows a much reduced use of supporting structure during production, apparently it is not able to produce a structure capable to cope adequately with the external loads. In this analysis, maximum displacement is the main term of comparison with the original roller holder and unless a very limited weight reduction is applied, the optimized shape is not good enough to justify the high increase in production cost due to AM technology applications. In this case it is possible to observe from Figure 34 that some of the original areas of stress concentration are retained, causing local peaks of stress which may be cause of rupture of the component.

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Figure 34: Optimal design for an extruded along Z-axis roller holder, SS316L.

5.5.3 Extrusion along Y-axis Optimization, Stainless Steel 316L The extrusion along Y-axis optimization has the objective to produce an optimal structure having a production direction that is directed along Y-axis, in this case production direction and load direction are both on the same axis; the aim of this analysis is to produce the stiffest structure possible. Three different results are reported, they correspond to a reduction of volume of 50%, 40% and 30% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 9 below whereas the images in Figure 35 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Extrusion along Y-axis

Optimization SS 316L

(Maximum Stiffness)

remaining 50% of total design space

volume

remaining 60% of total design space

volume

remaining 70% of total design space

volume

Max Displacement [mm] 0.460 0.402 0.357

Max von Mises Stress [MPa] 287 364 200

Min Safety Factor 1.5 2 2.5

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First 6 Normal Modes [Hz] F1=1407 F2=2904 F3=3908 F4=4220 F5=5310 F6=5791

F1=1472 F2=3108 F3=4151 F4=4436 F5=5440 F6=6297

F1=1440 F2=3188 F3=4447 F4=4793 F5=5432 F6=7405

Total Weight [Kg] 1.05 1.19 1.38

Table 9: Structural optimization results, extrusion along Y-axis, SS 316L. In this case a reduction of the 40% of the design space volume allows obtaining a total reduction of weight of 34% together with an overall performance similar to the original.

Figure 35: Optimal design for an extruded along Y-axis roller holder, SS316L. The production of the structure layer by layer along the Y-axis appears to be the optimal one, it could reduce significantly the time requested by the AM printer to build the piece; on the other hand the resulting component presents two supporting shelves for each ring that has to support the roller shaft: these areas are the weakest points of the structure that cause the final optimal design to have a low minimum safety factor. It is possible that increasing the minimum wall thickness, the two shelves increase their dimension and potentially their performance; however this eventuality is not explored in this analysis.

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5.5.4 Free Shape Optimization, CoCrMo Alloy The optimization process applied to the roller holder with the stainless steel 316L has been carried out in the same way with the CoCrMo alloy. CoCrMo alloy offers better performance than stainless steel but at a price: its density is higher and thus its specific weight. All the considerations on the results presented in the previous paragraphs still holds for the results of the analysis carried out using CoCrMo alloy. Three different results are reported, they correspond to a reduction of volume of 60%, 50% and 40% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 10 below whereas the images in Figure 36 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Free Direction of Optimization CoCrMo Alloy

(Maximum Stiffness)

remaining 40% of total design space

volume

remaining 50% of total design space

volume

remaining 60% of total design space

volume

Max Displacement [mm] 0.404 0.337 0.300

Max von Mises Stress [MPa] 288 243 217

Min Safety Factor Area 3.2 4 4.5

First 6 Normal Modes [Hz] F1=1726 F2=3786 F3=5035 F4=5302 F5=6096 F6=8054

F1=1751 F2=4013 F3=5174 F4=6177 F5=6399 F6=8621

F1=1728 F2=4066 F3=5175 F4=6295 F5=7201 F6=9082

Total Weight [Kg] 0.91 1.09 1.28

Table 10: Structural optimization results, free shape direction, CoCrMo Alloy. Even though the shapes obtained in this case are very similar to the ones using stainless steel 316L, several considerations can be made. CoCrMo alloy offers better performance than stainless steel 316L and than X20Cr13, in the face of higher density and a greater cost. A greater density implies that the same reduction of weight in percentage does not correspond to the same reduction obtained with stainless steel 316L but to a lower one; due to the greater specific weight of the alloy indeed. However the better performances granted by CoCrMo alloy allows to obtain an optimal design that is lighter and stiffer than those obtained with stainless steel 316L, the static performance in this case are very close to that of the original component and with a design that allows to save almost 29% of weight, which potentially is a remarkable result.

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Figure 36: Optimal design for a free shape roller holder, CoCrMo alloy.

5.5.5 Extrusion along Z-axis Optimization, CoCrMo alloy The extrusion along Z-axis optimization has the objective to produce an optimal structure having a production direction that is directed along Z-axis; the aim of this analysis is to produce the stiffest structure possible using this time the CoCrMo alloy. Three different results are reported, they correspond to a reduction of volume of 60%, 50% and 40% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 11 below whereas the images in Figure 37 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Extrusion along Z-axis Optimization CoCrMo alloy

(Maximum Stiffness)

remaining 40% of total design space

volume

remaining 50% of total design space

volume

remaining 60% of total design space

volume

Max Displacement [mm] 0.471 0.383 0.331

Max von Mises Stress [MPa] 277 390 300

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Min Safety Factor Area 3 3.5 4

First 6 Normal Modes [Hz] F1=1604 F2=3532 F3=4679 F4=4890 F5=5672 F6=7593

F1=1555 F2=3422 F3=4409 F4=5552 F5=6138 F6=7618

F1=1579 F2=3640 F3=4678 F4=5863 F5=6955 F6=8333

Total Weight [Kg] 0.91 1.11 1.3

Table 11: Structural optimization results, extrusion along Z-axis, CoCrMo alloy.

Also in this case, compared to the optimization analysis carried out for the stainless steel 316L, with the CoCrMo alloy, a better performing optimal structure comes at a lower weight: to obtain the same performance with stainless steel 316L only a lower weight reduction can consequently be achieved.

Figure 37: Optimal design for an extruded along Z-axis roller holder.

5.5.6 Extrusion along Y-axis Optimization, CoCrMo alloy The extrusion along Y-axis optimization has the objective to produce an optimal structure having a production direction that is directed along Y-axis, in this case production direction and load direction are both on the same axis; the aim of this analysis is to produce the stiffest structure possible, using this time CoCrMo alloy. Three different results are reported, they correspond to a reduction of volume of 50%, 40% and 30% applied to the design space only: this means that the final

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structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 12 below whereas the images in Figure 38 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Extrusion along Y-axis Optimization CoCrMo alloy

(Maximum Stiffness)

50% of total design space

volume

60% of total design space

volume

70% of total design space

volume

Max Displacement [mm] 0.366 0.329 0.292

Max von Mises Stress [MPa] 300 275 205

Min Safety Factor Area 3 4 4.5

First 6 Normal Modes [Hz] F1=1483 F2=3146 F3=4471 F4=4560 F5=5710 F6=6572

F1=1583 F2=3340 F3=4538 F4=4769 F5=5849 F6=6711

F1=1555 F2=3417 F3=4753 F4=5215 F5=5846 F6=7950

Total Weight [Kg] 1.12 1.3 1.44

Table 12: Structural optimization results, extrusion along Y-axis, CoCrMo alloy.

Figure 38: Optimal design for an extruded along Y-axis roller holder, CoCrMo alloy.

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In this case too, compared to the optimization analysis carried out for the stainless steel 316L, with the CoCrMo alloy, a better performing optimal structure comes at a lower weight. Moreover in this case the best results for the optimal design, corresponding to a reduction of the design space volume of 40%, are very close to those obtained for the extrusion along Z-axis;

5.5.7 Free Shape Optimization, Ti6Al4V-ELI Alloy The optimization process applied to the roller holder for the stainless steel 316L and CoCrMo alloy have been carried out in the same way for the Ti6Al4V-ELI alloy. Three different results are reported, they correspond to the original shape and to a reduction of volume of 50% and 40% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 13 below whereas the images in Figure 39 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Free Direction of Optimization

Ti6Al4V-ELI Alloy (Maximum Stiffness)

50% of total design space

volume

60% of total design space

volume

Original Shape

Max Displacement [mm] 0.639 0.571 0.537

Max von Mises Stress [MPa] 191 154 212

Min Safety Factor Area 4 5 5

First 6 Normal Modes [Hz] F1=1741 F2=3988 F3=5139 F4=6164 F5=6382 F6=8571

F1=1718 F2=4031 F3=5133 F4=6250 F5=7136 F6=9009

F1=1189 F2=2708 F3=3749 F4=4949 F5=5818 F6=7319

Total Weight [Kg] 0.68 0.78 1.15

Table 13: Structural optimization results, free shape direction, Ti6Al4V-ELI Alloy. Titanium alloys allows building strong lightweight high performance components but have three important drawbacks: they are difficult to be produced, it is very complicated to execute tooling operations on titanium components and this kind of alloys are very expensive. On the particular application of the roller holder, it has been interesting to test a FEA model of the original shape made of Ti6Al4V-ELI alloy to define a term of comparison in terms of performance and weight: the resulting structure will weigh little more than half the weight of the same component made of stainless steel X20Cr13 and will be very safe (the peak stress in the notch point is about a fifth of the titanium alloy yield stress, at least for a static load condition).

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Figure 39: Optimal design for a free shape roller holder, Ti6Al4V-ELI alloy. Starting from a much lighter structure, the optimized design presents very limited weight: a reduction of design space volume of 40% allows to obtain an optimal structure that weights about 56% less than the original structure; on the other hand its maximum displacement is almost doubled. Apparently this titanium alloy is not rigid enough due to its low value of elasticity modulus and if the value of maximum displacement for the original component is taken as a reference (it is of about 0.3 mm), the same component made out of titanium alloy will experience greater bending. On a side note, titanium alloys should be employed especially when weight saving is the main objective of the structural optimization research and in that case it is convenient to take advantage of titanium exceptional mechanical characteristics to build lightweight strong structures; anyway particularly high production costs have to be expected, especially if additive manufacturing technology is employed.

5.5.8 Extrusion along Z-axis Optimization, Ti6Al4V-ELI alloy The extrusion along Z-axis optimization has the objective to produce an optimal structure having a production direction that is directed along Z-axis; the aim of this analysis is to produce the stiffest structure possible, using this time Ti6Al4V-ELI alloy. Three different results are reported, they correspond to a reduction of volume of 50%, 40% and 25% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 14 below whereas the

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images in Figure 40 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Extrusion along Z-axis Optimization

Ti6Al4V-ELI alloy (Maximum Stiffness)

50% of total design space

volume

60% of total design space

volume

75% of total design space

volume

Max Displacement [mm] 0.727 0.625 0.551

Max von Mises Stress [MPa] 411 308 167

Min Safety Factor Area 3 4 5

First 6 Normal Modes [Hz] F1=1604 F2=3532 F3=4679 F4=4890 F5=5672 F6=7593

F1=1580 F2=3621 F3=4655 F4=5841 F5=6839 F6=8283

F1=1611 F2=3835 F3=4892 F4=5857 F5=7027 F6=8800

Total Weight [Kg] 0.69 0.79 0.94

Table 14: Structural optimization results, extrusion along Z-axis, Ti6Al4V-ELI alloy.

Figure 40: Optimal design for an extruded along Z-axis roller holder, Ti6Al4V-ELI alloy.

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Also in this case a strong lightweight structure is obtained but its bending performance appears to not be adequate for the job to which the component is destined to. However with a weight reduction of the 25% the results are closer to the ones of the unmodified structure and greater weight reductions cause even larger values of displacement.

5.5.9 Extrusion along Y-axis Optimization, Ti6Al4V-ELI alloy The extrusion along Y-axis optimization has the objective to produce an optimal structure having a production direction that is directed along Y-axis, in this case production direction and load direction are both on the same axis; the aim of this analysis is to produce the stiffest structure possible, this time Ti6Al4V-ELI alloy is employed. Three different results are reported, they correspond to a reduction of volume of 50%, 40% and 25% applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 15 below whereas the images in Figure 41 represent the best case among the three presented: in clockwise direction there are optimized shape, internal details of the optimized shape, displacement layout and von Mises stress layout.

Extrusion along Y-axis Optimization

Ti6Al4V-ELI alloy (Maximum Stiffness)

50% of total design space

volume

60% of total design space

volume

75% of total design space

volume

Max Displacement [mm] 0.743 0.624 0.544

Max von Mises Stress [MPa] 415 276 213

Min Safety Factor Area 3.5 4 5

First 6 Normal Modes [Hz] F1=1552 F2=3100 F3=4026 F4=4549 F5=5664 F6=6126

F1=1574 F2=3312 F3=4519 F4=4737 F5=5818 F6=6690

F1=1599 F2=3510 F3=4863 F4=5138 F5=5826 F6=8176

Total Weight [Kg] 0.66 0.76 0.91

Table 15: Structural optimization results, extrusion along Y-axis, Ti6Al4V-ELI alloy. Also in this case a strong lightweight structure is obtained but its bending performance appears to not be adequate for the job to which the component is destined to. As experienced for the other materials, also in the case of titanium alloy the performance of the component produced following an extrusion process along Y-axis are similar to the other two cases but the maximum stress is higher due to the stress concentration on the two shelves; the minimum safety factor is lower too for the same reason.

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Figure 41: Optimal design for an extruded along Y-axis roller holder, Ti6Al4V-ELI alloy.

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5.6 Minimize Mass - Results

5.6.1 Free Shape Optimization, Stainless Steel 316L

Even if the main purpose of this work is to produce a stiff lightweight component to replace the current roller holder, it is worth considering the how Inspire works when the aim is to produce a lightest structure possible. It will be seen that a too light structure is not suitable for the kind of work is requested to the roller holder but by introducing a displacement constraint to the analysis, it is possible to obtain a very similar result to the one of the analysis towards stiffness maximization. Also in this case, the free shape optimization has the objective to produce an optimal structure without following any particular direction of production. Four different results are reported, three of them correspond to a reduction of mass related to the following safety factors: 3, 4 and, 5; the last result is related to the eventuality in which among the constraints there is a displacement constraint paired to a safety factor of 3. Each safety factor is applied on the total stress level of the structure whereas the mass reduction is applied on the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together.

Free Direction of Optimization

SS 316L (Minimum Mass)

Safety Factor is 3

Safety Factor is 4

Safety Factor is 5

Safety Factor is 3 and

displacement constraint

Max Displacement [mm]

0.677 0.507 0.380 0.401

Max von Mises Stress [MPa]

258 184 150 194

Min Safety Factor Area

2.5 3 4 3.5

First 10 Normal Modes [Hz]

F1=1464 F2=1706 F3=2052 F4=2857 F5=4729 F6=4930 F7=5560 F8=5958 F9=6775 F10=8159

F1=1500 F2=2371 F3=2773 F4=2972 F5=5323 F6=5608 F7=6637 F8=7458 F9=8693 F10=9421

F1=1396 F2=2726 F3=3017 F4=4100 F5=5734 F6=6503 F7=8524 F8=8898 F9=9444

F10=12227

F1=1470 F2=2842 F3=3125 F4=4132 F5=5829 F6=6558 F7=8133 F8=8929 F9=9405

F10=12532

Total Weight [Kg] 0.6 0.78 1.10 1.06

Table 16: Structural optimization results, free shape direction, SS 316L.

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All the results are listed in the Table 16 below whereas the images in Figure 42 represent downward the optimized shapes corresponding to the listed results presented in Table 16, except for the last one (from the left side) that is represented in Figure 43.

Figure 42: Optimized shape obtained applying a safety factor of 3, 4 and 5 (from up to down).

With the minimum mass setting, Inspire tries to define the lightest structure able to cope with the applied constraints and loads: without setting a specific displacement constraint, very light structures are produced, on the other hand their rigidity will be limited, as it is shown by the results reported in Table 16. If a displacement constraint is applied, Inspire tries to reduce mass as much as possible and at the same time to respect the additional constraint: the resulting structure will have a similar topology to those produced by the maximum stiffness analysis, but poorer performance; after all, the main objective is mass minimization. It is important to underline here that the displacement constraints have been applied only to the roller shaft supports, but additional displacement constraints

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could have been applied to many other points of the roller holder structure; this configuration affects significantly the result, several displacement constraints lead the software toward a stiffer solution. All displacement constraints have the same direction of the applied loads and have upper and lower bounds: in this analysis their values are 0.35 mm and 0 respectively. The fact that displacement constraints have been applied only to two points is probably the reason why the results of the stiffness maximization and mass minimization are not coincident. In Figure 43 is represented the result of the last case described in Table 16.

Figure 43: Optimized topology of the roller holder, minimum mass objective with displacement constraint, SS316L.

5.6.2 Extrusion along Z- and Y-axes Optimization, Stainless Steel 316L The extrusion optimizations along Z-axis and Y-axis have the objective to produce an optimal structure having a production direction that is directed along the given axes; the aim of these analyses is to produce the lightest structure possible. For each direction of production, two different results are reported: they correspond to the minimum mass structure obtained with a safety factor of 4 and with a safety factor of 3 together with the displacement constraint described before. Also in this case the mass minimization process is applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the

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optimized design part together. All the results are listed in the Table 17 below whereas the images in Figure 44 represent the results obtained in all the four cases.

Extrusion Optimization

SS 316L (Minimum Mass)

Extrusion Z-axis

Safety Factor is 4

Extrusion Z-axis

Safety Factor is 3 and

displacement constraint

Extrusion Y-axis

Safety Factor is 4

Extrusion Y-axis

Safety Factor is 3 and

displacement constraint

Max Displacement [mm]

0.378 0.343 0.483 0.387

Max von Mises Stress [MPa]

147 134 180 163

Min Safety Factor Area

3.5 3 3 3

First 10 Normal Modes [Hz]

F1=1349 F2=3058 F3=3901 F4=5378 F5=6454 F6=7632 F7=8062 F8=8976 F9=9542

F10=11762

F1=1318 F2=3251 F3=4108 F4=5492 F5=7932 F6=8571 F7=8804 F8=9839

F9=11205 F10=12429

F1=1260 F2=2536 F3=2951 F4=3324 F5=5227 F6=7024 F7=7826 F8=8252

F9=10220 F10=10999

F1=1286 F2=2604 F3=2956 F4=3737 F5=5366 F6=7228 F7=8157 F8=8695

F9=11095 F10=11370

Total Weight [Kg] 1.35 1.58 1.01 1.23

Table 17: Structural optimization results, extrusion along Z- and Y-axes, SS 316L. In all the cases the resulting shapes are very similar to those obtained as results of the stiffness maximization process; their performance however are significantly different.

Figure 44a: Optimized shape having production direction along Z-axis, SS316L.

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Figure 44b: Optimized shape having production direction along Y-axis, SS316L (in both cases the shape influenced by the displacement constraint is on the right side). It is interesting to make a comparison with the results of the preceding analysis (safety factor of 4 and safety factor of 3 with displacement constraint), for what regards the mass minimization problems, the free direction of production allows to obtain a lighter and safer structure than to those obtained following a specific direction of production; this statement remains valid even when after the displacement constraint is added. The goodness of these results depends on the application to which the optimal structure is destined to; if the main goal is to save material, then a free direction of production appears to be the most convenient solution, even if it is still affected by all the production problems discussed in the preceding paragraph.

5.6.3 Free Shape Optimization, CoCrMo alloy

The previous analysis, which aim is to produce the lightest structure possible, has been repeated using CoCrMo alloy and results are presented in this paragraph and in the following one. Also in this case, the free shape optimization has the objective to produce an optimal structure without following any particular direction of production. Four different results are reported, four of them correspond to a reduction of mass related to the following safety factors: 3, 4 and, 5; the last result is related to the eventuality in which among the constraints there is a displacement constraint paired to a safety factor of 3. Each safety factor is applied on the total stress level of the structure whereas the mass reduction is applied on the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. The significant results are listed in Table 18 below.

Free Direction of Optimization CoCrMo alloy

(Minimum Mass)

Safety Factor is 3

Safety Factor is 4

Safety Factor is 5

Safety Factor is 3 and

displacement constraint

Max Displacement [mm]

0.766 0.592 0.468 0.385

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Max von Mises Stress [MPa]

416 373 191 209

Min Safety Factor Area

2 3 4 3.5

First 10 Normal Modes [Hz]

F1=1303 F2=1377 F3=1456 F4=2580 F5=3697 F6=4335 F7=4889 F8=5116 F9=5848 F10=7096

F1=1552 F2=1721 F3=2022 F4=2979 F5=4810 F6=5121 F7=5738 F8=6122 F9=6949 F10=8528

F1=1619 F2=2216 F3=2631 F4=3196 F5=5506 F6=5754 F7=6629 F8=7394 F9=8325 F10=9785

F1=1625 F2=2761 F3=3209 F4=3564 F5=5850 F6=6200 F7=7555 F8=8488 F9=9932

F10=10615

Total Weight [Kg] 0.49 0.59 0.72 0.89

Table 18: Structural optimization results, free shape direction, CoCrMo alloy. Compared to stainless steel, CoCrMo alloy has overall better performance and the results above show how a stronger material allows obtaining comparable results with a lower weight; the downside of using the CoCrMo alloy is by far the greater cost than stainless steel.

Figure 45: Optimized topology of the roller holder, minimum mass objective with displacement constraint, CoCrMo alloy.

5.6.4 Extrusion along Z- and Y-axes Optimization, CoCrMo alloy The extrusion optimization analysis, which aim is to produce the lightest structure possible, has been repeated using CoCrMo alloy and results are presented in this paragraph. The extrusion optimizations along Z-axis and Y-axis have the objective to produce an optimal structure having a production direction that is directed along the given axes; the aim of these analyses is to produce the lightest structure possible. For each direction of production, two different results are reported, they correspond to the minimum mass structure obtained with a safety factor of 4 and with a safety factor of 3 together with the displacement constraint described before.

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Also in this case the mass minimization process is applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 19 below.

Extrusion Optimization CoCrMo alloy

(Minimum Mass)

Extrusion Z-axis

Safety Factor is 4

Extrusion Z-axis

Safety Factor is 3 and

displacement constraint

Extrusion Y-axis

Safety Factor is 4

Extrusion Y-axis

Safety Factor is 3 and

displacement constraint

Max Displacement [mm]

0.362 0.297 0.504 0.366

Max von Mises Stress [MPa]

178 142 211 176

Min Safety Factor Area

4 4 3 3

First 10 Normal Modes [Hz]

F1=1404 F2=2856 F3=3683 F4=5608 F5=6184 F6=7300 F7=7846 F8=8633 F9=9686

F10=11384

F1=1430 F2=3348 F3=4242 F4=5908 F5=7919 F6=8625 F7=9045 F8=9861

F9=10989 F10=12842

F1=1315 F2=2243 F3=2704 F4=3070 F5=5442 F6=6673 F7=7295 F8=7396 F9=9628

F10=10258

F1=1399 F2=2668 F3=3070 F4=3399 F5=5676 F6=7536 F7=8227 F8=8573 F9=9886

F10=11959

Total Weight [Kg] 1.2 1.5 0.85 1.10

Table 19: Structural optimization results, extrusion along Z- and Y-axes, CoCrMo alloy.

Figure 47: Optimized shape having production direction along Z-axis, CoCrMo alloy.

Out of this analysis there is a particularly interesting result: the model obtained using CoCrMo alloy to produce an extruded component along Z-axis offers performances perfectly compatible with the original roller holder; this layout

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however presents a very limited weight reduction, it is 17% only; it is also necessary to consider that the CoCrMo alloy is little heavier than stainless steel X20Cr13. A detail of the optimized design is represented in Figure 47 above.

5.6.5 Free Shape Optimization, Ti6Al4V-ELI alloy The same analyses carried out for stainless steel 316L and CoCrMo alloy have been repeated using Ti6Al4V-ELI alloy, the results are presented in this paragraph and in the following one. Also in this case, the free shape optimization has the objective to produce an optimal structure without following any particular direction of production. Three different results are reported, four of them correspond to a reduction of mass related to the following safety factors: 4, 5 and 7; the last result is related to the eventuality in which among the constraints there is a displacement constraint paired to a safety factor of 4. Each safety factor is applied on the total stress level of the structure whereas the mass reduction is applied on the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the significant results are listed in Table 20 below.

Free Direction of Optimization

Ti6Al4V-ELI alloy (Minimum Mass)

Safety Factor is 4

Safety Factor is

5.5

Safety Factor is 7

Safety Factor is 4 and

displacement constraint

Max Displacement [mm]

1.43 1.16 0.875 0.538

Max von Mises Stress [MPa]

356 267 185 132

Min Safety Factor Area 3 4 5 5

First 10 Normal Modes [Hz]

F1=1360 F2=1430 F3=1564 F4=2579 F5=3945 F6=4456 F7=5058 F8=5292 F9=6185 F10=7518

F1=1528 F2=1764 F3=1950 F4=2933 F5=4704 F6=5001 F7=5604 F8=5983 F9=6833 F10=8498

F1=1591 F2=2364 F3=2762 F4=3202 F5=5584 F6=5823 F7=6887 F8=7749 F9=8607

F10=10176

F1=1510 F2=3099 F3=3525 F4=5517 F5=6192 F6=8047 F7=9325

F8=10409 F9=11119 F10=14350

Total Weight [Kg] 0.36 0.4 0.5 0.86

Table 20: Structural optimization results, free shape direction, Ti6Al4V-ELI alloy. The resulting optimized structures made out of titanium alloys are particularly lightweight and strong, but if the performance parameter is lateral bending, they

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bend a bit too much. Once the displacement constraint is added to the analysis, then a stiffer and heavier structure is produced, although much lighter and probably costly than those that would be obtained with the other two materials. The design corresponding to safety factors 5.5 and 7 and to the last analysis, where the safety factor is 4 and the displacement constraint is applied, are portrayed in Figure 48 below: it is possible to notice how the structure topology becomes less slim as the constraints imposed to the design becomes more restrictive.

Figure 44: Free direction optimized designs, Ti6Al4V-ELI alloy.

A little difference characterizes the optimized topology of all the mass minimization analysis using the titanium alloy: the displacement constraint of 0.35 mm used for the analysis with the other materials is too restrictive to be applied also to the titanium alloy. The software was not able to produce an optimal design that can

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respect the displacement constraint using this alloy; consequently the displacement constraint value has been increased to 0.45 mm allowing to maximize the stiffness of the resulting optimized structure.

5.6.6 Extrusion along Z- and Y-axes Optimization, Ti6Al4V-ELI alloy The extrusion optimization analysis, which aim is to produce the lightest structure possible, has been repeated using Ti6Al4V-ELI alloy and results are presented in this paragraph. The extrusion optimizations along Z-axis and Y-axis have the objective to produce an optimal structure having a production direction that is directed along the given axes; the aim of these analyses is to produce the lightest structure possible. For each direction of production, two different results are reported, they correspond to the minimum mass structure obtained with a safety factor of 7 and with a safety factor of 4 together with the displacement constraint presented before. Also in this case the mass minimization process is applied to the design space only: this means that the final structure will have a total weight made out of the non-design part and the optimized design part together. All the results are listed in the Table 21 below and two examples of optimized design are reported in Figure 45.

Figure 45: Optimized design obtained using titanium alloy.

Extrusion Optimization

Ti6Al4V-ELI alloy (Minimum Mass)

Extrusion Z-axis Safety

Factor is 7

Extrusion Z-axis

Safety Factor is 4 and

displacement constraint

Extrusion Y-axis Safety

Factor is 7

Extrusion Y-axis

Safety Factor is 4 and

displacement constraint

Max Displacement [mm]

0.634 0.505 0.833 0.529

Max von Mises Stress [MPa]

155 126 107 139

Min Safety Factor Area

5 5 5 5

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First 10 Normal Modes [Hz]

F1=1439 F2=3084 F3=3940 F4=5665 F5=6418 F6=7531 F7=8154 F8=8799 F9=9758

F10=12099

F1=1404 F2=3496 F3=4430 F4=5833 F5=8806 F6=9479 F7=9650

F8=11156 F9=12991 F10=13795

F1=1298 F2=2535 F3=2954 F4=3551 F5=5522 F6=7307 F7=8306 F8=8882

F9=10615 F10=11058

F1=1383 F2=2733 F3=3202 F4=4446 F5=5742 F6=7578 F7=9340 F8=9687

F9=10620 F10=13144

Total Weight [Kg] 0.79 1.05 0.61 0.91

Table 21: Structural optimization results, extrusion along Z- and Y-axes, Ti6Al4V-ELI alloy.

The most interesting result out of this last analysis consists in observing that the structure produced by extrusion along Z-axis is the stiffer and the one that weights more. When the displacement constraint is added to the analysis, the best resulting structure is obtained by extrusion along the Y-axis: even if the two have comparable bending performance, the second weight a bit less, resulting in a better overall performance.

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6. Conclusion

Out of the stiffness maximization analysis, the best results were obtained with a free shape production and have been chosen taking the minimum displacement as comparison parameter. These results are listed in Table 22 below for each material.

Stiffness

Maximization

Building direction

Displacement [mm]

Total Weight

[Kg]

Percentage of weight

reduction

Stainless Steel 316L Free 0.368 1.23 32%

CoCrMo alloy Free 0.3 1.28 29%

Ti6Al4V-ELI alloy Free 0.571 0.78 43%

Table 22: Best results of the stiffness maximization analysis.

From Table 22 it is deduced the best result: using CoCrMo alloy, a very close performance to the one of the original configuration is achieved together with a significant reduction of weight. For what regards production following an extrusion direction, for stainless steel 316L and Ti6Al4V-ELI alloy, extrusion along Y-axis is able to produce a structure that is lighter but performs in the same way of the structure produced following an extrusion along Z-axis; for what regards CoCrMo alloy the resulting structures, built following an extrusion along the two axes, are equivalent in weight and performance. Out of the mass minimization analysis, the best results were obtained with a free shape production, using a safety factor of 3 and the additional displacement constraint; they have been chosen taking the minimum displacement as comparison parameter and they are listed in Table 23 below for each material.

Mass

Minimization

Building direction

Displacement [mm]

Total Weight

[Kg]

Percentage of weight

reduction

Stainless Steel 316L Free 0.401 1.06 41%

CoCrMo alloy Free 0.385 0.89 50%

Ti6Al4V-ELI alloy Free 0.538 0.86 52%

Table 23: Best results of the mass minimization analysis. From Table 23 is deduced the best result: using CoCrMo alloy it is possible to obtain the lightest and stiffest structure; however in this case the stiffness performance is lower than the one of the original component. For what regards production following an extrusion direction, for stainless steel 316L, extrusion along Y-axis is able to produce a structure that is lighter but performs in the poorer way than the structure produced following an extrusion

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along Z-axis; for what regards CoCrMo alloy the resulting structures, built following an extrusion along the two axes, have a similar performance behaviour than stainless stell, although with lower weight and higher rigidity. Titanium Ti6Al4V-ELI alloy is the material that allows to obtain the highest weight reduction, however its performance on the configuration obtained by extrusion along one axis are very similar to those obtained for the free shape model but also very close to the results obtained for the stiffness maximization analysis. Out of the three materials tested numerically, CoCrMo alloy appears to be the most suitable one for the roller holder application, thanks to its high elasticity modulus and mechanical properties and thanks to its high corrosion resistance. As portrayed in the previous chapter, a structural optimization problem presents several different problems but addresses several different solutions. From the finite element analysis of the original structure, it has been possible to define the critical areas (so the most stressed areas) of the structure; out of the finite element analysis of the various optimized designs that have been produced, it has been possible to define how the critical areas change, due to the removal of areas that before were stress concentration areas or due to the creation of new ones as the result of the optimization process. The biggest limit to this work has been the limited amount of available information, this condition prevented the research to come to a possible design proposal of optimized structure. In order to define a full proposal of optimized structure out of the component object of this analysis, a fully detailed experimental campaign, focused on the research of the effective working conditions of the roller holder, should be undertaken. It is important to define the load history that characterizes the component working conditions, from the vibration loads and constraints applied to the component to its expected fatigue life; probably the most important aspect of the load history consists in the load and overload to which the component is subjected: only by defining the average loads and the eventual load peaks it is possible designing accurately the optimized structure to cope fully with the working conditions that the actual roller holder is able to bear. It is however known that is particularly difficult to record the exact load conditions and that the machine may experience different working conditions that consequently require several record processes. Another important aspect regards the Company requisite expected out of the machine and its components: at the time it is not known in detail the level of criticality for the machine and what requirements are expected out of it from the Company. A complete analysis necessarily requires to considers also all the Company requirements on quality and durability for the machine’s component. The research work is not entirely pointless but is able to fully portrait the possibilities that an optimization procedure applied to a mechanical component are currently available to all the industrial sectors; it has an important methodological value and presents in a detailed way two possible paths to design an optimized component out of an existing one. However, at the current state of the analysis, it is exactly unknown what is the expected performance for the actual roller holder; consequently the whole reasoning about the component performance is based on a very limited amount of information and may not be entirely correct; moreover it is referred only to a static load condition. It is important here to underline that a more in depth optimization analysis can be carried out with Optistruct, rather than the simplified process offered by Inspire: in

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this case it is possible to subdivide the component in criticality areas and proceed setting different optimization objectives for each area, in order to obtain the best possible result. However to proceed in such a specific way, it is even more necessary to known in detail the component and all its expected requirements. At this point it is necessary to spend little time to define the next and final steps of the optimization process: shape optimization and production. Shape optimization deserves particular attention, the reader has probably noticed how rough all the optimized design appeared when compared to the original design: shape optimization process is necessary to improve the aesthetics of the component but also to improve the critical areas of the optimized design by adding manually additional material upon those areas. In this way a good looking and well performing product is the final result of the process. This process of shape optimization has not been carried out in this work since a final design proposal has not been defined due to the reason mentioned before. Whether the objective of the optimization analysis is to maximize the structure stiffness or to minimize its mass, it has been pointed out that the optimal structure performance depends strictly on the material it has been chosen to employ. The basis for the realization of such optimized structures in this work is Additive Manufacturing technology and when it comes to material choice, at the current state of the technology, the material assortment is very limited: it is thus not possible to choose any material but the choice is limited to the materials proposed by AM machinery builders and producers. All the materials proposed in this work are suited to be employed to realize components with AM machinery but they differ for performance, availability and costs. A particular characteristic of AM produced components, is that the technology is not able to build an isotropic material structure: often in the direction of production, for example upward layer by layer, the mechanical characteristics of the material are poorer than on the building plane, this effect affects especially ultimate tensile strength, yield point, elasticity modulus, elongation and material hardness. However the possibility that additive manufacturing technology offers are remarkable: it allows building virtually any shape with the possibility of having large void areas on the inside of the structure or truss like structures; but together with great production costs come some additional drawbacks that makes the technology unripe. It is worth here mentioning some of the most influential limits of this revolutionary technology. It is commonly recognised that surface roughness is not good enough to cope with strict tolerances requested by precision mechanics, this has the obvious consequence to need surface finish processes to be applied on certain areas of the component; another example may regards threaded holes, even if it is possible to directly build the thread during the AM printing process, its tolerance and its surface finish may not be adequate and thus it may be necessary to invest further resources on other tooling operations to trace over the thread. When titanium alloys are chosen for the optimized component, tooling processes can be an additional issue due to the great performance of the alloy. As mentioned before, there is also the problem of supporting structures, they are often necessary to build certain internal shapes or cavities and in certain cases to obtain a better internal surface finish: supporting structures cause a greater material consumption and require to be removed once the building process is completed. If supporting structures are needed to support large internal cavities,

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the design process must consider also an efficient way to remove them once the component is built; the same reasoning must be done if large metallic powder quantities are trapped inside large internal cavities, a drain hole must be added in the design domain to remove this powder. Potentially in some cases it is possible to not remove supporting structures, this choice cause poorer mechanical performance, noise and also dust, since supporting structures are often very weak structures that might easily break during the component working life, their sole aim is to support suspended mass during building process. The last aspect regards sharp edges: they simply cannot be built using current additive manufacturing machinery so they have to be built later with further tooling operations. In conclusion, once all the limits of additive manufacturing technology, together with its still particularly high implementation costs, have been taken into consideration, it is reasonable resorting to it only if the optimized design of a component is able to maximize the performance of the component itself, whether it is weight saving, stiffness maximization or heat transfer maximization, just to mention the most interesting applications. The development of the current work has been made possible thanks to the collaboration between Politecnico di Torino University and the well know Company in the heavy industry sector Danieli & C. Officine Meccaniche; thanks to the special interest of Prof. Dr. Brusa Eugenio from the Politecnico side and of the V.P. De Luca Andrea from the Company side. The current work was produced for the Danieli’s 2015 Fabbricando Contest [15], which also financed my three months internship at the Danieli Research Centre in Buttrio, Udine as a preliminary experience to fulfil this work.

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