low cycle fatigue weld optimization using chaboche ...1237812/...examensarbete mechanical...

INOM EXAMENSARBETE MECHANICAL ENGINEERING,AVANCERAD NIVÅ, 30 HP

, STOCKHOLM SVERIGE 2018

Low Cycle Fatigue Weld Optimization using Chaboche Material Model

FILIPPA HALLBÄCK

LOUISE ROSENBLAD

KTHSCHOOL OF ENGINEERING SCIENCES

AbstractIn this master thesis a method for optimizing welds has been investigated. The method was de-veloped by the use of a finite element (FE)-model of a silencer. The silencer is exposed to bothdynamic and thermal loads. Focus has been on using topology optimization for the welds. Themain driver for developing a method for weld optimization is to investigate whether the stressesclose to the welds could be decreased if some weld material were to be removed. Another motivefor conducting the study is to understand the potential for decreased computation time for mod-eling welds (continuous welds) should the component have more intermittent welds. Given theloads that the component is subjected to, a plastic material model would be preferable. Howeveras of today, a material model in an optimization is limited to being linear elastic and hence it isnot possible to use a plastic material model in an optimization, even though that would betterto capture the real conditions of the silencer. Thus, as part of this thesis an investigation aimingto find a transfer function between a plastic material model and an elastic material model wasconducted. An important part of an optimization is to have a relevant requirement to optimizeagainst. This requirement could be calculated from the transfer function and then be used in anoptimization.

Summarizing the findings, a transfer function between a plastic and elastic material model wasidentified, but only for a specific model and position. The identified function can translate andenable the stricter conditions used in a plastic material model to be adapted to an elastic materialmodel. To get a functional transfer function the Super Neuber needs to be calculated for everyelement in a time efficient way. This might be done by finding a relation of the geometry and theSuper Neuber parameter but this will require more investigations. If the Super Neuber parame-ters for the model are found then the fatigue requirement can be translated to an elastic stressconstraint which will give a more accurate optimization.

The method for weld optimization has been evaluated but without a requirement calculated fromthe transfer function. By changing from continuous to intermittent welds the stresses caused bythe thermal load can be decreased.

AcknowledgmentsThis master thesis has been carried out at FS Dynamics in Solna, Sweden in collaboration withScania CV in Södertälje, Sweden. The master thesis is the last examination part of the Master’sprogramme Engineering Mechanics with specialization in Solid Mechanics at the department SolidMechanics, Royal Institute of Technology in Stockholm (KTH), Sweden.

We would like to thank all our colleagues at FS Dynamics for the support and helpful discussionsas well as a special thank to our supervisor Fredrik Södergren for his fully support during thisproject and through all the ups and downs that we have been challenged by.

We would also like to thank Andreas Rydin at Altair Engineering for answering all our questionsabout Altair software such as OptiStruct and HyperMesh. Also a big thanks to the group NXDS,specially Andreas Rietz, at Scania that have been a support during the whole project.

Lastly, we like to thank Jonas Faleskog, our supervisor at KTH, for helping us understanding thetheory behind this thesis.

Stockholm, June

Filippa Hallbäck

Louise Rosenblad

Contents1 Background 1

2 Theory 22.1 Structural design optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Formulate a design optimization problem . . . . . . . . . . . . . . . . . . . 32.2 Topology optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Manufacturability in topology optimization . . . . . . . . . . . . . . . . . . 52.2.2 Objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Solution method in OptiStruct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Constraint screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Post-processing with OSSmooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Coupled linear thermal structural analysis . . . . . . . . . . . . . . . . . . . . . . . 92.6 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6.1 Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.2 Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.3 Yield surfaces and flow rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.4 Temperature dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6.5 Shakedown and ratcheting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7.1 Strain-life approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7.2 Fatigue for different temperatures . . . . . . . . . . . . . . . . . . . . . . . 122.7.3 Fatigue in weldments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8 Transformation between plastic and elastic material model . . . . . . . . . . . . . . 14

3 Calculation models 163.1 Model 1: Optimization model L-welded plates . . . . . . . . . . . . . . . . . . . . . 163.2 Model 2: Inlet/outlet pipe welded on a plate . . . . . . . . . . . . . . . . . . . . . 173.3 Model 3: Silencer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Model 4: T-welded plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Model 5: L-welded plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6 Model 6: Intermittent L-welded plates . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Weld optimization 244.1 Weld optimization with topology optimization . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Parameter studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 Weld optimization of a silencer . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Transformation 445.1 Collecting results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 Ratcheting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3.1 Saturated curve in one direction . . . . . . . . . . . . . . . . . . . . . . . . 455.3.2 Saturated curve in a multi-axial system . . . . . . . . . . . . . . . . . . . . 47

5.4 Super Neuber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5 Fatigue calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.6 Temperature dependency in the transformation function . . . . . . . . . . . . . . . 58

6 The alternative method 596.1 Evaluation of the alternative method . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 Discussion 617.1 Weld Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.3 The alternative method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8 Conclusions 63

9 Future work 64

1 BackgroundA silencer, part of the exhaust aftertreatment system in an automotive vehicle, is to a large extentwelded. The welds are of great importance for one major reason; attach parts of the componentand make sure that it does not break during usage. On some parts the continuous welds also hasanother purpose in sealing the component. On other parts of the component the welds does nothave to be continuous, here the welds could for example be intermittent to allow for more expan-sion of the material and the decrease of stresses. Today the positions of the welds are decided by atrial-and-error procedure, which is time consuming for the engineer. If the positioning of the weldscould be determined faster and if the amount of weld material could be decreased it would savethe computational time as well as lower the production costs.The question then arises where the amount of weld could be decreased without significantly re-ducing the performance of the component. In the exhaust aftertreatment system the silencer isexposed to high temperatures from the exhaust and dynamic load from the operation of the vehicle.This means that the component is subjected to cyclic loading and plastic deformation is presentdue to the high temperatures. Today a bilinear material model is used which is a simplificationand cannot capture the nonlinear cyclic hardening of the material. One fatigue approximationfor the welds is an extrapolation from the high cycle region, where it is assumed that only elasticdeformation occurs. This will lose the effects from the plastic deformation that the componentis prone to. Due to the special load cases the component is exposed to, there is a risk that thematerial model and the fatigue requirements does not capture the real conditions.

Problem descriptionTo answer the question formulation described in the background a topology optimization on thewelds can be performed. In a topology optimization it is convenient to assume a linear isotropicmaterial with small deformation theory. Also, when doing a topology optimization it is necessaryto have a relevant requirement for the optimization to optimize against. This requirement shouldbe based on the limitation that the optimization requires; a linear elastic material model. However,this limited material model will not capture the described effects that the silencer is exposed to.Therefore, an investigation of a transfer function between a plastic material model to an elasticmaterial model will be performed. From this transfer function, an optimization requirement willthen be calculated.

ObjectivesThe purpose of this master thesis is to find a plastic material model that better can capture thecyclic behavior for the load cases that the component are subjected to. Then, using this plasticmaterial model the fatigue behavior of welds in the low cycle fatigue area will be investigated.From the plastic material model a relevant transfer function to an elastic material model will alsobe sought. If a transfer function is found a requirement for the optimization, or more specificallya stress constraint, can be calculated.

An investigation on how to perform a topology optimization in order to decrease the amount ofweld will as well be performed. The aim is to develop a method for topology optimization ofwelds. The used material data for all models were chosen to represent a realistic material that iscommon in the industry. The Finite Element modeling was done in the commercial pre-processingsoftware’s Altair HyperMesh v 2017.2 and Abaqus CAE v 2017. The optimizations were done inAltair OptiStruct v 2017.2.

LimitationsThe topology optimization will be limited to linear elastic material models. Today it is almostimpossible to perform a topology optimization with nonlinear material models. There will not beany material testing in order to find material parameters, instead all material data will be takenfrom earlier performed tests.

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2 TheoryThis chapter presents the theory behind this master thesis. First, structural design optimizationis described with emphasis put on topology optimization. Thereafter some common effects of theChaboche material model is described. Lastly, the chapter ends with the theory behind low andhigh cycle fatigue, and the transformation between plastic and elastic material models.

2.1 Structural design optimizationAccording to OptiStruct’s User Guide [1] structural design optimization means finding the optimaldesign for a component so that it sustain loads in the best way. Performing an optimization is aniterative process which hopefully ends up in a well-designed component that is lightweight, manu-facturable, and durable. The terminology for optimizations can differ between software. For mostcommercial software it includes several types of optimizations such as topology optimization, to-pography optimization, size optimization, and shape optimization. The software Altair OptiStructis a structural analysis solver for linear and nonlinear problems under several different types ofloadings. In this software the optimization techniques can be divided into two parts; concept de-sign and fine tuning design. In a classic design process there is usually endless of iterating loopsto end up with a design that fulfills all the requirements. With an optimization, included in thedesign process, the iterative loops can be much fewer. The outcome of the design process will bea model that meet all the requirements and the whole process will be less time consuming. Eachoptimization approach used in OptiStruct will be described shortly below.

Concept design - changing the overall design of a component, first step in an opti-mization:

• Topology optimization: Given a defined region (also called the design space) of a component,an optimum design is reached by modifying the material distribution within this region byremoving or adding material from the initial design. In the optimization process the newmaterial distribution is generated by changing the material density and by changing thestiffness of elements [2].

• Free-size optimization: The free-size optimization allows the thickness to vary in each ele-ment. This is done on shell cross-sections.

• Topography optimization: This approach is similar to topology optimization but instead ofusing density variables this one uses shape variables. In the defined design region an advancedform of shape optimization is performed by a pattern of shape variable-based reinforcements.The shape variables influence on the structure is calculated and optimized in an iterativeprocess. Then this allows for creating any reinforcements pattern within the design space.An example of topography optimization is stiffening beads in a shell model [1].

Fine tuning design - refine the optimized component further, next step in an opti-mization process:

• Size (parameter) optimization: In the size optimization the focus is on changing the compo-nents geometries such as thicknesses of the sheet metal components, or beam cross-sectionalproperties, typically to increase the stiffness or reduce vibration [2].

• Shape optimization: In this approach the component is modified by moving surface nodes todecrease local stress concentrations. Here the user needs to define a number of different shapesof the component and then an optimization is done by finding the optimum combination ofthose shapes to meet the requirements.

• Free-shape optimization: This optimization approach is similar to shape optimization buthere there is no predefined shape of the component. Instead given an outer boundary thenodes are moved and an optimal structure is found [1].

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2.1.1 Formulate a design optimization problem

A design optimization problem statement needs the following parameters; design variables, re-sponses, constraint functions, and objective function(s).

• The design variables define the parameters, properties, or elements that are allowed to varyduring the optimization. These variables constitutes the design space.

• The responses are the inputs to the optimization. Structural responses are for example staticstress, static displacement, force, strain, mass, volume, etc. and these are calculated in afinite element analysis (FEA). A response can either be global or load case related, the latestmeans that the response must be defined to a specific load case. A global response can forexample be mass and a load case response could be static displacement.

• The constraint functions are connected to the responses. They constitute boundaries forthe response functions and therefore they must be satisfied in order for the design to beacceptable.

• Lastly the objective function is the objective of the optimization. This one is connected toone of the responses, which is a function of the design variables. The most common setup isthat the objective function should be minimized or maximized. The objective function mustbe defined for one or several load cases [2].

In a design optimization when minimizing the objective function, the following problem is solved:

Objective function: min f(xxx) = f(x1, x2, ..., xn) (1)

Subject to:

Design variables: xLoweri ≤ xi ≤ xUpperi where i = 1, 2, ..., nDesign constraints: gj(xxx) ≤ 0 where j = 1, 2, ...,m

(2)

where the objective function is defined as f(xxx) and the constraint functions as gj(xxx). The con-straint functions are structural responses achieved from the finite element analysis. A constraint isfulfilled, considered active, if the equality is satisfied exactly (g = 0), if the equality is g < 0 thenit is inactive, and if g > 0 then it is violated. This is similarly done for maximizing the objectivefunction.

The choice of the vector of design variables xxx can differ between the different types of optimizations.In the topology optimization the element densities are the design variables. In size and free-sizeoptimizations they are the properties of the structural elements. In topography, shape, and free-shape optimizations are the factors in a linear combination of shape perturbations [1].

2.2 Topology optimizationA topology optimization is a mathematical technique that produces an optimal structure in thesense of shape and material distribution. This is done by discretizing the structure into a finiteelement mesh and then calculate material properties for each element. The material distributionis then alternated in order to satisfy the defined objective under given constraints.

During a topology optimization the material distribution can change in the design space (thedomain of the structure where the optimization is done). The design space can be the entirecomponent or parts of it. By finding a shape of the model and number of holes the optimal ma-terial distribution is determined. The shape and number of holes does not have any restrictions,it can take any shape within the design space. Therefore, the form of the structure can change alot and it could be hard to guess how the result from the optimization will look like before it is done.

Before doing a topology optimization the structure is divided into a design space and a non-designspace. A possible division is presented in Figure 1 [1].

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Figure 1: Meshed FE-model with design space (green area) and non-design space (blue area).

The FE-model is represented with a discretized finite element mesh. In a topology optimizationproblem the design variables are the element densities of each element in the design space. Thisresults in a discrete optimization problem. In OptiStruct this type of optimization problem issolved by a method called the density method, also known as the SIMP (Solid Isotropic Materialwith Penalization) method. The SIMP method handles discrete material distribution problems byconsidering a continuous optimization problem. This is because optimization of a large number ofdiscrete variables is computationally heavy. The used method then penalize or steer the intermedi-ate densities, which represents fictitious material. The method will be described in a more detailedform below. Define the design space as Ω, the goal is to find an optimal subset Ωmat within Ω. Aninteger formulation can be expressed as:

Eijkl = ρeE0ijkl, ρe =

1 if x ∈ Ωmat,0 if x ∈ Ω− Ωmat,

(3)

where E0ijkl is the stiffness tensor for the defined isotropic material. A volume constraint, Equation

4, also needs to be defined which describes the amount of material at our disposal, with V as theinitial volume of the design space ∫

Ω

ρedΩ ≤ V. (4)

The design variable x describes in which elements there should be material (solid), ρe = 1, andin which elements there should be no material (void), ρe = 0. In the optimization problem agradient based solution strategy is used. Therefore, in order for the density function to take onvalues between 0 and 1 the integer formulation, Equation 3, need to be formulated as a continuousfunction. The most common approach to solve this problem is to use the SIMP method, then thepenalized, proportional stiffness model is used:

Eijkl(x) = ρe(x)pE0ijkl, p > 1,∫

Ωρe(x)dΩ ≤ V ; 0 ≤ ρe(x) ≤ 1, x ∈ Ω,

(5)

where the "density" ρe(x) represent the design function and the factor p is called the penalty factorand it penalizes elements with intermediate densities to approach 0 or 1. Now Eijkl represent thepenalized stiffness and E0

ijkl the material properties of a given isotropic material. The density willbe interpolated between the following material properties [3]:

Eijkl(ρe = 0) = 0, Eijkl(ρe = 1) = E0ijkl. (6)

For five different values of the penalty factor, p, the function ρpe are plotted against the density ρeand presented in Figure 2. From Figure 2 it can also be seen that if p > 1 the stiffness will bedisproportionately low for density values between 0 and 1 (void and material, respectively). These

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values should be avoided because they do not represent an economical use of material. The figurealso shows that the SIMP method will force the topology optimization to create a 0 - 1 design [4].

Figure 2: Relative stiffness ratio as a function of the element density.

2.2.1 Manufacturability in topology optimization

From a topology optimization the result will not always be as good as one hope for, the optimizationcan lead to a design that is not manufacturable. To get the best design it is usually not enoughto perform an optimization with all parameters set to default values. Sometimes the optimizationhas to be repeated and some important parameters with appropriate manufacturing constraintsneed to be studied. Another issue that can cause bad result is that the solution of the optimizationcan be mesh dependent. For different sizes of the mesh, the optimization can give qualitativelydifferent solutions, even if the setup of the optimization is the same. The reason for this could bethat the penalized topology optimization problem does not have a unique solution to converge to.Parameters that can be useful to study in OptiStruct are the following:

• Member size control for topology optimization which are either minimum member size con-trol, called “MINDIM ” or maximum member size control, called “MAXDIM ”.

• Improved Discrete Topology Optimization Formulation, called “TOPDISC”.

• The penalty factor, called “p”. The penalty factor can be changed through the “DISCRETE”parameter or Discreteness Parameter in OptiStruct. The relation between the “DISCRETE”parameter and the penalty factor, p, is DISCRETE = p – 1 [1].

Minimum member size control, MINDIM

The minimum member size, MINDIM , is a manufacturing constraint in OptiStruct. It is im-portant to distinguish between a mathematical optimization constraint and a manufacturing con-straint. The latest can be violated even though the optimization tries to fulfill them. By settinga value for the minimum member size one can control the smallest structural features of materialto be used in the topology optimization. This parameter can be seen as a minimum diameter offeatures in two dimensions. When changing the minimum member size it has some restrictions, itis dependent on the mesh size and therefore it can only be varied in the following range [5]:

• MINDIM > 3× the average element size.

• MINDIM < 12× the average element size.

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To have a requirement on the allowed minimum member size is important because if the objectivefunction is for example to minimize the compliance (increase the stiffness) the topology optimiza-tion will aim for thinner structural members. Thinner structural members is better than fewerthicker structural members, from a numerical point of view. The objective function will be lowerif the structural members are thinner. Unfortunately, to have too many thin structural membersare usually not an acceptable structure. Therefore, to avoid this problem there is a requirementon the range of allowed minimum member size [6]. The average element size can be calculated bytaking the average square root of the area of the elements. The restrictions for MINDIM refersto the elements in the design space.

Another reason for using this parameter is that it can help in avoiding a checkerboard effect. Thecheckerboard problem can be described as when the element densities are alternating solid andvoid, or with high and low values of the element densities, in a checkerboard like pattern. Thisresult is usually something that should be avoided and instead strive for a more defined result withareas of either solid (1) or void (0) element densities. A disadvantage when using the parameterMINDIM is that the computation time could increase. When it is activated the maximum num-ber of iterations must be increased to 80, default is 30 [1].

Maximum member size control, MAXDIM

Similar to the MINDIM parameter the maximum member size control, MAXDIM , is also amanufacturing constraint that instead penalizes the creation of large features of material. Therestriction for the parameter is any of the following [1]:

• MAXDIM > 2× MINDIM or

• MAXDIM > 6× the average element size.

Penalty factor, p

According to S. Subramaniam [5] the discrete value should be in the range 0 < DISCRETE < 4 inOptiStruct to get a good result. This means that the penalty factor are in range 1 < p < 5. A toohigh value can lead to checkerboard effects. In OptiStruct the default value for the DISCRETEparameter in a model with shell elements is 1 and therefore the penalty factor is p = 2.If the member size control is activated an automatic increase of the penalty factor is present duringthe optimization process. This is to achieve a more discrete solution [1].

Improved discrete topology optimization formulation, TOPDISC

TOPDISC is a parameter in OptiStruct that can be used to improve the discreteness in a topologyoptimization result. When activating the TOPDISC parameter it also includes the manufacturingconstraint used in the optimization. When using manufacturing constraints, such as MINDIM ,the result from a topology optimization can have a semi-dense layer of elements and by usingTOPDISC this could be avoided [1].

2.2.2 Objective functions

There are several different objectives functions available and here are just some of them presented.

Static compliance

In a linear static analysis a common objective function for an optimization is to minimize ormaximize the compliance (or the strain energy). It was also a suitable objective function for thisthesis thus the performance of the component should not be deteriorate. The same as finding thestiffest design of a structure. The static compliance can either be calculated as:

C =1

2uuuTfff =

1

2

∫εεεTσσσdv (7)

or as, since the same equation can also be rewritten by using the relation KuKuKu = fff

6

where KKK is the stiffness matrix of the structure, uuu is the displacement vector, and fff is the appliedforce vector.

The compliance could either be minimized or maximized depending on what type of load case thestructure is exposed to. If the structure is exposed to an applied force f then the compliance C isthe inverse of the stiffness K. Then applied force f is the known factor, force controlled, and thedisplacement u is an unknown factor, as:

C =1

2uuuTfff =

1

2fffTKKK−1fff (8)

because the stiffness matrix is symmetric then KKKT = KKK. To get the maximum stiffness KKK, thecompliance C has to be minimized.

On the other hand, if the structure is exposed to a displacement u the compliance is proportionalto the stiffness K. Then the applied displacement u is the known factor, displacement controlled,and the force f the unknown factor, described in:

C =1

2uuuTfff =

1

2uuuTKuKuKu (9)

and to get the maximum stiffness KKK the compliance C has to be maximized. Therefore, withforce controlled low compliance equals small displacement and high stiffness. Minimization of thecompliance implies minimization of the structural strain energy. With displacement controlled thestrain energy is instead maximized [1].

Weighted compliance

Another possible objective function is the weighted compliance, Cw, which is same as the onepresented in latest section but used in an optimization with several load cases. Depending on theseverity on each load case, a weight wi, can be defined for each load case. According to OptiStruct’sUser Guide [1] the weighted compliance is calculated as:

Cw =∑

i

wiCi =1

2

∑i

wiuuuTi fff i. (10)

Volume fraction

The volume fraction is also a commonly used objective function, usually minimizing the volume ormass fraction is the target in a topology optimization. This response describes the fraction of theinitial design space and does not include the non-design space in the calculation. It is describedin:

Vf =Vtot − Vnon-design

Vdesign(11)

where Vf is the volume fraction (the response that are minimized or maximized in an objectivefunction), Vtot is the total volume at the present iteration in the optimization, Vnon-design is theinitial volume for the non-design area, and Vdesign is the initial volume for the design space area. Thevolume fraction response is only available to use in a topology optimization and is also commonlyused as a constraint if not as an objective function [1].In this thesis the volume fraction is only used as a constraint in combination with either staticcompliance or weighted compliance as objective function.

2.3 Solution method in OptiStructFor OptiStruct to solve an optimization problem a gradient-based algorithm is used. By defaultan algorithm called the Method of Feasible Directions (MFD) is set. This is a mathematicalprogramming method and OptiStruct only uses these types of algorithms. This master thesis willnot include any investigation in different algorithms and the default one will be used. This is astandard optimization algorithm which is described in a more detail form in OptiStruct’s UserGuide [1]. To find a solution during an optimization problem OptiStruct uses a technique calledthe local approximation method. This is an iterative process and it follows these steps:

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1. A finite element analysis is performed on the structure that should be optimized.

2. The analysis starts to converge or not.

3. During an iteration a response screening (or constraint screening) is done. This is to checkwhich responses that should be retained as active or not for that specific iteration.

4. For the retained responses from step 3 a design sensitivity analysis is done.

5. From the sensitivity information gained in step 4 an optimization of the explicit approximatedproblem is performed.

6. Go back to step 1 and repeat the steps.

During step 2 it is important to get a stable convergence. This is done by using a method calledmove limits, and it is based on that the design variables are limited to change within a narrowrange during each iteration. During the first few iterations the design variables are allowed to makethe biggest changes.

In step 3 and 4 the sensitivity analysis is calculating derivatives of the retained structural responsesas regards to the design variables. These are two important steps because this represent the changefrom a finite element analysis tool to an automated design optimization tool.

Then in step 5 the actual design update is achieved. Here OptiStruct chooses, depending on whattype of characteristics that are used in the optimization problem, between two methods: dual orprimal method. The dual method is more common to use in a topology or topography optimization.That method works best for an optimization with more design variables than constraints, which istypical for those optimizations. The primal method is best suited for optimization problems thatuses as many design variables as constraints, which is usual for a size or shape optimization.

There are three outcomes from an optimization, the design is feasible, the design is infeasible, orthe optimum design is reached. The feasible design imply that all constraints are satisfied but theobjective function is not finding the absolute minimum or maximum, depending on the definitionof it. The infeasible design imply that at least one constraint or more is not satisfied. Lastly, theoptimum design is a feasible design but then is also the objective function satisfied [1].

2.3.1 Constraint screening

During each iteration all the responses in an optimization problem, which are the objective func-tion(s) and all the constraints, are investigated. Using the setting, which is default in OptiStruct,constraint screening, the responses are instead cut down to a smaller set of representative ones.Constraint screening maintain the optimization problem size on a fair level but still representthe original optimization problem. This is possible, by removing constrained responses that areunimportant and will not affect the direction of the optimization problem; it could be constrainedresponses that are far away from the upper and/or lower bounding values (of course already onthe acceptable side), or those that are less critical compare to other constrained responses.

There are some disadvantages of not using the constraint screening function. If considering alldesign variables and responses the optimization can grow into a huge problem. It is usually notcommon for optimization algorithms to cope with a large number of design variables and at thesame time having a large number of constraints. It is usually just one of them that works. If alsoall the sensitivities need to be calculated for each response, that will contribute to an increase incomputation time.

Even if constraint screening has shown to be very useful; the computation time decreases, andusually gives reliable and understandable result, it has some drawbacks. Because it is not consid-ering all constrained responses it can take longer to converge to a solution. Also if the screening isconsidering too many responses it can result in longer time to reach a converged solution [1].

8

2.4 Post-processing with OSSmoothAfter receiving an optimization result it is possible to implement that on the initial FE-model byusing the software OSSmooth by Altair. This software is a semi-automated design interpretationthat can be used for three different purposes; OSSmooth for geometry, FE topology reanalysis,FE topography reanalysis. The first one is usually for changing the initial FE-model according tothe optimization result. The two last ones are for doing a FE reanalysis with the new geometry(generated from the optimization result) and with stated boundary conditions.

If the result is from a topology optimization the software can create an iso-density boundary surface(or also called iso-surface). Before running the OSSmooth for geometry a value for the densitythreshold should be defined by the user (default values is usually between 0.3 – 0.4). The generatedsurface from OSSmooth consider the values of the element densities and all the values that arebelow the chosen density threshold value are considered to be void and therefore they will not bepart of the generated surface. The ones that are in the surface plot are above the density thresholdvalue and therefore considered as solid [1].

2.5 Coupled linear thermal structural analysisFrom a heat transfer analysis the temperature field is calculated. That represent the thermal en-ergy in the structure. Conduction is the heat transfer between solids and convection is the heattransfer between solid and surrounding fluids, for example air. In a linear steady-state heat trans-fer analysis there is no time dependencies and the thermal material properties for conduction,convection and thermal expansion (a) are linear.

For conductive heat transfer analysis the Fourier’s equation

fff = −k(∂T

∂xxx

)(12)

is used, where fff is the heat flux, k is the heat conductivity, T is the temperature, and xxx is theposition of a material point. For convection Newton’s law

fff s = h(Tamb − Ts)nnn (13)

can be used, where fff s is the heat flux vector which is proportional to temperature gradient, h isthe heat transfer coefficient, Tamb is the ambient temperature, Ts is the surface temperature of thestructure, and nnn is the outward normal vector of the surface where convection occurs. The heatexpansion is presented in:

εthermal = a(T0 − Ts) = a∆T (14)

where T0 is the initial temperature of the structure. The finite element equation for this analysisis defined as:

(KKKC +HHH)TTT = fffT (15)

where KKKC is the conductivity matrix, HHH is the boundary convection matrix, TTT is the unknownnodal temperatures, and fffT is the thermal loading vector.

The temperature field in a structure can cause thermal stresses, due to thermal expansion orcontraction of the material. By coupling a heat transfer analysis and a structural analysis thesethermal stresses can be achieved. During the coupling analysis the heat transfer analysis is donefirst and then becomes input to the structural analysis. The finite element equation solved ispresented in:

KuKuKu = fff t + fff (16)

where KKK is the global stiffness matrix, uuu is the unknown displacement vector, fff t is a body forcedue to the change in temperature from a reference solution, and fff is the structural loading vector[7]. The thermal and mechanical strain are related as:

εtotal = εmechanical + εthermal. (17)

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2.6 Material modelThis theory part is based on the Chaboche formulation for cyclic behavior of metals using nonlinearhardening models. The Chaboche material model can handle tension-compression cyclic loadings.During these types of loadings the hardening properties are changing. Depending on what type ofmaterial, temperature, or initial states that are used, the material may soften or harden.

2.6.1 Hardening

When loading a component beyond the yield stress the material can harden, meaning that theelastic zone for the material is increasing or translates. The same applied force gives a decreasingstrain amplitude during cyclic loading. The phenomenon of hardening can be divided into totwo types, isotropic and kinematic hardening. In isotropic hardening, the hardening occurs in alldirections, meaning that the yield surface expands, as can be seen to the left in Figure 3. Kinematichardening means that the hardening occurs in one direction but softens in the other, making theyield surface constant, as can be seen to the right in Figure 3.

Figure 3: Isotropic and kinematic hardening

2.6.2 Softening

Another phenomena that can occur is softening. This happens when the yield surface reduces andthe elastic zone decreases or translates. The same applied force gives an increasing strain amplitudeduring cyclic loading. It is common for steel materials that if the loading is first in tension andthen in compression the material will first harden in tension and then soften in compression (orthe other way around if it starts in compression), which is a typical kinematic behavior.

2.6.3 Yield surfaces and flow rule

In order to model the behavior of most metals during cyclic loading kinematic hardening modelscan be used. These kinematic hardening models are pressure-independent plasticity models. Thismeans that the yielding of the material is independent of the hydrostatic pressure stress. For thenonlinear combined isotropic and kinematic model the von Mises yield surface can be used and isdefined as

F = f(σσσ −ααα)− σ0 = 0, (18)

where σσσ is the back stress, σ0 is the yield stress and f(σσσ − ααα) is the equivalent von Mises stress.The equivalent von Mises stress can be described as

f(σσσ −ααα) =

√3

2(SSS −αααdev)(SSS −αααdev) (19)

where the deviatoric stress tensor is SSS and the deviatoric part of the back stress tensor is describedwith αααdev. The yield stress depends directly on the isotropic hardening which is described with

σ0 = σ|0 +Q∞(1− e−bεpl

), (20)

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where σ|0 stands for the initial yield stress for the material. The parameters Q∞ and b arematerial constants that can be determined by material tests. εpl is the equivalent plastic strain.These kinematic hardening models follow the plastic flow defined as

εεεpl = ˙εpl∂F

∂σσσ(21)

where the rate of the plastic flow is εεεpl and the equivalent plastic strain rate is symbolized as ˙εpl

[8]. Both the nonlinear kinematic and the isotropic hardening law are contributing to the so calledevolution law. One component defines the translation of the center of yield surface, also calledback stress, and one component describes the change of equivalent stress defining the size of theyield surface, respectively. The evolution law with field variables dependencies are defined as

αααk = Ck1

σ0(σσσ −ααα) ˙εpl − γkαααk ˙εpl (22)

where, Ck and γk are material parameters and γkαααk ˙εpl (no summation) is called the “recall term”.The recall term introduces nonlinearity to the law. This evolution law is based on the basic Zieglerlaw [9]. To get a better result it is possible to using several material parameters, Ck and γk, andthen superpose several kinematic hardening components (back stresses) [8] with

ααα =N∑

k=1

αααk. (23)

A rule of thumb that is often used is to superposition three back stresses to capture the importantessence of a material.

2.6.4 Temperature dependency

To include the temperature dependency in the material model it can be assume that some thematerial parameters such as Young’s modulus, yield stress, C1−3 and γ1−3 can be described asa function of temperature [10]. To determine the functions the material parameters should beextracted from material tests for different temperatures, and then by an optimizing tool, minimizethe errors of the calculated parameters in the exponential function

P = xp1

(1− xp2e

(T

xp3

))(24)

where P is the material parameter that is temperature dependent and T is the temperature inKelvin [10]. The parameters xp1-p3 is the three material variables that describes the temperaturedependency for material parameter P .

2.6.5 Shakedown and ratcheting

One phenomena that can occur when the load is not purely alternating, during a non-symmetricforce-controlled test is ratcheting [11]. When the material undergoes a non-symmetric cyclic loadthe plastic deformation will be larger in either compression or tension and smaller in the other,making the strain wander at a constant strain increment for each cycle until it reaches a maximumand breaks. If the stress and strain curve stabilizes after a number of cycles, shakedown hasoccurred instead of ratcheting. Figure 4 shows the difference of ratcheting and shakedown in astress-strain plot. If shakedown occurs, normal fatigue calculations can be used.

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Figure 4: Ratcheting and shakedown [12].

2.7 FatigueAll good things must come to an end. By loading the silencer repeatably cracks and voids willform and after they grown big enough the component will break. It is crucial to predict life lengthwhen designing a component.

2.7.1 Strain-life approach

There are mainly two types of fatigue, high cycle fatigue (HCF) and low cycle fatigue (LCF). Thedifference between these two types is that in HCF only elastic deformation can occur, while in LCFboth elastic and plastic deformation takes place. For HCF it is common to use a stress-life curve,which can be directly translated to a strain-life curve since it is still in the elastic region (σ = Eε).When LCF occurs the stress cannot be directly translated to a strain-life curve and therefore thestrain measurement is used when LCF occurs [12]. Basquin described the stress-life relationshipat HCF as

σa = σ′f(2Nf)b (25)

where 2Nf is the number of reversals, σ′f and b are material constants, and σa is the stress amplitude.The corresponding relationship for strain becomes

εea =σ′fE

(2Nf)b (26)

where E is the elastic modulus and εea is the elastic strain amplitude. The relationship betweenthe plastic strain amplitude, εpa, and the number of cycles to failure can be described by theCoffin-Manson expression

εpa = ε′f(2Nf)C (27)

where ε′f and C are material constants. The total strain amplitude, εtota , can be calculated byadding the elastic strain and the plastic strain, which gives

εtota =σ′fE

(2Nf)b + ε′f(2Nf)

C (28)

and is used for LCF.

2.7.2 Fatigue for different temperatures

The life expectancy of a component is related to the temperature. As it has been discussedbefore the material parameters changes when the temperature increases. The fatigue curve atone temperature can be predicted by knowing the fatigue curve at another temperature and thescale factor for the specific material [13]. This method cannot be translated to the strain-lifecurve. The stress depends on the applied force and model, while the strain depends on both thestress and Young’s modulus. Young’s modulus is directly dependent on temperature (as it wasdiscussed in the Chapter 2.6 Material model) and for higher temperature, E will decrease, giving

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a comparatively decreased stress at a given strain. The yield stress is also directly dependenton the temperature. This means that at a given stress level, with an increased temperature aspecimen might go from the elastic region to the plastic region, further increasing the strain. Inother words, for a given stress amplitude an increase in temperature will lead to a decrease in lifeexpectancy, while for a given strain amplitude, an increase in temperature does not necessary leadto a lower life expectancy; even with a lower yield stress, the stresses equivalent to a certain strainwill be lower to begin with. According to Alain [14], the life expectancy will increase when thetemperature increases for a given strain amplitude until it reaches a maximum, see Figure 1 (e.g.around 300C for 316L steel). It is therefore hard to find how the LCF fatigue parameters dependon temperature.

Figure 5: Life expectancy versus temperature for given plastic strain amplitude [14].

Creep

When higher temperatures (temperatures above half the melting temperature in Kelvin, K) arepresent, creep can occur, which will lower the life expectancy [13]. Creep depends on temperature,time and stress. The material 316L stainless steel which is used in this report [9] has a meltingtemperature at 1670 K, meaning that only at the locations where the temperature is over 835 Kcreep has to be considered in the calculations. The maximum temperature in the silencer has beenmeasured to be 970 K meaning that creep will occur but not in a large scale. Previous studies haveshown that the places where the highest temperatures are reached is most often not the placeswhere the highest stresses is present, making the creep effects negligible from a fatigue perspective[13]. Creep has therefore not been accounted for in this report.

Dynamic strain aging

When the material is exposed to higher temperatures the characteristics of the material changesover time in a process called aging. Some materials are more sensitive than others. Stainless steel,which is commonly used in the industry, is known to be more resistant to aging, but since thewelds has a different micro structure dynamic strain aging will occur at around 770 K (for 316L(N)welds) according to an article published by Prasad [15]. This means that the high temperaturesin the silencer will affect the material quality through aging effects. The aging effect has not beenevaluated specifically and has only been included by using the life expectancy curves that comesfrom material tests at higher temperatures [16]. The tested curves will indirectly include the agingeffect without being specific of how much time the material is exposed to the heat.

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2.7.3 Fatigue in weldments

The weld quality differs a lot from the base material (the material of the plates that is weldedtogether, see Figure 6). Even though the weld material and base material have almost the samematerial properties the process of welding will reduce the quality of both the materials. The heatthat is needed to attach the base material with the weld material creates the heat affected zone(HAZ) where unwanted phase transition of the material and micro cracks occurs, illustrated inFigure 6. When calculating life expectancy of weldment, standards are used [17] where the qualityof the weldment, material curve and geometry decides the weld class (FAT-value) and thereforethe life expectancy. One problem is that only stress-life curves are used in the evaluation methodwhich creates difficulties when the welds plastically deforms in the load cycles. When plasticdeformations occurs in a component subjected to cyclic loading, LCF needs to be evaluated andstrain-life curves are needed to calculate the life expectancy.

Figure 6: Material components in a weldment.

Residual stresses

High temperatures are applied to the base material when welding components together and whenthe material later cools down, the weld material and the HAZ contracts, while the base materialkeeps a constant volume. This phenomenon creates residual stresses that are not evenly distributedand that are dependent on the geometry of the welds, the temperature during the welding and theconstraints on the component. Those effects will influence the life span for welds exposed to HCF,but can be neglected when cyclic plastic strains occurs [18]. The impact of the plastic deformationis more significant than of the residual stresses, therefore will these be excluded in this report.

2.8 Transformation between plastic and elastic material modelTo save computing time and extensive material tests simplifications has often been made in theindustry. Having large simplifications in the calculations makes the result unreliable and in worstcase, irrelevant. A compromise has to be done between computing time and reliability of theresults. If a transformation can be made between a plastic and an elastic material model thereliability of the design can remain high and the computing time still be acceptable. Neuber is amethod used when calculating notches and transform a calculation done with an elastic materialmodel to a plastic material model [12]. It is based on the stress concentration factor, Kf, beingdescribed as

Kf =

√∆σ∆εE

∆σ∞(29)

where ∆σ is the stress amplitude, ∆ε is the strain amplitude, E is the Young’s modulus and ∆σ∞

is the cyclic load amplitude. ∆σ and ∆ε depends on the material model used for cyclic loading.This means that

14

∆σ∆ε =(Kf∆σ

∞)2

E= Constant (30)

for a fixed load and geometry. The term ∆σ∆ε = Constant is the transfer function that comesfrom the Neuber correction. For elastic loading the following relation applies; ∆σ = ∆εE, but forcomponents cycled with a plastic material model an increase in stress (over the yield stress) willgive a disproportionate higher strain (comparing to an elastic material model). Neuber correctionpresents a transfer function that describes the shape of two material models, then one can transformthe results from one analysis to the other. Figure 7 illustrates the difference between using a linearmaterial model and Chaboche material model and illustrates the transfer function.

Figure 7: Transfer function between plastic and elastic material model.

The transfer function based on Neubers theorem has its weaknesses. At higher plastic deformationthe transformation will be incorrect. A concept based on Neubers correction is Super Neuber whichhas been proposed by L. Samuelsson [19], it is based on the super hyperbolic function as

∆σ =Cq

∆εq, (31)

where Cq is a constant and q is the Super Neuber parameter that can be fitted depending on thematerial. If q was set to 1, Neubers transfer function would be used. If q is set to infinity, a lineartransformation would be used, meaning that the elastic strain is constant for both material modelsat the same load. A relation that R. Adibi-Asl [20] has used is

σep

εep=

(σep

σe

)jσe

εe(32)

where σep and εep is the more advanced material stress and strain. The parameters σe and εe

is the elastic materials stress and strain and j is a transformation parameter which will have thesame function as the Super Neuber parameter. R. Adibi-Asl [20] changed the value j by suggestingthat it is dependent on the concentration factor of the geometry, Kf, the stress, σ and the yieldstress σ|0. The equation is formulated as

j =arctan(σ)− arctan(σ|0Kf

)

arctan(σ|0)− arctan(σ|0Kf)

+ 1. (33)

To use Equation 33, Kf has to be found for every elements in the evaluation domain. To find Kfin a complex model needs computations and might be as time effort as finding the j directly. Inthis report the Super Neuber method has been used. The goal is to evaluate to which extent q isdependent on the geometry of the model and where on the model it is evaluated.

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3 Calculation modelsThis chapter will describe the calculation models that were used throughout this master thesis.The material models with parameters, loads, and boundary conditions for each model are alsopresented here. Some models were only used in the topology optimizations which will also bestated. Models 1 - 3 were either modeled with only two-dimensional quadrilateral or with mixed;triangle and quadrilateral first-order shell finite elements. Models 4 - 6 were modeled with quadraticshell elements of the second-order. The load level of models 4 - 6 were reached by iteration thusto avoid total plasticity of the elements but still reach plasticity in the evaluation domain. Due tothe complexity in model 5 and 6 the highest stress and strain values of these models were detectedoutside of the evaluation domain. Which meant that a higher applied load than the ones usedwould give total plasticity in those elements and the results in that case would not be reliable.

3.1 Model 1: Optimization model L-welded platesThe model in Figure 8 was used in topology optimizations and is a linear elastic isotropic modelwith the material parameters presented in Table 1. The shell elements have a thickness of 3 mmfor the plates and 2 mm for the welded part. The average element size is 1 mm.

Figure 8: Model 1: L-welded plates.

Table 1: Material parameters for Model 1.

Young’s modulus E = 210 GPaPoisson’s ratio ν = 0.3

In Figure 9 the structural boundary conditions and load are defined. The nodes are fixed indifferent degrees of freedom (DOF) as described in Figure 9. The translational directions are 1, 2,and 3 and the rotational directions are 4, 5, and 6. The load is defined as a pressure load on oneof the plates with a magnitude of 10 MPa, in negative x-direction, see Figure 8.

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Figure 9: Model 1: Structural boundary conditions and load for model 1.

3.2 Model 2: Inlet/outlet pipe welded on a plateThe model in Figure 10 was also used in topology optimizations and is a linear elastic isotropicmodel with the material parameters presented in Table 2. The shell elements have a thickness of10 mm for the plate, 5 mm for the pipe and the welded part. The average element size is 1.97 mm.

Figure 10: Model 2: Inlet/Outlet pipe welded on a plate.

Table 2: Material parameters for Model 2.


Thermal Expansion a = 3·10−6 K−1

Thermal Conductivity k = 25 W/(m·K)Heat Transfer Coefficient h = 100 W/(m2·K)

In Figure 11 the structural boundary conditions and load are defined. The edge nodes of the plateare fixed in the DOF 1, 2, and 3. The static load is a force with magnitude of 4 kN applied in thepositive x-direction at the center node of edge of the pipe. The edge nodes are dependent on thecenter node.

17

Figure 11: Structural boundary conditions, to the left, and load, to the right, for model 2.

In Figure 12 the thermal boundary conditions are defined. The ambient temperatures of the pipeand plate are also defined.

Figure 12: Thermal boundary conditions for model 2.

This resulted in a coupled linear thermal structural analysis. Three load cases 1, 2, and 3 wereperformed.

1. Structural analysis with boundary conditions and load as in Figure 11 (at room temperature,20C) – called structural load case.

2. Heat transfer analysis (steady-state) with thermal boundary conditions as in Figure 12.

3. Structural analysis with the temperature field from load case 2 as input to the thermal loadand same boundary condition as in load case 1 (the structural load used in load case 1 wasnot included in this load case) – called thermal load case.

The heat transfer analysis was therefore calculated in each optimization iteration. This will bedescribed in a more detailed form as Option 1 in Chapter 4.

3.3 Model 3: SilencerThe model presented in Figure 13 is a model that should represent a Scania silencer, part of theexhaust aftertreament system. It should be noted that this calculation model is just a representa-tion of how a silencer could look like, it does not represent any existing design, and it have never

18

been produced by Scania. Despite this, a limited information regarding its material model andother relevant parameters will be presented in this master thesis.

Figure 13: Model 3: Silencer.

Model 3 was also used in topology optimizations and is a linear elastic isotropic model with thematerial parameters presented in Table 3. The welds, indicated in the figure, are continuous andare attaching the brown and orange plates to the middle part of the silencer. The shell elementshave a thickness of 1.5 mm for the indicated welds, and also 1.5 mm for the brown and orangeplates. For the whole model the average element size is 4.9 mm.First a heat transfer analysis (steady-state) was performed in Abaqus. This for receiving thetemperature field in all nodes that were used in a structural analysis. In the heat transfer analysissome material parameters are not constant, instead they are temperature dependent and these are;the Young’s modulus, thermal expansion, and conductivity. Due to confidentiality against Scaniathese parameters will not be stated in this master thesis. Only the parameters that were used forthe structural load case will be presented, Table 3, and also the constant thermal expansion. Thisone was needed in order to calculate the thermal stresses due to the temperature field.The temperature dependency of some material parameters, for example Young’s modulus, couldbe calculated with Equation 24, for this a material test for at least three different temperatures isrequired.

Table 3: Structural material parameters for Model 3.


Thermal Expansion a = 1·10−5 K−1

In Figure 14 the structural boundary conditions are defined. The nodes around the holes on thebrackets are fixed in all degrees of freedom. A static equivalent acceleration load is applied on themodel to represent the dynamic load the silencer is exposed to. From a Multi Body Simulation(MBS) the static equivalent acceleration load of 15g (g is the gravitational acceleration, approxi-mately g = 9.81m

s2 ) was calculated at Scania. The static equivalent acceleration load was appliedon the whole model at room temperature (20C) and is henceforth called the static accelerationload.

19

Figure 14: Structural boundary condition for model 3.

This resulted in two load cases, 1 and 2:

1. Structural analysis with the boundary conditions as in Figure 14 and the applied staticacceleration load of 15g in the z-direction (vertical direction) – called the static accelerationload case.

2. Structural analysis with temperature field as input, calculated in Abaqus, same boundaryconditions as in load case 1 (the static acceleration load used in load case 1 was not includedin this load case) – called thermal load case.

Observe that the thermal load case (load case 2) had the temperature field as a constant load.Hence, a heat transfer analysis is only performed once (in Abaqus) and not during each optimizationiteration. This is described in a more detailed form as option 2 in Chapter 4.

3.4 Model 4: T-welded platesModel 4 describes a plate welded on both sides to the middle of another plate, as is shown in Figure15, where the load, boundary conditions and evaluation element are marked. The shell elementshave a thickness of 5 mm for the plates and welded part. The average element size is 1 mm. Theforce is evenly distributed on the top and periodic loaded where the amplitude and mean load arepresented in Table 4. The Chaboche material model is based on the strain range that the evaluatedelement has. Meaning that the same material model is used for the whole model but have differentparameters between the load cases. The elastic material model is constant for all load cases. Theparameters used for the material models are presented in Table 5 and 6.

20

Figure 15: Load and boundary condition for model 4.

Table 4: Load cases for model 4.

Load case Mean force [N] Amplitude [N]1 0 44002 0 66003 0 88004 2200 4400

Table 5: Chaboche material parameters for model 4 [9].

Load case E [GPa] ν σ|0 [MPa] C1 C2 C3 γ1 γ2 γ3 Q∞ [MPa] b1,4 193 0.3 100 4·1011 1.4·1011 3.5·109 8000 1100 139 20 102 193 0.3 100 5·1011 1.1·1011 3.5·109 8000 740 130 40 73 193 0.3 100 3·1011 4.2·1011 1.0·108 3000 250 200 65 5

Table 6: Elastic material for model 4-6.


3.5 Model 5: L-welded platesThe model 5 is similar to Model 1. The difference is that there is no boundary condition in frontof the loaded plane, and the load is only applied on half of the plate (the part closest to theweld). The load has also been divided to 4 parts where load direction alternates between positiveand negative x-direction. For the Super Neuber analysis the load is periodic and has 50 loadingcycles. The pressure in the red areas in Figure 16 is in positive x-direction when the blue areasare loaded in negative x-direction and vice versa. The evaluation domain is the same as Model 1.The material used is presented in Table 7 for the Chaboche material and Table 6 for the elasticmaterial. The strain amplitude of the elements varies in the evaluation domain, making it hard

21

to fit the material parameters to a suitable strain interval, all elements has been associated withthe same material model, and is independent of load case. The evaluated load cases is presentedin Table 8, where the mean pressure for all load cases is zero.


Table 7: Chaboche material model for model 5 and 6 [9].

E [GPa] ν σ|0 [MPa] C1 C2 C3 γ1 γ2 γ3 Q∞ [MPa] b193 0.3 100 4·1011 1.4·1011 3.5·109 8000 1100 139 20 10


Load case Pressure amplitude [MPa]1 452 503 60

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3.6 Model 6: Intermittent L-welded platesThis model is based on an optimized L-welded plates, and its geometry is shown in Figure 17below. The load and boundary conditions are applied at the same way as for model 5, and isloaded with smaller loads, which is presented in Table 9, in all load cases was the mean pressurezero. Both elastic and Chaboche material parameters used is presented above in Tables 6 and 7.



Load case Pressure amplitude [MPa]1 302 35

23

4 Weld optimizationIn general, an optimization problem, is an iterative process and could be described briefly in Figure18. This chapter will among other things indicate why it is an iterative process. Usually it is notpossible to give guidelines for how just one optimization with immediately good result should bedone. Instead it is possible to give some starting guidelines and then perform several optimizationsto get understandable result.This chapter will also include two parameter studies that aims for finding optimizations parametersthat are practical to use in a method for optimization of welds, see Section 4.1.1. At the end ofthe chapter a weld optimization of a silencer will be performed in order to describe the method fortopology optimizations of welds, see Section, 4.1.2.The focus was mainly on finding a method for topology optimizations of welds. The optimizationconstraints were either based on only the FE-analysis results performed before the optimizationsor on recommendations from Scania. In Chapter 5 an investigation for finding a way to get a moreaccurate optimizations constraint will be performed. However, it was not used in the followingoptimizations.

Heat transfer analysis in optimizations - Option 1 and 2

The silencer used in the exhaust aftertreatment system is assembled with a large amount of weldmaterial. In an attempt to decrease the amount of weld material and the high stresses thatare found both in and close to the welds due to the thermal and static acceleration load severaloptimizations for different models were performed in the software OptiStruct. In the exhaust af-tertreatment system the component is exposed to high temperatures from the exhaust and dynamicloads from the operation of the vehicle.

The thermal load was initially calculated with Abaqus through a steady-state analysis. Thenthere were two options to include the thermal load in the FE-model in OptiStruct.

• Option 1 - Heat transfer analysis (steady-state) calculated in OptiStruct.

• Option 2 - Constant temperature field as a thermal load in OptiStruct.

In Option 1 the calculated Abaqus FE-model was converted back to an OptiStruct FE-model,where the optimization was performed. The steady-state analysis and therefore also the tempera-ture field were updated during each iteration in the optimization. To be able to combine a thermaland a structural load in the optimization it is common to do a thermal analysis and then thetemperature field (the result from the thermal analysis) will be input to the structural analysis.This is usually called a coupled linear thermal structural analysis. The linear heat transfer analysisis limited to a steady-state analysis in this thesis.In Option 2 only the Abaqus result from the steady-state analysis (temperature field, temperaturesin each node) was used and applied directly as a thermal load in OptiStruct. The drawback hereis that the temperature field was not updated during each iteration instead the nodes will have aconstant temperature during the whole optimization.

Lastly the structural load is applied. Note, when the structural load is applied the models arealways considered to be in room temperature, 20C. The material models in the optimizations arelimited to be linear elastic.

24

Figure 18: Optimization as an iteration process [5].

Shortly, the steps to execute an optimization is described. The first step is to build a FE-model.This can be done from a CAD-geometry or building the geometry directly in a pre-processing FEsoftware. In order to get a FE-model the geometry needs to be meshed, discretized into elements.Next step is to define appropriate material parameters. Thereafter properties are assigned to themodel, such as thickness. For the optimization the model needs to be divided into different parts,the design space and the non-design space. Then boundary conditions such as supports, and loadsare applied. Before doing an optimization, the model is analyzed with a finite element analysis.This step is important to check if the applied boundary conditions are relevant and if the modelis behaving as expected. After that an optimization problem is defined. First step is to definewhat type of optimization should be performed, topology optimization for example, and where itshould be performed. In case of a topology optimization the densities of the elements in the designspace will be the design variables. After that the responses are defined, for example static stress,volume, or compliance. On some of the responses constraints can be set. A constraint could forexample be that the static stress should be lower or equal a certain value. Lastly the objectivefunction is decided, usually minimize or maximize one of the earlier defined responses. Now theactual optimization can be performed. The result from the optimization can be post-processed andif the result is satisfied it can be implemented on the initial FE-model. It is now recommended todo a FE reanalysis with the new geometry to be able to compare with the result from the initialFE-analysis and see if the results are improved or not.

Structural responses

The used structural responses in all optimization for this thesis were the following.

• Static stress

• Volume fraction

• Static compliance or weighted compliance

The static stress and volume fraction were used as constraints, the static compliance and weightedcompliance were used as objective functions. Note that the static stress, compliance, and weightedcompliance are load case dependent. A load case must therefore be related for each of them. Thevolume fraction was defined for the welds or also called design space in an optimization.

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Stress and strain evaluation of the welds

When evaluating welds it is important to understand where the failure will occur. Two main cracksoccurs in the weldment, toe crack and root crack, see Figure 19. Which one that occurs dependson the geometry, applied load, weld type, and weld quality [21].

For root crack the nominal stresses or strain is used as a criterion, while for toe crack hot spotis a common evaluation method. In this report the welds have been dimensioned for toe cracks.This decision was based on the fact that the silencer is modeled with thin-walled plates makingit more likely that the weld is fully penetrated, which will lead to less sensitive root of the weld.Only the stresses and strains at the top layer of the plate, where toe cracks occurs have thereforebeen investigated.

Figure 19: A welded structure and the most vulnerable areas.

Due to high risk of singularities close to the weld toe the stress evaluation was done a distanceaway from the weld toe. This is used in a common method, the hot spot method. In general, thedistance from the weld toe is related to the thickness of the sheets in the joint.There are two limitations when introducing a stress constraint into an optimization problem. Thefirst one is that the stress constraint cannot be directly applied on the design space, in this case theweld, for a topology optimization. Instead the stress constraint can be applied in nearby elementsto the design space. Fortunately, this is in line with the above mentioned evaluations of the welds.The second limitation is that the constraints and therefore the evaluation can only be done onelements, not on nodes in OptiStruct.

Intermittent welds

There are some restrictions when it comes to removing weld material. Of course, it is not possibleto remove all welds because the purpose of them are partly to attach the thin-walled plates. Theother purpose of the welds used in a silencer is to seal and therefore no gases are allowed to escapeout from that area. There are some welds, which have the purpose to only attach the plates,and these are allowed to be removed but not fully removed. From being continuous they can bemodeled as intermittent welds (which can be suggested by the optimization). Intermittent weldsare not continuous, instead there is unwelded parts between the welds. In vehicle production it isnot recommended to have too short intermittent seam welds. According to a welding engineering,seam welds should not be shorter than 10 mm [22].Therefore, when doing the weld optimization of the silencer it was decided in this thesis thatthe welds should not be shorter than 20 mm and the distance between the welds should be at aminimum of 20 mm.

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4.1 Weld optimization with topology optimizationThe goal for these optimizations were to reduce the amount of welds but not impair the per-formance of the silencer. Therefore, it seemed reasonable to targeting the stiffest design of thecomponent by minimizing the compliance. Mainly two optimization formulations have been usedand are presented in Equations 34 - 37 [22]. In Equations 34 and 35 the optimization problem isa minimization of the compliance with a component subjected to a structural load. It is a similarformulation for maximization of the compliance with a component subjected to a thermal load. InEquations 36 and 37 the optimization problem is a minimization of the weighted compliance witha component subjected to both a structural and a thermal load.

Optimization formulation 1 - Equations 34 and 35,

minρe

C(ρρρ) = minρe

∫Ω

1

2fffuuuT dΩ (34)

Subject to:

max σ ≤ C1

Vmin ≤∫

ΩρdΩ ≤ Vmax

0 ≤ ρmin ≤ ρρρ ≤ 1.

(35)

Optimization formulation 2 - Equations 36 and 37,

minρe

Cw(ρρρ) = minρe

[ ∫Ω

wi1

2fffuuuT dΩ +

∫Γ

wj1

2fff tuuu

T dΓ

](36)

Subject to:

max σstructural ≤ C2

max σthermal ≤ C3

Vmin ≤∫

ΩρdΩ ≤ Vmax

0 ≤ ρmin ≤ ρρρ ≤ 1

(37)

where Γ is the domain where the temperature field is calculated. The constants C1, C2, and C3

constitutes the limits on stresses from the structural load and the thermal load. These were thestress constraints that are evaluated a distance from the weld toe which were discussed earlier inthis chapter. The design variables were the element densities, ρe, in the design space.

In optimization formulation 2 the two load cases that are combined, structural and thermal load,were contradictory in the minimization of the weighted compliance. Thus, the structural load casewas defined as force controlled. The compliance for the structural load case should therefore beminimized to reach the stiffest structure. For the structural load case, in order to achieve the stiffestcomponent, it will strive for keeping as much weld material as possible. Instead for the thermal loadcase, in order to decrease the thermal stresses, it will strive for removing as much weld materialas possible. The thermal load case is also defined as displacement controlled, then the stiffeststructure is attained by maximizing the compliance. Although this problem, the minimizationof the weighted compliance (combination of both load cases) seemed as the best choice to avoida decrease in performance of the silencer after an optimization. In this thesis the main purposeis not to maximize the stiffness of the silencer. Instead the minimization of the compliance wasused as a measure to ensure that the stiffness keeps approximately the same value as before theoptimizations.

4.1.1 Parameter studies

Two parameter studies were performed. This to get information and knowledge about whichparameters in OptiStruct that could be useful to use in order to get an understandable result froma weld optimization. The first study was only containing a structural load and the second one wasalso including a thermal load.

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Parameter study 1 – structural load

The first parameter study was conducted in form of a topology optimization to investigate howdifferent parameters in OptiStruct were affecting the optimization results. The studied parameterswereMINDIM , penalty factor p, and TOPDISC. In this study Model 1 was used, correspondingboundary conditions and loads can be found in Chapter 3.1. In Figure 20 the non-design space,design space, and evaluation domain are defined. The average element size is 1 mm and is thesame for the design space. Therefore, the allowed range for the parameterMINDIM was 3 mm <MINDIM < 12 mm.

Figure 20: Optimization model 1 used in parameter study 1.

In this study the Optimization formulation 1 described in Equations 34 and 35 was used. Theoptimization set-up is defined in Table 10 and was the same for all optimizations performed inthis parameter study. Before doing an optimization the maximum static stress (von Mises in thisstudy) in the evaluation domain was measured to be 134.90 MPa, see stress plots in Appendix B.The upper stress constraint value used in the optimizations was based on this value.

Table 10: Optimization set-up for parameter study 1.

Optimization set-upDesign variables Element densities in the weld (Design space)

ResponsesStatic stress – von Mises (Evaluation domain)

Volume fractionCompliance

Optimization constraint 1 Static stress ≤ 175 MPaOptimization constraint 2 Volume fraction ≤ 0.4

Objective function Minimize the compliance

In Table 11 the varied parameters are presented. The first optimization was seen as the defaultoptimization, because the default settings in OptiStruct for an optimization were used.

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Table 11: Parameters used in parameter study 1.

Optimization MINDIM Penalty factor, p TOPDISCcase [mm]

Default 0 p = 2 (DISCRETE = 1) NO

1 4 p = 2 (DISCRETE = 1) NO2 6 p = 2 (DISCRETE = 1) NO3 8 p = 2 (DISCRETE = 1) NO4 10 p = 2 (DISCRETE = 1) NO

5 0 p = 3 (DISCRETE = 2) NO6 0 p = 4 (DISCRETE = 3) NO7 0 p = 5 (DISCRETE = 4) NO

8 4 p = 3 (DISCRETE = 2) NO9 6 p = 3 (DISCRETE = 2) NO10 8 p = 3 (DISCRETE = 2) NO11 10 p = 3 (DISCRETE = 2) NO12 0 p = 2 (DISCRETE = 1) YES13 4 p = 2 (DISCRETE = 1) YES

In Figure 21 the optimization result from the default Case is presented. The red colour meansthat the element density is equal to solid, ρe = 1, and blue colour means that the element densityis equal to void, ρe = 0, or close to zero.

Figure 21: Optimization result for default Case.

Firstly, the parameter MINDIM was varied. This parameter controls the minimum diameter ofmembers in the design space or in a weld optimization; the minimum length of a weld. Figure 22presents optimization results from Cases 1 - 4.

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Figure 22: Optimization results from Cases 1 – 4, starting from the upper left corner.

Then the penalty factor, p, was studied. This factor penalizes intermediate element densities toeither take ρe = 1 or ρe = 0. The optimization results are presented in Figure 23 from Cases 5 - 7.


In Cases 8 - 11 the combination of different values of the parameter MINDIM and a penaltyfactor of p = 3 were studied. The results from that study are presented in Figure 24.

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Lastly, the parameter TOPDISC was studied. This parameter aims for producing a more discreteoptimization result. The result for Case 12 and 13 are presented in Figure 25. For the last Case(13) TOPDISC was used in combination with MINDIM .

Figure 25: Optimization results from Case 12 and 13, starting from left.

When comparing the results from the parameter study with the result from the default Case itwas obvious that it is necessary to at least have one parameter deviating from the default value.The result from the default optimization is not as intuitive as for example the result from Case2. A disadvantage when using other values than default ones for these three parameters wasthat the amount of iterations increased, this is presented in Table 12. From the beginning of anoptimization it is not always clear which parameter settings that should be used to get the bestunderstandable result. Therefore, it is usually best to start by doing a shorter parameter studyand see how the optimization result evolves. This study also showed that it does not have to besuch a comprehensive study to acquire improved results. It could be enough to just change oneparameter, for example the penalty factor from p = 2 to p = 4 as in Case 6. It also showed thatparameters such as the penalty factor, p, and MINDIM are useful and gives optimization resultsthat are more clear and understandable.

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Table 12: Result from parameter study 1.

Optimization Iterations Compliance at Compliance at last Objective functioncase iteration 0 [Nmm] iteration [Nmm] change

Default 5 87.85 84.53 -3.78%

1 11 87.85 83.97 -4.42%2 11 87.85 84.52 -3.79%3 11 87.85 84.69 -3.60%4 11 87.85 84.94 -3.31%

5 7 94.02 84.44 -10.19%6 6 99.63 84.75 -14.94%7 6 103.29 84.69 -18.01%

8 12 94.02 84.44 -10.19%9 12 94.02 84.50 -10.13%10 12 94.02 84.95 -9.65%11 13 94.02 85.11 -9.48%

12 13 87.85 84.04 -4.34%13 12 87.85 85.30 -2.90%

Note, the compliance at the start iteration is differing between the Cases. Thus, when the penaltyfactor is increased it will decrease the penalized stiffness (which is not the same as the real stiffness,see Equation 5 and Figure 2) as well as increase the compliance (the compliance is the inverse ofthe stiffness) at start iteration. However, the result for the compliance at the last iterations aresimilar in all Cases.

Parameter study 2 – structural and thermal load

The second parameter study were also conducted in form of a topology optimization to investigatehow different settings in OptiStruct were affecting the optimization results. The difference betweenthis study and the first one was primarily the load. In this study both a structural and thermalload were applied but on a different model. From the first parameter study it was shown that themore useful parameters were penalty factor, p, and MINDIM , therefore the focus were to usethese in this study. Also, an investigation of how the combination of a thermal and structural loadwere affecting the results of weld optimizations was performed.

The model that was used in this study should imitate a part of a silencer, an outlet or inlet, but doesnot represent any existing design. A similar model was used by Rietz [22], but the one used in thethesis was modified to a certain degree. In this study Model 2 was used, corresponding boundaryconditions and loads can be found in Chapter 3.2. In Figure 26 the non-design space, designspace, and evaluation domain are defined. The average elements size is 1.56 mm for the designspace. Therefore, the allowed range for the parameter MINDIM was 4.68 mm < MINDIM <18.72 mm.

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Figure 26: Optimization model 2 used in parameter study 2.

In this study the Optimization formulation 1 and 2 described in Equations 34 - 37 were used.The three used Optimization set-ups; 1, 2, and 3, are defined in Table 13 - 15. Before doingan optimization the maximum static stress (first principal stress in this study) in the evaluationdomain was measured to be 53.48 MPa for the structural load case and to 88.55 MPa for thethermal load case, see stress plots in Appendix B. The upper stress constraint values used in theoptimizations were based on these values. Top view of the model is presented in Figure 10.

Table 13: Optimization set-up 1 for parameter study 2.

Optimization set-up 1Design variables Element densities in the weld (Design space)

ResponsesStatic stress – First principalstress (Evaluation domain)


Optimization constraint 1 - Structural load case Static stress ≤ 80 MPaOptimization constraint 2 - Thermal load case Static stress ≤ 100 MPa

Optimization constraint 3 0.05 ≤ Volume fraction ≤ 0.4Objective function Minimize the compliance

for the structural load case

33






Optimization constraint 3 0.05 ≤ Volume fraction ≤ 0.4Objective function Maximize the compliance

for the thermal load case






Optimization constraint 3 0.05 ≤ Volume fraction ≤ 0.4Objective function Minimize the weighted compliance

(both load case)

34

In Table 16 the varied parameters and optimization set-ups are presented.

Table 16: Parameters used in parameter study 2.

Optimization MINDIM Penalty factor, p Weight factorscase (set-up) [mm]

1 (1) 0 p = 2 (DISCRETE = 1) -

2 (2) 0 p = 2 (DISCRETE = 1) -

3 (3) 0 p = 2 (DISCRETE = 1) wstructural = 1.0wthermal = 1.0








In the two first optimizations presented in Figure 27 the compliance was first minimized for thestructural load (left) with Optimization set-up 1 and then maximized for the thermal load (right)with Optimization set-up 2. As mentioned before, the red colour means that the element densityis equal to solid, ρe = 1, and blue colour means that the element density is equal to void, ρe = 0,or close to zero. The other colours are domains where it is not clear if the density is void or solid.The black arrows in Figures 27 - 30 are pointing at areas where the weld material should be kept,where ρe = 1.

From the optimization results achieved in Case 1 and 2, in Figure 27, one can see that the suggestionfor removal of weld material are contradictory. At domains where the minimization of the structuralcompliance (Case 1) suggest that it should be weld material instead the maximization of the thermalcompliance (Case 2) suggesting that it should not be any weld material, and the other way around.This problem was described earlier in this chapter. The structural compliance should be minimizedin order to get the highest stiffness and the thermal compliance should be maximized in order toget the highest stiffness. The structural load case will always try to find an optimization result withas much weld as possible, then the stiffest structure is reached. On the other hand, the thermalload case will always strive for less weld material, in order to decrease the thermal stresses. Thesuggestion was therefore to try a combination of both load cases by minimizing the weightedcompliance, see Section 2.2.2 for explanation of the weighted compliance.

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Figure 27: Optimization result from Case 1 and 2, starting from the left.

In Cases 3 – 5, presented in Figure 28, the combination of a structural and thermal load wasinvestigated with minimization of the weighted compliance (optimization set-up 3 in Table 15). Theweight factors were varied to see how the optimization results were affected. Firstly, if comparingthese optimization results with Case 1 and 2 the structural load case was definitely the dominatedone. The removal of weld material suggestion in Cases 3 - 5 are very similar to Case 1. Secondly,there is not much difference in the results when the weight factors were varied. For the upcomingoptimizations in this study the weight factors were decided to be 1.0 for structural load case and0.5 for the thermal load case. This is according to Rietz [22], the allowed stress constraint for thethermal load case is higher and that is because the number of load cycles is lower for thermal loadthan for the structural load. Therefore, the thermal load case should have a lower weight factorthan the structural load case.

Figure 28: Optimization result from Cases 3 - 5, starting from the left.

In Cases 6 – 8, presented in Figure 29, the parameter MINDIM were investigated. These opti-mizations gave more distinct domains of solid (red colour) and void (blue colour). However, thereis not much differences between the investigated values for MINDIM .

36

Figure 29: Optimization result from cases 6 - 8, starting from the left.

Lastly in Case 9 and 10, presented in Figure 30, an increase of penalty factor, p = 3, and theparameterMINDIM were investigated. There is some difference in the result for Case 10 compareto the earlier ones, more weld material is suggested to be removed.

Figure 30: Optimization result from case 9 and 10, starting from the left.

Even this study showed that parameters such as the penalty factor, p, and MINDIM were usefuland also for the combination of load cases, structural and thermal load. In Table 17 the iterationsand compliance for each case is presented. As before, if using parameters such as p andMINDIMthe amount of iterations were increased. Except for Case 9 where the amount of iterations actuallydecreased compare to Case 1. Also notice that even in the weighted compliance was minimized theoptimizations cannot achieve this and was focusing on finding a solution that was accomplishingother stated conditions in the problem, such as stress constraints. When including the thermalload case the compliance is increasing drastically, that can be seen in Case 2 and further. Thus,with a decreasing Young’s modulus the stiffness is also decreasing and therefore will the complianceincrease. In Appendix B the updated temperature field for each optimization in this parameterstudy can be found.

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Table 17: Results from parameter study 2.

Optimization Iterations Compliance at Compliance at last Objective functioncase iteration 0 [Nmm] iteration [Nmm] change1 6 124.96 110.17 -11.84%

2 6 21 480 22 550 4.98%

3 4 21 609 21 700 0.42%4 4 10 867 10 924 0.52%5 5 21 547 21 640 0.43%

6 7 10 867 11 025 1.45%7 7 10 867 11 021 1.42%8 7 10 867 11 034 1.54%

9 4 11 316 11 092 -1.98%10 9 11 316 11 089 -2.01%

4.1.2 Weld optimization of a silencer

The two parameter studies ended up in recommendation of suitable parameters and how to combineload cases which was useful to know before doing the final optimization trial; weld optimization ofa silencer. This final part of the chapter will show the method for topology optimizations of weldson a more realistic component, a silencer. The model used in this study should mimic a silencer,but does not represent any existing design. In this part Model 3 was used, corresponding boundaryconditions and loads can be found in Chapter 3.3. In Figure 31 the design spaces and evaluationdomains are defined. The non-design space constitutes all the parts that were not the design space.The average elements size is 2.67 mm for the design space. Therefore, the allowed range for theparameter MINDIM was 8.01 mm < MINDIM < 32.04 mm. The parameter MAXDIM hasnot been included in the earlier parameter studies but it was found out that it was necessary inthese optimizations. The allowed values for that one was 16.02 mm < MAXDIM .

Figure 31: Optimization model 3 used in weld optimization of a silencer.

In these optimizations only the Optimization formulation 2 described in Equations 36 - 37 was used.Before doing an optimization the maximum static stress (von Mises) in the evaluation domains wasmeasured and presented in Table 18, see Appendix B for stress plots. The upper stress constraint

38

values used in the optimizations was based on these values and also on recommendations fromScania. The optimization set-up is defined in Table 19 and was the same for all optimizationsperformed further. The same penalty factor has been used throughout the following optimizations,p = 3 (DISCRETE = 2).

Table 18: Stress results from analysis.

Load case Evaluation domain Maximum static stress(von Mises) [MPa]

Static acceleration load case 1 46.372 79.70

Thermal load case 1 14442 1432

For the same reason mentioned earlier the weight factors were chosen to be 1.0 for the staticacceleration load case and 0.5 for the thermal load case, as presented in Table 19.

Table 19: Optimization set-up for weld optimization of a silencer. Static acceleration is abbreviatedas st. acc. in the weight factors.

Optimization set-upDesign variables Element densities in the weld (Design space)

ResponsesStatic stress – von Mises stress

(Evaluation domain)Volume fractionCompliance

Optimization constraint 1 - Static stress ≤ C1Static acceleration load caseOptimization constraint 2 - Static stress ≤ C2

Thermal load caseOptimization constraint 3 0.05 ≤ Volume fraction ≤ C3

Objective function Minimize the weighted compliance for boththe static acceleration and thermal load cases

with weight factors wst. acc. = 1.0 and wthermal = 0.5

In Table 20 the varied parameters and constraint values are presented.

Table 20: Parameters and constraints used in weld optimization of a silencer.

Optimization Static stress Volume fraction MINDIM MAXDIMC1 and C2 [MPa] C3 [mm] [mm]

Opt. 1 135 and 800 0.30 30 130Opt. 2 60 and 900 0.40 30 130Opt. 3 135 and 800 0.45 30 130Opt. 4 135 and 800 0.40 30 130

Many different optimizations were performed for the investigation of weld optimization of a si-lencer. However, the outcome was that only one parameter, the volume fraction, actually affectedthe optimization results in a wider range and would also give more significant result from a re-analysis in Abaqus. Therefore, only the optimizations that differed more are presented here. InAppendix C one can find the other optimization results from weld optimization of a silencer. Theones presented in Table 20 were the ones that gave more significant results.

After performing this investigation it was noticed that the parameterMAXDIM was necessary toinclude in these types of optimizations. In those that it was not included some kept elements of the

39

weld material were destroyed (not fully meshed), those results are also included in the AppendixC. Yet, the optimizations 2 and 4 gave results that were very similar, these are presented herejust for showing that even if stress constraints were different the result could be the same for twooptimizations. The result from these two optimizations are presented in Figure 32. Therefore, onlyoptimization result from Opt. 2 will be used further, not Opt. 4.

Figure 32: To show the similarities between Opt. 2 and 4. Left: Optimization result from Opt. 2.Right: Optimization result from Opt. 4.

Henceforth, the results from the first three optimizations are presented. The optimizations resultsare presented by iso-density boundary surfaces (also called iso-surfaces). To create iso-surfaces oneneeds to define a density threshold value, the decided ones are presented in Table 21. To the leftin the following three figures are the suggestions from OptiStruct of the weld material that shouldbe kept and to the right is the optimized FE-model was reanalyzed in Abaqus.

In Figure 33 the optimized result and optimized FE-model are presented for the optimizationwith volume fraction as 0.30. By setting the volume fraction as example 0.30 means that theoptimization tries to remove 70% of the original design space (weld material) and keep 30%.

Figure 33: Left: Optimization result from OptiStruct with volume fraction as 0.30 (Opt. 1). Right:Optimized FE-model.

40

In Figure 34 the optimized result and optimized FE-model is presented for the optimization withvolume fraction as 0.40.

Figure 34: Left: Optimization result from OptiStruct for volume fraction as 0.40 (Opt. 2). Right:Optimized FE-model.

In Figure 35 the optimized result and optimized FE-model is presented for the optimization withvolume fraction as 0.45.

Figure 35: Left: Optimization result from OptiStruct for volume fraction as 0.45 (Opt. 3). Right:Optimized FE-model.

In Table 21 the results from the finite element analyses performed in Abaqus before and after theoptimizations are presented, the stress plots are found in Appendix C. These results were calculatedaccording the procedure in Figure 36. The more weld material that was removed the more thethermal stresses decreased. For example in optimization 1, where 70% of the weld material hasbeen removed then the thermal stresses have decreased the most. This is because during thethermal load case the structure want to expand and if more weld material is removed then thekept weld is allowed to expand even more. Unfortunately, it is not possible to distinguish an equallysimple relationship for the static acceleration load case. A theory was that if keeping more weldmaterial that would favour the static acceleration load case and hence would the corresponding

41

stresses decrease, but that is clearly not the situation here. Although this is not visible in theresults a way for decreasing the static acceleration stresses could be by introducing reinforcementson positions where the corresponding stresses are high. This could decrease the overall staticacceleration stresses and also help the optimization algorithm to not focusing to much on the mostsevere positions. If the stress constraints in the optimizations are trying to target the most criticalpositions of stresses nearby the weld then this could be a problem. It would be better if theoptimization could focus on other positions too, not so severe ones, and the most critical positionscould instead be solved by other solutions such as reinforcements.

Table 21: Results from FEA.

Density Static Thermal load Stress change: Stress change:FE-model threshold acceleration case [MPa] static acceler- thermal load

value load case [MPa] ation load case case- - 1 2 1 2 1 2 1 2

Full weld (100%) - 46.37 79.70 1444 1432 - - - -Opt. 1 (30%) 0.30 71.55 104.60 844.10 815.40 54.30% 31.42% -41.54% -43.06%Opt. 2 (40%) 0.40 67.46 105.10 1021 1021 45.48% 31.87% -29.29% -28.70%Opt. 3 (45%) 0.45 78.35 104.40 1138 1056 68.97% 30.99% -21.19% -26.26%

Figure 36: Procedure for FEA before and after optimizations.

Table 22 is presenting the result for the amount of iterations, computation time, compliance, andcompliance change (objective function change) for the three optimizations. For all optimizationsthe compliance has increased even if the objective function was defined as minimize the compliance.Here, OptiStruct has focused on the fulfillment of other constraints in the optimization set-up thanthe objective function. Still, the change in compliance is very small and as written earlier; that isnot the main target for these optimizations.

Table 22: Result from weld optimization of a silencer.

Computation time Compliance Compliance at ObjectiveOptimization Iterations [hh:mm:ss] and used at iteration last iteration function

number of CPU 0 [kNmm] [kNmm] changeprocessors

Opt. 1 6 01:02:09 32 965 32 968 0.008%CPU: 4

Opt. 2 6 00:12:50 32 962 32 965 0.007%CPU: 8

Opt. 3 6 00:58:30 32 961 32 963 0.005%CPU: 4

From the presented method of weld optimization one can see that the thermal stresses could bedecreased and the static acceleration stresses are still on a reasonable level. The weld optimizationssaves time for the engineers because the trial-and-error procedure to where weld material shouldbe kept or not are not necessary. Also, the weld optimization can handle several load cases. Theresults from optimizations probably also gives more accurate results than if an engineer would guesswhere weld material should be kept or not. From Table 22 it can be seen that the time for doing

42

such weld optimizations is quite low. If using a higher number of CPU processor the computationtime is only around 13 minutes or for lower amount of CPU processor around 1 hour. Some extratime for rendering the optimization result should be included also. In the thesis the post-processingtool OSSmooth was tested on the optimization results Opt. 1 - 3. Unfortunately, this did not givesufficiently good results. Some of the elements were not even fully meshed (destroyed) and/or reallybad re-meshed. Instead it was decided to manually remove weld elements according to how theoptimization results suggested the weld should be removed. An advantage by doing it manuallyis that it is possible to include some engineering knowledge of demands for intermittent welds(recommendation of minimum length of weld etcetera). From the three presented optimizationresults it is tough to recommend which of them that are the best one. If looking at the decreasein thermal stresses then first one would be best, but the second one also gives good results. Theoptimization Opt. 3 have some smaller areas of suggested weld material and that could requiresome more time for the post-processing in order to investigate which weld parts that are fulfillingthe demands for intermittent welds or not. From that perspective is probably the result from Opt.1 and 2 better.

43

5 TransformationThis chapter will discuss a method to collect the stresses and strains in a model, how to evaluatethe results in a proper way, how to find the relation between the plastic and elastic material andhow they can be used for fatigue evaluation.

5.1 Collecting resultsSmall increments has to be used when collecting a hysteresis (a typical hysteresis can be seen in Fig-ure 39). Model 4 was loaded with 50 cycles in 10 000 steps. The results were collected manually inHyperworks and only one element was evaluated due to limited time of the project. Collecting onlythe stress in one direction and in one node can take 1 hour in HyperWorks for this size of the model.

Models 5 and 6 were loaded cyclic 50 times and to get reliable results 20 000 steps were taken foreach loading case. The time for calculating model 5 with Chaboche material takes around 14 hoursfor each load case in Abaqus when 1 CPU was used and ODB-files has a size of 56 GB. The manualway is no longer possible if the hole evaluation domain should be evaluated. To minimize the postprocessing time the command *ELPRINT was used. The stresses, elastic strains and plastic strainswould now be printed in the DAT-file. Abaqus does not calculate element stresses and strains,instead the Gauss point values of the element were printed. All the Gauss points had two differentvalues depending on the two sides of the shell element (top layer and bottom layer) and only thetop layer were chosen to investigate (since it is there toe cracks occurs according to Chapter 4).To manually go through the DAT-file is not an alternative, especially since the file has a size of 3GB meaning that it cannot be opened in a standard editor (e.g. Notpad ++). Therefore a Matlabscript was created to go through the DAT-file and calculate the stress and strain in a form thatcould be used. The script translated each word or number as a row. By knowing the form that theresults are printed the stress and strains can be obtained for each element. The size of DAT-filewill correlate with the RAM-memory needed to read the file, which is in this case necessary topay attention to since a DAT-file of 3 GB uses a RAM-memory of 65 GB which can overload thecomputer. The time needed for translating the DAT-file to the strains and stress matrix can beshorten by finding patterns in the DAT-file. Going through a text file for Matlab takes a lot oftime and by letting Matlab skip reading unnecessary text, much computing time will be saved.The script used in this project went from 30 hours of computing time to 3 hours per DAT-file of3 GB by recognizing some patterns, and could probably be more efficient if more time were spenton the script. Observe that the script produced and is presented in abstract only work for thisspecific type of element and the order of the requested results is the same. If one of those thingschange, some alternations of the code has to be made. See Appendix D to see the used script.

5.2 RatchetingWhen ratcheting occurs, the softening that happens in compression will be smaller than the hard-ening in tension (or the other way around, depending on loading), meaning that the strain wanderswith a constant strain increment. The stress amplitude will reach one stabilized value while thestrain amplitude will be larger (or smaller) in tension than compression, see Figure 37. It is ofimportance to find out which strain amplitude should be used when transforming from Chabochematerial to elastic material even when ratcheting occurs. Model 4 was therefore loaded with thesame load amplitude, but one was loaded symmetrical, while the other one were not (Load case1 and 4). In the hystereses for Load case 4 it was obvious that ratcheting had occur. The strainamplitude in Load case 4 was separated into two parts, the largest possible strain amplitude, andthe shortest strain amplitude as shown in Figure 37. The amplitudes were compared to Loadcase 1 where shakedown had occurred to find a relation between them. The results shows thatthe long strain amplitude in Load case 4 is very similar to the strain amplitude in Load case 1(long strain amplitude is 0.00255 in Load case 4 and the strain amplitude is 0.00262 in load case1) and therefore should the long strain amplitude be used directly. This also makes sense in afatigue perspective, where the worst load makes the most damage. Observe that when ratchetingis present fatigue will occur sooner than calculated with Coffin-Manson, and life expectancy shouldtherefore be calculated separately. This paper does not evaluate where and when ratcheting occurs.It evaluates the real stress and strain amplitude equivalent in an elastic material. This means thateven if ratcheting occurs, we still want to interpreted it in a transformation point of view.

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Figure 37: Ratcheting.

5.3 Cyclic loadingThe plastic material behavior changes depending on if it is cyclic loaded compared to monotonicallyloaded, an example of that is illustrated in Figure 38 for uniaxial tension. To find the cyclic loadedmaterial curve, which is known as the saturated curve, each model was loaded 50 cycles. After thatit was assumed that the materials behavior will not change. Even though more cycle used wouldrepresent a more correct calculations, more cycles would take more time and not result in noticeablebetter result. Since Abaqus does not extract element strains and stresses, the element stress andstrains has been estimated as the mean stress and strain of the four gauss point extracted for thetop layer. The largest differences between the mean value and the gauss point was later evaluatedbut the calculations shows has no significant results and has therefore not been presented.

Figure 38: Saturated curve, which is the material curve when it is cyclic loaded and the monotoniccurve, which is the material curve for virgin material [9].

5.3.1 Saturated curve in one direction

Model 4 was tested (in Abaqus CAE) with 3 different load cases, and each load case were testedwith both elastic material and with Chaboche material. One element was evaluated and from eachtest the strain and stress were collected for all steps and plotted in a strain stress diagram. Only

45

the dominant direction y direction (see the coordinate system in Figure 15). The result is calledhysteresis and the amplitude from the strain and stress is then collected when it stabilizes, whichhas been assumed to be the last cycle of the hysteresis (see Figure 39 which illustrates a hysteresiscurve and shows how the stress and strain amplitude are collected). By changing the load one canmake a saturated curve from the stress and strain amplitude, as it has been done in Figure 40.The saturated curve describes the material when it is cyclically loaded.

Figure 39: Typical hystereses.

Figure 40: The last curves of the hysteresis of load case 1-3 of Model 4. The strain values of thehysteresis has been moved in such way that the mean value is located at the origin, making itpossible to use the collected stress and strain amplitude directly.

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5.3.2 Saturated curve in a multi-axial system

When the problem becomes multi-axial (one direction is not clearly dominate) is it hard to estimatethe hardening in one direction in Chaboche material. Stresses and strains from other directionswill influence the hysteresis and become hard to evaluate. The highest stress in the hystereses isnot necessary located at the same point as the highest strain, as can be seen in Figure 40 for loadcase 3. With more influence from other directions is it in in general impossible to collect a reliableresult from the hysteresis. This means that if the stress in one direction is not clearly dominantin the model an effective stress and strain should be used. The evaluation domain in Model 5and 6 contains stresses and strains from more than one direction and therefore evaluated with themulti-axial method. The effective stress and strain is calculated by following method: Quadraticshell elements are used in the models, meaning that plane stress is assumed. The stresses in theelements are σ11, σ22 and the in-plane shear stress τ12. von Mises effective stress, σe is chosen asthe effective stress since Chaboche material model uses mises for its calculations. The effectivestress can therefore be calculated with

σe =√σ2

11 + σ222 − σ11σ22 + 3τ2

12. (38)

The effective strain was more complicated to evaluate. Abaqus does not write out the strain inthe out of plane direction, and therefore it has to be calculated with

ε33 = −ν(εe11 + εe22)− 1

2(εp11 + εp22) (39)

where ν is passions ratio and where plastic incompressibility has been assumed. The strain com-ponents εe11 and εe22 are the elastic normal strains in the in-plane directions 1 and 2, respectively,and εp11 and εp22 are the plastic normal strains in in-plane directions 1 and 2, respectively. Theeffective strain can now be described with

εe = A

√ε2

11 + ε222 + ε2

33 − ε11ε22 − ε11ε33 − ε22ε33 + 3(γ12

2

)2

+ 3(γ13

2

)2

+ 3(γ23

2

)2

(40)

If the strains would be entirely plastic, A would take the value 23 . On the other hand if the strains

would be entirely elastic, A would take the value 11+ν . In the present work A is chosen as 1

1+ν .

An obvious problem using an effective stress and strain is that it cannot reach a negative valueand can therefore not give a hysteresis (as we usually think of one). If we return to the originalquestion, why do we want a hysteresis? The answer is to find a corresponding stress value from aknown strain amplitude. From the effective strain- stress plot (an example is shown in Figure 41of an element cycled 50 times, and in Figure 42 its last curve) one can with a little imaginationfind how the hysteresis would be formed if negative values was possible as it is shown in Figure 43.

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Figure 41: The effective stress and strain in one node in model 5, all cycles.

Figure 42: The effective stress and strain in one node in model 5, last cycle.

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Figure 43: The effective stress and strain in one element in model 5. The marked points have beenmirrored to create a typical hystereses.

Stress and strain values from points 1-6 were collected as local or global maximum or minimum inthe last loading cycle. To get the effective stress and strain amplitude (σeamp respective εeamp) ofthe effective hysteresis, equation 41 and 42 were used:

σeamp =σ1 + σ2 − σ3 − σ6

2(41)

εeamp =ε1 + ε2 − ε4 − ε5

2. (42)

Now every elements effective stress and strain amplitude in the evaluation domain can be calculatedfor a specific load case, and for both the elastic and the Chaboche material, respectively. Figure44 shows the evaluation domains elements effective strain amplitude of both elastic and Chabochematerial at the same load. It can be seen that the overall shape is similar to each other, but for2 different point with the same strain amplitude of the Chaboche material, does not necessaryhave the same strain amplitude of the elastic material. The main reason behind those errors hasbeen assumed to be that the mesh is to coarse and gives small errors in the calculations. Bylooking at an average effective strain amplitude for an interval where the stress is relatively highand singularities is not predicted, the two material can be compared with success. In Model 5 itwas decided to use the average effective stress and strain amplitude for element number 58-67, andfor Model 6 element number 63-67. From the average stress and strain amplitude saturated curvescan now be made as described in 5.3.1.

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Figure 44: Effective strain amplitude curve for Model 5 and Load case 2 to the right with thegeometry of Model 5 illustrated under, notice that the mean value of elements 58-67 are used.The effective strain amplitude curve for Model 6 with Load case 1 to the left with its geometryillustrated under. Element number 63-67 mean values are used.

5.4 Super NeuberFrom the saturated curve, a Super Neuber plot can now be made and a suitable q value can beiterative selected. Here the selection of q has been done by trying different values and see whichone that gave the best results. The q needs to minimize the difference between the known elasticdeformation with the calculated one with Super Neuber. Figure 45 and Table 23 shows the resultsof the Super Neuber plot of model 4 at one specific element at one direction. Figure 46 representsModel 5 and the effective amplitude values are presented in Table 24 while Figure 47 shows Model6 Super Neuber curve with its values presented in Table 25. What can be seen in Figures 45-47 isthat the parameter q changes depending on the model.

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Figure 45: Super Neuber curve for model 4, q = 0.4. The circles with numbers marks the effectivestrain and stress amplitude for each load case and the numbers marks which load case the circlerepresent. Where the Super Neuber curve crosses the elastic curve is where one can predict anelastic stress and strain amplitude equivalent to the Chaboche curve. A good transformation iswhen the Super Neuber curve crosses the elastic curve close to the circle that represents the sameload case. The circle at 100 MPa marks the yield stress.

Table 23: The measured stress and strain amplitude for model 4.

Load case Type Elastic material Chaboche material Predicted elasticmaterial (q = 0.4)

1 Stress amplitude [MPa] 352 301 351Strain amplitude [-] 1.83·10−3 0.258·10−3 1.79·10−3



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Figure 46: Super Neuber curve for model 5, q = 0.6. The stars with numbers marks the effectivestrain and stress amplitude for each load case and the numbers marks which load case the starsrepresent. Where the Super Neuber curve crosses the elastic curve is where one can predict anelastic stress and strain amplitude equivalent to the Chaboche curve. A good transformation iswhen the Super Neuber curve crosses the elastic curve close to the star that represents the sameload case. The star at 100 MPa marks the yield stress.






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2 Stress amplitude [MPa] 427 265 421Strain ampliude [-] 2.13·10−3 3.44·10−3 2.10·10−3

Figure 48 shows what happens when q is not fitted to the model, here Model 5’s results are usedwith Model 6’s q.

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In Model 4 it can be seen that a higher load amplitude generated a lower value of q. Model 5and 6 shows instead that their values of q gave a good approximation of the predicted elasticstrain amplitude (compared to the elastic amplitude). It should also be noted that Model 4 eval-uates much higher strain amplitudes (up to 1.3%) with a good transformation of the two firstvalues (εa = 0.26% and εa = 0.55%). A theory is that q depends on the load level in such waythat after a certain strain value q should be adjusted. The results from Model 4 - 6 indicates thatfor at least strain amplitudes under 0.6%, using the same q for the model will only give small errors.

Out of interest it have been controlled that the material curve is constant for model 5 and 6. Thecollected effective stress and strain amplitude from both models and all load cases were plotted inFigure 49 and it can be concluded that the material model is not dependent of geometry.

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Figure 49: Material curve for Chaboche material, both model 5 and 6.

5.5 Fatigue calculationsToday it is common to use a bilinear material model in the computations and the fatigue calcu-lations when small plastic deformations occurs is based of extrapolation of HCF, see Figure 50.One problem with that method is that it does not capture the behavior of plastic deformation. Itis therefore of interest to see the difference using the extrapolated HCF and LCF. To get a stressconstrain from a required life span in the LCF zone, the HCF curve is extrapolated.

To translate the stress-life curve to strain-life curve, Hooke’s law σ = εE is used. It is nowpossible to see the difference between using an extrapolated HCF curve and a Coffin-Manson’sLCF curve, as it is illustrated in Figure 50.

Figure 50: Extrapolated HCF to the left and the strain-life plot of Coffin-Manson’s equation tothe right.

It becomes obvious that when including plastic deformation, the allowed total strain (calculatedwith Coffin-Manson’s equation) is higher than the elastic strain within the LCF zone. One couldthen say that using extrapolated strain would therefore be conservative and give a trustworthyevaluation. But going back to the material model, it becomes more complicated since the strainamplitude depends on the material model, as it is shown in Figure 51.

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Figure 51: Material models for Chaboche and Elastic material.

The strain in the Chaboche model is higher than in the elastic material model. Therefore ex-trapolation HCF should not be seen as conservative. The used material data comes from Bahnu’sarticle [16] and is presented in Table 26 and with Equation 28 can the strain life plot in Figure 52be made. This material represent a weldment at 823 K, and is of type 304 stainless steel as basematerial and type 308 Stainless steel as weld material. The strain-life is plotted in Figure 52.

Table 26: Fatigue parameters for weldment in 823 K.

C ε′f b σ′f E-0.681 0.395 -0.146 899 193 [GPa]

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Figure 52: Strain-life plot of a weldment at 823 K, and is of type 304 stainless steel as base materialand type 308 Stainless steel as weld material.

To compare the elastic strain calculated using the elastic material model with the elastic strain usingChaboche material model Figure 53 can bee used. Observe that the material is compared to anelastic material, not a bilinear as mention before. This is because the elastic material equivalent tothe used Chaboche material is known, but not the bilinear one. The Chaboche material parametersused is not of the same material as the weldment, and the Chaboche material parameters in thetest has not been customized to the temperature (Chaboche material comes from 393K and theweldments at 873K). The results are therefore not trustworthy, but a difference of results betweenthe methods will still indicate that one method is better.

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Figure 53: Chart of finding the strain to be used as a constraint in an optimization. The methodto the left is the new proposed method to calculate the constraint for a component with a linearelastic material exposed to LCF.

Let’s say the requested life span is 4 000 cycles in model 5 the constraint ∆εelast-HCF(4000) (whichis a extrapolated HCF strain) and ∆εChaboche(4000) (which is a total strain from the Coffin-Manson curve) can be calculated to 2.51·10−3 respective 4.25·10−3 by using Equation 26 and 28.Transforming the total strain amplitude to elastic strain using Super Neuber transformation andsetting parameter Cq to constant in Equation 31 we get

∆σelast∆εqelast = ∆σChaboche∆ε

qChaboche (43)

meaning that

∆εq+1elast =

∆σchaboche∆εqchaboche

E(44)

where ∆σchaboche can be approximated from the material curve when ∆εchaboche = 4.25·10−3. Let’ssay it is 270 MPa, using q = 0.6 and E = 193 GPa as before, which given ∆εelast= 2.12·10−3.This is 18 % lower than the outcome from the extrapolated HCF (∆εelast-HCF(4000)). If that valuewould be used to calculate the life in an extrapolated HCF, the predicted life would be 12 600cycles, which is more than three times longer! Meaning that if the optimization with this fatigueparameters, a strain amplitude of 2.12·10−3 using extrapolated HCF the life expectancy would be12 600 cycles. With the Super Neuber method (with the conditions stated here) the life expectancywould be only 4000 cycles. Concluding the errors in the strain measurement leads to large errorsof life expectancy, and therefore should the strain amplitude be calculated carefully.

5.6 Temperature dependency in the transformation functionDue to time limitations the temperature dependency has not been included in the transformationcalculations. It is predicted that changing the material parameters according to chapter 2.6.4 willlead to different forms and size of the hysteresis, leading to other material curves. Knowing thefatigue parameters of the material, all the temperature dependency of the material, one couldcalculate the Super Neuber parameter at one point and use it in fatigue if no thermal stress andstrain occurs. The relation between thermal stress and strains is strongly dependent on the modeland load, and therefore effect the Super Neuber parameter q [19].

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6 The alternative methodIn this chapter the alternative method of weld evaluation will be discussed and evaluated.

6.1 Evaluation of the alternative methodSince only elements can be analyzed in the optimization the hot spot method is not possible,and an alternative evaluation method has to be used. This method is based on hot spot, butinstead of calculating the tangent from two nodes and extrapolate the stress, the element stressfor a specific distance is used and multiplied with a correction factor y. The correction factor isgeometry dependent, and can also be used to factor in the weld quality. The correction factor canbe calculated according to the equation in Figure 55 where εweld is the actual strain in the weldand εd is the calculated strain at the distance d from the weld toe. Figure 54 shows the principlefrom the traditional hot spot method, while Figure 55 presents the alternative evaluation method.

Figure 54: The hot spot method [17].

Figure 55: The alternative hot spot method.

Hobbacher [17] has one standard form of of the hot spot stress σHS as

σHS = 1.67σ0.4t − 0.67σ1.0t (45)

where σ0.4t and σ1.0t is the stress at 0.4 respective 1.0 weld thicknesses, t, out from the weld.Because of limed time of this project only two analyses has been done both of elastic material,

59

one on model 5, load case 1 and the other on model 6, load case 1. We are interested in the hotspot strain, εweld instead of the hot spot stress. The weld has a thickness of 3 mm (t=3), andthe element size is 1 mm making the reference points at ε0.33t and ε0.67t. The same factors as inEquation 45 has been used to evaluate the corresponding hot spot strain (which will give an overprediction of the hot spot value), giving

εweld = 1.67ε0.33t − 0.67ε0.67t (46)

and y from Figure 55 can now be obtained from the evaluation points (since εd is the calculatedstrain of the elements in the evaluation domain). Figure 56 shows y depending on element numberof the evaluation domain, where the number increases from left to the right side.

Figure 56: Model 5 load case 1 to the right, and model 6 load case 1 to the left.

The results shows that y is not a distinct number, and in those models varies around 1. Lookingat the distribution of the von Mises strain of the model when loaded, this results can be expectedsince the highest strain on the plate does not appear at the weld toe, which is confirmed whenlooking at the strain plot of model 5 in Figure 57. For this type of loading it is therefore notadequate to use an alternative method and multiply with a correction factor. Looking at Figure56 and comparing it to Figure 44 one can conclude two things specific for model 5 (the same canbe done for model 6), in this case the correction factor is highest when the strain load is not,meaning that the highest strain will not change significantly, and that the correction factor lies inan interval of 0.85-1.25 when the equation overestimates the strain in the weld. With a correctequation used, this interval can be covered by a safety factor.

Figure 57: The strain plot of model 5, elastic load case 1.

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7 DiscussionIn this section some thoughts about Chapters 4, 5, and 6 are discussed.

7.1 Weld OptimizationPerforming a topology optimization of welds requires some knowledge of optimization parametersand manufacturing constraints. This was concluded during the two parameter studies performedin Section 4.1.1. The optimization results will otherwise not be manufacturable as well as toughto evaluate. An interesting result that could be seen in the method for weld optimization, Sec-tion 4.1.2, was that varying the constraint for the volume fraction gave optimization results thatwere more distinguishably to each other. When the other parameters, such as MINDIM , p, andMAXDIM , were varied then the optimization results almost looked the same and it did not differmuch. A reason for this could be that changing those parameters can be neglected by OptiStruct,even though they are defined. OptiStruct has the possibility to not prioritize a manufacturingconstraint but a mathematical constraint, such as the volume fraction constraint, will always betried to be fulfilled. However, it was necessary to include those parameters in the optimizationotherwise the result was not acceptable.

It was also found that the optimization results require some engineering polishing. Hence, it isimportant to not fully trust the results. For example, some weld material that should be keptaccording to a topology optimization could be too short or some welds could be suggested to beplaced too close to each other. The optimization result has to be inspected before regarded as anacceptable result. The weld optimization could generate a sense of where the weld could be re-moved and where it should be kept, which can be tough to guess before performing an optimization.

It is worth noting that optimizing welds is not a typical optimization problem and this holds fortopology optimization of welds as well. A typical topology optimization problem is often definedfor bigger areas of a component, which welds does not constitutes, for finding the best path forthe applied load. The goal of topology optimization is to remove areas of the component thatdoes not contribute to the path of the applied load. This goal is coinciding with the goal for weldoptimization, but this type of optimization might not account for the fact that the weld is such asmall part of the whole component.

Another issue that was found during the optimizations was that the results from structural andthermal loads were contradictory. If the loads were combined one of them will probably be moredominating than the other, usually this depends on which values for the stress constraints that areapplied. The structural load, in order to minimize the compliance (increase the stiffness), seeks tokeep as much weld material as possible. The thermal load, trying to decrease the thermal stresseswhich is a constraint in the optimization, seeks to remove as much weld material as possible. Ther-mal stresses are present as the structure is expanding due to the temperature increase. Also, thethermal load prefer a compliant structure. Therefore, if more weld material is removed the thermalstresses will decrease if the structure is allowed to expand more, which is shown in the results inSection 4.1.2.

It is possible that the stress constraints in the optimizations were focusing too much on targetingonly critical positions, where the highest stresses were found. It would probably be better if theoptimizations could calculate the mean value stresses over a certain area and therefore not focusingtoo much on the severe positions. A theory is that this could be solved by including a mathematicalexpression in the responses which is connected to the constraints. This is however not investigatedin this thesis. Investigating this could help the optimizations to focus more on other positions too,even if they are not the most severe ones in that iteration.

7.2 TransformationFrom Chapter 5 about the transformation function can it be concluded that the Super Neuberparameter q is dependent on the geometry and/or how the load is applied on the model. Thecalculations to find q is the same calculations as finding the the correct stress and strain for thematerial and it bears no interest to find the transformation parameter. But one thing that is

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reveled is that for the same geometry the same q can be used for different loads, as it can be seenin Figure 46 and 47. One should be careful since it has been concluded that a higher load casebenefits more of a lower q. In the test done in this report this effect has only been shown in model4 (it is also the only model reaching a strain amplitude over 0.4% ) and therefore needs more teststo back up this theory. If q could be predicted, one can simply use the demanded strain amplitude(coming from the strain-life graph) and predict an elastic strain amplitude. This would shortenthe computing time drastically. Right now is it to hard to predict q by using geometry factors(for harder geometries), but we know the q can be predicted by testing the model with load andboundary condition with Chaboche. We do not know if the geometry and/or load dependency canbe detected with another plastic material model, but if this can be done, Super Neuber is possibleto use by only knowing the material parameters. The next step is to implement this process tothe optimization process, meaning the that the optimization can be done with an elastic materialwith elastic stress constraints based from an original calculated q’s and then controlled by findingq’s of the optimized model and calculate the maximum strain amplitude of an advance materialmodel. An other theory is that q depends on the amount of primary stresses, since q equal to zeromeans that only primary stress are present. This theory has not been tested due to time limitations.

When thermal stress and strain occurs in the model it is expected that q will change. How atthis point is unclear and is very important for the optimizations with temperature fields. Thepossibility to recognize the non proportional relation between thermal stress and strain as SuperNeuber makes finding a relation between the Chaboche material and elastic material might bepossible. Only using normal Neuber will, without hesitation, give large errors [19].

This report has not done any analysis on in which degree ratcheting will occur and how it willinfluence the life of the component. When ratcheting occurs the strain amplitude still will beconstant, which has been assumed in this report and from there can the component be design notto reach that strain amplitude. But when ratcheting transpires the total strain will increase everycycle until it reaches a maximum value and breaks. The normal strain-life curve can not longerbe used. Because of time limitations it has been assumed that ratcheting does not occur, eventhough some of the results indicates that it does happen in some locations. Finding where and theextent of the ratcheting should always be done when dimensioning for a certain life expectancysince otherwise can cause unexpected fractures.

The fatigue analysis that has been done in Chapter 5.5 concludes that extrapolated HCF shouldbe avoided when plastic deformations occurs. Due to limitation of resources no material test hasbeen executed. Therefore will the results not be reliable. The life expectancy is three times longerusing extrapolated HCF than the Super Neuber transformation gives at the given conditions inChapter 5.5 strongly indicates that large errors will occur by not using an adequate method. Morecalculations needs to be done to draw a definitive conclusion.

7.3 The alternative methodWhen doing an optimization values for the constraints must be decided, for this thesis the mostcritical ones were stress constraints. The method used in Chapter 4 for getting the values forthe static stress constraints was not a robust one and they were basically guessed based on theFEA results. The evaluation domain was one element row out from the weld so that singularitiescan be avoided. Basically this means that the correct stress and strain in the optimization wasnot evaluated. The error that this entails depends strongly on model and load. However, as it ismentioned in Section 6 can the error be capture with a safety factor for the optimization processand then controlled after with the hot spot method for better precision.

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8 ConclusionsThe most important conclusions for this thesis are:

• With the presented method in this thesis for topology optimization of welds the thermalstresses in the evaluation domain are decreased.

• As observed from the parameter studies and the evaluation of the weld optimization methodthe optimization parameters that are necessary to include in the method for weld optimizationare the following:

– MINDIM

– Penalty factor, p

– MAXDIM

It was also concluded that varying the upper constraint value for the volume fraction gaveoptimizations results that was more distinguished from each other. It is also possible tooptimize against several load cases, such as a structural load and a thermal load.

• It is possible to find a transfer function between Chaboche and elastic material, but with thismethod one have to do a full analyses with Chaboche material, which will take unreasonablytime. If it is possible to find a simpler way to estimate the transformation parameter, thismethod can be used in the industry. With a correct q the transformation can be done byusing the material curves of the Chaboche and elastic material.

• The new proposed method should be used to calculate the constraints in the optimization,and not a method based on extrapolated HCF. The super Neuber transformation is thereforeneeded to get the necessary elastic constraints needed in the optimization.

• The common methods based on hot spot cannot be used, since Optistruct does not estimatescalar measures of stresses and strains at nodes. The alternative method to evaluate the weldis therefore necessary. Errors can be found when comparing it to the hot spot method, andtherefore should the alternative method be used with caution.

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9 Future workIn this chapter some suggestion for future work are presented.

• Further investigation could be made into how the stresses could be decreased. An optioncould be to perform a topography optimization of the surrounding plates before performinga topology optimization , this could provide insights on the potential to decrease the stressesclose to the welds.

• In parameter study 2 performed in Chapter 4.1.1 it was possible to include a thermal heattransfer analysis that was updated during optimizations. This could be an option to includefor the weld optimization for the silencer, which were only performed with a constant tem-perature field during the optimizations. This could be performed to investigate whether thetemperature affects the optimization result or not.

• Evaluate if the results from the optimizations can be made more accurate by including arequirement, such as a stress constraint, which is calculated from a robust transfer function.

• Evaluate the weld optimization results further, are the optimizations focusing only on themost critical positions and how is this potentially too narrow focus avoided?

• Finding the Super Neuber parameter q is crucial when translating the plastic requirementsto an elastic model. If q is found for the elements in the evaluation domain the Chabochematerial curve can be used. It is therefore of interest to find a simpler way to find q thancyclically load the component 50 times with a Chaboche material. These are the three wayssuggested ways one might find it:

– Calculated q by using a simple plastic material model and find the relationship to theelastic model.

– Investigate if q depends on the amount of primary stresses that are present.

– See if q can be found by only a few cycles of the Chaboche material.

• Examen how q depends on the temperature is important if thermal stresses and strains occursin the model.

• Finding a way to implement the calculations of q in an optimization and include constraintsfrom real fatigue test.

• Perform cyclic material tests for high temperatures in order to find relevant material param-eters for the silencer.

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References[1] Design Optimization, Retrieved from OptiStruct User Guides (2018-05-23):

https : //altairhyperworks.com/hwhelp/Altair/2017/help/os/topics/solvers/design_optimization.htm

[2] Optimization Techniques, Retrieved from Abaqus Analysis User’s Guide (2018-05-23):http : //abaqus.software.polimi.it/v2016/book/usb/default.htm?startat=pt04ch13s02aus90.html

[3] M. P. Bendsoe, and O. Sigmund, Topology Optimization: Theory, Methods and Applications,Section Edition, Springer-Verlag, pp. 1 - 69, (2003)

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[8] Models for metals subjected to cyclic loading, Retrieved from Abaqus Theory Manual, (2018-05-24):https : //classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/stm/default.htm?startat = ch04s03ath107.html

[9] P. van Eeten, and F. Nilsson, Constant and Variable Amplitude Cyclic Plasticity in 316LStainless Steel, Journal of Testing and Evaluation, Vol. 34, pp. 298 - 310, (2006)

[10] E. Hosseini, S. R. Holdsworth, I. Kühn, and E. Mazza, Temperature Dependent Representationfor Chaboche Kinematic Hardening Model, Materials at High Temperatures, Vol. 32, pp. 404-412, (2015)

[11] J.Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, The press syndicate of theuniversity of Cambridge, Cambridge, (2000)

[12] S. Suresh, Fatigue of Materials, Second edition, The press syndicate of the university ofCambridge, Cambridge, (1998)

[13] nCode Software, Design of Components for High Temperatures and Thermo-Mechanical Fa-tigue (TMF). 2014, [Video] https://www.youtube.com/watch?v=lSIEVtD7Avs (2018-05-24)

[14] R. Alain *, P. Violan, and J. Mendez, Low Cycle Fatigue Behavior in Vacuum of a 316LType Austenitic Stainless Steel Between 20 and 600C Part I: Fatigue Resistance and CyclicBehavior, Materials Science and Engineering A229, pp. 87 - 94, (1997)

[15] G.V. Prasad Reddy, R. Sandhya, M. Valsan, and K. Bhanu Sankara Rao, Temperature Depen-dence of Low Cycle Fatigue of 316(N) Weld Metals and 316L(N)/316(N) Weld Joints, MaterialsScience and Technology, Vol. 26, pp. 1384 - 1392, (2013).

[16] K. Bhanu Sankara Rao, M. Valsan and S.L Mannan, Strain-controlled Low Cycle FatigueBehaviour of Type 304 Stainless Steel Base Material, Type 308 Stainless Steel Weld Metal and304-308 stainless Steel Weldments, Materials Science and Engineering A130, Vol. 130, pp.67 -82, (1990)

[17] A. F. Hobbacher, Recommendations for Fatigue Design of Welded Joints and Components,Second Edition, Springer International Publishing, Switzerland, (2016)

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[19] L. Samuelsson, Fatigue Analysis: The Super-Neuber Technique for Correction of Linear ElasticFE Results, in proceedings of 26th International Congress of the Aeronautical Sciences, 14 - 19September 2008, Anchorage, Alaska, USA, (2008)

[20] R. Adibi-Asl, and R. Seshadri, Unified Approach for Notch Stress Strain Conversion Rules,Journal of Pressure Vessel Technology - Transactions of the ASME, Vol. 135, (2013)

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Appendix A

Settings for topology optimization of parameter studies 1 and 2

Table 1: Settings for topology optimization of parameter studies 1 and 2.

Automatic Screening (Constraint Screening) ONMinimum Element Volume Fraction 1.0000E-02Topology Optimization Method Density Method

Convergence Tolerance 5.0000E-03Step Size 0.5000

Maximum Number of Iterations (without MINDIM) 30Maximum Number of Iterations (with MINDIM) 80

Checkerboard Control (without MINDIM) OFFCheckerboard Control (with MINDIM) On (1 – Global Averaging)

Settings for topology optimization of weld optimization of a silencer

Table 2: Settings for topology optimization of weld optimization of a silencer.

Automatic Screening (Constraint Screening) ONActivates Element Quality Checking, “CHECKEL” NO

Minimum Element Volume Fraction 1.0000E-02Topology Optimization Method Density Method

Convergence Tolerance 5.0000E-03Step Size 0.5000

Maximum Number of Iterations (without MINDIM) 30Maximum Number of Iterations (with MINDIM) 80

Checkerboard Control (without MINDIM) OFFCheckerboard Control (with MINDIM) On (1 – Global Averaging)

Appendix B

Results from finite element analysis (FEA) performed before optimiza-tionsIn this section the results from FE-analyses are presented for Models 1 - 3 used in the optimizations.

Model 1 – von Mises stress, [MPa], results before optimizations

Figure 1: Stress result for the whole model.

Figure 2: Stress result in the evaluation domain.

Model 2 – First principal stress, [Pa], and temperature, [C], results before optimiza-tions

Figure 3: Stress result from the structural load case.

Figure 4: Stress result in the evaluation domain for the structural load case.

Figure 5: Temperature field calculated in the heat transfer analysis (steady-state), by OptiStruct.

Figure 6: Stress result from the thermal load case.

Figure 7: Stress result in the evaluation domain for the thermal load case.

Calculated temperature field by OptiStruct, [C], updated during optimizations forModel 2

Figure 8: Temperature field for the last iteration for cases 1 and 2.

Figure 9: Temperature field for the last iteration for cases 3 − 5.

Figure 10: Temperature field for the last iteration for cases 6 − 8.

Figure 11: Temperature field for the last iteration for cases 9 and 10.

Model 3 – Temperature, [C], and von Mises stress, [MPa], results before optimiza-tions

Figure 12: Temperature field calculated in a heat transfer analysis (steady-state), by Abaqus.

Figure 13: Stress results in the investigated welds from the static acceleration load case (to theleft) and thermal load case (to the right).

Figure 14: Stress results in the evaluation domain 1 (to the left) and 2 (to the right) for the staticacceleration load case.

Figure 15: Stress results in the evaluation domain 1 (to the left) and 2 (to the right) for the thermalload case.

Appendix C

FEA results calculated after optimizations for weld optimization of asilencer (Opt. 1-3, Chapter 4.1.2)

Figure 16: Opt. 1 − Stress results in the evaluation domain 1 (to the left) and 2 (to the right) forthe static acceleration load case.

Figure 17: Opt. 1 − Stress results in the evaluation domain 1 (to the left) and 2 (to the right) forthe thermal load case.

More optimization results from weld optimization of a silencerThe following optimization results are presented here just to show how the varied manufacturingparameters did not affect the optimization result significantly and that some of these gave badresult such as destroyed elements. There was a lot of parameters that were varied but the resultswere very similar and almost the same in some cases. In Tables 3 and 4 the optimization set-upsare defined. The iso-surfaces in Figures 22 - 27 were created with a density threshold value of 0.4.

Table 3: Optimization set-up 1 for weld optimization of a silencer.




Optimization constraint 1 - Static stress ≤ 60 MPaStatic acceleration

Optimization constraint 2 - Static stress ≤ 900 MPaThermal load case

Optimization constraint 3 0.05 ≤ Volume fraction ≤ 0.40Objective function Minimize the weighted compliance for both

the static acceleration and thermal load case

Table 4: Optimization set-up 2 for weld optimization of a silencer.




Optimization constraint 1 - Static stress ≤ 40 MPaStatic acceleration

Optimization constraint 2 - Static stress ≤ 700 MPaThermal load case

Optimization constraint 3 0.05 ≤ Volume fraction ≤ 0.40Objective function Minimize the weighted compliance for both

the static acceleration and thermal load case

In Table 5 the varied parameters for the optimizations are presented. The two last optimizations, 10and 11, were just investigations of how the computation time is affected if the constraint screeningis not activated (constraint screening, OFF ) and also if the “Activates Element Quality Checking” ison (CHECKEL, YES ). Note that the constraint screening is again activated (constraint screening,ON ) in case 11 as it is in the other optimizations.

Table 5: Optimization parameters for weld optimization of a silencer. Static acceleration is abbre-viated as st. acc. in the weight factors.

Optimization MINDIM MAXDIM Penalty Weight(Set-up) [mm] [mm] factor, p factors

1 (1) 30 - p = 2 wst. acc. = 1.0(DISCRETE = 1) wthermal = 0.5

2 (1) 30 - p = 3 wst. acc. = 1.0(DISCRETE = 2) wthermal = 0.5

3 (1) 30 130 p = 2 wst. acc. = 1.0(DISCRETE = 1) wthermal = 0.5









Figure 22: Optimization results 1 and 2 (from left). These results included destroyed elements atmany areas of the kept weld material.

Figure 23: Optimization results 3 and 4 (from left). These results are very similar to the result forOpt. 2, see Section 4.1.2.

Figure 24: Optimization results 5 and 6 (from left). These results are identical to the result inOpt. 2, see Section 4.1.2.

Figure 25: Optimization results 7 and 8 (from left). Optimization 7 is similar to the result in Opt.2, see Section 4.1.2. The same goes for optimization 8 and for this one it is also included manysmall areas of weld material, which is not preferably (recommendation is that a weld should be atleast 20 mm).

Figure 26: Optimization result 9. Optimization 9 is similar to the result in Opt. 2, see Section4.1.2.

Figure 27: Optimization results 10 and 11 (from left). These results are identical to the result inOpt. 2, see Section 4.1.2.

These results are presented in appendix because they are very similar or even identical to anotherearlier presented optimization result. It was therefore decided that it is not time efficient to do are-analysis in Abaqus with these above presented results.In table 6 the result for the amount of iterations, computation time, compliance, and compliancechange (objective function change) for the above mentioned optimizations are presented.

Table 6: Result from weld optimization of a silencer.

Computation time Compliance Compliance at ObjectiveOptimization Iterations [hh:mm:ss] and used at iteration last iteration function

number of CPU 0 [kNmm] [kNmm] changeprocessors

1 5 00:10:55 32 960.6 32 963.1 0.001%CPU: 8

2 5 00:53:02 32 962.3 32 962.4 0.0003%CPU: 4

3 6 00:13:20 32 960.6 32 963.1 0.008%CPU: 8

4 6 00:13:21 32 964.8 32 965.7 0.003%CPU: 8

5 6 01:43:05 65 918.1 65 922.8 0.007%CPU: 4

6 6 00:58:53 65 920.3 65 924.9 0.007%CPU: 4

7 6 01:02:47 32 962.3 32 964.6 0.007%CPU: 4

8 6 01:02:15 32 962.3 32 964.2 0.006%CPU: 4

9 6 01:11:39 32 962.3 32 964.6 0.007%CPU: 4

10 6 00:58:17 32 962.3 32 964.6 0.007%CPU: 4

11 6 00:58:24 32 962.3 32 964.6 0.007%CPU: 4

Appendix Dclear all, close allformat longticfid = fopen(’sinus30_halv’,’r’);datacell = textscan(fid, ’%s’, ’HeaderLines’, 1, ’CollectOutput’, 1);fclose(fid);A.data = datacell1; text=A.data;toc

B=[25 27 28 37 39 40 73 75 76 85 87 88 217 219 221 229 231 233 269 271 272 281 283 291...292 425 427 428 437 439 440 473 475 476 485 487 489 617 619 620 629 631 632 669 671 672 ...681 682 685 687 825 827 828 837 839 840 873 875 876 885 887 888 1017 1019 1020 1029 1031...1032 1069 1071 1072 1081 1083 1085 1086 1225 1227 1228 1237 1239 1240 1273 1275 1276...1285 1287 1288 1417 1419 1420 1429 1431 1432 1469 1471 1472 1481 1483 1489 1491];%if model 5 is used, the vector is equal to A, otherwise, equal to B (This is the nodes number of the evaluation domain)A=[25 27 28 37 39 40 73 75 77 85 87 89 217 219 221 229 231 233 269 271 272 281 282 285...286 425 427 428 437 439 440 473 475 477 485 487 489 617 619 621 629 631 633 669 671 672...681 682 685 686 825 827 828 837 839 840 873 875 877 885 887 889 1017 1019 1021 1029 ...1031 1033 1069 1071 1072 1081 1082 1085 1086 1225 1227 1228 1237 1239 1240 1273 1275...1277 1285 1287 1289 1417 1419 1421 1429 1431 1433 1469 1471 1472 1481 1482 1485 1486];%if model 6 is used, the vector is equal to A, otherwise, equal to B (This is the nodes number of the evaluation domain)C=1081;ones=ones(1, length(A));five=ones.*5;comp=[A’ ones’ five’];

x=1;ee11max=zeros(1, length(A));ee22max=zeros(1, length(A));ee12max=zeros(1, length(A));pe11max=zeros(1, length(A));pe22max=zeros(1, length(A));pe12max=zeros(1, length(A));S11max=zeros(1, length(A));S22max=zeros(1, length(A));S12max=zeros(1, length(A));acmax=zeros(1, length(A));peeqmax=zeros(1, length(A));pemagmax=zeros(1, length(A));

b=1;j=1;ticc=1;for i=1:length(text)-2

w= strcmp(text(i),’EE11’);if w==1

p=1;endv= strcmp(text(i),’PE11’);

if v==1p=2;

endif i==c

c=i+1;if A(j)==str2double(texti)

search=[str2double(texti), str2double(texti+1), str2double(texti+2)];

if str2double(texti)==A(j)

search=[str2double(texti), str2double(texti+1), str2double(texti+2)];

if search==comp(j,:)

info1=[str2double(texti+3) str2double(texti+4) str2double(texti+5)...str2double(texti+6) str2double(texti+7) str2double(texti+8)];info2=[str2double(texti+21) str2double(texti+22) str2double(texti+23)...str2double(texti+24) str2double(texti+25) str2double(texti+26)];info3=[str2double(texti+39) str2double(texti+40) str2double(texti+41)...str2double(texti+42) str2double(texti+43) str2double(texti+44)];info4=[str2double(texti+57) str2double(texti+58) str2double(texti+59)...str2double(texti+60) str2double(texti+61) str2double(texti+62)];medel1=(info1+info2+info3+info4)./4;

if p==1EE11(x,j)=medel1(2);EE22(x,j)=medel1(1);EE12(x,j)=medel1(3);S11(x,j)=medel1(4);S22(x,j)=medel1(5);S12(x,j)=medel1(6);

HS(x,j)=1.67*(info1(4)^2+info1(5)^2-info1(4)*info1(5)+3*info1(6).^2)....^0.5 -0.67*(info2(4)^2+info2(5)^2-info2(4)*info2(5)+3*info2(6).^2).^0.5;

if j==length(A)x=x+1;

end

maxee11=max([abs(info1(1)) abs(info2(1)) abs(info3(1))...abs(info4(1))])-abs(medel1(1));maxee22=max([abs(info1(2)) abs(info2(2)) abs(info3(2))...abs(info4(2))])-abs(medel1(2));maxee12=max([abs(info1(3)) abs(info2(3)) abs(info3(3))...abs(info4(3))])-abs(medel1(3));maxS11=max([abs(info1(4)) abs(info2(4)) abs(info3(4))...abs(info4(4))])-abs(medel1(4));maxS22=max([abs(info1(5)) abs(info2(5)) abs(info3(5))...abs(info4(5))])-abs(medel1(5));maxS12=max([abs(info1(6)) abs(info2(6)) abs(info3(6))...abs(info4(6))])-abs(medel1(6));

if maxee11>=ee11max(j);ee11max(j)=maxee11;

endif maxee22>=ee22max(j);

ee22max(j)=maxee22;endif maxee12>=ee12max(j);



ee22max(j)=maxee22;endif maxS12>=ee12max(j);

ee12max(j)=maxee12;c=i+62;

end

if j==length(A) ;j=1;

else j=j+1;end

else

if p==2PE11(x,j)=medel1(2);PE22(x,j)=medel1(1);PE12(x,j)=medel1(3);Aceield(x,j)=medel1(4);PPEQ(x,j)=medel1(5);PEMAG(x,j)=medel1(6);

maxpe11=max([abs(info1(1)) abs(info2(1)) abs(info3(1))...abs(info4(1))])-abs(medel1(1));maxpe22=max([abs(info1(2)) abs(info2(2)) abs(info3(2))...abs(info4(2))])-abs(medel1(2));maxpe12=max([abs(info1(3)) abs(info2(3)) abs(info3(3))...abs(info4(3))])-abs(medel1(3));maxac=max([abs(info1(4)) abs(info2(4)) abs(info3(4))...abs(info4(4))])-abs(medel1(4));maxpeeq=max([abs(info1(5)) abs(info2(5)) abs(info3(5))...abs(info4(5))])-abs(medel1(5));maxpemag=max([abs(info1(6)) abs(info2(6)) abs(info3(6))...abs(info4(6))])-abs(medel1(6));

if maxpe11>=pe11max(j);pe11max(j)=maxpe11;

endif maxpe22>=pe22max(j);

pe22max(j)=maxpe22;endif maxpe12>=pe12max(j);

pe12max(j)=maxpe12;endif maxac>=acmax(j);

acmax(j)=maxac;end

if maxpeeq>=peeqmax(j);peeqmax(j)=maxpeeq;

endif maxS12>=pemagmax(j);

pemagmax(j)=maxpemag;c=i+62;

end

if j==length(A)j=1;

else j=j+1;end

end

endendend

endendendtoc

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