topology optimization as a conceptual tool for designing ... · the inertia relief method utilized...

Master Thesis in Mechanical Engineering

Topology Optimization as a Conceptual Toolfor Designing New Airframes

Joakim Torstensson

ISRN: LIU-IEI-TEK-A16/02590SE

Division of Solid Mechanics | Department of Management and EngineeringSpring 2016 | Linköping University, SE-581 83 Linköping, Sweden

Linköping University | 013-28 10 00 | www.liu.se

June 20, 2016

Master Thesis in Mechanical Engineering

Topology Optimization as a Conceptual Toolfor Designing New Airframes

Joakim Torstensson

Supervisors : Erik Holmberg, Saab ABAnders Klarbring, Linköping University

Examiner : Carl-Johan Thore, Linköping University

ISRN: LIU-IEI-TEK-A16/02590SE

Division of Solid Mechanics | Department of Management and EngineeringSpring 2016 | Linköping University, SE-581 83 Linköping, Sweden,

Linköping University | 013-28 10 00 | www.liu.se

June 20, 2016

Preface

This masters thesis concludes my studies at Linköping University and is the nal piece inobtaining my Master of Science in Mechanical Engineering.

I would like to express my gratitude towards my supervisor at Saab, Erik Holmberg, for thesupport, guidance and interesting discussions all along. I would also like to thank the peoplesupporting me at the Division of Solid Mechanics at Linköping University: my supervisorAnders Klarbring and examiner Carl-Johan Thore for help all along.

Additionally, the people at the Altair support deserve a mention for the help they have pro-vided related to the OptiStruct software.

Joakim Torstensson

Linköping, June 2016

I

Abstract

During the two last decades, topology optimization has grown to be an accepted and usedmethod to produce conceptual designs. Topology optimization is traditionally carried out ona component level, but in this project, the possibility to apply it to airframe design on a fullscale aeroplane model is evaluated.

The project features a conceptual ying-wing design on which the study is to be carried out.Inertia Relief is used to constrain the aeroplane instead of traditional single point constraintswith rigid body motion being suppressed by the application of accelerations instead of tradi-tional forces and moments. The inertia relief method utilized the inertia of the aeroplane toachieve a state of quasi-equilibrium such that static nite element analysis can be carried out.

Two load cases are used: a steep pitch-up manoeuvre and a landing scenario. Aerodynamicforces are calculated for the pitch-up load case via an in-house solver, with the pressure beingmapped to the nite element mesh via a Matlab-script to account for dierent mesh sizes.Increased gravitational loads are used in the landing load case to simulate the dynamic loadingcaused in a real landing scenario, which is unable to be accounted for directly in the topologyoptimization.

It can be concluded that the optimization is unable to account for one of the major designlimitations: buckling of the outer skin. Approaches to account for the buckling of the outerskin are introduced and analysed, with a focus on local compression constraints throughoutthe wing. The compression constraints produce some promising results but are not withoutmajor drawbacks and complications.

In general, a one-step topology optimization to produce a mature conceptual airframe designis not possible with optimization algorithms today. It may be possible to adopt a multiple-stepoptimization approach utilizing topology optimization with following size and shape optimiza-tion to achieve a design, which could be expanded on in a future project.

II

CONTENTS

Contents

1 Introduction 11.1 Project Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Saab AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Historical Background and Previous Research . . . . . . . . . . . . . . . . . . . 21.5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 The Aeroplane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Other Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theoretical background 52.1 The General Structural Optimization Problem . . . . . . . . . . . . . . . . . . . 52.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The Topology Optimization Problem Formulation . . . . . . . . . . . . . . . . . 62.4 Solving the Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Intermediate Element Densities and the SIMP-Method . . . . . . . . . . . . . . 82.6 Numerical Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6.1 Checkerboard-patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6.2 Mesh-dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.3 Local Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Inuence of Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 The Inertia Relief Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8.1 Interaction with Topology Optimization . . . . . . . . . . . . . . . . . . 162.9 Short on Aerodynamic Panel Methods and Aerodynamic Forces . . . . . . . . 17

3 Method 193.1 Geometry Clean-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Meshing Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Component Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Flight Manoeuvres and Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Pitch-up Manoeuvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.2 Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 The Optimization Problem, Constraints and Settings . . . . . . . . . . . . . . . 243.7 Result Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Models 274.1 Wing Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Simplied Aeroplane, Outer Shell Mesh . . . . . . . . . . . . . . . . . . . . . . 284.3 Simplied Aeroplane, Tetra-only Mesh . . . . . . . . . . . . . . . . . . . . . . . 29

5 Complications 305.1 Wing Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Non-symmetric Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

III

CONTENTS

5.3 Neglected Component Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Intermediate Element Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5 Poor Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Developed Programs 346.1 Modelling of Point Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Pressure Distribution to FE-mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 Compression Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Results 367.1 Tetrahedron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 Wing Only Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.3 Simplied Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.3.1 SPCs only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3.2 Inertia Relief only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3.3 Combined Inertia Relief with SPCs and Shell Thickness Inuence . . . . 407.3.4 Full Wing Compression Constraints . . . . . . . . . . . . . . . . . . . . 407.3.5 Compression Constraints only in Specied Points . . . . . . . . . . . . . 40

8 Discussion 428.1 Topology Optimization and Inertia Relief . . . . . . . . . . . . . . . . . . . . . 428.2 Lack of Material Placement in the Outer Wing . . . . . . . . . . . . . . . . . . 428.3 Other Optimization Formulations and Model Uncertainties . . . . . . . . . . . . 448.4 Ways to Guide the Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9 Conclusions 45

References 46

A Sensitivity Analysis 49

B Inertia Relief Example 50

C Inuence of Shell Thickness 52

IV

LIST OF TABLES

List of Figures

1 Simple topology optimization example . . . . . . . . . . . . . . . . . . . . . . . 22 Conceptual ying-wing unmanned aerial vehicle model . . . . . . . . . . . . . . 33 Overview of internal cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Solution with intermediate densities . . . . . . . . . . . . . . . . . . . . . . . . 96 Checkerboard visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Mesh size dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Gravitational eects on topology optimization results . . . . . . . . . . . . . . . 139 Gravity eects on the TO results using a minimum lower volume constraints

and point masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410 Optimization problem using SPCs . . . . . . . . . . . . . . . . . . . . . . . . . 1611 Showcasing the eect inertia relief on the optimal design . . . . . . . . . . . . . 1712 Problematic geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1913 Geometry clean-up of the aeroplane nose . . . . . . . . . . . . . . . . . . . . . . 2014 Meshing aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2015 Pressure distribution (in [Pa]) on the upper part of the body . . . . . . . . . . 2216 Pressure distribution (in [Pa]) on the lower part of the body . . . . . . . . . . . 2217 SPC locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2318 Constrained nodes for the inertia relief analysis . . . . . . . . . . . . . . . . . . 2419 Dierent optimal designs by density cut-o thresholds . . . . . . . . . . . . . . 2620 Wing-only model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2721 Wing-only model with rod elements positions . . . . . . . . . . . . . . . . . . . 2822 The simplied model used for proof-of-concept testing . . . . . . . . . . . . . . 2823 Non-symmetric result from an optimization . . . . . . . . . . . . . . . . . . . . 3124 Same problem as in Figure 23, solved with an added symmetry constraint . . . 3225 Optimized topology of the tetrahedron model . . . . . . . . . . . . . . . . . . . 3626 Inuence of full wing compression constraint . . . . . . . . . . . . . . . . . . . . 3727 Inuence of compression constraint. 26 constraints used throughout the wing . 3828 Simplied model using SPCs, no displacement constraint . . . . . . . . . . . . . 3929 Simplied model using only inertia relief . . . . . . . . . . . . . . . . . . . . . . 3930 Simplied model, no displacement constraint, 2mm shell thickness . . . . . . . 4031 Simplied model with displacement constraint on the whole wing . . . . . . . . 4032 Displacement constraint, only in specic points . . . . . . . . . . . . . . . . . . 41

List of Tables

1 Optimization data, full wing constraints . . . . . . . . . . . . . . . . . . . . . . 372 Optimization data, 26 constraints throughout the wing . . . . . . . . . . . . . . 38

V

LIST OF TABLES

Nomenclature

Abbreviations

TO Topology OptimizationFE Finite ElementSIMP Solid Isotropic Material with PenalizationQUAD4 4-node Quadrilateral elementCST 3-node Constant Strain Triangle elementSPC Single Point ConstraintsDOF Degrees of Freedom

Mathematical Notations

Ω Design Domainx Set of all design variablesxe Design variable (element density) for element e

Ke Non-penalized element Stiness MatrixK(x) Global Stiness MatrixKe(x) Element Stiness MatrixM(x) Global Mass Matrixu(x) Nodal displacement vectoru(x) Nodal acceleration vectorf(x) Nodal force vectorfe(xe) Element force vector due to gravitational loadingf Nodal force vector for the inertia relief caseI Identity matrixm Number of design variablesc Number of constraintsN Number of elements in the nite element mesho Number of compression constraintsgi Constraint functionsC(x) Complianceε Lower element density limitVi, Ve Element volumeV0 Maximum allowed total volumesmax Maximum allowed strain in rod elementssmin Minimum allowed strain in rod elementsp SIMP penalization factorrmin Filter radiusdmin Minimum member sizedist(i, k) Distance between center of element i and kΩk Set of elements within distance rmin of element kρe Element specic material densityρ Isotropic material densityα Scalar load factor

VI

LIST OF TABLES

g Gravitational acceleration vectorcp Pressure coecientP Pressure where cp is evaluatedP∞ Surrounding airstream pressureρ∞ Surrounding airstream densityv Aeroplane speed relative surrounding airstreamα Angle of attackq Dynamic pressure

VII

1 INTRODUCTION

1 Introduction

With rapid technological advancements keeping product development cycles as short as pos-sible is of utmost importance for companies in a competitive market. Traditional productdevelopment is an iterative process where a design is created manually based on experienceand evaluated to ensure that the design satises certain design criteria. This traditional ap-proach can be very time-consuming and will often result in a design that satises the designcriteria, but still has lots of room for improvement.

Using structural optimization in the design process can increase performance and reduce de-velopment time signicantly. More so, the nal design can be specically optimized for aparticular objective, such as weight minimization. This is particularly useful in the automo-tive and aeronautics industry, where lower weight leads to better performance regarding e.g.lower fuel consumption, longer range, increased manoeuvrability or the ability to carry morepayload.

1.1 Project Description

The project has been carried out at the section Structural Dynamics, Aeroelasticity and StoresSeparation at Saab AB (2016) in Linköping during the rst half of 2016.

The goal of this master thesis has been to evaluate the possibility to use Topology Optimization(TO) on a full aeroplane model in an early stage of the design process. The assignment was,given an aerodynamic outer structure, to nd an optimized inner structure given several loadcases, geometric, and manufacturing constraints and computational cost restrictions. Thingsto consider included studying the level of required detail in the Finite Element (FE) model,evaluation of which structural requirements the design has to satisfy, which optimizationformulation, parameters and settings to consider, boundary conditions for the model andevaluation of the optimization results.

1.2 Saab AB

Saab AB (2016) was founded as Svenska Aeroplan AB in 1937 in Trollhättan, Sweden. Sincethen the company has branched out in several markets and provides products such as missilesystems, surveillance systems, submarines and most famously known, the Gripen-system. Saabis also active in the civilian market, having previously produced civilian aircraft as the Saab340 and Saab 2000, and now serving as a subcontractor to leading aircraft manufacturers suchas Boeing and Airbus.

1.3 Structural Optimization

The eld of structural optimization can be divided into three major categories: size, shapeand topology optimization. Size and shape optimization is usually performed on more maturedesigns and does not introduce any major design changes. For example, size optimization canbe used to determine the optimal cross-section area of a beam or the radius of a hole, whereas

1

1 INTRODUCTION

shape optimization can be used to determine the optimum shape of a cross section or whethera hole should be circular or elliptical.

TO is seldom performed on an already existing design. TO begins with a predened designspace, denoted Ω, in which a certain amount of material can freely be placed to nd anoptimized structure given specic load cases and boundary conditions. This makes TO agreat tool early in the design process where it can be used to produce a conceptual design.

Traditional TO is usually done by minimizing the compliance of the structure while con-straining the volume fraction allowed in the optimization. For a given load case, complianceis dened as the inverse of the stiness so minimizing the compliance maximizes the stinessof the structure.

Illustrated in Figure 1 is an example of a simple TO solution. The objective is to minimizethe compliance while only being allowed to ll 40% of the original design space. The designspace is subject to a force on the right-hand side while the left-hand side is xed, as seenin Figure 1a. Displayed in Figure 1b is the optimized structure obtained via TO, given thespecic optimization settings and boundary conditions.

Ω

(a) Design space with boundary conditions (b) Result of the TO

Figure 1: Example of a simple TO problem: minimize compliance formulation using 40% ofthe design space

The minimum compliance formulation is great for nding the most ecient load paths inthe design space but might lead to a design which could experience problems such as stressconcentrations, a risk of buckling, dangerous eigenfrequencies and fatigue related problems.These problems can be considered in the TO but due to increased computational time andthe introduction of even more diculties and uncertainties associated with each respectivearea, this is usually not done. Instead, size and shape optimization can be performed on theoptimal design acquired from TO in order to eliminate the previously mentioned problems.This thesis will, however, only cover the TO part of the design process.

1.4 Historical Background and Previous Research

Structural TO as it is used today was introduced by Bendsøe (1989). TO is usually conductedon a component level and not on a global scale, which is to be attempted in this project.While TO with large FE-models has been carried out, no examples of TO on the scale of afull aeroplane model has been found in the studied literature.

2

1 INTRODUCTION

Since its introduction TO has been used in many industrial applications (Bendøse and Sig-mund, 2003). A well known industrial example is the use of topology, size and shape opti-mization when designing conceptual wing box ribs for the Airbus A380 aircraft (Krog et al.,2002). The use of this optimization focused methodology has reportedly reduced the totalweight of the aeroplane by over a thousand kilos (Krog et al., 2004).

More closely related to this thesis; research has been done by Quinn (2010) on large-scale TOproblem using a combination of several weighed load cases and inertia relief to avoid rigidbody modes. While the work carried out by Quinn (2010) was done on a car chassis, manysimilarities exist with this project. Modelling was done on a full vehicle model with criticalcomponents modelled as point masses, which basically is the same approach used in this thesis.Cavazzuti et al. (2011) also studied a similar problem without the use of point masses.

Also studied by Luo et al. (2006) is the interaction between TO and inertia relief methodswhen performing TO on a missile body. Several load cases were considered which combinedto a single optimized body. All these studies show promising behaviour when coupling inertiarelief and TO.

1.5 Software

The Altair HyperWorks package (Altair, 2016a), including the pre-processor HyperMesh, thesolver OptiStruct (Altair, 2016b) and the post-processor HyperView have been used through-out this project. Matlab has been used to simplify and automate certain processes.

1.6 The Aeroplane Model

The aeroplane model which was to be used during the project is a conceptual ying-wingunmanned aerial vehicle design, showcased in Figure 2. The model contains approximately140 internal components which have to be taken into consideration in the TO.

Figure 2: Conceptual ying-wing unmanned aerial vehicle model

The conceptual design also features a number of internal cavities, seen in Figure 3, such asair intake to the engines, fuel tanks and landing gears, with some of those internal cavitiesaccommodating internal components.

3

1 INTRODUCTION

Figure 3: Overview of internal cavities

1.7 Other Considerations

No gender or environmental questions have been considered. The project has been carriedout at Saab AB which main area of operation is in the defence industry, but no ethicalconsiderations regarding this have been made.

4

2 THEORETICAL BACKGROUND

2 Theoretical background

The following chapter serves as an introduction to the eld of TO and inertia relief, as well ashort introduction to aerodynamic panel methods and aerodynamic forces.

2.1 The General Structural Optimization Problem

A general structural optimization problem (SO) can be described as in equation (2.1) Chris-tensen and Klarbring (2008). The problem consists of a objective function go(x,y), e.g. mass,displacement etc., which is minimized. Maximization problems can also be considered, butare usually rewritten as minimization problems.

(SO)

min g0(x,y)

s.t.

behavioral constraints on y

design constraints on x

equilibrium constraints.

(2.1)

Design constraints constitute constraints on the design variables x, which could representa minimum allowed thickness of a plate or the maximum length of a rod. The variablesy are state variables which represent the response of a structure. Considering structuraloptimization, this response is usually mass, displacement, force, stress etc. The equilibriumconstraint reads

K(x)u = f(x) (2.2)

where K(x) is the global stiness matrix, u is the nodal displacement vector and f(x) isthe nodal force vector. The force vector may be design variable independent, i.e. f , in caseswhere no internal body forces such as gravity, centrifugal forces or inertia forces are taken intoaccount (Zheng et al., 2009). More regarding body forces will be discussed later in section 2.7.

Concerning structural optimization it is very frequent that the equilibrium constraint uniquelydenes the state variable y. For this case the state variable, the displacement u, is implicitlydependent on the equilibrium constraint such that u = u(x) = K(x)−1f(x). The problemcan then be formulated, on nested form, as

(SO)nf

minx∈Rm

g0 (x)

s.t.

gi (x) ≤ gi, i = 1, . . . , c

¯xe ≤ xe ≤ xe, e = 1, . . . ,m

(2.3)

where m is the number of design variables contained in x ∈ Rm with c being the number ofconstraints. gi is a constraint function which must be below a dened upper limit gi.

¯xe and

xe are upper and lower bounds, respectively, on the e-th design variable.

5


2.2 Finite Element Discretization

In order to nd the optimal placement of material in the design domain Ω via structural TOthe domain is discretized into a FE-mesh consisting of N elements, illustrated in Figure 4. Theoptimization determines which elements to keep (representing material) and which elementsto remove (representing void). Traditionally TO is carried out using the lowest order niteelements, i.e. 3-node CST-triangles and 4-node Quadrilaterals in two dimensions and either6-node Tetrahedron or 8-node Hexahedron elements in three dimensions. The reason thesesimple elements are used, when a more accurate result probably could be an achieved usinghigher-order elements is simply the large increase in computational time by using higher-orderelements.

Ω

(a) Design space Ω before nite element discretiza-tion

xe

(b) FE mesh after discretization

Figure 4: FE discretization of the design space Ω with each element given a designateddesign variable xe

Associated with each element in the design domain is a design variable xe, which determineswhether the element should be included in the nal design. A value of 1 implies that theelement should be kept in the nal design and 0 (or very close to 0, more on that later)implies that the element should not be included.

2.3 The Topology Optimization Problem Formulation

The most common TO problem formulation is the minimum compliance formulation. Com-pliance, here denoted C(x), is dened as the inverse of the stiness, as that minimizing thecompliance returns a structure as sti as possible given the boundary conditions and opti-mization settings. The TO problem for minimizing the compliance can be formulated as

(TO)

minx∈Rm

C (x) =1

2fT (x)u(x)

s.t.

m∑e=1

Vexe ≤ V0

0 < ε ≤ xe ≤ 1, e = 1, . . . ,m.

(2.4)

6


The design domain is subject to a volume constraint, where the currently lled volume of thedesign domain, calculated by the sum of all individual element volumes Ve times the designvariable xe has to be lower or equal to a predened maximum allowed volume V0. It shouldbe noted that this volume constraint also can be dened over the entire volume rather thanjust the design domain, making the constraint read

∑Ni=1 Vixi ≤ V0. The design variables xe

are constrained to vary continuously between a small number ε and 1. The lower bound ε isintroduced in order to avoid zero-stiness diagonal terms in the stiness matrixK(x), makingit singular, and thus making the nite element equilibrium equation (2.2) non-solvable.

Several other problem formulations can be formulated, including objective functions such asstructural mass, maximum displacements (Sigmund, 1997), stress (Holmberg et al., 2013),eigenfrequency (Pedersen, 2000), fatigue life (Holmberg et al., 2014) and buckling (Gao andMa, 2015). Constraints and objective functions are mostly interchangeable: functions thatcan be used as objective functions can be used as constraints and vice verse.

2.4 Solving the Optimization Problem

Due to the non-convex nature of most structural optimization problems and the diculties as-sociated, it is not feasible to solve large-scale optimization problems directly. Instead, explicitconvex approximations of the general optimization problem are generated and subsequentlysolved instead (Christensen and Klarbring, 2008).

The optimization algorithms used to solve these subproblems are commonly of the rst order,meaning that the highest order of derivatives used is 1; implying that gradients of the objectiveand constraint functions have to be calculated. Common algorithms include The Methodof Moving Asymptotes (MMA) (Svanberg, 1987), Convex Linearization (CONLIN) (Fleury,1989) and Sequential Linear Programming (SLP), among others.

Solving the optimization problem is an iterative process and can be described by the followingsteps:

1. Start with an initial guess x0. Set iteration counter k = 0.

2. Calculate the displacement vector u(xk) by the nite element equilibrium equation (2.2).

3. Calculate objective and constraint function values and respective gradients given thecurrent design xk.

4. Formulate and solve an explicit convex approximation of the optimization problem toachieve a new design xk+1.

5. Update k = k + 1. Control if a convergence criteria is satised; if not, return to step 2.

Convergence criteria are most commonly formulated in either of two ways: a maximal allowedchange of the objective function value between two iterations or a maximum dierence indesign variables between iterations.

7


2.4.1 Sensitivity analysis

Calculation of gradients is referred to as sensitivity analysis. Two types of methods exist tocalculate the sensitivities: numerical and analytic methods. Numerical methods are based onnite dierences which may be inaccurate and computationally expensive and are thereforeseldom used in TO. These methods still have their uses due to the ease of implementation andcan give an estimate whether an analytic implementation has been done correctly or not, butfor pure calculations, analytic methods are used almost exclusively.

Analytic methods are based on the dierentiation of the objective and constraint functionwith respect to the design variable xe. Due to the nature of the problems solved in structuralTO the gradients are almost exclusively solved using an adjoint method where a second set oflinear equations is introduced and solved each iteration. Refer to Appendix A for more detailsregarding this.

2.5 Intermediate Element Densities and the SIMP-Method

As previously mentioned, a design variable xe is allowed to vary continuously between a smallnumber ε and 1. This is, however, contradictory to the fact that the nal solution is onlysupposed to contain elements with design variable values of either ε or 1, representing voidor material. If a normal nite element problem is considered, the global stiness matrix K isassembled from all element stiness matrices Ke, according to Cook et al. (2007), as

K =N∑e=1

Ke. (2.5)

This standard formulation considers all elements in the FE-mesh to have a stiness whichis not governed by the design variable. As such, even elements with a design variable valueof ε will give a full stiness contribution to the global stiness matrix. Because of this, thestiness of each design element is said to vary linearly with the design variable xe, such thatthe global stiness matrix is assembled by

K(x) =

N∑e=1

Ke(x), (2.6)

where K(x) is the global stiness matrix and Ke(x) is the (penalized) element stinessmatrix. The element stiness matrix for design elements is calculated as

Ke(x) = xeKe, e = 1, . . . ,m, (2.7)

where Ke is the non-penalized stiness matrix. Note that for problems with only designelements m = N holds.

No physical representation of the design variable has been done up until this point otherthan ε representing void and 1 representing material. While a physical representation is notnecessary since the nal design only should consist of void and solid elements, this can bedone by thinking of the design variable as a thickness in a two-dimensional problem or, as itwill be referenced throughout this report, a density in a three-dimensional problem.

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The solution of an optimization problem using the global stiness matrix formulation in (2.6)may show large areas consisting of elements with intermediate element densities, as showcasedin Figure 5. While a physical representation of these intermediate design variables has beenintroduced, it is rarely feasible to manufacture products with a continuously varying thicknessor density and thus the existence of intermediate element densities should be minimized.

Ω

(a) The design problem (b) Solution with intermediate element densities

Figure 5: Showcase of a solution with intermediate element densities, represented by greyareas in the nal design. Minimize compliance formulation with mass constraint of 50%.

In order to eliminate intermediate element densities in the nal design a penalization scheme isused. The Solid Isotropic Material with Penalization (SIMP) method was rst introduced byBendsøe (1989) and is still the most commonly used approach. Instead of the design elementstiness varying linearly with the density as in (2.7) a power-law penalization scheme is used:

Ke(x) = xpeKe, e = 1, . . . ,m. (2.8)

With a value of p > 1 (usually p = 3 (Bendøse and Sigmund, 2003)) the optimization willfavor non-intermediate density elements. Intermediate densities gain an uneconomically lowstiness compared to the amount of volume they occupy in the design space and thereforetend towards a density of ε or 1. A nal solution consisting of only void and solid elements isreferred to as a black-and-white solution.

2.6 Numerical Instabilities

A number of numerical problems arise during the TO process. Sigmund and Petersson (1998)divide these problem into three categories; checkerboard patterns, mesh dependency and theexistence of local minima. These problems are discussed in the following chapter.

2.6.1 Checkerboard-patterns

The appearance of checkerboard-like patterns (as shown in Figure 6) is common when theSIMP-method is used. These patches of altering void and solid elements appear due to numer-ical approximations introduced in the FE formulation which causes the checkerboard patternto have a higher stiness than a solid block of material (Diaz and Sigmund, 1995).

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Ω

(a) The design problem (b) Solution showing checkerboard behavior

Figure 6: Showcase of checkerboard pattern. Minimize compliance formulation with massconstraint of 50%.

The problem with checkerboard pattern can partially be avoided by using higher order niteelements in combination with a lower SIMP-penalization factor (Diaz and Sigmund, 1995),but the use of higher order elements drastically increase computation time. With TO runsalready being computationally demanding, the use of higher order nite elements is usuallyavoided.

In order to circumvent these problems dierent ltering techniques are used. First introducedby Sigmund (1994) is a sensitivity lter which modies the design sensitivity of a specicelement based on a weighted average of nearby elements. The method is purely heuristic buthas shown to produce good results with no major increase in computation time, while beingsimple and thus easy to implement. Another approach is a density slope algorithm introducedby Petersson and Sigmund (1998). The algorithm enforces a certain local gradient on theslope of element densities such that element densities of adjacent elements are not allowed tovary largely.

Since many of these techniques also deal with problems associated with the detail of theFE-mesh, they will be covered in detail later on in section 2.6.2.

2.6.2 Mesh-dependency

With an increasingly ne FE-mesh more thin structural members will exist in the optimalsolution. This behaviour can be seen in Figure 7 where two FE discretizations of 1800 and20000 quadrilateral elements respectively are used to solve the same problem as in Figure 6a.As clearly seen, more detail is emerging in the solution using the ner mesh.

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(a) Result using a 60×30 mesh (b) Result using a 200×100 mesh

Figure 7: Minimize compliance formulation with mass constraint of 50%. Showcasing thedierences between dierent mesh resolutions

Therefore, for an innite detailed mesh an innite number of structural members will exist.This is of course not a wanted behavior and instead, an optimal structure of the same shapeshould be produced regardless of the detail of the nite element model. The two lteringtechniques introduced in 2.6.1 both deal with this problem. The sensitivity lter modies thesensitivity of the objective function in (2.3) according to

∂f

∂xk= (xk)−1

1∑Ni=1Hi

N∑i=1

Hi xi∂f

∂xi, (2.9)

in which the weight factor Hi is equal to

Hi = rmin − dist(i, k), i ∈ N | dist(i, k) ≤ rmin.

A minimum member size dmin = rmin/2 is dened such that the formation of increasingly thinmembers is suppressed. The lter modied the sensitivity with respect to design variable xkbased on a weighted average of design variables associated with elements within the lteringradius rmin. Ωk denote the set of elements inside this radius with dist(i, k) being the distancebetween the center of element k and i. ∂f/∂xk is the updated sensitivity of element k, whichis used in the optimization instead of the original sensitivity ∂f/∂xk. It should be noted thatthe weight operator Hi is equal to zero outside the ltering radius as it decays linearly withthe distance between the elements.

While the sensitivity lter has shown to be very ecient, numerical diculties might arisewhen the method is used in conjunction with multiple constraints in the optimization problem.The quality of the constraint approximation might be heavily inuenced by the ltering andas such satisfying behavioural constraints might be dicult. (Zhou et al., 2001)

Zhou et al. (2001) also suggest a new algorithm - a modication of the previous mentioneddensity slope method. This algorithm is implemented in OptiStruct and works by constrainingthe lower bound of the element density by

xi ≥ max[ε, xj − (1− ε) dist(i, j)/rmin] (2.10)

where element j is the highest density element adjacent (inside the radius rmin) to element iat the previous iteration, as

xj = max(xk | k ∈ Ωi).

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Both these presented approaches for eliminating checkerboards and mesh dependency intro-duce a new problem: transition areas consisting of intermediate density elements appear atthe border of created structural members. The most common way to combat this behaviouris a several-step optimization scheme. For the modied density slope algorithm, a three-stepscheme is suggested:

Step I: Perform an initial optimization with density slope control according to (2.10)using a default SIMP-penalization parameter p

Step II: Once Step 1 has converged, increase the SIMP-penalty exponent to p = p + 1and run the optimization again

Step III: The density slope constraints on the boundary between void and solid elementsare relaxed so the solution can reach a black-and-white design

This method might cause slight checkerboard-pattern while solving three-dimensional prob-lems. The method is, regardless of this, used in commercial TO software with great results.

2.6.3 Local Minima

Local minima refer to the fact that the same optimization problem might produce dierentsolutions depending on initial optimization parameters such as x0, optimization settings, etc.This is due to the non-convexity introduced with the SIMP-penalization, such that minimafound in the optimization are stationary points and not necessarily global minima.

This problem can partially be avoided by using so-called continuation methods, where theinitial convex problem gradually is transformed to the non-convex problem. Sigmund andTorquato (1997) suggest when using a sensitivity lter, a ltering radius rmin that is startedat a large value and gradually decreased throughout the optimization. For the modied densityslope algorithm, the three-step optimization scheme described earlier reduces the problem.

2.7 Inuence of Body Forces

The introduction of body forces, i.e., gravitational, centrifugal or inertial forces, introducesnew diculties into the optimization problem. Since these forces are dependent on the massof specic elements, the resulting force will change throughout the optimization (Bruyneeland Duysinx, 2005). Only gravitational forces will be studied in this project, so the modiedFE-equilibrium equation reads (Lee et al., 2012):

K(x)u = f(x),

with the contributing element force fe(xe) being given by

fe(xe) = xeρeVeαg

where ρe is the element density, α is a load factor used to scale the gravitational accelerationg. By only considering homogeneous isotropic material throughout the design domain thedensity ρe can be written without element index, i.e. only ρ, giving the element force equation

fe(xe) = xeρVeαg. (2.11)

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A lower bound on the volume is needed when only gravitational forces are considered sincethe optimal solution will be the removal of almost all material. With element density ap-proaching the lower bound ε the ratio between the body force and the SIMP-penalized sti-ness approaches innity. This will lead to a solution that converges to a design with lots ofintermediate density elements (Holmberg et al., 2015). As such, a modication of the SIMP-penalization scheme has to be used. In what way this is done in OptiStruct is kept a companysecret, and thus can not be covered here.

To exemplify this, consider the problem seen in Figure 8a. Acting on the body is a gravitationalacceleration of 9810mm/s2 in negative z-direction. The compliance is minimized with amaximum allowed usable volume fraction of 25% of the design space volume and minimummember size control via the modied density slope algorithm. Visualized in Figure 8b is theproblem associated with a lack of lower volume constraint; the optimization simply removesthe majority of the material.

(a) Problem visualization. Gravity acting in nega-tive z-direction

(b) No minimum volume constraint

Figure 8: Gravity eects on the TO result

With an added lower bound of 20% on the volume constraint the optimized design seen inFigure 9a is found. OptiStruct experiences problems converging to a black-and-white solution,most likely due to the nature of the modied density slope algorithm. If a point mass with aweight equal to 25% of the total weight of the design domain is added, the solution in Figure9b is found. The optimization now opts to use the full allowed volume fraction of 25%.

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(a) Added minimum volume constraint of 20% (b) Added point mass to the top of the design do-main

Figure 9: Gravity eects on the TO results using a minimum lower volume constraints andpoint masses

2.8 The Inertia Relief Method

In order to perform static FE analysis on a body, the body is required to be in (quasi)-equilibrium. Traditionally, this is done using Single Point Constraints (SPC) which restricttranslational and rotational movement by applying reactional forces and moments in specicnodes, causing the body to be in static equilibrium.

Using normal SPC all externally applied forces will ultimately be reacted in the constrainednodes, with load paths between the external load and the SPC. In reality, these load paths donot always naturally exist. Rather, it is possible that in some cases the external forces often arereacted via local deformations and accelerations (Christensen et al., 2012). This makes normalSPCs not always applicable, and special techniques are needed when studying unconstrainedbodies subject to external loads such as aeroplanes and satellites. Stress concentrations dueto SPCs are removed when inertia relief is used, which is exemplied in Appendix B.

The Inertia Relief Method works by restricting rigid body motion by application of calculatedtranslational and rotational accelerations (Liao, 2011). With the applied inertia loads the bodyis found to be in a state of quasi-equilibrium and static analysis can be performed withoutthe creation of non-naturally existing load paths (Quinn, 2010).

To start, the inertia relief accelerations are calculated from the dynamic equilibrium equationwithout damping terms

Mu+Ku = f , (2.12)

which is an extension of the static equilibrium equation (2.2). M is the mass matrix and uthe nodal acceleration vector. The displacement vector u may be written as

u = ur + ud (2.13)

where ur is the rigid body displacement and ud the local deformation of the body. Combining(2.12) and (2.13) gives

Mur +Mud +Kud = f . (2.14)

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Inertia from local deformations may be neglected, so that (2.14) can be reduced to

Mur +Kud = f , (2.15)

where ur has to be eliminated in order to allow static FE-analysis to be carried out. The rigidbody displacement satises

Kur = 0

and is an eigenvector associated with the zero-eigenvalue of K. By collecting the m zero-eigenvectors in an orthonormal matrix R ∈ Rn×m it is possible to express the rigid bodydisplacements as

ur = Rc (2.16)

in which c(t) ∈ Rm. With R being independent of time and by combining (2.15) and (2.16):

MRc+Kud = f .

Pre-multiplication with RT gives

RTMRc+RTKud = RTf . (2.17)

The symmetry of K allows the second term on the left hand side of (2.17) to be re-written as

RTKud = (KR)Tud = 0, (2.18)

and (2.15) can thus be simplied as

RTMRc = RTf ,

from which c can be solved as

c = (RTMR)−1RTf . (2.19)

Substituting (2.19) in (2.16) gives the rigid body accelerations according to

ur = R(RTMR)−1RTf . (2.20)

Combining (2.15) and (2.20) gives the nal FE equilibrium equation as

Kud = f −MR(RTMR)−1RTf , (2.21)

which for simplicity is expressed as

Kud = f (2.22)

where f is the inertia relief force vector. The system still has to be constrained to removerigid body motion, i.e by locking six degrees of freedom for the three-dimensional case. Thiswill however, in contrast to SPCs, not introduce any stresses caused by the locked DOFs.

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2.8.1 Interaction with Topology Optimization

The major dierence when considering TO with inertia relief in contrast with SPC is theinuence of mass. The mass will have no eect on a optimization assuming no body forces,but the inertia associated with the mass will impact the result signicantly when inertia reliefis used.

Furthermore, body forces are not compatible with inertia relief analysis. The (for example)gravitational accelerations are simply nullied by the inertia relief acceleration acting in oppo-site direction. As such, a model analyzed using inertia relief under only gravitational loadingwill not be deformed or be subject to any stresses, other than what is caused by numericalerrors. Mathematically, the sensitives of the displacement and force vectors are dierent usinginertia relief; Pagaldipti and Shyy (2004) and the MSC Nastran (2012) Design Sensitivity andOptimization User's Guide cover this more thoroughly and it will as such not be discussedfurther here.

To visualize the eects of inertia relief in TO lets study an example. As a reference, consider theproblem in Figure 10a with a load acting on top of the design space as well as xed supports ineach corner, with the corresponding optimized topology in Figure 10b. Optimization settingsinclude a compliance minimization with a volume fraction constraint of 25% and minimummember size control via the modied density slope algorithm.

(a) Standard problem with SPCs (b) Solution of 10a

Figure 10: TO problem and solution using normal SPCs

Figure 11a shows the solution of a slight modication of the problem in Figure 10a; the SPCin Figure 10a are removed and replaced with inertia relief. In this case there are no load pathsto the bottom corners of the model and as such no material is distributed towards to corners.If instead masses are added to the bottom corner nodes the program can utilize the addedinertia by connecting the masses to the point where the external load is applied. The resultof this is seen in Figure 11b.

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(a) Automatic inertia relief control (b) Automatic inertia relief control with pointmasses added to each lower corner

Figure 11: Showcasing the eect of inertia relief on the TO result, both with and withoutadded point masses

2.9 Short on Aerodynamic Panel Methods and Aerodynamic Forces

Aerodynamic potential-ow methods, also called Panel methods, are methods to evaluate uidvelocity around an object. Knowing the uid velocity pressure coecients can be calculatedand subsequently the pressure distribution. No eort will be made throughout this report todescribe the theory behind these methods, but the interested reader can consult an introduc-tion made by Erickson (1990).

However, some aerodynamic relations which are used are worth mentioning. The relationshipbetween pressure coecient cp and the surrounding pressure can be described by

cp =P − P∞

q, (2.23)

where P is the pressure at the point where the pressure coecient is being evaluated, P∞ isthe far-away pressure and q is the dynamic pressure calculated by

q =1

2ρ∞v

2, (2.24)

given the density of the surrounding uid ρ∞ and the velocity v of the object relative to theuid.

Given specic ight conditions (such as cruise) the aeroplane is supposed to be in staticequilibrium. In this state forces and moments acting on the plane are zero with ruddersbeing used to balance the moments. The dominating moment, and the only moment takeninto account here is the pitch moment. Other ying-wing aeroplanes use a pitching so-called"beaver-tail" to balance this moment, but for simplicity the moment will be countered bymodelling aperons (a combination of aps and ailerons) at the trailing edge of the wing.

To nd the angle of attack α needed to achieve the required lift force a sweep is done over theangle of attack. For example, the aerodynamic problem is solved for an incremental increase

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of angle of attack from 0 to 10. From this the angle of attack needed to achieve the requiredlift force can be determined, and subsequently, the pressure distribution given the specicangle of attack.

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3 Method

The following chapter describes a suggested work ow when considering structural TO of inter-nal aeroplane structures. The suggested approach utilizes an aerodynamic solver to calculatepressure distributions over the wing given specic ight manoeuvres. An interpolation schemeis then created based on the pressure distribution such that the pressure can be mapped todierent FE-meshes easily. With the pressure mapped to the nite element mesh, classicalTO can be carried out.

3.1 Geometry Clean-up

Before the model can be meshed it has to be simplied so that the model can be meshed inan ecient way. It is desirable to have a mesh that is as homogeneous as possible, whichan original model might not produce. Problematic areas include duplicate geometry lines orgeometry lines very close to each other (Figure 12a), thin edges (Figure 12b), intersectingsurfaces (Figure 12c) or unnecessarily complex geometries (Figure 13).

(a) Duplicate lines (b) Thin edges (c) Intersecting surfaces

Figure 12: Visualization of common areas that needs to be xed before meshing

Geometry lines in HyperMesh determine the way the FE mesh is created. The FE meshwill follow all geometry lines but lines can be suppressed such that they are not taken intoconsideration when a mesh is created. Suppression of lines should not necessarily be restrictedto suppressing duplicate or adjacent lines as an excess of unnecessary lines will make it harderfor HyperMesh to create a good mesh.

Other problematic areas require more complex ways to x and are therefore more easily dealtwith by modifying the model in a CAD-software instead of in HyperMesh. It is possible tosimplify the model by clever use of functions found in HyperMesh, but it would be preferredto have a detailed ready-to-mesh model directly from the CAD-software.

Due to the absence of CAD-software, all geometry clean-up throughout the project has beencarried out in HyperMesh. The thin edge seen in Figure 12b have been given a thickness andthe intersecting surfaces in Figure 12c have been removed by slightly moving surfaces around.Also modied is the complex nose cone seen in Figure 13, where the "bubbles" have beenreplaced with a at surface.

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(a) Nose before geometry clean-up (b) Nose after geometry clean-up

Figure 13: Close up shot of the geometry clean-up done on the aeroplane nose. It is nowpossible to mesh the nose without incorporating lots of ill-conditioned elements

3.2 Meshing

While it is desired to have a homogeneous element size throughout mesh it is not alwayspossible. By geometry clean-up most areas can be modied such that a homogeneous meshcan be created, but modifying the geometry too much might result in information being lostand thus give a misleading result. Therefore, dierent mesh sizes can be necessary. For thismodel, this is done along the engine exhaust as seen in Figure 14a. Also note the addedthickness along the edge, which was added during the geometry clean-up.

Also, it might not be necessary to include the whole model in the design domain. Areas wherea very ne mesh is needed to keep element quality high are preferably removed from the designdomain. This is done along the trailing edge of the aeroplane model, as seen in Figure 14b,where the area in red is removed from the design domain and thus not included in the mesh.If previous optimization runs have shown large areas where no material is placed, the emptyareas can also be removed from the design domain beforehand to reduce computational time.

(a) Dierent mesh size around the engine exhaust (b) Trailing edge (in red) not being meshed

Figure 14: Dierent aspects of mesh creation

3.2.1 Meshing Approach

The general approach to meshing the FE-models has been with an outer layer of shell elementsand an internal solid mesh. The outer shell layers, consisting of QUAD4 and CST elements,

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serves as non-design space and creates the outer shape of the aeroplane. It is on this shell meshthat aerodynamic pressures will be applied. The internal space of the aeroplane is meshedwith tetrahedron elements. The shell and solid layers share FE-nodes, so forces applied onthe outer shell are transferred to the solid mesh.

Additionally, to help HyperMesh create a good tetrahedron mesh internal cavities can alsobe meshed with shell elements. The shell mesh is used to guide the creation of the internaltetrahedron mesh so specically in areas with complex geometries this is very favourable todo. Also, the tetrahedron mesh can be optimized for factors such as element size, aspect ratioand similar to give a better result.

It is possible that the optimization is heavily governed by the thickness of the outer shelllayer. If the outer shell layer is too thin, no load will be taken by the shell layer, and if it istoo thick, all load will be taken by the outer shell. How much load the shell takes directlytranslates to how much load the internal structure has to absorb, which directly will aectthe optimization result.

3.3 Component Modelling

Internal components such as engines, landing gears etc. are modelled as point masses con-nected to the FE-mesh. In cases where the components are far away from the mesh, rigidelements are created to connect the mass to the FE-mesh. The program described in section6.1 is used to speed up this process.

Flaperons are modelled with rigid elements connecting to a node in which forces are applied.The point where the aps are localized on the aeroplane has not yet been specied in thedesign process and therefore their position in the FE-mesh is somewhat ctional.

While in a real scenario the internal components would occupy volume, this is mostly notconsidered as it would make the model overly complicated as it would introduce more uncer-tainties and restrictions on the design space. For example, which attachment points to usewould have to be evaluated for each component. Some major components, such as the engineswith respective air intakes and landing gears have predened cavities.

3.4 Flight Manoeuvres and Load Cases

Several load cases have to be considered in the optimization to achieve a robust design. Ifonly a single load case is considered, a design that is well suited for the specic load case willbe found, but the design might perform poorly in other load cases.

The aerodynamic forces due to the dierent manoeuvres are calculated using an in-houseMatlab-code at Saab. In the program, an approximation of the ying wing geometry ismodelled, and aerodynamic panel methods are used to calculate output data such as pressurecoecient distribution, lift force and pitch moment. This pressure is then mapped to theFE-model via a Matlab-script, described in section 6.2.

Two load cases are used during the optimization. These are obtained from a steep pitch-up manoeuvre scenario and a landing scenario, both of which are described in the following

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sections. The reasoning behind choosing those two load cases is pretty simple; the aeroplaneis simply not designed to perform any advanced manoeuvres and because of this the landingand pitch-up manoeuvre load cases are among the most demanding. Further, the two loadcases will cause wing bending in dierent directions, upwards during the pitch-up manoeuvreand downwards during the landing.

3.4.1 Pitch-up Manoeuvre

During the pitch-up manoeuvre load case only aerodynamic forces are considered due to theincompatibility between the inertia relief analysis and the gravitational load. The pressuredistribution over the aeroplane can be seen in Figures 15 and 16.

Components in this load case are modelled as forces. The forces are supposed to act in aglobal z-direction directly towards the centre of the earth to simulate gravitational pull. Theaeroplane, however, is not perpendicular to the ground as it ies with an angle of attack toincrease lift, and therefore the forces are angled to be perpendicular to the ground. Flightdata used for the pitch-up manoeuvre case is a ight height of 3 km and a speed of 400 km/h.

Figure 15: Pressure distribution (in [Pa]) on the upper part of the body

Figure 16: Pressure distribution (in [Pa]) on the lower part of the body

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3.4.2 Landing

Landing simulations are normally time-dependent problems which are far to complicated tobe considered in the TO. To simulate landing, increased gravitational loads are used, i.e. theload factor α in (2.11) is larger than 1. Further, SPCs are used with no aerodynamic loadsbeing considered.

3.5 Boundary Conditions

Boundary conditions are set up for the two load cases in either of two ways:

Landing, normal SPCs: Constrained xyz-directions in landing gear positions

Pitch-up, Inertia Relief: Constrained nodes along the symmetry plane of the aeroplaneto avoid introducing non-symmetric loading

The SPC locations are chosen based on the location of components associated with the landinggears, and can be seen in Figure 17. While in a real landing the plane usually lands withthe two back landing gears touching the ground rst it is assumed for simplicity that alllanding gears touch the ground at the same time. Further, in a real scenario the plane willobviously roll forward while being able to move sideways, thus only really being constrainedin the z-direction. For simplicity this is not taken into consideration either and all DOF areconstrained.

Figure 17: SPC locations

For the inertia relief method to work all six degrees of freedom of the model have to beconstrained. It is also important not to introduce any non-symmetric boundary conditionswhich would introduce any non-realistic loads into the structure. Because of this, the inertiarelief reference points are chosen along the aeroplane symmetry plane as seen in Figure 18.

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Figure 18: Constrained nodes for the inertia relief analysis

Constraint number 1 in Figure 18 represents a constraint in x-, y- and z-directions, number2 is a constraint in y- and z-directions and number 3 in only y. This set-up of constraintswill remove all three translational DOF and all three rotational DOF of the system withoutintroducing any unwanted stresses or unsymmetrical behaviour.

3.6 The Optimization Problem, Constraints and Settings

The compliance minimization problem (2.4) with a volume fraction constraint is slightly mod-ied to account for the two load cases, the pitch-up manoeuvre and landing load cases. Themodied optimization problem reads

(TO)

minx∈Rm

C (x) =1

2

(w1f

T1 ud,1(x) + w2f

T2 (x)u2(x)

)s.t.

m∑e=1

Vexe ≤ V0

0 < ε ≤ xe ≤ 1, e = 1, . . . ,m,

(3.1)

where w1 and w2 are weight factors, u1 solves the inertia relief equilibrium equation

K1(x)ud,1(x) = f1,

and u2 solves the static FE equilibrium equation with body forces

K2(x)u2(x) = f2(x).

The index on the stiness matrices, the force and displacement vectors represent the twoload cases, with 1 representing the pitch-up manoeuvre case and 2 the landing load case.

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Constraints to reduce compression of the wing has also been tested, making the optimizationproblem read

(TO)

minx∈Rm

C (x) =1

2

(w1f

T1 ud,1(x) + w2f

T2 (x)u2(x)

)

s.t.

m∑e=1

Vexe ≤ V0

0 < ε ≤ xe ≤ 1, e = 1, . . . ,m

smin < so(x) < smax, o = 1, . . . , n,

(3.2)

in which so(x) is the strain in rod element o, smin and smax are bounds on minimum andmaximum allowed strain and n is the number of strain constraints. The reason why the rodelements are needed is discussed in 5.1 and how the constraints are implemented and appliedis covered more thoroughly in 6.3. Symmetry constraints on the design variable are also usedalong the symmetry plane of the aeroplane to achieve a symmetric design, more on this in 5.2.

Apart from objective function and constraints, there is an incredible amount of optimizationsettings in OptiStruct and it has not been feasible to try out every combination available. Thesettings most commonly used are:

Minimum Member Size dmin: Three times the average element size

Penalization factor p: 3. The variable in OptiStruct, DISCRETE, corresponds to p− 1

Initial density fraction: Equal to the allowed mass fraction

Minimum element density ε: 0.001

Optimization convergence tolerance: 0.005

Max allowed iterations: 200, which should never be reached assuming standard compli-ance minimization

3.7 Result Interpretation

Optimal designs acquired via TO are often complex and need to be manually rened. Thedesigns may contain geometries which are not manufacturable or geometries which would betoo expensive to produce. Also, some creativity from the designer might be needed wheninterpreting the TO result.

An acquired design from TO will, in cases considering compliance minimization, be a rathernon-mature design regarding e.g. stress evaluation and buckling. The design will have toundergo testing to make sure that it satises all design requirements, either via other formsof optimization (size or shape) or by static analyses.

When interpreting the optimization results in OptiStruct, it is possible to get dierent levelsof details depending on which element density threshold is used to illustrate which elementsto keep in a nal design, as shown in Figure 19.

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(a) Element densities >0.5 shown (b) Element densities >0.25 shown

(c) Element densities >0.1 shown (d) Element densities >0.01 shown

Figure 19: Dierent topologies depending on density threshold

Altair claim this is a feature in the software, allowing the user to 'choose his design'. Due to thepenalization elements with a low density will contribute with almost zero stiness to the naldesign, so it is uncertain why these elements are kept. The problem with kept intermediateelement densities is further discussed in 5.4.

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4 MODELS

4 Models

A number of simplied FE models have mostly been used throughout the project. The modelshave been used to test how the optimization behaves with dierent settings, including, butnot limited to: inertia relief, manufacturing constraints, dierent load cases and boundaryconditions, and optimization settings.

4.1 Wing Only

The main use of the wing model, seen in Figure 20, has been testing of compression constraintson the wing. The wing model consists of an outer shell layer mesh with 2350 elements, aninternal tetrahedron mesh of 9011 elements and 1166 rod elements connecting the top andbottom shell layer.

Figure 20: Wing-only model

The wing is subject to a pressure acting on the outer shell layer taken from the pitch-up loadcase. The pressure should, therefore, be the same as the pressure which would be acting onthe wing had it been attached to the aeroplane. Further, components in the wing are alsothe same as those that should be included in the full model. To simulate the wing connectionto the aeroplane the root of the wing is constrained with SPCs. Gravitational loads are alsoapplied to account for the landing load case. In short, the wing is subject to the same forcesas it would have been when connected to the aeroplane.

In some optimization runs, several or the majority of the rod elements are removed to give arepresentation of how the displacement constraints associated with these rod elements aectcomputational times. Some rod elements are kept along the thick line shown in Figure 21with remaining elements being equally distributed along the thick lines to get compressionconstraints evenly distributed along the wing.

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4 MODELS

Figure 21: Wing-only model, rod element positions along the thick lines and correspondingcompression constraints

4.2 Simplied Aeroplane, Outer Shell Mesh

The simplied aeroplane model in Figure 22 has mostly been used throughout the project.The idea is that this model can be used for proof-of-concept testing and when done, the meshdensity can be increased to achieve a better, more detailed result. The model features amesh that gives a good trade-o between accuracy and run times, making it great for testingdierent optimization settings.

The mesh consists of an outer shell layer of 12616 combined CST and QUAD4 elementsand internal tetrahedron mesh consisting of 73616 elements. Due to meshing diculties thefrontal nose and the tail section are not included. These areas include structures that are farto detailed to model using a rough FE-mesh. Including these areas into the FE-mesh wouldrequire ner elements and therefore the-the number of elements would be increased by tens ofthousands.

Figure 22: The simplied model used for proof-of-concept testing

The outer shell layer is subject to a pressure load calculated for the take-o-loading case. Thepressure loading is balanced by inertia relief with gravitational loads being considered in aseparate load case balanced by SPC. Components are modelled as point masses and therebytaken into consideration via the gravitational load.

28

4 MODELS

4.3 Simplied Aeroplane, Tetra-only Mesh

An even simpler model was used to test the inuence of the modelled components. The modelonly consists of an internal tetrahedron mesh and does not contain any non-design elementswhere pressure loading can not be applied in a good way and thus only the loading due tointernal components is considered. The model uses xed SPC in the positions of the landinggear, located at the bottom of the plane.

29

5 COMPLICATIONS

5 Complications

Some complications have been reoccurring during the project. These include problems re-lated to poor material placement in the outer wing of the aeroplane, i.e. no safety againstwing compression, non-symmetric results, neglected component loads, intermediate elementdensities and poor convergence. These problems are introduced in the following sections.

5.1 Wing Compression

A major problem throughout the project has been the lack of material in the outer part ofthe wing. This is troublesome as one of the main tasks of the airframe of the wing is to cancelbuckling of the outer skin and due to the absence of material in the outer wing, it is safe toassume that the structure will buckle.

The reason for the lack of material in the outer wing is thought to be due to small compressionof the wing relative to the bending of the whole wing. As such, the optimization favoursreducing the amount of wing ex rather than limiting the compression of the wing. Whilemathematically correct due to the global compliance dependency of the displacements, thebehaviour is not wanted.

Ideally, buckling could be taken into consideration in the optimization problem, but currentversions of OptiStruct does not support this for meshes containing solid elements. Instead,the amount of compression of the wing, i.e. how much the upper shell layer and the bottomshell layer are allowed to move towards each other, can be measured and constrained.

These constraints are implemented in problem 3.2 by creating one-dimensional rod elementsconnecting nodes on the top shell layer to the bottom shell layer of the wing. The strainof these elements can be used as a design response which in turn can be constrained. Forexample, constraining the strain of the rod elements to 1% allows the wing to compress 1%.The rod elements do not introduce any additional DOF to the system so the FE-equilibriumequation should take approximately the same time to solve as before. The assembly process ofthe stiness matrix will become slightly longer, but the increase in time should be negligible.However, this method introduces a lot of constraints to the optimization problem. Theseconstraints have to be evaluated, including evaluation of the sensitivities for each constraint,which will result in a massive increase of computational time.

The easiest way to avoid this massive increase in computational time is to introduce compres-sion constraints at specic points of the wing, instead of the whole wing. While this reducesthe time it takes to solve the problem, the constraints are very local, and several questionshave to be considered:

Where to place these constraints?

How many constraints should be included?

What distance is required between the constraints?

What percentage of compression is allowed?

30

5 COMPLICATIONS

Should all compression constraints be of the same magnitude, i.e. same values of smin

and smax throughout the wing?

How does the number of constraints needed correspond to the detail in the FE-mesh?

It has to be noted that the optimization algorithms used to solve TO problems are specicallydesigned to solve problems with a large amount of design variables and few constraints. Theincrease in computational time is therefore expected, and not much can be done about it.

5.2 Non-symmetric Results

It has been seen throughout the project that a symmetric model and symmetric load mightnot always produce a symmetric optimized design. The dierence in those cases is often notmajor and can often be neglected, but if a fully symmetric problem is solved a fully symmetricsolution is probably wanted. These small changes in the topology may be caused by numericalerrors in the FE-calculations which may grow as the optimization proceeds.

To counteract this symmetry planes can be dened on the design variables in OptiStruct. Byforcing the optimization to balance element densities on both sides of the symmetry plane, anoptimal, symmetric, design is found.

Symmetry constraints can also be used to reduce the complexity of the optimization problemby reducing the number of constraints and design variables. Figure 23 and 24 show an exampleof this. The model features constraints on the wing compression on the right wing seen inthe gures. The application of this constraint on one wing obviously causes a non-symmetricdesign, as Figure 23 shows.

Figure 23: Non-symmetric result from an optimization

However, if the symmetry constraint is added to the aeroplane symmetry plane the topologyin Figure 24 is acquired. So, instead of introducing tens or hundreds of additional constraints,the same eect can be achieved by symmetry constraints.

31

5 COMPLICATIONS

Figure 24: Same problem as in Figure 23, solved with an added symmetry constraint

It would be, given that the load cases used are symmetric, possible to only include half theaeroplane in the FE-mesh. While the load cases used throughout the project are symmetric,other real load cases are likely to be anti-symmetric, such as the loads caused by a turn.

5.3 Neglected Component Loads

Another recurring problem has been components "oating" in the design space, without anymaterial being distributed towards them to connect them to the shell layer. There are somepossible causes to this:

The loads associated with the components are small relative to the other loads causingthe optimization to focus on countering the pressure or gravitational loading

The loads generated by the components are so small that they are carried by the ε-densityelements

The rough mesh used does not allow the optimization to create small enough structuralmembers to carry the component load

It might be possible to avoid this problem by scaling up the forces due to the components tothe same magnitude as the external pressure, but it would not be a good representation ofreality. While not being optimal, it might be necessary to do so to get material distribution tothe components. A better approach would probably be a rened FE-mesh but with increasedrun times as a result.

5.4 Intermediate Element Densities

While intermediate element densities are to be expected at the boundary layer between solidand void elements due to the density slope algorithm, the optimization often favours largeareas of intermediate element densities and does not converge to a black-and-white solution.These elements will, after penalization, provide close to zero stiness to the model.

32

5 COMPLICATIONS

The reason for this behaviour is not clear. It might be caused by a conservative mass fractionconstraint leaving the optimization with too little material to account for all loading in themodel. Simple tests have however shown this not be the case. Another consideration couldbe a penalization factor that is too low, but tests have also been conducted which showno signicant dierence in the number of intermediate density elements with an increasedpenalization factor.

Another cure to the problem could be a stricter convergence tolerance. This has shown tohave some promising behaviour, but with increased computational time as a side eect. Theincrease of black-and-white structure is as best minimal whereas the amount of iterationsusually is at least double, so overall this approach is not eective either. The upper limit onnumber of iterations might also have to be increased to reach convergence with the strictertolerance.

5.5 Poor Convergence

Problems related to poor optimization convergence or failure to converge are, in this case,mostly connected to the use of compression constraints on the wing. The cause of this couldbe two-fold: too restrictive constraints will make it impossible for the optimization to nda feasible design and OptiStruct will run until a maximum number of iterations is reachedwithout converging. Changing the value of the constraint could be a simple x if it is allowedfrom a design point of view.

The other reason is related to a large number of constraints in the optimization problem. Op-tiStruct only evaluates a maximum number of constrains in each iteration. Which constraintsthat are to be evaluated is based on how much they violate the constraint or how close theyare to being active. Constraints which violate the boundary the most will be evaluated rst,with constraints being the closest to the constraint boundary will be evaluated second-hand.

By only evaluating a set amount of constraints each iteration while having to satisfy a largenumber of constraints, the optimization is forced to do a larger amount of iterations to evaluateand satisfy all constraints.

33

6 DEVELOPED PROGRAMS

6 Developed Programs

Three Matlab-programs have been developed during the project. The idea is that timespent doing monotonic, repetitive work is reduced to a minimum so time can be spent onmore productive things. The programs are developed mostly with regards to changes in theFE-mesh, such that changes in the mesh do not induce a large amount of manual modellingwork. Some of the things carried out by the programs, such as the pressure mapping, wouldnot even be possible to do manually.

The data input to OptiStruct is supplied in dierent les for ease of use. These les are asfollows:

1. A main-le including the FE-mesh with SPC and inertia relief boundary conditions

2. A component le containing nodes, masses and rigid elements associated with the com-ponents

3. A pressure le including the pressure loading on the outer shell

4. A compression constraint le, including rod elements and constraint denitions

The developed programs automate the creation of the components, pressure and compressionconstraint data-les.

6.1 Modelling of Point Masses

The rst program deals with modelling of the components described in section 3.3. Thesecomponents are modelled as concentrated masses connected to the FE-mesh. In some cases,these mass elements are connected via rigid elements. Required input to the program is nodecoordinates of the components with a respective weight of each component.

Also required is the FE-mesh in which the components are to be placed in. The program readsan OptiStruct (.fem) le, but without any major changes, it should be possible to modify theprogram to read data and write data for use with other FE-solvers. The output is written toa new OptiStruct le such that no changes are made to the original le.

The program determines the closest node in the FE-mesh to the node coordinates of everycomponent. If the closest node in the FE-mesh is within a certain distance of point mass nodecoordinates, a mass element is created in the calculated node.

If the distance to the closest node is outside the tolerance, a new FE-node is created. Theprogram then connects this newly created node to a specied number of nodes in the FE-meshvia rigid elements. These connector nodes are chosen as the nodes closest to the newly creatednode. Finally, a mass element is created in the new node.

6.2 Pressure Distribution to FE-mesh

In order to translate the pressure load from the in-house aerodynamic solver to the FE-meshanother Matlab-script was written.

34

6 DEVELOPED PROGRAMS

This script imports the pressure coecient distribution over the wing from the in-house aero-dynamic solver and calculates the pressure distribution over the wing using equations (2.23)and (2.24). The air density ρ∞ and the surrounding airstream pressure P∞ is determined viaan atmospheric in-house-tool with the velocity v being known from the used load case.

The program creates an interpolation scheme based on x and y-coordinates from the in-houseaerodynamic solver. The outer shell mesh from the FE-model is also imported from which thecentroid coordinates of each element can be calculated. These element coordinates are usedto calculate the element specic pressure via the interpolation scheme.

Output is written to an OptiStruct le in the same way as the program used to model compo-nents. It should be noted that the interpolation scheme may introduces errors, which mightgive a slightly dierent pressure distribution over the FE-mesh compared to the pressure fromthe aerodynamic solver.

6.3 Compression Constraints

The third program is developed to handle the creating of displacement constraints on thewing according to (3.2). The program reads node data of the top and bottom shell layer andidenties node pairs consisting of one node from the top layer and one from the bottom layerthat have the most similar xy-coordinates. A rod element is then created between these twonodes.

The program also writes necessary code to handle the constraint on the allowed compressionof the wing. The program works in the same way as the other programs in the way that a newOptiStruct is created, which can be used separately. Also, slight modication of the originalmodel le has to be made in order for the optimization to consider the constraint.

35

7 RESULTS

7 Results

The following section contains results from a wide range of optimization runs on the previouslymentioned models. The sections aim to visualize the dierences caused by inertia relief,manufacturing constraints, dierent load cases and boundary conditions, and optimizationsettings. Optimization settings used are, unless otherwise specied, those found in section3.6. The weight factors in (3.1) and (3.2) are set to 1. Material used is strictly isotropic withan Young's modulus of E = 71000 MPa and a Poisson's ratio of ν = 0.3.

7.1 Tetrahedron Model

The tetrahedron optimization result, featuring a compliance minimization as in (2.4) with amass fraction constraint of 15% of the full model volume, can be seen in Figure 25. The modelfeatures a gravitational loading into the picture, resulting in forces due to the components andthe solid mesh, with SPCs being used to balance these forces. The optimization behaves asexpected, placing material from the SPC to the simulated components.

Figure 25: Optimized topology of the tetrahedron model

7.2 Wing Only Model

The following optimization runs aim to show the dierence on the wing topology dependingon the compression constraints. The optimization objective for all models in Figure 26 hasbeen a compliance minimization with a volume fraction constraint of 15% of the full modelvolume and compression constraints applied between all nodes on the top and bottom shelllayer, totalling 1166 constraints. Data from the optimization runs is summarized in Table 1.Note that the optimization run with a compression constraint of 0.175% does not converge.

36

7 RESULTS

(a) No compression constraint (b) 0.175% compression allowed

(c) 0.225% compression allowed (d) 0.25% compression allowed

Figure 26: Visualization of the inuence of compression constraints on the whole wing

Compression constraint Compliance Run time Iterations Time/iteration

None 2.65× 106 00:00:27 43 0.63 s0.250% 2.61× 106 00:00:58 47 1.23 s0.225% 3.06× 106 00:27:39 160 10.4 s0.175% 4.30× 106 02:17:02 200 (Max) 41.1 s

Table 1: Optimization data, full wing constraints

The topologies seen in Figure 27 are optimization results featuring 26 displacement constraintsequally distributed along the thick lines, as seen in Figure 21. The optimization runs are a

37

7 RESULTS

compliance minimization with a volume fraction constraint of 15 %. A summation of opti-mization data, such as run time and compliance is found in Table 2.

(a) 0.05% compression allowed (b) 0.1% compression allowed

(c) 0.25% compression allowed (d) 0.5% compression allowed

Figure 27: Visualization of the inuence of compression constraint. 26 constraints usedthroughout the wing

Compression constraint Compliance Run time Iterations Time/iteration

0.01% 4.33× 106 00:21:21 60 21.4 s0.05% 4.66× 106 00:12:02 38 19.0 s0.10% 3.75× 106 00:07:47 33 14.1 s0.25% 3.73× 106 00:04:26 36 7.40 s

Table 2: Optimization data using 26 compression constraints throughout the wing

38

7 RESULTS

7.3 Simplied Model

The following optimization runs on the simplied aeroplane model are carried out using slightmodications of the problem formulations in (3.1) and (3.2). What diers is the choice ofboundary conditions, where some models feature only SPCs, some only inertia relief andsome a combination of both. Also, topologies from optimization run utilizing compressionconstraints on the whole wing or only in specic points are presented.

7.3.1 SPCs only

In order to exemplify the dierence between an inertia relief solution and one using only SPCs,the simplied model was optimized using only SPCs. The result is seen in Figure 28, using ashell thickness of 2mm.

Figure 28: Simplied model using SPCs, no displacement constraint

7.3.2 Inertia Relief only

When only considering inertia relief the gravitational load has the be removed due to the in-compatibility between the two. Figure 29 show the optimized topology when only consideringthe aerodynamic loads with inertia relief boundary conditions. Shell thickness is 2mm.

Figure 29: Simplied model using only inertia relief

39

7 RESULTS

7.3.3 Combined Inertia Relief with SPCs and Shell Thickness Inuence

Tests were carried out on the simplied model, without compression constraints, to test thedierences caused by the thickness of the outer shell. Refer to Appendix C for pictures of thedierent topologies. In Figure 30 the optimized topology using a shell thickness of 2 mm ispresented.

Figure 30: Simplied model, no displacement constraint, 2mm shell thickness

7.3.4 Full Wing Compression Constraints

Introducing compression constraints on the whole wing produce a result that can be seen inFigure 31. The optimization utilizes inertia relief to counteract the aerodynamic load caseand SPCs to balance the gravitational load.

Figure 31: Simplied model with displacement constraint on the whole wing

7.3.5 Compression Constraints only in Specied Points

Below in Figure 32 is a close up view of the wing from an optimization run of the simpliedmodel using local compression constraints. Notice the "islands" of material in the outerpart of the wing. These islands appear where the compression constraints are applied. Theoptimization creates a zone of material connecting the top and bottom of the wing in a verylocal fashion to satisfy the constraint.

40

7 RESULTS

Figure 32: Close up of the wing on the simplied model, with displacement constraint inonly specic points

41

8 DISCUSSION

8 Discussion

This chapter aims to discuss the dierent obtained results, encountered complications andobservations that have been made. Also discussed are ideas that could be expanded to furtherprojects.

8.1 Topology Optimization and Inertia Relief

The coupling between TO and Inertia relief is a rather unexplored subject. Some papers havestudied the interaction but often without providing mathematical or theoretical proof of howthe inertia relief analysis is carried out and how it aects the optimization result.

Resulting designs from TO may be unintuitive, even more so when inertia relief is used. Ithas been seen that many optimization runs using inertia relief place a patch of material in thewing tip, seen in for example Figure 31. The reason behind this is not clear, but it is coupledwith the inertia relief analysis, as the material is not placed in the wing tip when only SPCsare used.

When comparing optimization results between inertia relief and SPC some similarities can beseen, mostly in what could resemble a wing spar. The optimization using inertia relief placedpatches of material at the leading edge of the wing and the wing tip, although the amount ofmaterial placed at the wing tip signicantly reduced with an increased shell thickness, as seenin Appendix C.

The inertia relief only solution in Figure 29 is also interesting as it shows material placementalong the leading edge of the aeroplane. The pressure at the leading edge is higher than overthe rest of the wing, which might be the cause of the material placement. However, somematerial is placed at the root of the wing, most likely to reduce the bending of the wing.

While SPCs produce more intuitive and simpler designs, the use of inertia relief to obtainequilibrium is the correct way to constrain the model. Further investigation on how inertiarelief couples with large-scale TO may be warranted.

8.2 Lack of Material Placement in the Outer Wing

The lack of material placement in the wing (other than the wing tip) has been troublesomethroughout the project, and it is safe to assume that the outer layer of the wing will bucklewhen no material is distributed throughout the outer part of the wing. The optimizationfavours a signicant amount of material placement at the root of the wing to reduce thebending of the wing while allowing the wing to compress freely.

While this behaviour seems to be correct due to compliance being a global measure based onglobal displacements, some constraint was needed to simulate a buckling constraint, whichcurrently is not available for solid element meshes in OptiStruct. In some way, this makessense, as models which are made up of solid elements prone to buckle most likely could bemodelled using shell or beam elements instead, which are elements that OptiStruct supportbuckling responses for.

42

8 DISCUSSION

However, the models used throughout the project consist of a mix of solid and shell elementswhere the solid elements are supposed to counteract the buckling, not the shells themselves.While one could think that it would be possible only to introduce buckling constraints on theouter shells, it is not feasible either. OptiStruct just refuses to run optimization with bucklingconstraints when solid elements are present in the FE-mesh.

Attempts to counteract this behaviour by introducing constraints on how much the wing isallowed to compress has in some ways been successful, but the method has large drawbackswhen it comes to computational time. Results from optimization runs featuring this constraint,summarized in section 7.2, has lead to some interesting observations:

As expected, introducing additional constraints will lead to a worse result in terms ofcompliance as the compliance increases with more restricting constraints

No compression constraint and a not-so restricting constraint will provide the same topol-ogy and approximately the same compliance, although the optimization with constraintswill take longer time to solve

When many constraints are active the optimization run time increases incredibly fast,both with regards to an increase in number of iterations and time spent per iteration

It is hard to nd a suitable magnitude of the constraint so that the optimization stillcan nd a topology that satises all constraints

The use of compression constraint seem to create very localized zones of material onlyconnecting the top and bottom shell meshes

The very localized placement of material in the wing makes sense as it is required to satisfythe compression constraints, but it does not lead to a continuous design starting at the wingroot extending out towards the end of the wing as it does in almost every aeroplane today.This would require the outer shell to take a lot of the bending load of the wing, which maybe feasible if the shell is thick enough.

The impact the thickness of the outer shell has on the nal design is interesting. Having athick outer shell causes the shell to take a signicant amount of load usually taken by theinternal structure, while a thin shell makes the internal structure take almost all load. Inreality, the outer shell thickness of an aeroplane varies throughout the wing and body withthe thickness usually being determined rather late in the design process. This causes problemswhen TO of the internal structure is to be done as the thickness of the outer shell is coupledwith the amount of internal structure needed. Utilizing a constant shell thickness, as it hasbeen used in this project, will most likely lead to a misguiding design.

It would be possible to introduce thickness optimization of the outer shell into the optimizationproblem as well. Some simple attempts have been made regarding this, but it most often resultsin the optimization utilizing the full allowed thickness of the shell, as it is the most ecientway to reduce the bending of the wing. Thickness optimization of the shell would probablybe more ecient in a later stage of the design process when the airframe has been designed.

Further, isotropic material has been used almost exclusively throughout the project. In anewer military aeroplane, the outer shell is mostly made up of composite material with areassuch as the leading edges and nose cone being reinforced by stronger materials. It is thoughtthat these decisions also are taken into consideration at a slightly later stage in the design

43

8 DISCUSSION

process, but introducing a simple composite lay-up of the outer shell is not all too complicatedand might provide a more accurate result. This composite lay-up could also be coupled witha thickness optimization of the individual plies, but while interesting, would be far too earlyin the design process to carry through.

It could also be discussed if all compression constraints should be of the same magnitude.Due to the constant thickness of the wing and higher pressure at the leading edge, it can beseen in Figure 26b and 26c that the optimization favours material placement at the leadingedge when compression constraints are considered. To get a homogeneous material placementthroughout the wing, it might be required to allow more wing compression at the leading edge,or less throughout the rest of the wing.

8.3 Other Optimization Formulations and Model Uncertainties

Another approach that could have been taken to the optimization problem would have beenmass minimization. Minimizing the mass of the aeroplane makes sense from a design pointof view as a low mass leads to increased performance. By minimizing the mass problemsassociated with infeasible designs caused by strict compression constraints would be eliminatedbut, however, additional constraints to limit the amount of allowed wing bending would haveto be introduced via either displacement or compliance constraints.

It should also be noted that the two load cases which are used most likely are too few to give areal robust design. Including more aerodynamic load cases, e.g., unsymmetrical loads causedby a turn, in the optimization would lead to a more robust design.

Further, the traditional way of designating space for fuel tanks in an aeroplane is by lling allremaining void space in the aeroplane with fuel. The position of the fuel will have an eecton the TO result while the position of the fuel will be governed by the TO. This creates adependency that is hard to account for.

8.4 Ways to Guide the Optimization

It could also be possible to provide guidance to the optimization by including non-design wingspars in the FE-model. These spars would counteract the global bending of the wing suchthat the optimization might create wing-rib-like structures based on the local deformation ofthe wing. While interesting results might be acquired, the choice of number of wing spars,their position, length and shape would signicantly impact the optimization result.

It could be argued that the wing spar positions could be determined via an initial TO, butthe goal of this thesis has been to evaluate the possibility to utilize TO to design an entireairframe, not only the wing spar positions and the option has thus not been explored further.

Another approach that could be tested would be the introduction of an arbitrary compressionload of the wing load case. By introducing the arbitrary load case, the optimization wouldbe forced to take the local deformation of the wing into account, which may produce a resultwith material throughout the outer part of the wing. This has not been tested thoroughly,but a further investigation could be warranted. While obvious, the resulting topology wouldnot be based on the real loading of the wing so that the result may be slightly misleading.

44

9 CONCLUSIONS

9 Conclusions

It can be concluded that TO is not mature enough to produce a conceptual airframe design ina one-step optimization scheme. The optimization favours the reduction of the global displace-ment of the wing and neglects local deformations due to the global displacement dependencyof the compliance formulation. Also, problems associated with major dimensioning factorssuch as buckling can not be taken into consideration in the optimization.

Combining inertia relief and TO has shown some interesting result Results obtained using in-ertia relief are more unintuitive than respective results using only SPCs, which is no surprise.While the use of inertia relief is not thoroughly tested and understood, it is the correct theo-retical way to constrain the rigid motion of the aeroplane as it ies in non-static-equilibrium.

Buckling constraints have simulated by introducing local compression constraints on the outerpart of the wing. These constraints show some promising results when considering large-scaleproblems, but are not very useful due to a massive increase in computational time. The designis heavily governed by the position of these constraints, and freedom is taken away from theoptimization and placed in the hand of the designer once again.

In general, there are many things to take into consideration that will aect how the optimiza-tion behaves. These include the position of internal components, the thickness of the outershell and which load cases to consider. A multiple-step optimization scheme including sizeand shape optimization might be more applicable but would also require the supervision ofan experienced airframe designer.

45

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REFERENCES

Zhou, M., Shyy, Y., and Thomas, H. (2001). Checkerboard and minimum member size controlin topology optimization. Structural and Multidisciplinary Optimization, 21(2):152158.

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A SENSITIVITY ANALYSIS

A Sensitivity Analysis

Due to the nested formulation an arbitrary function can be written as gj(x) = gj(x,u(x)).Dierentiating this with respect to xe and utilizing the chain and product rule the derivativereads

∂gj(x)

∂xe=∂gj(x,u(x))

∂xe+∂gj(x,u(x))

∂u(x)

∂u(x)

∂xe. (A.1)

The derivative ∂u(x)/∂xe is found by dierentiation of the state equation (2.2):

∂K(x)

∂xeu(x) +K(x)

∂u(x)

∂xe=∂f(x)

∂xe,

which can be rewritten as

∂u(x)

∂xe= K−1(x)

[∂f(x)

∂xe− ∂K(x)

∂xeu(x)

]. (A.2)

By combining (A.1) and (A.2)

∂gj(x)

∂xe=∂gj(x,u(x))

∂xe+∂gj(x,u(x))

∂u(x)K−1(x)

[∂f(x)

∂xe− ∂K(x)

∂xeu(x)

], (A.3)

and introducing the adjoint vector λj as

λTj =

∂gj(x,u(x))

∂u(x)K−1(x)

the adjoint problem formulation is found as

∂gj(x)

∂xe=∂gj(x,u(x))

∂xe+ λT

j

[∂f(x)

∂xe− ∂K(x)

∂xeu(x)

]. (A.4)

Equation (A.3) is referred to as a direct method while (A.4) is referred to as an adjointmethod. The adjoint method is preferred when the optimization problem consists of moredesign variables than constraints, and as such the direct method is preferred when the numberof constraints is larger than the number of design variables. Since the number of designvariables almost always is greater than the number of constraint in in structural TO, adjointmethods are naturally used.

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B INERTIA RELIEF EXAMPLE

B Inertia Relief Example

Let us consider the paper plane model seen in Figure B.1. The model consists of an FE-shellmesh subject to a local force acting in the rear to simulate engine thrust. In order to performstatic FE analysis of the model it has to be constrained in order to avoid rigid body motion,which is done in either of two ways:

Normal SPC in the nose

Automatic inertia relief

Static FE-analysis is carried out on the two models, with resulting stress plots seen in FigureB.2 and B.3, using a normal SPC and inertia relief, respectively. Notice the stress concen-tration at the SPC location in the rst stress plot. If the plane would be ying this stressconcentration would not be present, it is purely an artifact of the SPC. The inertia relief so-lution in Figure B.3 is a more accurate description of reality. While not shown in the picture,the stress where the load is applied is the same in both models.

Figure B.1: Original problem using SPCs

Figure B.2: Resulting stress using SPCs

50

B INERTIA RELIEF EXAMPLE

Figure B.3: Resulting stress using inertia relief

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C INFLUENCE OF SHELL THICKNESS

C Inuence of Shell Thickness

Figure C.4: Shell thickness of 0.1mm

Figure C.5: Shell thickness of 0.5mm

Figure C.6: Shell thickness of 2mm

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C INFLUENCE OF SHELL THICKNESS




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topology optimization as a conceptual tool for designing ... · the inertia relief method utilized...

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