Topology Optimization using a Topology Description Function Approach

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Topology Optimization using a Topology

Description Function Approach

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 13 december 2005 om 13.00 uur door

Marten Jan DE RUITER,

wiskundig ingenieur, master of technological design, geboren te Bunschoten.

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Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. F. van Keulen.

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. ir. F. van Keulen, Technische Universiteit Delft, promotor Prof. dr. ir. M. J. L. van Tooren, Technische Universiteit Delft

Prof. dr. Z. G¨urdal, Technische Universiteit Delft Prof. dr. V. V. Toropov, University of Bradford Prof. Dr.-Ing. habil. F. J. Barthold, University of Bradford Prof. dr. ir. G. Lodewijks, Technische Universiteit Delft

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4.3 Update strategies . . . 77 4.3.1 Accumulating strategy . . . 78 4.3.2 Resetting strategy . . . 79 4.3.3 Halving strategy . . . 80 4.3.4 Generalization . . . 80 4.4 Numerical examples . . . 80 4.4.1 MBB beam example . . . 81 4.4.2 Crane example . . . 91 4.5 Conclusions . . . 99 5 Design sensitivities 103 5.1 Design sensitivities . . . 103

5.2 Computation of design sensitivities . . . 104

5.3 Existence of design sensitivities . . . 105

5.4 Sensitivity of the isoline with respect to the width . . . 108

5.5 Different methods of computation . . . 109

5.5.1 Partially filled elements . . . 111

5.5.2 Relocation of nodes . . . 112

5.6 Quality of the sensitivities . . . 113

5.6.1 Strip example . . . 113

5.6.2 Cantilever beam example . . . 115

6 Topological derivative 121 6.1 Conventional and topological sensitivities . . . 122

6.2 Computation of topological derivatives . . . 123

6.3 Creation of a hole in the TDF approach . . . 125

6.4 Uniformly stressed disk with a hole . . . 126

6.4.1 Analytical evaluation . . . 126

6.4.2 Numerical evaluation . . . 128

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7 Gradient-based optimization methods 133

7.1 Gradient-based optimization methods . . . 133

7.2 Steepest descent . . . 134

7.3 Method of moving asymptotes . . . 136

7.4 Multipoint approximation method . . . 138

7.5 Optimality criteria method . . . 139

7.5.1 Traditional optimality criteria method . . . 139

7.5.2 Modified optimality criteria method . . . 143

7.6 Numerical results . . . 147

7.6.1 Steepest descent . . . 150

7.6.2 Method of moving asymptotes . . . 151

7.6.4 Modified optimality criteria method . . . 157

7.6.5 Work in progress . . . 160

8 Conclusions and recommendations 171 8.1 Recommendations . . . 172

Appendices

A Level set methods 175 B Sensitivities 179 B.1 Bubble method and topological derivative . . . 179

B.2 Relation to Sigmunds’ article . . . 179

B.3 Topological derivative in a uniformly stressed infinite plate . . . 180

C Miscellaneous 183 C.1 Definition of compliance . . . 183

D Glossary 185

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List of symbols

In this thesis, a vector will be denoted by a lowercase bold character, e.g. u, and a matrix as an uppercase bold character, e.g. K.

·0 _{the derivative / design sensitivity of ·}

·i the i-th component of a vector ·

·,i the derivative / design sensitivity of · with respect to the i-th

com-ponent / design variable

¯· the approximation of a function ·

d·

dx the total derivative of · with respect to x ∂·

∂x the partial derivative of · with respect to x

∇· the gradient of ·

·i _{the value of · at the i-th iteration/step}

∆· the (finite) change of ·

η a step size parameter used in the optimality criteria method θ the angle in the polar coordinate system

λ a Lagrange multiplier ν Poisson’s ratio

φ a level set function / density function χ a characteristic function

ρ the (normalized) density

ω the (infinitely small) hole that is removed from a structure Ω the reference domain

ai coefficients

a coefficients vector, adjoint vector

A the subset of the reference domain consisting of material A? _{the optimal subset A}

A the area of the structure c the compliance fT_u

d the typical distance between the centers of the basis functions D· the topological derivative of ·

Dk· the topological derivative of · at element k

E Young’s modulus

E0 Young’s modulus of the solid in A

f the (nodal) force vector f the objective function

g the constraint function (g < 1)

hi a design variable, the ‘amplitude’ of the i-th basis function

hs_i a basis function parametrized by s_i

K the stiffness matrix

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Ke an element stiffness matrix

` the location where the TDF cuts the cut-off level m the fraction of material used

M the mesh size

N the number of design variables Nel the number of finite elements

Nbf the number of basis functions

p a penalty

r the distance to a point from a center (either the origin of the coor-dinate system or the center of a basis function)

R the (true) response function ¯

R the model response function s a design variable

si the vector of design variables xi, yi, hi, wi of the i-th basis function

T the topology description function u the (nodal) displacement vector

wi a design variable, the ‘width’ of the i-th basis function

W the total weight in weighted sums/integrals

xi, yi the design variables, the ‘center’ of the i-th basis function; xi

some-times also serves as ‘any’ design variable

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Chapter 1 Introduction

1.1 Topology optimization

Structural optimization is the process of searching better designs of structures. The aim can be to find the optimal design, or an improved design or just a hard-to-find feasible design. This makes structural optimization a field of engineering which is very important.

Topology optimization more specifically deals with the search for the best topology of a structure. Here, topology refers to layout and connectivity of a design. This makes it more general than shape optimization, where the topology is fixed, and size optimization, where both topology and shape are fixed, see Figure 1.1.

(a) Size optimization (b) Shape optimization (c) Topology optimization

Figure 1.1: Several classes of structural optimization, of which sizing optimization is the most and topology optimization the least constrained. For sizing optimization, the shape of a design is given and only the sizes of the design have to be tuned for optimal performance. Shape optimization additionally allows the shape to be modified, and topology optimization extends that with the connectivity.

The present section elaborates on the topology optimization problem and its rele-vance. The problem that is adressed in the present thesis is formulated in Section 1.2. The topological description function (TDF) is presented as a solution to this prob-lem. The literature that inspired the TDF is presented in Section 1.3, and the TDF approach is explained. Finally, the TDF approach is placed in the context of topology

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2 Chapter 1. Introduction

(a) Discrete topology optimization (b) Continuous topology optimization

Figure 1.2: The two kinds of topology optimization are continuous and discrete topology optimization. In discrete topology optimization, the structure consists of discrete members, e.g. trusses. In continuous topology optimization, the structure consists of a continuum.

optimization in Section 1.4. At that point, an overview of the thesis can be given. A short summary of the chapters is given, and their interdependence clarified.

Following the review of Eschenauer and Olhoff (2001), there are two kinds of topol-ogy optimization, namely discrete and continuous topoltopol-ogy optimization. Discrete topology optimization refers to optimization of structures with discrete members, e.g. truss structures, where the number of trusses, their connectivity and sizes must be determined. Continuous topology optimization refers to the optimization of the shape and connectivity of the material, see Figure 1.2.

This thesis focuses on continuous topology optimization. For extensive references, the reader is referred to the books of Bendsøe (1995), Bendsøe and Sigmund (2003) and the review of Eschenauer and Olhoff (2001). Furthermore, a small review on level sets is given in Appendix A of the present thesis.

In mathematics, the topology of a domain refers to the connectivity of that do-main. Two domains are topologically equivalent if the domains can be mapped to each other, i.e. if a homeomorphism between the two domains exists. From an engi-neering perspective, imagine that the domains are made out of very flexible rubber. If one domain can be deformed into the other without cutting or gluing, then they are topologically equivalent. The usual example is that a donut and a coffee cup (with ear) are topologically equivalent, because by flattening the cup and rounding the ear the cup can be transformed into a donut, and vice-versa.

The knowledge of the topology of a structure is not sufficient to describe its me-chanical properties. Among other things, knowledge of its shape and size is required. Hence, it must be clear that in the context of structural optimization, continuous topology optimization also involves the optimization of the shape and size of a struc-ture, and is not restricted to the topology on its own. The term topology optimization is used to distinguish it from fixed-topology shape optimization. As topology opti-mization is more general than shape optiopti-mization, it is also referred to as generalized shape optimization. Another term that appears in literature is layout optimization. In earlier references these terms were used distinctively (see e.g. Kirsch 1992, Rozvany, Zhou, Birker, and Sigmund 1992), but later on the terms are used interchangeably (Eschenauer and Olhoff 2001).

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1.1. Topology optimization 3

Ω

Reference domain

A

sub-domain

Figure 1.3: The sub-domain A in the reference domain Ω. The aim of topology optimization is to find the best material layout.

A geometry of a structure can also be regarded as how the material is distributed in space. For a different geometry, the material is distributed differently. E.g. for a square tube the material is located differently in space than for a round one. Looking to it the other way around, it can be said that the distribution of the material deter-mines the geometry. Thus, the topology optimization problem can also be viewed as determining how to distribute the material over space, see Figure 1.3. This prompts the term material distribution problem, and is the basis for the topology optimization problem definition as stated by Haber and Bendsøe (1998):

Let Ω be the given candidate design region (or reference domain) in which the structure should “live”, and let A ⊂ Ω. Then the characteristic func-tion corresponding to A is χA: Ω → {0, 1} such that χA(x) = 1 ∀x ∈ A

and χA(x) = 0 otherwise. In general, topology design involves finding

the solid sub-domain A? _{⊂ Ω, or equivalently χ}

A? that solves a given optimization problem.

Notice that the reference domain Ω is assumed to be given, and that the range of the characteristic function, or material indicator function, χA is discrete.

The purpose of this thesis is to describe the results of the research on a specific way to describe the geometry of A, and a way to perform topology optimization using this description. In real life, a structure usually involves multiple materials. However, in this thesis the research is limited to a single material. The methods presented here for a single material can be extended to multiple materials in the same manner as Wang and Wang (2003a)1 _{have done.}

1.1.1 Practical applications

A few examples will be given in order to show the potential of topology optimization. The first example is a support beam used in an aircraft that is produced by Messerschmidt-B¨olkow-Blohm GmbH, see Figure 1.4. The beam supports the floor

1

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in the fuselage of an Airbus passenger carrier. The example stems from Olhoff, Bendsøe, and Rasmussen (1991), and is known literature as the MBB beam problem. In 1994, Haber, Jog, and Bendsøe (1994) treated the example for different levels of complexity. Here, the objective is to minimize the compliance, where 40% of the support beam may contain material.

PSfrag replacements

F = 20 kN

(a) problem layout

(b) compliance = 0.1285 (c) compliance = 0.1300

(d) compliance = 0.1312 (e) compliance = 0.1317

Figure 1.4: The MBB beam problem. Structural solutions with different complexi-ties. Images courtesy of Haber, Jog, and Bendsøe (1994).

The following example is the development of a small satellite structure, see Fig-ure 1.5. The example is due to Sigmund (2000). To observe gamma ray bursts, a satellite is developed that must detect a burst, and turn its telescope to the burst as fast as possible. For that purpose, the satellite is equipped with four wide-angle cameras that scan all directions. Once a burst is detected, a telescope is aimed at the source. Also the instruments for control and communications, batteries and solar panels must be included. Now the question is to design a frame that is able to carry those instruments even during launch, while its weight is below 12 kg, and has no eigenfrequencies below 35 Hz. Furthermore, it must be possible to attach two hooks for ground handling. Topology optimization came up with a potential design.

Currently, a lot of attention is given to Micro Electro Mechanical Systems (MEMS, see e.g. Gad-El-Hak 2001). These are microscopical devices that are coupled with electronic circuits. Their small size is clearly an advantage. Examples are acceler-ation meters used in airbag systems in cars, or flow measuring devices. MEMS are manufactured in a way similar to microchips, and can be mass-produced. As another example, techniques are developed to store information by moving atoms from one position to another on a microchip. In this way, an information storage device can be

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Figure 1.5: Design of a small satellite. (a) Design domain and instrumentation; (b) Topology-optimized support structure; (c) Support structure with instrumenta-tion. Example courtesy of Sigmund (2000).

created, similar to the current hard disk, only with much denser storage. To manip-ulate the atoms, a microscopic pickup must be moved by some miniature robot arm. In Jonsmann, Sigmund, and Bouwstra (1999), the objective is to construct a Micro Electro Mechanical System (MEMS) that can serve as such an arm. The thermal ex-pansion is used to obtain the desired displacement. The required temperature change is obtained by sending an electrical current through the device, where the electrical resistance causes the heating. In Figure 1.6 the resulting design is depicted.

1.1.2 Ill-posedness of the problem

As described previously, topology optimization aims at finding the optimal layout. Unfortunately, the topology optimization problem is not always well-posed, see e.g. Cheng and Olhoff (1981) and Kohn and Strang (1986). There are, for instance, cases of compliance optimization where an optimal design does not exist. In those cases, a design can always be improved by a design with more detail. This can for instance be observed in Figure 1.4, where the design depicted in Subfigure (b) performs the best, since it is able to represent more detail, i.e. finer members, compared to the other designs.

This so-called ill-posedness can be overcome by modifying the optimization prob-lem (Haber and Bendsøe 1998). The probprob-lem can be relaxed or regularized by allevi-ating the discreteness of the characteristic function χA. This can be done by replacing

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Figure 1.6: Design of a MEMS. The layout of the structure is such that appli-cation of a voltage causes a displacement at the right tip. Example courtesy of Jonsmann, Sigmund, and Bouwstra (1999).

can be interpreted physically as the result of a homogenization procedure. Often, φ is interpreted as a local density and denoted using the symbol ρ.

Another way to overcome the ill-posedness is to restrict the problem. This limits the complexity of the designs. Examples of restriction methods are the application of an upper bound on the perimeter of the designs, or limiting the member width e.g. by using filtering (see e.g. Haber et al. 1994; Sigmund 1997). The results in Figure 1.4 are obtained by limiting the perimeter.

1.1.3 Microstructure and macrostructure approaches

Eschenauer and Olhoff (2001) roughly distinguish two classes of approaches, the mate-rial or microstructure approaches, and the geometrical or macrostructure approaches, see Figure 1.7. In the microstructure approach, the design domain is meshed, and the material properties of each element are manipulated to find the optimal mate-rial distribution. In the macrostructure approach, the geometry of a structure is manipulated.

In the microstructure approach, typically, the design variables are the densities ρ of the elements. The value of ρ varies continuously from 0 for an empty element, to 1

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PSfrag replacements

F F

(a) Microstructure approach

PSfrag replacements

F F

(b) Macrostructure approach

Figure 1.7: In the microstructure approach, the constitution of the elements is adapted. In the macrostructure approach, the geometry of the structure is adapted.

for a solid element. The densities are mapped to the element properties. A popular approach is to use the SIMP power law, where SIMP stands for Solid Isotropic Ma-terial with Penalization, see e.g. Bendsøe (1989), Mlejnek and Schirrmacher (1993), and Rozvany et al. (1994). The SIMP approach models the stiffness of the element material by multiplying the Young’s modulus of the solid material with ρp_{, where p}

is a penalization parameter, see Figure 1.8. Good results can be obtained if the value if p is chosen larger or equal to 3. This power law has the property that elements with intermediate densities are relatively compliant. In other words, the stiffness of elements with intermediate densities is expensive in terms of “material cost”. This penalizes for intermediate densities, resulting in a preference for designs with ele-ment densities close to 0 and 1. Although the power law seems artificial, it has been shown that materials with properties satisfying the power law can be created using microstructures (Bendsøe 1999).

In the macrostructure approach, only solid materials are used. The topology can be changed in several manners. One way is to change the structure by growing and degenerating material, as e.g. done in evolutionary strategies (Querin et al. 1998; Querin et al. 2000). The other is by inserting holes at places, followed by shape optimization, as e.g. done in the bubble method (Eschenauer et al. 1994).

The microstructure approaches have a number of attractive features. Typically, a regular rectangular finite element grid is used, and the design variables are directly linked to the contents of the elements. This setup is computationally efficient. More-over, using a fixed mesh, with the contents of the elements as design variables, design sensitivities are comparatively easy and cheap to compute.

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8 Chapter 1. Introduction 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 PSfrag replacemen ts ρ ρ2 ρ3 ρ4 ρ E /E 0

Figure 1.8: In the SIMP approach, the Young’s modulus is given by E = ρpE0. The

relative stiffness is plotted for several penalization values p.

rather let the geometry determine the mesh. This may require changes of the mesh during the optimization steps, and therefore be more expensive in computational cost. On the other hand, the mesh can be adapted to the accuracy requirements, e.g. refined at places where stress concentrations occur to get a more accurate solution.

Although the microstructure approaches are very popular to obtain designs, they have a number of drawbacks. The design variables are coupled one-to-one to the elements, i.e. the discretization of the problem. Hence the number of design variables is proportional to the number of elements, and consequently very large. Furthermore, the accuracy of the finite element calculations depends on the element size. The elements must be small enough to represent the geometry and the corresponding response fields accurately, hence requiring a large number of elements.

Furthermore, in the microstructure approaches, the value of the density ranges between 0 and 1. For intermediate densities, modeling of the physical properties is not straightforward, especially when multiple physical phenomena are involved, e.g. electricity, heat, magnetism and elasticity. Another issue is the interpretation of the intermediate densities, when the model is translated in a manufacturable design. This is not a problem for the macro structure approach.

As mentioned before, the topology optimization problem may be ill-posed. This reveals itself by lack of convergence or by mesh dependence. The problem may become well-posed by relaxing or restricting the problem.

1.2 Problem formulation

Haber and Bendsøe (1998) suggest one should use an geometry representation that is fully independent of the finite element discretization. Furthermore, they formulate

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1.3. Literature 9

some properties of an ideal geometry description:

• Geometric efficiency, which refers to the ability to represent complex geometries using relatively few design parameters.

• Topological robustness refers to the ability to represent arbitrary geometries and to the ability to evolve gracefully between different topologies.

• Continuous parameterization allows design sensitivities and avoids the problem of optimization with discrete variables.

• Frame and mesh independence refer to the invariance of the geometric proper-ties with respect to the choice of coordinate frame and mesh orientation. • Compatibility with automatic grid generation and adaptive refinement software

means that standard operations required for mesh generation and adaptive refinement are available. Examples of these operations are the calculation of intersections and normal directions.

• Shape sensitivity capability refers to the availability of the sensitivities of the nodal coordinates with respect to perturbations of the design variables.

Haber and Bendsøe (1998) propose some geometry modeling methods that may perform successfully. These are the level set method by Osher and Sethian (1988), the skin and body model by Edelsbrunner (1999) and the bubble method by Eschenauer, Kobelev, and Schumacher (1994). The call for geometry models that are independent of the finite element discretization and are purely solid-void is repeated by Bendsøe (1999).

The present thesis proposes a method that fulfills these requirements. It is called the topology description function (TDF) approach. Before the method is exposed, an overview of the literature will be given. This overview serves to indicate the line of thought leading to the TDF approach.

1.3 Literature

A brief overview of the literature will be given here. It is used to introduce the ideas behind the topology description function. This overview is not intended to give a rigorous review of the topology optimization field. For that, the review paper of Eschenauer and Olhoff (2001), and the books of Bendsøe (1995) and Bendsøe and Sigmund (2003) are recommended.

Eschenauer, Kobelev, and Schumacher (1994) worked on the bubble method. This method starts optimizing the shape of a topology, and then changes to another topol-ogy by inserting a new bubble, i.e. a new hole in the structure, after which the shape

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is optimized again. The insertion of the hole is determined by a so-called charac-teristic function2_{. In Appendix B.1 of this thesis, more information is given on the}

relation between the characteristic function and the topological derivative.

88 H. Edelsbrunner

Fig. 1. The skin surface defined by nine weighted points inR3_{, eight at the corners and one at the center of}

a cube.

of the spheres is the zero-set. We call this envelope the skin and the subset ofR3_bounded

by the envelope the body of the set of weighted points, see Fig. 1. Second we consider the construction of f and the deformation of the skin by taking a continuous sequence of preimages f−1(τ ). The flexibility in choosing different maps f with the same zero-set translates into the freedom of deforming the skin in different ways. The topology of the skin changes when τ passes a critical value.

Summary of Results. Whether or not skin and body are indeed useful concepts in

geometric modeling depends on the existence of efficient algorithms. We describe a discrete framework and combinatorial algorithms constructing the skin as a collection of quadratic patches. The framework is based on Voronoi, Delaunay, and Alpha complexes of a finite set of points with weights [7]. The geometric and computational properties of skin and body are summarized in the following nonexhaustive list of informal claims:

S1.decomposability: skin consists of a finite number of degree-2 patches (Theo-rem 13, Section 6),

S2.constructibility: there are fast combinatorial algorithms constructing skin (Sec-tion 6),

S3.symmetry: skin can be defined from the inside as well as the outside (Theorem 14, Section 6),

S4.smoothness: in the nondegenerate case skin is everywhere tangent continuous (Theorem 16, Section 7),

S5.economy: even a small number of weighted points can generate fairly compli-cated skin,

S6.universality: every orientable closed surface has a skin representation,

S7.deformability: topology changes of skin can be efficiently computed (Theo-rem 18, Section 7),

S8.continuity: skin varies continuously with points and weights (Section 8).

Figure 1.9: Example of a skin surface defined by nine weighted points in R3_{, eight}

at the corners and one at the center of a cube. It is constructed using the algebra of spheres, see H. Edelsbrunner (1999)

Edelsbrunner (1999) describes deformable smooth surface design using the algebra of spheres. It is a relatively complex way of describing a geometry: A geometry is written as a collection of spheres, see Figure 1.9. These spheres are characterized by their centers and weights (which determine the radius). The shape of the geometry between two adjacent balls is determined by a level set function determined by the parameters of those balls. For details, the reader is referred to Edelsbrunner (1999). Sienz (1994) uses the hard-kill method. The Young’s modulus of each element of the finite element discretization is adapted depending on the stress an element has to cope with. If the stress is less than a certain threshold value, then the material in the element is considered superfluous and the Young’s modulus for the material in that element is scaled down to ²E; if the stress is above a certain threshold value, then the element must be reinforced, and the Young’s modulus of the adjacent elements is increased to E. Typically, ² should be very small, e.g. ² = 10−6_.

Kumar and Gossard (1996) use a so-called shape density function φ to determine the geometry of the structure. The values of φ are computed in the nodes of the finite element discretisation of the reference domain. The use of φ is twofold: First, the Young’s modulus and Poisson’s ratio are related to φ, and in that manner φ determines the properties of the elements of the finite element model of the structure. Since φ typically takes values between 0 and 1, a penalization of the form φ(1 − φ) is added to avoid intermediate values. Using a modified form of sequential linear programming, φ is optimized. Second, φ is used to determine the geometry of the structure using a threshold level φth. The material is in the parts of the reference

domain where φ exceeds φth.

Maute and Ramm (1995) also use a density function, see Figure 1.10(a). They search for the optimal topology, i.e. density distribution, in a given mesh, using the same method as Bendsøe and Kikuchi (1988). Using the isolines of the density

2

This characteristic function has nothing to do with the material indicator function χAmentioned

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1.3. Literature 11

(a) The first optimization cycle (b) The subsequent optimization cycles

Figure 1.10: In the approach of Maute and Ramm (1995), the isolines of the opti-mized density are used to generate a new finite element model (a). In this model, the densities can be optimized again. This cycle is then iterated (b). Images used with permission.

distribution, the new shape of the structure is determined. A new mesh is created, and the procedure is repeated, see Figure 1.10(b).

Osher and Sethian (1988) use a level set function f that returns, e.g., the time a flame arrives at a certain spot. The geometry of the flame at a certain time consists of all points that ignite at, or before, that time, see Figure 1.113 Notice that the ignition time separates ‘burnt’ space from ‘fuel’: for a certain time t, the ‘burnt’ part of the space is characterized by f (x) < t, which is the part of the cone below t.

The common aspect in the papers mentioned before is that a function on the reference domain determines the material properties in the reference domain. To obtain the shape of a design, a threshold value is applied. This is very similar to level set methods, see Appendix A. The crucial notion is that a function can be defined independently of a spatial discretization. Hence, a function description can

3

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12 Chapter 1. Introduction ignition time 12 9 6 3 -15 -10 -5 0 5 10 15-15 -10 -5 0 5 10 15 0 2 4 6 8 10 12 14 PSfrag replacemen ts time x y

Figure 1.11: If the function f used by Osher (1988) is used to describe the propaga-tion of a flame, then its value can be interpreted as the ignipropaga-tion time. For an artificial example, the graph is given here. The level sets of f , depicted in the bottom plane of the graph, indicate how the flame propagates outward.

be given without creating a mesh first. A second aspect is that by using a cut-off level, a function can divide a space in parts, namely parts below the cut-off level, and parts above the cut-off level, which can be used to represent void and material, respectively.

Of course, if a level set function is represented by its values in the elements of the finite element discretization of the reference domain, a huge amount of data is used for its representation. This may not be very efficient. A more efficient way of representing a function is by using a mathematical expression. An efficient way to describe a family of functions is to select a basis of functions, and to use linear combinations of basis functions to express members of the family. Examples of such bases are Legendre polynomials and goniometric functions (think of Fourier series).

The common way to depict a function defined in two dimensions is to depict a surface, i.e. a landscape where the height z represents the function value of the coordinates in the (x, y)-plane. One may regard the landscape as a collection of hills and valleys. The combination of a number of hills and valleys enables the creation of complex landscapes, while a hill or valley can be a relatively simple function of only a few parameters.

Alternatively, the surface can be regarded as a combination of waves, like the surface of the sea. The differences in the approach is that hills and valeys have a local character, whereas waves have a global character. Consequently, modifying a basis function slightly has a local or a global effect. Since it is easier to keep track of local changes, it is prefered that the basis functions have a local nature. However, there is no principle reason why a goniometric basis would not work; on the contrary,

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1.3. Literature 13

the orthogonality of the basis functions may be an advantage. The application of goniometric functions for a basis is beyond the scope of this thesis.

Recapitulating, the combination of simple functions can yield a complex function. Using that function as a level set function, it can divide space. This space division can be considered as the division between material and void. Hence, the variables used to describe the simple functions can be used to determine the geometry of a structure. This leads to the topological description function (TDF) approach. Basis functions described by relatively few parameters combine to form a more complex TDF. The TDF determines a geometry using a cut-off level. For topology optimization, the geometry is evaluated, and the result is used in an optimization program that tunes the parameters, see Figure 1.12.

1 0.5 0 y 2 1 0 -1 -2 x 2 1 0 -1 -2 1 0.5 0 y 2 1 0 -1 -2 x 2 1 0 -1 -2 1 0.5 0 y 2 1 0 -1 -2 x 2 1 0 -1 -2 Response values Design sensitivities

Finite element computations

Solver Preprocessor Postprocessor TDF Design variables x,y,h,w Cut−off level Geometry Topology description function approach

Optimization program Gradient free algorithms Intuitive optimization algorithms Gradient based algorithms

Figure 1.12: In the optimization loop, the optimization program controls the values of the design variables. These determine the contributions of the individual basis functions to the TDF. The TDF determines the geometry of a structure using a cut-off level. That structure may be evaluated using a finite element model. The resulting response values are returned to the optimization program, closing the loop.

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1.4 Outline of the topology optimization

proce-dure

In this section, an overview of the topology optimization procedure will be given. The details will be elaborated in later sections.

As mentioned, the topic for the present thesis is the topological description func-tion (TDF) approach. This approach is a way to describe the geometry of a structure. It is intended to solve some of the problems mentioned in the previous sections.

What distinguishes the TDF approach from conventional approaches is that the modeling of the geometry and the modeling of the evaluation are separated. In most conventional approaches, the finite element model used for the evaluation is also the model used to describe the geometry, leading to a strong link between the finite element model and the geometry.

For topology optimization, the TDF approach is embedded in an optimization loop as depicted in Figure 1.12. Using the TDF approach, the design parameters and design variables4 _{determine the geometry of a structure. To evaluate the}

per-formance of this geometry, a finite element (FE) model of the structure is created. Subsequently, a FE analysis is used to compute the objective value, the constraint values and the sensitivities. Those are returned to the optimization program, which uses the information to iteratively search for an optimal set of values for the design variables.

The optimization loop can be regarded as several modules coupled together: a geometry modeling module, an evaluation module, and an optimization module. The chapters of the present thesis follow the modular approach.

In the present thesis, the geometry modeling module is limited to the TDF ap-proach, since the TDF approach is the topic of this thesis. In Chapter 2, the details of the topological description function will be given. This comprises the definition of the TDF, the advantages and disadvantages, the choice of the design parameters, implementation issues, and a discussion and comparison with other methods from literature.

The evaluation module is not given attention explicitly. Currently, an in-house FE code is used to evaluate the designs, because of its ease of use and availability. From the view of modularity, it should not differ if this FE code were replaced by analytical evaluations, or by mesh-less methods, or any other method. Although each of these has advantages and disadvantages.

The optimization module has received most of the attention. The optimization problem is very difficult to solve, since it may have many local optima. Most of the present thesis is dedicated to the optimization method. The material is classified

4

The design variables are the variables that can be modified by the optimization program. The

design parameters can not be modified. As an example, often the locations and widths of the basis

functions are fixed for simplicity, and the optimal heights are sought. The heights then are the design variables, and the locations and widths the design parameters. The number of basis functions, and the cut-off level also are design parameters.

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1.4. Outline of the topology optimization procedure 15

in chapters based on the type of information used. Several gradient-free methods are covered in Chapter 3. These are the genetic algorithm and a response surface method. Both require only response values and turn out to be quite computationally expensive, because they need many evaluations.

In Chapter 4, intuitive methods are applied. For compliance optimization, for example, a structure needs to be reinforced at locations where stresses are high, and material is wasted at lowly stressed regions. Hence, improvement may be made by moving material from lowly stressed areas to highly stressed areas.

However, as soon as the objective becomes more difficult, intuitive updating rules become almost impossible to conceive, e.g., in the case of maximization of the lowest frequency. This prompts the development of more advanced mathematical tech-niques. These are based on design sensitivities, in quite a broad sense. In Chapter 5, the design sensitivities are covered. Chapter 6 elaborates on the topological deriva-tive.

The gradient information is incorporated in the optimization procedures as de-scribed in Chapter 7. The gradient-based method are the steepest descent method, the method of moving asymptotes (Svanberg 1987), the multipoint approximation method (Toropov 1992), and an optimality criteria method (Haftka and G¨urdal 1999) extended with information based on the topological derivative.

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Chapter 2 Topology Description Function

approach

2.1 Topology description function approach

The topological description function approach is an approach to describe the geome-try of a structure. The general idea of the topological description function approach is to define a function T that assigns a value to each point of the reference domain, and that by comparison with a threshold value, the value of T separates material parts of the reference domain from voids. The level set of T for the threshold value then is the set of points on the boundary of the structure, and the structure itself consists of the points with a function value equal to or exceeding the threshold level: A = x ∈ Ω : T (x) ≥ C. The remaining space, where the function is below the thresh-old level, is void. In this manner, T determines the geometry of a structure, that is, it describes both the topology and shape of a structure. Therefore T is called the topology description function (TDF). Furthermore the threshold level is also called cut-off level and is referred to using the symbol C. This is because the areas in the reference domain for which the TDF does not attain the threshold value are cut off from the structure.

An easy way to grasp the idea is to think of the topological description function T as a function that determines the ‘altitude’ of a point in the reference domain. For simplicity, this reference domain is two-dimensional, but the method can be extended easily to three-dimensional reference domains. The threshold has the role of ‘sea-level’. A plot of T gives a ‘landscape’. Points below ‘sea-level’ are ‘inundated’, and points above ‘sea-level’ are ‘dry’. Drawing a map of the reference domain then automatically yields the topology of the design: the ‘land’ corresponds to material, and the ‘water’ to void. The level set can be interpreted as the ‘coastline’.

In the present thesis, the TDF is chosen to be defined as a superposition of smooth basis functions in the shape of a bell. In one dimension, such a basis function can be characterized by three values: the location x, the height h, and the width w, see Figure 2.1. In probability theory, this bell-shaped function is known as the

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18 Chapter 2. Topology Description Function approach 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -3 -2 -1 0 1 2 3 PSfrag replacemen ts e−x2 h (x ) h w x y

(a) A basis function (1-D)

-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 PSfrag replacemen ts e−(x2+y2) x y h(x, y) (b) A basis function (2-D)

Figure 2.1: Basis functions are the building components to compose a TDF. For the cases depicted above, the height and width are both set to 1, and the location of the center is in the origin.

probability density function of the normal (or Gaussian) distribution. The location is called the mean or expectation, and the width is related to the standard deviation. In probability theory, the height is determined by the condition that the probability of all possible outcomes together must be one. The mathematical expression for the bell function is hi = hie−(x−xi)

2_/w2

i. Here h

i is the height of the basis function, wi is

the width and xi is the location. The index i is used since multiple basis functions

will be combined to construct the TDF. In Figure 2.1, the graph of a Gaussian bell is depicted for both the 1-D and the 2-D case. Of course, the basis function can easily

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2.1. Topology description function approach 19 -3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3 -6 -4 -2 0 2 4 PSfrag replacemen ts TDF of 3 basis functions T (x, y) x y

Figure 2.2: As an example of a slightly more complicated TDF, three 2-D basis functions are used to compose the depicted TDF.

be generalized to 3-D, but that is difficult to depict graphically.

Several basis functions can be combined to create a more complex TDF. As an example, in Figure 2.2, a TDF composed of three basis functions is depicted. A basis function with height 1 and width 1 is centered at (−2, −2). The second basis function, with height 2 and width 1 is centered at (−2, 2). The third basis function, with (negative!) height −6 and width 1/3 is located at (1, 0).

Now the TDF is to be defined more formally. It is written as a sum of Nbf basis

functions, i.e.: Ts(x, y) = Nbf X i=1 hs_i(x, y). (2.1)

where s is the vector of design variables, and (x, y) refers to a location in the reference domain. For the basis function the following function has been chosen:

hs_i(x, y) = h_ie

−(x−xi)2+(y−yi)2

w2_i _. _(2.2)

Here the vector of design variables si consist of (xi, yi, hi, wi)T. The design variables

(xi, yi) determine the location of the center of the basis function. The design variable

hi scales the height, and wi the width. The entire vector s of design variables consists

of the design variables of all the basis functions put together.

Having defined the TDF, it has to be explained how the TDF determines a struc-ture. For simplicity, it is assumed that the reference domain contains a structure consisting of only one material1, i.e. each point of the reference domain is either a

1

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20 Chapter 2. Topology Description Function approach

material point, if it is part of the structure, or a void point otherwise. The corre-spondence between a TDF and a structure is done using a cut-off level. Wherever the TDF exceeds the cut-off level in the reference domain, there is the material, and on the other places, the reference domain is void. For example, in Figure 2.3, the TDF is chosen as 2 exp(−x2_{), and the cut-off level is 1. The interval where the TDF}

exceeds the cutoff-level is h₋pln(2), pln(2)i. In this interval the TDF determines material, and in the rest of the reference domain void.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -3 -2 -1 0 1 2 3 PSfrag replacemen ts h( x ) x void void material C 2e−x2

Figure 2.3: Application of the cut-off level in one dimension. The reference domain is a line parametrized from −3 to 3. Comparison of the TDF 2e−x2 to the cut-off level 1 determines the part of the reference domain that contains material.

Some statements on the behavior of the design variables can be made. Increasing the width variable of a basis function causes the value of the basis function to increase everywhere (except at the center). Increasing the height of the basis function causes the value of the basis function to increase everywhere. Increasing a single basis function translates into increasing the TDF, and this implies a larger structure.

To see this, notice that every material point corresponds to a TDF value above the cut-off level, and stays above the cut-off level if the TDF value is increased. Now consider a point at the boundary (‘skin’) of the structure, see Figure 2.4. Increasing the TDF value in that point causes the TDF to exceed the cut-off level in that point. Due to the continuity of the TDF, now there is a neighborhood of that point for which the TDF exceeds the cut-off level. Thus, the void points that were located next to the boundary have become material points. Similarly, decreasing the height or the width causes the structure to become smaller. Of course, if the entire reference domain is solid, it cannot grow bigger, and no material can be removed if the reference domain is void. A similar argument can be given to demonstrate that a reduction or increment of the cut-off level makes the corresponding structure larger or smaller,

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2.1. Topology description function approach 21

respectively.

Original structural boundary Incremented TDF

Cut−off level

Increment of the TDF value Material neigbourhood TDF

x

Original TDF

Figure 2.4: Increasing the TDF yields a bigger structure as a neighborhood of a boundary point becomes entirely solid.

However, this way a TDF maps to a structure is not one-to-one. Every TDF determines a single structure, but in general not the other way around. For a given structure, multiple TDFs can be determined. Two examples are shown in Figure 2.5. In Figure 2.3 the function 2 exp(−x2_{) determines an interval. However, the functions}

3 exp(−x2_{ln(3)/ ln(2)) and 4 exp(−2x}2_{) exceed the cut-off level in precisely the same}

interval. A similar example is constructed for two basis functions.

Two TDFs that determine the same structure are called synonyms. The occur-rence of synonyms in one dimension has been shown in the previous examples. For 2-D and 3-D it is also easy to construct synonyms for TDFs with only a single basis function. For multiple basis functions it becomes quite difficult.

0 0.5 1 1.5 2 2.5 3 3.5 4 -3 -2 -1 0 1 2 3 PSfrag replacemen ts T (x ) x (a) 2 exp(−x2 ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -3 -2 -1 0 1 2 3 PSfrag replacemen ts T (x ) x (b) Two basisfunctions

Figure 2.5: A TDF is not unique for a structure. TDFs that determine the same structure are called synonymous.

To summarize the TDF approach, a 2D example of the concept is depicted in Figure 2.6. The design variables determine the locations, heights and widths of the

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22 Chapter 2. Topology Description Function approach 1 0.5 0 y 2 1 0 -1 -2 x 2 1 0 -1 -2 (a) (b) (c) (d)

Figure 2.6: The concept of a TDF. A basis function (hill) is determined by the location of its center, its height and its width (a). Superposition of a number of hills enables the description of a function (b). Using a cut-off level the domain is separated in a material part (where the function exceeds the cut-off level) and a void part, thus determining a geometry (c). This geometry can be discretized to obtain a finite element model (d).

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2.2. Advantages and disadvantages 23

basis functions (a). Summing a number of basis functions together, a function with a more complicated shape can be constructed, the so-called TDF (b). Comparison with a cut-off level determines which part of the reference domain is solid and what part void (c). Finally, a finite element representation of that geometry can be constructed to enable the evaluation of the objective and constraint functions (d).

As mentioned before, the decomposition of the TDF is not limited to bell-shaped basis functions. For example, polynomial basis functions (Legendre), or goniometric basis functions (Fourier) could be applied. However, the convenience of the bell-shaped basis functions is that the effect of a basis function on the geometry is localized around its center.

One idea that has not been pursued yet is to extend the basis functions in such a way that it has a sense of direction. In the present thesis, the level set of a single basis function is a circle, and that could be extended to ellipse shapes. This could be advantageous. Often, the result of a topology optimization shows a structure containing beam-like members, where each beam is composed by a number of basis functions in a row. These could be replaced by relatively few basis functions with a properly oriented ellipse shape.

2.2 Advantages and disadvantages

The most significant feature of the TDF approach is the separation of the description of the geometry and the FE modeling. The advantage of this is that the designer can tune the detailedness of the structure and the accuracy of the FE model inde-pendently. By choosing the number and locations of the basis functions the designer determines how much detail can be represented in the geometry, and where that detail must be applied. By using more basis functions, the designer can represent more detailed geometries. If a part of the reference domain requires more detail, the designer can cover this part with more basis functions2_{. This is similar to the use of}

mesh refinement at places where more accurate data is required.

The formulation using TDFs has the following advantages (compare with the properties that Haber and Bendsøe formulated and that are repeated in Section 1.2): • The description uses relatively few parameters to describe a topology. This is based on a comparison of the number of design variables as used by the different approaches, see also Section 2.6. The number of basis functions is small compared to the number of elements used in the FE analysis of the structure.

• The description can describe any geometry within the resolution of the model. To see this, draw a map of an arbitrary structure, fill in the isolines, and the landscape can be envisaged. Then decompose the landscape into hills to obtain

2

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the description of that structure. That any landscape can be decomposed into a sum of hill functions can be proven by the theorem of Stone-Weierstraß (see any book on (functional) analysis, e.g. Brown and Page 1970), in a manner similar to the proof of the Fourier Theorem. But like the Fourier series, an infinite number of basis functions is needed to approximate any continuous function arbitrarily well. Using a finite number of basis functions limits the accuracy of the approximation. Notice however, that a limited number of basis functions limits the detailedness of a structure, and is required to avoid the ill-posedness of the discrete material distribution problem.

• The design variables are continuous, and the resulting geometry is discrete. The continuity of the design variables avoids the integer programming problem. It also allows the computation of sensitivities. The discreteness of the geometry leads to a clear definition of the model. This makes subsequent modeling and manufacturing straightforward.

• The description is mesh independent, so mesh refinement does not induce a change of the optimal solution. The design determined by a TDF is solely determined by the level set, and though mesh refinement can serve as a method to get a better FE representation of the level set (i.e. the geometry), it has nothing to do with the design. Hence, choosing a different mesh with respect to element size or orientation, or choosing a regular or an irregular grid, does not affect the problem at hand.

• Efficient shape design sensitivity. In Chapter 5 and Chapter 6, the computation of design sensitivities and topological derivatives are discussed.

Haber and Bendsøe also mentioned compatibility with automatic grid generation and adaptive mesh refinement software. They explain that the geometry model should support the standard operations (calculations of intersections, projections, normal directions etc.) required in mesh generation and adaptive mesh refinement (AMR) algorithms. In the present thesis, little attention is given to mesh generation and AMR. To obtain FE models, the properties of the elements of a fixed FE mesh repre-sentation of the reference domain are adapted. This will be described in Section 2.4. An important issue of the topology function approach is that, although the de-scription of a design is mesh independent, the results of the finite element calculations are mesh dependent. This is because the response of the design may be represented more adequately in one mesh than in another. As an example of this, see Section 2.5.2 and Section 4.4.2 (Page 94).

A drawback of the TDF approach is that the designs that can be represented are limited by the chosen basis functions. If the design should consist of slender members, then many basis functions will be necessary for an adequate representation. Clearly, the number may still be small as compared to the number of elements. Notice that the number of elements required for a geometry in 2D has an inversely quadratic

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2.3. Design parameter choice 25

relation to the length scale of the elements, i.e. to halve the typical length scale of an element quadruples the number of elements required. The same holds for the number of basis functions in the TDF approach. In 3D, the relation is an inversely cubic.

Moreover, if there is no limit on the number of basis functions, then the problem still may be ill-posed. If there is no limit on the number of basis functions, then still an arbitrary amount of detail can be represented. For that reason, the number of functions is fixed. Another consideration is of more practical nature: the geometry must be evaluated. To that end, a FE model must be able to represent the geometry. Hence, it is useless to allow more detail in the TDF than the FE mesh can represent. Another issue is the redundancy of the model. E.g. multiplying the cut-off level and all the heights of the basis functions does not change the design; it just scales the TDF. Also the order in which the basis functions are added to the TDF does not matter3_{. Hence, (s}

1, s2)T represents exactly the same TDF as (s2, s1)T. Furthermore,

if two basis functions have the same center (i.e. they are on top of each other) and width, then they can be replaced by a single one.

2.3 Design parameter choice

At this stage, the TDF under consideration is composed of a number of basis func-tions. Each of these basis functions is a bell-shaped function characterized by its parameters describing location, height and width. To obtain a optimization problem that can be handled, a more specific description is required.

T = Nbf X i=1 hie −(x−xi)2+(y−yi)2 w2_i _(2.3)

The design parameter choice concerns the number of basis functions, the ranges for the parameters of the basis functions, and the cut-off level.

2.3.1 Number of basis functions

The first issue to be addressed is also the most difficult one, namely the number of basis functions Nbf required. Of course, the more basis functions are used, the

more flexible the TDF is to represent structures. If too few basis functions are used, a feasible design might even not exist. On the other hand, using too many basis functions enlarges the dimension of the design space too much.

In the present thesis, the basis functions are arranged in a regular grid on the reference domain. This is done by restricting the location of each basis function to a small part of the reference domain, as depicted in Figure 2.7(a). Later on, the

3

Since the basis functions are arranged in a fixed grid in the examples in the present thesis, these issues do not occur; however, if one wants to use the locations of the basis functions as design variables, these issues must be taken into consideration

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(a) Restricted grid

PSfrag replacements d

(b) Fixed grid

Figure 2.7: For simplicity, the basis functions are arranged in a regular grid. One way is to restrict each basis function to a small area (a). Another is to fix the location (b).

location has been fixed, as depicted in Figure 2.7(b). In both ways, the number of basis functions is fixed. Furthermore, some of the redundancy mentioned in the previous section is avoided.

The grid can be characterized by the distance between the centers of the basis functions. The distance will be denoted by d. It introduces a length scale, because it determines the minimum size of the details of the geometry. In this manner, the complexity of the design is restricted. Obviously, d and Nbf are closely related.

Choosing the proper distribution distance d is up to the designer.

2.3.2 Parameters of the basis functions

The second issue is to select reasonable ranges for the parameters of the basis func-tions. In this manner the design space can be limited. To keep the problem simple, only the height of the basis function is considered as a design variable. The loca-tion and width are fixed. For further analysis, at present, a 1-D example will be considered. Later on, the obtained data will be extended to 2-D.

It has been mentioned that the distance d introduces a length scale. Assume that the distribution of the basis functions is regular, and set the distance d as the unit length. Then a TDF is a linear combination of the basis functions depicted in Figure 2.8. Three more parameters have to be selected to define the design space: the upper and lower bound for the height design variables, denoted hmax and hmin,

and the width design parameter w.

A few requirements are formulated to choose proper values for the design parame-ters. First, the significant range of a basis function may not be too large or too small. The significant range is the distance at which a change in the basis function parame-ters yields a significant change in the corresponding geometry. If the significant range is too large, then it becomes impossible to generate details. If it is too small, parts of the geometry cannot be controlled by the TDF. For example, if the range of the basis

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2.3. Design parameter choice 27 0 0.5 1 1.5 2 -3 -2 -1 0 1 2 3 PSfrag replacemen ts x h (x )

Figure 2.8: The set of basis functions used in the analysis of the properties. Here all the heights are set to 1, the widths to 0.5 and the basis functions are distributed regularly using a distance 1.

functions in Figure 2.8 would be 3, then a change in the parameters would affect the TDF in the entire depicted range of x. If the range would be 0.2, then the TDF between the basis functions is not controlled well. In that case, at x = 0.5, the TDF would consist of contributions of the almost zero and nearly horizontal ‘tails’ of the basis functions resulting in an almost horizontal TDF. This implies that the location of the intersection of the cut-off level and the TDF is very sensitive to small changes, which complicates optimization.

Second, a basis function must be able to fill its neighborhood. This means that it must be able to elevate the TDF above the cut-off level in its neighborhood. In Figure 2.8, this means that the basis function at x = 0 must be able to fill at least the domain x = [−0.5, 0.5], even if the adjacent basis functions are in their worst possible state with respect to this goal. This requirement ensures that a line of basis functions can represent a solid bar in a void surrounding.

Third, and similar to the second requirement: a basis function must be able to empty its neighborhood. This requirement ensures that holes can be represented in a solid piece of material.

As an example for study, a TDF is defined as the sum of all the basis functions on the x-axis. Heights and widths are considered equal for all basis functions. This TDF can be written as T = ∞ X xi=−∞ he(x−xi)2/w2 _(2.4)

To study the influence of the width w, assume that the cut-off level is 1, and set the heights of the basis functions to 2 for now. When the widths are varied, the TDF changes accordingly. The graphs for w = 0.02, w = 0.2, w = 0.4 and w = 0.6 are

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28 Chapter 2. Topology Description Function approach 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 3 PSfrag replacemen ts x T (x ) (a) width=0.02 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 3 PSfrag replacemen ts x T (x ) (b) width=0.2 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 3 PSfrag replacemen ts x T (x ) (c) width=0.4 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 3 PSfrag replacemen ts x T (x ) (d) width=0.6

Figure 2.9: The influence of the widths of the basis functions. A small width gener-ates a TDF with spikes, where the local contributions dominate the TDF. Increasing the width, the contribution of basis functions becomes less local, ultimately exceed-ing the cut-off level everywhere.

depicted in Figure 2.9. It can be observed that the width must exceed 0.4 if the TDF has to represent more than isolated dots of material. Alternatively, a larger height can be chosen.

The height and width are not uncoupled. Given an arbitrary width w, the TDF that represents a solid domain can be constructed by choosing a proper height, as illustrated in Figure 2.10. The relation between the chosen width and the required height can be analyzed as follows. The design is entirely solid if the minimum of the TDF exceeds the cut-off level. In this example, the minima are located at x = {. . . , −1.5, −0.5, 0.5, 1.5, . . .}, and the value of the TDF at the minimum can be written as 2P∞

i=1he−(i−1/2)

2_/w2

. Putting this equal to the cut-off level 1, and solving h for given w, and plotting the results yields the graph depicted in Figure 2.10(b).

To incorporate solid domains in the design space for given widths, the heights must be able to at least attain the proper value. In Figure 2.10(a), TDFs are depicted for several combinations of w and h satisfying the equation precisely. To conclude, it is obvious that hmax≥ h(w) to allow solid designs, and hmin ≤ h(w) to allow holes.

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2.3. Design parameter choice 29 0 0.5 1 1.5 2 2.5 3 3.5 4 -3 -2 -1 0 1 2 3 PSfrag replacemen ts w h (w ) x T (x ) required heigh t cut-off level w = 0.3 w = 0.4 w = 0.6 w = 0.8

(a) TDFs representing solid designs

0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 PSfrag replacemen ts w h (w ) x T (x ) required height cut-off lev el w = 0 .3 w = 0 .4 w = 0 .6 w = 0 .8

(b) The relation between w and h

Figure 2.10: Solid designs can be obtained if the height of the basis function is large enough. For several w and corresponding h(w), the TDFs are depicted in (a). The graph of the relation h(w) is depicted in (b).

cannot be chosen independently. To investigate the relation between the parameters further, a two-hole example is constructed, see Figure 2.11. The objectives of this example are to make sure that a TDF can represent:

• A larger solid part at the left and right hand of the two holes. • A larger void part covering both x = 0 and x = 1.

• A solid piece surrounded by void at x = 2. • A void surrounded by material at x = 3.

To this end, a TDF is defined in the following manner: A series of basis functions is constructed, with the locations at x ∈ {. . . , −2, −1, 0, 1, 2, . . .}. All the heights are set to the maximum height hmax = 2, except for three basis functions: at x = 0,

x = 1 and x = 3 the heights are set to a parameter value h.

In Figure 2.12 the graphs of the TDFs for several combinations of heights and widths h and w are depicted in a table. The TDF for h = −2 and w = 0.6 is satisfactory, because TDF represents a solid structure with two holes. For the first two columns, with widths 0.2 and 0.4, the structure is not solid, but consists of isolated pieces of material. This is caused by a too small range of influence, due to too small a width w. A too large width also causes trouble: In the column for w = 1.6, the contributions of the holes at x = 0, x = 1 and x = 3 are so large that the solid part at x = 2 cannot be represented. The TDFs in the first row, for h = 1, are also not satisfactory: The TDF exceeds the cut-off level at x = 0, x = 1 and x = 3, where it should represent a hole. Here clearly the minimum hmin level has

been chosen too large. Looking at the column for w = 1.2, it can be observed that for h = −2 and h = −1.5, the top at x = 2 is pulled down too much by the holes next to it. For h = −0.5 to h = 1, the hole at x = 3 cannot cope with the contribution of the tops next to it. At h = −1 it is not clear from the graph. These considerations

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30 Chapter 2. Topology Description Function approach -2 -1 0 1 2 3 -4 -2 0 2 4 6 PSfrag replacemen ts x T (x ) w w w w h

Figure 2.11: A TDF is sought that represents a solid structure with a large hole at x = 0 to x = 1, and a small hole at x = 3. To obtain that TDF, basis functions with a height of 2 are distributed regularly. The height of the three basis functions at x = 0, x = 1 and x = 3 are set to h. The width of all the basis functions is set to w. The cut-off level is 1. The case that h = −1 and w = 0.2 is depicted.

lead to a domain in which h and w lead to satisfactory designs. This domain has been depicted in Figure 2.13(d).

In the two-hole example, the maximum height hmax was arbitrarily set to 2. For

other maximum heights a table similar to Figure 2.12 can be computed. The domains of satisfactory designs are depicted in Figure 2.13. In the gray areas the combination of h, hmax and w is such that the TDF can represent a structure with two holes

satisfactory.

In Figure 2.14 the graphs for three different values of hmax are depicted. These

graphs also explain the results for Figure 2.13. For hmax= 0.7, the TDF is very close

to the cut-off level. Increasing hmin almost immediately pushes the valley of the TDF

at x = 2 above the cut-off level. Decreasing hmax almost immediately pulls the top at

x = 2 below the cut-off level. Hence the domain of satisfactory designs is very small. When the width is increased, the graph tends to become higher and more flat. Comparing Figure 2.14(a) to Figure 2.14(b), the size of the design space is very small for the first. However, a tiny variation in height causes a large change in the corresponding structure. Besides, the wrinkle in the graph is quite large compared to the trend of the TDF. In this respect, Figure 2.14(b) looks more robust, although it has a smaller w. This does not contradict the conclusion that increasing the width w flattens the TDF. The wrinkles in Figure 2.14(a) are flatter, but the variation in the general trend of the TDF is comparatively small. Comparison of Figure 2.14(b) and Figure 2.14(c) does not yield a preferable choice. Both are able to represent satisfactory designs.

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2.3. Design parameter choice 31 h w = 0.2 w = 0.4 w = 0.6 w = 0.8 w = 1.2 w = 1.6 1 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 0.5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 0 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 −0.5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 −1 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 −1.5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 −2 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5

Figure 2.12: The two-hole example. The parameters for a solid design containing two holes cannot be chosen arbitrarily. Using a given maximum height of 2, the re-quirements on the minimum height h and width w are investigated. Problems using the wrong combination of values include incoherent designs, inability to represent the two holes and inability to represent the material between the holes.

likely result in severe difficulties for the optimization program to find feasible designs. This is not that severe for larger hmax. For the current 1-D setup, good settings are

to let, for example, w = 0.65, and to let h range between 0.7 and 1. Notice, this analysis assumes a cut-off level 1. However, other cut-off levels are also available.

Topology Optimization using a Topology Description Function Approach