Acta Mechanica Solida Sinica, Vol. 19, No. 2, June, 2006 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-006-0611-y

ISSN 0894-9166

TOPOLOGY DESCRIPTION FUNCTION BASED METHOD FOR MATERIAL DESIGN  Cao Xianfan

Liu Shutian1

(State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China)

Received 22 December 2005; revision received 15 May 2006

ABSTRACT The purpose of this paper is to investigate the application of topology description function (TDF) in material design. Using TDF to describe the topology of the microstructure, the formulation and the solving technique of the design problem of materials with prescribed mechanical properties are presented. By presenting the TDF as the sum of a series of basis functions determined by parameters, the topology optimization of material microstructure is formulated as a size optimization problem whose design variables are parameters of TDF basis functions and independent of the mesh of the design domain. By this method, high quality topologies for describing the distribution of constituent material in design domain can be obtained and checkerboard problem often met in the variable density method is avoided. Compared with the conventional level set method, the optimization problem can be solved simply by existing optimization techniques without the process to solve the ‘Hamilton-Jacobi-type’ equation by the difference method. The method proposed is illustrated with two 2D examples. One gives the unit cell with positive Poisson’s ratio, the other with negative Poisson’s ratio. The examples show the method based on TDF is effective for material design.

KEY WORDS topology optimization, topology description function, material design

I. INTRODUCTION Topology optimization is generally considered as one of the most challenging projects in the field of optimization, and in the past twenty years it has fascinated many workers in this field[1] . At the same time, advanced structures require materials with specific properties. For example, materials with prescribed properties are needed in some fields, such as a material with zero thermal expansion coefficients, negative Poisson’s ratio etc. Hence this needs to go beyond the limit of natural materials and to design new materials with prescribed properties. Composites consist of several materials, whose properties are determined by the distribution and properties of each constituent material. The effective properties of a composite can be changed by adjusting its microstructure (properties of every constituent material, distribution, volume fraction, etc.). Hence materials with prescribed properties can be obtained by designing microstructures. This problem can be formulated as one of finding the distribution of the components in micro scale domain. This idea is similar to that of topology optimization of structures. Thus material design can be implemented by use of the techniques developed for structural topology 

Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No.10332010), the Innovative Research Team Program (No.10421202), the National Basic Research Program of China (No. 2006CB601205) and the Program for New Century Excellent Talents in Universities of China (2004). 

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optimization. There are some examples of successful design of materials by topology optimization in literatures[2−−7] . Frequently used methods for material design include the microstructure-based method (Homogenization method[3] , variable density method or SIMP[4−−6] ) and the boundary evolution method (e.g. Level set method[7] ). In the homogenization method and SIMP method, the relative density of material at an arbitrary point in a design domain is taken as a design variable. The material properties of every point in a design domain are determined by the relative density of materials and can be obtained by the homogenization method in a homogenization based topology optimization method or is expressed as Ei = ρηi E0 in the SIMP method, where Ei are the properties of an arbitrary point in the design domain, η is a penalization factor, and E0 represents the properties of solid material. This kind of methods has become the most popular method for topology optimization. With them, a large group of topology optimization problems has been solved successively. However, some disadvantages often occur in the topology optimization process. First, checkerboard patterns usually occur, which are more prone to appear when using lower order finite elements[5, 8] . The reason for the occurrence of the checkerboard pattern and the method to overcome it are studied in literatures[8−−11] . Second, the number of design variables depends on the finite elements. For complicated structures, very fine grids are needed, and a large number of elements means the number of design variables is large, which makes solving the problem very difficult if not impossible. The third problem is mesh dependency, which refers to the non-convergence of a solution with mesh refinement. The level set method[7, 12] is a new approach to topology optimization, in which topological change is performed through the movement, expansion, merger and evolution of the zero level set (boundary). The level function is often set to be the signed distance function of an arbitrary point in the design domain. Regions with different level values of functions are placed with different materials, according to which the material distribution or the topology is described. In the traditional level set method, the evolution of the boundaries is driven by the moving speed controlled by a ‘Hamilton-Jacobi’ equation. Solving the Hamilton-Jacobi equation with finite difference method is often time consuming, which restricts the efficiency of optimization. Instead of this disadvantage, the indirect independence of the design variables of the finite meshes has overcome to a certain extent the difficulties encountered in the variable density method. Some researchers have investigated the combination of the level set method with other methods for topology optimization. In Ref.[13], the function values of nodes of elements in a design domain are used as variables and the implicit topology description function is obtained by the interpolation of a shape function. Based on this idea, the topology optimization has been performed to obtain a geometry with distinct and smooth boundaries. But the variables are related to the nodes of element, so the model of optimization still depends on the mesh. In Ref.[14], the TDF is presented as the sum of a series of basis functions determined by parameters, and the values of TDF in a design domain can be changed by adjusting the corresponding parameters of basis functions. By introducing the cut-off level, the point whose value of TDF is higher than the cut-off level is placed by solid material, or void. Then the topology of the design domain is described by using the corresponding parameters of the basis functions. Based on this idea a topology optimization is performed. Based on the idea presented in Ref.[14], a topology description function based method for designing materials with prescribed properties is proposed in this paper. The material is constructed by periodically repeating a unit cell consisting of several solid component phases and a void phase. The effective properties of materials are related to the distribution of component phases (the topology of the microstructure) in the domain of the unit cell using the homogenization method. Defining a level function as a series of basis functions of which the shape and values are governed by its representative parameters, and based on the concept of the level set method to describe the distribution of materials in the design domain, the microstructure (distribution of component phase in unit cell) design problem is formulated as a parameter optimization problem, in which the parameters are the representative parameters of basis functions. Two kinds of materials with prescribed elastic properties with positive and negative Poisson’s ratio respectively are designed by use of the proposed method. The topologies of the microstructures are clear and of high quality, which verifies that the method is effective.

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II. TDF BASED METHOD FOR MATERIAL DESIGN 2.1. Topology Description Function (TDF) The material considered in the material design process is a composite composed of several solid component phases and void phase constructed by periodically repeating a unit cell. The aim of material design is to find an optimal microstructure, i.e., the optimal distribution of components in the design domain (the unit cell domain). The main idea to describe the microstructure by TDF is as follows. In the design domain, a function T (x, y, z) called TDF is defined and the projection of its level set in the design domain forms several regions with different level values of the function. Regions with specified level values are placed by specified materials. In this way, the material distribution in the design domain is described by TDF. In order to fulfill topology optimization, TDF should be related to some parameters (design variables) and the change of TDF is controlled by changing these parameters. The parameterization of TDF is realized by constructing TDF as the superposition of basis functions hsi , each with its own parameters si = {xi , yi , hi , wi }T T (x, y) =

N 

hsi (x, y)

(1)

i=1

where the basis functions are defined as



(x − xi )2 + (y − yi )2 hsi (x, y) = hi exp − wi2

 (2)

Each of the basis functions can be considered as a ‘hill’, whose configuration is similar to that of Gauss probability distribution with center (xi , yi ), height hi and width wi , see Fig.1(a). In Fig.1(b), several basis functions are superposed. Using several cut-off levels, the geometry can be obtained by mapping TDF onto the design domain in the following manner. The regions of different value levels are placed different solid materials or void. In this way, the microstructure (the distribution of materials in the

Fig. 1. The basic concept of a TDF.

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design domain) is completely controlled by parameters of basis functions, see Fig.1(c). The finite element model of microstructure is shown as Fig.1(d). If the material designed consists of a solid material phase and a void phase, the elastic properties of the material at an arbitrary point (x, y) can be expressed as D(x, y) = D 0 H [T (x, y) − T0 ]

(3)

where D 0 is the elastic matrix of the solid material phase, T (x, y) denotes the value of TDF at point (x, y), and T0 is the value of the cut-off level. H (t) is modified Heaviside function defined by ⎧ α, t < −Δ ⎪ ⎪ ⎪ ⎪ ⎨

3 1+α H (t) = 3 (1 − α) t − t (4) , −Δ ≤ t < Δ + ⎪ 3 ⎪ 4 Δ 3Δ 2 ⎪ ⎪ ⎩ 1, t≥Δ where Δ and α are small positive numbers. Here, α is introduced to avoid stiffness singularity by assigning a small elastic stiffness to the void phase, and 2Δ denotes the width of ‘gray material’ in boundary. As shown in the Eqs.(4), T (x, y) > T0 + Δ, the point (x, y) is placed the solid material; T (x, y) < T0 − Δ is void; else is ‘gray material’. In this paper, the effective elastic properties of materials are obtained using the homogenization method. Based on the basic idea of homogenization theory[15] , the effective elastic properties of the materials are presented as  1 H D = [D − Dεy (Φ)] dy (5) |Y | Y

where, Y denotes the design domain, D is the elastic matrix at an arbitrary point. Generalized dis 11 22 12 j  j j  placement vector, Φ = Φ , Φ , Φ , Φ = Φ1 , Φ2 , j = (11, 22, 12), is the periodic solution for the following problem  εT ∀v∈Z (6) y (v) [D − Dεy (Φ)] dy = 0 Y

where Z is the space of the periodic vector defined on unit cell Y , εy (·) is the matrix of the strain operator with ⎡ ⎤ ∂ ∂ T 0 ⎢ ∂y1 ∂y2 ⎥ ⎥ εy (·) = ⎢ (7) ⎣ ∂ ⎦ ∂ 0 ∂y2 ∂y1 2.2. Formulation of Material Design Problem 2.2.1. Design variables Design variables are parameters of basis functions. Every basis function defined in the paper has four parameters. Although all of them can be selected as design variables, some of them may be fixed in the design process in order to simplify the problem. In this paper, the widths and coordinates of the central point of the basis functions are fixed, which means that only the heights of the basis functions are selected as design variables. 2.2.2. Objective function The objective function is shown as follows:  H        ¯ H 2 + ξ2 DH − D ¯ H 2 + ξ3 DH − D ¯ H 2 + ξ4 DH − D ¯H 2 f (s) = ξ1 D11 −D 11 22 22 12 12 33 33

(8)

H H H H ¯H , D ¯H, D ¯H, D ¯ H are desired , D22 , D12 , D33 are the elements of the effective elastic matrix, D where D11 11 22 12 33 values of effective properties, and ξ1 , ξ2 , ξ3 , ξ4 are weighting factors, which are used to adjust all desired values of effective properties.

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2.2.3. Constraint function When the design material has the prescribed properties, the volume fraction of solid material need be restricted, which is shown as V low ≤ V ≤ V up (9) where V is the volume fraction of solid material and V low and V up are the lower and upper limits respectively. The volume fraction of solid material is expressed as  1 1 V = H [T (x, y) − T0 ]dy (10) V0 |Y | Y

The design problem of material with prescribed elastic properties can be formulated as To find: X = {h1 , h2 , · · · , hn }T  H        ¯ H 2 + ξ2 DH − D ¯ H 2 + ξ3 DH − D ¯ H 2 + ξ4 DH − D ¯H 2 minimize: f (X) = ξ1 D11 −D 11 22 22 12 12 33 33

(11)

s.t: V low ≤ V ≤ V up where n is the number of the basis functions. 2.3. Sensitivity Analysis The sensitivities of the objective function and the volume fraction with respect to a design variable hi can be expressed as H H  H   H  ∂f (X) H ∂D11 H ∂D22 ¯ 11 ¯ 22 = 2ξ1 D11 −D + 2ξ2 D22 −D ∂hi ∂hi ∂hi H H  H    ∂D 12 H ¯H ¯ H ∂D33 D + 2ξ3 D12 − D + 2ξ − D (12) 4 12 33 33 ∂hi ∂hi ⎤ ⎡   N ∂H [T (x, y) − T0 ] ∂V 1 1 1 1 ∂ ⎣ = dy = δ [T (x, y) − T0 ] hsj (x, y)⎦ dy (13) ∂hi V0 |Y | ∂hi V0 |Y | ∂hi j=1 Y

Y

From Eqs.(5) and (6), the sensitivity of the effective elastic matrix can be determined by  ∂D 1 ∂D H = [I − εy (Φ)]T [I − εy (Φ)] dy ∂hi |Y | ∂hi

(14)

Y

where ∂H [T (x, y) − T0 ] ∂D ∂T = D0 = D 0 δ [T (x, y) − T0 ] ∂hi ∂hi ∂hi ⎤ ⎡ N ∂ ⎣ = D0 δ [T (x, y) − T0 ] hsj (x, y)⎦ ∂hi j=1 δ (t) is a modified Dirac delta function defined by ⎧ 0, t < −Δ ⎪ ⎪ ⎪ ⎪ ⎨

2 δ (t) = 3 (1 − α) 1 − t , −Δ ≤ t < Δ ⎪ ⎪ 4 Δ Δ3 ⎪ ⎪ ⎩ 0, t≥Δ

(15)

(16)

As shown in Eqs.(16), only ‘gray material’ takes effect on the sensitivities of the objective function and constraint function.

III. EXAMPLES In order to verify that the TDF based method is effective for designing material with prescribed properties, two examples are given. One of them is the design of a material with positive Poisson’s ratio, and the other, one with negative Poisson’s ration. The design domain is shown in Fig.2.

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Fig. 2. Design domain.

3.1. Example 1: Material with Positive Poisson’s Ratio In order to compare the result obtained by the TDF suggested in the paper with that of Ref.[4], the microstructure of the materials with the following prescribed elastic properties are designed first. ⎡ ⎤ 1 0.75 0 ⎢ ⎥ 1 0 ⎦ × 9 GPa (17) ⎣ 0.75 0

0

0.125

The volume fraction of the solid material is 28%, which is performed by setting V low = V up = 0.28. The design domain is meshed by 40 × 40 quadrilateral isoparametric elements. The elastic modulus of the solid material is 91 GPa, the Poisson’s ratio is 0.3. The parameter α of Heaviside function is 1.0 × 10−5 . The weight factors are: ξ1 = ξ2 = ξ4 = 1.0, ξ3 = 1000.0. 12 × 12 basis functions are equally placed in the design domain. Owing to the symmetry of the problem, the heights of basis functions placed in the left bottom 1/4 part of domain are design variables. So there are 36 design variables while in the corresponding variable density method there are 400 design variables. When a material with prescribed properties is designed with a variable density approach, it is difficult to obtain desired geometry if the initial design variables are uniform[6] . The same problem is encountered if the initial heights are uniform. Hence the heights are initialized by random numbers between 800 and 1200. The initial widths of basis functions remains uniform, 6.0. After 76 iteration steps the final geometry is shown in Fig.3(a) and (b). The geometry from Ref.[4] is shown in Fig.3(c). By the comparison between two results, it is evident that they have similar geometry, which shows that the proposed method can give distinct topology of the microstructure.

Fig. 3. Microstructure materials with prescribed elastic properties given by Eq.(17), whose Poisson’s ratio is positive, the black regions consist of solid materail.

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3.2. Example 2: Material with Negative Poisson’s Ratio As for the second example, let the design of the material with prescribed elastic properties be given as follows. In that case, the Poisson’s ratio desired is negative. ⎡ ⎤ 1 −0.6 0 ⎢ ⎥ 1 0 ⎦ × 6 GPa (18) ⎣ −0.6 0 0 0.8 The volume fraction of the solid material is 59%. 12 × 12 basis functions are uniformly placed in the design domain. The heights of basis functions placed in the bottom half domain are design variables. Other information is the same as Example 1. So there are 72 design variables while in the corresponding variable density method there are 800 design variables. After 117 iteration steps the final geometry is shown as Fig.4(a) and (b). The geometry from Ref.[4] is shown in Fig.4(c) and (d). Comparison shows that the design topology obtained by the proposed method is almost the same as that presented in Ref.[4]. Besides, the former has more distinct boundary than the latter.

Fig. 4. Microstructure with certain coefficients, whose Poisson’s ratio is negative, the black regions consist of solid material.

IV. CONCLUSION In this paper, a topology description function (TDF) based method for material design is proposed. The TDF is used to describe the topology of the microstructure (the distribution of the component materials in a microscale domain) by placing different materials in regions with different value levels of the TDF. The TDF is expressed as the superposition of a series of basis functions with their own parameters; the topology of the microstructure is controlled by the parameters of all the basis functions. In this way, the topology optimization problem of material design is formulated as a parameter optimization one, which makes the problem solving easy and able to be solved by existing optimization methods. The design variables of the TDF based optimization model of material design is decoupled with the finite element mesh, implying that refinement of the mesh for improving the analysis accuracy does not influence the design space. Like other methods based on the concept of the level set, the TDF based

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method can determine unambiguously the points of the design domain that has become void or the individual material without checkerboard-like numerical instabilities, and there is no need to solve the Hamilton-Jacobi equation for determining the boundary moving speed which is done in the conventional level set method. Two kinds of materials with prescribed elastic properties including the materials with negative Poisson’s ratio are designed with the proposed method, the design results verifying the efficiency of this method for material design.

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