Journal of Computer-Aided Materials Design (2004) 11: 117–138 DOI 10.1007/s10820-005-3169-y
© Springer 2005
Synthesis of shape and topology of multi-material structures with a phase-field method MICHAEL YU WANG∗ and SHIWEI ZHOU Department of Automation & Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong Received 1 February 2004; Accepted 17 July 2004 Abstract. In this paper, we present a phase-field method to the problem of shape and topology synthesis of structures with three materials. A single phase model is developed based on the classical phase-transition theory in the fields of mechanics and material sciences. The multi-material synthesis is formulated as a continuous optimization problem within a fixed reference domain. As a single parameter, the phase-field model represents regions made of any of the three distinct material phases and the interface between the regions. The Van der Waals–Cahn-Hilliard theory is applied to define a dynamic process of phase transition. The -convergence theory is used for an approximate numerical solution to this free-discontinuity problem without any explicit tracking of the interface. Within this variational framework, we show that the phase-transition theory leads to a well-posed problem formulation with the effects of “domain regularization” and “region segmentation” incorporated naturally. The proposed phase-field method is illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods. It is further suggested that such a phase-field approach may represent a promising alternative to the widely-used homogenization models for the design of heterogeneous materials and solids, with a possible extension to a general model of multiple material phases. Keywords: interface evolution, multi-material structures, phase field method, structure optimization
1. Introduction A heterogeneous or multi-material object is referred to as a solid object made of different constituent materials. The object may be synthesized with continuously varying material properties satisfying prescribed material conditions on a finite collection of material features. The material composition has a discontinuous change across the interface of material regions in the solid. The material interface in the solid is usually used to explicitly describe the shape and topology of a structural object. The continuously varying material composition produces gradation in material properties, as they are often known as functionally gradient materials. The use of heterogeneous objects has increased rapidly in the past decade due to the rise of emerging technologies [1]. Objects with varying material properties across spatial domain are common in biomedical, geophysical and nano-scale modeling. Rapid prototyping techniques have been developed to allow material composition to be varied by region, by layer, or point-wise. Novel nano-technologies are emerging with new materials synthesized by manipulating their atomic structures. Applications ∗
To whom correspondence should be addressed. E-mail:
[email protected]
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of heterogeneous materials range from bio-medical products, aerospace structures, to meta-materials, with demonstrated performance improvements in structural, mechanical, electronic, and/or sensing and control properties. In these applications, the performance goals are achieved because of the capability of varying material properties globally and locally across the spatial domain and/or the scale domain. In order to take full advantage of the greatest potential of heterogeneous objects, one must have matching capabilities for their computer modeling, analysis and design optimization. The representation and modeling schemes have been a primary focus of the research on heterogeneous object design. A typical class of representation schemes is based on the subdivision concept in which the solid model is discretized into sub-regions and analytical blending functions are assigned for each region to describe material composition [1, 2]. These schemes have many theoretical and practical complications and are extensive to compute [2]. Methods for the design of heterogeneous objects have been extensively studied in the context of shape and topology optimization in structural design. Powerful analysis and optimization methods are available now for topology optimization of continuum structures with multiple physics and/or multiple materials [3–5], particularly, using homogenization methods [3, 4], material interpolation schemes [6–8], or levelset models [9, 10]. These approaches have addressed some fundamental issues such as well-posedness, topological flexibility, fidelity of geometric representation, and computational efficiency [3, 11, 12]. The homogenization-based methods have been a main approach to structural optimization [3,4,13], in which the domain of the structure to be optimized is fixed in a reference domain and is discretized with the elements for finite element analysis. In this so-called “raster” geometric model, the design variables are the artificial densities of the elements and their material properties are modeled in terms of a set of material interpolation functions such that the intermediate properties are penalized, usually by using a “power-law” applied on the relative material density. For a structure of multi-phase materials (with “void” as one phase), this type of method requires a “rule of mixture” to formulate the overall material properties. For the simple case of two-phase material (e.g., “solid” and “void‘’), the overall modulus, for example, is described artificially as E(ρ) = ρ p E1 + (1 − ρ p )E2 , where ρ ∈ [0, 1] is the density variable of a non-void phase, E1 and E2 are the respective moduli of the material phases, and p > 1 is the penalization parameter for the intermediate densities in the final design. The “void” phase can be incorporated in the model by letting E2 = 0. In that case, it has been shown that there exist physical microstructures for the power-law models when p ≥ 3 [6]. In fact, such a physical model can be directly confirmed by using an approximate linear analysis of honeycomb structures [14]. For the three-phase material design with two solid materials and void, an extended power-law interpolation scheme has been widely employed with two design variables ρ1 ∈ [0, 1] and ρ2 ∈ [0, 1] p p p E(ρ1 , ρ2 ) = ρ1 ρ2 E1 + 1 − ρ2 E2 . Thus, the number of design variables is doubled in the three-phase material model compared to the two-phase material interpolation. Generally, the material mixture
Synthesis of shape and topology 119 method requires (n − 1) design variables for n distinct material phases, as all of the phases comprise the fixed design domain. In other words, the number of design variables is linear to the number of material phases. Recently, other interpolation schemes were proposed with the use of a peak function without increasing the number of the design variables in the optimization procedure [7, 8]. Furthermore, the interpolation model of the multiple phases can become quite complicated [8]. While the above interpolation schemes are based on the common Voigt model, the rule of mixture can be certainly modified to favor certain characteristics of the “phase mixtures” [15]. For example, the well-known Halpin–Tsai composite model and the weighted average over the Voigt–Reuss bounds or over the Hashin–Shtrikman bounds have been used in the literature. Although the rule of mixture is an artificial construction in the context of the topology optimization and a successful power-law would result in a single, distinct material phase within each design element at the final optimal solution, the interpolation model usually has a strong influence on the course of the optimization process and on the final design (due to the existence of local optima). An appropriate “mixing” model is yet to be developed for a general interpolation of the physical properties of multiple material phases (n > 3). Another key issue in the homogenization and the power-law approaches is related to the fundamental fact that the problem of topology optimization as stated is an illposed problem in its mathematical theory and numerical methods (cf. [16, 17]). The relaxation procedure of the homogenization method makes the problem well-posed; but it has a “side-effect” that the optimal solutions generated commonly have perforated microstructures in the resulting design, as expected in consistence with the relaxation [18]. Unfortunately, perforated microstructures are difficult to manufacture. Various suppression-based regularization approaches have been suggested, including adding more constraints into the problem such as perimeter controls [11] and slope constraints [19], and employing filters for chattering solutions [5, 19, 20]. Fundamentally, however, the suppressions do not directly address the chattering problem underlying the relaxation concept. In this paper, we present an alternative method for shape and topology synthesis of structures of multiple materials based on a phase field model. We adapt the theory of phase transitions from mechanics and material sciences, where the problem of stability of systems containing several instable components has been extensively studied over years. These components may be liquid phases having different levels of density distribution [21, 22] or solid phases under solidification [23]. Through the Van der Waals–Cahn-Hilliard theory of phase transitions, the stable configuration is proven to be a compound of homogeneous regions separated by sharp interfaces having minimal length. The theory also leads to the analysis of the interface between the phases during the process of phase transition to the system stability. In our approach proposed here, we formulate the shape and topology optimization problem as a phase transition problem and extend the mechanics results to a multimaterial domain. Similar to the setting of material distribution approach, the design is restricted within a fixed reference domain. A continuous phase variable θ(x) is used within the design domain x ∈ such that it leads to distribution of the multiple materials in the final designed structure. As in the basic topology optimization problem, this basic problem here is also ill-posed. Therefore, we will apply a variational
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regularization theory for a well-posed problem formulation. The role of this variational model is to partition the reference domain into distinct regions, with each region being made of a single material phase. In addition, the interface between the regions is characterized by a sharp transitions of the phase variable θ(x) across x. In other words, the model induces an “edge-preserving” solution of a “region segmentation” of the design domain. The phase-field model presented here incorporates three distinct material phases and is an extension of the basic phase-field method [24] for structures of bi-material phases of solid. In the following, we first define the variational problem of shape and topology optimization. A phase-field model is then described for diffusion-based regularization and phase transition, based on the Van der Waals–Cahn-Hilliard theory of phase transitions in mechanics. For three different material phases, we present a periodic phase-field such that three materials are represented without any specific order between their possible transitions. The interface between any two different phases is described by a diffuse-interface (or “soft”-interface). We then discuss a method of numerical solution to the variational problem based on the theory of -convergence in the field of free-discontinuity problems. The solution amounts to solving a partial differential equation. Finally, the proposed variational method is illustrated with 2D examples of mean compliance minimization typically used in the literature of topology optimization, especially in the homogenization based methods. 2. The shape and topology optimization problem In this paper, we use a linear elastic structure to describe the problem of structural optimization. Conceptually, the approach presented here would apply to a general structure model. Let ⊆ R d (d = 2 or 3) be an open and bounded set occupied by a linear elastic structure. The boundary of consists of three parts: = ∂ = 0 ∪ 1 ∪ 2 , with Dirichlet boundary conditions on 1 and Neumann boundary conditions on 2 . It is assumed that the boundary segment 0 is traction free. The displacement field u in is the unique solution of the linear elastic system −div σ (u) = f u = u0 σ (u) · n = h
in , on 1 , on 2 ,
(1)
where the strain tensor ε and the stress tensor σ at any point x ∈ are given in the usual form as ε(u) =
1 ∇u + ∇uT , 2
σ (u) = E : ε(u),
with E as the elasticity tensor, u0 the prescribed displacement on 1 , f the applied body force, h the boundary traction force applied on 2 such as an external pressure load exerted by a fluid, and n the outward normal to the boundary. In this paper, we consider to use three different material phases to partition the whole material domain of the structure into regions = (1 , 2 , 3 ) for each distinct material phase. This partition leads to
Synthesis of shape and topology 121 =
3
i
and
i ∩ j = Ø,
i = j.
(2)
i=1
The boundary of i is specified ∂i is specified by and the interface between ∂i and ∂j is given by ∂i ∩ ∂j . The material properties in each region are given by the corresponding material phase. For example, the elasticity tensor in is given by E(x) = E 1 , E 2 , or E 3 depending on if x ∈ 1 , 2 , or 3 , respectively. Thus, the “basic” problem of structure optimization is specified with respect to a specific objective function described by F (u) such that min imize J (u, ) = F (u)d,
subject to : where
G(u, ) ≤ 0, 3 = i .
(3)
i=1
The variational equation of the linear elasticity is written as Eε(u) : ε(v)d = f · v d + h · v d for all v ∈ U, 2 U = u : u ∈ H 1 (); u = u0 on 1 ,
(4)
with U denoting the space of kinematically admissible displacement fields and ‘:’ representing the second order tensor operator. The system constraints are described by functions G(u, ), including, for example, the limit on the amount of each material phase in terms of its maximum admissible volume. The goal of optimization is to find a minimizer such that the design has the following characteristic functions [3, 4, 6, 13]: 1 for x ∈ i , χi (x) : i → {0, 1} such that χi (x) = (5) 0 for x ∈ \i , with the domain partition conditions (2) satisfied. The basic optimization problem as stated is known to be ill-posed as it involves the “free-discontinuities” of the boundaries ∂i of i . 3. The phase-field model of three materials We now introduce a phase-field model based on the phase-transition theory in mechanics and in image processing. The theory of phase transitions has been developed for the study of mechanical systems of a material compound of instable phases [21, 22, 25]. The material may be solid or liquid of two of more phases. The common problem is to characterize the stability of such a system and to describe the interface between the phases while the system undergoes a physical process (e.g., solidification) to reach its stability. From this process point of view, there is a similarity between our basic topology optimization problem (3) and a phase transition system of three material phases.
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3.1. The phase-field parameter In an unstable configuration of a multi-phase mixture, there are two kinds of distinct quantities to consider: the distinct material phases and their interface. The interface can be described as an evolving surface whose motion is controlled according to the physical models consistent with the mechanism of transformation. The interface is simply a mathematical surface with no width or structure; it is said to be a sharp-interface. The phase interface, however, may have a complex topology, and the need to track this boundary has made sharp-interface models difficult to use and to implement with good numerical properties. These difficulties associated with topological changes in the problems of phase transitions such as alloy solidification are essentially identical to those of the problem of topology optimization. The phase-field method is an alternative to the sharp-interface model widely used in the fields of mechanics and material sciences. In the phase-field method, the state of the entire multi-phase system is represented continuously by a single variable θ known as the phase parameter. For example, θ = 1, θ = 0, and 0 < θ < 1 represent the two distinct phases of solid and fluid and the interface, respectively. The interface is therefore located by the region over which θ changes from, for example, the liquid phase to the solid phase [22]. The range over which it changes is the width of the interface. Therefore, the interface is also regarded as a diffuse-interface [22, 25]. The set of values of the order parameter over the whole system domain is the phase field. Such a phase-field model eliminates the assumption of a sharp interface and the concomitant need to track its motion. Indeed, if a theory can be created to describe the dynamics of transitions between the phases in terms of the phase parameter, then we no longer need to track the interface. Instead we can follow the evolution of the phase field. Phase-field methods have been extensively studied for more than two decades (cf. [26, 27]), and the reader is referred to the recent reviews of their applications to solidifications [28] and to a wide range of other material processes [29]. It is straightforward to adapt such a phase field model for the case of single material structures, i.e., of two phases of solid and void. We may simply set the density of material phase ρ(x) as the phase parameter, i.e., ρ = θ, while ρ = 1 representing solid and ρ = 0 the void. Then, the intermediate values of ρ (0 < ρ < 1) have a new meaning – they indicate the interface between solid and void or the boundary of the structure being optimized. This approach is developed in [24], showing its potential in application to structure design. In the phase transition theory, a single phase field is sufficient to represent the domain partition into n multiple phases if the phases are ordered. In this case, a phase is allowed to change to a specific phase only, but it cannot be arbitrarily transformed to any other phase. The phase-field becomes an order parameter and it assumes n distinct values to represent the material phases [30]. For example, in the model presented in [17], the three material phases of a structure of a water dam are liquid, solid and void, and they are specified at the values of θ = 1, −1, and 0. Physically, it is not meaningful for a liquid region to be connected to a void region. Thus, the three phases are ordered as so. In general, if the phases are not ordered, then it requires n phase field parameters (or a vector phase field) to represent n material phases [30].
Synthesis of shape and topology 123 For our basic topology optimization problem (3) the material phases are assumed generally not ordered. If we use the general vector phase field model for n material phases, we would have n phase-field parameters, thus substantially increasing the complexity of the optimization solution. Here, we consider the case of three material phases and present a model using a single phase parameter defining a periodic field, in an effort to reduce the number of variables in the optimization process. As in the classical phase-field theory, we will represent the domain partition by a single phase-filed θ ∈ R which takes three values 0, (2/3)π, (4/3)π when corresponding to the three material phases. The volume fractions of the material phases at the point x, ρ1 (θ(x)), ρ2 (θ(x)) and ρ3 (θ(x)), are controlled by the phase-field θ(x) such that 1−sin(3/4θ )2 2kπ ≤ θ ≤ 2/3π + 2kπ, 1+2 sin(3/4θ )2 , (6) ρ1 (θ) = 0, 2/3π + 2kπ ≤ θ ≤ 4/3π + 2kπ, 1−cos(3/4θ )2 4/3π + 2kπ < θ ≤ 2π + 2kπ, 2, 1+2 cos(3/4θ2) 1−cos(3/4θ ) 2kπ ≤ θ ≤ 4/3π + 2kπ, 2, ρ2 (θ) = 1+2 cos(3/4θ ) (7) 0, 4/3π + 2kπ < θ ≤ 2π + 2kπ, 0, 2kπ ≤ θ < 2/3π + 2kπ, ρ3 (θ) = 1−sin(3/4θ )2 (8) , 2/3π + 2kπ ≤ θ ≤ 2π + 2kπ, 1+2 sin(3/4θ )2 where k is an integer. Clearly, the densities of the three phases are periodic functions of the phase field parameter −∞ < θ < ∞ as shown in Figure 1. Only at the values (0, (2/3)π, (4/3)π) of θ ∈ [0, 2π), only one of the three volume fractions (ρ1 , ρ2 , ρ3 ) assumes value 1 while the other two both assume value 0 (see Figure 1). This corresponds to the ideal model of domain partition with distinct material phases. In the approximate models, however, the intermediate values of ρi (0 < ρi < 1, i = 1, 2, 3) will occur, satisfying
Figure 1. Volume fractions of the three materials as a periodic function of the phase-field for 0 ≤ θ ≤ 2π (top) and for −∞ < θ < ∞ periodically (bottom).
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ρ1 (θ) + ρ2 (θ) + ρ3 (θ) = 1 for x ∈ .
(9)
As −∞ < θ < ∞ varies, the composition of the material phases may change from one particular phase to either other phase. Thus, the single phase-field parameter can accommodate three non-ordered material phases. The intermediate density values have a new meaning – they indicate the interface between the regions of distinct material phases of the structure being optimized. Physical and material properties of the structure are specified in terms of the volume fractions of the material phases. For example, the mass density and the elastic modulus of the structure at the point x are given, respectively, as m(x) = ρ1 (θ)m1 + ρ2 (θ)m2 + ρ3 (θ)m3 , E(x) = ρ1 (θ)E 1 + ρ2 (θ)E 2 + ρ3 (θ)E 3
(10) (11)
with the corresponding mass densities and elastic moduli of the three material phases denoted by mi and E i (i = 1, 2, 3). The goal of optimization is to find a minimizer such that
m(x) =
3
mi χi (x).
(12)
i=1
3.2. The phase transition theory Strategically, what we need to describe our variational approach with the phase-field model is to define an optimization process with the phase-field variable θ(x) in our system. In particular, we must first formulate the dynamics of the phase transition. The dynamic model we consider is based on the phase transition theory of Van der Waals and Cahn-Hilliard in mechanics (cf. [21, 22, 25, 30]). The evolution of a three-phase system is assumed to be governed by a generalized free energy in the following form: |∇θ (x)|2 dx +
Eε (θ, ε) = ε
η ε
W (θ(x))dx.
(13)
The first term depends only on the gradient of θ(x) and hence is non-zero only in the interfacial region. Thus, it represents the interfacial energy of the system and it acts to stabilize the interfacial transition region. It can also be viewed as regularization term to overcome an ill-posed problem of optimization. Term |∇θ|2 dx is widely used in the field of image processing as Thikonov regularization [16, 31]. This is known as isotropic smoothing in the field of variational approaches [32]. Thikonov regularization is equivalent to linear (Gauss) filtering of the solution field, but it does not preserve edges. In our topology optimization problem, “edges” would represent the interface between the material phases. Edges are the most important features in our problem, and they are defined as sharp transitions of the density level.
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Figure 2. The potential-well function W (θ) for 0 ≤ θ ≤ 2π (top) and for −∞ < θ < ∞ periodically (bottom).
Thus, the regularization method should be able to yield sharp material transitions, or “edge-preserving”. Following the variational analyses of anisotropic smoothing developed in the field of image processing [32], we may replace Thikonov regularization term |∇θ|2 dx with ϕ(|∇θ (x)|)dx, where function ϕ is regular and continuously differentiable with certain properties required for an “edge”-preserving effect [16, 32]. Some widely used edge-preserving functions include ϕ(s) = |s| (total variation), ϕ(s) = s 2 /(1 + s 2 ), and ϕ(s) = log(1 + s 2 ), to name a few. The second term W is a potential inducing a “region classification” constraint. This means that the optimization process remains to produce a crisp design, where the whole material domain of the structure will be partitioned into distinct regions of material phases, and every point of a partitioned region belongs to the same “phase class” of a material. In other words, for every point x ∈ we must be able to determine if x is a boundary point on an interface between regions and which material phase it belongs to. Function W (θ) has as many minima as the number of phases (i.e., 3 in this model) and imposes a level constraint on the solution. Here, η > 0 is the weight in the variational model. This is a model adapted from the phase-field theory of phase transition problems studied in mechanics. As in the classical Cahn-Hilliard theory [25], W (θ) is taken to be a double-welled potential so that the system separates into three distinct phases when θ(x) takes any value of (0, (2/3)π, (4/3)π) such that W (0) = W ((2/3)π) = W ((4/3)π) = 0. Furthermore, it is assumed that W is quadratic around these values. Therefore, one may use the following function for W (as shown in Figure 2):
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0 ≤ θ < π/3, π/3 ≤ θ < π, (14) π ≤ θ < 5/3π, 5π/3 ≤ θ ≤ 2π.
The Van der Waals–Cahn-Hilliard theory of Eq. (13) has let to the introduction of the perturbation term ε|∇θ|2 , with small ε. The presence of the coefficient ε is necessary in order for the diffuse-interface system to converge to the sharp-interface system, in the limit as ε → 0. The two generalized free energy terms together keep the thickness of the interfacial region proportional to ε, and the asymptotic behavior of the model as ε → 0 allows us to solve for stable configurations characterized by min W (θ(x))dx, ρi (x)dx = Vi (i = 1, 2, 3), θ
where Vi is the volume ratio of the material phase. This technique belongs to the general theory of -convergence, and it has been extensively studied as the subject of gradient theory of phase transitions [33]. The numerical analyses for the variational problem of minimizing (13) concern a sequence of minimizers of Eε (ε, θ) as ε → 0 [33, 34]. -convergence implies existence of a minimum solution and the convergence of the minimizer sequence to the minimum solution [34]. A general analysis has been extended to multiple wells of a vector phase-field (see p. 221, [16]). 3.3. The phase-field model of three material phases Based on the theories on model regularization and phase-transition, it is straightforward to transport these ideas into a computational model for a continuous variational solution of the optimization problem (3). The phase-field model relies on the minimization of the following generalized “energy” function: η Jε (θ, u) = F (u, θ)dx + εµ ϕ(|∇θ(x)|)dx + W (θ(x))dx. (15) ε The corresponding topology optimization problem is to find a solution θ ∗ such that θ ∗ = lim [arg min Jε (θ, u)], ε→0
θ
(16)
subject to all given constraints. As before in (13), ϕ(s) might be taken as an edge-preserving function. When ϕ is a convex function, the inclusion of the objective functional F (u)dx has no effect on the conclusions of the -convergence theory stated before [16, 32]. Thus, as ε → 0 there exists a sequence of minimizers θ of Jε converging to a solution of distinct regions of different materials. solid and void separated by sharp interface with regularized properties, When ε → 0 it can be seen that the contributions of the two energy integral terms have the same order for the minimizing sequence. When ε is not too small (ε 0), the
Synthesis of shape and topology 127 level constraint of the third term of Jε is quite insignificant; only smoothing process occurs with the second term. As ε further decreases, the diffusion process gets progressively “softened”, while the “classification” process get stronger in the third term. This would “sharpen” the interface and force the solutions to form distinct phase regions. The role of the term W is clear: it forces the stable solution to take one of the three phase field values (0, (2/3)π, (4/3)π), while the effect of the gradient energy |∇θ|2 term is to penalize unnecessary interfaces. In fact, it is proven that the -convergence sequence results in a minimal perimeter of the interface between the material phases [34]. Thus, the generalized free energy has a regularization effect on the solution θ(x) by avoiding formation of any singularities and by restricting the space of solutions. While the mathematical analysis is yet to complete for a general objective function F (u) rather than the structure’s mean compliance, various numerical experiences seem to confirm that the variational formulation provides a well-behaved framework for seeking meaningful optimal solutions, particularly when the models of the structure have finite perimeter. 4. Minimization solutions Based on the previous discussions, we now define the general objective functional of our variational model for problem (3) as follows: η Lε (θ, u) = F (u, θ)dx + εµ ϕ(|∇θ(x)|)dx + W (θ(x))dx + λT G(u, θ) (17) ε where Lagrange multipliers λ are used to incorporate the given constraint functions G, without loss of generality. The elastic displacement field u must satisfy the bi-linear equation (4) of the linear elasticity system. The periodic phase-field variable θ(x) is defined within −∞ < θ (x) < ∞, but practically it is sufficient that θ(x) ∈ [0, 2π]. The corresponding topology optimization problem is to find a solution θ ∗ such that θ ∗ = lim [arg min Lε (θ, u)] ε→0
θ
(18)
and a set of Lagrange multipliers satisfying the Kuhn–Tucker conditions associated with the objective functional Lε . The variational solution to minimize (17) would yield a numerical algorithm for a minimizer. The necessary condition required is the Kuhn–Tucker condition, which is derived from the Euler–Lagrange equation to compute the derivative of the functional Lε with respect to θ. This leads to the following system of partial differential equations: ϕ (|∇θ|) η
F (u, θ) − εµdiv ∇θ + W (θ(x)) + λT G (u, θ) = 0 for x ∈ , |∇θ| ε (19) ∂θ(x) + = ∇θ · n = 0 on ∂, ∂n+ where n+ is the outward normal to the boundary of the reference domain , div = ∇· denotes the divergence operator, and F (u, θ) and G (u, θ) denote the respective Euler derivative of F and G with respect to the phase-field variable θ . It should be pointed out that generally F may not depend explicitly on the phase-field θ but rather on the
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Figure 3. A bridge type structure with multiple loads.
equilibrium displacement u and the interpreted material properties such as the elastic modulus E(θ ) (Eq. (11)) associated with the design. Thus, the Euler derivative may be obtained by using the general shape derivative theory. The theory has been well established (cf. [35, 36]), and the shape derivatives are also described in the recent work on level set methods for topology optimization [9]. We shall omit the details of its derivation here. For our problem of topology optimization, the partial differential equations (19) are generally non-linear, owning to a complex Euler derivative of F (u) and/or an edge-preserving regularization function ϕ(s). 5. Numerical examples Numerical examples are presented in this section for mean compliance optimization problems that have been widely studied in the relevant literature (e.g., [4, 20]). The objective function of the problem is the strain energy of the structure while the portions of the three material phases are specified as constraints, J (u, ) = Eε(u) : ε(u)d, Gi (θ) = ρi (θ)dx − Vi = 0 (i = 1, 2, 3),
(20)
where Vi is the volume ratio of the given material phase such that V1 + V2 + V3 = 1. Note that void is considered as a material phase with negligible material properties as typically dealt with in the homogenization based methods for shape and topology optimization. For clarity in presentation, the examples are in 2D under plane stress condition. 5.1. Bridge structure A bridge type structure is considered first. A rectangular design domain of L long and H high with a ratio of L : H = 12 : 6 is loaded vertically at its bottom with multiple loads P1 = 20 N and P2 = 10 N as shown in Figure 3. The left bottom corner of the beam is fixed, while it is simply supported at the right bottom corner. Two materials are used with the modulus of elasticity of E 1 = 200 GPa and E 2 = 100 GPa separately and with the same Poisson’s ratio of ν = 0.3. The third material phase is void, E 3 ≈ 0. Volume ratios of the three materials are specified as V1 = 0.2, V2 = 0.1 and V3 = 0.7. A mesh of 100 × 50 quadrilateral elements is used for the discrete analysis and optimization. In this example, Thikonov regularization function ϕ(s) = s 2 is used, and we set µ = 0.32 , η = 0.12 .
Synthesis of shape and topology 129
Figure 4. The first 18 solution steps of the bridge example (in red (black), blue (grey) and white color for the three phases, respectively) with m being the step number of -convergence.
For all three examples presented here, the three material phases of E 1 , E 2 , and E 3 , respectively, are illustrated in respective red, blue and white color in the figures of the obtained optimal designs. In the initial design for each of the three examples, the phase-field variable is set to be θ = 6 such that initially ρ1 = 0.96, ρ2 = 0, and ρ3 = 0.04. A selective set of intermediate results are shown in Figure 4, as the sequence of optimal solutions converge to the final design, with m being the step number of convergence as ε → 0. Within each step of a small value of ε, the optimization problem for (17) is solved with the method of moving asymptotes (MMA) as an iterative process [3]. For this example, it is found that it is sufficient to use 100 iterations to obtain a solution for (17) in the -convergence sequence of ε → 0. Changes in mean compliance (J1 ), the regularization energy term J2 = εµ φ (|∇θ |)dx and the potential energy term J3 = (η/ε) W (θ)dx during the process are shown in Figure 5 in terms of the total number of iterations to illustrate its conver-
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Figure 5. Changes in the mean compliance, regularization, and potential energy for the bridge structure in the total number of iteration.
Figure 6. The MBB structure with fixed-simple supports.
gence. It is noticed that the regularization and potential energy terms undergo a rapid change at the beginning of each segment in the sequence of -convergence when ε is set to a smaller value in the process of ε → 0. 5.2. MBB beam This example is known as MBB beam related to a problem of designing a floor panel of a passenger airplane in Germany. The floor panel is loaded with a unit concentrated vertical force P = 1 N at the center of the top edge. It is has a fixed support and a simple support at its bottom corners, respectively (Figure 6). The design domain has a length to height ratio of 4:1. The material properties and volume ratios are the same with the first example. We use 60 × 30 quadrilateral elements to model right half of the structure due to the geometric symmetry. Using Thikonov regularization function ϕ(s) = s 2 , we obtain the optimization sequence as shown in Figure 7 for the first 45 steps of convergence. In this example, µ = 0.22 , and η = 0.052 . Changes in mean compliance (J1 ), the regularization energy (J2 ) and the potential energy (J3 ) during the process are shown in Figure 8. For this example, it is found that it is sufficient to use 50 iterations to obtain a solution for (17) in the -convergence sequence of ε → 0. 5.3. Cantilever beam The last example is a cantilever beam with a unit concentrated vertical force P = 1 N at the bottom of its free vertical edge. The design domain has a length to height ratio
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Figure 7. The first 45 solution steps of the MBB beam (in red (black), blue (grey) and white color for the three phases, respectively) with m being the step number of -convergence.
of 2:1 (Figure 9). The material properties and volume ratios are the same with the above two examples. We use 60 × 30 quadrilateral elements with Thikonov regularization function and µ = 0.32 , η = 0.12 . The results of optimization sequence are shown in Figure 10 for the first 45 steps of -convergence, while changes in mean compliance (J1 ), the regularization energy (J2 ) and the potential energy (J3 ) during the pro-
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Figure 8. Changes in the mean compliance, regularization, and potential energy for the MBB beam in the total iteration number.
Figure 9. The cantilever beam.
cess are shown in Figure 11. For this example, 50 iterations are used in each step of the -convergence. 5.4. Discussions of the examples Here we present further discussions of the numerical examples presented above. First, we examine the issue of grid-dependency of the proposed method with the third example of the cantilever beam. In the result given in Figure 10, a 60 × 30 rectilinear grid is used. The optimization design problem is also solved with a denser grid of 80 × 40 and 100 × 50, respectively. The final results of the optimization are shown in Figure 12. It is evident that the topology of the structure remains the same among these three different meshes and the fine discretization of the 100 × 50 grid appears to result in changes only in small geometric details of material distribution. For comparison with Figures 10 and 11, the first 18 steps of -convergence for the case of 100 × 50 grid mesh are shown in Figure 13, while changes in mean compliance (J1 ), the regularization energy (J2 ) and the potential energy (J3 ) during the process are shown in Figure 14. Next, these examples are compared with those obtained from using the SIMP method based on homogenization theory [3]. We employ the computer program provided in [3] and utilize the material interpolation scheme of power-law penalization
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Figure 10. The first 45 solution steps of the cantilever beam (in red (black), blue (grey) and white color for the three phases, respectively.) with m being the step number of -convergence.
as discussed in the introduction section of the paper. Figure 15 shows the designs obtained with the SIMP method for the three examples of the bridge structure, MBB beam, and cantilever beam, with all design parameters identical. Clearly, the optimal solutions of the SIMP method and of our phase-field method are very different for all of the three examples. In Table 1, we provide a comparison between the solutions for the mean compliance of the final design and the computational time of the methods. It is clearly shown that the phase-field method yields
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Figure 11. Changes in the mean compliance, regularization, and potential energy for the cantilever beam in the total iteration number.
Figure 12. Results of different mesh grid for the cantilever beam example.
a better solution for the bridge structure and the MBB beam. The value of mean compliance is 23% less for the bridge structure example and 11.7% less for the MBB beam example. For the cantilever beam example, however, the phase-field method yields a design with a slightly higher value of the mean compliance, about 4% higher than that of the solution from the SIMP method. Given the nature of the topology optimization problems, these solutions are all likely to be sub-optimal solutions. Generally speaking, it is difficult to make a direct comparison between these two methods. It should be further pointed out that the phase-field method has a much higher level of computational cost, about in one order of magnitude of that of the SIMP implementation. The computation times listed in Table 1 are based on our numerical computations on a PC with a 3.2 GHz CPU and 1 GB memory module. 6. Conclusions and Discussions In this paper, we have presented a phase-field method to address the shape and topology optimization of structures with multiple materials. In contrast to the widely-used material distribution approach based on homogenization, there is no distinction made between the material phases and their interface in the phase-field model. We have extended this concept to a three-phase model. A single field parameter is used to describe three different materials without any constraint on the order of their transition. This allows the entire design domain to be treated simultaneously, without any explicit tracking of the interface. Thus, the basic formulation of the free boundary problem of the topology optimization is replaced by a system of phase transition. We then apply the Van der Waals–Cahn-Hilliard theory to define the variational topology optimization as a dynamic process of phase transition to the system stability. Within this variational framework, the problem of synthesis of heterogeneous structure becomes a well-posed problem, allowing us to adapt the -convergence theory
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Figure 13. The solution steps of the cantilever beam for 100 × 50 grid.
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Figure 14. Changes in the mean compliance, regularization, and potential energy for the cantilever beam for 100 × 50 grid.
Figure 15. Optimal designs of the examples using the SIMP method (in red (black), blue (grey) and white color for the three phases, respectively).
Table 1. Comparison between the phase-field method and the SIMP method for the three examples. Bridge structure (100 × 50) SIMP Mean compliance Computing time Average time/iteration (s) Iterations (total)
155.5486 809.0 8.1 100
Phase-field 137.2827 7266.1 4.04 18 (each of 100)
MBB beam (60 × 30)
Mean compliance Computing time (s) Average TIME/iteration (s) Iterations (total)
SIMP
Phase-field
90.5044 594.7 5.95 100
69.3761 4731.7 2.10 45 steps (each of 50)
Cantilever beam (60 × 30)
Mean compliance Computing time (s) Average time/iteration (s) Iterations (total)
SIMP
Phase-field
79.9936 574.09 5.74 100
83.2081 4420.2 1.96 45 (each of 50)
Synthesis of shape and topology 137 for a numerical solution scheme. The proposed phase-field method is illustrated with several 2D examples that have been extensively used in the recent literature of topology and material optimization, especially in the homogenization based methods. While we have demonstrated the method only with examples of mean compliance optimization in two dimensions, this is mainly for convenience. Other regularization functions rather than Tikhonov function can also be employed. Currently, we are extending our approach to a general multi-phase model [23] for problems of optimization of heterogeneous materials and/or graded materials. The results are to be reported separately. When comparing the results obtained here with those of other methods, especially the homogenization-based and level-set methods, it is clear that these very different approaches produce very different optimal designs. It is clear that the final optimal topology of a structure depends heavily on the formulation of the problem, the geometric model, and the nature of regularization that changes the problem from being ill-posed to well-posed. In view of the recent discussions on the challenges and future of the variable-topology optimization (cf. [4]), the phase-transition approach proposed here seems to be a promising alternative to the widely-used material distribution model [3]. Its further development could possibly yield optimization procedures that have more desirable features in the sense of having geometric models independent of the finite element discretization, using a relatively small number of continuous design variables, while maintaining a high flexibility in handling topological changes and mechanical analyses. Acknowledgement This research work is supported in part by the Research Grants Council of Hong Kong SAR (Project No. CUHK4164/03E) and the Natural Science Foundation of China (NSFC) (Grants No. 50128503 and No. 50390063). References 1. 2. 3. 4.
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