On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization
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Abstract
This paper deals with the classical problem of material distribution for minimal compliance of two-dimensional structures loaded in-plane. The main objective of the research consists in investigating the properties of the exact solution to the minimal compliance problem and incorporating them into an approximate solid-void interpolation model. Consequently, a proposition of a constitutive relation for a porous material arise. The non-smoothness of stress energy functional known from the approach based on homogenization may be thus avoided which is beneficial for the numerical implementation of the scheme. It is next shown that the simplified variant of the proposed formula justifies and generalizes the RAMP (Rational Approximation of Material Properties) interpolation model of Stolpe and Svanberg (Struct Multidisc Optim 22:116–124). In the second part of the paper, explicit formulae for function θ: Ω → [0, 1] describing the distribution of basic isotropic material in the design space Ω ∈ ℝ^{2} are derived for various interpolation schemes by the requirement of optimality imposed at each x ∈ Ω. Theoretical considerations are illustrated by a code written in MATLAB for typical optimization problems of a cantilever and MBB beam.
Keywords
Topology optimization Minimal compliance Two-dimensional elasticity Interpolation scheme Effective properties1 Introduction
The classical optimization problem of minimal structural compliance was set in the late 1960s. Roughly speaking, the goal is to search the set of characteristic functions χ: Ω → {0, 1}, where Ω denotes the design area, for a minimizer χ_{opt} such that the compliance J = J(χ) of a structure subjected to a given load achieves its minimal value \(\underline{J}{}=J(\chi_{{\rm opt}})\). At the early stage of research the above-mentioned task had proven to be badly posed due to the general non-convergence of sequences {χ_{n}} in the standard norm of L^{ ∞ }(Ω, {0, 1}) (see e.g. Kozłowski and Mróz 1969; Rozvany et al. 1982), hence χ_{opt} could not be computed. This phenomenon, often referred to as “non-existence of classical solutions”, revealed a need for regularizing the optimal design problem.
One of the regularization methods assumes replacing the classical design set L^{ ∞ }(Ω, {0, 1}) with its larger counterpart L^{ ∞ }(Ω, [0, 1]), i.e. the set of generalized designs θ whose main property is that these functions can take any value between 0 and 1. From the mechanical point of view, the extension of this type results in allowing the microstructural composites of basic material and void in the analysis of the problem. The mathematical foundation of such method, known as the homogenization theory, is being developed simultaneously with its mechanical applications from 1970s. Detailed exposition of this topic lays outside of scope of the paper, hence we refer the reader to Cherkaev (2000), Lewiński and Telega (2000), Milton (2002) and Tartar (2000) for further references.
It turns out that the shape optimization problem posed in the linear elasticity setting in two space dimensions has an explicit and accurate solution in terms of principal values N_{I}(x), N_{II}(x), x ∈ Ω of the force resultant tensor N. Namely, one may analytically determine the material distribution function θ_{opt} = θ_{opt}(x) realizing \(\underline{J}{}=J(\theta_{{\rm opt}})\) as well as the local microstructural geometrical properties. Consequently, the macroscopic effective constitutive characteristics of a composite made of basic material and void locally mixed in proportions θ_{opt}(x) and 1 − θ_{opt}(x) respectively can also be computed. This feature is thoroughly dealt with in Allaire (2002). Nevertheless, the numerical implementation of a homogenization-based results is hampered by the non-smoothness of stress energy functional \(W(\boldsymbol{N},\theta)\) for N_{I}N_{II} = 0. Namely, if this is the case, then the constitutive formula is not unique, hence it cannot be directly inverted into the stress-displacement form which serves as a natural base for a numerical implementation by FEM. As a possible remedy, one may either make use of the solution to the problem of optimal layout formulated for a two-material structure, see Lewiński (2004), or seek the smooth, thus invertible, approximation of \(W(\boldsymbol{N},\theta)\). In the sequel, the latter option is investigated.
According to the homogenization approach, θ_{opt}(x) ranges from 0 to 1 in the design area Ω hence the generalized material distribution function may serve as a reference solution to certain suboptimal, though practical and manufacturable, “0–1” designs stemming from the simplified material-void interpolation models. The most popular SIMP (Solid Isotropic Material with Penalization) scheme emerged from the papers of Bendsøe (1989) or Zhou and Rozvany (1991) and it is now widely used in solving various engineering optimization problems, see (Bendsøe and Sigmund 2003) for their extensive review. The mathematical foundation of SIMP is also subject to continuous research, see e.g. Rietz (2001), Martínez (2005), Almeida et al. (2010), Amstutz (2011) and Azegami et al. (2011). Another power law-like model of the material-void composite constitutive behavior, called RAMP (Rational Approximation of Material Properties), was proposed by Stolpe and Svanberg (2001b) following Rietz (2001). Both SIMP and RAMP interpolation schemes incorporate a certain real parameter which can be adjusted during the optimization procedure to penalize the intermediate densities of the basic material in the effective composite thus the almost “0–1” designs can be created. Deep discussion of problems corresponding to penalization and numerous filtering techniques is a subject of e.g. Sigmund and Petersson (1998), Bourdin (2001), Bruns and Tortorelli (2001), Stolpe and Svanberg (2001a), Rietz (2007) and Sigmund (2007).
The constitutive properties of a composite and, consequently, the value of the energy accumulated in a particle of an effective medium depend on the material interpolation scheme. Hence, the range of the SIMP or RAMP parameters is restricted by the requirement W_{opt} ≤ W_{app}, where W_{opt} denotes the stress energy related to the optimal solution based on the homogenization approach and W_{app} stands for its approximation corresponding to the chosen interpolation scheme. Even more restrictive is the requirement of the macrostructural isotropy of the solid-void composite. Bounds on the effective isotropic properties of a two-material mixture were introduced by Hashin and Shtrikman (1963) and improved by Cherkaev and Gibiansky (1993). It is worth pointing out, however, that both results coincide for the mixture of material and void hence in the sequel we will write W_{HS} ≤ W_{app} to denote the requirement of effective isotropy.
The paper is organized as follows: The notation necessary for the presentation of the topic is introduced in Section 2 together with a brief recall of optimal solution to the minimal compliance problem for material and void in a two-dimensional setting. Section 3 contains a derivation of an energy-based solid-void interpolation scheme in its general two-parameter form related to the homogenization approach. It is shown that a certain, simplified variant of the scheme allows for uniform estimation of the optimal stress energy functional and justifies the RAMP interpolation function of Stolpe and Svanberg (2001b). Hence, the proposed solid-void constitutive formula is nicknamed as generalized RAMP (GRAMP). Thus obtained scheme is next confronted with the Hashin-Shtrikman and SIMP estimates. Section 4 is devoted to the derivation of the explicit formulae for material distribution functions corresponding to various interpolation models. Numerical examples obtained with the help of GRAMP are presented in Section 5 and compared with the exact solutions and those based on SIMP and classical RAMP schemes.
2 Background of the research
2.1 Notation
From now on, Greek indices take values 1 or 2 while the Latin ones range from 1 to 3 and usual summation convention applies. Moreover, in the sequel we write \(f_{,\alpha}=\partial f/\partial(x_\alpha)\) for the differential operator; \({\bf A}\boldsymbol{B}\) for the contraction of the Hooke tensor A (bold upright) and symmetric second-order tensor B (bold italic) and \(\boldsymbol{A}\cdot\boldsymbol{B}\) for the scalar product of two symmetric second-order tensors.
Formula \(\boldsymbol{\varepsilon}={\bf A}\boldsymbol{N}\) links the force resultant tensor with the deformation tensor \(\boldsymbol{\varepsilon}(\boldsymbol{u})\) whose components are derived from the kinematically admissible displacement vector \(\boldsymbol{u}\) by 2ε_{αβ} = u_{α,β} + u_{β,α}.
It is worth pointing out that all considerations in the sequel remain valid if one swaps N_{I} with N_{II} in (9) thus redefining ζ and the expressions in (10) accordingly. This operation is possible due to the assumed isotropy of A(θ) and allows for the straightforward application of the subsequent results in the limiting cases of N_{I} = 0 and N_{II} = 0. We assume that N_{I} = N_{II} = 0 and W = 0 if θ = 0.
As this Theorem is crucial for further considerations, let us discuss its conditions in optimal design setting. First, note that the existence of a minimizer \(\theta_{{\rm min}}\in L^\infty(\Omega,[0,1])\) in (12) is provided by the homogenization theory, see Allaire (2002, Th. 4.2.6). Moreover, structural compliance calculated for any material distribution is represented by a finite number, hence \(\underline{J}{}<+\infty\) for given θ_{min}. By this, one of the Rockafellar’s Theorem assumptions is fulfilled.
In the original formulation of the Theorem it is expected that the minimizing function θ_{min} is looked for in the decomposable linear space of measurable functions. Therefore, the set L^{ ∞ }(Ω, [0, 1]) should be extended to L^{ ∞ }(Ω, ℝ) thus admitting θ_{min}(x) < 0 and θ_{min}(x) > 1 as possible solutions in (13). The extension, however, is formal and pose no additional problems. Indeed, by the inspection of results in the sequel one may check that the values θ_{min}(x) and − θ_{min}(x) saturate the extremum of the integrand in (13) at x ∈ Ω, but the minimizer is always given by θ_{min}(x) > 0. On the other hand, it is always possible to set θ_{min}(x) = 1 if the upper bound of the [0,1] interval is violated but such restriction has to be compensated by adjusting the Lagrange multiplier ℓ which is necessary to retain the isoperimetric condition (6).
The last two assumptions sufficient for (13) to hold are that the integrand is lower semicontinuous in θ for any x ∈ Ω and that its epigraph is a measurable function. The former condition is fulfilled as a(θ) and b(θ) defining U(ζ, θ), see (10), are continuous in θ for all cases considered in the remainder of this paper. Consequently, the latter condition is also satisfied.
2.2 Properties of the explicit solution to the problem of material layout for minimal compliance
3 Solid-void interpolation scheme based on the properties of optimal stress energy
3.1 Derivation of the interpolation function
In this way, the two-parameter solid-void interpolation scheme is established with the approximate stress energy function W_{app} defined through (10) and (26). It follows that U_{app} shares the common tangent with U_{opt} and U_{app} = U_{opt} at two distinct points iff q_{1} = q_{2} = 1.
3.2 Relation to the Hashin-Shtrikman bounds and SIMP model
Let us emphasize that both U_{HS} and U_{opt} establish certain lower limits on the stress energy accumulated in a particle of a structure. The difference in their formulae are, roughly speaking, due to the fact that the calculations of U_{HS} incorporate the control over the isotropy of the effective material which is not the case for U_{opt}.
Indeed, the derivation of Hashin-Shtrikman bounds consists of two independent steps. Application of a hydrostatic field \(\boldsymbol{N}=N_{{\rm H}}\boldsymbol{E}_1\) allows for defining the minimal value of K(θ), while obtaining the minimal value of L(θ) requires subjecting a composite to the deviatoric field \(\boldsymbol{N}= N_{{\rm D}}(\boldsymbol{E}_2+\boldsymbol{E}_3)\).
Conversely, U_{opt} measures the energy of a composite subjected to an arbitrary field \(\boldsymbol{N}\). However, if we assume that \(\boldsymbol{e}_\alpha\) stand for the eigenbasis of some N at given x ∈ Ω, and if we set N_{I} = N_{II} = N (i.e. ζ = 1) then we obtain U_{HS} = U_{opt}, as \(\boldsymbol{N}=(\sqrt{2}N)\boldsymbol{E}_1\) is a hydrostatic field. On the contrary, setting N_{I} = − N_{II} = N (i.e. ζ = − 1) does not lead to a similar conclusion on the equality of energies because applying the deviatoric field \(\boldsymbol{N}=(\sqrt{2}N)\boldsymbol{E}_2\) and controlling the response of a composite medium in the direction E_{3} at the same time is impossible.
Requirements on the parameters in GRAMP, RAMP and SIMP interpolation schemes imposed by the conditions of optimality (U_{app} ≥ U_{opt}) or material isotropy (U_{app} ≥ U_{HS})
| U_{app} = U_{GRAMP}^{a} | U_{app} = U_{RAMP}^{b} | U_{app} = U_{SIMP}^{b} |
---|---|---|---|
U_{app} ≥ U_{opt} | q ≥ 1 | \(q\geq\max\left\{\dfrac{1+\nu_0}{1-\nu_0}, \dfrac{1-\nu_0}{1+\nu_0}\right\}\) | \(p\geq\max\left\{\dfrac{2}{1-\nu_0}, \dfrac{2}{1+\nu_0}\right\}\) |
[8pt] U_{app} ≥ U_{HS} | q ≥ 3 | \(q\geq\max\left\{\dfrac{1+\nu_0}{1-\nu_0}, \dfrac{3-\nu_0}{1+\nu_0}\right\}\) | \(p\geq\max\left\{\dfrac{2}{1-\nu_0}, \dfrac{4}{1+\nu_0}\right\}\) |
Constitutive properties predicted by Hashin-Shtrikman approach are realizable on certain isotropic microstructures like 3rd rank sequential laminates (Francfort and Murat 1986) or coated circles (Hashin 1962; Grabovsky and Kohn 1995a). On the other hand, composites of minimal compliance can be arranged as orthotropic 2nd rank orthogonal sequential laminates or, if \(\det\boldsymbol{N}>0\), in a form of the Vigdergauz microstructures (Vigdergauz 1989; Grabovsky and Kohn 1995b) or 4th rank sequential laminates (Allaire and Aubry 1999).
In light of the studies in this section it seems that one may neglect the requirement of the macrostructural isotropy if the best possible macroscale material distribution is considered as a primal goal of the optimization problem. Nevertheless, thus obtained solution may serve as a good starting point of the continuation method where the change of a parameter in the interpolation scheme (like e.g. q in RAMP or GRAMP) reflects in the tendency to the pure “0-1” distribution of a basic isotropic material in the domain of an analyzed structure. Problems related to the microscopic layout of material and void in optimized structures are detailed at length in e.g. Allaire (2002), Cherkaev (2000) and Bendsøe and Sigmund (2003), see also references therein.
4 Material distribution functions for minimal compliance
Our next objective is to derive the material distribution functions that correspond to various interpolation schemes. Repeated application of (14) where \(W(\boldsymbol{N}, \theta)\) is given by (10) and appropriate expressions for a(θ) and b(θ) enables us to write the sought formulae in their explicit forms similarly to (21).
It follows that functions \(\theta(\boldsymbol{N})\) in SIMP, RAMP and Hashin-Shtrikman interpolation schemes are sensitive to the sign of ζ and ν_{0} which is not the case for the material distribution based on GRAMP or exact homogenization approach.
Explicit formulae for material distribution can be directly applied in numerical codes for compliance minimization problem in this way allowing to avoid the heuristic density updating schemes.
5 Examples of material layouts
5.1 General comments on numerical implementation
All calculations are coded in MATLAB and performed by FEM using the mesh of 200×100 square, four-node elements with two degrees of freedom per node. The idea of programming is to supplement the applications published at the TopOpt research group website (www.topopt.dtu.dk) hence the codes presented in the Appendices have a structure based on the program written by Andreassen et al. (2011) for topology optimization in 2D.
Typical, k-th step of the optimization loop consists of two substeps. First, the FE analysis is performed for fixed vector \(\boldsymbol{x}^{k-1}\) representing the values of θ obtained in the previous run. Next, vectors of principal values of stress resultant \(\boldsymbol{N}_{{\rm I}}^k\) and \(\boldsymbol{N}_{{\rm II}}^k\) are determined. Components of each vector are calculated either at the Gauss points or in the middle of each element, see Sections 5.2 and 5.3 for further details.
Here we set ϵ = 10^{ − 2} but this value can be adjusted to meet the required accuracy. It has to be pointed out that the convergence measured by (40) is fairly slow. Some authors prefer to rewrite this criterion in terms of the material density vector \(\boldsymbol{x}\), see the numerical codes in Bendsøe and Sigmund (2003) and Andreassen et al. (2011).
Minimal compliance problem tackled in this paper falls into the category of constrained optimization. The necessary conditions for optimum are thus obtained by applying the Karush-Kuhn-Tucker (KKT) Theorem, see e.g Haftka and Kamat (1985) for further details in context of structural optimization. Discussed formulation (12) is a double minimization with respect to N and θ. The KKT conditions of optimality admit the Lagrange multiplier ℓ for the isoperimetric constraint (6). The requirement of statical admissibility imposed on N by (5) can be accounted for in a similar way with the Lagrange multiplier assumed in a functional form. Both constraints are to be satisfied at the stationary point of an objective function. In this way, the stopping criterion of the numerical algorithm is provided.
In the corresponding numerical algorithm the unknowns N and θ are respectively referred to as vectors N and \(\boldsymbol{x}\). Iterative numerical implementation of (12) with alternating minimization in both variables guarantees that the objective function converges to a stationary point. Indeed, determining \(\boldsymbol{N}^k\) for fixed \(\boldsymbol{x}^{k-1}\) reduces to solving the linear elasticity problem. Consequently, the stress energy is minimized hence the value of the objective function decreases. Upon updating the stress resultant vector, the objective function is minimized explicitly by \(\boldsymbol{x}^k\) obtained through (21) or one of (36)–(39) depending on the solid-void interpolation model. The Lagrange multiplier ℓ is adjusted in order to enforce the corresponding KKT condition. Formula (40) serves as the stopping condition of a numerical algorithm. Its fulfilment shows the convergence of N and the convergence of the objective function follows.
Computational details of the solutions to the half-MBB beam problem shown in Fig. 8
5.2 Comments on the homogenization-based approach
5.3 Comments on the GRAMP approach
5.4 Comments on the filtered designs
For practical purposes it is desirable that the intermediate densities are filtered out from the final design. Application of such filter allows not only for obtaining nearly black-and-white material layouts but also helps in avoiding the checkerboard instability of numerical calculations. Roughly speaking, filtering the density θ(x) at given x ∈ Ω works by averaging it over the neighborhood of given radius r. By this, the tendency to reproduce the fine-scale arrangement of solid and void is bypassed and composite regions are introduced in Ω. In turn, these regions are penalized by proper parameter adjustment in the material interpolation formula. As a result, the topology of final design becomes nearly “0-1” and the width of transition between void and solid areas depends on the filtering radius r.
According to e.g. Bendsøe and Sigmund (2003), implementing a filtering technique in the formulation of optimal design problem at hand does not impose any additional limit on the space L^{ ∞ }(Ω, [0, 1]), i.e. the space of material distribution functions. Density filter can be applied at each x ∈ Ω independently as a part of the material redistribution procedure.
Numerous filtering methods are reviewed in Sigmund (2007) and some issues related to the programming in MATLAB are tackled in Bendsøe and Sigmund (2003) and Andreassen et al. (2011). In the present paper we make use of the built-in MATLAB convolution function conv2, see MathWorks (2011), adjusted to the cone-shaped density filter of Bruns and Tortorelli (2001) and Bourdin (2001).
For the layouts related to the original RAMP and SIMP schemes, shown in Fig. 8d, f, the \(\mathtt{topgramp1}\) code requires slight modifications whose summary is given in Appendix B.
6 Conclusions
Considering the formula \(W_{{\rm opt}}(\boldsymbol{N}, \theta)\) for stress energy of a material-void mixture as a reference expression in a research for its approximation is motivated by the non-smoothness of W_{opt} for \(\det\boldsymbol{N}=0\). Consequently, a differentiable, two-parameter estimate \(W_{{\rm app}}(\boldsymbol{N}, \theta)\) is obtained. Additional requirement of approximation uniformity for different signs of \(\det\boldsymbol{N}\) and arbitrary θ ∈ [0,1] leads to the simplified variant of a scheme. It is nicknamed GRAMP, as it justifies and generalizes the RAMP model of Stolpe and Svanberg (2001b) in a simple way, easy to implement in numerical subroutines. Moreover, besides the simplified composite solutions to the minimal compliance problem, GRAMP is able to produce the almost “0-1” layouts of basic material as a result of the continuation method possibly combined with certain design smoothing routines. However, application of these techniques are dealt with in this paper only in limited scope.
Exact formulae determining material distribution functions are explicitly derived for various interpolation models, like SIMP, original RAMP and the one based on the Hashin-Shtrikman requirement of macrostructural isotropy of a composite material. By this, the heuristic material density updating schemes may be avoided in the development of numerical codes solving the compliance minimization problem.
Acknowledgement
The paper was prepared within the Research Grant no N506 071338, financed by the Polish Ministry of Science and Higher Education, entitled: Topology Optimization of Engineering Structures. Simultaneous shaping and local material properties determination.
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