Connections between topological sensitivity analysis and material interpolation schemes in topology optimization

Abstract

Material interpolation schemes, like SIMP, are very popular in topology optimization. They convert the difficult 0-1 problem into a nonlinear programming problem defined over a convex set by involving an interpolation (or penalization) function, usually constructed in rather empirical ways. This paper gives an insight into such methods with the help of the notion of topological sensitivity, and in particular provides some arguments for the choice of the penalization function. A simple algorithm based on these concepts is proposed and illustrated by numerical experiments.

Appendix A: Mathematical complements

The goal of this appendix is to prove Theorem 4.2. We begin by two preliminary lemmas.

Lemma A.1

Let $\mathcal{S}$ be the set of polarization matrices generated by arbitrary ellipses, i.e,

$$ \mathcal{S}=\left\{R Q_{r,e} R^T, R \in \mathcal{U}, 0<e \leq 1\right\}. $$

Then$\mathcal{S}$ is the set of all 2 × 2 symmetric positive definite matrices P verifying

$$ \label{trace} \mbox{trace} \ (P^{-1}) = 1+r , \qquad \mbox{trace} \ (P) < 1+\frac{1}{r}. $$

(26)

Proof

If $P \in \mathcal{S}$, one easily checks (26). Suppose that P satisfies (26) and denote by λ ₁,λ ₂ the eigenvalues of P, which satisfy

$$ \label{lambda} \frac{1}{\lambda_1} + \frac{1}{\lambda_2} = 1+r, \qquad \lambda_1+\lambda_2 < 1+\frac{1}{r}. $$

(27)

We deduce from (27) that

$$ \lambda_1>\frac{1}{1+r}, $$

$$ \lambda_1+\lambda_2-\left(1+\frac{1}{r}\right)=\left(1+\frac{1}{r}\right)\frac{(1-r\lambda_1)(1-\lambda_1)}{(1+r)\lambda_1-1}<0, $$

which entails

$$ (1-\lambda_1)(r\lambda_1-1)>0. $$

Hence $\hat{e}:=(1-\lambda_1)/(r\lambda_1-1)>0$. If $\hat{e}\leq 1$, we set $e=\hat{e}$, so that λ ₁ = (1 + e)/(1 + re) and, using (27), λ ₂ = (1 + e)/(e + r). If $\hat{e}>1$, we set $e=1/\hat{e}$, so that λ ₁ = (1 + e)/(e + r) and, using (27), λ ₂ = (1 + e)/(1 + re). In both cases P has the same eigenvalues as Q _r,e, thus there exists $R \in \mathcal{U}$ such that $P=RQ_{r,e}R^T$. □

Lemma A.2

The closure of $\mathcal{S}$ is the set of all symmetric positive definite matrices P verifying

$$ \label{tracecl} \mbox{trace} \ (P^{-1}) = 1+r , \qquad \mbox{trace} \ (P) \leq 1+\frac{1}{r}. $$

(28)

Proof

Clearly, (28) is fulfilled by all P belonging to the closure of $\mathcal{S}$, denoted by $\mbox{cl} (\mathcal{S})$. In addition, every eigenvalue λ of a matrix $P \in \mathcal{S}$ satisfies λ ≥ 1/(1 + r). Hence each $P \in \mbox{cl} (\mathcal{S})$ is symmetric positive definite. Suppose now that P is a symmetric positive definite matrix verifying

$$ \label{tracebo} \mbox{trace} \ (P^{-1}) = 1+r , \qquad \mbox{trace} \ (P) = 1+\frac{1}{r}. $$

(29)

Using (29) we obtain that the eigenvalues of P are 1 and 1/r. Therefore there exists $R \in \mathcal{U}$ such that

$$ P=R \begin{pmatrix} 1 & 0 \\ 0 & 1/r \end{pmatrix} R^T. $$

Let (α _n ) be a sequence of positive numbers such that

$$ \alpha_n > \frac{1}{1+r} \qquad \forall n \in \mathbb N, \qquad \lim_{n \to \infty} \alpha_n \to 1, $$

and define

$$ \beta_n = \frac{\alpha_n}{\alpha_n(1+r)-1}>0, $$

$$ P_n=R \begin{pmatrix} \alpha_n & 0 \\ 0 & \beta_n \end{pmatrix} R^T. $$

By construction, we have for all n

$$ \frac{1}{\alpha_n}+\frac{1}{\beta_n}=1+r, $$

$$ a_n:=\alpha_n+\beta_n-\left(1+\frac{1}{r}\right) = \left(1+\frac{1}{r}\right) \frac{(1-\alpha_n)(1-r\alpha_n)}{\alpha_n(1+r)-1}. $$

We choose (α _n ) such that

$$ \begin{array}{lll} \alpha_n < 1 \quad \mbox{if} \quad r>1, \\ \alpha_n > 1 \quad \mbox{if} \quad r<1. \end{array} $$

In each case, for n large enough, a _n  < 0, hence $P_n \in \mathcal{S}$. As lim_{n → ∞} P _n  = P, we conclude that $P \in \mbox{cl} (\mathcal{S})$. □

We are now in position to prove Theorem 4.2. First, we easily check that, in the nontrivial case where U,V ≠ 0, Λ ⁺ and Λ − given by (20) are eigenvalues of M associated with the eigenvectors U/|U| + V/|V| and U/|U| − V/|V|, respectively. Suppose that $\gamma \in \mathcal{E}$ is a local minimizer of j with respect to every elliptic inclusion, and choose $x \in D_\gamma^-$. According to Lemma 2.3, we have g _γ,ω(x) ≥ 0 for any ellipse ω. Then the condition (22)(a) will be fulfilled if we set

$$ g_\gamma^\star(x) = \inf \{g_{\gamma,\omega}(x), \omega \ \mbox{ellipse} \}. $$

We now check that the above infimum satisfies (21)(a). For each ellipse ω, we write (18) in the form

$$ g_{\gamma,\omega}(x) = P_{\omega,r}:M(x)+\ell. $$

With the notation of Lemma A.1, we have

$$ g_\gamma^\star(x) = \inf \{P:M+\ell, P \in \mathcal{S}\}. $$

As $\mathcal{S}$ is bounded, we also have

$$ g_\gamma^\star(x) = \min \{P:M+\ell, P \in \mbox{cl}(\mathcal{S})\}. $$

Let us consider an arbitrary $P \in \mbox{cl}(\mathcal{S})$, which we write in the form

$$ \begin{array}{lll} P=R\hat{P}R^T, \quad \hat{P}=\begin{pmatrix} \lambda^+ & 0 \\ 0 & \lambda^- \end{pmatrix}, \\ R \in \mathcal{U}, \quad \lambda^+ \geq \lambda^->0. \end{array} $$

Likewise, there exists $S \in \mathcal{U}$ such that

$$ M=S\hat{M}S^T, \qquad \hat{M}=\begin{pmatrix} \Lambda^+ & 0 \\ 0 & \Lambda^- \end{pmatrix}. $$

Set T = S ^T R. As $T \in \mathcal{U}$, there exists φ ∈ ℝ such that

$$ T=\begin{pmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{pmatrix} \ \mbox{or} \ T=\begin{pmatrix} \cos\phi & \sin\phi \\ \sin\phi & -\cos\phi \end{pmatrix}. $$

We arrive in both cases at

$$ \begin{array}{rll} P:M &=& \left(T\hat{P}T^T\right):\hat{M} = \left(\lambda^+-\lambda^-\right)\left(\Lambda^+-\Lambda^-\right)\\ &&\times\,\cos^2\phi+\Lambda^+\lambda^-+\Lambda^-\lambda^+. \end{array}$$

(30)

Arguing as in Lemma A.1 with the characterization of $\mbox{cl}(\mathcal{S})$ given by Lemma A.2, we obtain that

$$ \left(1-\lambda^\pm\right)\left(r\lambda^\pm-1\right) \geq 0. $$

Since $x \in D_\gamma^-$, we have r = γ ⁺/γ − > 1, hence the above inequality implies that 1/r ≤ λ − ≤ λ ⁺ ≤ 1. Using that Λ − ≤ 0 ≤ Λ ⁺ we derive

$$ P:M \geq \frac{\Lambda^+}{r} + \Lambda^-. $$

In addition, this bound is attained for cosφ = 0, λ ⁺ = 1, λ − = 1/r, which provides (21)(a). The case where $x \in D_\gamma^+$ can be treated in a similar way, with $\inf$ replaced by sup and r = γ − /γ ⁺ < 1.