Topology optimization of freely vibrating continuum structures based on nonsmooth optimization

Abstract

The non-differentiability of repeated eigenvalues is one of the key difficulties to obtain the optimal solution in the topology optimization of freely vibrating continuum structures. In this paper, the bundle method, which is a very promising one in the nonsmooth optimization algorithm family, is proposed and implemented to solve the problem of eigenfrequency optimization of continuum. The bundle method is well-known in the mathematical programming community, but has never been used to solve the problems of topology optimization of continuum structures with respect to simple or multiple eigenfrequencies. The advantage of this method is that the specified information of iteration history may be collected and utilized in a very efficient manner to ensure that the next stability center is closer to the optimal solution, so as to avoid the numerical oscillation in the iteration history. Moreover, in the present method, both the simple and multiple eigenfrequencies can be managed within a unified computational scheme. Several numerical examples are tested to validate the proposed method. Comparisons with nonlinear semidefinite programming method and 0–1 formulation based heuristic method show the advantages of the proposed method. It is showed that, the method can deal with the nonsmoothness of the repeated eigenvalues in topology optimization in a very effective and efficient manner without evaluating the multiplicity of the eigenvalues.

Appendix

THEOREM Given two positive definite matrices $\mathbf{A}\left(\mathbf{x}\right),\mathbf{B}\left(\mathbf{x}\right)\in {\mathbb{S}}_{++}^{ndof}$ which are smooth functions (the smoothness implies the continuity and differentiability) of the independent variables $\mathbf{x}\in {\mathtt{\mathbb{R}}}^n$, λ is the largest eigenvalue of (A, B) with multiplicity t, the corresponding eigenvectors are the columns of Φ = [φ ₁, ⋯, φ _t ] ∈ ℝ^{ndof × t}, then the subdifferential of λ(x) is the set

$$ \partial \lambda \left(\mathbf{x}\right)=\left\{\mathbf{v}\in {\mathbb{R}}^n\left|{v}_i=\right.\left\langle {\boldsymbol{\Phi}}^{\mathrm{T}}\left(\frac{\partial \mathbf{A}}{\partial {x}_i}-\lambda \frac{\partial \mathbf{B}}{\partial {x}_i}\right)\boldsymbol{\Phi}, \mathbf{U}\right\rangle \right\} $$

(39)

for some U ∈ ℝ^t, tr U = 1, U ≽ 0.

Proof. The generalized eigenvalue problem is given by

$$ \lambda =\underset{\boldsymbol{\upvarphi} \in {\mathbb{R}}^{ndof}}{ \sup}\left\{{\boldsymbol{\upvarphi}}^{\mathrm{T}}\mathbf{A}\boldsymbol{\upvarphi } \left|{\boldsymbol{\upvarphi}}^{\mathrm{T}}\mathbf{B}\boldsymbol{\upvarphi } =1\right.\right\} $$

(40)

It is useful to define another auxiliary eigenvalue problem corresponding to the original generalized eigenvalue problem,

$$ \begin{array}{c}\lambda =\underset{\mathbf{q}\in {\mathbb{R}}^{ndof}}{ \sup}\left\{\left\langle \mathbf{q},\mathbf{S}\mathbf{q}\right\rangle \left|{\mathbf{q}}^{\mathrm{T}}\mathbf{q}=1\right.\right\}\\ {}=\underset{\mathbf{q}\in {\mathbb{R}}^{ndof}}{ \sup}\left\{\left\langle \mathbf{q}{\mathbf{q}}^{\mathrm{T}},\mathbf{S}\right\rangle \left|{\mathbf{q}}^{\mathrm{T}}\mathbf{q}=1\right.\right\}\end{array} $$

(41)

where, S and q are related to A and φ by

$$ \begin{array}{l}\mathbf{S}={\mathbf{G}}^{-1}\mathbf{A}{\mathbf{G}}^{-\mathrm{T}}\\ {}\mathbf{q}={\mathbf{G}}^{\mathrm{T}}\boldsymbol{\upvarphi} \end{array} $$

(42)

G comes from the Cholesky factorization of matrix B,

$$ \mathbf{B}=\mathbf{G}{\mathbf{G}}^{\mathrm{T}} $$

(43)

where G is a lower triangular matrix.

First we consider the subdifferential of the eigenvalue λ with respect to S by (41),

$$ \partial \lambda \left(\mathbf{S}\right)=\mathrm{conv}\left\{\mathbf{q}{\mathbf{q}}^{\mathrm{T}}\left|{\mathbf{q}}^{\mathrm{T}}\mathbf{q}=1,\mathbf{Sq}=\lambda \mathbf{q}\right.\right\} $$

(44)

where, conv(⋅) denotes the convex hull of a set.

Next we compute $\frac{\partial \mathbf{S}}{\partial {x}_i}$ which will be used afterwards,

$$ \begin{array}{l}\kern1em \frac{\partial \mathbf{S}}{\partial {x}_i}=\frac{\partial \left({\mathbf{G}}^{-1}\mathbf{A}{\mathbf{G}}^{-\mathrm{T}}\right)}{\partial {x}_i}\\ {}=-{\mathbf{G}}^{-1}\frac{\partial \mathbf{G}}{\partial {x}_i}\mathbf{S}+{\mathbf{G}}^{-1}\frac{\partial \mathbf{A}}{\partial {x}_i}{\mathbf{G}}^{-\mathrm{T}}-\mathbf{S}\frac{\partial {\mathbf{G}}^{\mathrm{T}}}{\partial {x}_i}{\mathbf{G}}^{-\mathrm{T}}\end{array} $$

(45)

The inner product of qq ^T and $\frac{\partial \mathbf{S}}{\partial {x}_i}$ gives,

$$ \begin{array}{l}\left\langle \frac{\partial \mathbf{S}}{\partial {x}_i},\mathbf{q}{\mathbf{q}}^{\mathrm{T}}\right\rangle =\left\langle \frac{\partial \mathbf{S}}{\partial {x}_i}\mathbf{q},\mathbf{q}\right\rangle \\ {}=-2\lambda {\mathbf{q}}^{\mathrm{T}}{\mathbf{G}}^{-1}\frac{\partial \mathbf{G}}{\partial {x}_i}\mathbf{q}+{\mathbf{q}}^{\mathrm{T}}{\mathbf{G}}^{-1}\frac{\partial \mathbf{A}}{\partial {x}_i}{\mathbf{G}}^{-\mathrm{T}}\mathbf{q}\end{array} $$

(46)

By using the chain rule, we have

$$ \frac{\partial \lambda}{\partial {x}_i}=\left\langle \frac{\partial \lambda}{\partial \mathbf{S}},\frac{\partial \mathbf{S}}{\partial {x}_i}\right\rangle $$

(47)

Using (44) and (46), after some trivial algebra, we have

$$ \frac{\partial \lambda}{\partial {x}_i}=\mathrm{conv}\left\{{\boldsymbol{\upvarphi}}^{\mathrm{T}}\frac{\partial \mathbf{A}}{\partial {x}_i}\boldsymbol{\upvarphi} -\lambda {\boldsymbol{\upvarphi}}^{\mathrm{T}}\frac{\partial \mathbf{B}}{\partial {x}_i}\boldsymbol{\upvarphi} \right\} $$

(48)

Equation (48) can be further simplified to drop the convex hull operator,

$$ \begin{array}{c}\kern1em \frac{\partial \lambda}{\partial {x}_i}\\ {}=\mathrm{conv}\left\{\left\langle \frac{\partial \mathbf{A}}{\partial {x}_i}-\lambda \frac{\partial \mathbf{B}}{\partial {x}_i},\boldsymbol{\Phi} \boldsymbol{\upomega} {\boldsymbol{\upomega}}^{\mathrm{T}}{\boldsymbol{\Phi}}^{\mathrm{T}}\right\rangle \right\}\\ {}=\left\{\left\langle \frac{\partial \mathbf{A}}{\partial {x}_i}-\lambda \frac{\partial \mathbf{B}}{\partial {x}_i},\boldsymbol{\Phi} \mathbf{U}{\boldsymbol{\Phi}}^{\mathrm{T}}\right\rangle \right\}\\ {}=\left\{\left\langle {\boldsymbol{\Phi}}^{\mathrm{T}}\left(\frac{\partial \mathbf{A}}{\partial {x}_i}-\lambda \frac{\partial \mathbf{B}}{\partial {x}_i}\right)\boldsymbol{\Phi}, \mathbf{U}\right\rangle \right\}\end{array} $$

(49)

where,

$\boldsymbol{\upomega} \in {\mathbb{R}}^t,{\boldsymbol{\upomega}}^{\mathrm{T}}\boldsymbol{\upomega} =1$ ;Φ = [φ ₁, ⋯, φ _t ] ;U ∈ ℝ ^t, tr U = 1, U ≽ 0;

t is the multiplicity of the maximum eigenvalue;

The first equation holds because of inner product’s properties and φ = Φω; the second equation holds because of the lemma (Overton 1992)

$$ \begin{array}{l}\kern1.5em \mathrm{conv}\left\{\boldsymbol{\upomega} {\boldsymbol{\upomega}}^{\mathrm{T}}\left|{\boldsymbol{\upomega}}^{\mathrm{T}}\boldsymbol{\upomega} =1\right.\right\}\\ {}=\left\{\left.\mathbf{U}\right|\mathbf{U}\in {\mathbb{S}}^t,\mathrm{tr}\kern0.5em \mathbf{U}=1,\mathbf{U}\succcurlyeq 0\right\}\end{array} $$

(50)