A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration

Abstract

We present a level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration. Considering that a simple eigenvalue is Fréchet differentiable with respect to the boundary of a structure but a repeated eigenvalue is only Gateaux or directionally differentiable, we take different approaches to derive the boundary variation that maximizes the first eigenvalue. In the case of simple eigenvalue, material derivative is obtained via adjoint method, and variation of boundary shape is specified according to the steepest descent method. In the case of N-fold repeated eigenvalue, variation of boundary shape is obtained as a result of a N-dimensional algebraic eigenvalue problem. Constraint of a structure’s volume is dealt with via the augmented Lagrange multiplier method. Boundary variation is treated as an advection velocity in the Hamilton–Jacobi equation of the level set method for changing the shape and topology of a structure. The finite element analysis of eigenvalues of structure vibration is accomplished by using an Eulerian method that employs a fixed mesh and ersatz material. Application of the method is demonstrated by several numerical examples of optimizing 2D structures.

Appendix A

For an N-fold repeated first eigenvalue, the linear combination, (29), of N linear independent eigenvectors is also an eigenvector

$$ \tilde{u}=\sum\limits_{j=1}^{N} \beta_{j}u_j $$

(29)

where β _j are unknown coefficients to be determined.

Directly differentiating the weak form of the eigenvalue problem, we have

$$ a^\prime(u,v) = \lambda^\prime_1 b(u,v)+\lambda_1 b^\prime(u,v),\quad \forall v\in U \label{e:direct_diff} $$

(30)

Note that $\lambda_1^\prime$ here denotes directional derivative. Substituting $\tilde{u}$ and N linear independent eigenvectors u _i into (30), we get

$$ a^\prime(\tilde{u},u_i) = \lambda^\prime_1 b(\tilde{u},u_i)+\lambda_1 b^\prime(\tilde{u},u_i),\quad i=1\ldots N $$

(31)

Substituting (29) into (31), and noting that u _i , u _j are mass-othonormalized, i.e., b(u _i , u _j ) = δ _ij , we have

$$ \sum\limits_{j=1}^{N} \beta_{j}\left[\,a^\prime(u_i,u_j) \!-\! \lambda_1 b^\prime(u_i,u_j) \!-\! \lambda^\prime_1 \delta_{ij}\,\right]\!=\!0,\quad i\!=\!1\ldots N \label{e:linear_system_0} $$

(32)

Nontrivial solution to the linear equations (32) exists only if the determinant is equal to zero, i.e.,

$$ \mathrm{det} \left[\,a^\prime(u_i,u_j) - \lambda_1 b^\prime(u_i,u_j) - \lambda^\prime_1 \delta_{ij}\,\right]=0 \label{e:det_0} $$

(33)

Finally, substitute (8) and (9) into (33) and noting that $a(u_i,u_j^\prime)=\lambda_1 b(u_i,u_j^\prime)$ and $a(u_i^\prime,u_j)=\lambda_1 b(u_i^\prime,u_j)$, we can get

$$ \mathrm{det} \left[\,\mathcal{M}_{ij} - \lambda^\prime_1 \delta_{ij}\,\right]=0 $$

(34)

where $\mathcal{M}_{ij}$ are given by

$$ \mathcal{M}_{ij} = \int_{\Gamma_N} G^{ij}\,V_n\mathrm{d} s,\quad i,j=1,2 \label{e:sensitivity_matrix} $$

(35)

This is the result of (18).