Local optimum in multi-material topology optimization and solution by reciprocal variables

Abstract

It is revealed that the local optimum is particularly prone to occur in multi-material topology optimization using the conventional SIMP method. To overcome these undesirable phenomena, reciprocal variables are introduced into the formulation of topology optimization for minimization of total weight with the prescribed constraint of various structural responses. The SIMP scheme of multi-phase materials is adopted as the interpolation of the elemental stiffness matrix, mass matrix and weight. The sensitivities of eigenvalue and weight with respect to design variables are derived. Explicit approximations of natural eigenvalue and weight are given with the help of the first and second order Taylor series expansion. Thus, the optimization problem is solved using a sequential quadratic programming approach, by setting up a sub-problem in the form of a quadratic program. The filtering technique by solving the Helmholtz-type partial differential equation is performed to eliminate the checkerboard patterns and mesh dependence. Numerical analysis indicates that it is beneficial to avoid the local optimum by using the reciprocal SIMP formulation. Besides, the structure composed of multi-materials can achieve a lighter design than that made from the exclusive base material. The effectiveness and capability of the proposed method are also verified by nodal displacement constraint and multiple constraints.

Appendix

The evaluation of matrix B and H in (14) can be expressed as.

$\mathbf{B}=\left[\frac{\partial W}{\partial {x}_i},\frac{\partial W}{\partial {x}_i}\right]$, $\mathbf{H}=\left[\begin{array}{cc}\hfill \frac{\partial^2W}{\partial {x}_i^2}\hfill & \hfill \frac{\partial^2W}{\partial {x}_i^2}\hfill \\ {}\hfill \frac{\partial^2W}{\partial {x}_i^2}\hfill & \hfill \frac{\partial^2W}{\partial {x}_i^2}\hfill \end{array}\right]$.

with

$$ \frac{\partial W}{\partial {x}_i}=-\gamma {x}_i^{-\left(\gamma +1\right)}\left[{w}^{\mathrm{I}\mathrm{I}}+{y}_i^{-\gamma}\left({w}^{\mathrm{I}}-{w}^{\mathrm{I}\mathrm{I}}\right)\right] $$

$$ \frac{\partial W}{\partial {y}_i}=-\gamma {x}_i^{-\gamma }{y}_i^{-\left(\gamma +1\right)}\left({w}^{\mathrm{I}}-{w}^{\mathrm{I}\mathrm{I}}\right) $$

$$ \frac{\ {\partial}^2W}{\partial {x}_i^2}=\gamma \left(\gamma +1\right){x}_i^{-\left(\gamma +2\right)}\left[{w}^{\mathrm{I}\mathrm{I}}+{y}_i^{-\gamma}\left({w}^{\mathrm{I}}-{w}^{\mathrm{I}\mathrm{I}}\right)\right] $$

$$ \frac{\ {\partial}^2W}{\partial {y}_i^2}=\gamma \left(\gamma +1\right){x}_i^{-\gamma }{y}_i^{-\left(\gamma +2\right)}\left({w}^{\mathrm{I}}-{w}^{\mathrm{I}\mathrm{I}}\right) $$

$$ \frac{\ {\partial}^2W}{\partial {x}_i\partial {y}_i}={\gamma}^2{x}_i^{-\left(\gamma +1\right)}{y}_i^{-\left(\gamma +1\right)}\left({w}^{\mathrm{I}}-{w}^{\mathrm{I}\mathrm{I}}\right) $$

When the off-diagonal elements are set to be zeros, the Hessian matrix can be always positive definite. More importantly, we can easily compute the inverse of the Hessian matrix.