Robust and stochastic compliance-based topology optimization with finitely many loading scenarios

Abstract

In this paper, the problem of load uncertainty in compliance problems is addressed where the uncertainty is described in the form of a set of finitely many loading scenarios. Computationally, more efficient methods are proposed to exactly evaluate and differentiate: (1) the mean compliance or (2) any scalar-valued function of the individual load compliances such as the weighted sum of the mean and standard deviation. The computational time complexities of all the proposed algorithms are analyzed, compared with the naive approaches and then experimentally verified. Finally, a mean compliance minimization problem, a risk-averse compliance minimization problem, and a maximum compliance-constrained problem are solved to showcase the efficacy of the proposed algorithms. The maximum compliance-constrained problem is solved using the augmented Lagrangian method and the method proposed for handling scalar-valued functions of the load compliances, where the scalar-valued function is the augmented Lagrangian function.

Appendix: A partial derivative of the inverse quadratic form

In this section, it will be shown that the ith partial derivative of

$$\begin{aligned} f(\varvec{x})&= \varvec{v}^T (\varvec{A}(\varvec{x}))^{-1} \varvec{v} \end{aligned},$$

(12)

is

$$\begin{aligned} \frac{\partial f}{\partial x_i}&= -\varvec{y}^T \frac{\partial \varvec{A}}{\partial x_i} \varvec{y}^T \end{aligned},$$

(13)

where $\varvec{A}$ is a matrix-valued function of $\varvec{x}$, $\varvec{v}$ is a constant vector, and $\varvec{y} = \varvec{A}^{-1} \varvec{v}$ is a an implicit function of $\varvec{x}$ because $\varvec{A}$ is a function of $\varvec{x}$.

$$\begin{aligned} \varvec{v}&= \varvec{A} \varvec{y} \end{aligned},$$

(14)

$$\begin{aligned} \varvec{0}&= \varvec{A} \frac{\partial \varvec{y}}{\partial x_i} + \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} \end{aligned},$$

(15)

$$\begin{aligned} \frac{\partial y}{\partial x_i}&= - \varvec{A}^{-1} \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} \end{aligned},$$

(16)

$$\begin{aligned} f(\varvec{x})&= \varvec{v}^T \varvec{A}^{-1} \varvec{v} \end{aligned},$$

(17)

$$\begin{aligned}&= \varvec{y}^T \varvec{A} \varvec{y} \end{aligned},$$

(18)

$$\begin{aligned} \frac{\partial f}{\partial x_i}&= 2 \varvec{y}^T \varvec{A} \frac{\partial \varvec{y}}{\partial x_i} + \varvec{y}^T \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} \end{aligned},$$

(19)

$$\begin{aligned}&= - 2 \varvec{y}^T \varvec{A} \varvec{A}^{-1} \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} + \varvec{y}^T \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} \end{aligned}$$

(20)

$$\begin{aligned}&= - 2 \varvec{y}^T \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} + \varvec{y}^T \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} \end{aligned},$$

(21)

$$\begin{aligned}&= - \varvec{y}^T \frac{\partial \varvec{A}}{\partial x_i} \varvec{y} \end{aligned},$$

(22)