A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems

TL;DR

This paper introduces a novel computational framework combining polynomial dimensional decomposition and topology derivatives to efficiently evaluate stochastic topology sensitivities in high-dimensional complex systems, aiding robust and reliability-based topology optimization.

Contribution

It presents a new framework that analytically and semi-analytically computes stochastic topology sensitivities for high-dimensional systems, including benchmark examples for validation.

Findings

Accurate analytical solutions for low-dimensional cases.

Semi-analytical solutions for high-dimensional failure probabilities.

The method outperforms traditional approaches in accuracy for sensitivities.

Abstract

In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.