Efficient multi-partition topology optimization

TL;DR

This paper introduces an efficient method for solving multi-partition topology optimization problems, significantly reducing computational effort especially for large-scale problems with similar partitions, by using static condensation.

Contribution

The paper presents a novel static condensation approach for multi-partition topology optimization, enabling faster response and sensitivity calculations compared to traditional methods.

Findings

Substantial computational gains for large-scale problems.

Effective for problems with few degrees of freedom and similar partitions.

Reduces the need for large adjoint analyses.

Abstract

In topology optimization, the state of structures is typically obtained by numerically evaluating a discretized PDE-based model. The degrees of freedom of such a model can be partitioned in free and prescribed sets to define the boundary conditions. A multi-partition problem involves multiple partitions of the same discretization, typically corresponding to different loading scenarios. As a result, solving multi-partition problems involves multiple factorization/preconditionings of the system matrix, requiring a high computational effort. In this paper, a novel method is proposed to efficiently calculate the responses and accompanying design sensitivities in such multi-partition problems using static condensation for use in gradient-based topology optimization. A main problem class that benefits from the proposed method is the topology optimization of small-displacement multi-input-multi-output compliant mechanisms. However, the method is applicable to any linear problem. We present its formulation and an algorithmic complexity analysis to estimate computational advantages for both direct and iterative solution methods to solve the system of equations, verified by numerical experiments. It is demonstrated that substantial gains are achievable for large-scale multi-partition problems. This is especially true for problems with both a small set of number of degrees of freedom that fully describes the performance of the structure and with large similarities between the different partitions. A major contribution to the gain is the lack of large adjoint analyses required to obtain the sensitivities of the performance measure.