Optimisation of spatially varying orthotropic porous structures based on conformal mapping

TL;DR

This paper introduces a novel compliance minimisation method for designing spatially varying orthotropic porous structures using conformal mapping, simplifying the design variables and ensuring integrability and feature size control.

Contribution

The method reduces design variables to boundary values, naturally resolves integrability issues, and exploits symmetry to lower computational costs in designing orthotropic porous structures.

Findings

Homogenised results converge to fine-scale results

Maximum principle allows explicit feature size monitoring

Analytical solutions available for rectangular domains

Abstract

In this article, a compliance minimisation scheme for designing spatially varying orthotropic porous structures is proposed. With the utilisation of conformal mapping, the porous structures here can be generated by two controlling field variables, the (logarithm of) the local scaling factor and the rotational angle of the matrix cell, and they are interrelated through the Cauchy-Riemann equations. Thus the design variables are simply reduced to the logarithm values of the local scaling factor on selected boundary points. Other attractive features shown by the present method are summarised as follows. Firstly, with the condition of total differential automatically met by the two controlling field variables, the integrability problem which necessitates post-processing treatments in many other similar methods can be resolved naturally. Secondly, according to the maximum principle for harmonic functions, the minimum feature size can be explicitly monitored during optimisation. Thirdly, the rotational symmetry possessed by the matrix cell can be fully exploited in the context of conformal mapping, and the computational cost for solving the cell problems for the homogenised elasticity tensor is maximally abased. In particular, when the design domain takes a rectangle shape, analytical expressions for the controlling fields are available. The homogenised results are shown, both theoretically and numerically, to converge to the corresponding fine-scale results, and the effectiveness of the proposed work is further demonstrated with more numerical examples.