A level set-based structural optimization code using FEniCS

Abstract

This paper presents an educational code written using FEniCS, based on the level set method, to perform compliance minimization in structural optimization. We use the concept of distributed shape derivative to compute a descent direction for the compliance, which is defined as a shape functional. The use of the distributed shape derivative is facilitated by FEniCS, which allows to handle complicated partial differential equations with a simple implementation. The code is written for compliance minimization in the framework of linearized elasticity, and can be easily adapted to tackle other functionals and partial differential equations. We also provide an extension of the code for compliant mechanisms. We start by explaining how to compute shape derivatives, and discuss the differences between the distributed and boundary expressions of the shape derivative. Then we describe the implementation in details, and show the application of this code to some classical benchmarks of topology optimization. The code is available at http:// antoinelaurain.com/compliance.htm, and the main file is also given in the appendix.

Structural optimization under internal porosity constraints using topological derivatives

“…The following result, proved in e.g. [8,18], supplies the expression of the shape derivative of J 2 (Ω).…”

“…Therefore, a high level of porosity in the final manufactured design can completely ruin their optimality. In this third experiment, inspired by [3,7,10,18], we address this issue by incorporating our porosity constraint into the shape and topology optimization formulation.…”

A new stabilisation approach for level-set based topology optimisation of hyperelastic materials

“…-The code used to run the numerical examples in Section 5 is comprised of the following standard ingredients: (i) level-set code (Laurain, 2018); (ii) nonlinear continuum mechanics code (Bonet et al, 2016b); (iii) incrementally linearised solver (Poya et al, 2016); (iv) a bisection algorithm to determine the stabilisation required (Burden et al, 2015). -Equilibrium paths displayed for the final designs have been obtained by means of an standard arclength technique (Bonet et al, 2016b).…”

Fully and semi-automated shape differentiation in NGSolve

“…For more details on the implementation of this example, we refer to Online Resource 1 and for more details on shape optimisation in a space-time setting, we refer the interested reader to Köthe (2020).…”