Fireshape: a shape optimization toolbox for Firedrake

Abstract

We introduce Fireshape, an open-source and automated shape optimization toolbox for the finite element software Firedrake. Fireshape is based on the moving mesh method and allows users with minimal shape optimization knowledge to tackle with ease challenging shape optimization problems constrained to partial differential equations (PDEs).

Control of Bifurcation Structures using Shape Optimization

“…branch of solutions arises at a specific value \lambda = \lambda \star . Additionally, we release an open-source implementation of the algorithm built on Firedrake [58], PETSc [7], and Fireshape [54]. While this paper focuses on controlling simple pitchfork and fold bifurcation points with respect to the shape of the domain, we expect that the ideas developed here also apply to other settings, such as the control of the bifurcation diagram with respect to a parameter \lambda 1 \in \BbbR as another parameter \lambda 2 in some (possibly infinitedimensional) parameter space is varied, or the control of other kinds of bifurcations such as Hopf bifurcations.…”

“…In this section, we give a brief introduction to PDE-constrained shape optimization. A more detailed exposition of the mathematical and implementation techniques employed here is given in [55,54].…”

“…Downloaded 01/20/22 to 94.4.252.85 Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms A61 There are competing approaches to define the set of admissible domains \scrU , such as the phase-field method [8,30], the level-set method [3,45], and the moving mesh method [4,55], among others. In this work, we consider the moving mesh method because its neat integration with standard finite element software allows the automation of several key steps in shape optimization [54]. The main idea of the moving mesh method is to construct the set of admissible domains \scrU by applying diffeomorphisms to an initial domain \Omega 0 \subset \BbbR d , that is,…”

“…For the numerical simulations in section 5 we employ the shape optimization software Fireshape [54], which is based on the moving mesh method. This choice offers numerous convenient features, including automated shape differentiation in the Unified Form Language [34], automated adjoint computation via pyadjoint [19,26], integration with the finite element software Firedrake [58] for efficient linear solvers (e.g., block preconditioners and geometric multigrid) from PETSc [7], and access to numerous optimization algorithms via the Rapid Optimization Library [44], including augmented Lagrangian algorithms [51,Chap.…”

“…Once \Omega 0 and (u 0 , \lambda 0 , \phi 0 ) have been determined, we use Fireshape [54] to solve (4.1) iteratively with the trust-region optimization algorithm [16] and the inner solver limited memory BFGS [46] implemented in the Rapid Optimization Library [44]. To ensure that the computational mesh is not tangled, we reject optimization steps for which the minimum of the determinant of the Jacobian of the transformation T defined in (3.1) becomes negative.…”