Scalable solvers for nonlinear hyperelastic topology optimisation

Develop scalable solvers for the resolution of the Neo-Hookean hyperelastic symmetric cantilever topology optimisation problem formulated as finding u in [H^1_{Γ_D}(Ω)]^3 such that the nonlinear weak form R(u, v) = 0 holds for all v in [H^1_{Γ_D}(Ω)]^3, where R(u, v) = ∫_Ω S(u) : dE(u, v) d x − ∫_{Γ_N} g · v d s and S(u) denotes the second Piola–Kirchhoff tensor. The objective is to achieve efficient, scalable solution of this nonlinear PDE-constrained optimisation problem, which the authors state remains an open area of research.

Background

The paper extends a three-dimensional minimum elastic compliance problem to a hyperelastic setting using a Neo-Hookean constitutive law. The resulting state equations are nonlinear and are solved via a Newton–Raphson method within the GridapTopOpt framework, with PETSc-based linear solvers used inside Newton iterations and for adjoint computations.

In discussing this extension, the authors explicitly state that the development of scalable solvers for resolving the nonlinear hyperelastic weak form remains an open area of research. They also note that the current implementation is suited to small to moderate strains due to the ersatz material approximation and that efficient nonlinear solvers are outside the scope of the article, highlighting ongoing challenges in achieving robust and scalable performance for such problems.