Orientation preserving mesh optimisation and preconditioning

Abstract

Summary A robust mesh optimisation method is presented that directly enforces the resulting deformation to be orientation preserving. Motivated by aspects from mathematical elasticity, the energy functional of the mesh deformation can be related to a stored‐energy functional of a hyperelastic material. Formulating the functional in the principal invariants of the deformation gradient allows fine‐grained control over the resulting deformation. Solution techniques for the arising nonconvex and highly nonlinear system are presented. As existing preconditioners are not sufficient, a partial differential equation–based preconditioner is developed.

Rational Bézier Guarding

“…Polynomial, without both Guarantees. There is quite a list of methods to generate a mesh of higher‐order polynomial triangles that follow an indirect approach: first generate a mesh with linear elements, approximating the curve constraints, then perform an incremental deformation procedure with the goal of making the then curved elements conform the given curves [SP02,LSO∗04,SFJ∗05, Oli08,PP09,RGPS11,GB12,GPRPS13,XSHM13,TGRL13,XC14, RGSR16,MEK∗16,PSG16,FP16,TPM18,Pau18,HSG∗19]. While the deformation can be constrained to prevent the introduction of any degeneracies or inversions, guarantees of convergence to a conforming state cannot be given.…”