2022/11/18 by Tommaso Taddei, Taddei, Tommaso
Engineering · Physics and Astronomy · #3D Shape Modeling and Analysis #41A45 #65K10 #65M32 #65M50 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2211.10275
openalex publication_date 2022/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop and assess an optimization-based approach to parametric geometry reduction. Given a family of parametric domains, we aim to determine a parametric diffeomorphism Φ that maps a fixed reference domain Ω into each element of the family, for different values of the parameter; the ultimate goal of our study is to determine an effective tool for parametric projection-based model order reduction of partial differential equations in parametric geometries. For practical problems in engineering, explicit parameterizations of the geometry are likely unavailable: for this reason, our approach takes as inputs a reference mesh of Ω and a point cloud \yi\rm raw\i=1Q that belongs to the boundary of the target domain V and returns a bijection Φ that approximately maps Ω in V. We propose a two-step procedure: given the point clouds \xj\j=1N⊂ ∂ Ω and \yi\rm raw\i=1Q ⊂ ∂ V, we first resort to a point-set registration algorithm to determine the displacements \ vj \j=1N such that the deformed point cloud \yj:= xj+vj \j=1N approximates ∂ V; then, we solve a nonlinear non-convex optimization problem to build a mapping Φ that is bijective from Ω in ℝd and (approximately) satisfies Φ(xj) = yj for j=1,…,N.We present a rigorous mathematical analysis to justify our approach; we further present thorough numerical experiments to show the effectiveness of the proposed method.