An optimization-based registration approach to geometry reduction

TL;DR

This paper introduces an optimization-based method for parametric geometry reduction that constructs bijective mappings between a reference domain and target geometries using point cloud registration and nonlinear optimization, aiding model order reduction.

Contribution

The paper presents a novel two-step approach combining point-set registration and nonlinear optimization to approximate geometry mappings without explicit parameterizations.

Findings

The method effectively constructs bijective mappings for complex geometries.

Numerical experiments demonstrate the approach's accuracy and robustness.

The approach is suitable for parametric model order reduction in engineering applications.

Abstract

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 $\Phi$ that maps a fixed reference domain $\Omega$ 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 $\Omega$ and a point cloud $\{y_i^{\rm raw}\}_{i=1}^Q$ that belongs to the boundary of the target domain $V$ and returns a bijection $\Phi$ that approximately maps $\Omega$ in $V$. We propose a two-step procedure: given the point clouds $\{x_j\}_{j=1}^N\subset \partial \Omega$ and $\{y_i^{\rm raw}\}_{i=1}^Q \subset \partial V$, we first resort to a point-set registration algorithm to determine the displacements $\{ v_j \}_{j=1}^N$ such that the deformed point cloud $\{y_j:= x_j+v_j \}_{j=1}^N$ approximates $\partial V$; then, we solve a nonlinear non-convex optimization problem to build a mapping $\Phi$ that is bijective from $\Omega$ in $\mathbb{R}^d$ and (approximately) satisfies $\Phi(x_j) = y_j$ for $j=1,\ldots,N$.We present a rigorous mathematical analysis to justify our approach; we further present thorough numerical experiments to show the effectiveness of the proposed method.