An optimization-based registration approach to geometry reduction

Overview & Synopsis

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 {y raw i } Q i=1 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} N j=1 ⊂ ∂Ω and {y raw i } Q i=1 ⊂ ∂V , we first resort to a point-set registration algorithm to determine the displacements {vj} N j=1 such that the deformed point cloud {yj := xj + vj} N j=1 approximates ∂V ; then, we solve a nonlinear non-convex optimization problem to build a mapping Φ that is bijective from Ω in R d 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.