Compositional Maps for Registration in Complex Geometries

We develop and analyze a parametric registration procedure for manifolds associated with the solutions to parametric partial differential equations in two-dimensional domains. Given the domain Ω ⊂ ℝ2 and the manifold )Formular Presented) associated with the parameter domain P ⊂ ℝP and the parametric field (Formular Presented), our approach takes as input a set of snapshots from M and returns a parameter-dependent mapping ϕ: Ω X P → Ω, which tracks coherent features (e.g., shocks, shear layers) of the solution field and ultimately simplifies the task of model reduction. We consider mappings of the form ϕ = N(a), where N: ℝM → Lip(Ω; ℝ2) is a suitable linear or nonlinear operator; then we state the registration problem as an unconstrained optimization statement for the coefficients a. We identify minimal requirements for the operator N to ensure the satisfaction of the bijectivity constraint; we propose a class of compositional maps that satisfy the desired requirements and enable nontrivial deformations over curved (nonstraight) boundaries of Ω; we develop a thorough analysis of the proposed ansatz for polytopal domains, and we discuss the approximation properties for general curved domains. We perform numerical experiments for a parametric inviscid transonic compressible flow past a cascade of turbine blades to illustrate the many features of the method.