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COMPOSITIONAL MAPS FOR REGISTRATION IN COMPLEX GEOMETRIES
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DOI：10.1137/23M1597393.png)
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Abstract
En 中文
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 Q \subset R2 and the manifold M = {u\mu : \mu \in P\} associated with the parameter domain P \subset RP and the parametric field\mu H- u\mu \in L2(Q), our approach takes as input a set of snapshots from M and returns a parameter-dependent mapping \Phi : Q \times P- Q, 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 \Phi = N(a), where N: RM- Lip(Q; 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 Q; 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.
Keywords:
parameterized partial differential equations
registration in bounded domains
model order reduction
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