Elasticity-based morphing technique and application to reduced-order modeling

TL;DR

Elasticity-based morphing provides a boundary-parameter-free registration between domains of identical topology by iteratively solving linear-elasticity problems on the evolving domain. The method supports enforcing boundary feature correspondences and tangential boundary motion, and it is embedded into an offline/online reduced-order modeling pipeline to handle geometric variability efficiently. In 2D tests on Tensile2D and AirfRANS, the approach achieves robust boundary alignment and substantial online speedups (up to hundreds of times faster than high-fidelity morphing) while enabling accurate learning of scalar outputs via Gaussian process regression. The work lays a foundation for geometry-agnostic, topology-preserving morphing with broad applicability to model-order reduction and geometry-aware prediction, with promising paths to 3D extensions and optimized mode management.

Abstract

The aim of this article is to introduce a new methodology for constructing morphings between shapes that have identical topology. The morphings are obtained by deforming a reference shape, through the resolution of a sequence of linear elasticity equations, onto every target shape. In particular, our approach does not assume any knowledge of a boundary parametrization, and the computation of the boundary deformation is not required beforehand. Furthermore, constraints can be imposed on specific points, lines and surfaces in the reference domain to ensure alignment with their counterparts in the target domain after morphing. Additionally, we show how the proposed methodology can be integrated in an offline and online paradigm, which is useful in reduced-order modeling involving variable shapes. This framework facilitates the efficient computation of the morphings in various geometric configurations, thus improving the versatility and applicability of the approach. The robustness and computational efficiency of the methodology is illustrated on two-dimensional test cases, including the regression problem of the drag and lift coefficients of airfoils of non-parameterized variable shapes.

Figures (22)

• Figure 1: Reference domain $\Omega_0$ with two samples $\Omega_i$ and $\Omega_j$ from the target dataset.
• Figure 2: Example of reference domain $\Omega_0$, target domain $\Omega$, and intermediate domain $\boldsymbol{\phi}^{(m)}(\Omega_0$).
• Figure 3: Visual representation of \ref{['vectDistance']}. $\boldsymbol{D}^{\partial\Omega}_{\boldsymbol{\phi}}(\boldsymbol{x})$ is the vector that points from $\boldsymbol{x}\in \boldsymbol{\phi}(L_i^0)$ to its projection on $L_i$.
• Figure 4: Illustration of the final correction step.
• Figure 5: Reference and target domains, with the partition used on the boundary of the reference domain.

Theorems & Definitions (12)

• Lemma 1
• proof
• Proposition 1
• Lemma 2
• Proposition 2
• proof
• Remark 1: Comparison with de2016optimization
• Remark 2: Parameter $\gamma^{(m)}$
• Remark 3: Gradient descent
• Remark 4: Alternative definition