Multi-patch isogeometric neural solver for partial differential equations on computer-aided design domains

TL;DR

This work introduces a multi-patch isogeometric neural solver that unifies physics-informed neural networks with patchwise isogeometric analysis to solve PDEs on CAD geometries. By operating patchwise on reference domains and employing dedicated interface networks, the method achieves strong Dirichlet enforcement and solution continuity across nonconforming patch interfaces within a variational energy-minimization framework. Demonstrations on 2D magnetostatics and 3D nonlinear solid mechanics with contact show close agreement with high-fidelity FEM solutions, validating the approach for CAD-native PDEs and complex engineering problems. The framework paves the way for CAD-to-simulation integration using neural solvers, with potential benefits in parametric design and rapid prototyping of engineering systems directly within their design models.

Abstract

This work develops a computational framework that combines physics-informed neural networks with multi-patch isogeometric analysis to solve partial differential equations on complex computer-aided design geometries. The method utilizes patch-local neural networks that operate on the reference domain of isogeometric analysis. A custom output layer enables the strong imposition of Dirichlet boundary conditions. Solution conformity across interfaces between non-uniform rational B-spline patches is enforced using dedicated interface neural networks. Training is performed using the variational framework by minimizing the energy functional derived after the weak form of the partial differential equation. The effectiveness of the suggested method is demonstrated on two highly non-trivial and practically relevant use-cases, namely, a 2D magnetostatics model of a quadrupole magnet and a 3D nonlinear solid and contact mechanics model of a mechanical holder. The results show excellent agreement to reference solutions obtained with high-fidelity finite element solvers, thus highlighting the potential of the suggested neural solver to tackle complex engineering problems given the corresponding computer-aided design models.

Figures (15)

• Figure 1: Illustration of the -based geometry map from the reference to the physical domain. We have $\mathbf{x}_k = \bm{f}(\hat{\mathbf{x}}_k)$, $k=1,...,4$. The normal $\hat{\mathbf{n}}$ is transformed to $\mathbf{n}$.
• Figure 2: Example of a two-dimensional domain represented by three patches, $\Omega_1$, $\Omega_2$, $\Omega_3$. The reference domain $\hat{\Omega}$ and the mapping of the reference coordinate system onto each patch are also shown (red and blue dotted lines). The Dirichlet boundary $\Gamma_{\mathrm{D}}$ is marked in purple. The point $\bm{\xi}_{123} \in \Omega$ belongs to all three patches simultaneously.
• Figure 3: Illustration of the construction of interface ansatz functions. At the point where multiple subdomains meet, indicated by the red circle, the interface functions take zero values. Continuity is enforced at the interfaces indicated by the colored lines (yellow, cyan, and purple) in the reference and physical domains.
• Figure 4: Domain geometry of the simple quadrupole model. The iron yoke is shown in gray, the current excitations in yellow, and the air gap in white. The purple shading indicates the model section in the symmetry planes. Left: Full cross-section. Right: One-eighth of cross-section, exploiting rotational symmetry. The domain is partitioned into patches $\Omega_i$, $i=1, \dots,4$.
• Figure 5: Relative $L^2$ error during training for the entire computational domain and for each subdomain and the simple quadrupole model.