isogeometric physics-informed solver

This work addresses the inefficiency and geometric inaccuracy inherent in traditional CAD/CAE workflows for shell structures by proposing a novel integration of isogeometric analysis and deep learning. The method employs watertight T-splines to construct a geometrically exact neural network architecture, uniquely embedding Bézier extraction and Bernstein polynomials as activation functions within the DeepONet framework. Coupled with the Kirchhoff–Love shell formulation, this approach enables seamless unification of CAD and CAE within a single neural network. Without requiring repeated modeling or retraining, the model efficiently and accurately predicts mechanical responses across diverse complex shell geometries, substantially enhancing both computational efficiency and solution accuracy.

Solving partial differential equations (PDEs) on complex CAD geometries remains challenging due to difficulties in mesh generation, enforcing strong boundary conditions, and ensuring inter-patch continuity. This work proposes a variational framework integrating multi-patch isogeometric analysis with physics-informed neural networks (PINNs). Leveraging NURBS-based multi-patch representations, it constructs patch-local PINNs and introduces dedicated interface networks to enforce C⁰ continuity across patches. A variational energy minimization loss, combined with a Dirichlet-constrained output layer, enables end-to-end solutions that are geometrically faithful, boundary- and interface-consistent, and highly accurate. To our knowledge, this is the first approach unifying patch-local neural networks with variational PINNs to systematically address strong Dirichlet enforcement and multi-patch coordination in realistic engineering CAD models. Numerical experiments—including 2D static magnetic fields in a quadrupole magnet and 3D nonlinear contact mechanics—demonstrate excellent agreement with high-fidelity finite element benchmarks, validating the method’s efficacy and generalizability in practical engineering applications.

Membrane locking severely degrades accuracy in Kirchhoff–Love thin-shell analysis, particularly within isogeometric analysis (IGA) frameworks. Method: This paper proposes a mixed discretization scheme that eliminates membrane locking without introducing auxiliary degrees of freedom (DOFs). Retaining the original NURBS surface geometry, it couples Lagrange-type surface discretizations to construct mixed shell elements—preserving the sparsity pattern and bandwidth of the stiffness matrix and requiring no modifications to existing IGA solvers. Contribution/Results: The method achieves, for the first time, zero DOF increase, zero bandwidth expansion, and zero code restructuring while suppressing membrane locking in both linear and nonlinear problems. Stress recovery is direct and computationally efficient, yielding significantly improved membrane stress accuracy. Numerical experiments on classical benchmark problems demonstrate complete elimination or substantial mitigation of membrane locking, with optimal convergence rates preserved across all cases.

This work addresses the membrane locking phenomenon in isogeometric analysis of the Kirchhoff–Love shell model. To resolve this issue, we propose a novel, locking-free, high-order accurate discretization method grounded in the Hellinger–Reissner variational principle. The strain field is independently approximated using splines of one degree lower than the displacement field. Crucially, we introduce for the first time a combination of approximate dual splines with row- and lumped-mass projection techniques to construct an approximately diagonalized strain projection matrix, enabling efficient static condensation. The method eliminates membrane locking while rigorously preserving optimal convergence rates and high-order accuracy. Numerical experiments on Euler–Bernoulli beams and classical shell benchmark problems demonstrate exceptional computational efficiency, accuracy, and robustness—fully alleviating membrane locking without compromising solution quality or asymptotic convergence behavior.

Physics-informed neural networks (PINNs) face two key bottlenecks in solid mechanics: the mismatch between infinite solution domains and finite structural boundaries, and the inability of Euclidean solution spaces to represent complex geometries. To address these, we propose a Euclidean–topological joint solution space, enabling intrinsic adaptation to arbitrary bounded domains and irregular geometries via topological embedding mappings. We further introduce stress–displacement dual-field decoupled parameterization to enhance physical consistency, and integrate PDE-constrained loss with FEM-inspired boundary treatment. The method enables mesh-free, small-data 2D/3D forward and inverse problem solving, achieving accuracy comparable to FEM while significantly improving convergence speed, generalization capability, and robustness in inverse parameter identification. Our contributions are threefold: (1) the first formulation of a Euclidean–topological joint solution space for solid mechanics; (2) a geometry–physics co-design modeling paradigm; and (3) a novel dual-field decoupled PINN framework.

This study addresses the limited accuracy of neural solvers and the insufficient error control in traditional isogeometric analysis (IGA) by proposing a gradient-free closed-form fitting mechanism. Specifically, it enhances IGA solutions via local Kolmogorov–Arnold Networks (KANs), employing hybrid basis functions to integrate strong-form equations with boundary data, thereby improving accuracy without requiring optimizers. Furthermore, a maximum-principle-based a posteriori safeguarding strategy is incorporated to reconcile computational efficiency with precision. Experimental results demonstrate that the proposed approach reduces L2 and H1 errors by 4.2 to 220 times compared to KANs trained from scratch, while achieving a 167-fold improvement in source term recovery accuracy for inverse problems.

This study addresses the challenge of efficiently integrating GPU acceleration with differentiable computing in isogeometric analysis. We present an open-source framework built upon JAX that decouples mesh construction from computational tracing, enabling just-in-time compilation and reverse-mode automatic differentiation for geometric evaluation, system assembly, and solving. Methodologically, we introduce a unified energy density paradigm compatible with Galerkin, energy minimization, and collocation formulations, supporting hierarchical extraction for high-order local refinement. Sensitivities are efficiently computed via the adjoint method. Experimental results demonstrate a 24-fold speedup in GPU-based assembly, reducing total solution time from 31.8 seconds to 1.38 seconds while achieving an accuracy of 5e-11, thereby significantly lowering overall computational costs.

This study addresses the limitations in accuracy and efficiency associated with computing near-crack fracture quantities—such as dynamic stress intensity factors (DSIFs) and local stress fields—in dynamic crack propagation problems. To overcome these challenges, the authors propose a hybrid s-version isogeometric analysis method (hS-IGA), which employs B-spline basis functions over the global domain while retaining Lagrangian meshes in the crack-tip region. This approach effectively eliminates the accuracy loss typically caused by discontinuous coupling integrals between global and local domains in conventional s-methods. Notably, the proposed strategy avoids recursive mesh refinement and remains compatible with standard Gaussian quadrature, substantially enhancing computational efficiency. Numerical experiments in both two and three dimensions demonstrate that the method reduces the number of coupling integration points by approximately 81% and 95.6%, respectively, while accurately capturing DSIFs and local stress fields, thereby confirming its high efficiency and reliability.

This study addresses the challenge of identifying nonlinear, spatially heterogeneous constitutive parameters of solid materials when only surface displacements and contact forces are available. To this end, an efficient inverse method based on isogeometric finite element model updating (FEMU) is proposed. By employing low-order Lagrangian interpolation—decoupled from the analysis mesh—to represent spatially varying material fields, and integrating analytical gradient computation, a material-parameter continuation strategy, and a trust-region reflective optimization algorithm, the approach enables non-destructive, high-fidelity parameter identification. Numerical experiments successfully reconstruct heterogeneous material distributions in three-dimensional hyperelastic solids and thin shells, demonstrating the method’s effectiveness and computational efficiency for applications in soft tissue biomechanics and advanced material characterization.

This work addresses the limitations of traditional approaches to nonlinear elasticity on curved surfaces, which rely on symmetry assumptions and require repeated solves for each combination of geometric and material parameters, thereby struggling with continuously varying parameters or broken symmetries. The authors propose a unified solution framework based on physics-informed neural networks (PINNs), wherein the governing equations of nonlinear elasticity—formulated using differential geometry—and associated boundary conditions are hard-encoded into the loss function. This enables a single model to generalize across a continuous parameter space. Notably, the method achieves the first unified modeling of nonlinear elastic systems with fivefold defects on spherical surfaces across varying parameters, accurately reproducing known analytical and numerical solutions while successfully extrapolating to parameter combinations beyond the training range, thus overcoming key constraints of conventional solvers.