implement boundary element methods

A long-standing interoperability barrier exists between CAD design and electrostatic boundary element method (BEM) analysis due to incompatible geometric representations and meshing requirements. Method: This paper proposes a high-order, non-conforming BEM–CAD tightly coupled framework that abandons conventional conforming mesh constraints. It employs NURBS-based exact geometric representation and non-uniform boundary discretization, enabling fully automated mapping from CAD models to simulation meshes via a custom-developed CAD plugin. Contribution/Results: The framework is the first to systematically preserve sensitivity to the original CAD parametric representation within electrostatic BEM, while supporting arbitrary-order geometric and field reconstruction. Numerical experiments demonstrate one- to two-order-of-magnitude reduction in error compared to conventional low-order conforming BEM, and over fivefold improvement in modeling efficiency—significantly enhancing fidelity and iteration speed for virtual prototyping of electrical equipment.

In two-dimensional smoothed particle hydrodynamics (SPH), boundary integrals are notoriously difficult to evaluate analytically, severely limiting both accuracy and computational efficiency. Method: This paper proposes a closed-form area integration method for SPH kernel functions and their gradients over triangular boundaries. We derive, for the first time, a general analytical integral formula for arbitrary-degree compactly supported polynomials over arbitrary triangular domains, enabling numerically stable evaluation via Chebyshev polynomials and the Gaussian hypergeometric function $_2F_1$. By discretizing boundaries into triangular elements and embedding differentiable analytical integration modules, the method supports high-order boundary conditions and flexible coupling with SPH–mesh hybrid schemes. Contribution/Results: Experiments demonstrate a five-order-of-magnitude speedup in integral and gradient evaluations over conventional numerical quadrature, achieving machine precision. The implementation is open-source, significantly enhancing boundary treatment efficiency in SPH and enabling robust multi-paradigm method integration.

This work challenges the conventional wavelength-dependent meshing criterion (e.g., λ/10) for high-frequency boundary element methods (BEM). Through systematic spectral analysis of the discretized 3D boundary integral operator matrices, we demonstrate that their spectral deviation from the continuous operator’s spectrum grows significantly with increasing frequency, leading to deteriorating solution accuracy. Integrating matrix spectral analysis, high-frequency asymptotic theory, and operator approximation error modeling, we rigorously establish—via spectral convergence analysis—that traditional meshing strategies fail to satisfy spectral consistency at high frequencies. This fundamentally questions long-standing empirical guidelines in computational electromagnetics and acoustics. Our analysis quantifies the discrete error’s power-law growth with frequency and provides a new theoretical foundation for adaptive mesh design and error-controlled high-frequency BEM simulations.

This study addresses the numerical pollution problem arising from boundary element method (BEM) discretization in high-frequency two-dimensional electromagnetic scattering. It reveals the fundamental limitations of the Nyquist sampling criterion in the discretization of electromagnetic integral equations. Through rigorous spectral analysis and boundary integral operator theory, the work establishes— for the first time—that pollution effects are ubiquitous in both well-conditioned and ill-conditioned integral equations, stemming from systematic spectral distortion induced by discretization on the composition of integral operators. Based on this insight, a novel operator-spectrum correction framework is proposed to quantitatively suppress dispersion errors introduced by discretization. Numerical experiments demonstrate that the approach significantly enhances the accuracy and stability of high-frequency BEM solutions while ensuring controllable dispersion characteristics. This work provides a theoretically grounded, robust paradigm for reliable high-frequency BEM applications.

This work addresses the high computational cost of traditional finite element methods in mesh deformation and the difficulty of existing neural operators in effectively enforcing vector-field Dirichlet boundary conditions. The authors formulate mesh deformation as a linear elastic boundary value problem and introduce a boundary integral representation based on a Dirichlet-type Green’s tensor, which enables exact reconstruction of the interior displacement field from boundary displacements alone. By integrating this boundary integral formulation with neural operators, they develop a geometry- and material-aware Green traction kernel learning framework that decouples physical constraints from geometric representation. The method rigorously satisfies linearity and superposition principles, demonstrating high accuracy, strong generalization across boundary conditions, adaptability to diverse geometries—including large deformations of flexible beams and rigid-body motions of NACA airfoils—while preserving mesh quality and computational efficiency.

This work addresses the challenges of complex boundary condition handling, code redundancy, and poor distributed efficiency in partial differential equation solvers on block-structured grids. The authors propose a unified modeling approach that expresses user-defined boundary conditions as affine sparse linear operators and, for the first time, systematically reformulates them into sparse matrix-vector multiplication (SpMV) form. Leveraging a domain-specific language (DSL) and compiler techniques—combined with multi-stage programming and polyhedral analysis—the framework automatically generates highly optimized matrix-free or sparse matrix kernels while optimizing communication scheduling and reuse. The method achieves substantial performance gains, demonstrating 72%–88% strong scaling efficiency on 1,344 CPU cores, up to 7.6× acceleration in boundary computation kernels, and a reduction of over 70% in code size.

This work addresses the well-known ill-conditioning of the electric field integral equation (EFIE) at low frequencies, high frequencies, and fine discretizations, which hinders efficient numerical solution. The authors propose a unified preconditioning strategy based on shifted Helmholtz operator regularization, integrating a novel preconditioner design with a fast matrix-vector product algorithm of quasi-linear complexity. This approach effectively overcomes the limitations of conventional pseudo-inverse methods in handling the shift operator. The resulting solver exhibits remarkably stable iteration counts across a wide range of frequencies and mesh resolutions, achieving—for the first time—a unified, efficient treatment of all three canonical ill-conditioned regimes while attaining quasi-linear computational complexity for EFIE solutions.

This work addresses a key limitation of conventional LLM-based PDE solvers, which implicitly embed numerical strategies within generated code, making pre-execution validation and post-failure correction challenging. To overcome this, the authors propose AutoPDE, the first framework to explicitly model solution strategies as revisable, decoupled objects separate from implementation code. AutoPDE employs a three-stage pipeline—PDE type identification, numerical method selection, and adaptive parameter tuning—augmented by low-overhead trial solves and a reusable skill library to construct and refine strategies prior to code generation. Evaluated on the PDE Agent Bench, AutoPDE achieves a 54.5% pass rate, outperforming the strongest baseline by 14.2 percentage points, thereby substantially improving both the reliability and interpretability of AI-driven PDE solving.

This work addresses the high memory and computational complexity typically associated with solving three-dimensional partial differential equations on Cartesian grids. By exploiting tensor-product structure, the proposed method decomposes the 3D operator into one-dimensional banded kernels aligned with coordinate axes, thereby avoiding explicit assembly of the global matrix and enabling a matrix-free solution strategy. Within a unified framework that integrates diverse numerical approaches—including Kronecker product algebra, compact finite differences, isogeometric analysis, and direct diagonalization—the study systematically identifies three key techniques: multi-right-hand-side reshaping, sum factorization, and pencil-style MPI decomposition. These innovations collectively enhance hardware affinity and parallel scalability, reducing algorithmic complexity to O(N) and storage requirements to O(Nₓ + Nᵧ + N_z), thus enabling efficient large-scale 3D PDE simulations.