implement finite element methods

Modeling coupled multiphysics PDEs via finite element discretization faces challenges in representing composite function spaces, rigid degree-of-freedom (DoF) numbering schemes, and poor interoperability with linear algebra backends. Method: This paper introduces a tree-based abstraction for function spaces, wherein product spaces are represented hierarchically as trees; a unified multi-index mechanism generates diverse DoF numbering strategies—supporting heterogeneous data layouts such as block-structured and interleaved formats. Contribution/Results: Implemented in the dune-functions module of the DUNE framework, the approach significantly enhances interoperability with algebraic solvers, sparse matrix formats, and parallel data structures. Experiments demonstrate efficient modeling of canonical multiphysics problems—including Taylor–Hood discretizations of the Stokes equations—while maintaining scalability and flexibility across diverse solvers (e.g., UMFPACK, PETSc) and storage layouts.

This work addresses the challenge of mesh compatibility arising from the coupled discretization of interfaces and bulk domains in contact mechanics. The authors propose a decoupled discretization approach that employs a NURBS-based boundary layer mesh, constructed directly from CAD boundary representations, to accurately capture the contact interface, while the bulk domain is discretized using a structured Cartesian grid. Non-matching meshes are coupled across scales via mortar-type embedded constraints. This method represents the first formulation that decouples isogeometric boundary layers from structured volume meshes, enabling independent optimization of element type and resolution for both interface and bulk domains. Consequently, it preserves high-order smoothness along the contact surface while significantly enhancing geometric accuracy, computational efficiency, and modeling flexibility.

This work proposes an end-to-end automated framework that generates compliant engineering reports directly from a single image of a mechanical component. The approach employs a solver-agnostic multi-agent system operating within a shared contextual space, leveraging a quality-gated conditional iteration mechanism to collaboratively perform geometric reconstruction, material inference, adaptive mesh generation, multi-case finite element analysis, and code compliance assessment. A unified uncertainty quantification framework is innovatively formulated by integrating interval analysis, probability density functions, and fuzzy logic, complemented by task-dependent conservativeness criteria to reconcile conflicting multi-limit-state requirements. Demonstrated on a single photograph of an L-shaped steel bracket, the system autonomously produced a 171,504-node mesh, executed seven analyses, and delivered a complete report—including failure diagnosis and redesign recommendations—without any human intervention.

This study addresses the lack of a rigorous formal definition of "finite elements" in the finite element method by proposing a formal framework within the Rocq proof assistant based on record types, wherein a finite element is modeled as a structure comprising geometric data and validity proofs. The work presents the first complete formalization of simplicial Lagrange finite elements of arbitrary dimension and polynomial degree in a proof assistant, and rigorously verifies their unisolvence property using foundational theories of finite families, affine spaces, and multivariate polynomials. This achievement yields a general definition and correctness proof for simplicial Lagrange finite elements with uniform nodal distributions, thereby establishing a formal foundation for the verification of scientific computing software.

This work proposes the first autonomous simulation system that integrates an agent-based architecture with domain-finetuned large language models (LLMs) to enable end-to-end modeling and solution of solid mechanics, fluid dynamics, and multiphysics problems. Addressing the limitations of conventional LLMs—which often hallucinate, lack awareness of variational structures, and fail to close the loop from problem description to verified solutions—the system incorporates retrieval-augmented multi-LLM code generation and filtering, finetuned models spanning 3B to 120B parameters, multi-agent collaboration, and runtime feedback mechanisms. A high-quality corpus of over a thousand FEniCS codes was curated to support training and evaluation. On a benchmark suite of 39 nonlinear elasticity, plasticity, and non-Newtonian fluid problems, the GPT OSS 120B model achieved a code generation success rate of 71.79%, substantially outperforming non-agent-based approaches.

Although topology optimization has matured, its reliance on manual intervention—such as modeling, meshing, and boundary condition specification—hinders accessibility for non-experts. This work proposes the first conversational framework based on a large language model (LLM) agent that enables end-to-end topology optimization through natural language instructions and optional inputs (e.g., images, geometry, or meshes), automatically invoking finite element solvers and optimization tools. The approach integrates multi-load structural and thermal optimization, handles stress constraints, and employs few-shot prompting strategies, successfully reproducing benchmark cases while solving complex engineering problems and autonomously generating optimized structures, field distributions, and convergence curves. Ablation studies confirm that prompt design critically enhances system robustness, substantially lowering the usability barrier without compromising numerical reliability.

This work proposes a measurement-driven, constrained natural language interface architecture to reduce manual configuration overhead in finite element simulations while mitigating the risk of unreliable code generation by large language models (LLMs) in critical solver stages. The approach confines the LLM to front-end tasks—such as prompt parsing and Gmsh script generation for non-standard geometries—while a deterministic scheduler orchestrates verified FEniCS/UFL templates for core computations across five multiphysics problem classes: linear elasticity, hyperelasticity, elastoplasticity, thermomechanical coupling, and phase-field fracture. Experimental results demonstrate 100% prompt parsing success, 97.1% field extraction accuracy, and 90% success rate in custom geometry generation. Simulation accuracy reaches sub-percent levels for smooth problems, with errors in nonlinear cases maintained within 2–5%.

This work addresses the high computational cost of traditional finite element analysis and the limited generalizability of existing machine learning surrogates to varying geometries and loading conditions. The authors propose a Mesh Graph Network (MGN) that efficiently predicts von Mises stress fields for arbitrary two-dimensional structures with holes by encoding node types, relative edge features, and global load information. The model inherently satisfies translational and rotational invariance and generalizes to unseen combinations of geometry and loading without retraining. On test cases, it achieves an R² as high as 0.97, substantially outperforming current machine learning approaches, which report R² values ranging from approximately 0.01 to 0.86, thereby demonstrating superior generalization capability and strong potential for practical engineering applications.

This work addresses the challenges of high-fidelity simulation in fusion energy systems—specifically, geometric modeling, multiphysics coupling, and the integration of particle and continuum methods—by proposing a unified framework that seamlessly combines commercial CAE software with existing fusion codes. The framework enables accurate representation of complex geometries, automatic generation of unstructured meshes, and efficient coupling between particle transport and continuum solvers. The resulting simulation workflow significantly enhances geometric fidelity for critical components and strengthens capabilities in multiscale, multiphysics co-simulation, while maintaining strong scalability and computational efficiency.