finite element analysis

To address the computational inefficiency and trade-off between accuracy and efficiency in traditional adaptive finite element method (FEM) mesh generation for solving partial differential equations (PDEs) on complex 3D geometries—such as turbine blade scans—this paper proposes an end-to-end learning-based adaptive meshing framework. Our method employs a lightweight neural network to directly regress a spatial size field from sparse Monte Carlo (MC) solution estimates, replacing iterative optimization with a single forward inference pass. The pipeline integrates Monte Carlo sampling, neural regression, size-field-driven tetrahedral mesh generation, and FEM solving. Evaluated across diverse 3D shapes and boundary conditions, our approach achieves 2–4× speedup over conventional adaptive FEM and MC-based methods while maintaining comparable solution accuracy and demonstrating strong generalization capability.

Existing CAD generation models struggle to emulate engineers’ iterative design processes and lack the capability to validate physical and structural compliance. This work proposes an industry-native CAD generation framework that produces complete multi-part STEP files from engineering text and, for the first time, integrates finite element analysis (FEA) into the generative loop to verify structural plausibility. The approach leverages structured blueprint descriptions and 21-view image renderings as dual supervisory signals to guide large language model agents—such as GPT-5.5 and Claude Code—toward self-improving generation. Evaluated on the S2O and Fusion360 datasets, the method significantly enhances geometric reconstruction quality and engineering compliance, improving Box-IoU from 0.444 to 0.592 and from 0.397 to 0.505, respectively.

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 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.

This study addresses the loss of precision caused by rounding errors in finite element computations, which remains difficult to analyze a priori. We propose the first automated a posteriori rounding error estimation framework that leverages running error analysis to track numerical and error propagation in real time. Built upon the FEniCS Form Compiler, this method achieves the first automated error estimation within finite element kernels through a C++ backend, custom arithmetic types, and templated kernel generation techniques. Experimental results demonstrate that the framework successfully detects catastrophic cancellation with only a 2–4× performance overhead. By effectively supporting mixed-precision design and numerical debugging, this work establishes a new paradigm for ensuring reliability in scientific computing.

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 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 a key limitation of conventional neural PDE surrogates, which model field evolution on fixed grids and thereby overlook the critical role of mesh design in allocating spatial resolution and spectral bandwidth. The study introduces, for the first time, adaptive discretization as a physics-constrained conditional generation task, proposing a two-stage diffusion framework: it first generates an r-adaptive displacement grid conditioned on observed dynamics and then predicts solution evolution on this adaptive mesh. By incorporating physics-aware regularization, geometric validity constraints, and local spectral concentration, the method achieves learnable, interpretable, and numerically stable mesh adaptation. Extensive experiments across five classes of PDE problems demonstrate substantial improvements over traditional adaptive and reduced-order methods, with particularly notable gains in complex domains.

This work addresses the challenge that existing automatic code generation methods often produce structurally invalid or physically inconsistent models, which are unsuitable for engineering simulation. To ensure physical consistency and simulatability, the authors propose a procedural modeling framework that integrates domain knowledge injection, constraint-guided fine-tuning, and closed-loop simulation validation. Key contributions include CivilInstruct—the first instruction-following dataset tailored for structural engineering—along with a two-stage fine-tuning strategy and MBEval, a validation-driven evaluation benchmark. Experimental results demonstrate that the proposed approach significantly outperforms baseline methods across multiple rigorous metrics, effectively suppressing hallucinations and constraint violations, and enabling the direct use of generated models in structural dynamics simulations.