fem solver implementation

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 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%.

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.

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.

Current LLM evaluations lack scientifically valid benchmarks tailored to computational mechanics, hindering rigorous assessment of their ability to generate physically consistent and numerically reliable finite element method (FEM) code. To address this, we propose FEM-Bench—the first structured scientific reasoning benchmark that systematically integrates physical principles (e.g., partial differential equations, constitutive relations) with numerical constraints (e.g., mesh discretization, element assembly), covering core modeling tasks including geometric representation, material behavior, and boundary condition specification. We introduce a novel “physics–mathematics” dual-constraint criterion for verifiable scientific code generation, enabling progressive difficulty scaling and objective quantification. Coupled with an automated unit-testing framework, FEM-Bench evaluates 33 nontrivial introductory tasks: Gemini 3 Pro achieves a 78.8% full-pass rate in function generation, while GPT-5 attains a 73.8% pass rate in joint testing—revealing critical limitations of current LLMs in rigorous scientific programming.

This work addresses the absence of a standardized benchmark for evaluating code generation targeting partial differential equation (PDE) solvers, particularly with respect to numerical accuracy, computational efficiency, and compatibility with mainstream finite element libraries. To bridge this gap, the authors introduce the first multi-metric, multi-library benchmark for PDE solver generation, comprising 645 structured instances spanning six mathematical problem types and eleven PDE classes. The benchmark supports three major finite element frameworks—DOLFINx, Firedrake, and deal.II—and incorporates a staged evaluation framework that holistically assesses code executability, numerical correctness, and performance. Experimental results demonstrate that while current large language models can produce executable code, their success rate drops substantially when stringent accuracy and efficiency requirements are imposed, thereby underscoring the necessity and effectiveness of the proposed benchmark in advancing reliable and efficient automated PDE solver generation.

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 addresses the complexity and expert dependency of traditional finite element analysis by proposing the first end-to-end automation framework capable of processing both image and text inputs. The approach introduces a multi-agent system grounded in ReAct-style reasoning, integrating vision-language understanding, collaborative task planning, and a verification-first code generation mechanism. To ensure physical validity, the framework incorporates self-debugging and fallback strategies. Evaluated across diverse engineering mechanics scenarios, the method substantially outperforms existing large language model baselines, demonstrating high success rates and robustness in generating complete, correct, and physically consistent simulation models.

This work proposes AbaqusAgent, the first end-to-end, natural language–driven multi-agent framework for finite element analysis (FEA) in solid mechanics, designed to lower the barrier to entry and reduce reliance on expert knowledge. The framework integrates six collaborative modules—Interpreter, Architect, Input Generator, Runner, Reviewer, and Visualizer—to automatically translate user-provided natural language instructions into a complete Abaqus workflow encompassing pre-processing, solving, and post-processing. Evaluated on 50 solid mechanics problems, the system achieves an 86% success rate, significantly enhancing simulation efficiency. Furthermore, it advances human–computer interaction paradigms and enables seamless integration with AI-driven optimization and material characterization pipelines.

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.