arXiv subjects

Mingjiao Yan

Publications and source records attributed to Mingjiao Yan.

9 recordsLinked to original sources

arXiv2026

OptiXDE: A fast optical-inspired solver for differential equations

OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform--operator--inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr"odinger equation, viscous Burgers dynamics, the two-dimensional Allen--Cahn equation and incompressible flows from the Taylor--Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require \(\mathcal{O}(N\log N)\) work and \(\mathcal{O}(N)\) memory. Device-resident transform workloads reach \(94.9\times\) GPU acceleration, and the complete embedded-cylinder solver achieves a \(42.1\times\) CPU--GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations.

math.NA↗

arXiv2026

Steady-State and Transient Heat Conduction Analysis Using a Polygonal Cell-Based Smoothed Finite Element Method

This paper presents a polygonal cell-based smoothed finite element method (CS-FEM) for two-dimensional steady-state and transient heat-conduction analysis. In the proposed formulation, Wachspress shape functions are employed to construct the temperature approximation over general polygonal elements, and the smoothed temperature gradient is evaluated through boundary integration over cell-based smoothing domains. The resulting formulation is implemented in ABAQUS through the user-defined element (UEL) interface, enabling heat-conduction analysis using polygonal meshes within a commercial finite element environment. Several numerical examples, including a linear patch test, steady-state benchmark problems, and transient heat-conduction problems with different boundary conditions, are investigated to verify the accuracy, convergence behavior, and robustness of the proposed method. The numerical results show good agreement with analytical or reference solutions. Compared with conventional FEM using triangular and quadrilateral elements, the proposed polygonal CS-FEM exhibits favorable accuracy and convergence performance, while providing greater flexibility in mesh generation for complex geometries. The proposed framework therefore offers an accurate and robust numerical approach for steady-state and transient heat-conduction analysis.

math.NT↗

arXiv2026

Nonlinear Geotechnical Analysis Using a Polygonal Cell-Based Smoothed Finite Element Framework

Nonlinear geotechnical analysis often involves complex geometries, staged construction, local failure, and mesh-dependent stress and plastic strain responses. This study develops a polygonal cell-based smoothed finite element method (CS-FEM) for nonlinear geotechnical analysis and implements it in ABAQUS through the user element subroutine. The proposed method combines Wachspress interpolation with cell-based strain smoothing, in which the smoothed strain--displacement matrix is evaluated by boundary integration over polygonal smoothing subcells. This formulation avoids direct calculation of shape-function derivatives inside polygonal elements and enables standard polygonal meshes and hybrid quadtree meshes with hanging nodes to be handled in a unified framework. Nonlinear geomaterial behavior is incorporated through incremental elasto-plastic constitutive updates, including the Mohr--Coulomb model and the Duncan--Chang model. Several benchmark and engineering examples, including a perforated plate, strip footing, core rockfill dam, tunnel excavation, and slope stability problems, are presented for verification. The results show that the proposed method accurately predicts displacement, stress, plastic strain, bearing capacity, and factor of safety, while providing improved mesh flexibility and computational efficiency for nonlinear geotechnical analysis.

math.NA↗

arXiv2026

Two-dimensional seepage analysis using a polygonal cell-based smoothed finite element method

This study develops a polygonal cell-based smoothed finite element method (CSFEM) for two-dimensional seepage analyses in porous media, covering steady-state, transient, and free-surface problems. Wachspress interpolation on convex polygonal elements is combined with cell-based gradient smoothing, so that element matrices are assembled using boundary integrals only, avoiding in-element derivatives and improving robustness on distorted and locally refined meshes. To improve efficiency, a solution-driven adaptive refinement strategy is employed to concentrate resolution near steep hydraulic gradients and evolving wet-dry interfaces. Free-surface seepage is handled by a fixed-mesh iterative scheme that updates the wetted region and boundary conditions to track the phreatic surface. Benchmark tests validate the formulation against analytical solutions and high-fidelity FEM references. In steady seepage examples, the proposed polygonal CSFEM reproduces linear hydraulic-head fields to near machine precision and yields smaller head errors than conventional FEM at the same characteristic mesh size. In transient problems, accurate head evolution and stable time responses are obtained, while adaptive refinement efficiently resolves localized high-gradient zones. For free-surface cases, the method captures the phreatic-surface profile and seepage-face development reliably without remeshing. The quadtree refinement and adaptivity provide substantial efficiency gains in degrees of freedom and runtime for a prescribed accuracy level.

math.NA↗

arXiv2025

Elasto-plastic cell-based smoothed finite element method solving geotechnical problems

This work develops an elasto-plastic cell-based smoothed finite element method (CSFEM) for geotechnical analysis. The formulation incorporates a smoothed strain field into the standard elasto-plastic framework based on the Mohr-Coulomb criterion and is implemented in ABAQUS through a user-defined element (UEL). A UEL-UMAT data-transfer strategy is introduced to enable post-processing of stress and strain in ABAQUS. The method is assessed using several benchmark problems, including three classical examples, a tunnel excavation, and a slope stability analysis. The results show that the CSFEM achieves accuracy comparable to or slightly better than the conventional FEM and matches analytical or reference solutions for the examined cases. These findings indicate that the proposed CSFEM provides a practical and robust alternative for routine elasto-plastic analyses in geotechnical engineering.

math.NA↗

arXiv2025

Three dimensional seepage analysis using a polyhedral scaled boundary finite element method

This work presents a polyhedral scaled boundary finite element method (PSBFEM) for three dimensional seepage analysis. We first derive the scaled boundary formulation for 3D seepage problems, and subsequently incorporate Wachspress shape functions to construct shape functions over arbitrary polygonal elements, thereby establishing the foundation of the proposed polyhedral SBFEM. The method combines the semi-analytical nature of the SBFEM with the geometric flexibility of polyhedral and octree meshes, making it well-suited for complex seepage simulations. The PSBFEM is implemented within the ABAQUS UEL framework to facilitate steady-state, transient, and free-surface seepage analyses. A series of numerical examples are conducted to verify the accuracy, efficiency, and convergence properties of the proposed approach, including benchmark tests and applications with intricate geometries. The results demonstrate that the PSBFEM achieves higher accuracy and faster convergence than conventional FEM, particularly when using hybrid octree meshes with local refinement. This framework provides a robust and efficient computational tool for three-dimensional seepage analysis in geotechnical and hydraulic engineering applications.

math.NA↗

arXiv2025

Steady-state and transient thermal stress analysis using a polygonal finite element method

This work develops a polygonal finite element method (PFEM) for the analysis of steady-state and transient thermal stresses in two dimensional continua. The method employs Wachspress rational basis functions to construct conforming interpolations over arbitrary convex polygonal meshes, providing enhanced geometric flexibility and accuracy in capturing complex boundary conditions and heterogeneous material behavior. A quadtree-based acceleration strategy is introduced to significantly reduce computational cost through the reuse of precomputed stiffness and mass matrices. The PFEM is implemented in ABAQUS via a user-defined element (UEL) framework. Comprehensive benchmark problems, including multi-scale and non-matching mesh scenarios, are conducted to verify the accuracy, convergence properties, and computational efficiency of the method. Results indicate that the proposed PFEM offers notable advantages over conventional FEM in terms of mesh adaptability, solution quality, and runtime performance. The method shows strong potential for large-scale simulations involving thermal-mechanical coupling, complex geometries, and multi-resolution modeling.

math.NA↗

arXiv2024

A polyhedral scaled boundary finite element method solving three-dimensional heat conduction problems

In this study, we derived a three-dimensional scaled boundary finite element formulation for heat conduction problems. By incorporating Wachspress shape functions, a polyhedral scaled boundary finite element method (PSBFEM) was proposed to address heat conduction challenges in complex geometries. To address the complexity of traditional methods, this work introduced polygonal discretization techniques that simplified the topological structure of the polyhedral mesh and effectively integrated polyhedral and octree meshes, thereby reducing the number of element faces and enhancing mesh efficiency to accommodate intricate shapes. The developed formulation supported both steady-state and transient heat conduction analyses and was implemented in ABAQUS through a user-defined element (UEL). Through a series of numerical examples, the accuracy and convergence of the proposed method were validated. The results indicated that the PSBFEM consistently achieved higher accuracy than the FEM as the mesh was refined. The polyhedral elements offered a computationally efficient solution for complex simulations, significantly reducing computational costs.Additionally, by utilizing the octree mesh parent element acceleration technique, the computational efficiency of PSBFEM surpassed that of the FEM.

math.NA↗

arXiv2023

A novel solution for seepage problems using physics-informed neural networks

A Physics-Informed Neural Network (PINN) provides a distinct advantage by synergizing neural networks' capabilities with the problem's governing physical laws. In this study, we introduce an innovative approach for solving seepage problems by utilizing the PINN, harnessing the capabilities of Deep Neural Networks (DNNs) to approximate hydraulic head distributions in seepage analysis. To effectively train the PINN model, we introduce a comprehensive loss function comprising three components: one for evaluating differential operators, another for assessing boundary conditions, and a third for appraising initial conditions. The validation of the PINN involves solving four benchmark seepage problems. The results unequivocally demonstrate the exceptional accuracy of the PINN in solving seepage problems, surpassing the accuracy of FEM in addressing both steady-state and free-surface seepage problems. Hence, the presented approach highlights the robustness of the PINN and underscores its precision in effectively addressing a spectrum of seepage challenges. This amalgamation enables the derivation of accurate solutions, overcoming limitations inherent in conventional methods such as mesh generation and adaptability to complex geometries.

cs.CE↗