arXiv subjects

Zongliang Zhang

Publications and source records attributed to Zongliang Zhang.

13 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

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↗

arXiv2026

Multi-instance robust fitting for non-classical geometric models

Most existing robust fitting methods are designed for classical models, such as lines, circles, and planes. In contrast, fewer methods have been developed to robustly handle non-classical models, such as spiral curves, procedural character models, and free-form surfaces. Furthermore, existing methods primarily focus on reconstructing a single instance of a non-classical model. This paper aims to reconstruct multiple instances of non-classical models from noisy data. We formulate this multi-instance fitting task as an optimization problem, which comprises an estimator and an optimizer. Specifically, we propose a novel estimator based on the model-to-data error, capable of handling outliers without a predefined error threshold. Since the proposed estimator is non-differentiable with respect to the model parameters, we employ a meta-heuristic algorithm as the optimizer to seek the global optimum. The effectiveness of our method are demonstrated through experimental results on various non-classical models. The code is available at https://github.com/zhangzongliang/fitting.

cs.CV↗

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↗

arXiv2025

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↗

arXiv2021

A novel method in solving seepage problems implementation in Abaqus based on the polygonal scaled boundary finite element method

The scaled boundary finite element method (SBFEM) is a semi-analytical computational scheme, which is based on the characteristics of the finite element method (FEM) and combines the advantages of the boundary element method (BEM). This paper integrates the scaled boundary finite element method (SBFEM) and the polygonal mesh technique into a new approach to solving the steady-state and transient seepage problems. The proposed method is implemented in Abaqus using a user-defined element (UEL). The detailed implementations of the procedure, defining the UEL element, updating the RHS and AMATRX, and solving the stiffness/mass matrix by the eigenvalue decomposition are presented. Several benchmark problems from seepage are solved to validate the proposed implementation. Results show that the polygonal element of PS-SBFEM has a higher accuracy rate than the standard FEM element in the same element size. For the transient problems, the results between PS-SBFEM and the FEM are in excellent agreement. Furthermore, the PS-SBFEM with quadtree meshes shows a good effect for solving complex geometric in the seepage problem. Hence, the proposed method is robust accurate for solving the steady-state and transient seepage problems. The developed UEL source code and the associated input files can be downloaded from https://github.com/yangyLab/PS-SBFEM.

math.NA↗

arXiv2021

A polygonal scaled boundary finite element method for solving heat conduction problems

This paper presents a steady-state and transient heat conduction analysis framework using the polygonal scaled boundary finite element method (PSBFEM) with polygon/quadtree meshes. The PSBFEM is implemented with commercial finite element code Abaqus by the User Element Sub-routine (UEL) feature. The detailed implementation of the framework, defining the UEL element, and solving the stiffness/mass matrix by the eigenvalue decomposition are presented. Several benchmark problems from heat conduction are solved to validate the proposed implementation. Results show that the PSBFEM is reliable and accurate for solving heat conduction problems. Not only can the proposed implementation help engineering practitioners analyze the heat conduction problem using polygonal mesh in Abaqus, but it also provides a reference for developing the UEL to solve other problems using the scaled boundary finite element method.

math.NA↗

arXiv2018

Geometric Multi-Model Fitting by Deep Reinforcement Learning

This paper deals with the geometric multi-model fitting from noisy, unstructured point set data (e.g., laser scanned point clouds). We formulate multi-model fitting problem as a sequential decision making process. We then use a deep reinforcement learning algorithm to learn the optimal decisions towards the best fitting result. In this paper, we have compared our method against the state-of-the-art on simulated data. The results demonstrated that our approach significantly reduced the number of fitting iterations.

cs.CV↗

arXiv2016

Partial Procedural Geometric Model Fitting for Point Clouds

Geometric model fitting is a fundamental task in computer graphics and computer vision. However, most geometric model fitting methods are unable to fit an arbitrary geometric model (e.g. a surface with holes) to incomplete data, due to that the similarity metrics used in these methods are unable to measure the rigid partial similarity between arbitrary models. This paper hence proposes a novel rigid geometric similarity metric, which is able to measure both the full similarity and the partial similarity between arbitrary geometric models. The proposed metric enables us to perform partial procedural geometric model fitting (PPGMF). The task of PPGMF is to search a procedural geometric model space for the model rigidly similar to a query of non-complete point set. Models in the procedural model space are generated according to a set of parametric modeling rules. A typical query is a point cloud. PPGMF is very useful as it can be used to fit arbitrary geometric models to non-complete (incomplete, over-complete or hybrid-complete) point cloud data. For example, most laser scanning data is non-complete due to occlusion. Our PPGMF method uses Markov chain Monte Carlo technique to optimize the proposed similarity metric over the model space. To accelerate the optimization process, the method also employs a novel coarse-to-fine model dividing strategy to reject dissimilar models in advance. Our method has been demonstrated on a variety of geometric models and non-complete data. Experimental results show that the PPGMF method based on the proposed metric is able to fit non-complete data, while the method based on other metrics is unable. It is also shown that our method can be accelerated by several times via early rejection.

cs.GR↗