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SOPTX: A High-Performance Multi-Backend Framework for Topology Optimization

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arXiv 2025年

作者： He, Liang Wei, Huayi Tian, Tian School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Mathematics and Computational Science Xiangtan University National Center of Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China

In recent years, topology optimization (TO) has gained widespread attention as a powerful structural design method. However, its application remains challenging due to the deep expertise and extensive development effort required. Traditional TO methods, tightly coupled with computational mechanics like finite element method (FEM), result in intrusive algorithms demanding a comprehensive system understanding. This paper presents SOPTX, a TO package based on FEALPy, which implements a modular architecture that decouples analysis from optimization, supports multiple computational backends (NumPy, PyTorch, JAX), and achieves a non-intrusive design paradigm. Core innovations include: (1) cross-platform design that supports multiple computational backends, enabling efficient algorithm execution on central processing units (CPUs) and flexible acceleration using graphics processing units (GPUs), while leveraging automatic differentiation (AD) technology for efficient sensitivity computation of objective and constraint functions;(2. fast matrix assembly techniques that overcome the performance bottlenecks of traditional numerical integration methods, significantly accelerating finite element computations and enhancing overall efficiency;(3) a modular framework supporting TO problems for arbitrary dimensions and meshes, allowing flexible configuration and extensibility of optimization workflows through a rich library of composable components. Using the density-based method for the classic compliance minimization problem with volume constraints as an example, numerical experiments demonstrate SOPTX’s high efficiency in computational speed and memory usage, while showcasing its strong potential for research and engineering *** Codes 74P15 (Primary) 68N30, 65N30 (Secondary) Copyright © 2.2., The Authors. All rights reserved.

关键词： Shape optimization

Hm-CONFORMING VIRTUAL ELEMENTS IN ARBITRARY DIMENSION

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arXiv 2021年

作者： Huang, Xuehai Chen, Chunyu Wei, Huayi Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Mathematics Shanghai University of Finance and Economics Shanghai200433 China

The Hm-conforming virtual elements of any degree k on any shape of polytope in n with m, n ≥ 1 and k ≥ m are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest degree case k = m, the set of degrees of freedom only involves function values and derivatives up to order m − 1 at the vertices of the polytope. The inverse inequality and several norm equivalences for the Hm-conforming virtual elements are rigorously proved. The Hm-conforming virtual elements are then applied to discretize a polyharmonic equation with a lower order term. With the help of the interpolation error estimate and norm equivalences, the optimal error estimates are derived for the Hm-conforming virtual element *** Codes 65N12. 65N15, 65N2., 65N30 Copyright © 2.2., The Authors. All rights reserved.

关键词： Degrees of freedom (mechanics)

NUMERICAL SOLUTIONS OF NONAUTONOMOUS STOCHASTIC DELAY DIFFERENTIAL EQUATIONS BY DISCONTINUOUS GALERKIN METHODS

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Journal of computational Mathematics 2019年第3期37卷 419-436页

作者： Xinjie Dai Aiguo Xiao School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105China School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China

This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential *** prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square *** addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential *** tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.

关键词： Discontinuous Galerkin method Wong-Zakai approximation Nonautonomous Stratonovich stochastic delay differential equation

Hybridization of evolutionary algorithm and deep reinforcement learning for multi-objective orienteering optimization

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arXiv 2022年

作者： Liu, Wei Wang, Rui Zhang, Tao Li, Kaiwen Li, Wenhua Ishibuchi, Hisao The College of System Engineering National University of Defense Technology Changsha410073 China The Hunan Key Laboratory of Multi-Energy System Intelligent Interconnection Technology HKLMSI2T Changsha410073 China Guangdong Provincial Key Laboratory of Braininspired Intelligent Computation Department of Computer Science and Engineering Southern University of Science and Technology Shenzhen518055 China

Multi-objective orienteering problems (MO-OPs) are classical multi-objective routing problems and have received a lot of attention in the past decades. This study seeks to solve MO-OPs through a problem-decomposition framework, that is, a MO-OP is decomposed into a multi-objective knapsack problem (MOKP) and a travelling salesman problem (TSP). The MOKP and TSP are then solved by a multi-objective evolutionary algorithm (MOEA) and a deep reinforcement learning (DRL) method, respectively. While the MOEA module is for selecting cities, the DRL module is for planning a Hamiltonian path for these cities. An iterative use of these two modules drives the population towards the Pareto front of MO-OPs. The effectiveness of the proposed method is compared against NSGA-II and NSGA-III on various types of MO-OP instances. Experimental results show that our method exhibits the best performance on almost all the test instances, and has shown strong generalization ability. Copyright © 2.2., The Authors. All rights reserved.

关键词： Multiobjective optimization

Invariant-preserving schemes with arbitrarily high-order accuracy for the two-component Camassa–Holm system via Hamiltonian boundary value methods

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Communications in Nonlinear science and Numerical simulation 2025年 150卷

作者： Qian Luo Aiguo Xiao Min Li Xiaoqiang Yan School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan 411105 China School of Mathematics and Physics China University of Geosciences Wuhan 430074 China National Center for Applied Mathematics in Hunan Xiangtan University Hunan 411105 China

In this paper, we present an arbitrarily high-order numerical scheme for the two-component Camassa–Holm system, ensuring the preservation of three invariants: energy and two Casimir functions. The spatial discretization is achieved using Fourier–Galerkin methods, resulting in a semi-discrete system which retains a Hamiltonian structure and approximates the invariants of the original continuous system. Subsequently, an energy-preserving Runge–Kutta method, such as Hamiltonian boundary value methods, is applied to the semi-discrete system, yielding a fully discrete scheme of arbitrarily high order. The proposed scheme is validated through numerical simulations, demonstrating its accuracy and capability in capturing different types of solutions of the two-component Camassa–Holm equation.

关键词： Blended iteration Fourier–Galerkin methods Hamiltonian boundary value methods Invariant-preserving scheme Two-component Camassa–Holm system

High-order energy stable schemes of incommensurate phase-field crystal model

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arXiv 2020年

作者： Jiang, Kai Si, Wei School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the L2.gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures. Copyright © 2.2., The Authors. All rights reserved.

关键词： Numerical methods

An adaptive virtual element method for the polymer self-consistent field theory

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arXiv 2020年

作者： Wei, Huayi Wang, Xin Chen, Chunyu Jiang, Kai School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan411105 China

In this paper, we develop a high-order adaptive virtual element method (VEM) to simulate the self-consistent field theory (SCFT) model in arbitrary domains. The VEM is very flexible in handling general polygon elements and can treat hanging nodes as polygon vertices without additional processing. Besides, to effectively simulate the phase separation behavior in strong segregation systems, an adaptive method on polygonal mesh equipped with a new marking strategy is developed. This new marking strategy will indicate the times of marked elements to be refined and coarsened, making full use of the information contained in the current numerical results. Using the halfedge data structure, we can apply the adaptive method to the arbitrary polygonal mesh. Numerical results demonstrate that the developed method is efficient in simulating polymers’ phase behavior in complex geometric domains. The accuracy is consistent with theoretical results. The adaptive method can greatly reduce computational costs to obtain prescribed numerical accuracy for strong segregation systems. Copyright © 2.2., The Authors. All rights reserved.

关键词： Mean field theory

High accuracy techniques based adaptive finite element methods for elliptic PDEs

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Journal of computational and Applied Mathematics 2026年 472卷

作者： Jingjing Xiao Ying Liu Nianyu Yi School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 PR China School of Science Xi’an University of Technology Xi’an 710048 PR China Hunan National Center for Applied Mathematics； Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan 411105 PR China

This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection refinement has been well established, in practical computations, the use of non-asymptotic exactness of error estimator and the excessive number of adaptive iteration steps often lead to inefficiency of the adaptive algorithm. We propose an efficient adaptive finite element method based on high-accuracy techniques including the superconvergence recovery technique and high-quality mesh optimization. The centroidal Voronoi Delaunay triangulation mesh optimization is embedded in the mesh adaptation to provide a high-quality mesh, and then ensure the superconvergence property of the recovered gradient and the asymptotic exactness of the error estimator. A tailored adaptive strategy, which can generate high-quality meshes with a target number of vertices, is developed to ensure the adaptive computation process terminates within 7 steps. The effectiveness and robustness of the adaptive algorithm are numerically demonstrated.

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An Interpretable MRI Reconstruction Network with Two-grid-cycle Correction and Geometric Prior Distillation

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arXiv 2022年

作者： Fan, Xiaohong Yang, Yin Chen, Ke Zhang, Jianping Dong, Ke The School of Mathematics and Computational Science Xiangtan University Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Xiangtan411105 China The School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China Centre for Mathematical Imaging Techniques Department of Mathematical Sciences The University of Liverpool Merseyside LiverpoolL6972L United Kingdom The School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Xiangtan411105 China Department of Radiation Oncology Xiangtan Central Hospital Xiangtan411101 China

Although existing deep learning compressed-sensing-based Magnetic Resonance Imaging (CS-MRI) methods have achieved considerably impressive performance, explainability and generalizability continue to be challenging for such methods since the transition from mathematical analysis to network design not always natural enough, often most of them are not flexible enough to handle multi-sampling-ratio reconstruction assignments. In this work, to tackle explainability and generalizability, we propose a unifying deep unfolding multi-sampling-ratio interpretable CS-MRI framework. The combined approach offers more generalizability than previous works whereas deep learning gains explainability through a geometric prior module. Inspired by the multigrid algorithm, we first embed the CSMRI-based optimization algorithm into correction-distillation scheme that consists of three ingredients: pre-relaxation module, correction module and geometric prior distillation module. Furthermore, we employ a condition module to learn adaptively step-length and noise level, which enables the proposed framework to jointly train multi-ratio tasks through a single model. The proposed model not only compensates for the lost contextual information of reconstructed image which is refined from low frequency error in geometric characteristic k-space, but also integrates the theoretical guarantee of model-based methods and the superior reconstruction performances of deep learning-based methods. Therefore, it can give us a novel perspective to design biomedical imaging networks. Numerical experiments show that our framework outperforms state-of-the-art methods in terms of qualitative and quantitative evaluations. Our method achieves 3.18 dB improvement at low CS ratio 10% and average 1.42.dB improvement over other comparison methods on brain dataset using Cartesian sampling mask. Copyright © 2.2., The Authors. All rights reserved.

关键词： Distillation