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A novel numerical method for infinite domain potential problems

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Chinese Science Bulletin 2010年第16期55卷 1598-1603页

作者： CHEN Wen FU ZhuoJia center for Numerical simulation software in engineering and sciences Department of Engineering Mechanics College of Civil EngineeringHohai University Nanjing 210098 China

The infinite domain potential problems arise in many branches of scientific and engineering fields, which by now still pose a great challenge to scientific computing *** study proposes a novel meshless singular boundary method (SBM) to solve infinite domain potential *** SBM is mathematically simple, easy-to-program, meshless and *** guar-antee the uniqueness of numerical solutions, this article adds a constant term into the SBM approximate *** effi-ciency and accuracy of the proposed technique are tested to the three infinite domain potential problems.

关键词：数值方法无限域边界无单元法管理无界域一体化无网格数值解

FAST MULTIPOLE SINGULAR BOUNDARY METHOD FOR LARGE-SCALE PLANE ELASTICITY PROBLEMS

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Acta Mechanica Solida Sinica 2015年第6期28卷 626-638页

作者： Wenzhen Qu Wen Chen State Key Laboratory of Hydrology-Water Resources and Hydraulic engineering Center for Numerical Simulation Software in Engineering and SciencesCollege of Mechanics and MaterialsHohai University Department of Mathematics University of North Carolina

The singular boundary method （SBM） is a recent meshless boundary collocation method that remedies the perplexing drawback of fictitious boundary in the method of fundamental solutions （MFS）. The basic idea is to use the origin intensity factor to eliminate singularity of the fundamental solution at source. The method has so far been applied successfully to the potential and elasticity problems. However, the SBM solution for large-scale problems has been hindered by the operation count of O（N^3） with direct solvers or O（N^2） with iterative solvers, as well as the memory requirement of O（N^2）. In this study, the first attempt was made to combine the fast multipole method （FMM） and the SBM to significantly reduce CPU time and memory requirement by one degree of magnitude, namely, O（N）. Based on the complex variable represen- tation of fundamental solutions, the FMM-SBM formulations for both displacement and traction were presented. Numerical examples with up to hundreds of thousands of unknowns have successfully been tested on a desktop computer. These results clearly illustrated that the proposed FMM-SBM was very efficient and promising in solving large-scale plane elasticity problems.

关键词： fast multipole method singular boundary method meshless boundary collocation method elasticity problems

A Truly Boundary-Only Meshfree Method Applied to Kirchhoff Plate Bending Problems

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Advances in Applied Mathematics and Mechanics 2009年第3期1卷 341-352页

作者： Zhuojia Fu Wen Chen center for Numerical simulation software in engineering and sciences Department of Engineering MechanicsCollege of Civil EngineeringHohai University No.1 XiKang RoadNanjingJiangsu 210098China

The boundary particle method(BPM)is a truly boundary-only collocation scheme,whose basis function is the high-order nonsingular general solution or singular fundamental solution,based on the recursive composite multiple reciprocity method(RC-MRM).The RC-MRM employs the high-order composite differential operator to solve a much wider variety of inhomogeneous problems with boundary-only collocation nodes while significantly reducing computational cost via a recursive *** this study,we simulate the Kirchhoff plate bending problems by the BPM based on the *** results show that this approach produces accurate solutions of plates subjected to various loadings with boundary-only discretization.

关键词： Boundary particle method recursive composite multiple reciprocity method Kirchhoff plate boundary-only meshfree

A Homogenized Function to Recover Wave Source by Solving a Small Scale Linear System of Differencing Equations

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Computer Modeling in engineering & sciences 2016年第5期111卷 421-435页

作者： Chein-Shan Liu Wen Chen Ji Lin center for Numerical simulation software in engineering and sciences College of Mechanics and MaterialsHohai UniversityNanjingJiangsu 210098China Department of Civil engineering National Taiwan UniversityTaipei 106-17Taiwan

In order to recover unknown space-dependent function G(x)or unknown time-dependent function H(t)in the wave source F(x;t)=G(x)H(t),we develop a technique of homogenized function and differencing equations,which can significantly reduce the difficulty in the inverse wave source recovery problem,only needing to solve a few equations in the problem domain,since the initial condition/boundary conditions and a supplementary final time condition are satisfied *** a consequence,the eigenfunctions are used to expand the trial solutions,and then a small scale linear system is solved to determine the expansion coefficients from the differencing *** the ill-posedness of the inverse wave source problem is greatly reduced,the present method is accurate and stable against a large noise up to 50%,of which the numerical tests confirm the observation.

关键词： Wave source recovery problem Eigenfunctions Homogenized function Differencing equations

On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline Space S_(3)^(1,2)(∆_(mn)^((2)))

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Numerical Mathematics(Theory,Methods and Applications) 2014年第1期7卷 1-22页

作者： Jiang Qian Fan Wang College of sciences Hohai UniversityNanjing 210098China center for Numerical simulation software in engineering and sciences Department of Engineering MechanicsHohai UniversityNanjing 210098China College of engineering Nanjing Agricultural UniversityNanjing 210031China

In this paper,based on the basis composed of two sets of splines with distinct local supports,cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 *** variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points,which can reproduce any polynomial of nearly best *** by means of the modulus of continuity,the estimation of the operator approximating a real sufficiently smooth function is reviewed as ***,the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 *** then the convergence results are worked out.

关键词： Bivariate splines conformality of smoothing cofactor method nonuniform type-2 triangulation quasi-interpolation modulus of continuity

Localized space-tim e method of fundamental solutions for three-dimensional transient diffusion problem

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Acta Mechanica Sinica 2020年第5期36卷 1051-1057,I0002页

作者： L.Qiu J.Lin Q.-H.Qin W.Chen international center for simulation software in engineering and sciences College of Mechanics and MaterialsHohai UniversityNanjing 210098China College of engineering and Computer Science Australian National UniversityCanberraACT 2601Australia State Key Laboratory of Mechanical Behavior and System Safety of Traffic engineering Structures Shijiazhuang Tiedao UniversityShijiazhuang 050043China State Key Laboratory of Acoustic Institute of AcousticsChinese Academy of SciencesBeijing 100190China

A localized space-time method of fundamental solutions(LSTMFS)is extended for solving three-dimensional transient diffusion problems in this *** interval segmentation in temporal direction is developed for the accurate simulation of long-time-dependent diffusion *** the LSTMFS,the whole space-time domain with nodes arranged i divided into a series of overlapping subdomains with a simple *** each subdomain,the conventional method of fundamental solutions is utilized and the coefficients associated with the considered domain can be explicitly *** calculating a combined sparse matrix system,the value at any node inside the space-time domain can be *** experi-ments demonstrate that high accuracy and efficiency can be simultaneously achieved via the LSTMFS,even for the problems defined on a long-time and quite complex computational domain.

关键词： Localized spacetime method of fundamental solutions Meshless method Long-time evolution Transient diffusion

An Efficient Approach for Solving Optimization over Linear Arithmetic Constraints

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Journal of Computer Science & Technology 2016年第5期31卷 987-1011页

作者： Li Chen State Key Laboratory of Computer Science Institute of Software Chinese Academy of Sciences Beijing 100190 ChinaUniversity of Chinese Academy of Sciences Beijing 100190 China National engineering Research center of Fundamental software Institute of Software Chinese Academy of Sciences Beijing 100190 China

Satisfiability Modulo Theories （SMT） have been widely investigated over the last decade. Recently researchers have extended SMT to the optimization problem over linear arithmetic constraints. To the best of our knowledge, SYMBA and OPT-MATHSAT are two most efficient solvers available for this problem. The key algorithms used by SVMBA and OPT-MATHSAT consist of the loop of two procedures： 1） critical finding for detecting a critical point, which is very likely to be globally optimal, and 2） global checking for confirming the critical point is reMly globally optimal. In this paper, we propose a new approach based on the Simplex method widely used in operation research. Our fundamental idea is to find several critical points by constructing and solving a series of linear problems with the Simplex method. Our approach replaces the Mgorithms of critical finding in SYMBA and OPT-MATHSAT, and reduces the runtime of critical finding and decreases the number of executions of global checking. The correctness of our approach is proved. The experiment evaluates our implementation against SYMBA and OPT-MATHSAT on a critical class of problems in real-time systems. Our approach outperforms SYMBA on 99.6% of benchmarks and is superior to OPT-MATHSAT in large-scale cases where the number of tasks is more than 24. The experimental results demonstrate that our approach has great potential and competitiveness for the optimization problem.

关键词： constrained optimization Satisfiability Modulo Theories linear programming

Domain-Decomposition Localized Method of Fundamental Solutions for Large-Scale Heat Conduction in Anisotropic Layered Materials

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Advances in Applied Mathematics and Mechanics 2022年第3期14卷 759-776页

作者： Shuainan Liu Zhuojia Fu Yan Gu center for Numerical simulation software in engineering and sciences College of Mechanics and MaterialsHohai UniversityNanjingJiangsu 211100China School of Mathematics and Statistics Qingdao UniversityQingdaoShandong 266071China

The localized method of fundamental solutions(LMFS)is a relatively new meshless boundary collocation *** the LMFS,the global MFS approxima-tion which is expensive to evaluate is replaced by local MFS formulation deﬁned in a set of overlapping *** LMFS algorithm therefore converts differential equations into sparse rather than dense matrices which are much cheaper to *** paper makes theﬁrst attempt to apply the LMFS,in conjunction with a domain-decomposition technique,for the numerical solution of steady-state heat con-duction problems in two-dimensional(2D)anisotropic layered ***,the layered material is decomposed into several subdomains along the layer-layer inter-faces,and in each of the subdomains,the solution is approximated by using the LMFS *** the subdomain interface,compatibility of temperatures and heatﬂuxes are *** numerical experiments illustrate that the proposed domain-decomposition LMFS algorithm is accurate,stable and computationally efﬁcient for the numerical solution of large-scale multi-layered materials.

关键词： Meshless method localized method of fundamental solutions heat conduction prob-lems layered materials large-scale problems

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

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Advances in Applied Mathematics and Mechanics 2012年第5期4卷 519-542页

作者： Zhuo-Jia Fu Wen Chen Qing-Hua Qin center for Numerical simulation software in engineering and sciences Department of Engineering MechanicsHohai UniversityNanjingJiangsuP.R.China Research School of engineering Building 32Australian National UniversityCanberraACT 0200Australia

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials(FGMs).The three methods are,respectively,the method of fundamental solution(MFS),the boundary knot method(BKM),and the collocation Trefftz method(CTM)in conjunction with Kirchhoff transformation and various variable *** the analysis,Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent *** proposed MFS,BKM and CTM are mathematically simple,easyto-programming,meshless,highly accurate and *** numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered,and the results are compared with those from meshless local boundary integral equation method(LBIEM)and analytical solutions to demonstrate the effi-ciency of the present schemes.

关键词： Method of fundamental solution boundary knot method collocation Trefftz method Kirchhoff transformation Laplace transformation meshless method