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New Perspective to Isogeometric Analysis:Solving Isogeometric Analysis Problem by Fitting Load Function

Computer Modeling in Engineering & Sciences 2023年第9期136卷 2957-2984页

作者： Jingwen Ren Hongwei Lin School of Mathematical Sciences Zhejiang UniversityHangzhou310058China State Key Lab.of CAD&CG Zhejiang UniversityHangzhou310058China

Isogeometric analysis(IGA)is introduced to establish the direct link between computer-aided design and *** is commonly implemented by Galerkin formulations(isogeometric Galerkin,IGA-G)through the use of nonuniform rational B-splines(NURBS)basis functions for geometric design and *** promising approach,isogeometric collocation(IGA-C),working directly with the strong form of the partial differential equation(PDE)over the physical domain defined by NURBS geometry,calculates the derivatives of the numerical solution at the chosen collocation *** a typical IGA,the knot vector of the NURBS numerical solution is only determined by the physical domain.A new perspective on the IGAmethod is proposed in this study to improve the accuracy and convergence of the *** the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry(right-hand side)with derivatives of the NURBS numerical solution(left-hand side).Moreover,the design of the knot vector has a close relationship to theNURBS functions to be fitted in the area of data fitting in geometric ***,the detected feature points of the load function are integrated into the initial knot vector of the physical domainto construct thenewknot vector of thenumerical ***,they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency,which we call isogeometric collocation by fitting load function(IGACL).In numerical experiments,we implement our method to solve 1D,2D,and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C *** also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method,which we call isogeometric Galerkin by fitting load function(IGA-GL).

关键词： Isogeometric analysis collocation methods feature point detection knot vector

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On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations

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Numerical Mathematics(Theory,methods and Applications) 2022年第1期15卷 91-124页

作者： Hui Liang School of Science Harbin Institute of TechnologyShenzhen518055China

collocation and Galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces,in short,denoted as DC,CC,DG and CG methods respectively,are employed to solve second-kind Volterra integral equations(VIEs).It is proved that the quadrature DG and CG(QDG and QCG)methods obtained from the DG and CG methods by approximating the inner products by suitable numerical quadrature formulas,are equivalent to the DC and CC methods,*** addition,the fully discretised DG and CG(FDG and FCG)methods are equivalent to the corresponding fully discretised DC and CC(FDC and FCC)*** convergence theories are established for DG and CG methods,and their semi-discretised(QDG and QCG)and fully discretized(FDG and FCG)*** particular,it is proved that the CG method for second-kind VIEs possesses a similar convergence to the DG method for first-kind *** examples illustrate the theoretical results.

关键词： Volterra integral equations collocation methods Galerkin methods discontinuous Galerkin methods convergence analysis

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The extended cubic B-spline algorithm for a modified regularized long wave equation

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Chinese Physics B 2013年第4期22卷 75-80页

作者： I.Dag D.Irk M.SarI Faculty of Science and Arts Department of Mathematics and Computer Sciences Eskis ehir Osmangazi niversitesi Eskis ehir Turkey Faculty of Science and Arts Department of Mathematics Pamukkale niversitesi Denizli Turkey

A collocation method based on an extended cubic B-spline function is introduced for the numerical solution of the modified regularized long wave equation. The accuracy of the method is illustrated by studying the single solitary wave propagation and the interaction of two solitary waves of the modified regularized long wave equation.

关键词： collocation methods solitary waves

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Efficient collocational Approach for Parametric Uncertainty Analysis

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Communications in Computational Physics 2007年第2期2卷 293-309页

作者： Dongbin Xiu Department of Mathematics Purdue UniversityWest LafayetteIN 47906USA.

A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is *** uncertain parameters are modeled as random variables,and the governing equations are treated as *** solutions,or quantities of interests,are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters.A high-order stochastic collocation method is employed to solve the solution statistics,and more importantly,to reconstruct the polynomial *** retaining the high accuracy by polynomial expansion,the resulting“pseudo-spectral”type algorithm is straightforward to implement as it requires only repetitive deterministic *** estimate on error bounded is presented,along with numerical examples for problems with relatively complicated forms of governing equations.

关键词： collocation methods pseudo-spectral methods stochastic inputs random differential equations uncertainty quantification.

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Mixed Generalized Jacobi and Chebyshev collocation Method for Time-Fractional Convection-Diffusion Equations

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Journal of Mathematical Research with Applications 2016年第5期36卷 608-620页

作者： Tao SUN School of Statistics and Mathematics Shanghai Lixin University of Accounting and Finance

In this paper,we study an efficient higher order numerical method to timefractional partial differential equations with temporal Caputo derivative.A collocation method based on shifted generalized Jacobi functions is taken for approximating the solution of the given time-fractional partial differential equation in time and a shifted Chebyshev collocation method based on operational matrix in *** derived numerical solution can approximate the non-smooth solution in time of given equations *** numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.

关键词： time-fractional convection-diffusion equations collocation methods shifted generalized Jacobi functions shifted Chebyshev polynomials

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A New Technique for Solving Fractional Order Systems: Hermite collocation Method

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Applied Mathematics 2016年第18期7卷 2307-2323页

作者： Nilay Akgonullu Pirim Fatma Ayaz Ankara Turkey Department of Mathematics Gazi University Ankara Turkey

In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.

关键词： Fractional Order Differential Equations Hermite Polynomials Hermite Series collocation methods

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