A preliminary study on the meshless local exponential basis functions method for nonlinear and variable coefficient PDEs

Purpose The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential equations (PDEs). Design/methodology/approach The original version of MLEBF method is limited to linear, constant coefficient PDEs. The reason is that exponential bases which satisfy the homogeneous operator can only be determined for this class of problems. To extend this method to the general case of linear PDEs, the variable coefficients along with all involved derivatives are first expanded. This expanded form is evaluated at the center of each cloud, and is assumed to be constant over the entire cloud. The solution procedure is followed as in the former version. Nonlinear problems are first converted to a succession of linear, variable coefficient PDEs using the Newton-Kantorovich scheme and are subsequently solved using the aforementioned approach until convergence is achieved. Findings The results obtained show good performance of the method as solution to a wide range of problems. The results are compared with the well-known methods in the literature such as the finite element method, high-order finite difference method or variants of the boundary element method. Originality/value The MLEBF method is a simple yet effective tool for analyzing various kinds of problems. It is easy to implement with high parallelization potential. The proposed method addresses the biggest limitation of the method, and extends it to linear, variable coefficient PDEs as well as nonlinear ones.

[1]  Carlos J. S. Alves,et al.  The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates , 2009 .

[2]  Farshid Mossaiby,et al.  A meshless method using local exponential basis functions with weak continuity up to a desired order , 2014 .

[3]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[4]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[5]  YuanTong Gu,et al.  MESHFREE METHODS AND THEIR COMPARISONS , 2005 .

[6]  B. Boroomand,et al.  On using exponential basis functions for laminates modeled by CLPT, FSDT and TSDT: Further tests and results , 2012 .

[7]  B. Boroomand,et al.  Explicit relations for the solution of laminated plates modeled by a higher shear deformation theory: Derivation of exponential basis functions , 2013 .

[8]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[9]  Ismael Herrera,et al.  Trefftz method: Fitting boundary conditions , 1987 .

[10]  Farshid Mossaiby,et al.  Implementation of a generalized exponential basis functions method for linear and non‐linear problems , 2016 .

[11]  Bijan Boroomand,et al.  Exponential Basis Functions in solution of laminated structures , 2011 .

[12]  O. C. Zienkiewicz,et al.  Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions , 1989 .

[13]  Somchart Chantasiriwan,et al.  Methods of fundamental solutions for time‐dependent heat conduction problems , 2006 .

[14]  F. Bobaru,et al.  Studies of dynamic crack propagation and crack branching with peridynamics , 2010 .

[15]  John W. Stephenson,et al.  High‐order methods for elliptic equations with variable coefficients , 1987 .

[16]  Soheil Soghrati,et al.  Exponential basis functions in solution of problems with fully incompressible materials: A mesh-free method , 2012, J. Comput. Phys..

[17]  Bangti Jin,et al.  The method of fundamental solutions for inverse boundary value problems associated with the steady‐state heat conduction in anisotropic media , 2006 .

[18]  M. A. AL-Jawary,et al.  Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods , 2012, Int. J. Comput. Math..

[19]  Pizhong Qiao,et al.  An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis , 2015 .

[20]  B. Boroomand,et al.  The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions , 2014 .

[21]  S. Silling,et al.  A meshfree method based on the peridynamic model of solid mechanics , 2005 .

[22]  Mirco Zaccariotto,et al.  Crack propagation with adaptive grid refinement in 2D peridynamics , 2014, International Journal of Fracture.

[23]  S. H. Hashemi,et al.  Exponential basis functions in space and time: A meshless method for 2D time dependent problems , 2013, J. Comput. Phys..

[24]  Marc Duflot,et al.  Meshless methods: A review and computer implementation aspects , 2008, Math. Comput. Simul..

[25]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[26]  Soheil Soghrati,et al.  A mesh-free method using exponential basis functions for laminates modeled by CLPT, FSDT and TSDT - Part II: Implementation and results , 2011 .

[27]  E. Kita,et al.  Trefftz method: an overview , 1995 .

[28]  O. C. Zienkiewicz,et al.  Application of the Trefftz method in plane elasticity problems , 1990 .

[29]  Farshid Mossaiby,et al.  Extending the method of exponential basis functions to problems with singularities , 2015 .

[30]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[31]  Y. T. Lee,et al.  On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations , 2007, Comput. Math. Appl..

[32]  Wing Kam Liu,et al.  Wavelet and multiple scale reproducing kernel methods , 1995 .

[33]  Graeme Fairweather,et al.  The method of fundamental solutions for axisymmetric potential problems , 1999 .

[34]  M. Ganesh,et al.  The numerical solution of a nonlinear hypersingular boundary integral equation , 2001 .

[35]  B. Boroomand,et al.  Exponential basis functions in solution of static and time harmonic elastic problems in a meshless style , 2010 .

[36]  Farshid Mossaiby,et al.  A simple meshless method for challenging engineering problems , 2015 .

[37]  A. Cheng,et al.  Trefftz and Collocation Methods , 2008 .

[38]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[39]  Soheil Soghrati,et al.  A Trefftz method in space and time using exponential basis functions: Application to direct and inverse heat conduction problems , 2013 .

[40]  Y. Hon,et al.  Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators , 2007 .

[41]  Q. Qin,et al.  A meshless method for generalized linear or nonlinear Poisson-type problems , 2006 .

[42]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[43]  Bijan Boroomand,et al.  Nonlocal elasticity defined by Eringen’s integral model: Introduction of a boundary layer method , 2014 .

[44]  Olaf Steinbach,et al.  Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements , 2007 .

[45]  E. Oñate,et al.  A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW , 1996 .

[46]  Ying-Te Lee,et al.  Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept. , 2009 .

[47]  A. Zieliński.  Trefftz method: elastic and elasto-plastic problems , 1988 .

[48]  B. Boroomand,et al.  Exponential basis functions in the solution of laminated plates using a higher-order Zig–Zag theory , 2013 .

[49]  P. Ramachandran.  Method of fundamental solutions: singular value decomposition analysis , 2002 .

[50]  Soheil Soghrati,et al.  Exponential basis functions in solution of incompressible fluid problems with moving free surfaces , 2012, J. Comput. Phys..

[51]  O. C. Zienkiewicz,et al.  Solution of Helmholtz equation by Trefftz method , 1991 .

[52]  I. Argyros.  Convergence and Applications of Newton-type Iterations , 2008 .

[53]  D. L. Young,et al.  The method of fundamental solutions for inverse 2D Stokes problems , 2005 .

[54]  B. Boroomand,et al.  On definition of clamped conditions in TSDT and FSDT: The use of exponential basis functions in solution of laminated composites , 2013 .

[55]  Soheil Soghrati,et al.  A mesh-free method using exponential basis functions for laminates modeled by CLPT, FSDT and TSDT – Part I: Formulation , 2011 .

[56]  Hui Wang,et al.  Fundamental Solution Based Graded Element Model for Steady-State Heat Transfer in FGM , 2012 .