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3 Computer Assisted Mechanics and Engineering Sciences

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Transient heat conduction by different versions of the Method of Fundamental Solutions – a comparison study

Kołodziej J. A. J. A., Mierzwiczak M.

Computer Assisted Mechanics and Engineering Sciences

2010

Vol. 17, no. 1

75-88

The computational accuracy of three versions of the method of fundamental solutions (MFS) is compared. The first version of MFS is based on the Laplace transformation of the governing differential equations and of the boundary conditions. The second version of MFS is based on the fundamental solution of the governing differential equation and discretization in time. The third method approximates the temperature time derivative by finite difference scheme. As the test problems the 2D boundary-initial-value problems (2D_BIVP) in square rectangular region ? with known exact solutions are considered. Our numerical experiments show that all discussed methods achieve relatively accurate approximate solution but the third one offers less computational complexity and better efficiency.

Optimal parameters of method of fundamental solutions for Poisson problems in heat transfer by means of genetic algorithms

Kołodziej J. A., Klekiel T.

Computer Assisted Mechanics and Engineering Sciences

2008

Vol. 15, No. 2

99-112

This paper describes the application of the method of fundamental solutions to the solution of the boundary value problems of the two-dimensional steady heat transfer with heat sources. For interpolation of an inhomogeneous term in Poisson equation the radial basis functions are used. Three cases of boundary value problems are solved and five cases of radial basis functions are used. For comparison purposes the boundary value problems for which exact solution exists were chosen. Application of method of fundamental solutions with boundary collocation and radial basis function for solution of inhomogeneous boundary value problems introduces some number of parameters related with these tools. For optimal choosing of these parameters the genetic algorithm is used. The results of numerical experiences related to optimal parameters are presented.

A meshless method for non-linear Poisson problems with high gradients

Algahtani H. I.

Computer Assisted Mechanics and Engineering Sciences

2006

Vol. 13, No. 3

367-377

A meshless method for the solution of linear and non-linear Poisson-type problems involving high gradients is presented. The proposed method is based on collocation with 3rd order polynomial radial basis function coupled with the fundamental solution. The linear problem is solved by satisfying the boundary conditions and the governing differential equations over selected points over the boundary and inside the domain, respectively. In the case of the non-linear case, the resulted equations are highly non-linear and therefore, they are solved using an incremental-iterative procedure. The accuracy and efficiency of the method is verified through several numerical examples.