GMRES Solver for MLPG Method Applied to Heat Conduction

2021 ◽  
Author(s):  
Krishna Singh ◽  
Abhishek Kumar Singh
Keyword(s):  
Heat Conduction ◽  
Mlpg Method

Stabilised MLS in MLPG method for heat conduction problem

2019 ◽  
Vol 36 (4) ◽  
pp. 1323-1345
Author(s):  
Rituraj Singh ◽  
Krishna Mohan Singh
Keyword(s):  
Heat Conduction ◽  
Design Methodology ◽  
Heat Conduction Problem ◽  
Moving Least Squares ◽  
Moment Matrix ◽  
Content Type ◽  
Patch Tests ◽  
Mlpg Method ◽  
The One ◽  
Method Analysis

Purpose The purpose of this paper is to assess the performance of the stabilised moving least squares (MLS) scheme in the meshless local Petrov–Galerkin (MLPG) method for heat conduction method. Design/methodology/approach In the current work, the authors extend the stabilised MLS approach to the MLPG method for heat conduction problem. Its performance has been compared with the MLPG method based on the standard MLS and local coordinate MLS. The patch tests of MLS and modified MLS schemes have been presented along with the one- and two-dimensional examples for MLPG method of the heat conduction problem. Findings In the stabilised MLS, the condition number of moment matrix is independent of the nodal spacing and it is nearly constant in the global domain for all grid sizes. The shifted polynomials based MLS and stabilised MLS approaches are more robust than the standard MLS scheme in the MLPG method analysis of heat conduction problems. Originality/value The MLPG method based on the stabilised MLS scheme.

scholarly journals INverse problem of identifying a time-dependent coefficient and free boundary in heat conduction equation by using the meshless local Petrov-Galerkin (MLPG) method via moving least squares approximation

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3319-3337
Author(s):  
Akbar Karami ◽  
Saeid Abbasbandy ◽  
Elyas Shivanian
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Heat Conduction Equation ◽  
Time Dependent ◽  
Moving Least Squares ◽  
Fixed Boundary ◽  
Mlpg Method

In this paper we investigated the inverse problem of identifying an unknown time-dependent coefficient and free boundary in heat conduction equation. By using the change of variable we reduced the free boundary problem into a fixed boundary problem. In direct solver problem we employed the meshless local Petrov-Galerkin (MLPG) method based on the moving least squares (MLS) approximation. Inverse reduced problem with fixed boundary is nonlinear and we formulated it as a nonlinear least-squares minimization of a scalar objective function. Minimization is performed by using of f mincon routine from MATLoptimization toolbox accomplished with the Interior - point algorithm. In order to deal with the time derivatives, a two-step time discretization method is used. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

GMRES Solver for MLPG Method Applied to Heat Conduction

2020 ◽  
Author(s):  
Abhishek Kumar Singh ◽  
Krishna Mohan Singh
Keyword(s):  
Heat Conduction ◽  
Three Dimensional ◽  
Computation Time ◽  
Symmetric System ◽  
Algebraic Equations ◽  
Relaxation Methods ◽  
Convergence Behaviour ◽  
Mlpg Method ◽  
Steady State Heat ◽  
Successive Over Relaxation

Abstract In recent years, meshless local Petrov-Galerkin (MLPG) method has emerged as the promising choice for solving variety of scientific and engineering problems. MLPG formulation leads to a non-symmetric system of algebraic equations. Iterative methods (such as GMRES and BiCGSTAB methods) are more competent than the direct solvers for solving a general linear system of larger size (order of millions or billions). This paper presents the use of GMRES solver with MLPG method for the very first time. The restarted version of the GMRES method is applied in connection with the interpolating MLPG method, to solve steady-state heat conduction in three-dimensional regular geometry. The performance of GMRES solver (with and without preconditioner) has been compared with the preconditioned BiCGSTAB method in terms of computation time and convergence behaviour. Jacobi and successive over-relaxation methods have been used as preconditioners in both the solvers. The results show that GMRES solver takes about 18 to 20% less CPU time than the BiCGSTAB solver along with better convergence behaviour.

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