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ABSTRACT
Meshless local Petrov–Galerkin (MLPG) method is a promising meshfree method for continuum problems in complex domains, especially for large deformation, moving boundary and phase change problems. For large-scale problems, iterative methods for solving the discretized equations are more suitable than direct methods. Krylov subspace solvers of conjugate gradient type are the most preferred iterative solvers. The convergence rate of these methods depends on preconditioner used. Recently, proposed schedule relaxation Jacobi (SRJ) method can be used as a stand-alone solver and as a preconditioner. In the present work, the SRJ method is tested as a stand-alone solver and as a preconditioner for BiCGSTAB solver in the MLPG method, and its performance has been compared with successive overrelaxation (k) preconditioner. Two-dimensional linear steady-state heat conduction in complex shape geometry has been used as the model test problem.
Nomenclature
| aj | = | nonconstant coefficients in MLS |
| k | = | number of iterations in preconditioner call |
| m | = | number of terms in basis |
| M | = | size of iteration cycle for SRJ method |
| n | = | grid size of iteration cycle |
| N | = | grid size |
| ns | = | number of nodes in support domain |
| p | = | level of SRJ scheme |
| = | specific heat flux | |
| Qg | = | heat generation W/m3 |
| Q | = | heat flux at Neumann boundary condition W/m2 |
| rq | = | radius of quadrature domain, m |
| rs | = | radius of support domain, m |
| rw | = | radius of weight domain, m |
| Th | = | moving least square approximant |
| = | specified temperature at Dirichlet boundary | |
| v | = | test function for MLPG method |
| w | = | weight function used in MLS approximation |
| z | = | index in SRJ iteration cycle |
| Greek symbols | = | |
| αq | = | dimensionless parameter of quadrature domain |
| αs | = | dimensionless parameter of support domain |
| Γ1 | = | global domain boundary for Dirichlet boundary condition |
| Γ2 | = | global domain boundary for Neumann boundary condition |
| Γ | = | global boundary |
| κ | = | thermal conductivity, W/m°C |
| ϕI | = | MLS shape function |
| Ω | = | two-dimensional domain |
| ΩQ | = | local domain |
| ∂Ω | = | boundary of local domain |
| ω | = | relaxation factor for SOR |
| Superscript | = | |
| h | = | approximated variable |
| Subscripts | = | |
| g | = | heat generation |
| i,I j,k,l | = | indices |
| q | = | quadrature domain |
| Q | = | local domain |
| S | = | support domain |
| w | = | weight function |
Nomenclature
| aj | = | nonconstant coefficients in MLS |
| k | = | number of iterations in preconditioner call |
| m | = | number of terms in basis |
| M | = | size of iteration cycle for SRJ method |
| n | = | grid size of iteration cycle |
| N | = | grid size |
| ns | = | number of nodes in support domain |
| p | = | level of SRJ scheme |
| = | specific heat flux | |
| Qg | = | heat generation W/m3 |
| Q | = | heat flux at Neumann boundary condition W/m2 |
| rq | = | radius of quadrature domain, m |
| rs | = | radius of support domain, m |
| rw | = | radius of weight domain, m |
| Th | = | moving least square approximant |
| = | specified temperature at Dirichlet boundary | |
| v | = | test function for MLPG method |
| w | = | weight function used in MLS approximation |
| z | = | index in SRJ iteration cycle |
| Greek symbols | = | |
| αq | = | dimensionless parameter of quadrature domain |
| αs | = | dimensionless parameter of support domain |
| Γ1 | = | global domain boundary for Dirichlet boundary condition |
| Γ2 | = | global domain boundary for Neumann boundary condition |
| Γ | = | global boundary |
| κ | = | thermal conductivity, W/m°C |
| ϕI | = | MLS shape function |
| Ω | = | two-dimensional domain |
| ΩQ | = | local domain |
| ∂Ω | = | boundary of local domain |
| ω | = | relaxation factor for SOR |
| Superscript | = | |
| h | = | approximated variable |
| Subscripts | = | |
| g | = | heat generation |
| i,I j,k,l | = | indices |
| q | = | quadrature domain |
| Q | = | local domain |
| S | = | support domain |
| w | = | weight function |
Acknowledgments
The authors gratefully acknowledge the funding to Rituraj Singh from IIT-Roorkee through MHRD research fellowship and computational resources provided by CFD Laboratory, MIED, IIT-Roorkee.