Documents published in Scipedia

• Solution of the Navier-Stokes equations using projection method and preconditioned conjugated gradient with multigrid and ILU solver

  M. Anunciação, M. Augusto Villela Pinto, R. Neundorf

  Rev. int. métodos numér. cálc. diseño ing. (2020). Vol. 36, (1), 17

  
  

  Abstract

  Among the main problems in the field of Computational Fluid Dynamics (CFD), there is the two-dimensional laminar flow in a transient regime of an incompressible fluid modeled by Navier-Stokes equations. Among the decoupled solutions for this equation system, that is, solutions for velocity regardless to pressure, there are the Projection methods, which separate the solution in three parts, solved in each time steps applying an iterative process regardless to the iterative process for the other variables. It may result, according to the Projection method applied, in two Reaction-Diffusion equations and one Poisson equation to be solved in each time step. This paper sought to develop an algorithm to solve the Navier-Stokes equation, applying the Finite Volume Method (FVM) with second order approximation scheme (CDS), beside a Projection method with incremental pressure-correction scheme, so that each Reaction-Diffusion and the Poisson equation are solved efficiently. Therefore, several solvers were tested for each equation, resulting in an algorithm with the combination that achieved the best result for each equation, with the preconditioned Conjugate Gradient method (PCG) with the Multigrid method (MG) and ILU solver (Incomplete LU factorization) being the methodology used in the whole problem solving process. The geometric Multigrid with V cycle, the correction scheme (CS), the full weighting restriction, the prolongation through bilinear interpolation and the maximum number of levels for the studied cases were utilized. The results achieved were satisfactory, since the proposed methodology accelerated the iterative process considerably in relation to the classical methods available in the literature.  

  Abstract
  Among the main problems in the field of Computational Fluid Dynamics (CFD), there is the two-dimensional laminar flow in a transient regime of an incompressible fluid modeled [...]

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• On the robustness of the multigrid method combining ILU and Partial Weight applied in an orthotropic diffusion problem

  G. Vassoler Rutz, M. Augusto Villela Pinto, S. de Fátima Tomazzoni Gonçalves

  Rev. int. métodos numér. cálc. diseño ing. (2019). Vol. 35, (1), 23

  
  

  Abstract

  This paper proposes an efficient and robust algorithm for solving a physical orthotropy problem. The algorithm is based on choosing the most efficient restriction operator and on an incomplete LU decomposition suited for each orthotropy direction. Local Fourier Analysis (LFA) is carried out in order to increase the efficiency of the multigrid method. Pure diffusion with orthotropy aligned to the coordinate axis x is the model considered. Equations are discretized by Finite Difference Method with uniform grid and second-order numerical scheme. Problems are solved with geometric multigrid method, correction scheme, V-cycle and standard coarsening ratio. The asymptotic convergence factor is calculated for different multigrid components, such as restriction operators, prolongation operators and solvers. Based on the optimum components obtained by LFA, we carried out experiments to analyze the complexity and computational cost of the algorithm proposed. The main conclusion is that the methodology proposed is efficient for the resolution of problems with strong orthotropy.

  Abstract
  This paper proposes an efficient and robust algorithm for solving a physical orthotropy problem. The algorithm is based on choosing the most efficient restriction operator [...]

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Articles (Documents not available in Scipedia)

• A multigrid waveform relaxation method for solving the poroelasticity equations

  S. R. Franco, C. Rodrigo, F. J. Gaspar, M. A. V. Pinto

  https://doi.org/10.1007/s40314-018-0603-9

• Multigrid method based on a space-time approach with standard coarsening for parabolic problems

  Sebastião Romero Franco, Francisco José Gaspar, Marcio Augusto Villela Pinto, Carmen Rodrigo

  http://dx.doi.org/10.1016/j.amc.2017.08.043

• On the robustness of ILU smoothers on triangular grids

  M.A.V. Pinto, C. Rodrigo, F.J. Gaspar, C.W. Oosterlee

  https://doi.org/10.1016/j.apnum.2016.02.007

• Polynomial interpolation with repeated Richardson extrapolation to reduce discretization error in CFD

  MARCHI, Carlos Henrique ; MARTINS, MÁRCIO ANDRÉ ; NOVAK, LEANDRO ALBERTO ; ARAKI, Luciano Kiyoshi ; PINTO, MÁRCIO AUGUSTO VILLELA ; GONÇALVES, SIMONE DE FÁTIMA TOMAZZONI ; MORO, DIEGO FERNANDO ; FREITAS, INAJARA DA SILVA

  https://doi.org/10.1016/j.apm.2016.05.029

• Simulation and optimization of Al-Fe aerospace alloy processed by laser surface remelting using geometric Multigrid solver and experimental validation

  PARIONA, MOISÉS MEZA ; DE OLIVEIRA, FABIANE ; TELEGINSKI, VIVIANE ; MACHADO, SILIANE ; PINTO, MARCIO AUGUSTO VILLELA

  DOI 10.1007/s00231-015-1619-y

• Repeated Richardson extrapolation applied to the two-dimensional Laplace equation using triangular and square grids

  Marchi, C.H. ; Araki, L.K. ; Alves, A.C. ; Suero, R. ; Gonçalves, S. F. T.

  https://doi.org/10.1016/j.apm.2012.09.071

• Analysis of algebraic multigrid parameters for two-dimensional steady-state heat diffusion equations

  Suero, R. ; PINTO, Marcio A. V. ; Marchi, C.H. ; Araki, L.K. ; Alves, A.C.

  https://doi.org/10.1016/j.apm.2011.09.088

• Optimized Partial Semicoarsening Multigrid Algorithm for Heat Diffusion Problems and Anisotropic Grids

  OLIVEIRA, Fabiane de ; PINTO, Marcio A. V. ; MARCHI, Carlos Henrique ; ARAKI, Luciano Kiyoshi

  https://doi.org/10.1016/j.apm.2011.11.084

Conference papers and presentations (Documents not available in Scipedia)

• ANALYSIS OF PARAMETERS OF GEOMETRIC MULTIGRID FOR THE CPU TIME FOR THREE-DIMENSIONAL POISSON EQUATION.

  OLIVEIRA, Fabiane de ; FRANCO, S. R. ; PINTO, M. A. V.

  Proceedings of the XXXVIII Iberian Latin-American Congress on Computational Methods in Engineering, 2017.

• UM ESTUDO TEÓRICO DO FATOR DE CONVERGÊNCIA ASSINTÓTICA VIA LFA DOS SOLVERS LINHA EM UM PROBLEMA DE DIFUSÃO ANISOTRÓPICO

  OLIVEIRA, M. L. ; PINTO, Marcio A. V. ; Gonçalves, S. F. T. ; RUTZ, G. V.

  Proceedings of the XXXVIII Iberian Latin-American Congress on Computational Methods in Engineering, 2017.

• ANÁLISE DE PARÂMETROS DO MÉTODO DO GRADIENTE CONJUGADO PRÉ- CONDICIONADO COM MUTIGRID PARA A EQUAÇÃO DE POISSON BIDIMENSIONAL

  ANUNCIACAO, M. A. M. ; PINTO, Marcio A. V. ; NEUNDORF, R. L. A.

  Proceedings of the XXXVIII Iberian Latin-American Congress on Computational Methods in Engineering, 2017.

• USO DO MÉTODO MULTIGRID NA RESOLUÇÃO DO PROBLEMA DE DIFUSÃO DE CALOR 2D EM MALHAS ESTRUTURADAS NÃO-ORTOGONAIS

  ZANATTA, D. C. ; ARAKI, Luciano Kiyoshi ; PINTO, Marcio A. V.

  Proceedings of the XXXVIII Iberian Latin-American Congress on Computational Methods in Engineering, 2017.

• COMPARISON BETWEEN EXTRAPOLATION TECHNIQUES ASSOCIATED TO THE MULTIGRID METHOD APPLIED IN THE NAVIER-STOKES EQUATIONS

  RUTYNA, B. B. ; PINTO, Marcio A. V. ; NEUNDORF, R. L. A. ; ANUNCIACAO, M. A. M. ; MARTINS, M. A.

  Proceedings of the XXXVIII Iberian Latin-American Congress on Computational Methods in Engineering, 2017