Leonardo Muniz de Lima

A Nonlinear Multiscale Viscosity Method to Solve Compressible Flow Problems

In this work we present a nonlinear multiscale viscosity method to solve inviscid compressible flow problems in conservative variables. The basic idea of the method consists of adding artificial viscosity adaptively in all scales of the discretization. The amount of viscosity added to the numerical model is based on the YZ\(\beta \) shock-capturing parameter, which has the property of being mesh and numerical solution dependent. The subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. This new numerical formulation can be considered a free parameter and self adaptive method. Performance and accuracy comparisons with the well known method SUPG combined with shock capturing operators are conducted based on benchmark 2D problems.

A Nonlinear Finite Element Formulation Based on Multiscale Approach to Solve Compressible Euler Equations

This work presents a nonlinear finite element method for solving compressible Euler equations. The formulation is based on the strategy of separating scales – the core of the variational multiscale (finite element) methodology. The proposed method adds a nonlinear artificial viscosity operator that acts only on the unresolved mesh scales. The numerical model is completed by adding the YZ\(\beta \) shock-capturing operator to the resolved scale, taking into account the Mach number. We evaluate the efficiency of the new formulation through numerical studies, comparing it with other methodologies such as the SUPG combined with a shock-capturing operator.

Evaluation of Two Parallel Finite Element Implementations of the Time-Dependent Advection Diffusion Problem: GPU versus Cluster Considering Time and Energy Consumption

We analyze two parallel finite element implementations of the 2D time-dependent advection diffusion problem, one for multi-core clusters and one for CUDA-enabled GPUs, and compare their performances in terms of time and energy consumption. The parallel CUDA-enabled GPU implementation was derived from the multi-core cluster version. Our experimental results show that a desktop machine with a single CUDA-enabled GPU can achieve performance higher than a 24-machine (96 cores) cluster in this class of finite element problems. Also, the CUDA-enabled GPU implementation consumes less than one twentieth of the energy (Joules) consumed by the multi-core cluster implementation while solving a whole instance of the finite element problem.

A Multiscale Finite Element Formulation for the Incompressible Navier-Stokes Equations

In this work we present a variational multiscale finite element method for solving the incompressible Navier-Stokes equations. The method is based on a two-level decomposition of the approximation space and consists of adding a residual-based nonlinear operator to the enriched Galerkin formulation, following a similar strategy of the method presented in [1, 2] for scalar advection-diffusion equation. The artificial viscosity acts adaptively only onto the unresolved mesh scales of the discretization. In order to reduce the computational cost typical of two-scale methods, the subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Accuracy comparisons with the streamline-upwind/Petrov-Galerkin (SUPG) formulation combined with the pressure stabilizing/Petrov-Galerkin (PSPG) method are conducted based on 2D benchmark problems.

A Trade-off Analysis of the Parallel Hybrid SPIKE Preconditioner in a Unique Multi-core Computer

In this paper we apply the parallel hybrid SPIKE algorithm as a preconditioner for a nonstationary iterative method to solve large sparse linear systems. In order to obtain a good preconditioner, we employ several strategies solving combinatorial problems such as matching, reordering, partitioning, and quadratic knapsack. Our SPIKE implementation combines MPI and OpenMP paradigms in a unique multi-core computer. The computational experiments show the influence of each strategy evaluating the number of iterations and CPU time of the iterative solver in a set of large systems from miscellaneous application areas. The experiments suggest that the SPIKE preconditioner is very advantageous when a suitable set of parameters is chosen. The choice of the number of MPI ranks and OpenMP threads is not an easy task, because the SPIKE algorithm can increase the number of iterations when the number of MPI ranks grows. Moreover, the increase in the number of threads does not ensure a better performance.

Comparative Studies to Solve Compressible Problems by Multiscale Finite Element Methods

In this work we evaluate two multiscale methodologies to solve compressible flow problems, named, Dynamic Diffusion (DD) and Nonlinear Multiscale Viscosity (NMV), using the well know predictor-multicorrector time integration scheme. The subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. The time integration schemes assume that the resolved coarse scale advances in time by second order approximation and the unresolved scale can advance by first and second order approximations. Performance and accuracy comparisons are conducted based on benchmark 2D problems.

Estudo do comportamento da análise isogeométrica para o problema de Hemker

Este trabalho apresenta um estudo numerico de solucoes aproximadas para o problema de Hemker utilizando os conceitos da Analise Isogeometrica (AIG) e a ferramenta GeoPDEs. O estudo avalia a influencia da disposicao das patches e dos knots na qualidade da solucao aproximada, alem de apresentar a influencia da continuidade das funcoes base no tamanho e na esparsidade do sistema linear associado e no seu tempo de resolucao.

An Alternative Approach of the SPIKE Preconditioner for Finite Element Analysis

This paper describes an alternative approach to using the SPIKE preconditioner in the context of finite element analysis. Commonly, SPIKE operations evaluate finite element matrices in sequential form with the resulting matrices distributed in parallel partitions. The goal is to obtain these matrices literally in parallel, avoiding excessive communication at the beginning of the process. This study offers a comparison of two SPIKE versions. One version considers a domain decomposition of the mesh as a preprocess for the finite element calculations. The other version considers a domain decomposition of the resulting linear system. With numerical experiments, this paper shows that the first version demonstrates advantages related to CPU time and memory usage.

Técnicas de otimização combinatória aplicadas a um precondicionador paralelo baseado no algoritmo SPIKE

Neste trabalho, utilizamos o algoritmo paralelo hibrido SPIKE como um precondicionador para um metodo iterativo nao-estacionario combinando as arquiteturas de memoria distribuida e compartilhada MPI/OpenMP. A fim de obter um bom precondicionador resolvemos um conjunto de problemas combinatorios como reordenamento, particionamento, matching e o problema quadratico da mochila. Apresentamos os resultados avaliando o speedup e escalabilidade em sistemas resultantes de formulacoes de elementos finitos.