Lucia Catabriga

Implicit SUPG solution of Euler equations using edge-based data structures

In this work we present an implicit, edge-based implementation of the semi-discrete SUPG formulation with shock-capturing for the Euler equations in conservative variables. By disassembling the resulting finite element matrices into their edge contributions, sparse matrix coefficients, residuals and matrix-vector products needed in Krylov-update techniques are computed based on edge data structures. The resulting solution method requires less memory and CPU time than element-based implementations.

Three-Dimensional Edge-Based SUPG Computation of Inviscid Compressible Flows With YZβ Shock-Capturing

The streamline-upwind/Petrov-Galerkin (SUPG) formulation of compressible flows based on conservation variables, supplemented with shock-capturing, has been successfully used over a quarter of a century. In this paper, for inviscid compressible flows, the YZβ shock-capturing parameter, which was developed recently and is based on conservation variables only, is compared with an earlier parameter derived based on the entropy variables. Our studies include comparing, in the context of these two versions of the SUPG formulation, computational efficiency of the element- and edge-based data structures in iterative computation of compressible flows. Tests include ID, 2D, and 3D examples.

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.

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

Abstract In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.

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.

A non-speculative parallelization of reverse cuthill-McKee algorithm for sparse matrices reordering

This work presents a new parallel non-speculative implementation of the Unordered Reverse Cuthill-McKee algorithm. Reordering quality (bandwidth reduction) and reordering performance (CPU time) are evaluated in comparison with a serial implementation of the algorithm made available by the state-of-the-art mathematical software library HSL. The bandwidth reductions reached by our parallel RCM were more than 90% for several large matrices out of the ones tested, and the time reordering improvement was up to 57.82%. Speedups higher than 3.0X were achieved with the parallel RCM. The underlying parallelism was supported by the OpenMP framework and three strategies for reducing idle threads were incorporated into the algorithm.

An Implementation of the Unordered Parallel RCM for Bandwidth Reduction of Large Sparse Matrices

This paper describes an implementation of the Unordered Parallel Reverse Cuthill-McKee algorithm which is compared with its well-known serial version. The OpenMP framework is used for supporting the parallelism and a strategy for reducing lazy threads is evaluated. Large sparse matrices are used to test sequential and parallel approaches. The computational cost reduction and the quality of matrices bandwidth minimization are validated by CPU time and speedup.