Solução ponta a ponta em GPU do método sem malha local Petrov-Galerkin (MLPG)

Abstract

This work presents an end-to-end massively parallelized procedure for the solution of boundary value problems on Graphics Processing Units (GPU). The proposal is an integrated strategy that not only entails the calculation of nodal contributions, and the stiffness matrix assembly using the Meshless Local Petrov Galerkin Method (MLPG) but also the iterative solution of the system of algebraic equations in combination with methods from the Conjugate Gradient (CG) family.

The most important contribution is a complete end-to-end MLPG implementation in GPU taking full advantage of the device's Single Instruction Multiple Thread (SIMT) architecture. From initial calculations to solution delivery for the problem addressed, the integrated steps are computed exclusively in the GPU to avoid costs related to data movement and conversion. The proposed solution takes advantage of the parallel nature of the MLPG by holding each cloud node in a thread on the device, until the final computation of the last process step is completed with subsequent delivery of the values of interest to the respective cloud nodes. Thus minimal auxiliary structures and few synchronization points are required.

Different solution variations are tested to assess the effective impact on the GPU, supporting more accurate decisions. Among others, the tests include variations on the initial node position in the domain, the MLPG form function techniques and four different solvers of the Conjugate Gradient family, namely: (i) BiConjugate Gradients - BICG, (ii) Quadratic Conjugate Gradients - CGS, (iii) BiConjugate Gradients Stabilized - BICG-Stab and (iv) BiConjugate Gradients Enhanced - I-BICG-Stab.

To evaluate the solution viability and performance metrics, the two parallel plate capacitor problem is used. Despite the simplicity of the problem, the application of the proposed algorithms to more complex problems is straightforward, and a sevenfold speedup is observed in the end-to-end solution. This number can be even more significant with cloud node growth during domain discretization.