Towards a complete FEM-based simulation toolkit on GPUs: Unstructured grid finite element geometric multigrid solvers with strong smoothers based on sparse approximate inverses

Geveler, Markus; Ribbrock, Dirk; Göddeke, Dominik; Zajac, Peter; Turek, Stefan

doi:10.1016/j.compfluid.2012.01.025

Cited by 26 publications

(19 citation statements)

References 16 publications

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“…In continuation of the first project phase, we enhanced the assembly of sparse approximate inverses (SPAI), a kind of preconditioner that we had shown to be very effective within the DUNE solver before [26,9]. Concerning the assembly of such matrices we have investigated three strategies regarding their numerical efficacy (that is their quality in approximating A −1 ), the computational complexity of the actual assembly and ultimately, the total efficiency of the amortised assembly combined with all applications during a system solution.…”

Section: Strong Smoothers On the Gpu: Fast Approximate Inverses With mentioning

confidence: 99%

EXA-DUNE: Flexible PDE Solvers, Numerical Methods and Applications

Bastian

Engwer

Göddeke

et al. 2014

Lecture Notes in Computer Science

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In the E -D project we have developed, implemented and optimised numerical algorithms and software for the scalable solution of partial differential equations (PDEs) on future exascale systems exhibiting a heterogeneous massively parallel architecture. In order to cope with the increased probability of hardware failures, one aim of the project was to add flexible, application-oriented resilience capabilities into the framework. Continuous improvement of the underlying hardwareoriented numerical methods have included GPU-based sparse approximate inverses, matrix-free sum-factorisation for high-order discontinuous Galerkin discretisations as well as partially matrix-free preconditioners. On top of that, additional scalability is facilitated by exploiting massive coarse grained parallelism offered by multiscale and uncertainty quantification methods where we have focused on the adaptive choice of the coarse/fine scale and the overlap region as well as the combination of local reduced basis multiscale methods and the multilevel Monte-Carlo algorithm. Finally, some of the concepts are applied in a land-surface model including subsurface flow and surface runoff.

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Section: Strong Smoothers On the Gpu: Fast Approximate Inverses With mentioning

confidence: 99%

EXA-DUNE: Flexible PDE Solvers, Numerical Methods and Applications

Bastian

Engwer

Göddeke

et al. 2014

Lecture Notes in Computer Science

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“…A GPU approach for geometric multigrid solvers on finite elements for unstructured grid problems was performed by Geveler et al . , in which their GPU implementation is based on cascades of sparse matrix‐vector multiplication by applying strong smoothers. Komatitsch et al have used CUDA to speedup numerical simulation of seismic wave propagation resulting from earthquakes.…”

Section: Related Work On Graphics Processor Units and High Performancmentioning

confidence: 99%

Mapping Cohesive Fracture and Fragmentation Simulations to Graphics Processor Units

Alhadeff

Celes

Paulino

2015

Numerical Meth Engineering

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SUMMARYA graphics processor units(GPU)-based computational framework is presented to deal with dynamic failure events simulated by means of cohesive zone elements. The work is divided into two parts. In the first part, we deal with pre-processing of the information and verify the effectiveness of dynamic insertion of cohesive elements in large meshes in parallel. To this effect, we employ a novel and simplified topological data structure specialized for meshes with triangles, designed to run efficiently and minimize memory occupancy on the GPU. In the second part, we present a parallel explicit dynamics code that implements an extrinsic cohesive zone formulation where the elements are inserted 'on-the-fly', when needed and where needed. The main challenge for implementing a GPU-based computational framework using an extrinsic cohesive zone formulation resides on being able to dynamically adapt the mesh, in a consistent way, by inserting cohesive elements on fractured facets. In order to handle that, we extend the conventional data structure used in the finite element method (based on element incidence) and store, for each element, references to the adjacent elements. This additional information suffices to consistently insert cohesive elements by duplicating nodes when needed. Currently, our data structure is specialized for triangular meshes, but an extension to tetrahedral meshes is feasible. The data structure is effective when used in conjunction with algorithms to traverse nodes and elements. Results from parallel simulations show an increase in performance when adopting strategies such as distributing different jobs among threads for the same element and launching many threads per element. To avoid concurrency on accessing shared entities, we employ graph coloring. In a pre-processing phase, each node of the dual graph (bulk elements of the mesh as graph nodes) is assigned a color different from the colors assigned to adjacent nodes. In that fashion, elements of the same color can be processed in parallel without concurrency. All the procedures needed for the insertion of cohesive elements along fracture facets and for computing nodal properties are performed by threads assigned to triangles, invoking one kernel per color. Computations on existing cohesive elements are also performed based on adjacent bulk elements. Experiments show that GPU speedup increases with the number of nodes and bulk elements.

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“…Additionally, the Jacobi method depends on the sparse matrix-vector multiplication operation, which has low computational density and is generally memory bandwidth bounded. In [31, 32, 33], the authors introduce GPU-based linear solvers with multigrid methods. The solvers use the ELLpack sparse matrix data structure for their specific problems, which is not efficient when number of non-zero entries per row varies largely.…”

Section: Previous Workmentioning

confidence: 99%

Architecting the finite element method pipeline for the GPU

Lewis

Kirby

et al. 2014

Journal of Computational and Applied Mathematics

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The finite element method (FEM) is a widely employed numerical technique for approximating the solution of partial differential equations (PDEs) in various science and engineering applications. Many of these applications benefit from fast execution of the FEM pipeline. One way to accelerate the FEM pipeline is by exploiting advances in modern computational hardware, such as the many-core streaming processors like the graphical processing unit (GPU). In this paper, we present the algorithms and data-structures necessary to move the entire FEM pipeline to the GPU. First we propose an efficient GPU-based algorithm to generate local element information and to assemble the global linear system associated with the FEM discretization of an elliptic PDE. To solve the corresponding linear system efficiently on the GPU, we implement a conjugate gradient method preconditioned with a geometry-informed algebraic multi-grid (AMG) method preconditioner. We propose a new fine-grained parallelism strategy, a corresponding multigrid cycling stage and efficient data mapping to the many-core architecture of GPU. Comparison of our on-GPU assembly versus a traditional serial implementation on the CPU achieves up to an 87 × speedup. Focusing on the linear system solver alone, we achieve a speedup of up to 51 × versus use of a comparable state-of-the-art serial CPU linear system solver. Furthermore, the method compares favorably with other GPU-based, sparse, linear solvers.

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Towards a complete FEM-based simulation toolkit on GPUs: Unstructured grid finite element geometric multigrid solvers with strong smoothers based on sparse approximate inverses

Cited by 26 publications

References 16 publications

EXA-DUNE: Flexible PDE Solvers, Numerical Methods and Applications

EXA-DUNE: Flexible PDE Solvers, Numerical Methods and Applications

Mapping Cohesive Fracture and Fragmentation Simulations to Graphics Processor Units

Architecting the finite element method pipeline for the GPU

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