Computational performance of a parallelized three-dimensional high-order spectral element toolbox

Bosshard, Christoph; Bouffanais, Roland; Deville, Michel; Gruber, R.; Lätt, Jonas

doi:10.1016/j.compfluid.2010.11.014

Cited by 23 publications

(5 citation statements)

References 12 publications

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“…Although the LW-ACM was carefully validated in [3], no high-resolution validation study of the LW-ACM has been published so far to the best of our knowledge. For this purpose, we generated highly accurate reference data of the lid-driven cubic cavity at Reynolds number Re = 2000, using a spectral and mortar element analysis program developed using the OpenSPECULOOS toolbox [1,5]. We performed the simulation on 8 3 elements with Gauss-Lobato-Legendre polynomials of degree 12 and a dimensionless time step duration of 10 −3 .…”

Section: Reference Datamentioning

confidence: 99%

High-performance implementations and large-scale validation of the link-wise artificial compressibility method

Obrecht

Asinari

Kuznik

et al. 2014

Journal of Computational Physics

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performance implementations and large-scale validation of the link-wise artificial compressibility method. AbstractThe link-wise artificial compressibility method (LW-ACM) is a recent formulation of the artificial compressibility method for solving the incompressible Navier-Stokes equations. Two implementations of the LW-ACM in three dimensions on CUDA enabled GPUs are described. The first one is a modified version of a stateof-the-art CUDA implementation of the lattice Boltzmann method (LBM), showing that an existing GPU LBM solver might easily be adapted to LW-ACM. The second one follows a novel approach, which leads to a performance increase of up to 1.8× compared to the LBM implementation considered here, while reducing the memory requirements by a factor of 5.25. Large-scale simulations of the lid-driven cubic cavity at Reynolds number Re = 2000 were performed for both LW-ACM and LBM. Comparison of the simulation results against spectral elements reference data shows that LW-ACM performs almost as well as multiple-relaxation-time LBM in terms of accuracy.

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Section: Reference Datamentioning

confidence: 99%

High-performance implementations and large-scale validation of the link-wise artificial compressibility method

Obrecht

Asinari

Kuznik

et al. 2014

Journal of Computational Physics

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“…For the CG solver (without a sophisticated pre-conditioner), the number of iterations required to achieve convergence (close to machine zero) is proportional to the initial residue and the cubic root of the total number of unknowns [25]. Remarking that the initial residue is proportional to the time step size, thus…”

Section: à3mentioning

confidence: 99%

“…It is considered very good scalability if a linear (ideal) speedup is obtained, i.e., S p = p. In practical tests, T 1 is sometimes replaced by n 0 Á T n 0 , where T n 0 is the execution time while the parallel code is running on a small number of processes (n 0 ) [25]. Two tests are conducted in this work: a problem with the mesh size of 32 3 running on 3-63 processes and a problem with the mesh size of 256 3 running on 127-2047 processes.…”

Section: D Lid-driven Cavity Flowmentioning

confidence: 99%

Parallel computing strategy for a flow solver based on immersed boundary method and discrete stream-function formulation

Wang

Zhang

2013

Computers & Fluids

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a b s t r a c tThe development of a parallel immersed boundary solver for flows with complex geometries is presented. The numerical method for incompressible Navier-Stokes equations is based on the discrete streamfunction formulation and unstructured Cartesian grid framework. The code parallelization is achieved by using the domain decomposition (DD) approach and Single Program Multiple Data (SPMD) programming paradigm, with the data communication among processes via the MPI protocol. A 'gathering and scattering' strategy is used to handle the force computing on the immersed boundaries. Three tests, 3D lid-driven cavity flow, sedimentation of spheres in a container and flow through and around a circular array of cylinders are performed to evaluate the parallel efficiency of the code. The speedups obtained in these tests on a workstation cluster are reasonably good for the problem size up to 10 million and the number of processes in the range of 16-2048.

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“…This opens the door for fully explicit time integration schemes and is a well-known property of GLL basis functions in conjunction with Gauss-Lobatto quadrature [1,20,32,[47][48][49].Collocation methods that use the Kronecker ı property of Gauss-Lobatto nodes in conjunction with corresponding nodal basis functions are not new, and many instantiations of this concept have been developed under different names in the past, for example, the C 0 -collocation-Galerkin method [50-53], the differential quadrature method [54,55], the Galerkin/spectral element methods with numerical integration [11,12,56], multidomain spectral or pseudospectral elements [12,31,32,36,57], or hp-FEM with Gauss-Lobatto basis functions [58,59]. Beyond the straightforward implementation of collocated FEA advocated in the present paper, advanced implementation technologies for collocation methods have been developed, which are documented in particular in the spectral element literature [11,12,20,32,48,60,61]. In this context, we would be very happy if the presented material implicitly triggered the interest of standard finite element analysts in collocation type methods that have reached a very mature state in spectral elements.…”

mentioning

confidence: 99%

“…the C 0 -collocation-Galerkin method [50][51][52][53], the differential quadrature method [54,55], the G-NI or SEM-NI methods (Galerkin/spectral element methods with numerical integration) [11,12,56], multidomain spectral or pseudospectral elements [12,31,32,36,57] or hp-FEM with Gauss-Lobatto basis functions [58,59]. Beyond the straightforward implementation of hp-collocation advocated in the present paper, advanced implementation technologies for collocation methods have been developed, which are documented in particular in the spectral element literature [11,12,20,32,48,60,61]. In this context, we would be very happy if the presented material implicitly triggered the interest of standard finite element analysts in collocation type methods that have reached a very mature state in spectral elements.…”

Section: Introductionmentioning

confidence: 99%

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Schillinger

Evans

Frischmann

et al. 2014

Numerical Meth Engineering

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We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. AbstractWe demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, i.e. weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocation and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established.

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Computational performance of a parallelized three-dimensional high-order spectral element toolbox

Cited by 23 publications

References 12 publications

High-performance implementations and large-scale validation of the link-wise artificial compressibility method

High-performance implementations and large-scale validation of the link-wise artificial compressibility method

Parallel computing strategy for a flow solver based on immersed boundary method and discrete stream-function formulation

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

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Computational performance of a parallelized three-dimensional high-order spectral element toolbox

Cited by 23 publications

References 12 publications

High-performance implementations and large-scale validation of the link-wise artificial compressibility method

High-performance implementations and large-scale validation of the link-wise artificial compressibility method

Parallel computing strategy for a flow solver based on immersed boundary method and discrete stream-function formulation

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

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A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics