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Porting of finite element integration algorithm to Xeon Phi coprocessor-based HPC architectures

Krużel Filip, Banaś Krzysztof, Iacomo Mauro

Computer Assisted Methods in Engineering and Science

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2023

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Vol. 30, no. 4

427--459

EN

In the present article, we describe the implementation of the finite element numerical integration algorithm for the Xeon Phi coprocessor. The coprocessor was an extension of the many-core specialized unit for calculations, and its performance was comparable with the corresponding GPUs. Its main advantages were the built-in 512-bit vector registers and the ease of transferring existing codes from traditional x86 architectures. In the article, we move the code developed for a standard CPU to the coprocessor. We compareits performance with our OpenCL implementation of the numerical integration algorithm, previously developed for GPUs. The GPU code is tuned to fit into a coprocessor by ourauto-tuning mechanism. Tests included two types of tasks to solve, using two types of approximation and two types of elements. The obtained timing results allow comparing the performance of highly optimized CPU and GPU codes with a Xeon Phi coprocessor performance. This article answers whether such massively parallel architectures perform better using the CPU or GPU programming method. Furthermore, we have compared the Xeon Phi architecture and the latest available Intel’s i9 13900K CPU when writing this article. This comparison determines if the old Xeon Phi architecture remains competitive in today’s computing landscape. Our findings provide valuable insights for selectingthe most suitable hardware for numerical computations and the appropriate algorithmic design.

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Efficient simulations of large-scale convective heat transfer problems

Goik Damian, Banaś Krzysztof, Bielański Jan, Chłoń Kazimierz

Computer Science

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2021

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T. 22 (4)

517--538

EN

We describe an approach for efficient solution of large-scale convective heat transfer problems that are formulated as coupled unsteady heat conduction and incompressible fluid-flow equations. The original problem is discretized over time using classical implicit methods, while stabilized finite elements are used for space discretization. The algorithm employed for the discretization of the fluid-flow problem uses Picard’s iterations to solve the arising nonlinear equations. Both problems (the heat transfer and Navier–Stokes equations) give rise to large sparse systems of linear equations. The systems are solved by using an iterative GMRES solver with suitable preconditioning. For the incompressible flow equations, we employ a special preconditioner that is based on an algebraic multigrid (AMG) technique. This paper presents algorithmic and implementation details of the solution procedure, which is suitably tuned – especially for ill-conditioned systems that arise from discretizations of incompressible Navier–Stokes equations. We describe a parallel implementation of the solver using MPI and elements from the PETSC library. The scalability of the solver is favorably compared with other methods, such as direct solvers and the standard GMRES method with ILU preconditioning.

3

Plant replacement trend in soft-water lakes with isoetids

Ronowski Rafał P., Banaś Krzysztof, Merdalski Marek, Szmeja Józef

Oceanological and Hydrobiological Studies

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2020

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Vol. 49, No. 2

157--167

EN

A number of small, shallow inland lakes in the Northern Hemisphere are acidic, soft-water water bodies, mostly fed by precipitation. Some of these lakes are home to Lobelia dortmanna and other aquatic macrophytes from the isoetid group. The present study identifies a plant species replacement trend along the main environmental gradient in such lakes, i.e. pH of water. In 70 lakes covered by the study and ordered along the gradient of increasing alkalinity, bryophytes gradually decline. Their decreasing frequency is 86.8% (pH 4.0), 32.5% (pH 6.5) and <10% (pH 6.6–9.5). At the same time, the frequency of vascular plants increases from 13.2% in strongly acidic lakes up to 92.8% in alkaline ones. The majority of isoetids usually occur at pH 5.5–6.7, but one of them (Littorella uniflora) often (frequency 40.5%) occurs also in eutrophic, alkaline lakes (pH 9.0–9.5), where it forms large populations. In lakes ranked according to the decreasing alkalinity gradient, bryophyte populations have been found to be smaller, while vascular plants become more abundant. The decreasing water acidity leads to partial or total elimination of bryophytes, which are replaced by eutrophilic vascular plants, frequently accompanied by some stonewort species.

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Implementation of numerical integrationto high-order elements on the GPUs

Krużel Filip, Banaś Krzysztof, Nytko Mateusz

Computer Assisted Methods in Engineering and Science

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2020

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Vol. 27, no. 1

3--26

EN

This article presents ways to implement a resource-consuming algorithm on hardware with a limited amount of memory, which is the GPU. Numerical integration for higher-order finite element approximation was chosen as an example algorithm. To perform compu- tational tests, we use a non-linear geometric element and solve the convection-diffusion- reaction problem. For calculations, a Tesla K20m graphics card based on Kepler archi- tecture and Radeon r9 280X based on Tahiti XT architecture were used. The results of computational experiments were compared with the theoretical performance of both GPUs, which allowed an assessment of actual performance. Our research gives sugges- tions for choosing the optimal design of algorithms as well as the right hardware for such a resource-demanding task.