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ISI TAŞINIMI İÇİN YAPISAL OLMAYAN AĞLARDA BİR HIZLANDIRILMIŞ SÜREKSİZ GALERKİN METDU: FORMÜLASYONU VE DOĞRULANMASI

Yıl 2022, Cilt: 42 Sayı: 1, 91 - 100, 30.04.2022
https://doi.org/10.47480/isibted.1107459

Öz

Bu çalışmada, büyük ölçekli, birleştirilmiş sıkıştırılamaz akış ve ısı transferi problemleri için GPU ile hızlandırılmış bir yöntem sunulmuştur. Yapılandırılmamış üçgen ağlar üzerinde tanımlayıcı denklemlerini ayrıklaştırmak için yüksek dereceli, nodal süreksiz Galerkin yöntemi kullanılmıştır. Zaman ayrıklaştırması, taşınım terimlerinin açık bir şekilde ele alındığı, Stokes operatörlerinin ise örtük olarak çözüldüğü bir yarı-örtük şema kullanılarak elde edilmiştir. Basınç sistemi, tamamen GPU ile hızlandırılmış bir çoklu-ağ ön koşullandırıcının kullanıldığı, eşlenik gradyan yöntemiyle çözülmüştür. Kod, yüksek dereceli ayrıklaştırmalar için yüksek performanslı kerneller sağlayan, ölçeklenebilir kütüphane olan libParanumal üzerine yazılmıştır. Platformdan bağımsız performans taşınabilirliği OCCA (open concurrent compute abstraction) dili ile elde edilmiştir. Yapılan serbest ve karışık taşınım problemlerini içeren bir dizi sayısal test ile sunulan metodun deneysel olarak beklenen, spektral doğruluğa ulaştığını gösterilmiştir.

Kaynakça

  • Adams, M., Brezina, M., Hu, J., and Tuminaro, R., 2003, Parallel multigrid smoothing: polynomial versus Gauss–Seidel, Journal of Computational Physics, 188(2), 593–610.
  • Arnold, D., 1982, An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis, 19(4), 742–760.
  • Chalmers, N., Karakus, A., Austin, A. P., Swirydowicz, K., and Warburton, T., 2020, libParanumal: a performance portable high-order finite element library. Release 0.4.0.
  • Chan, J., Wang, Z., Modave, A., Remacle, J. F., and Warburton, T., 2016, GPU-accelerated discontinuous Galerkin methods on hybrid meshes, Journal of Computational Physics, 318, 142–168.
  • Darekar, R. M. and Sherwin, S. J., 2001, Flow past a square-section cylinder with a wavy stagnation face, Journal of Fluid Mechanics, 426, 263–295.
  • De Vahl Davis, G., 1983, Natural convection of air in a square cavity: A bench mark numerical solution, International Journal for Numerical Methods in Fluids, 3(3), 249–264.
  • Ferrer, E. and Willden, R. H. J., 2011, A high order discontinuous Galerkin finite element solver for the incompressible Navier–Stokes equations, Computers & Fluids, 46(1), 224–230.
  • Gandham, R., Esler, K., and Zhang, Y., 2014, A GPU accelerated aggregation algebraic multigrid method, Computers & Mathematics with Applications, 68(10), 1151–1160.
  • Gandham, R., Medina, D., and Warburton, T., 2015, GPU accelerated discontinuous Galerkin methods for shallow water equations, Communications in Computational Physics, 18(1), 37–64.
  • Hesthaven, J. S. and Warburton, T., 2008, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Springer.
  • Hossain, M. Z., Cantwell, C. D., and Sherwin, S. J., 2021, A spectral/hp element method for thermal convection, International Journal for Numerical Methods in Fluids, 93(7), 2380–2395.
  • Karakus, A., Chalmers, N., Hesthaven, J. S., and Warburton, T., 2019a, Discontinuous Galerkin discretizations of the Boltzmann–BGK equations for nearly incompressible flows: Semi-analytic time stepping and absorbing boundary layers, Journal of Computational Physics, 390, 175–202.
  • Karakus, A., Chalmers, N., Swirydowicz, K., and Warburton, T., 2019b, A GPU accelerated discontinuous Galerkin incompressible flow solver, Journal of Computational Physics, 390, 380–404.
  • Karakus, A., Warburton, T., Aksel, M. H., and Sert, C., 2016a, A GPU-accelerated adaptive discontinuous Galerkin method for level set equation, International Journal of Computational Fluid Dynamics, 30(1), 56–68.
  • Karakus, A., Warburton, T., Aksel, M. H., and Sert, C., 2016b, A GPU accelerated level set reinitialization for an adaptive discontinuous Galerkin method, Computers & Mathematics with Applications, 72(3), 755–767.
  • Kumar, A. and Pothérat, A., 2020, Mixed baroclinic convection in a cavity, Journal of Fluid Mechanics, 88. Publisher: Cambridge University Press.
  • Medina, D. S., St-Cyr, A., and Warburton, T., 2014, OCCA: A unified approach to multi-threading languages. arXiv:1403.0968.
  • Modave, A., St-Cyr, A., and Warburton, T., 2016, GPU performance analysis of a nodal discontinuous Galerkin method for acoustic and elastic models, Computers & Geosciences, 91, 64–76.
  • Notay, Y., 2006, Aggregation-based algebraic multilevel preconditioning, SIAM journal on matrix analysis and applications, 27(4), 998–1018.
  • Notay, Y., 2010, An aggregation-based algebraic multigrid method, Electronic transactions on numerical analysis, 37(6), 123–146.
  • Roca, X., Nguyen, N. C., and Peraire, J., 2011, GPU-accelerated sparse matrix-vector product for a hybridizable discontinuous Galerkin method. In Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics, AIAA 2011–687.
  • Saha, S., Klewicki, J. C., Ooi, A. S. H., and Blackburn, H. M., 2015, Comparison of thermal scaling properties between turbulent pipe and channel flows via DNS, International Journal of Thermal Sciences, 89, 43–57.
  • Shahbazi, K., 2005, An explicit expression for the penalty parameter of the interior penalty method, Journal of Computational Physics, 205(2), 401–407.
  • Shahbazi, K., Fischer, P. F., and Ethier, C. R., 2007, A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations, Journal of Computational Physics, 222(1), 391– 407.
  • Stokos, K., Vrahliotis, S., Pappou, T., and Tsangaris, S., 2015, Development and validation of an incompressible Navier-Stokes solver including convective heat transfer, International Journal of Numerical Methods for Heat & Fluid Flow, 25(4), 861–886.
  • Warburton, T., 2006, An explicit construction of interpolation nodes on the simplex, Journal of Engineering Mathematics, 56(3), 247–262.
  • Wheeler, M. F., 1978, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15(1), 152–161.
  • Swirydowicz, K., Chalmers, N., Karakus, A., and Warburton, T., 2019, Acceleration of tensor-product ´ operations for high-order finite element methods, The International Journal of High Performance Computing Applications, 33(4).

AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION

Yıl 2022, Cilt: 42 Sayı: 1, 91 - 100, 30.04.2022
https://doi.org/10.47480/isibted.1107459

Öz

We present a GPU-accelerated method for large scale, coupled incompressible fluid flow and heat transfer problems. A high-order, nodal discontinuous Galerkin method is utilized to discretize governing equations on unstructured triangular meshes. A semi-implicit scheme with explicit treatment of the advective terms and implicit treatment of the split Stokes operators are used for time discretization. The pressure system is solved with a conjugate gradient method together with a fully GPU-accelerated multigrid preconditioner. The code is built on scalable libParanumal solver which is a library of high-performance kernels for high-order discretizations. Performance portability is achieved by using the open concurrent compute abstraction, OCCA. A set of numerical experiments including free and mixed convection problems indicate that our approach experimentally reaches design order of accuracy.

Kaynakça

  • Adams, M., Brezina, M., Hu, J., and Tuminaro, R., 2003, Parallel multigrid smoothing: polynomial versus Gauss–Seidel, Journal of Computational Physics, 188(2), 593–610.
  • Arnold, D., 1982, An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis, 19(4), 742–760.
  • Chalmers, N., Karakus, A., Austin, A. P., Swirydowicz, K., and Warburton, T., 2020, libParanumal: a performance portable high-order finite element library. Release 0.4.0.
  • Chan, J., Wang, Z., Modave, A., Remacle, J. F., and Warburton, T., 2016, GPU-accelerated discontinuous Galerkin methods on hybrid meshes, Journal of Computational Physics, 318, 142–168.
  • Darekar, R. M. and Sherwin, S. J., 2001, Flow past a square-section cylinder with a wavy stagnation face, Journal of Fluid Mechanics, 426, 263–295.
  • De Vahl Davis, G., 1983, Natural convection of air in a square cavity: A bench mark numerical solution, International Journal for Numerical Methods in Fluids, 3(3), 249–264.
  • Ferrer, E. and Willden, R. H. J., 2011, A high order discontinuous Galerkin finite element solver for the incompressible Navier–Stokes equations, Computers & Fluids, 46(1), 224–230.
  • Gandham, R., Esler, K., and Zhang, Y., 2014, A GPU accelerated aggregation algebraic multigrid method, Computers & Mathematics with Applications, 68(10), 1151–1160.
  • Gandham, R., Medina, D., and Warburton, T., 2015, GPU accelerated discontinuous Galerkin methods for shallow water equations, Communications in Computational Physics, 18(1), 37–64.
  • Hesthaven, J. S. and Warburton, T., 2008, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Springer.
  • Hossain, M. Z., Cantwell, C. D., and Sherwin, S. J., 2021, A spectral/hp element method for thermal convection, International Journal for Numerical Methods in Fluids, 93(7), 2380–2395.
  • Karakus, A., Chalmers, N., Hesthaven, J. S., and Warburton, T., 2019a, Discontinuous Galerkin discretizations of the Boltzmann–BGK equations for nearly incompressible flows: Semi-analytic time stepping and absorbing boundary layers, Journal of Computational Physics, 390, 175–202.
  • Karakus, A., Chalmers, N., Swirydowicz, K., and Warburton, T., 2019b, A GPU accelerated discontinuous Galerkin incompressible flow solver, Journal of Computational Physics, 390, 380–404.
  • Karakus, A., Warburton, T., Aksel, M. H., and Sert, C., 2016a, A GPU-accelerated adaptive discontinuous Galerkin method for level set equation, International Journal of Computational Fluid Dynamics, 30(1), 56–68.
  • Karakus, A., Warburton, T., Aksel, M. H., and Sert, C., 2016b, A GPU accelerated level set reinitialization for an adaptive discontinuous Galerkin method, Computers & Mathematics with Applications, 72(3), 755–767.
  • Kumar, A. and Pothérat, A., 2020, Mixed baroclinic convection in a cavity, Journal of Fluid Mechanics, 88. Publisher: Cambridge University Press.
  • Medina, D. S., St-Cyr, A., and Warburton, T., 2014, OCCA: A unified approach to multi-threading languages. arXiv:1403.0968.
  • Modave, A., St-Cyr, A., and Warburton, T., 2016, GPU performance analysis of a nodal discontinuous Galerkin method for acoustic and elastic models, Computers & Geosciences, 91, 64–76.
  • Notay, Y., 2006, Aggregation-based algebraic multilevel preconditioning, SIAM journal on matrix analysis and applications, 27(4), 998–1018.
  • Notay, Y., 2010, An aggregation-based algebraic multigrid method, Electronic transactions on numerical analysis, 37(6), 123–146.
  • Roca, X., Nguyen, N. C., and Peraire, J., 2011, GPU-accelerated sparse matrix-vector product for a hybridizable discontinuous Galerkin method. In Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics, AIAA 2011–687.
  • Saha, S., Klewicki, J. C., Ooi, A. S. H., and Blackburn, H. M., 2015, Comparison of thermal scaling properties between turbulent pipe and channel flows via DNS, International Journal of Thermal Sciences, 89, 43–57.
  • Shahbazi, K., 2005, An explicit expression for the penalty parameter of the interior penalty method, Journal of Computational Physics, 205(2), 401–407.
  • Shahbazi, K., Fischer, P. F., and Ethier, C. R., 2007, A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations, Journal of Computational Physics, 222(1), 391– 407.
  • Stokos, K., Vrahliotis, S., Pappou, T., and Tsangaris, S., 2015, Development and validation of an incompressible Navier-Stokes solver including convective heat transfer, International Journal of Numerical Methods for Heat & Fluid Flow, 25(4), 861–886.
  • Warburton, T., 2006, An explicit construction of interpolation nodes on the simplex, Journal of Engineering Mathematics, 56(3), 247–262.
  • Wheeler, M. F., 1978, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15(1), 152–161.
  • Swirydowicz, K., Chalmers, N., Karakus, A., and Warburton, T., 2019, Acceleration of tensor-product ´ operations for high-order finite element methods, The International Journal of High Performance Computing Applications, 33(4).
Toplam 28 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Makine Mühendisliği
Bölüm Araştırma Makalesi
Yazarlar

Ali Karakus Bu kişi benim 0000-0002-9659-6712

Yayımlanma Tarihi 30 Nisan 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 42 Sayı: 1

Kaynak Göster

APA Karakus, A. (2022). AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION. Isı Bilimi Ve Tekniği Dergisi, 42(1), 91-100. https://doi.org/10.47480/isibted.1107459
AMA Karakus A. AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION. Isı Bilimi ve Tekniği Dergisi. Nisan 2022;42(1):91-100. doi:10.47480/isibted.1107459
Chicago Karakus, Ali. “AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION”. Isı Bilimi Ve Tekniği Dergisi 42, sy. 1 (Nisan 2022): 91-100. https://doi.org/10.47480/isibted.1107459.
EndNote Karakus A (01 Nisan 2022) AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION. Isı Bilimi ve Tekniği Dergisi 42 1 91–100.
IEEE A. Karakus, “AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION”, Isı Bilimi ve Tekniği Dergisi, c. 42, sy. 1, ss. 91–100, 2022, doi: 10.47480/isibted.1107459.
ISNAD Karakus, Ali. “AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION”. Isı Bilimi ve Tekniği Dergisi 42/1 (Nisan 2022), 91-100. https://doi.org/10.47480/isibted.1107459.
JAMA Karakus A. AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION. Isı Bilimi ve Tekniği Dergisi. 2022;42:91–100.
MLA Karakus, Ali. “AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION”. Isı Bilimi Ve Tekniği Dergisi, c. 42, sy. 1, 2022, ss. 91-100, doi:10.47480/isibted.1107459.
Vancouver Karakus A. AN ACCELERATED NODAL DISCONTINUOUS GALERKIN METHOD FOR THERMAL CONVECTION ON UNSTRUCTURED MESHES: FORMULATION AND VALIDATION. Isı Bilimi ve Tekniği Dergisi. 2022;42(1):91-100.