Alternating directions parallel hybrid memory iGRM direct solver for non-stationary simulations

Maciej Woźniak; Anna Janina Bukowska

doi:10.7494/csci.2020.21.4.3834

Authors

Maciej Woźniak AGH University of Science and Technology https://orcid.org/0000-0002-5576-5671
Anna Janina Bukowska AGH University of Science and Technology https://orcid.org/0000-0002-3179-7863

DOI:

https://doi.org/10.7494/csci.2020.21.4.3834

Keywords:

isogeometric finite element method, integration, shared memory, parallel

Abstract

The three-dimensional isogeometric analysis (IGA-FEM) is a modern method for simulation. The idea is to utilize B-splines or NURBS basis functions for both computational domain descriptions and the engineering computations. Refined isogeometric analysis (rIGA) employs a mixture of patches of elements with B-spline basis functions, and $C^0$ separators between them. It enables a reduction of the computational cost of direct solvers. Both IGA and rIGA come with challenging sparse matrix structure, that is expensive to generate. In this paper, we show a hybrid parallelization method to reduce the computational cost of the integration phase using hybrid-memory parallel machines. The two-level parallelization includes the partitioning of the computational mesh into sub-domains on the first level (MPI), and loop parallelization on the second level (OpenMP). We show that hybrid parallelization of the integration reduces the contribution of this phase significantly. Thus, alternative algorithms for fast isogeometric integration are not necessary.

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Author Biographies

Maciej Woźniak, AGH University of Science and Technology

Maciej Wozniak is post-doc at the Department of Computer Science at AGH University, Krakow, Poland.
His research interests include optimization of solver algorithms for different architectures of parallel
machines, isogeometric analysis as well as tree algorithms.

Anna Janina Bukowska, AGH University of Science and Technology

Anna Bukowska is a masters degree student of Computer Science at AGH University, Krakow, Poland.
Her research interests include concurrency theory, isogeometric analysis and adaptive algorithms.

References

Http://www.cyfronet.krakow.pl/computers/15226,artykul,prometheus.html.

Amestoy P.R., Du_ I.S., L'Excellent J.Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. In: Computer Methods in Applied Mechanics and Engineering, vol. 184, pp. 501-520, 2000.

Amestoy P.R., Du_ I.S., L'Excellent J.Y., Koster J.: A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling. In: SIAM Journal of Matrix Analysis and Applications, vol. 23(1), pp. 15-41, 2001.

Amestoy P.R., Guermouche A., L'Excellent J.Y., Pralet S.: Hybrid scheduling for the parallel solution of linear systems. In: Parallel Computing, vol. 32, pp. 136-156, 2006.

Balay S., Abhyankar S., Adams M.F., Brown J., Brune P., Buschelman K., Dalcin L., Dener A., Eijkhout V., Gropp W.D., Karpeyev D., Kaushik D., Knepley M.G., May D.A., McInnes L.C., Mills R.T., Munson T., Rupp K., Sanan P., Smith B.F., Zampini S., Zhang H., Zhang H.: PETSc Web page. https://www.mcs.anl.gov/petsc, 2019. URL https://www.mcs.anl.gov/petsc.

Balay S., Abhyankar S., Adams M.F., Brown J., nd Kris Buschelman P.B., Dalcin L., Dener A., Eijkhout V., Gropp W.D., Karpeyev D., Kaushik D., Knepley M.G., May D.A., McInnes L.C., Mills R.T., Munson T., Rupp K., nd Barry F. Smith P.S., Zampini S., Zhang H., Zhang H.: PETSc Users Manual, 2020. URL https://www.mcs.anl.gov/petsc.

Balay S., Gropp W.D., McInnes L.C., Smith B.F.: Efficient Management of Parallelism in Object Oriented Numerical Software Libraries. In: , pp. 163–202, 1997.[8] Barton M., Calo V.: Optimal quadrature rules for isogeometric analysis. In: , 2015. ArXiv:1511.03882.

Bazilevs Y., Calo V., Cottrell J., Evans J., Lipton S., Scott M., Sederberg T.: Isogeometric analysis using T-splines. In: Computer Methods in Applied Mechanics and Engineering, vol. 199, pp. 229-263, 2010.

Beirão da Veiga L., Buffa A., Sangalli G., Vázquez R.: Analysis-suitable T-splines of arbitrary degree: definition, linear independence and approximation properties. In: Mathematical Models and Methods in Applied Sciences, vol. 23(11), pp. 1979–2003, 2013.

Benson D., Bazilevs Y., Hsu M.C., Hughes T.: A large-deformation, rotation-free isogeometric shell. In: Computer Methods in Applied Mechanics and Engineering, vol. 200, pp. 1367-1378, 2011.

Bubak M., Kitowski J., Wiatr K., eds.: eScience on Distributed Computing Infrastructure: Achievements of PLGrid Plus Domain-Specific Services and Tools, vol. 8500. Springer, 2014. ISBN 978-3-319-10894-0.

Calabrò F., Sangalli G., Tani M.: Fast formation of isogeometric Galerkin matrices by weighted quadrature. In: Computer Methods in Applied Mechanics and Engineering, vol. 316, pp. 606–622, 2017.

Calo V., Brasher N., Bazilevs Y., Hughes T.: Multiphysics model for blood ow and drug transport with application to patient-specific coronary artery ow. In: Computational Mechanics, vol. 43, pp. 161-177, 2008.

Chang K., Hughes T., Calo V.: Isogeometric variational multiscale large-eddy simulation of fully-developed turbulent ow over a wavy wall. In: Computers and Fluids, vol. 68, pp. 94-104, 2012.

Collier N., Pardo D., Dalcin L., Paszyński M., Calo V.: The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers. In: Computer Methods in Applied Mechanics and Engineering, vol. 213-216, pp. 353-361, 2012.

Cottrell J., Hughes T., Bazilevs Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. 2009. ISBN 978-0-470-74873-2.

Dalcin L., Collier N., Vignal P., Côrtes A., Calo V.: PetIGA: A framework for high-performance isogeometric analysis. In: Computer Methods in Applied Mechanics and Engineering, vol. 308, pp. 151-181, 2016.

Dedè L., Borden M., Hughes T.: Isogeometric Analysis for Topology Optimization with a Phase Field Model. In: Archives of Computational Methods in Engineering, vol. 19, pp. 427-465, 2012.

Duddu R., Lavier L., Hughes T., Calo V.: A Finite strain Eulerian formulation for compressible and nearly incompressible hyperelasticity using high-order Bspline finite elements. In: International Journal of Numerical Methods in Engineering, vol. 89, pp. 762-785, 2012.

Garcia D., Pardo D., Dalcin L., Paszyński M., Collier N., Calo V.: The value of continuity: Refined isogeometric analysis and fast direct solvers. In: Computer Methods in Applied Mechanics and Engineering, vol. 316, pp. 586-605, 2017. DOI: 10.1016/j.cma.2016.08.017.

Gómez H., Hughes T., Nogueira X., Calo V.: Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations. In: Computer Methods in Applied Mechanics and Engineering, vol. 199, pp. 1828-1840, 2010.

Hossain S., Hossainy S., Bazilevs Y., Calo V., Hughes T.: Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient specific coronary artery walls. In: Computational Mechanics, vol. 49, 2012. Doi: 10.1007/s00466-011-0633-2.

Łoś M., Muñoz-Matute J., Muga I., Paszyński M.: Isogeometric Residual Minimization Method (iGRM) for Stokes and Time-Dependent Stokes Problems. In: . ArXiv:2001.00178 [math.NA].

Łoś M., Muñoz-Matute J., Muga I., Paszyński M.: Isogeometric Residual Minimization Method (iGRM) with direction splitting for non-stationary advection-diffusion problems. In: Computers & Mathematics with Applications, vol. 79, pp. 213-229, 2020.

Łoś M., Woźniak M., Paszyński M., Lenharth A., Hassaan M., Pingali K.: IGAADS: Isogeometric analysis FEM using ADS solver. In: Computer Physics Communications, vol. 217, pp. 99-116, 2017.

M.-C. Hsu I. Akkerman Y.B.: High-performance computing of wind turbine aero-dynamics using isogeometric analysis. In: Computers and Fluids, vol. 49, pp. 93-100, 2011.

Paszyński M.: On the Parallelization of Self-Adaptive hp-Finite Element Methods Part I. Composite Programmable Graph Grammar Model. In: Fundamenta Informaticae, vol. 93(4), pp. 411-434, 2009.

Paszyński M.: Fast solvers for mesh-based computations. CRC Press, Taylor & Francis, 2016. ISBN 978-1498754194.

Paszyński M., Paszyńska A.: Graph Transformations for Modeling Parallel hp-Adaptive Finite Element Method. In: Lecture Notes in Computer Science, vol. 4967, pp. 1313-1322, 2008.

Paszyński M., Siwik L., Woźniak M.: Concurrency of three-dimensional refined isogeometric analysis. In: Parallel Computing, vol. 80, pp. 1-22, 2018.

Piegl L., Tiller W.: The NURBS Book (Second Edition). Springer-Verlag New York, Inc., 1997. ISBN 978-3-642-59223-2.

Siwik L., Woźniak M., Trujillo V., Pardo D., Calo V., Paszyński M.: Parallel Refined Isogeometric Analysis in 3D. In: IEEE Transactions on Parallel and Distributed Systems, vol. 30, pp. 1134-1142, 2019.

Terpstra D., Jagode H., You H., Dongarra J.: Collecting Performance Data with PAPI-C. In: Tools for High Performance Computing 2009, Springer Berlin / Heidelberg, 3rd Parallel Tools Workshop, Dresden, Germany, pp. 57-173, 2010.

Woźniak M., Kuźnik K., Paszyński M., Calo V., Pardo D.: Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers. In: Computers & Mathematics with Applications, vol. 67, pp. 1864-1883, 2014.

Woźniak M., Łoś M., Paszyński M., Dalcin L., Calo V.: Parallel fast isogeometric solvers for explicit dynamic. In: Computing and Informatics, vol. 36(2), pp. 423-448, 2017.

Woźniak M., Paszyński M., Pardo D., Dalcin L., Calo V.: Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines. In: Computers Methods in Applied Mechanics and Engineering, vol. 284, pp. 971-987, 2015.