Advertisement

Automatic Algorithms for the Construction of Element Partition Trees for Isogeometric Finite Element Method

  • Bartosz Janota
  • Anna PaszyńskaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 659)

Abstract

The IGA-FEM is a modern method for simulation of different engineering and biomedical problems by using B-spline basis functions. To allow for real time interaction between human and computer while performing numerical simulations, there is a need for extremely fast solvers solving systems of linear equations generated while performing simulations. In this paper, an algorithm for construction of the ordering that controls the execution of the multi-frontal direct solver algorithm is presented. The ordering prescribes the permutation of the computational matrix in order to minimize the computational cost of the direct solver. We show that execution of our algorithm generating the recursive partitions of the h-adaptive grids with B-spline basis functions allows reducing the computational cost of the IGA-FEM simulations.

Keywords

Simulations on biomedicine Graph algorithms Topology of computational meshes Isogeometric finite element method 

References

  1. 1.
    Aboueisha, H., Calo, V.M., Jopek, K., Moshkov, M., Paszyńka, A., Paszyński, M., Skotniczny, M.: Element partition trees for \(h\)-refined meshes to optimize direct solver performance. Part I. Dynamic Programming. Int. J. Appl. Math. Comput. Sci. 27(2), 351–365 (2017)CrossRefzbMATHGoogle Scholar
  2. 2.
    Amestoy, P., Guermouche, A., L’Excellent, J.Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Amestoy, P., Duff, I., L’Excellent, J.Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184, 501–520 (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Amestoy, P., Duff, I., L’Excellent, J.Y., Koster, J.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bazilevs, Y., Calo, V., Cottrell, J., Hughes, T., Reali, A., Scovazzi, G.: Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Eng. 197(1), 173–201 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bazilevs, Y., Calo, V.M., Zhang, Y., Hughes, T.J.R.: Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput. Mech. 38(4–5), 310–322 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benson, D., Bazilevs, Y., Hsu, M.C., Hughes, T.: A large deformation, rotation-free, isogeometric shell. Comput. Methods Appl. Mech. Eng. 200(13), 1367–1378 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Calo, V.M., Brasher, N.F., Bazilevs, Y., Hughes, T.J.R.: Multiphysics model for blood flow and drug transport with application to patient-specific coronary artery flow. Comput. Mech. 43(1), 161–177 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chang, K., Hughes, T., Calo, V.: Isogeometric variational multiscale large-eddy simulation of fully-developed turbulent flow over a wavy wall. Comput. Fluids 68, 94–104 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Unification of CAD and FEA. Wiley, Chichester (2009)CrossRefGoogle Scholar
  11. 11.
    Dedè, L., Borden, M.J., Hughes, T.: Isogeometric analysis for topology optimization with a phase field model. Technical report, The Institute for Computational Engineering and Sciences (2011)Google Scholar
  12. 12.
    Dedè, L., Hughes, T., Lipton, S., Calo, V.: Structural topology optimization with isogeometric analysis in a phase field approach. In: USNCTAM 2010, State College, PA, USA (2010)Google Scholar
  13. 13.
    Demkowicz, L.: Computing with \(hp\)-Adaptive Finite Elements, vol. I. One and Two Dimensional Elliptic and Maxwell Problems. Chapman and Hall/CRC, Boca Raton (2006)Google Scholar
  14. 14.
    Demkowicz, L., Kurtz, J., Pardo, D., Paszyński, M., Rachowicz, W., Zdunek, A.: Computing with hp-Adaptive Finite Elements, Vol. II. Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications. Chapman & Hall/CRC, Boca Raton (2007)Google Scholar
  15. 15.
    Duddu, R., Lavier, L., Hughes, T., Calo, V.: A finite strain Eulerian formulation for compressible and nearly incompressible hyper-elasticity using high-order NURBS elements. Int. J. Numer. Methods Eng. 89(6), 762–785 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Garcia, D., Pardo, D., Dalcin, L., Paszyński, M., Collier, N., Calo, V.: The value of continuity: refined isogeometric analysis and fast direct solvers. Comput. Methods Appl. Mech. Eng. 316, 586–605 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gómez, H., Calo, V., Bazilevs, Y., Hughes, T.: Isogeometric analysis of the cahn-hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197(49–50), 4333–4352 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gomez, H., Hughes, T., Nogueira, X., Calo, V.: Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations. Comput. Methods Appl. Mech. Eng. 199(25–28), 1828–1840 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hossain, S., Hossainy, S.A., Bazilevs, Y., Calo, V., Hughes, T.R.: Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls. Comput. Mech. 49(2), 213–242 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hsu, M., Akkerman, I., Bazilevs, Y.: High-performance computing of wind turbine aerodynamics using isogeometric analysis. Comput. Fluids 49(1), 93–100 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hughes, T., Cottrell, J., Bazilevs, Y.: Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Janota, B.: Algorithms for construction of elimination tree for multi-frontal solver of isogeometric finite element method. Master’s thesis, AGH University, Poland (2016)Google Scholar
  23. 23.
    Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li, X., Demmel, J., Gilbert, J., Grigori, I., Shao, M., Yamazaki, I.: Superlu users’ guide. Technical report, Ernest Orlando Lawrence Berkeley National Laboratory (1999)Google Scholar
  25. 25.
    Li, X.S.: An overview of SuperLU: algorithms, implementation, and user interface. ACM Trans. Math. Softw. 31(3), 302–325 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, J.: The role of elimination trees in sparse factorization. SIAM J. Matrix Anal. Appl. 11(1), 134–172 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hénon, P., Ramet, P., Roman, J.: PaStiX: a high-performance parallel direct solver for sparse symmetric positive definite systems. Parallel Comput. 28(2), 301–321 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Paszyńska, A.: Volume and neighbors algorithm for finding elimination trees for three dimensional-adaptive grids. Comput. Math. Appl. 68(10), 1467–1478 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Paszyńska, A., Paszyński, M., Jopek, K., Woźniak, M., Goik, D., Gurgul, P., AbouEisha, H., Moshkov, M., Calo, H., Lenharth, A., Nguyen, D., Pingali, K.: Quasi-optimal elimination trees for 2D grids with singularities. Sci. Programm. 2015, 1–18 (2015)Google Scholar
  30. 30.
    Paszyński, M.: Fast Solvers for Mesh-Based Computations. CRC Press, Boca Raton (2016)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceAGH University of Science and TechnologyKrakówPoland
  2. 2.The Faculty of Physics, Astronomy and Applied Computer ScienceJagiellonian UniversityKrakówPoland

Personalised recommendations