Applications of alternating direction solver for simulations of time-dependent problems

• Autorzy: Łoś M. , Paszyński M.

Identyfikatory

DOI

10.7494/csci.2017.18.2.117

• Języki publikacji: EN

Abstrakty

EN

This paper deals with the application of an Alternating Direction Solver (ADS) to a non-stationary linear elasticity problem solved with the isogeometric finite element method (IGA-FEM). Employing a tensor product B-spline basis in isogeometric analysis under some restrictions leads to a system of linear equations with a matrix possessing a tensor product structure. The ADI algorithm is a direct method that exploits this Kronecker product structure to solve the system in O (N), where N is the number of degrees of freedom (basis functions). This is asymptotically faster than state-of-the-art, general-purpose, multi-frontal direct solvers when applied to explicit dynamics. In this paper, we also present a complexity analysis of the ADS incorporating dependence on the B-spline basis of order p.

Słowa kluczowe

EN

finite element method; isogeometric analysis; Alternating Direction Solver

• Wydawca: Wydawnictwa AGH
• Czasopismo: Computer Science
• Rocznik: 2017
• Tom: Vol. 18 (2)
• Strony: 117--128
• Opis fizyczny: Bibliogr. 21 poz., rys.

Twórcy

autor

Łoś M.

• AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Department of Computer Science, Krakow, Poland

autor

Paszyński M.

• AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Department of Computer Science, Krakow, Poland

Bibliografia

• [1] Anderson E., Bai Z., Dongarra J., Greenbaum A., McKenney A., Croz J.D., Hammerling S., Demmel J., Bischof C., Sorensen D.: LAPACK: A portable linear algebra library for high-performance computers. In: Proceedings of the 1990 ACM/IEEE Conference on Supercomputing , Supercomputing ’90, IEEE Computer Society Press, pp. 2–11, Los Alamitos, CA, USA, 1990. http://dl . acm . org/ citation . cfm?id=110382 . 110385 .
• [2] Birkhoff G., Varga R.S., Young D.: Alternating Direction Implicit Methods, Advances in Computers , vol. 3, pp. 189–273, 1962.
• [3] Collier N., Pardo D., Dalcin L., Paszyński M., Calo V.M.: The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers, Computer Methods in Applied Mechanics and Engineering , vol. 213–216, pp. 353–361, 2012.
• [4] Cottrell J.A., Hughes T.J.R., Bazilevs Y.: Isogeometric Analysis: Toward Integration of CAD and FEA , Wiley Publishing, 1st ed., 2009.
• [5] Dalcin L., Collier N., Vignal P., Cortes A.M.A., Calo V.M.: PetIGA: High-Performance Isogeometric Analysis. In: arxiv , (1305.4452), 2013. http://arxiv . org/ abs/1305 . 4452 .
• [6] Douglas J., Rachford H.: On the numerical solution of heat conduction problems in two and three space variables, Transactions of American Mathematical Society , vol. 82, pp. 421–439, 1956.
• [7] Duff I.S., Reid J.K.: The Multifrontal Solution of Indefinite Sparse Symmetric Linear, ACM Transactions on Mathematical Software (TOMS) , vol. 9(3), pp. 302–325, 1983.
• [8] Gao L., Calo V.M.: Fast isogeometric solvers for explicit dynamics, Computer Methods in Applied Mechanics and Engineering , vol. 274, pp. 19–41, 2014.
• [9] Gao L., Calo V.M.: Preconditioners based on the Alternating-Direction-Implicit algorithm for the 2D steady-state diffusion equation with orthotropic heterogeneous coefficients, Journal of Computational and Applied Mathematics , vol. 273, pp. 274–295, 2015.
• [10] Golub G.G., Loan C.F.V.: Matrix Computations , The Johns Hopkins University Press, 4th ed., 2013.
• [11] Hawkins-Daarud A., Prudhomme S., Zee van der K.G., Oden J.T.: Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth, Journal of Mathematical Biology , vol. 67(6), pp. 1457–1485, 2013. http://dx . doi . org/10 . 1007/s00285-012-0595-9 .
• [12] Hawkins-Daarud A., Zee van der K.G., Tinsley Oden J.: Numerical simulation of a thermodynamically consistent four-species tumor growth model, International Journal for Numerical Methods in Biomedical Engineering , vol. 28(1), pp. 3–24, 2012, http://dx . doi . org/10 . 1002/cnm . 1467 .
• [13] Hughes T.J.R., Cottrell J.A., Bazilevs Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering , vol. 194(39–41), pp. 4135–4195, 2005.
• [14] Loś M., Woźniak M., Paszyński M., Dalcin L., Calo V.M.: Parallel alternating direction preconditioner for isogeometric simulations of explicit dynamics , 1st Pan-American Congress on Computational Mechanics, Buenos Aires, April 27–29.
• [15] Marsh D.: Applied Geometry for Computer Graphics and CAD , Springer Under- graduate Mathematics Series, Sprlinger-Verlag London, 2005.
• [16] Peaceman D., Rachford H.: The numerical solution of parabolic and elliptic differential equations, Journal of Society of Industrial and Applied Mathematics , vol. 3(1), pp. 28–41, 1955.
• [17] Piegl L., Tiller W.: The NURBS Book , 2nd ed., Springer-Verlag, New York, 1997.
• [18] Saad Y.: Iterative Methods for Sparse Linear Systems , 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2003.
• [19] Wachspress E., Habetler G.: An alternating-direction-implicit iteration technique, Journal of Society of Industrial and Applied Mathematics , vol. 8(2), pp. 403–424, 196.
• [20] Woźniak M., Loś M., Paszyński M., Dalcin L., Calo V.M.: Dynamics with matrices possesing Kronecker product structure, Procedia Computer Science , vol. 51, pp. 286–295, 2015.
• [21] Woźniak M., Loś M., Paszyński M., Dalcin L., Calo V.M.: Parallel fast isogeometric solvers for explicit dynamic, Computing and Informatics , vol. 32, pp. 1001–1026, 2015.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).