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Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method∗

Published online by Cambridge University Press:  11 March 2014

Ivo Babuška ,
Xu Huang  and
Robert Lipton
Ivo Babuška
Affiliation:
Institute for Computational Engineering and Science and Department of Aerospace Engineering, University of Texas, Austin, TX 78712, USA. [email protected]
Xu Huang
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA; [email protected]
Robert Lipton
Affiliation:
Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA; [email protected]
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Abstract

A multiscale spectral generalized finite element method (MS-GFEM) is presented for the solution of large two and three dimensional stress analysis problems inside heterogeneous media. It can be employed to solve problems too large to be solved directly with FE techniques and is designed for implementation on massively parallel machines. The method is multiscale in nature and uses an optimal family of spectrally defined local basis functions over a coarse grid. It is proved that the method has an exponential rate of convergence. To fix ideas we describe its implementation for a two dimensional plane strain problem inside a fiber reinforced composite. Here fibers are separated by a minimum distance however no special assumption on the fiber configuration such as periodicity or ergodicity is made. The implementation of MS-GFEM delivers the discrete solution operator using the same order of operations as the number of fibers inside the computational domain. This implementation is optimal in that the number of operations for solution is of the same order as the input data for the problem. The size of the MS-GFEM matrix used to represent the discrete inverse operator is controlled by the scale of the coarse grid and the convergence rate of the spectral basis and can be of order far less than the number of fibers. This strategy is general and can be applied to the solution of very large FE systems associated with the discrete solution of elliptic PDE.

Keywords

Generalized finite elementsmultiscale methodspectral methodheterogeneous mediafiber reinforced composites
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 493 - 515
Copyright
© EDP Sciences, SMAI, 2014

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References

Arbogast, T. and Boyd, K.J., Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44 (2006) 1150–1171. Google Scholar
I. Babuška, Homogenization and Its Application. Mathematical and Computational Problems. SYNSPADE 1975, Numer. Solution Part. Differ. Eqs. lll, edited by B. Hubbard. Academic Press (1976) 89–116.
Babuška, I., Anderson, B., Smith, P. and Levin, K., Damage analysis of fiber composites, Part I Statistical analysis on fiber scale. Comput. Methods Appl. Mech Engrg. 172 (1999) 27–77. Google Scholar
Babuška, I., Banerjee, U. and Osborn, J., Generalized Finite Element Methods-Main Ideas, Results and Perspective. Int. J. Comput. Methods 1 (2004) 67–103. Google Scholar
Babuška, I. and Banerjee, U., Stable Generalized Finiter Element Methods (SGFEM). Comput. Meth. Appl. Mech. Eng. 201-204 (2012) 91–111. Google Scholar
Babuška, I., Caloz, G. and Osborn, J.E., Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945–981. Google Scholar
Babuška, I. and Lipton, R., Optimal local approximation spaces for Generalized Finite Element Methods with application to multiscale problems. Multiscale Model. Simul., SIAM 9 (2011) 373–406. Google Scholar
Babuška, I. and Lipton, R., L 2 global to local projection: an approach to multiscale analysis. M3AS 21 (2011) 2211–2226. Google Scholar
Babuška, I. and Melenk, J., The Partition of Unity Finite Element Method. Internat. J. Numer. Methods Engrg. 40 (1997) 727–758. Google Scholar
Babuška, I., Osborn, J, E., Generalized finite element methods:Their performance and their relation to the mixed methods. SIAM, J. Numer. Anal. 20 (1983) 510–536. Google Scholar
I. Babuška and J.E. Osborn, Eigenvalue Problems. Handbook of Numerical Analysis, Finite Element Methods (Part 1), Vol. II, edited by P.G. Ciarlet and J.L. Lions. Elsevier Science Publishers, Amsterdam (1991).
N.S. Bakhvalov and G. Panasenko, Homogenization Processes in Periodic Media. Nauka, Moscow (1984).
R.M. Barrer, Diffusion and permeation in heterogenous media. Diffusion in Polymers, edited by J. Crank, G.S. Park. Academic Press (1968).
Bebendorf, M. and Hackbusch, W., Existence of ℋ-matrix approximants to the inverse FE-matrix of elliptic operators with L ∞ coefficients. Numer. Math. 95 (2003) 1–28. Google Scholar
Berlyand, L. and Owhadi, H., Flux norm approach to finite dimensional homogenization approximations with nonseparated length scales and high contrast. Arch. Rat. Mech. Anal. 198 (2010) 177–221. Google Scholar
A. Besounssan, J.L. Lions and G.C. Papanicolau, Asymptotic Analysis for Periodic Structures. North Holland Pub., Amsterdam (1978).
Burchuladze, T. and Rukhadze, R., Asymptotic distribution of eigenfunctions and eigenvalues of the basic boundary-contact oscillation problems of the classical theory of elasticity. Georgian Math. J. 6 (1999) 107–126. Google Scholar
Chams, C.C. and Sendeckij, G.P., Critique on theories predicting thermoelastic properties of fibrous composites. J. Comput. Mat. 2 (1968) 332–358. Google Scholar
Engquist, W. E, B., The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. Google Scholar
Ming, Weinan E, P. and Zhang, P., Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc. 18 (2005) 121–156. Google Scholar
Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods. Springer (2009).
Efendiev, Y.R., Hou, T.Y. and Wu, X.H., Convergence of a nonconforming mutiscale finite element method. SIAM J. Numer. Anal. 37 (2000) 888–910. Google Scholar
Efendiev, Y. and Hou, T., Multiscale finite element methods for porous media flows and their applications. Appl. Numer. Math. 57 (2007) 577–596. Google Scholar
Engquist, B. and Souganidis, P.E., Asymptotic and numerical homogenization. Acta Numer. 17 (2008) 147–190. Google Scholar
Fish, J. and Huan, Z., Multiscale enrichment based on partition unity. Int. J. Num. Mech. Eng. 62 (2005) 1341–1359. Google Scholar
S.K. Garg, V. Svalbonas and G.A. Gurtman, Analysis of Structural Composite Materials. Marcel Dekker, New York (1973).
Grasedyck, L., Greff, I. and Sauter, S., The AL basis for the solution of elliptic problems in heterogeneous media. Multiscale Model. Simul. 10 (2012) 245–258. Google Scholar
L.J. Gurtman and R.H. Krock, Composite Materials, in vol II of Mechanics of composite materials, edited by G.P. Sendeckyj. Academic Press (1974).
Z. Hashin, Theory of Fiber reinforced materials, NASA Report CR-1974 (1972) 1–704.
Hou and Xiao-Hui Wu, T.Y., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. Google Scholar
Hou, T.Y., Wu, Xiao-Hui and Zhang, Yu, Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Commun. Math. Sci. 2 (2004) 185–205. Google Scholar
Hughes, T.J.R., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulaion, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401. Google Scholar
Hughes, T.J.R., Feijoo, G.R., Mazzei, L. and Quincy, J.B., The variational multiscale method. A Paradigm for computational mechanics. Comput. Meth. Appl. Mech. Eng. 166 (1998) 3–24. Google Scholar
Lichtenecker, K., Die Electrizitatskonstante naturlicher und kustlicher Mischkorper. Phys. Zeitschr. XXVII (1926) 115–158. Google Scholar
Malquist, A., Multiscale methods for elliptic problems. Multiscale Model. Simul. 9 (2011) 1064–1086. Google Scholar
Masotti, G.F., Discussione analitica sul influenze che L’azione di mezo dialettrico hu sulla distribuziione dell’ electtricita alla superficie di pin corpi ellecttici diseminati in esso. Mem.Di Math. et di Fisica in Modena 24 (1850) 49. Google Scholar
G. Maxwell, Trestise on Electricity and Magnetisum, vol. 1. Oxford Univ. Press (1873) 62.
Melenk, J. and Babuška, I., The Partion of Unity Method Basic Theory and Applications, Comput. Meth. Appl. Mech. Eng. 139 (1996) 289–314. Google Scholar
Melenk, J.M., On n-widths for elliptic problems. J. Math. Anal. Appl. 247 (2000) 272–289. Google Scholar
G.W. Milton. The Theory of Composites. Cambridge University Press, Cambridge (2002).
F. Murat, H-convergence, Séminaire d’Analyse Fonctionelle et Numérique de l’Université d’Alger, mimeographed notes, 1978. L. Tartar Cours Peccot, College de France (1977). Translated into English as F. Murat L. Tartar, H- convergence, in Topics in the Mathematical Modeling of Composite Materials, Progress in Nonlinear Differential Equations and their Applications, in vol. 31, edited by A.V. Cherkaev, R.V. Kohn. Birkhäuser, Boston (1997) 21–43.
Nolen, J., Papanicolaou, G. and Pironneau, O., A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul. 7 (2008) 171–196. Google Scholar
Owhadi, H. and Zhang, L., Metric-based upscaling. Commun. Pure Appl. Math. 60 (2007) 675–723. Google Scholar
Owhadi, H. and Zhang, L., Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast. Multiscale Model. Simul. 9 (2011) 1373–1398. Google Scholar
A. Pinkus, n-Widths in Approximation Theory. Springer-Verlag, Berlin, Heidelberg, New York 7 (1985).
S.D. Poisson, Second mem. sur la theorie de magnetism, Mem. de L Acad. de France (1822) 5.
Rayleigh, J.W., On the influence of obstacles in rectangular order upon the properties of the medium. Philos. Mag. 50 (1892) 481. Google Scholar
E. Sanchez-Palencia. Non-Homogeneous Media and Vibration Theory, in vol. 127 of Lecture Notes in Physics. Springer-Verlag (1980).
Spagnolo, S., Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore. Ann Scu. Norm. Pisa 21 (1967) 657–699. Google Scholar
Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa 22 (1968) 517–597. Google Scholar
S. Spagnolo, Convergence in Energy for Elliptic Operators, edited by B. Hubbard. Numer. Solutions Partial Differ Eqs. III, (Synspade 1975, College Park, Maryland 1975). Academic Press, New York (1975).
W. Streider and R. Aris, Variational Methods Applied to Problems of Diffusion and Reaction, Springer Tracts in Natural Philosophy. Springer-Verlag (1973).
Strouboulis, T., Zhang, L. and Babuška, I, Generalized finite element method using mesh-based handbooks application to problem in domains with many voids. Comput. Methods Appl. Mechanics Engrg. 192 (2003) 3109–3161. Google Scholar
Strouboulis, T., Babuška, I. and Copps, K., The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Engrg. 181 (2001) 43–69. Google Scholar
Strouboulis, T., Zhang, L. and Babuška, I., p-version of generalized FEM using mesh based handbooks with applications to multiscale problems. Int. J. Num. Meth. Engrg. 60 (2004) 1639–1672. Google Scholar
S. Torquato, Random Heterogeneous Materials, Microstructure and Macroscopic Properties. Springer, New York (2002).