Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation

Zhu, Shengfeng; Dedè, Luca; Quarteroni, Alfio

doi:10.1051/m2an/2016056

Zhu, Shengfeng ¹ ; Dedè, Luca ² ; Quarteroni, Alfio ^2,³

¹ Department of Mathematics and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China.
² CMCS-Chair of Modeling and Scientific Computing, MATHICSE-Mathematics Institute of Computational Science and Engineering, EPFL-École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland.
³ MOX-Modeling and Scientific Computing, Department of Mathematics, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milano, Italy.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1197-1221.

Résumé

Discretization methods such as finite differences or finite elements were usually employed to provide high fidelity solution approximations for reduced order modeling of parameterized partial differential equations. In this paper, a novel discretization technique-Isogeometric Analysis (IGA) is used in combination with proper orthogonal decomposition (POD) for model order reduction of the time parameterized acoustic wave equations. We propose a new fully discrete IGA-Newmark-POD approximation and we analyze the associated numerical error, which features three components due to spatial discretization by IGA, time discretization with the Newmark scheme, and modes truncation by POD. We prove stability and convergence. Numerical examples are presented to show the effectiveness and accuracy of IGA-based POD techniques for the model order reduction of the acoustic wave equation.

Reçu le : 2016-01-26
Accepté le : 2016-08-26

MR Zbl

DOI : 10.1051/m2an/2016056

Classification : 65M12, 65M15, 65M60
Mots clés : Isogeometric analysis, proper orthogonal decomposition, reduced order modeling, acoustic wave equation

Affiliations des auteurs :

Zhu, Shengfeng ¹ ; Dedè, Luca ² ; Quarteroni, Alfio ^{2, 3}

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     title = {Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1197--1221},
     publisher = {EDP-Sciences},
     volume = {51},
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Zhu, Shengfeng; Dedè, Luca; Quarteroni, Alfio. Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1197-1221. doi : 10.1051/m2an/2016056. http://archive.numdam.org/articles/10.1051/m2an/2016056/

Bibliographie
Cité par

R. Abgrall and D. Amsallem, Robust model reduction by $L$ -norm minimization and approximation via dictionaries: application to linear and nonlinear hyperbolic problems. Preprint (2015). | arXiv

D. Amsallem and U. Hetmaniuk, Error estimates for Galerkin reduced-order models of the semi-discrete wave equation. ESAIM: M2AN 48 (2014) 135–163. | DOI | Numdam | MR | Zbl

F. Auricchio, F. Calabrò, T.J.R. Hughes, A. Reali and G. Sangalli, A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 249/252 (2012) 15–27. | DOI | MR | Zbl

A. Bartezzaghi, L. Dedè and A. Quarteroni, Isogeometric analysis of high order partial differential equations on surfaces. Comput. Methods Appl. Mech. Engrg. 295 (2015) 446–469. | DOI | MR | Zbl

Y. Bazilevs, L. Beirão De Veiga, J.A. Cottrell, T.J.R. Hughes and G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for $h$ -refined meshes. Math. Models Methods Appl. Sci. 16 (2006) 1031–1090. | DOI | MR | Zbl

L. Beirão De Veiga, A. Buffa, J. Rivas and G. Sangalli, Some estimates for $h$ - $p$ - $k$ -refinement in isogeometric analysis. Numer. Math. 118 (2011) 271–305. | DOI | MR | Zbl

G. Berkooz, P. Holmes and J.L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1993) 539–575. | DOI | MR

A. Buffa, G. Sangalli and R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation. J. Comput. Phys. 199 (2010) 1143–1152. | MR | Zbl

D. Chapelle, A. Gariah and J. Sainte-Marie, Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples. ESAIM: M2AN 46 (2012) 731–757. | DOI | Numdam | MR | Zbl

J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons (2009). | MR

J.A. Cottrell, T.J.R. Hughes and A. Reali, Studies of refinement and continuity in Isogeometric structural analysis. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4160−4183. | DOI | Zbl

W. Dahmen, C. Plesken and G. Welper, Double greedy algorithms: reduced basis methods for transport dominated problems. ESAIM: M2AN. 48 (2014) 623–663. | DOI | Numdam | MR | Zbl

R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Evolution Problems I. Vol. 5 and II. Vol. 6. Springer Verlag, Berlin (1992). | MR | Zbl

L. Dedè, M.J. Borden and T.J.R. Hughes, Isogeometric analysis for topology optimization with a phase field model. Arch. Comput. Methods Eng. 19 (2012) 427–465. | DOI | MR | Zbl

L. Dedè, C. Jäggli and A. Quarteroni, Isogeometric numerical dispersion analysis for elastic wave propagation. Comput. Methods Appl. Mech. Engrg. 284 (2015) 320–348. | DOI | MR | Zbl

J.A. Evans, Y. Bazilevs, I. Babuška and T.J.R. Hughes, $n$ -widths, sup-infs, and optimality ratios for the $k$ -version of the isogeometric finite element method. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1726–1741. | DOI | MR | Zbl

C. De Falco, A. Reali and R. Vàzquez, GeoPDEs: a research tool for isogeometric analysis of PDEs. Adv. Eng. Softw. 42 (2011) 1020–1034. | DOI | Zbl

C. Grossmann, H.G. Roos and M. Stynes, Numerical Treatment of Partial Differential Equations. Springer-Verlag, Berlin (2007). | MR | Zbl

M.D. Gunzburger, J. Peterson and J.N. Shadid, Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1030–1047. | DOI | MR | Zbl

S. Herkt, M. Hinze and R. Pinnau, Convergence analysis of Galerkin POD for linear second order evolution equations. Electron. Trans. Numer. Anal. 40 (2013) 321–337. | Zbl

T. Henri and J.P. Yvon, Stability of the POD and convergence of the POD Galerkin method for parabolic problems, IRMAR No. 02-40 (2002).

P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Univ. Press, New York (1996). | Zbl

T.J.R. Hughes, J. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Engrg. 194 (2005) 4135–4195. | DOI | Zbl

T.J.R. Hughes, J.A. Evans and A. Reali, Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Engrg. 272 (2014) 290–320. | DOI | Zbl

T.J.R. Hughes, A. Reali and G. Sangalli, Efficient quadrature for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg. 199 (2010) 301–313. | DOI | Zbl

T. Iliescu and Z. Wang, Variational multiscale proper orthogonal decomposition: convection dominated convection-diffusion-reaction equations. Math. Comp. 82 (2013) 1357–1378. | DOI | Zbl

T. Iliescu and Z. Wang, Are the snapshot difference quotients needed in the proper orthogonal decomposition? SIAM J. Sci. Comput. 36 (2014) A1221–A1250. | DOI | MR | Zbl

K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117–148. | DOI | MR | Zbl

K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer Anal. 40 (2002) 492–515. | DOI | Zbl

S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods. Springer (2008). | Zbl

Z. Luo, J. Chen, I.M. Navon and X. Yang, Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary Navier–Stokes equations. SIAM. J. Numer. Anal. 47 (2008) 1–19. | DOI | Zbl

S. Morganti, F. Auricchio, D.J. Benson, F.I. Gambarin, S. Hartmann, T.J.R. Hughes, A. Reali, Patient-specific isogeometric structural analysis of aortic valve closure. Comput. Methods Appl. Mech. Engrg. 284 (2015) 508–520. | DOI | Zbl

L. Piegl and W. Tiller, The NURBS book. Springer-Verlag, New York (1997).

A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1 (2011) 1–44. | Zbl

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Problems. Springer Verlag, Berlin (1997).

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations. Vol. 92, of Unitext. Springer (2016). | Zbl

D. Schillinger, L. Dedè, M.A. Scott, J.A. Evans, M.J. Borden, E. Rank and T.J.R. Hughes, An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput. Methods Appl. Mech. Engrg. 249-252 (2012) 116–150. | DOI | Zbl

D. Schillinger, S.J. Hossain, T.J.R. Hughes, Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 277 (2014) 1–45. | DOI | Zbl

A. Schmidt, A. Potschka, S. Körkel and H.G. Bock, Derivative-extended POD reduced-order modeling for parameter estimation. SIAM J. Sci. Comput. 35 (2013) A2696–A2717. | DOI | Zbl

J. Singler, New POD error expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs. SIAM J. Numer. Anal. 52 (2014) 852–876. | DOI | Zbl

A. Tagliabue, L. Dedè and A. Quarteroni, Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. Comput. Fluids 102 (2014) 277–303. | DOI | Zbl

T. Taddei, S. Perotto and A. Quarteroni, Reduced basis techniques for nonlinear conservation laws. ESAIM: M2AN 49 (2015) 787–814. | DOI | Numdam | Zbl

S. Volkwein, Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling. Lecture Notes, Universität Konstanz (2013).

K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition. AIAA 40 (2002) 2323–2330. | DOI

S. Zhu, L. Dedè and A. Quarteroni, Isogeometric analyis and proper orthogonal decomposition for parabolic problems. Numer. Math. 135 (2017) 333–370. | DOI | Zbl

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