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January  2021, 26(1): 61-79. doi: 10.3934/dcdsb.2020351

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

Gang Bao 1,, , Mingming Zhang 1, , Bin Hu 1, and  Peijun Li 2,

1. 

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
2. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

* Corresponding author: Gang Bao

Received  July 2020 Revised  October 2020 Published  January 2021 Early access  November 2020

Fund Project: The first author is supported in part by an NSFC Innovative Group Fund (No.11621101). The last author is supported in part by the NSF grant DMS-1912704

This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Keywords: Acoustic scattering problem, adaptive finite element method, transparent boundary condition, a posteriori error estimates.
Mathematics Subject Classification: Primary: 65N30, 78A45, Secondary: 35Q60, 78M10.
Citation: Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
References:
[1] I. Babuška and A. Aziz, Survey Lectures on mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Application to the Partial Differential Equations, ed. by A. Aziz, Academic Press, New York, 1972. 
[2]

G. Bao, R. Delgadillo, G. Hu, D. Liu and S. Luo, Modeling and computation of nano-optics, CSIAM Trans. Appl. Math..

[3]

G. Bao, Y. Gao and P. Li, Time-domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 229 (2018), 835-884.  doi: 10.1007/s00205-018-1228-2.

[4]

G. Bao, G. Hu and D. Liu, An h-adaptive finite element solver for the calculations of the electronic structures, J. Comput. Phys., 231 (2012), 4967-4979.  doi: 10.1016/j.jcp.2012.04.002.

[5]

G. Bao, P. Li and H. Wu, An adaptive edge element method with perfectly matched absorbing layers for wave scattering by periodic structures, Math. Comp., 79 (2010), 1-34.  doi: 10.1090/S0025-5718-09-02257-1.

[6]

G. Bao and H. Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations, SIAM J. Numer. Anal., 43 (2005), 2121-2143.  doi: 10.1137/040604315.

[7]

A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math., 33 (1980), 707-725.  doi: 10.1002/cpa.3160330603.

[8]

J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200.  doi: 10.1006/jcph.1994.1159.

[9]

Z. Chen and X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), 645-671.  doi: 10.1137/040610337.

[10]

Z. Chen and H. Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41 (2003), 799-826.  doi: 10.1137/S0036142902400901.

[11]

F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), 2061-2090.  doi: 10.1137/S1064827596301406.

[12]

D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983.

[13]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition, Springer, Berlin, New York, 1998. doi: 10.1007/978-3-662-03537-5.

[14]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.  doi: 10.2307/2005997.

[15]

Q. Fang, D. P. Nicholls and J. Shen, A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering, J. Comput. Phys., 224 (2007), 1145-1169.  doi: 10.1016/j.jcp.2006.11.018.

[16]

C. Geuzaine and J.-F. Remacle, GMSH: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009), 1309-1331.  doi: 10.1002/nme.2579.

[17]

M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243.  doi: 10.1006/jcph.1995.1210.

[18]

M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys., 201 (2004), 630-650.  doi: 10.1016/j.jcp.2004.06.012.

[19]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.  doi: 10.1017/S0962492900002890.

[20]

I. Harari and T. J. R. Hughes, Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. Methods Appl. Mech. Engrg., 97 (1992), 103-124.  doi: 10.1016/0045-7825(92)90109-W.

[21]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. Math., 121 (1985), 463-488.  doi: 10.2307/1971205.

[22]

X. Jiang, P. Li, J. Lv and W. Zheng, An adaptive finite element method for the wave scattering with transparent boundary condition, J. Sci. Comput., 72 (2017), 936-956.  doi: 10.1007/s10915-017-0382-2.

[23]

X. Jiang, P. Li and W. Zheng, Numerical solution of acoustic scattering by an adaptive DtN finite element method, Commun. Comput. Phys., 13 (2013), 1227-1244.  doi: 10.4208/cicp.301011.270412a.

[24]

J. Jin, The Finite Element Method in Electromagnetics, Second edition. New York, 2002.

[25]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, Springer International Publishing, 2015. doi: 10.1007/978-3-319-11086-8.

[26]

P. Li and X. Yuan, Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures, Comput. Methods Appl. Mech. Engrg., 360 (2020), 112722. doi: 10.1016/j.cma.2019.112722.

[27]

Y. Li, W. Zheng and X. Zhu, A CIP-FEM for high-frequency scattering problem with the truncated DtN boundary condition, CSIAM Trans. Appl. Math., 1 (2020), 530-560.  doi: 10.4208/csiam-am.2020-0025.

[28] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[29]

J.-C. Nédélec, Acoustic and Electromagnetic Equations Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001.

[30]

A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), 959-962.  doi: 10.2307/2005357.

[31]

E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math., 27 (1998), 533-557.  doi: 10.1016/S0168-9274(98)00026-9.

[32]

Z. Wang, G. Bao, J. Li, P. Li and H. Wu, An adaptive finite element method for the diffraction grating problem with transparent boundary condition, SIAM J. Numer. Anal., 53 (2015), 1585-1607.  doi: 10.1137/140969907.

[33]

X. Yuan, G. Bao and P. Li, An adaptive finite element DtN method for the open cavity scattering problems, CSIAM Trans. Appl. Math., 1 (2020), 316-345.  doi: 10.4208/csiam-am.2020-0013.

show all references

References:
[1] I. Babuška and A. Aziz, Survey Lectures on mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Application to the Partial Differential Equations, ed. by A. Aziz, Academic Press, New York, 1972. 
[2]

G. Bao, R. Delgadillo, G. Hu, D. Liu and S. Luo, Modeling and computation of nano-optics, CSIAM Trans. Appl. Math..

[3]

G. Bao, Y. Gao and P. Li, Time-domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 229 (2018), 835-884.  doi: 10.1007/s00205-018-1228-2.

[4]

G. Bao, G. Hu and D. Liu, An h-adaptive finite element solver for the calculations of the electronic structures, J. Comput. Phys., 231 (2012), 4967-4979.  doi: 10.1016/j.jcp.2012.04.002.

[5]

G. Bao, P. Li and H. Wu, An adaptive edge element method with perfectly matched absorbing layers for wave scattering by periodic structures, Math. Comp., 79 (2010), 1-34.  doi: 10.1090/S0025-5718-09-02257-1.

[6]

G. Bao and H. Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations, SIAM J. Numer. Anal., 43 (2005), 2121-2143.  doi: 10.1137/040604315.

[7]

A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math., 33 (1980), 707-725.  doi: 10.1002/cpa.3160330603.

[8]

J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200.  doi: 10.1006/jcph.1994.1159.

[9]

Z. Chen and X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), 645-671.  doi: 10.1137/040610337.

[10]

Z. Chen and H. Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41 (2003), 799-826.  doi: 10.1137/S0036142902400901.

[11]

F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), 2061-2090.  doi: 10.1137/S1064827596301406.

[12]

D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983.

[13]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition, Springer, Berlin, New York, 1998. doi: 10.1007/978-3-662-03537-5.

[14]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.  doi: 10.2307/2005997.

[15]

Q. Fang, D. P. Nicholls and J. Shen, A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering, J. Comput. Phys., 224 (2007), 1145-1169.  doi: 10.1016/j.jcp.2006.11.018.

[16]

C. Geuzaine and J.-F. Remacle, GMSH: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009), 1309-1331.  doi: 10.1002/nme.2579.

[17]

M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243.  doi: 10.1006/jcph.1995.1210.

[18]

M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys., 201 (2004), 630-650.  doi: 10.1016/j.jcp.2004.06.012.

[19]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.  doi: 10.1017/S0962492900002890.

[20]

I. Harari and T. J. R. Hughes, Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. Methods Appl. Mech. Engrg., 97 (1992), 103-124.  doi: 10.1016/0045-7825(92)90109-W.

[21]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. Math., 121 (1985), 463-488.  doi: 10.2307/1971205.

[22]

X. Jiang, P. Li, J. Lv and W. Zheng, An adaptive finite element method for the wave scattering with transparent boundary condition, J. Sci. Comput., 72 (2017), 936-956.  doi: 10.1007/s10915-017-0382-2.

[23]

X. Jiang, P. Li and W. Zheng, Numerical solution of acoustic scattering by an adaptive DtN finite element method, Commun. Comput. Phys., 13 (2013), 1227-1244.  doi: 10.4208/cicp.301011.270412a.

[24]

J. Jin, The Finite Element Method in Electromagnetics, Second edition. New York, 2002.

[25]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, Springer International Publishing, 2015. doi: 10.1007/978-3-319-11086-8.

[26]

P. Li and X. Yuan, Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures, Comput. Methods Appl. Mech. Engrg., 360 (2020), 112722. doi: 10.1016/j.cma.2019.112722.

[27]

Y. Li, W. Zheng and X. Zhu, A CIP-FEM for high-frequency scattering problem with the truncated DtN boundary condition, CSIAM Trans. Appl. Math., 1 (2020), 530-560.  doi: 10.4208/csiam-am.2020-0025.

[28] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[29]

J.-C. Nédélec, Acoustic and Electromagnetic Equations Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001.

[30]

A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), 959-962.  doi: 10.2307/2005357.

[31]

E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math., 27 (1998), 533-557.  doi: 10.1016/S0168-9274(98)00026-9.

[32]

Z. Wang, G. Bao, J. Li, P. Li and H. Wu, An adaptive finite element method for the diffraction grating problem with transparent boundary condition, SIAM J. Numer. Anal., 53 (2015), 1585-1607.  doi: 10.1137/140969907.

[33]

X. Yuan, G. Bao and P. Li, An adaptive finite element DtN method for the open cavity scattering problems, CSIAM Trans. Appl. Math., 1 (2020), 316-345.  doi: 10.4208/csiam-am.2020-0013.

Figure 1.  A schematic of octree data structure
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Figure 2.  Two geometries to avoid hanging points. (left) Twin-tetrahedron geometry. (right) Four-tetrahedron geometry
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Figure 3.  Mesh refinement on the surface (red points are redefined midpoints on the boundary)
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Figure 4.  Example 1: (left) initial mesh on the $ x_1 x_2 $-plane. (right) adaptive mesh on the $ x_1 x_2 $-plane
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Figure 5.  Example 1: quasi-optimality of the a priori and a posteriori error estimates
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Figure 6.  Example 2: (left) an adaptively refined mesh with 63898 elements. (right) quasi-optimality of the a posteriori error estimate
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Table 1.  The adaptive FEM-DtN algorithm
1 Given a tolerance $ \varepsilon > 0 $;
2 Choose $ R $, $ R' $ and $ N $ such that $ \varepsilon_{N}<10^{-8} $;
3 Construct an initial tetrahedral partition $ \mathcal{M}_h $ over $ \Omega $ and compute error estimators;
4 While $ \eta_K>\varepsilon $, do
5     mark $ K $, refine $ \mathcal{M}_h $, and obtain a new mesh $ \hat{\mathcal{M}}_h $.
6     solve the discrete problem on the $ \hat{\mathcal{M}}_h $.
7     compute the corresponding error estimators;
8 End while.
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Download as excel

1 Given a tolerance $ \varepsilon > 0 $;
2 Choose $ R $, $ R' $ and $ N $ such that $ \varepsilon_{N}<10^{-8} $;
3 Construct an initial tetrahedral partition $ \mathcal{M}_h $ over $ \Omega $ and compute error estimators;
4 While $ \eta_K>\varepsilon $, do
5     mark $ K $, refine $ \mathcal{M}_h $, and obtain a new mesh $ \hat{\mathcal{M}}_h $.
6     solve the discrete problem on the $ \hat{\mathcal{M}}_h $.
7     compute the corresponding error estimators;
8 End while.
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