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A Second Order Finite-Difference Ghost-Point Method for Elasticity Problems on Unbounded Domains with Applications to Volcanology

Published online by Cambridge University Press:  03 June 2015

Armando Coco ,
Gilda Currenti ,
Ciro Del Negro  and
Giovanni Russo
Armando Coco*
Affiliation:
Dipartimento di Scienze della Terra e Geoambientali, Università di Bari Aldo Moro, Bari, Italy Bristol University, Queens Road, Bristol BS8 1RJ, United Kingdom
Gilda Currenti*
Affiliation:
Istituto Nazionale di Geofisica e Vulcanologia, Italy
Ciro Del Negro*
Affiliation:
Istituto Nazionale di Geofisica e Vulcanologia, Italy
Giovanni Russo*
Affiliation:
Dipartimento di Matematica e Informatica, Università di Catania, Catania, Italy
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains. The technique is based on a smooth coordinate transformation, which maps an unbounded domain into a unit square. Arbitrary geometries are defined by suitable level-set functions. The equations are discretized by classical nine-point stencil on interior points, while boundary conditions and high order reconstructions are used to define the field variables at ghost-points, which are grid nodes external to the domain with a neighbor inside the domain. The linear system arising from such discretization is solved by a multigrid strategy. The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources. The method is suitable to treat problems in which the geometry of the source often changes (explore the effects of different scenarios, or solve inverse problems in which the geometry itself is part of the unknown), since it does not require complex re-meshing when the geometry is modified. Several numerical tests are successfully performed, which asses the effectiveness of the present approach.

Keywords

74B0565N0665N5574S2074G15Linear elasticityCauchy-Navier equationsground deformationunbounded domaincoordinate transformation methodCartesian gridghost pointslevel-set methodsmultigrid
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 4 , October 2014 , pp. 983 - 1009
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Alshin, A. B.Application of quasi-uniform grids for numerical solution of initial boundary value problems in unbounded domains. In Proceedings of the International Conference on Differential Equations, pages 1024–1026, Hasselt, Belgium, July 1997. yorld Scientific.Google Scholar
[2]Alshina, E., Kalitkin, N., and Panchenko, S.Numerical solution of boundary value problem in unlimited area. Math. Modelling, 14:10–22, 2002. (in Russian).Google Scholar
[3]Boyd, J. P.The optimization of convergence for Chebyshev polynomial methods in an unbounded domain. Journal of Computational Physics, 45:43–79, 1982.Google Scholar
[4]Brandt, A.Guide to multigrid developments. In Hackbusch, W. and Trottenberg, U., editors, Multigrid Methods, Lectures Notes in Mathematics, volume 960, pages 220–312, Berlin, 1982. ypringer.Google Scholar
[5]Briggs, W. L., Henson, V. E., and McCormick, S. F.A Multigrid Tutorial. SIAM, 2000.Google Scholar
[6]Burger, M. and Osher, S.A survey on level set methods for inverse problems and optimal design. European Journal of Applied Mathematics, 16:263–301, 2005.Google Scholar
[7]Castillo, J. E.Mathematical Aspects of Numerical Grid Generation. Frontiers in Applied Mathematics, 1991.Google Scholar
[8]Cayol, V. and Cornet, H. C.Effects of topography on the interpretation of the deformation field of prominent volcanoes - Application to Etna. Geophys. Res. Lett., 25:1979–1982, 1998.Google Scholar
[9]Coco, A. and Russo, G.A Ghost-Cell Finite-Difference Multigrid approach for systems of PDE’s in arbitrary domains. Submitted.Google Scholar
[10]Coco, A. and Russo, G.Finite-Difference Ghost-Point Multigrid Methods on Cartesian Grids for Elliptic Problems in Arbitrary Domains. Journal of Computational Physics, 241:464–501, 2013.CrossRefGoogle Scholar
[11]Coco, A., Russo, G., and Semplice, M.Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction. Submitted.Google Scholar
[12]B., Comsol AComsol multiphysics 4.3, 2012. ytockholm, Sweden.Google Scholar
[13]Currenti, G., Negro, C. Del, Stefano, A. Di, and Napoli, R.Numerical simulation of stress induced piezomagnetic fields at Etna volcano. Geophysical Journal International, 179:1469–1476, 2009.Google Scholar
[14]Currenti, G., Negro, C. Del, and Ganci, G.Modelling of ground deformation and gravity fields using finite element method: an application to Etna volcano. Geophysical Journal International, 169:775–786, 2007.Google Scholar
[15]Currenti, G., Negro, C. Del, Ganci, G., and Scandura, D.3D numerical deformation model of the intrusive event forerunning the 2001 Etna eruption. Physics of the Earth and Planetary Interiors, 168:88–96, 2008.CrossRefGoogle Scholar
[16]Currenti, G., Negro, C. Del, Ganci, G., and Williams, C. A.Static stress changes induced by the magmatic intrusions during the 2002-2003 Etna eruption. Journal of Geophysical Research, 113, 2008.Google Scholar
[17]de Hoog, F. R. and Weiss, R.An approximation theory for boundary value problems on infinite intervals. Computing, 24:227–239, 1980.Google Scholar
[18]Negro, C. Del, Currenti, G., and Scandura, D.Temperature-dependent viscoelastic modeling of ground deformation: application to Etna volcano during the 1993-1997 inflation period. Physics of the Earth and Planetary Interiors, 172:299–309, 2009.CrossRefGoogle Scholar
[19]Chéné, A. du, Min, C., and Gibou, F.Second-Order Accurate Computation of Curvatures in a Level Set Framework Using Novel High Order Reinitialization Schemes. Journal of Scientific Computing archive, 35:114–131, 2008.Google Scholar
[20]Fazio, R. and Jannelli, A.Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals. Journal of Computational and Applied Mathematics, 269:14–23, 2014.Google Scholar
[21]Fung, Y.Boundations of solid Mechanics. Prentice-Hall, Englewood Cliffs, 1965.Google Scholar
[22]Givoli, D.Numerical methods for problems in infinite domains. Elsevier, Amsterdam, 1992.Google Scholar
[23]Grosch, C. E. and Orszag, S. A.Numerical solution of problems in unbounded regions: coordinate transforms. Journal of Computational Physics, 25:273–295, 1977.CrossRefGoogle Scholar
[24]Guo, B.-Y.Jacobi spectral approximations to differential equations on the half line. Journal of Computational Mathematics, 18:95–112, 2000.Google Scholar
[25]Han, D. and Huang, Z.A class of artificial boundary conditions for heat equation in unbounded domains. Comp. Math. Appl., 43:889–900, 2002.Google Scholar
[26]Hend Ben Ameur, M. B., and Hackl, B.Level set methods for geometric inverse problems in linear elasticity. Inverse Problems, 20:673–696, 2004.Google Scholar
[27]Jaeger, J., Cook, N., and Zimmerman, R.Fundamentals of Rock Mechanics (4th Edition). Blackwell Publishing, Oxford, 2007.Google Scholar
[28]Kearey, P. and Brooks, M.An introduction to geophysical exploration. Second edition. Blackwell Scientific Publications, Oxford, 1991.Google Scholar
[29]Koleva, M. N.Numerical Solution of the Heat Equation in Unbounded Domains Using Quasi-uniform Grids. Large-Scale Scientific Computing, Lecture Notes in Computer Science, 3743:509–517, 2006.Google Scholar
[30]Lentini, M. and Keller, H. B.Boundary value problems on semi-infinite intervals and their numerical solutions. SIAM J. Numer. Anal., 17:577–604, 1980.Google Scholar
[31]Lohner, R., Cebral, J. R., Camelli, F. E., Appanaboyina, S., Baum, J. D., Mestreau, E. L., and Soto, O. A.Adaptive embedded and immersed unstructured grid techniques. Comput. Methods Appl. Mech. Engrg., 197:217–297, 2008.CrossRefGoogle Scholar
[32]Maleki, M., Hashim, I., and Abbasbandy, S.Analysis of IVPs and BVPs on Semi-Infinite Domains via Collocation Methods. Journal of Applied Mathematics, 2012:21 pages, 2012.Google Scholar
[33]Markowich, P. A.A theory for the approximation of solution of boundary value problems on infinite intervals. SIAM J. Math. Anal., 13:484–513, 1982.Google Scholar
[34]Markowich, P. A.Analysis of boundary value problems on infinite intervals. SIAM J. Math. Anal., 14:11–37, 1983.CrossRefGoogle Scholar
[35]Masterlark, T., Feigl, K. L., Haney, M., Stone, J., Thurber, C., and Ronchin, E.Nonlinear estimation of geometric parameters in FEMs of volcano deformation: Integrating tomography models and geodetic data for Okmok volcano, Alaska. J. Geophys. Res., 117:B02407, 2012.Google Scholar
[36]Min, C. and Gibou, F.A second order accurate level set method on non-graded adaptive cartesian grids. Journal of Computational Physics, 225:300–321, 2007.Google Scholar
[37]Osher, S. and Fedkiw, R.Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag New York, Applied Mathematical Sciences, 2002.Google Scholar
[38]Osher, S. and Sethian, J.Fronts Propagating with Curvature Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations. Journal of Computational Physics, 79:12–49, 1988.Google Scholar
[39]Quang, D. and Hung, T. D.Method of Infinite System of Equations for Problems in Unbounded Domains. Journal of Applied Mathematics, 2012:17 pages, 2012.Google Scholar
[40]Russo, G. and Smereka, P.A remark on computing distance functions. Journal of Computational Physics, 163:51–67, 2000.Google Scholar
[41]Sethian, J.Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge University Press, 1999.Google Scholar
[42]Sussman, M., Smereka, P., and Osher, S.A level set approach for computing solutions to incompressible 2-phase flow. Journal of Computational Physics, 114:146–159, 1994.CrossRefGoogle Scholar
[43]Trasatti, E., Giunchi, C., and Agostinetti, N. P.Numerical inversion of deformation caused by pressure sources: application to Mount Etna (Italy). Geophys. J. Int., 172:873–884, 2007.Google Scholar
[44]Tsynkov, S.Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math., 27:465–532, 1989.CrossRefGoogle Scholar
[45]Trottemberg, U., Oosterlee, C., and Schuller, A.Multigrid. Academic Press, 2000.Google Scholar
[46]Williams, C. and Wadge, G.The effects of topography on magma chamber deformation models: Application to Mt. Etna and radar interferometry. Geophys. Res. Lett., 25:1549–1552, 1998.Google Scholar
[47]Williams, C. A. and Wadge, G.An accurate and efficient method for including the effects of topography in three-dimensional elastic models of ground deformation with applications to radar interferometry. J. Geophys. Res., 105:103–120, 2000.Google Scholar
[48]Wu, X. and Sun, Z.Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions. Appl. Numer. Math., 50:261–277, 2004.CrossRefGoogle Scholar
[49]Yukutake, T. and Tachinaka, H.Geomagnetic variation associated with stress change within a semi-infinite elastic Earth caused by a cylindrical fource source. Bull. Earthquake Res. Inst., Tokyo, 45:785–798, 1967.Google Scholar
[50]Zienkiewicz, O., Emson, C., and Bettess, P.A Novel Boundary Infinite Element. International Journal for Numerical Methods in Engineering, 19:393–404, 1983.Google Scholar