Commun. Comput. Phys., 16 (2014), pp. 983-1009.


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

Armando Coco 1*, Gilda Currenti 2, Ciro Del Negro 2, Giovanni Russo 3

1 Dipartimento di Scienze della Terra e Geoambientali, Universita di Bari Aldo Moro, Bari, Italy; Bristol University, Queen's Road, Bristol BS8 1RJ, United Kingdom.
2 Istituto Nazionale di Geofisica e Vulcanologia, Italy.
3 Dipartimento di Matematica e Informatica, Universita di Catania, Catania, Italy.

Received 21 July 2013; Accepted (in revised version) 1 April 2014
Available online 8 August 2014
doi:10.4208/cicp.210713.010414a

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.

AMS subject classifications: 74B05, 65N06, 65N55, 74S20, 74G15

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Key words: Linear elasticity, Cauchy-Navier equations, ground deformation, unbounded domain, coordinate transformation method, Cartesian grid, ghost points,level-set methods, multigrid.

*Corresponding author.
Email: Armando.Coco@bristol.ac.uk (A. Coco), gilda.currenti@ct.ingv.it (G. Currenti), ciro.delnegro@ct.ingv.it (C. Del Negro), russo@dmi.unict.it (G. Russo)
 

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