zbMATH — the first resource for mathematics

Cell-centered finite-volume method for elastic deformation of heterogeneous media with full-tensor properties. (English) Zbl 07143638
Summary: We propose a cell-centered finite-volume method for the heterogeneous anisotropic linear elasticity problem. The internal traction vector is represented as a discrete flux on the interface. This ‘elastic’ flux is decomposed into a discrete two-point flux approximation and a semi-discrete transversal part. We introduce an interpolation method in the presence of discontinuity in the full-tensor material properties. The scheme yields an accurate reconstruction of the gradient of the displacement in each cell. The formulation is based on the assumption of linearity of the displacement, and we enforce continuity of the internal traction vector and the displacement at the interface. The treatment of the boundary conditions can be complicated in finite-volume methods. Here, we describe a general treatment of boundary conditions that does not entail the introduction of additional degrees of freedom. The finite-volume method is tested for a series of challenging elasticity problems.

74 Mechanics of deformable solids
65 Numerical analysis
Full Text: DOI
[1] Carcione, José M., Wave propagation in anisotropic, saturated porous media: Plane-wave theory and numerical simulation, J. Acoust. Soc. Am., 99, 5, 2655-2666 (1996)
[2] Shapiro, Serge A.; Dinske, Carsten; Kummerow, Jörn, Probability of a given-magnitude earthquake induced by a fluid injection, Geophys. Res. Lett., 34, 22, n/a-n/a (2007), L22314
[3] Zienkiewicz, Olgierd Cecil; Taylor, Robert Leroy; Taylor, Robert Lee, The Finite Element Method, vol. 3 (1977), McGraw-hill London · Zbl 0974.76003
[4] Demirdzic, Ismet; Martinovic, Dunja; Ivankovic, Alojz, Numerical simulation of thermal deformation in welded workpiece, Zavarivanje, 31, 5, 209-219 (1988)
[5] Demirdžić, Ismet; Muzaferija, Samir, Finite volume method for stress analysis in complex domains, Internat. J. Numer. Methods Engrg., 37, 21, 3751-3766 (1994) · Zbl 0814.73075
[6] Tuković, Željko; Ivanković, Alojz; Karač, Aleksandar, Finite-volume stress analysis in multi-material linear elastic body, Internat. J. Numer. Methods Engrg., 93, 4, 400-419 (2013) · Zbl 1352.74010
[7] Lemoine, Grady I.; Ou, M. Y.vonne; LeVeque, Randall J., High-resolution finite volume modeling of wave propagation in orthotropic poroelastic media, SIAM J. Sci. Comput., 35, 1, B176-B206 (2013) · Zbl 1342.74173
[8] LeVeque, Randall J., Finite-volume methods for non-linear elasticity in heterogeneous media, Internat. J. Numer. Methods Fluids, 40, 1-2, 93-104 (2002) · Zbl 1024.74044
[9] Nordbotten, Jan Martin, Cell-centered finite volume discretizations for deformable porous media, Internat. J. Numer. Methods Engrg., 100, 6, 399-418 (2014) · Zbl 1352.76072
[10] Nordbotten, Jan Martin, Convergence of a cell-centered finite volume discretization for linear elasticity, SIAM J. Numer. Anal., 53, 6, 2605-2625 (2015) · Zbl 1330.74171
[11] Eirik Keilegavlen, Jan Martin Nordbotten, Finite volume methods for elasti with weak symmetry, arXiv preprint arXiv:1512.01042, 2015. · Zbl 1327.76138
[12] Da Veiga, Lourenco Beirão, A mimetic discretization method for linear elasticity, ESAIM Math. Model. Numer. Anal., 44, 2, 231-250 (2010) · Zbl 1258.74206
[13] Da Veiga, Lourenco Beirão; Brezzi, Franco; Marini, Luisa Donatella, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal., 51, 2, 794-812 (2013) · Zbl 1268.74010
[14] Felippa, Carlos A.; Oñate, Eugenio, Stress, strain and energy splittings for anisotropic elastic solids under volumetric constraints, Comput. Struct., 81, 13, 1343-1357 (2003)
[15] Agélas, Léo; Eymard, Robert; Herbin, Raphaele, A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media, C. R. Acad. Sci., Paris I, 347, 11-12, 673-676 (2009) · Zbl 1166.65051
[16] Lauga, Eric; Brenner, Michael; Stone, Howard, Microfluidics: the no-slip boundary condition, (Springer Handbook of Experimental Fluid Mechanics (2007), Springer), 1219-1240
[17] Terekhov, Kirill M.; Mallison, Bradley T.; Tchelepi, Hamdi A., Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem, J. Comput. Phys., 330, 245-267 (2017) · Zbl 1380.65335
[18] Sleijpen, Gerard L. G.; Van der Vorst, Henk A.; Fokkema, Diederik R., BiCGstab (l) and other hybrid Bi-CG methods, Numer. Algorithms, 7, 1, 75-109 (1994) · Zbl 0810.65027
[19] Kaporin, Igor E., High quality preconditioning of a general symmetric positive definite matrix based on its UTU+UTR+RTU-decomposition, Numer. Linear Algebra Appl., 5, 6, 483-509 (1998) · Zbl 0969.65037
[20] Le Potier, Christophe, Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés, C. R. Acad. Sci., Paris I, 341, 12, 787-792 (2005) · Zbl 1081.65086
[21] Russo, Remigio, Mathematical Problems in Elasticity, vol. 38 (1996), World Scientific · Zbl 0838.00006
[22] Gain, Arun L.; Talischi, Cameron; Paulino, Glaucio H., On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 282, 132-160 (2014) · Zbl 1423.74095
[23] Barber, James R., Elasticity (1992), Springer · Zbl 0787.73001
[24] Alberty, Jochen; Carstensen, Carsten; Funken, Stefan A.; Klose, Roland, Matlab implementation of the finite element method in elasticity, Computing, 69, 3, 239-263 (2002) · Zbl 1239.74092
[25] Bramwell, Jamie; Demkowicz, Leszek; Gopalakrishnan, Jay; Qiu, Weifeng, A locking-free hp DPG method for linear elasticity with symmetric stresses, Numer. Math., 122, 4, 671-707 (2012) · Zbl 1283.74094
[26] Nečas, Jindřich; Štípl, Miloš, A paradox in the theory of linear elasticity, Apl. Mat., 21, 6, 431-433 (1976) · Zbl 0398.73013
[27] Štípl, Miloš, On the maximum principle in the linear-elasticity theory, Acta Univ. Carolin. Math. Phys., 19, 2, 65-68 (1978) · Zbl 0404.73004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.