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Integral equation methods for elliptic problems with boundary conditions of mixed type. (English) Zbl 1177.65176
Summary: Laplace’s equation with mixed boundary conditions, that is, Dirichlet conditions on parts of the boundary and Neumann conditions on the remaining contiguous parts, is solved on an interior planar domain using an integral equation method. Rapid execution and high accuracy is obtained by combining equations which are of Fredholm’s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique. Then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results. This opens up for the rapid and accurate solution of several elliptic problems of mixed type.
MSC:
65N38Boundary element methods (BVP of PDE)
References:
[1]Atkinson, K. E.: The numerical solution of integral equations of the second kind, (1997)
[2]Brenner, S. C.; Scott, L. Ridgway: The mathematical theory of finite element methods, (2008)
[3]Englund, J.: Stable algorithm for the stress field around a multiply branched crack, Int. J. Numer. meth. Eng. 63, No. 6, 926-946 (2005) · Zbl 1084.74058 · doi:10.1002/nme.1311
[4]Greengard, L.; Rokhlin, V.: A fast algorithm for particle simulations, J. comput. Phys. 73, No. 2, 325-348 (1987) · Zbl 0629.65005 · doi:10.1016/0021-9991(87)90140-9
[5]Greengard, L.; Kropinski, M. C.; Mayo, A.: Integral equation methods for Stokes flow and isotropic elasticity in the plane, J. comput. Phys. 125, No. 2, 403-414 (1996) · Zbl 0847.76066 · doi:10.1006/jcph.1996.0102
[6]Helsing, J.: On the numerical evaluation of stress intensity factors for an interface crack of a general shape, Int. J. Numer. meth. Eng. 44, No. 5, 729-741 (1999) · Zbl 0931.74078 · doi:10.1002/(SICI)1097-0207(19990220)44:5<729::AID-NME529>3.0.CO;2-A
[7]Helsing, J.; Ojala, R.: On the evaluation of layer potentials close to their sources, J. comput. Phys. 227, No. 5, 2899-2921 (2008) · Zbl 1135.65404 · doi:10.1016/j.jcp.2007.11.024
[8]Helsing, J.; Ojala, R.: Corner singularities for elliptic problems: integral equations, graded meshes, quadrature, and compressed inverse preconditioning, J. comput. Phys. 227, No. 20, 8820-8840 (2008) · Zbl 1152.65114 · doi:10.1016/j.jcp.2008.06.022
[9]Helsing, J.; Peters, G.: An efficient numerical algorithm for cracks partly in frictionless contact, SIAM J. Appl. math. 61, No. 2, 551-566 (2000) · Zbl 0990.74054 · doi:10.1137/S0036139999356934
[10]Ioakimidis, N. I.; Papadakis, K. E.; Perdios, E. A.: Numerical evaluation of analytic functions by Cauchy’s theorem, BIT numer. Math. 31, No. 2, 276-285 (1991) · Zbl 0737.65011 · doi:10.1007/BF01931287
[11]Martı&acute, A. E.; Nez-Castro; Gallego, R.: Tangential residual as error estimator in the boundary element method, Comput. struct. 83, No. 10/11, 685-699 (2005)
[12]Martinsson, P. G.; Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions, J. comput. Phys. 205, No. 1, 1-23 (2005) · Zbl 1078.65112 · doi:10.1016/j.jcp.2004.10.033
[13]Mikhlin, S. G.: Integral equations and their applications to certain problems in mechanics, Mathematical physics and technology (1964)
[14]Muskhelishvili, N. I.: Some basic problems of the mathematical theory of elasticity, (1953) · Zbl 0052.41402
[15]Muskhelishvili, N. I.: Singular integral equations, (1953) · Zbl 0051.33203
[16]Natroshvili, D.; Zazashvili, S.: Mixed type boundary value problems in the linear theory of elastic mixtures for bodies with interior cuts, Mem. differ. Eq. math. Phys. 42, 69-91 (2007) · Zbl 1147.35028
[17]Noda, N. A.; Xu, C. H.: Controlling parameter of the stress intensity factors for a planar interfacial crack in three-dimensional bimaterials, Int. J. Solids struct. 45, No. 3/4, 1017-1031 (2008) · Zbl 1167.74433 · doi:10.1016/j.ijsolstr.2007.09.013
[18]Ottosen, N.; Petersson, H.: Introduction to the finite element method, (1992) · Zbl 0806.73001
[19]Saad, Y.; Schultz, M. H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. stat. Comput. 7, No. 3, 856-869 (1986) · Zbl 0599.65018 · doi:10.1137/0907058
[20]Sethian, J. A.; Wilkening, J.: A numerical model of stress driven grain boundary diffusion, J. comput. Phys. 193, No. 1, 275-305 (2004) · Zbl 1117.74302 · doi:10.1016/j.jcp.2003.08.015
[21]D.I. Sherman, On the problem of plane strain in non-homogeneous media, in: W. Olszag (Ed.), Non-Homogeneity in Elasticity and Plasticity, Proceedings of I.U.T.A.M. Symposium, Warsaw, 1958, Pergamon Press, London, 1959. · Zbl 0097.17502
[22]Tlupova, S.; Cortez, R.: Boundary integral solutions of coupled Stokes and Darcy flows, J. comput. Phys. 228, No. 1, 158-179 (2009) · Zbl 1188.76232 · doi:10.1016/j.jcp.2008.09.011
[23]Tran-Cong, T.; Nguyen-Thien, T.; Phan-Thien, N.: Boundary element method based on a new second kind integral equation formulation, Eng. anal. Bound. elem. 17, No. 4, 313-320 (1996)
[24]Wei, Y. J.; Bower, A. F.; Gao, H. J.: Recoverable creep deformation and transient local stress concentration due to heterogeneous grain-boundary diffusion and sliding in polycrystalline solids, J. mech. Phys. solids 56, No. 4, 1460-1483 (2008) · Zbl 1171.74318 · doi:10.1016/j.jmps.2007.08.007