Abstract
A new approach (Domain-Element Local Integro-Differential-Equation Method -- DELIDEM) is developed and implemented for the solution of 2-D potential problems in materials with arbitrary continuous variation of the material parameters. The domain is discretized into conforming elements for the polynomial approximation and the local integro-differential equations (LIDE) are considered on subdomains determined by domain elements and collocated at interior nodes. At the boundary nodes, either the prescribed boundary conditions or the LIDE are collocated. The applicability and reliability of the method is tested for several numerical examples.
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T. Hirai, Functionally gradient materials, In: R.J. Brook(ed.), Material Science and Technology, Vol. 17B, Processing of Ceramics, Part 2. Weinheim, Germany (1993) pp. 292–341.
G.H. Paulino, Z.H. ** and R.H. Dodds Jr., Failure of functionally graded materials. In: B.L. Karihaloo and W.G. Knauss (eds.), Comprehensive Structural Integrity, Vol. 2, Chap. 13. Oxford: Elsevier Science Ltd. (2003) in print.
S. Suresh A. Mortensen (1998) Fundamentals of Functionally Graded Materials The Institute of Materials London 165
Y. Miyamoto W.A. Kaysser B.H. Rabin A. Kawasaki R.G. Ford (1999) Functionally Graded Materials,Design,Processing and Applications Kluwer Academic Publishers Dordrecht 330
J.H. Kim G.H. Paulino (2002) ArticleTitleIsoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials J. Appl. Mech. Trans. ASME 69 502–514 Occurrence Handle1110.74509 Occurrence Handle10.1115/1.1467094
M. Tanaka K. Tanaka (1980) ArticleTitleTransient heat conduction problems in inhomogeneous media discretized by means of boundary-volume element Nucl. Eng. Des. 60 381–387
V. Sladek J. Sladek I. Markechova (1993) ArticleTitleAn advanced boundary element method for elasticity in nonhomogeneous media Acta Mech. 97 71–90 Occurrence Handle0763.73070 Occurrence Handle1196536
V. Sladek J. Sladek (2003) ArticleTitleA new formulation for solution of boundary value problems using domain-type approximations and local integral equations Electr. J. Bound. Elem. 1 132–153 Occurrence Handle2164148
D.L. Clements (1980) ArticleTitleA boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients J. Aust. Math. Soc. Ser. B 22 218–228 Occurrence Handle0452.65070 Occurrence Handle594006
A.H.-D. Cheng (1984) ArticleTitleDarcy’s flow with variable permeability – a boundary integral solution Water Resour. Res. 20 980–984 Occurrence Handle10.1029/WR020i007p00980 Occurrence Handle1984WRR....20..980C
A.H.-D. Cheng, Heterogeneities in flows through porous media by boundary element method. In: Topics in Boundary Element Research Vol. 4. Applications to Geomechanics. (1987) pp. 1291–1344.
R.P. Shaw N. Makris (1992) ArticleTitleGreen’s functions for Helmholtz and Laplace equations in heterogeneous media Eng. Anal. Bound. Elem. 10 179–183
R.P. Shaw (1994) ArticleTitleGreen’s functions for heterogeneous media potential problems Eng. Anal. Bound. Elem. 13 219–221
W.T. Ang J. Kusuma D.L. Clements (1986) ArticleTitleA boundary element method for a second order elliptic partial differential equation with variable coefficients Eng. Anal. Bound. Elem. 18 311–316
L.J. Gray T. Kaplan J.D. Richardson G.H. Paulino (2003) ArticleTitleGreen’s functions and boundary integral analysis for exponentially grded materials: Heat conduction J. Appl. Mech. Trans. ASME 70 543–549 Occurrence Handle1110.74461 Occurrence Handle10.1115/1.1485753
P.A. Martin J.D. Richardson L.J. Gray J.R. Berger (2002) ArticleTitleOn Green’s function for a three-dimensional exponentially-graded elastic solid Proc. Roy. Soc. London A 458 1931–1947 Occurrence Handle1056.74017 Occurrence Handle1921946 Occurrence Handle2002RSPSA.458.1931A Occurrence Handle10.1098/rspa.2001.0952
Y.-S. Chan L.J. Gray T. Kaplan G.H. Paulino (2003) ArticleTitleGreen’s function for a two-dimensional exponentially-graded elastic medium Proc. Roy. Soc. London A 460 1689–1706 Occurrence Handle2067557 Occurrence Handle2004RSPSA.460.1689C
J. Sladek V. Sladek S.N. Atluri (2000) ArticleTitleLocal boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties Comp. Mech. 24 456–462 Occurrence Handle0961.74073 Occurrence Handle1830639 Occurrence Handle2000CompM..24..456S
J. Sladek, and V. Sladek, Local boundary integral equation method for heta conduction problem in an anisotropic medium. In: C.A. Herrera, (ed.), Advances in Computational and Experimental Engineering & Sciences CD – Proc. Conf. ICCES 2003, Corfu, Greece (2003).
J. Sladek V. Sladek Ch. Zhang (2003) ArticleTitleA local BIEM for analysis of transient heat conduction with nonlinear source terms in FGMs Eng. Anal. Bound. Elem. 28 1–11
J. Sladek V. Sladek Ch. Zhang (2003) ArticleTitleTransient heat conduction analysis in functionally graded materials by meshless local boundary integral equation method Comput. Mater. Sci. 28 494–504
J. Sladek V. Sladek J. Krivacek Ch Zhang (2003) ArticleTitleLocal BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids Comp. Mech. 32 169–176 Occurrence Handle1038.80510 Occurrence Handle2003CompM..32..169S
J. Sladek V. Sladek Ch Zhang (2003) ArticleTitleApplication of meshless Petrov-Galerkin (MLPG) method to elastodynamic problems in continuously nonhomogeneous solids Comp. Model. Eng. Sci. 4 637–647 Occurrence Handle1064.74178
V. Sladek J. Sladek (2000) ArticleTitleGlobal and local integral equations for potential problems in non-homogeneous media CTU Report 4 133–138
V. Sladek, J. Sladek and R. Van Keer: New integral equation approach to solution of diffusion equation. In: V. Kompis, M. Zmindak and E.A.W. Maunder, (eds.), CD-Proc. of the 8th Int. Conf. on Numerical Methods in Continuum Mechanics, Univ. of Zilina, Slovakia (2000).
V. Sladek J. Sladek R. Van (2002) ArticleTitleKeer, New integral equation approach to solution of diffusion equation Comp. Assist. Mech. Engn. Sci. 9 555–572 Occurrence Handle1044.65076
V. Sladek, J. Sladek and Ch. Zhang, Local integral equations treatment of potential problems in non-homogeneous media. In: V. Kompis and M. Zmindak, (eds.), Numerical Methods in Continuum Mechanics. CD-Proc. of the 9th Int. Conf. NMCM 2003, Zilina, Slovakia, ISBN 80-968823-4-1 (2003).
S.E. Mikhailov (2002) ArticleTitleLocalized boundary-domain integral formulations for problems with variable coefficients Eng. Anal. Bound. Elem. 26 681–690 Occurrence Handle1016.65097
S.E. Mikhailov and I.S. Nakhova, Numerical solution of a Neumann problem with variable coefficients by the localized boundary-domain integral equation method. In: Sia Amini (ed.), Fourth UK Conf. on Boundary Integral Methods. Salford University, UK, ISBN 0-902896-40-7, (2003) 175–184.
L.C. Wrobel (2002) The Boundary Element Method, Vol. 1: Applications in Thermo-Fluids and Acoustics John Wiley & Sons Chichester 451 Occurrence Handle0994.74002
M. Abramowitz I.A. Stegun (1964) Handbook of Mathematical Functions, Applied Mathematics Series Dover Publications New York 1046
W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press (1988–1992) 1032 pp.
M.H. Aliabadi (2002) The Boundary Element Method, Vol. 2: Applications in Solids and Structures John Wiley & Sons Chichester 580 Occurrence Handle0994.74003
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Sladek, V., Sladek, J. & Zhang, C. Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients. J Eng Math 51, 261–282 (2005). https://doi.org/10.1007/s10665-004-3692-y
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DOI: https://doi.org/10.1007/s10665-004-3692-y