Approximated Fundamental Solutions Based on Levi Functions

Authors

  • Rafael Gallego Department of Structural Mechanics and Hydraulic Engineering, School of Civil Engineering, University of Granada, Fuentenueva Campus, Spain
  • Esther Puertas García Department of Structural Mechanics and Hydraulic Engineering, School of Civil Engineering, University of Granada, Fuentenueva Campus, Spain

Keywords:

Fundamental Solution, Levi Function, Functionally Graded Materials.

Abstract

Fundamental Solutions (FS) are useful for numerical methods, specifically in applications such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Obtaining analytically the FS for a given problem is frequently unfeasible since it entails the solution of a complex system of differential equations. In this paper, a novel method for the computation of approximate FS based in enhanced Levi Function is presented. The first terms of the enhanced Levi Function are computed analytically by an iterative procedure until the residual is regular at the collocation point. The last step in the computation of the FS is completed approximating the residual by Modified Radial Base Functions that are particular solutions of the problem equations.

The method is applied to potential problems with variable coefficients, and validated comparing to available analytically computed Fundamental Solutions.

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Author Biographies

Rafael Gallego, Department of Structural Mechanics and Hydraulic Engineering, School of Civil Engineering, University of Granada, Fuentenueva Campus, Spain

Rafael Gallego is full professor at the University of Granada since 1995. He received his Bachelor and Master degrees in Structural and Mechanical Engineering at the University of Sevilla (1987) and got his PhD at the same University in 1990; he was granted a Fulbright postdoctoral scholarship (1990–92) at the Brown University. Among other academic posts, he has been chairman of the Department of Structural and Hydraulic Engineering at the University of Granada since 2000 to 2008 and since 2016 to date. His research work has focused in developing computational models based on Boundary Integral Equations for solid mechanics problem, mainly related to wave propagation in solids, dynamic fracture mechanics and dynamic soil-structure interaction, including resolution of inverse problems to match computational results and experimental ones.

Esther Puertas García, Department of Structural Mechanics and Hydraulic Engineering, School of Civil Engineering, University of Granada, Fuentenueva Campus, Spain

Esther Puertas Garc´ıa is permanent lecturer at the University of Granada. She received her Bachelor and Master degrees in Civil Engineering at the University of Granada, and PhD degrees in Civil Engineering from the University of Granada in 2014. She has an extensive teaching, research and professional experience in the Mechanics of Continuous Media and Theory of Structures topics. Her research work concerns the development of computational toolkits for continuum media dynamics. She belongs to the Research Group of Mechanics of Solids and Structures. Her lines of work are based on the study of Integral Equations Methods for advanced applications in Solid Mechanics. She has developed techniques for the analysis of wave propagation problems in two and a half domains.

References

Brebbia, C. A. and Domnguez, J. (1989). Boundary elements: An

introductory course. McGraw Hill Book Co.

Kausel, E. (2006). Fundamental Solutions in Elastodynamics. A compendium.

Cambridge University Press.

Clements, D. L. (1998) Fundamental solutions for second order linear

elliptic partial differential equations. Computational Mechanics, 22,

–31.

Shaw, R. P. (1994). Green’s functions for heterogeneous media potential

problems. Engineering Analysis with Boundary Elements, 13, 219–221.

Nardini, D and Brebbia, C. A.(1983). A new approach to free vibration

analysis using boundary elements. Applied Mathematical Modelling,

(3), 157–162.

Gao, X. W. (2002). The radial integration method for evaluation

of domain integrals with boundary-only discretization. Engineering

Analysis with Boundary Elements, 26 905–916.

Gao, X. W. Zhang, Ch. and Guo, L. (2007). Boundary-only element

solutions of 2D and 3D nonlinear and nonhomogeneous elastic problems.

Engineering Analysis with Boundary Elements, 31(12), 947–982.

Katsikadelis, J.T. (1994). The Analog Equation Method: A Powerful

BEM-based Solution Technique for Solving Linear and Nonlinear Engineering

Problems, In: Boundary Element Method XVI, ed. C.A. Brebbia,

Computational Mechanics Publications, 167–182.

Riveiro, M.A. and Gallego, R.(2013). Boundary elements and the

analog equation method for the solution of elastic problems in 3-D

non-homogeneous bodies. Comput. Meth. Appl. Mech. Eng., 263, 12–19.

Levi, E. E. (1909). I problemi dei valori al contorno per le equazioni

lineari totalmente ellittiche alle derivate parziali. Memorie della Societa

Italiana di Scienze XL, 16, 1–112.

Al-Jawary, M. A. and Wrobel, L. C. (2011). Numerical solution of twodimensional

mixed problems with variable coefficients by the boundarydomain

integral and integro-differential equation methods. Engineering

Analysis with Boundary Elements, 35, 1279–1287.

Mikhailov, S. E. (2015). Analysis of Segregated Boundary-Domain Integral

Equations forVariable-Coefficient Dirichlet and Neumann Problems

with General Data. arXiv:1509.03501.

Pomp, A. (1998). The Boundary-Domain Integral Method for Elliptic

Systems. Springer-Verlag, Berlin.

Buhmann, M. D. (2003). Radial Basis Functions, Theory and Implementations.

Cambridge University Press.

Chen,W. and Tanaka, M. (2002). A meshless, exponential convergence,

integration-free, and boundary only RBF technique. Computers and

Mathematics with Applications, 43, 379–391.

Golberg, M. A. and Chen, C. S. (1998). The method of fundamental

solutions for potential, Helmholtz and diffusion problems. In M. A. Golberg,

editor, Boundary Integral Methods: Numerical and Mathematical

Aspects,

–176. WIT Press.

Kansa, E. J. (1990). Multiquadrics – a scattered data approximation

scheme with applications to computational fluid dynamics – I. Computers

and Mathematics with Application, 19 (8/9), 127–145.

Kansa, E. J. (1990). Multiquadrics – a scattered data approximation

scheme with applications to computational fluid dynamics – II.

Computers and Mathematics with Application, 19 (8/9), 147–161.

Kupradze, V. D. and Aleksidze, M. A. (1964). The method of functional

equations for the approximate solution of certain boundary value problems.

U.S.S.R. Computational Mathematics and Mathematical Physics,

, 82–126.

Gupta, A. and Talha, M. (2015). Recent development in modeling

and analysis of functionally graded materials. Progress in Aerospace

Sciences, 79, 1–14.

Jha, D. K., Kant, T. and Singn, R. K. (2013). A critical review of

recent research on functionally graded plates. Composite Structures, 96,

–849.

Pomp, A. (1998). Levi Functions for linear elliptic systems with variable

coefficients including shell equations. Computational Mechanics, 22,

–99.

Dumont, N. A. Chaves, R. A. P. and Paulino, G. H. (2002). The hybrid

boundary element method applied to functionally graded materials.

In C. A. Brebbia, A. Tadeu, V. Popov, editors, Boudary Elements

XXIV.

Sutradhar, A. and Paulino, G. H. (1994). A symple boundary element

method for problems of non-homogeneous media. International Journal

for numerical Methods in Engineering, 60, 2203–2230.

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Published

2019-08-07

How to Cite

Gallego, R., & García, E. P. (2019). Approximated Fundamental Solutions Based on Levi Functions. European Journal of Computational Mechanics, 28(1-2), 31–50. Retrieved from https://journals.riverpublishers.com/index.php/EJCM/article/view/927

Issue

2019: Vol28 Iss 1-2

Section

Original Article