Extending the Exponential Basis Functions Meshless Method to Solve Singular Plate Problems
Authors
Abstract
: Existence of singular points inside the solution domain or on its boundary deteriorates the accuracy and convergence rate of numerical methods. This phenomenon usually happens due to discontinuities in the boundary conditions or abrupt changes in the domain shape. This study has focused on the solution of singular plate problems using the exponential basis functions method. In this method, unknown functions are considered as a linear combination of exponential basis functions and the coefficients are calculated by approximate satisfaction of the boundary conditions. To increase the accuracy and convergence rate in problems with singular points, a series of singular, quasi-exponential functions are added to the method’s exponential basis functions. These functions have proper discontinuity in location of the singular points and satisfy the homogenous differential equation. The results obtained from the solution of three cracked plate problems show considerable increase in the accuracy and convergence rate of the proposed method compared with the exponential basis functions method without any noticeable increase in the computational cost.
Keywords
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2. Gingold, R. A., and Monaghan, J. J., “Smoothed Particle Hydrodynamics-Theory and Application to Non-Spherical Stars”, Monthly Notices of the Royal Astronomical Society, Vol. 181, pp. 375-389, 1977.
3. Belytschko, T., Lu, Y. Y., and Gu, L., “Element Free Galerkin Methods”, International Journal for Numerical Methods in Engineering, Vol. 37, No. 2, pp. 229-256, 1994.
4. Atluri, S., and Zhu, T., “A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics”, Computational Mechanics, Vol. 22, No. 2, pp. 117-127, 1998.
5. Boroomand, B., Soghrati, S., and Movahedian, B., “Exponential Basis Functions in Solution of Static and Time Harmonic Elastic Problems in a Meshless Style”, International Journal for Numerical Methods in Engineering, Vol. 81, No.8, pp. 971-1018, 2010.
6. Shamsaei, B., and Boroomand, B., “Exponential Basis Functions in Solution of Laminated Structures”, Composite Structures, Vol. 93, No. 8, pp. 2010-2019, 2011.
7. Zandi, S., Boroomand, B., and Soghrati, S., “Exponential Basis Functions in Solution of Problems with Fully Incompressible Materials: A Mesh-Free Method”, Journal of Computational Physics, Vol. 231, No. 21, pp. 7255-7273, 2012.
8. Shahbazi, M., Boroomand, B., and Soghrati, S., “A Mesh-Free Method Using Exponential Basis Functions for Laminates Modeled by CLPT, FSDT and TSDT–Part I: Formulation”, Composite Structures, Vol. 93, No. 12, pp. 3112-3119, 2011.
9. Shahbazi, M., Boroomand, B., and Soghrati, S., “A Mesh-Free Method Using Exponential Basis Functions for Laminates Modeled by CLPT, FSDT and TSDT–Part II: Implementation and Results”, Composite Structures, Vol. 94, No. 1, pp. 84-91, 2011.
10. Zandi, S., Boroomand, B., and Soghrati, S., “Exponential Basis Functions in Solution of Incompressible Fluid Problems with Moving Free Surfaces”, Journal of Computational Physics, Vol. 231, No. 2, pp. 505-527, 2012.
11. Hashemi, S., Boroomand, B., and Movahedian, B., “Exponential Basis Functions in Space and Time: a Meshless Method for 2D Time Dependent Problems”, Journal of Computational Physics,Vol. 241, pp. 526-545, 2013.
12. Azhari, F., Boroomand, B., and Shahbazi, M., “Exponential Basis Functions in the Solution of Laminated Plates Using a Higher-order Zig-Zag Theory”, Composite Structures, Vol. 105, pp. 398-407, 2013.
13. Boroomand, B., Azhari, F., and Shahbazi, M., “On Definition of Clamped Conditions in TSDT and FSDT; the Use of Exponential Basis Functions in Solution of Laminated Composites”, Composite Structures,Vol. 97, pp. 129-135, 2013.
14. Shahbazi, M., Boroomand, B., and Soghrati, S., “On Using Exponential Basis Functions for Laminates Modeled by CLPT, FSDT and TSDT: Further Tests and Results”, Composite Structures,Vol. 94, No. 7, pp. 2263-2268. 2012.
15. Movahedian, B., and Boroomand, B., “The Solution of Direct and Inverse Transient Heat Conduction Problems with Layered Materials Using Exponential Basis Functions”, International Journal of Thermal Sciences,Vol. 77, pp. 186-198, 2014.
16. Movahedian, B., Boroomand, B., and Soghrati, S., “A Trefftz Method in Space and Time Using Exponential Basis Functions: Application to Direct and Inverse Heat Conduction Problems”, Engineering Analysis with Boundary Elements, Vol. 37, No. 5, pp. 868-883, 2013.
22. Li, Z. C., Huang, J., and Huang, H. T., “Stability Analysis of Method of Fundamental Solutions for Mixed Boundary Value Problems of Laplace’s Equation”, Computing, Vol. 88, No. 1-2, pp. 1-29, 2010.
23. Whiteman, J., “Finite-Difference Techniques for a Harmonic Mixed Boundary Problem Having a Reentrant Boundary”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, Vol. 323, No. 1553, pp. 271-276, 1971.
24. Ingham, D. B., and Kelmanson, M. A., Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems, Springer-Verlag, Berlin, 1984.
25. Rao, B., and Rahman, S., “An Efficient Meshless Method for Fracture Analysis of Cracks”, Computational Mechanics,Vol. 26, No. 4, pp. 398-408, 2000.
26. Motz, H., “The Treatment of Singularities of Partial Differential Equations by Relaxation Methods”, Quarterly of Applied Mathematics, Vol. 4, pp. 371-377, 1946.
27. Igarashi, H., and Honma, T., “A boundary Element Method for Potential Fields with Corner Singularities”, Applied Mathematical Modelling, Vol. 20, No. 11, pp. 847-852, 1996.
28. Hu, H. Y., Li, Z. C. and Cheng, A. H. D., “Radial Basis Collocation Methods for Elliptic Boundary Value Problems”, Computers & Mathematics with Applications, Vol. 50, No. 1, pp. 289-320, 2005.
29. Karageorghis, A., “Modified Methods of Fundamental Solutions for Harmonic and Biharmonic Problems with Boundary Singularities”, Numerical Methods for Partial Differential Equations, Vol. 8, No. 1, pp. 1-19, 1992.
30. Mossaiby, F., Bazrpach, M., and Shojaei, A., “Extending the Method of Exponential Basis Functions to Problems with Singularities”, Engineering Computations, Vol. 32, No. 2, pp. 406-423, 2015.
31. Li, Z. C., Lu, T. T., and Hu, H. Y., “The Collocation Trefftz Method for Biharmonic Equations with Crack Singularities”, Engineering Analysis with Boundary Elements, Vol. 28, No. 1, pp. 79-96, 2004.
32. Elliotis, M., Georgiou, G., and Xenophontos, C., “The Singular Function Boundary Integral Method for Biharmonic Problems with Crack Singularities”, Engineering Analysis with Boundary Elements, Vol. 31, No. 3, pp. 209-215, 2007.
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