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ARTICLE
Material discontinuity problems solved by a meshless method based on variably scaled discontinuous radial functions
1
Cracow University of Technology, Institute of Computer Science, Cracow, Poland
These authors had equal contribution to this work
Submission date: 2023-10-10
Final revision date: 2023-12-14
Acceptance date: 2023-12-14
Online publication date: 2024-03-06
Publication date: 2024-04-30
Corresponding author
Artur Krowiak
Institute of Computer Science, Cracow University of Technology, Al. Jana Pawła II 37, 31-864, Kraków, Poland
Institute of Computer Science, Cracow University of Technology, Al. Jana Pawła II 37, 31-864, Kraków, Poland
Journal of Theoretical and Applied Mechanics 2024;62(2):241-251
KEYWORDS
TOPICS
ABSTRACT
The paper presents a meshless method based on global interpolation with radial base functions
and shows its application in solving interface problems. Such problems arise when two
or more different materials are used to construct the element under consideration. Across the
interface between the materials, a discontinuity of material parameters arises. To solve the
problem, the radial basis function-based collocation method is applied. To properly reflect
the discontinuity, the base functions are modified. In this paper, the method is applied to
solve problems described by elliptic equations. Using these examples, the accuracy, stability
and convergence of the method are examined.
REFERENCES (20)
1.
Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P., 1996, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 139, 3-47.
2.
Chen W., Fu Z.J., Chen C.S., 2014, Recent Advances in Radial Basis Function Collocation Methods, Springer.
3.
De Marchi S., Erb W., Marchetti F., Perracchione E., Rossini M., 2020, Shape-driven interpolation with discontinuous kernels: error analysis, edge extraction and applications in MPI, SIAM Journal on Scientific Computing, 42, 2, B472-B491.
4.
Fasshauer G.E., 2007, Meshfree Approximation Methods with Matlab, World Scientific, Singapore.
5.
Ferreira A.J.M, Fasshauer G.E., 2007, Analysis of natural frequencies of composite plates by an RBF-pseudospectral method, Composite Structures, 79, 202-210.
6.
Fornberg B., 1996, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge.
7.
Hon Y.C., Schaback R., 2001, On nonsymmetric collocation by radial basis functions, Applied Mathematics and Computation, 119, 177-186.
8.
Kansa E., 1990, Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics II: Solutions to parabolic, hyperbolic, and elliptic partial differential equations, Computers and Mathematics with Applications, 19, 147-161.
9.
Krowiak A., 2008, Methods based on differential quadrature in vibration analysis of plates, Journal of Theoretical and Applied Mechanics, 46, 1, 123-139.
10.
Krowiak A., 2018, Domain-type RBF collocation methods for biharmonic problems, International Journal of Computational Method, 15, 1850078-1 – 1850078-20.
11.
LeVeque R.J., Li Z., 1994, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analysis, 31, 4, 1019-1044.
12.
Li Z., 2003, An overview of the immersed interface method and its applications, Taiwanese Journal of Mathematics, 7, 1, 1-49.
13.
Liu G.R., 2003, Mesh-free Methods, Moving Beyond the Finite Element Method, CRC Press, Boca Raton.
14.
Martin B., Fornberg B., 2017, Using radial basis function-generated finite differences (RBF-FD) to solve heat transfer equilibrium problems in domains with interfaces, Engineering Analysis with Boundary Elements, 79, 38-48.
15.
Rippa, S., 1999, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Advances in Computational Mathematics, 11, 193-210.
16.
Stevens D., Power H., 2015, The radial basis function finite collocation approach for capturing sharp fronts in time dependent advection problems, Journal of Computational Physics, 298, 423-445.
17.
Trefethen L.N., 2000, Spectral Methods in MATLAB, 3rd. repr. ed., SIAM, Philadelphia.
18.
Wu Z., 1992, Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approximation Theory and its Applications, 8, 1-10.
19.
Yang Q., Zhang X., 2016, Discontinuous Galerkin immersed finite element methods for parabolic interface problems, Journal of Computational and Applied Mathematics, 299, 127-139.
20.
Yoon Y.-C., Song J.-H., 2014, Extended particle difference method for weak and strong discontinuity problems: Part I. Derivation of the extended particle derivative approximation for the representation of weak and strong discontinuities, Computational Mechanics, 53, 1087-1103.
| eISSN: | 2543-6309 |
| ISSN: | 1429-2955 |
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