Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

Document Type : Article

Authors

1 Department of Civil & Environmental Engineeri ng , Amirkabir university of technology

2 Department of Civil & Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

3 School of Civil Engineering, Iran University of Science and Technology Narmak, Tehran, P.O. Box 16765-163, Iran ‎

Abstract

A Mixed formulation of Discrete Least Squares Meshless (MDLSM) as a truly meshfree method is presented in this paper for solving both linear and non-linear propagation problems. In DLSM method, the irreducible formulation was deployed which needs to calculate the costly second derivatives of the MLS shape functions. In the proposed MDLSM method, the complex and costly second derivatives of shape functions are not required. Furthermore, using the mixed formulation, both unknown parameters and their gradients are simultaneously obtained circumventing the need for post-processing procedure performed in irreducible formulation to calculate the gradients. Therefore, the accuracy of gradients of unknown parameters is increased. In MDLSM method, the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional defined as the sum of squared residuals of the differential equation and its boundary condition. The proposed method automatically leads to symmetric and positive-definite system of equations and, therefore, is not subject to the Ladyzenskaja-Babuska-Brezzi (LBB) condition. The proposed MDLSM method is validated and verified by a set of benchmark problems. The results indicate the ability of proposed method to efficiently and effectively solve the linear and non-linear propagation problems.
 

Keywords

Main Subjects


References
1. Shao, S. and Lo, E.Y. \Incompressible SPH method for simulating Newtonian and non- Newtonian flows with a free surface", Advances in Water Resources, 26(7),
pp. 787-800 (2003).
2. Khanpoura, M., Zarratia, A.R., Kolahdoozana, M., Shakibaeiniab, A., and Amirshahia, S.M. \Mesh-free
SPH modeling of sediment scouring and
ushing",
Computers & Fluids, 129, pp. 67-78 (2016).
3. Chen, Z., Zong, Z., Liu, M.B., Zou, L., Li, H.T.,
and Shu, C. \An SPH model for multiphase
ows
with complex interfaces and large density di erences",
Journal of Computational Physics, 283, pp. 169-188
(2015).
4. Koshizuka, S. and Oka, Y. \Moving-particle semiimplicit
method for fragmentation of incompressible

uid", Nuclear Science and Engineering, 123(3), pp.
421-434 (1996).
5. Macia, F., Souto-Iglesias, A., Gonzalez, L.M., and
Cercos-Pita, J.L. \<MPS>=<SPH>", 8th International
SPHERIC Workshop (2013).
6. Ataie-Ashtiani, B. and Farhadi, L. \A stable movingparticle
semi-implicit method for free surface
ows",
Fluid Dynamics Research, 38(4), pp. 241-256 (2006).
7. Sun, Z., Djidjeli, K., Xing, J.T., and Cheng, F.
\Coupled MPS-modal superposition method for 2D
nonlinear
uid-structure interaction problems with free
surface", Journal of Fluids and Structures, 61, pp. 295-
323 (2016).
8. Shakibaeinia, A. and Jin, Y.-C. \MPS mesh-free particle
method for multiphase
ows", Computer Methods
in Applied Mechanics and Engineering, 229, pp. 13-26
(2012).
9. Kolahdoozan, M., Ahadi, M., and Shirazpoor, S.
\E ect of turbulence closure models on the accuracy of
moving particle semi-implicit method for the viscous
free surface
ow", Scientia Iranica. Transactions A,
Civil Engineering, 21(4), p. 1217 (2014).
10. Monaghan, J., Kos, A. and Issa, N. \Fluid motion
generated by impact", Journal of Waterway, Port,
Coastal, and Ocean Engineering, 129(6), pp. 250-259
(2003).
11. Shakibaeinia, A. and Jin, Y.C. \A weakly compressible
MPS method for modeling of open-boundary freesurface

ow", International Journal for Numerical
Methods in Fluids, 63(10), pp. 1208-1232 (2010).
12. Khayyer, A. and Gotoh, H. \Modi ed moving particle
semi-implicit methods for the prediction of 2D wave
impact pressure", Coastal Engineering, 56(4), pp. 419-
440 (2009).
13. Liu, C.-S. and Young, D. \A multiple-scale Pascal
polynomial for 2D stokes and inverse Cauchy-Stokes
problems", Journal of Computational Physics, 312,
pp. 1-13 (2016).
14. Liu, C.-S. \Homogenized functions to recover
H(t)=H(x) by solving a small scale linear system of
di erencing equations", International Journal of Heat
and Mass Transfer, 101, pp. 1103-1110 (2016).
15. Belytschko, T., Lu, Y.Y., and Gu, L. \Element-free
Galerkin methods", International Journal for Numerical
Methods in Engineering, 37(2), pp. 229-256 (1994).
16. Liu, L., Chua, L., and Ghista, D. \Element-free
Galerkin method for static and dynamic analysis of
spatial shell structures", Journal of Sound and Vibration,
295(1), pp. 388-406 (2006)
17. Deng, Y., Liu, C., Peng, M., and Cheng, Y. \The
interpolating complex variable element-free Galerkin
method for temperature eld problems", International
Journal of Applied Mechanics, 7(02), p. 1550017
(2015).
18. Zhang, Z., Liew, K.M., Cheng, Y., and Lee, Y.Y.
\Analyzing 2D fracture problems with the improved
element-free Galerkin method", Engineering Analysis
with Boundary Elements, 32(3), pp. 241-250 (2008).
19. Singh, A., Singh, I.V., and Prakash, R. \Meshless
element free Galerkin method for unsteady nonlinear
heat transfer problems", International Journal of Heat
and Mass Transfer, 50(5), pp. 1212-1219 (2007).
20. Belytschko, T., Gu, L., and Lu, Y. \Fracture and crack
growth by element free Galerkin methods", Modelling
and Simulation in Materials Science and Engineering,
2(3A), p. 519 (1994).
S. Faraji Gargari et al./Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 565{578 577
21. Atluri, S. and Zhu, T. \A new meshless local Petrov-
Galerkin (MLPG) approach in computational mechanics",
Computational Mechanics, 22(2), pp. 117-127
(1998).
22. Atluri, S., Liu, H., and Han, Z. \Meshless local Petrov-
Galerkin (MLPG) mixed nite di erence method for
solid mechanics", Computer Modeling in Engineering
and Sciences, 15(1), p. 1 (2006).
23. Enjilela, V. and Arefmanesh, A. \Two-step Taylorcharacteristic-
based MLPG method for
uid
ow and
heat transfer applications", Engineering Analysis with
Boundary Elements, 51, pp. 174-190 (2015).
24. Luan, T. and Sun, Y. \Solving a scattering problem in
near eld optics by a least-squares method", Engineering
Analysis with Boundary Elements, 65, pp. 101-111
(2016).
25. Arzani, H. and Afshar, M. \Solving Poisson's equations
by the discrete least square meshless method", WIT
Transactions on Modelling and Simulation, 42, pp. 23-
31 (2006).
26. Firoozjaee, A.R. and Afshar, M.H. \Discrete least
squares meshless method with sampling points for
the solution of elliptic partial di erential equations",
Engineering Analysis with Boundary Elements, 33(1),
pp. 83-92 (2009).
27. Afshar, M., Lashckarbolok, M., and Shobeyri, G.
\Collocated discrete least squares meshless (CDLSM)
method for the solution of transient and steadystate
hyperbolic problems", International Journal for
Numerical Methods in Fluids, 60(10), pp. 1055-1078
(2009).
28. Afshar, M. and Lashckarbolok, M. \Collocated discrete
least-squares (CDLS) meshless method: Error estimate
and adaptive re nement", International Journal
for Numerical Methods in Fluids, 56(10), pp. 1909-
1928 (2008).
29. Afshar, M., Amani, J., and Naisipour, M. \A node enrichment
adaptive re nement in discrete least squares
meshless method for solution of elasticity problems",
Engineering Analysis with Boundary Elements, 36(3),
pp. 385-393 (2012).
30. Kazeroni, S.N. and Afshar, M. \An adaptive node
regeneration technique for the ecient solution of elasticity
problems using MDLSM method", Engineering
Analysis with Boundary Elements, 50, pp. 198-211
(2015).
31. Shobeyri, G. and Afshar, M. \Simulating free surface
problems using discrete least squares meshless
method", Computers & Fluids, 39(3), pp. 461-470
(2010)
32. Shobeyri, G. and Afshar, M. \Adaptive simulation of
free surface
ows with discrete least squares meshless
(DLSM) method using a posteriori error estimator",
Engineering Computations, 29(8), pp. 794-813 (2012)
33. Pehlivanov, A., Carey, G., and Lazarov, R. \Leastsquares
mixed nite elements for second-order elliptic
problems", SIAM Journal on Numerical Analysis,
31(5), pp. 1368-1377 (1994).
34. Jiang, B.-N., Lin, T., and Povinelli, L.A. \Large-scale
computation of incompressible viscous
ow by leastsquares
nite element method", Computer Methods in
Applied Mechanics and Engineering, 114(3), pp. 213-
231 (1994).
35. Faraji, S., Afshar, M., and Amani, J. \Mixed discrete
least square meshless method for solution of quadratic
partial di erential equations", Scientia Iranica, 21(3),
pp. 492-504 (2014).
36. Amani, J., Afshar, M., and Naisipour, M. \Mixed
discrete least squares meshless method for planar
elasticity problems using regular and irregular nodal
distributions", Engineering Analysis with Boundary
Elements, 36(5), pp. 894-902 (2012).
37. Kumar, R., A Least-Squares/Galerkin Split Finite
Element Method for Incompressible and Compressible
Navier-Stokes Equations, ProQuest (2008).
38. Shcherbakov, V. and Larsson, E. \Radial basis function
partition of unity methods for pricing vanilla
basket options", Computers & Mathematics with Applications,
71(1), pp. 185-200 (2016).
39. Hon, Y.-C., Sarler, B., and Yun, D.-f. \Local radial
basis function collocation method for solving thermodriven

uid-
ow problems with free surface", Engineering
Analysis with Boundary Elements, 57, pp. 2-8
(2015).
40. Dehghan, M. and Abbaszadeh, M. \Two meshless procedures:
moving Kriging interpolation and elementfree
Galerkin for fractional PDEs", Applicable Analysis,
pp. 1-34 (2016).
41. Peco, C., Millan, D., Rosolen, A., and Arroyo, M. \Ef-
cient implementation of Galerkin meshfree methods
for large-scale problems with an emphasis on maximum
entropy approximants", Computers & Structures, 150,
pp. 52-62 (2015).
42. Arroyo, M. and Ortiz, M. \Local maximum-entropy
approximation schemes: a seamless bridge between
nite elements and meshfree methods", International
Journal for Numerical Methods in Engineering,
65(13), pp. 2167-2202 (2006)
43. Wang, J. and Liu, G. \Radial point interpolation
method for elastoplastic problems", ICSSD 2000, 1st
Structural Conference on Structural Stability and Dynamics
(2000).
44. Tanaka, Y., Watanabe, S., and Oko, T. \Study of
eddy current analysis by a meshless method using
RPIM", IEEE Transactions on Magnetics, 51(3), pp.
1-4 (2015).
45. Liu, G.-R., Meshfree Methods: Moving Beyond the
Finite Element Method, CRC press (2009).
46. Li, X. and Zhang, S. \Meshless analysis and applications
of a symmetric improved Galerkin boundary node
578 S. Faraji Gargari et al./Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 565{578
method using the improved moving least-square approximation",
Applied Mathematical Modelling, 40(4),
pp. 2875-2896 (2016)
47. Li, X. \A meshless interpolating Galerkin boundary
node method for Stokes
ows", Engineering Analysis
with Boundary Elements, 51, pp. 112-122 (2015)
48. Li, X. \Meshless Galerkin algorithms for boundary
integral equations with moving least square approximations",
Applied Numerical Mathematics, 61(12), pp.
1237-1256 (2011).
49. Li, X. \Error estimates for the moving least-square
approximation and the element-free Galerkin method
in n-dimensional spaces", Applied Numerical Mathematics,
99, pp. 77-97 (2016).
50. Liu, G.-R. and Gu, Y.-T., An Introduction to Meshfree
Methods and Their Programming, Springer Science &
Business Media (2005).
51. Li, X., Zhang, S., Wang, Y., and Chen, H. \Analysis
and application of the element-free Galerkin method
for nonlinear sine-Gordon and generalized sinh-Gordon
equations", Computers & Mathematics with Applications,
71(8), pp. 1655-1678 (2016)
52. Li, X., Chen, H., and Wang, Y. \Error analysis
in Sobolev spaces for the improved moving leastsquare
approximation and the improved element-free
Galerkin method", Applied Mathematics and Computation,
262, pp. 56-78 (2015)
53. Van de Vosse, F. and Minev, P. \Spectral elements
methods: Theory and applications", EUT Report, 96-
W-001 ISBN 90-236-0318-5, Eindhoven University of
Technology (1996).
54. Coppoli, E.H.R., Mesquita, R.C., and Silva, R.S.
\Periodic boundary conditions in element free Galerkin
method", COMPEL-The International Journal for
Computation and Mathematics in Electrical and Electronic
Engineering, 28(4), pp. 922-934 (2009).
55. Moin, P., Fundamentals of Engineering Numerical
Analysis, Cambridge University Press (2010).
56. Chen, S., Wu, X., Wang, Y., and Kong, W. \High accuracy
time and space transform method for advectiondi
usion equation in an unbounded domain", International
Journal for Numerical Methods in Fluids,
58(11), pp. 1287-1298 (2008).

Volume 25, Issue 2
Transactions on Civil Engineering (A)
March and April 2018
Pages 565-578
  • Receive Date: 09 January 2016
  • Revise Date: 24 September 2016
  • Accept Date: 05 December 2016