A boundary-only meshless method for numerical solution of the Eikonal equation

Computational Mechanics

Volumn 47, Issue 3, 2011, Pages 283-294

  Dehghan, Mehdi ^a   Salehi, Rezvan ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Analog equation method; Boundary knot method; Boundary type meshless methods; Eikonal equation; Hamilton Jacobi equations; Radial basis function

Indexed keywords

BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; FUNCTIONS; GEOMETRICAL OPTICS; PARTIAL DIFFERENTIAL EQUATIONS; RADIAL BASIS FUNCTION NETWORKS;

ANALOG EQUATION METHODS; BOUNDARY KNOT METHOD; EIKONAL EQUATION; HAMILTON - JACOBI EQUATIONS; MESH-LESS METHODS; RADIAL BASIS FUNCTIONS;

NUMERICAL METHODS;

EID: 79954618933     PISSN: 01787675     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00466-010-0547-4     Document Type: Article

References (66)

1. M Bardi F Dalio 1997 On the Bellman equation for some unbounded control problems Nonlinear Differ Equ Appl 4 491 510 1485734 0894.49017 10.1007/s000300050027 (Pubitemid 127621084)
2. T Belytschko YY Lu L Gu 1994 Element-free Galerkin methods Int J Numer Meth Eng 37 229 256 1256818 0796.73077 10.1002/nme.1620370205
3. M Boue P Dupuis 1992 Markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control SIAM J Numer Anal 36 667 695 1681057 10.1137/S0036142997323521
4. AR Bruss 1982 The Eikonal equation: some results applicable to computer vision J Math Phys 23 890 896 655907 0502.35079 10.1063/1.525441
5. S Bryson D Levy 2003 High-order central WENO schemes for multidimensional Hamilton-Jacobi equations SIAM J Numer Anal 41 1339 1369 2034884 1050.65076 10.1137/S0036142902408404
6. T Cecil J Qian S Osher 2004 Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions J Comput Phys 196 327 347 2054346 1053.65086 10.1016/j.jcp.2003.11.010 (Pubitemid 38521673)
7. Y Cheng CW Shu 2007 A discontinuous Galerkin finite element method for directly solving Hamilton-Jacobi equations J Comput Phys 223 398 415 2314396 1124.65090 10.1016/j.jcp.2006.09.012
8. JS Chen C Pan CT Wu WK Liu 1996 Reproducing kernel particle methods for large deformation analysis of nonlinear structures Comput Methods Appl Mech Eng 139 195 228 1426009 0918.73330 10.1016/S0045-7825(96)01083-3
9. W Chen M Tanaka 2000 New insights in boundary-only and domain-type RBF methods Int J Nonlinear Sci Numer Simulation 1 145 152 1766092 0954.65084 (Pubitemid 33220663)
10. W Chen M Tanaka 2002 A meshfree, integration-free, and boundary-only RBF technique Comput Math Appls 43 379 391 1883574 0999.65142 10.1016/S0898-1221(01) 00293-0 (Pubitemid 34104975)
11. W Chen LJ Shen ZJ Shen GW Yuan 2005 Boundary knot method for Poisson equations Engrg Anal Bound Elem 29 756 760 1182.74250 10.1016/j.enganabound. 2005.04.001 (Pubitemid 41013886)
12. W Chen 2002 Symmetric boundary knot method Engrg Anal Bound Elem 26 489 494 1006.65500 10.1016/S0955-7997(02)00017-6 (Pubitemid 34498236)
13. MG Crandall PL Lions 1983 Viscosity solution of Hamilton-Jacobi equations Trans Amer Math Soc 227 1 42 690039 10.1090/S0002-9947-1983-0690039-8
14. P Danielsson 1980 Euclidean distance mapping Comput Graph Image Process 14 227 248 10.1016/0146-664X(80)90054-4
15. Deschamps T, Cohen LD (2002) Fast extraction of tubular and tree 3D surfaces with front propagation methods. Proc Intl Conf Pattern Recognition 731-734 (Pubitemid 44845101)
16. M Dehghan 2006 Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices Math Comput Simul 71 16 30 2206903 1089.65085 10.1016/j.matcom.2005.10.001 (Pubitemid 43243672)
17. Dehghan M, Hosseinzadeh H (2010) Development of circular arc boundary elements method. Eng Anal Bound Elem. doi: 10.1016/j.enganabound.2010.08.014
18. M Dehghan A Shokri 2008 A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions Math Comput Simul 79 700 715 2477558 1155.65379 10.1016/j.matcom.2008.04.018
19. M Dehghan R Salehi 2010 A seminumeric approach for solution of the Eikonal partial differential equation and its applications Numer Meth Partial Diff Equ 26 702 722 2656308 1189.65237
20. Dehghan M, Salehi R (2009) The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem. Commun Numer Meth Eng. doi: 10.1002/cnm.1315A
21. CA Duarte JT Oden 1996 H-p clouds-an hp meshless method Numer Method Partial Diff Eq 12 673 705 1419770 0869.65069 10.1002/(SICI)1098-2426(199611)12: 6<673::AID-NUM3>3.0.CO;2-P
22. B Engquist O Runborg 1996 Multi-phase computations in geometric optics J Comput Appl Math 74 175 192 1430373 0947.78001 10.1016/0377-0427(96)00023-4
23. YC Hon W Chen 2003 Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry Int J Numer Meth Engrg 56 1931 1948 1072.76048 10.1002/nme.642 (Pubitemid 36405756)
24. C Hu CW Shu 1999 A discontinuous Galerkin finite element method for Hamilton-Jacobi equations SIAM J Sci Comput 21 666 690 1718679 0946.65090 10.1137/S1064827598337282
25. GS Jiang DP Peng 2000 Weighted ENO schemes for Hamilton-Jacobi equations SIAM J Sci Comput 21 2126 2143 1762034 0957.35014 10.1137/S106482759732455X (Pubitemid 32076060)
26. S Jin Z Xin 1998 Numerical passage from systems of conservation laws to Hamilton-Jacobi equations and relaxation schemes SIAM J Numer Anal 35 2385 2404 1655852 0921.65063 10.1137/S0036142996314366 (Pubitemid 128464113)
27. B Jin Y Zheng 2005 Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation Eng Anal Bound Elem 29 925 935 1182.65179 10.1016/j.enganabound.2005.05.005 (Pubitemid 41455091)
28. B Jin W Chen 2006 Boundary knot method based on geodesic distance for anisotropic problems J Comput Phys 215 614 629 2219875 1093.65116 10.1016/j.jcp.2005.11.032 (Pubitemid 43623598)
29. Katsikadelis JT (1994) The analog equation method-a powerful BEM-based solution technique for solving linear and nonlinear engineering problems. In: Brebbia CA (eds) Boundary Element Method XVI. CLM Publications, Southampton, p 167
30. JT Katsikadelis MS Nerantzaki 1999 The boundary element method for nonlinear problems Eng Anal Bound Elem 23 365 373 0945.65132 10.1016/S0955-7997(98)00093-9
31. JT Katsikadelis MS Nerantzaki 2001 A Boundary element solution to the soap bubble problem Comput Mech 27 154 159 0980.65142 10.1007/s004660000224 (Pubitemid 32325751)
32. JT Katsikadelis 2002 The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies Int J Theor Appl Mech 27 13 38 2015530 1051.74052 10.2298/TAM0227013K
33. JT Katsikadelis GC Tsiatas 2003 Nonlinear dynamic analysis of heterogeneous orthotropic membranesby the analog equation method Eng Anal Bound Elem 27 115 124 1080.74559 10.1016/S0955-7997(02)00089-9
34. JT Katsikadelis 2008 The 2D elastostatic problem in inhomogeneous anisotropic bodies by the meshless analog equation method(MAEM) Eng Anal Bound Elem 32 997 1005 10.1016/j.enganabound.2007.10.016
35. JT Katsikadelis 2009 The meshless analog equation method. I. Solutionvof elliptic partial differential equations Arch Appl Mech 79 557 578 10.1007/s00419-008-0294-6
36. R Kimmel JA Sethian 2001 Optimal algorithm for shape from shading and path planning J Math Imaging Vis 14 237 244 1847813 0995.68101 10.1023/A:1011234012449 (Pubitemid 32638801)
37. R Kimmel AM Bruckstein 1993 Shape offsets via level sets Comput Aided Des 25 154 162 0771.65007 10.1016/0010-4485(93)90040-U
38. R Kimmel D Shaked N Kiryati A Bruckstein 1995 Skeletonization via distance maps and level sets Comput Vis Image Unders 62 382 391 10.1006/cviu.1995.1062
39. DW Kim WK Liu 2006 Maximum principle and convergence analysis for the meshfree point collocation method SIAM J Numer Anal 44 515 539 2218958 1155.65090 10.1137/04060809X (Pubitemid 46622335)
40. DW Kim YC Yoon WK Liu T Belytschko 2007 Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative J Comput Phys 221 370 394 2290576 1110.65110 10.1016/j.jcp.2006.06.023
41. DW Kim WK Liu YC Yoon T Belytschko SH Lee 2007 Meshfree point collocation method with intrinsic enrichment for interface problems Comput Mech 40 1037 1052 2335902 1161.74053 10.1007/s00466-007-0162-1 (Pubitemid 47292879)
42. F Li CW Shu YT Zhao H Zhao 2008 A second order discontinuous Galerkin fast sweeping method for Eikonal equations J Comput Phys 227 8191 8208 2442450 1151.65092 10.1016/j.jcp.2008.05.018
43. WK Liu S Jun Y Zhang 2005 Reproducing kernel particle method Int J Numer Methods Fluids 20 1081 1106 1333917 10.1002/fld.1650200824
44. WK Liu S Jun S Li J Adee T Belytschko 1995 Reproducing kernel particle methods for structural dynamics Int J Numer Methods Eng 38 1655 1679 1331505 0840.73078 10.1002/nme.1620381005
45. Li S, Liu WK (2004) Meshfree particle methods. Springer, Berlin
46. WK Liu S Li T Belytschko 1997 Moving least squares reproducing kernelm ethods, Part I. methodology and convergence Comput Methods Appl Mech Eng 143 113 154 1442392 0883.65088 10.1016/S0045-7825(96)01132-2 (Pubitemid 127411648)
47. G Li NR Aluru 2002 Boundary cloud method: a combined scattered point/boundary integral approach for boundary-only analysis Comput Meth Appl Mech Engrg 191 2337 2370 1903147 0995.65124 10.1016/S0045-7825(01)00415-7 (Pubitemid 34294695)
48. GR Liu YT Gu 2001 A point interpolation method for two dimensional solid Int J Numer Meth Engrg 50 937 951 1050.74057 10.1002/1097-0207(20010210)50: 4<937::AID-NME62>3.0.CO;2-X (Pubitemid 32192399)
49. LB Lucy 1977 A numerical approach to the testing of the fission hypothesis Astron J 82 12 1013 1024 10.1086/112164
50. JM Melenk I Babuska 1996 The partition of unity finite element method: basic theory and applications Comput Methods Appl Mech Engrg 139 289 314 1426012 0881.65099 10.1016/S0045-7825(96)01087-0 (Pubitemid 126382701)
51. D Mirzaei M Dehghan 2010 A meshless based method for solution of integral equations Appl Numer Math 60 245 262 2602676 1202.65174 10.1016/j.apnum.2009. 12.003
52. IM Mitchell S Sastry 2003 Continuous path planning with multiple constraint IEEE 5 5502 5507
53. YX Mukherjee S Mukherjee 1997 The boundary node method for potential problems Int J Numer Meth Engrg 40 797 815 0885.65124 10.1002/(SICI)1097- 0207(19970315)40:5<797::AID-NME89>3.0.CO;2-# (Pubitemid 127549025)
54. B Nayroles G Touzot P Villon 1992 Generalizing the finite element method: Diffuse approximation and diffuse elements Comput Mech 10 307 318 0764.65068 10.1007/BF00364252 (Pubitemid 23570159)
55. S Osher 1993 A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations SIAM J Math Anal 24 1145 1152 1234009 0804.35021 10.1137/0524066
56. S Osher J Sethian 1988 Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations J Comput Phys 79 12 49 965860 0659.65132 10.1016/0021-9991(88)90002-2
57. S Osher CW Shu 1991 High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations SIAM J Numer Anal 28 907 922 1111446 0736.65066 10.1137/0728049 (Pubitemid 21682139)
58. E Rouy A Tourin 1992 A viscosity solutions approach to shape-from-shading SIAM J Numer Anal 29 867 884 1163361 0754.65069 10.1137/0729053
59. N Rawlinson M Sambridge 2005 The fast marching method: an effective tool for tomographics imaging and tracking multiple phases in complex layered media Explor Geophys 36 341 350 10.1071/EG05341
60. Shu C-W (2004) High order numerical methods for time dependent Hamilton-Jacobi equations. WSPC/Lecture Notes Series
61. N Sukumar B Moran T Belytschko 1998 The natural element method in solid mechanics Int J Numer Meth Eng 43 839 887 1650918 0940.74078 10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R
62. M Tatari M Dehghan 2009 On the solution of the non-local parabolic partial differential equations via radial basis functions Appl Math Modell 33 1729 1738 2478588 1168.65403 10.1016/j.apm.2008.03.006
63. Wang F, Chen W, Jiang X (2009) Investigation of regularized techniques for boundary knot method. Commun Numer Method Eng doi: 10.1002/cnm.1275
64. T Zhu JD Zhang SN Atluri 1998 A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approaches Comput Mech 21 223 235 1633729 0920.76054 10.1007/s004660050297 (Pubitemid 128566412)
65. YT Zhang HK Zhao J Qian 2006 High order fast sweeping methods for static Hamilton-Jacobi equations J Sci Comput 29 25 56 2266807 10.1007/s10915-005-9014- 3 (Pubitemid 44631999)
66. YT Zhang CW Shu 2003 High order WENO schemes for Hamilton-Jacobi equations on triangular meshes SIAM J Sci Comput 24 1005 1030 1950522 1034.65051 10.1137/S1064827501396798