1396:
investigations are made on the convergence of BKM in the analysis of homogeneous
Helmholtz, modified Helmholtz and convection-diffusion problems; in the BKM is employed to deal with complicated geometry of two and three dimension Helmholtz and convection-diffusion problems; in membrane vibration under mixed-type boundary conditions is investigated by symmetric boundary knot method; in the BKM is applied to some inverse Helmholtz problems; in the BKM solves Poisson equations; in the BKM calculates Cauchy inverse inhomogeneous Helmholtz equations; in the BKM simulates the anisotropic problems via the geodesic distance; in relationships among condition number, effective condition number, and regularizations are investigated; in heat conduction in nonlinear functionally graded material is examined by the BKM; in the BKM is also used to solve nonlinear Eikonal equation.
1166:
1388:, thanks to its dimensional reducibility. The major bottlenecks of BEM, however, are computationally expensive to evaluate integration of singular fundamental solution and to generate surface mesh or re-mesh. The method of fundamental solutions (MFS) has in recent decade emerged to alleviate these drawbacks and getting increasing attentions. The MFS is integration-free, spectral convergence and meshfree.
57:
equation leads to a boundary-only formulation for the homogeneous solution. Without the singular fundamental solution, the BKM removes the controversial artificial boundary in the method of fundamental solutions. Some preliminary numerical experiments show that the BKM can produce excellent results with relatively a small number of nodes for various linear and nonlinear problems.
871:
1161:{\displaystyle {\begin{aligned}&g\left(x_{k},y_{k}\right)=\sum \limits _{i=1}^{N}\alpha _{i}\phi \left(r_{i}\right),\qquad k=1,\ldots ,m_{1}\\&h\left(x_{k},y_{k}\right)=\sum \limits _{i=1}^{N}\alpha _{i}{\frac {\partial \phi \left(r_{i}\right)}{\partial n}},\qquad k=m_{1}+1,\ldots ,m\\\end{aligned}}}
1395:
The BKM has since been widely tested. In, the BKM is used to solve
Laplace equation, Helmholtz Equation, and varying-parameter Helmholtz equations; in by analogy with Fasshauer’s Hermite RBF interpolation, a symmetric BKM scheme is proposed in the presence of mixed boundary conditions; in, numerical
1391:
As its name implies, the fundamental solution of the governing equations is used as the basis function in the MFS. To avoid singularity of the fundamental solution, the artificial boundary outside the physical domain is required and has been a major bottleneck for the wide use of the MFS, since such
56:
The BKM is basically a combination of the distance function, non-singular general solution, and dual reciprocity method (DRM). The distance function is employed in the BKM to approximate the inhomogeneous terms via the DRM, whereas the non-singular general solution of the partial differential
47:
in that the former does not require special techniques to cure the singularity. The BKM is truly meshfree, spectral convergent (numerical observations), symmetric (self-adjoint PDEs), integration-free, and easy to learn and implement. The method has successfully been tested to the
Helmholtz,
355:
690:
795:
242:
513:
565:
1571:
149:
1335:
1249:
876:
826:
251:
458:
428:
1668:
1362:
1392:
fictitious boundary may cause computational instability. The BKM is classified as one kind of boundary-type meshfree methods without using mesh and artificial boundary.
859:
398:
594:
585:
378:
31:
is not trivial especially for moving boundary, and higher-dimensional problems. The boundary knot method is different from the other methods based on the
1478:
W. Chen and Y.C. Hon, Numerical convergence of boundary knot method in the analysis of
Helmholtz, modified Helmholtz, and convection-diffusion problems,
1364:
can be uniquely determined by above Eq. (6). And then the BKM solution at any location of computational domain can be evaluated by the formulation (4).
698:
2138:
1759:
158:
1709:
463:
1685:
1659:
518:
1983:
1753:
2053:
1910:
1765:
72:
23:
Recent decades have witnessed a research boom on the meshfree numerical PDE techniques since the construction of a mesh in the standard
1978:
1961:
1491:
Y.C. Hon and W. Chen, Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems with complicated geometry,
2112:
1898:
1405:
1254:
40:
1504:
X.P. Chen, W.X. He and B.T. Jin, Symmetric boundary knot method for membrane vibrations under mixed-type boundary conditions,
1879:
1868:
1845:
1632:
1174:
1851:
1968:
1337:
denotes the collocation points located at
Dirichlet boundary and Neumann boundary respectively. The unknown coefficients
1933:
1973:
1627:
2133:
1652:
1543:
B.T. Jin, Y. Zheng, Boundary knot method for the Cauchy problem associated with the inhomogeneous
Helmholtz equation,
2090:
1439:
R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions,
2075:
1951:
1717:
1699:
1737:
1410:
350:{\displaystyle {\frac {\partial u}{\partial n}}=h\left(x,y\right),\ \ h\left(x,y\right)\in \partial \Omega _{N}}
2060:
1946:
1676:
1597:
Z.J. Fu; W. Chen, Q.H Qin, Boundary knot method for heat conduction in nonlinear functionally graded material,
1452:
W. Chen and M. Tanaka, A meshfree, exponential convergence, integration-free, and boundary-only RBF technique,
1420:
1415:
44:
1722:
800:
1517:
B.T. Jing and Z. Yao, Boundary knot method for some inverse problems associated with the
Helmholtz equation,
2102:
2080:
2065:
2048:
1956:
1941:
1857:
1373:
36:
28:
32:
2022:
1793:
1645:
433:
403:
2070:
1916:
1832:
1610:
D. Mehdi and S. Rezvan, A boundary-only meshfree method for numerical solution of the
Eikonal equation,
1377:
24:
1637:
2107:
1780:
1381:
1874:
1788:
1569:
F.Z. Wang, W. Chen, X.R. Jiang, Investigation of regularized techniques for boundary knot method.
2097:
2038:
1340:
1556:
B.T. Jin and W. Chen, Boundary knot method based on geodesic distance for anisotropic problems,
685:{\displaystyle u^{*}\left(x,y\right)=\sum \limits _{i=1}^{N}\alpha _{i}\phi \left(r_{i}\right)}
1727:
835:
48:
diffusion, convection-diffusion, and
Possion equations with very irregular 2D and 3D domains.
383:
2043:
2033:
1922:
1890:
1385:
20:
is proposed as an alternative boundary-type meshfree distance function collocation scheme.
2085:
2028:
2017:
1863:
1810:
864:
By employing the collocation technique to satisfy the boundary conditions (2) and (3),
570:
363:
1530:
W. Chen, L.J. Shen, Z.J. Shen, G.W. Yuan, Boundary knot method for
Poisson equations,
2127:
1732:
1904:
1821:
1798:
1584:
F.Z. Wang, Leevan L, W. Chen, Effective condition number for boundary knot method.
790:{\displaystyle r_{i}=\left\|\left(x,y\right)-\left(x_{i},y_{i}\right)\right\|_{2}}
1815:
1693:
237:{\displaystyle u=g\left(x,y\right),\ \ \left(x,y\right)\in \partial \Omega _{D}}
508:{\displaystyle \partial \Omega _{D}\cup \partial \Omega _{N}=\partial \Omega }
560:{\displaystyle \partial \Omega _{D}\cap \partial \Omega _{N}=\varnothing }
2001:
1840:
1572:
International Journal for Numerical Methods in Biomedical Engineering
567:. The BKM employs the non-singular general solution of the operator
460:
denote the Dirichlet and Neumann boundaries respectively, satisfied
144:{\displaystyle Lu=f\left(x,y\right),\ \ \left(x,y\right)\in \Omega }
1506:
International Journal of Nonlinear Science and Numerical Simulation
1995:
1989:
1804:
1641:
1519:
International Journal for Numerical Methods in Engineering
1493:
International Journal for Numerical Methods in Engineering
1330:{\displaystyle \left(x_{k},y_{k}\right)|_{k=m_{1}+1}^{m}}
1384:(FVM) for infinite domain, thin-walled structures, and
1244:{\displaystyle \left(x_{k},y_{k}\right)|_{k=1}^{m_{1}}}
1480:
Computer Methods in Applied Mechanics and Engineering
1343:
1257:
1177:
874:
838:
803:
701:
597:
573:
521:
466:
436:
406:
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366:
254:
161:
75:
1669:
Numerical methods for partial differential equations
2010:
1932:
1889:
1831:
1779:
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1708:
1684:
1675:
1633:
Examplary Matlab codes and geometric configurations
1356:
1329:
1243:
1160:
853:
820:
789:
684:
587:to approximate the numerical solution as follows,
579:
559:
507:
452:
422:
392:
372:
349:
236:
143:
1653:
8:
1599:Engineering Analysis with Boundary Elements
1545:Engineering Analysis with Boundary Elements
1532:Engineering Analysis with Boundary Elements
1467:Engineering Analysis with Boundary Elements
1454:Computers and Mathematics with Applications
1681:
1660:
1646:
1638:
1586:CMC: Computers, Materials, & Continua
1465:W. Chen, Symmetric boundary knot method,
1348:
1342:
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821:{\displaystyle \phi \left(\cdot \right)}
1432:
554:
400:represents the computational domain,
7:
1911:Moving particle semi-implicit method
1822:Weighted essentially non-oscillatory
453:{\displaystyle \partial \Omega _{N}}
423:{\displaystyle \partial \Omega _{D}}
1048:
920:
631:
1760:Finite-difference frequency-domain
1441:SIAM Journal on Numerical Analysis
1376:(BEM) is an alternative method to
1107:
1081:
828:is the general solution satisfied
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14:
65:Consider the following problems,
2139:Numerical differential equations
1558:Journal of Computational Physics
797:denotes the Euclidean distance,
2113:Method of fundamental solutions
1899:Smoothed-particle hydrodynamics
1406:Method of fundamental solutions
1368:History and recent developments
1119:
974:
41:method of fundamental solutions
1754:Alternating direction-implicit
1293:
1213:
777:
717:
380:is the differential operator,
16:In numerical mathematics, the
1:
1766:Finite-difference time-domain
1805:Advection upstream-splitting
1372:It has long been noted that
1816:Essentially non-oscillatory
1799:Monotonic upstream-centered
1411:Regularized meshfree method
1357:{\displaystyle \alpha _{i}}
2155:
2076:Infinite difference method
1694:Forward-time central-space
1495:, 1931-1948, 56(13), 2003.
18:boundary knot method (BKM)
1979:Poincaré–Steklov operator
1738:Method of characteristics
1575:, 26(12), 1868–1877, 2010
1996:Tearing and interconnect
1990:Balancing by constraints
1560:, 215(2), 614–629, 2006.
1421:Singular boundary method
1416:Boundary particle method
854:{\displaystyle L\phi =0}
45:singular boundary method
2103:Computer-assisted proof
2081:Infinite element method
1869:Gradient discretisation
1612:Computational Mechanics
1601:, 35(5), 729–734, 2011.
1534:, 29(8), 756–760, 2005.
1482:, 192, 1859–1875, 2003.
1469:, 26(6), 489–494, 2002.
1374:boundary element method
393:{\displaystyle \Omega }
37:boundary element method
29:boundary element method
2091:Petrov–Galerkin method
1852:Discontinuous Galerkin
1521:, 62, 1636–1651, 2005.
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650:
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351:
238:
145:
2071:Isogeometric analysis
1917:Material point method
1378:finite element method
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33:fundamental solutions
25:finite element method
2108:Integrable algorithm
1934:Domain decomposition
1628:Boundary knot method
1614:, 47, 283–294, 2011.
1588:, 12(1), 57–70, 2009
1547:, 29, 925–935, 2005.
1456:, 43, 379–391, 2002.
1382:finite volume method
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159:
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1952:Schwarz alternating
1875:Loubignac iteration
1508:, 6, 421–424, 2005.
1326:
1240:
2134:Numerical analysis
2098:Validated numerics
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234:
141:
2121:
2120:
2061:Immersed boundary
2054:Method of moments
1969:Neumann–Dirichlet
1962:abstract additive
1947:Fictitious domain
1891:Meshless/Meshfree
1775:
1774:
1677:Finite difference
1114:
580:{\displaystyle L}
373:{\displaystyle L}
308:
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273:
198:
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115:
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2066:Analytic element
2049:Boundary element
1942:Schur complement
1923:Particle-in-cell
1858:Spectral element
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1443:, 638–650, 1977.
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1386:inverse problems
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2086:Galerkin method
2029:Method of lines
2006:
1974:Neumann–Neumann
1928:
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1794:High-resolution
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231:
227:
223:
220:
216:
212:
209:
206:
202:
192:
188:
184:
181:
178:
174:
170:
167:
164:
152:
151:
140:
137:
133:
129:
126:
123:
119:
109:
105:
101:
98:
95:
91:
87:
84:
81:
78:
62:
59:
53:
50:
13:
10:
9:
6:
4:
3:
2:
2151:
2140:
2137:
2135:
2132:
2131:
2129:
2114:
2111:
2109:
2106:
2104:
2101:
2099:
2096:
2092:
2089:
2088:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2055:
2052:
2051:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2024:
2021:
2019:
2016:
2015:
2013:
2009:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1980:
1977:
1975:
1972:
1970:
1967:
1963:
1960:
1958:
1955:
1954:
1953:
1950:
1948:
1945:
1943:
1940:
1939:
1937:
1935:
1931:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1896:
1894:
1892:
1888:
1881:
1878:
1876:
1873:
1870:
1867:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1842:
1839:
1838:
1836:
1834:
1830:
1823:
1820:
1817:
1814:
1812:
1809:
1806:
1803:
1800:
1797:
1795:
1792:
1790:
1787:
1786:
1784:
1782:
1781:Finite volume
1778:
1767:
1764:
1761:
1758:
1755:
1752:
1751:
1749:
1745:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1715:
1713:
1711:
1707:
1701:
1698:
1695:
1692:
1691:
1689:
1687:
1683:
1680:
1678:
1674:
1670:
1663:
1658:
1656:
1651:
1649:
1644:
1643:
1640:
1634:
1631:
1629:
1626:
1625:
1621:
1613:
1607:
1604:
1600:
1594:
1591:
1587:
1581:
1578:
1574:
1573:
1566:
1563:
1559:
1553:
1550:
1546:
1540:
1537:
1533:
1527:
1524:
1520:
1514:
1511:
1507:
1501:
1498:
1494:
1488:
1485:
1481:
1475:
1472:
1468:
1462:
1459:
1455:
1449:
1446:
1442:
1436:
1433:
1426:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1403:
1399:
1397:
1393:
1389:
1387:
1383:
1379:
1375:
1367:
1365:
1349:
1345:
1322:
1317:
1314:
1309:
1305:
1301:
1298:
1287:
1281:
1277:
1273:
1268:
1264:
1259:
1234:
1230:
1224:
1221:
1218:
1207:
1201:
1197:
1193:
1188:
1184:
1179:
1151:
1148:
1145:
1142:
1139:
1136:
1131:
1127:
1123:
1120:
1116:
1110:
1101:
1096:
1092:
1088:
1084:
1073:
1069:
1063:
1058:
1055:
1052:
1044:
1040:
1034:
1030:
1026:
1021:
1017:
1012:
1008:
998:
994:
990:
987:
984:
981:
978:
975:
971:
967:
962:
958:
954:
950:
945:
941:
935:
930:
927:
924:
916:
912:
906:
902:
898:
893:
889:
884:
880:
867:
866:
865:
848:
845:
842:
839:
831:
830:
829:
814:
811:
808:
804:
782:
772:
766:
762:
758:
753:
749:
744:
740:
736:
732:
729:
726:
722:
712:
707:
703:
678:
673:
669:
665:
661:
656:
652:
646:
641:
638:
635:
627:
623:
619:
616:
613:
609:
603:
599:
590:
589:
588:
574:
551:
546:
535:
530:
496:
491:
480:
475:
445:
415:
367:
342:
331:
327:
323:
320:
317:
313:
309:
300:
296:
292:
289:
286:
282:
278:
275:
269:
261:
247:
246:
229:
218:
214:
210:
207:
204:
200:
190:
186:
182:
179:
176:
172:
168:
165:
162:
154:
153:
135:
131:
127:
124:
121:
117:
107:
103:
99:
96:
93:
89:
85:
82:
79:
76:
68:
67:
66:
60:
58:
51:
49:
46:
42:
38:
34:
30:
26:
21:
19:
1905:Peridynamics
1723:Lax–Wendroff
1611:
1606:
1598:
1593:
1585:
1580:
1570:
1565:
1557:
1552:
1544:
1539:
1531:
1526:
1518:
1513:
1505:
1500:
1492:
1487:
1479:
1474:
1466:
1461:
1453:
1448:
1440:
1435:
1394:
1390:
1371:
1170:
863:
694:
359:
64:
55:
22:
17:
15:
2039:Collocation
61:Formulation
52:Description
2128:Categories
1728:MacCormack
1710:Hyperbolic
1427:References
1380:(FEM) and
35:, such as
2044:Level-set
2034:Multigrid
1984:Balancing
1686:Parabolic
1346:α
1146:…
1108:∂
1085:ϕ
1082:∂
1070:α
1049:∑
988:…
951:ϕ
942:α
921:∑
843:ϕ
812:⋅
805:ϕ
741:−
662:ϕ
653:α
632:∑
604:∗
555:∅
543:Ω
539:∂
536:∩
527:Ω
523:∂
503:Ω
500:∂
488:Ω
484:∂
481:∪
472:Ω
468:∂
442:Ω
438:∂
412:Ω
408:∂
388:Ω
339:Ω
335:∂
332:∈
267:∂
259:∂
226:Ω
222:∂
219:∈
139:Ω
136:∈
2018:Spectral
1957:additive
1880:Smoothed
1846:Extended
1400:See also
778:‖
718:‖
2002:FETI-DP
1882:(S-FEM)
1801:(MUSCL)
1789:Godunov
2011:Others
1998:(FETI)
1992:(BDDC)
1864:Mortar
1848:(XFEM)
1841:hp-FEM
1824:(WENO)
1807:(AUSM)
1768:(FDTD)
1762:(FDFD)
1747:Others
1733:Upwind
1696:(FTCS)
1171:where
695:where
360:where
307:
304:
197:
194:
114:
111:
2025:(DVR)
1986:(BDD)
1925:(PIC)
1919:(MPM)
1913:(MPS)
1901:(SPH)
1871:(GDM)
1860:(SEM)
1818:(ENO)
1756:(ADI)
1907:(PD)
1854:(DG)
1251:and
591:(4)
515:and
430:and
248:(3)
155:(2)
69:(1)
43:and
27:and
868:(6)
832:(5)
2130::
39:,
1661:e
1654:t
1647:v
1350:i
1323:m
1318:1
1315:+
1310:1
1306:m
1302:=
1299:k
1294:|
1288:)
1282:k
1278:y
1274:,
1269:k
1265:x
1260:(
1235:1
1231:m
1225:1
1222:=
1219:k
1214:|
1208:)
1202:k
1198:y
1194:,
1189:k
1185:x
1180:(
1152:m
1149:,
1143:,
1140:1
1137:+
1132:1
1128:m
1124:=
1121:k
1117:,
1111:n
1102:)
1097:i
1093:r
1089:(
1074:i
1064:N
1059:1
1056:=
1053:i
1045:=
1041:)
1035:k
1031:y
1027:,
1022:k
1018:x
1013:(
1009:h
999:1
995:m
991:,
985:,
982:1
979:=
976:k
972:,
968:)
963:i
959:r
955:(
946:i
936:N
931:1
928:=
925:i
917:=
913:)
907:k
903:y
899:,
894:k
890:x
885:(
881:g
849:0
846:=
840:L
815:)
809:(
783:2
773:)
767:i
763:y
759:,
754:i
750:x
745:(
737:)
733:y
730:,
727:x
723:(
713:=
708:i
704:r
679:)
674:i
670:r
666:(
657:i
647:N
642:1
639:=
636:i
628:=
624:)
620:y
617:,
614:x
610:(
600:u
575:L
552:=
547:N
531:D
497:=
492:N
476:D
446:N
416:D
368:L
343:N
328:)
324:y
321:,
318:x
314:(
310:h
301:,
297:)
293:y
290:,
287:x
283:(
279:h
276:=
270:n
262:u
230:D
215:)
211:y
208:,
205:x
201:(
191:,
187:)
183:y
180:,
177:x
173:(
169:g
166:=
163:u
132:)
128:y
125:,
122:x
118:(
108:,
104:)
100:y
97:,
94:x
90:(
86:f
83:=
80:u
77:L
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