In this paper, the four integral identities satisfied by the fundamental solution for elastostatic problems are reviewed and slightly different forms of the third and fourth identities are presented. Two new identities, namely the fifth and sixth identities, are derived. These integral identities can be used to develop weakly singular and nonsingular forms of the boundary integral equations (BIEs) for elastostatic problems. They can also be employed to show the nonuniqueness of the solution of the traction (hypersingular) BIE for an elastic body on a multiconnected domain. This nonuniqueness is shown in a general setting in this paper. It is shown that the displacement (singular) BIE does not allow any rigid-body displacement terms, while the traction BIE can have arbitrary rigid-body translation and rotation terms, in the BIE solutions on the edge of a hole or surface of a void. Therefore, the displacement solution from the traction BIE is not unique. A remedy to this nonuniqueness solution problem with the traction BIE is proposed by adopting a dual BIE formulation for problems with multiconnected domains. A few numerical examples using the 2D elastostatic boundary element method for domains with holes are presented to demonstrate the uniqueness properties of the displacement, traction and the dual BIE solutions for multiconnected domain problems.

References

1.
Rizzo
,
F. J.
,
1967
, “
An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics
,”
Q. Appl. Math.
,
25
, pp.
83
95
.
2.
Liu
,
Y. J.
,
Mukherjee
,
S.
,
Nishimura
,
N.
,
Schanz
,
M.
,
Ye
,
W.
,
Sutradhar
,
A.
,
Pan
,
E.
,
Dumont
,
N.A.
,
Frangi
,
A.
, and
Saez
,
A.
,
2011
, “
Recent Advances and Emerging Applications of the Boundary Element Method
,”
ASME Appl. Mech. Rev.
,
64
, pp.
1
38
.10.1115/1.4005491
3.
Chen
,
J. T.
,
Kou
,
S. R.
, and
Lin
,
J. H.
,
2002
, “
Analytic Study and Numerical Experiments for Degenerate Scale Problems in the Boundary Element Method for Two-Dimensional Elasticity
,”
Int. J. Numer. Methods Eng.
,
54
, pp.
1669
1681
.10.1002/nme.476
4.
Krishnasamy
,
G.
,
Rizzo
,
F. J.
, and
Liu
,
Y. J.
,
1994
, “
Boundary Integral Equations for Thin Bodies
,”
Int. J. Numer. Methods Eng.
,
37
, pp.
107
121
.10.1002/nme.1620370108
5.
Liu
,
Y. J.
,
1998
, “
Analysis of Shell-Like Structures by the Boundary Element Method Based on 3-D Elasticity: Formulation and Verification
,”
Int. J. Numer. Methods Eng.
,
41
, pp.
541
558
.10.1002/(SICI)1097-0207(19980215)41:3<541::AID-NME298>3.0.CO;2-K
6.
Liu
Y. J.
, and
Rudolphi
,
T. J.
,
1991
, “
Some Identities for Fundamental Solutions and Their Applications to Weakly-Singular Boundary Element Formulations
,”
Eng. Anal. Boundary Elements
,
8
, pp.
301
311
.10.1016/0955-7997(91)90043-S
7.
Liu
Y. J.
, and
Rudolphi
,
T. J.
,
1999
, “
New Identities for Fundamental Solutions and Their Applications to Non-Singular Boundary Element Formulations
,”
Comput. Mech.
,
24
, pp.
286
292
.10.1007/s004660050517
8.
Liu
,
Y. J.
,
2000
, “
On the Simple-Solution Method and Non-Singular Nature of the BIE/BEM—A Review and Some New Results
,”
Eng. Anal. Boundary Elements
, 24(10), pp. 789–795.10.1016/S0955-7997(00)00061-8
9.
Frangi
,
A.
, and
Novati
,
G.
,
1996
, “
Symmetric BE Method in Two-Dimensional Elasticity: Evaluation of Double Integrals for Curved Elements
,”
Comput. Mech.
,
19
, pp.
58
68
.10.1007/BF02757784
10.
Perez-Gavilan
,
J. J.
, and
Aliabadi
,
M. H.
,
2001
, “
Symmetric Galerkin BEM for Multi-Connected Bodies
,”
Commun. Numer. Methods Eng.
,
17
, pp.
761
770
.10.1002/cnm.444
11.
Liu
,
Y. J.
,
2008
, “
A New Fast Multipole Boundary Element Method for Solving 2-D Stokes Flow Problems Based on a Dual BIE Formulation
,”
Eng. Anal. Boundary Elements
,
32
, pp.
139
151
.10.1016/j.enganabound.2007.07.005
12.
Liu
,
Y. J.
,
2009
,
Fast Multipole Boundary Element Method—Theory and Applications in Engineering
,
Cambridge University
,
Cambridge
.
13.
Cruse
,
T. A.
,
1974
, “
An Improved Boundary-Integral Equation Method for Three Dimensional Elastic Stress Analysis
,”
Comput. Struct.
,
4
, pp.
741
754
.10.1016/0045-7949(74)90042-X
14.
Rizzo
,
F. J.
, and
Shippy
,
D. J.
,
1977
, “
An Advanced Boundary Integral Equation Method for Three-Dimensional Thermoelasticity
,”
Int. J. Numer. Methods in Eng.
,
11
, pp.
1753
1768
.10.1002/nme.1620111109
15.
Rudolphi
,
T. J.
,
1991
, “
The Use of Simple Solutions in the Regularization of Hypersingular Boundary Integral Equations
,”
Math. Comput. Modell.
,
15
, pp.
269
278
.10.1016/0895-7177(91)90071-E
16.
Liu
,
Y. J.
, and
Rizzo
,
F. J.
,
1993
, “
Hypersingular Boundary Integral Equations for Radiation and Scattering of Elastic Waves in Three Dimensions
,”
Comput. Methods Appl. Mech. Eng.
,
107
, pp.
131
144
.10.1016/0045-7825(93)90171-S
17.
Burton
,
A. J.
, and
Miller
,
G. F.
,
1971
, “
The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary-Value Problems
,”
Proc. R. Soc. London, Ser. A
,
323
, pp.
201
210
.10.1098/rspa.1971.0097
18.
Liu
,
Y. J.
, and
Rizzo
,
F. J.
, “
A Weakly-Singular Form of the Hypersingular Boundary Integral Equation Applied to 3-D Acoustic Wave Problems
,”
Comput. Methods Appl. Mech. Eng.
,
96
, pp.
271
287
.10.1016/0045-7825(92)90136-8
19.
Liu
,
Y. J.
, and
Chen
,
S. H.
,
1999
, “
A New Form of the Hypersingular Boundary Integral Equation for 3-D Acoustics and its Implementation With C0 Boundary Elements
,”
Comput. Methods Appl. Mech. Eng.
,
173
, pp.
375
386
.10.1016/S0045-7825(98)00292-8
20.
Liu
,
Y. J.
, and
Shen
,
L.
,
2007
, “
A Dual BIE Approach for Large-Scale Modeling of 3-D Electrostatic Problems With the Fast Multipole Boundary Element Method
,”
Int. J. Numer. Methods Eng.
,
71
, pp.
837
855
.10.1002/nme.2000
21.
Liu
,
Y. J.
, and
Rizzo
,
F. J.
,
1997
, “
Scattering of Elastic Waves From Thin Shapes in Three Dimensions Using the Composite Boundary Integral Equation Formulation
,”
J. Acoust. Soc. Am.
,
102
(
2
), pp.
926
932
.10.1121/1.419912
22.
Aliabadi
,
M. H.
,
2002
,
The Boundary Element Method—Vol. 2 Applications in Solids and Structures
,
Wiley
,
Chichester
, UK.
23.
Liu
,
Y. J.
,
2006
, “
Dual BIE Approaches for Modeling Electrostatic MEMS Problems With Thin Beams and Accelerated by the Fast Multipole Method
,”
Eng. Anal. Boundary Elements
,
30
, pp.
940
948
.10.1016/j.enganabound.2006.04.010
24.
Frangi
,
A.
,
2005
, “
A Fast Multipole Implementation of the Qualocation Mixed-Velocity-Traction Approach for Exterior Stokes Flows
,”
Eng. Anal. Boundary Elements
,
29
, pp.
1039
1046
.10.1016/j.enganabound.2005.05.010
You do not currently have access to this content.