# Boundary Element Method

• Qing-Hua Qin
Chapter

## Abstract

In Chapter 2 Green’s functions in piezoelectric materials were described. Applications of these Green’s functions to the boundary element method (BEM) are discussed in this chapter. In contrast to the finite element method (FEM), BEM involves only discretization of the boundary of the structure due to the governing differential equation being satisfied exactly inside the domain leading to a relatively smaller system size with sufficient accuracy. This is an important advantage over domain-type solutions such as FEM or the finite difference method. During the past two decades several BEM techniques have been successfully developed for analyzing structure performance with piezoelectric materials [1–4]. Lee and Jiang [1] derived the boundary integral equation of piezoelectric media by the method of weighted residuals for plane piezoelectricity. Lu and Mahrenholtz [5] presented a variational boundary integral equation for the same problem. Ding, Wang, and Chen [6] developed a boundary integral formulation that is efficient for analyzing crack problems in piezoelectric material. Rajapakse [7] discussed three boundary element methods (direct boundary method, indirect boundary element method, and fictitious stress-electric charge method) in coupled electroelastic problems.

## Keywords

Boundary Element Method Piezoelectric Material Boundary Integral Equation Electric Displacement Boundary Integral Formulation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Lee JS, Jiang LZ (1994) A boundary integral formulation and 2D fundamental solution for piezoelectric media. Mech Res Comm 21: 47–54
2. 2.
Qin QH (2001) Fracture Mechanics of Piezoelectric Materials. WIT Press, SouthamptonGoogle Scholar
3. 3.
Qin QH (2004) Material Properties of Piezoelectric Composites by BEM and Homogenization Method. Composite Struc 66: 295–299
4. 4.
Qin QH (2007) Green’s Function and Boundary Elements in Multifield Materials. Elsevier, OxfordGoogle Scholar
5. 5.
Lu P, Mahrenholtz O (1994) A variational boundary element formulation for piezoelectricity. Mech Res Comm 21: 605–611
6. 6.
Ding HJ, Wang GP, Chen WQ (1998) Boundary integral formulation and 2D fundamental solutions for piezoelectric media. Comp Meth Appl Mech Eng 158: 65–80
7. 7.
Rajapakse RK (1997) Boundary element methods for piezoelectric solids. Procs of SPIE, Mathematics and Control in Smart Structures 3039: 418–428Google Scholar
8. 8.
Xu XL, Rajapakse RKND (1998) Boundary element analysis of piezoelectric solids with defects. Composites Part B: Eng 29: 655–669
9. 9.
Rajapakse RKND, Xu XL (2001) Boundary element modeling of cracks in piezoelectric solids. Eng Anal Boun Elem 25: 771–781
10. 10.
Liu YJ, Fan H (2001) On the conventional boundary integral formulation for piezoelectric solids with defects or of thin shapes. Eng Anal Boun Elem 25: 77–91
11. 11.
Pan E (1999) A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids. Eng Anal Boun Elem 23: 67–76
12. 12.
Denda M, Lua J (1999) Development of the boundary element method for 2D piezoelectricity. Composites: Part B 30: 699–707
13. 13.
Davi G, Milazzo A (2001) Multidomain boundary integral formulation for piezoelectric materials fracture mechanics. Int J Solids Struct 38: 7065–7078
14. 14.
Groh U, Kuna M (2005) Efficient boundary element analysis of cracks in 2D piezoelectric structures. Int J Solids Struct 42: 2399–2416
15. 15.
Khutoryaansky N, Sosa H, Zu WH (1995) Approximate Green’s functions and a boundary element method for electroelastic analysis of active materials. Comput Struct 66: 289–299
16. 16.
Chen T, Lin FZ (1995) Boundary integral formulations for three–dimensional anisotropic piezoelectric solids. Comput Mech 15: 485–496
17. 17.
Schclar NA (1994) Anisotropic Analysis Using Boundary Elements. Computational Mechanics, Southampton
18. 18.
Beer G (2001) Programming the Boundary Element Method. John Wiley & Sons, ChichesterGoogle Scholar
19. 19.
Stroud AH, Secrest D (1966) Gaussian Quadrature Formulas. Prentice-Hall, New York
20. 20.
Diego N (2002) Thermoelastic Fracture Mechanics Using Boundary Elements. WIT Press, Southampton
21. 21.
Garcia-Sanchez F, Saez A, Dominguez J (2005) Anisotropic and piezoelectric materials fracture analysis by BEM. Comput Struct 83: 804–820
22. 22.
Lachat JC, Watson JO (1976) Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics. Int J Num Meth Eng 10: 991–1005
23. 23.
Rizzo FJ, Shippy DJ (1977) An advanced boundary integral equation method for three-dimensional thermoelasticity. Int J Num Meth Eng 11: 1753–1768
24. 24.
Nardini D, Brebbia CA (1982) A new approach to free vibration analysis using boundary elements. In: Boundary Element Methods in Engineering. Springer, BerlinGoogle Scholar
25. 25.
Partridge PW, Brebbia CA, Wrobel LC (1992) The Dual Reciprocity Boundary Element Method. Computational Mechanics, Southampton
26. 26.
Kogl M, Gaul L (2000) A boundary element method for transient piezoelectric analysis. Eng Anal Boun Elements 24: 591–598
27. 27.
Atkinson KE (1985) The numerical evaluation of particular solutions for Poisson’s equation. IMAJ Numer Anal 5: 319–338
28. 28.
Golberg MA (1995) The numerical evaluation of particular solutions in the BEM-a review. Bound Elem Commun 6: 99–106Google Scholar
29. 29.
Grundemann H (1989) A general procedure transferring domain integrals onto boundary integrals in BEM. Eng Anal Boun Elem 6: 214–222