Abstract: To improve the computing accuracy and application ability for practical structures with complex boundaries, the discontinuous linear and quadratic boundary elements were discussed. The conventional boundary integral equation method (CBIE) was used for the numerical solution, the performance of different types of boundary elements were presented and compared, the effect of the nodal position at discontinuous elements on the computing accuracy was studied and the optional nodal position was obtained. But the CBIE may produce the non-uniqueness problems, Burton-Miller method (BM) could be used to overcome this problem. The performance of different types of boundary elements based on BM was presented and compared. The value of the optimal nodal position for discontinuous elements was very close to the zeroes of Legendre polynomial for CBIE. However, different performance was found when BM was used, and the empirical values for optimal nodal positions for BM were concluded.