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However, these special log-type schemes cannot be so readily used on other types of fundamental solution so for a general purpose implementation, Gaussian quadrature is still the norm. 6: Illustration of the decrease in ri as node i approaches element ,j . subelement. 11). 8. It should be noted that research still continues in an attempt to find more efficient ways of evaluating the boundary element integrals. 6 The Three-Dimensional Boundary Element Method The three-dimensional boundary element method is very similar to the two-dimensional boundary element method discussed above.
7) dx dx where u = 'n un and ! = 'm . Since 'n and 'm are both functions of the derivatives with respect to x need to be converted to derivatives with respect to . 8) d dx d dx 0 d is dx 1 evaluated by substituting the finite element approximation x = 'n :Xn . In this case x = or 3 d = 3 and the Jacobian is J = dx = 1 . The term multiplying the nodal parameters u is called n dx d 3 the element stiffness matrix, Emn Notice that un has been taken outside the integral because it is not a function of .
14) ,," ," We must now investigate this equation as lim"0 . There are 4 integrals to consider, and we look at each of these in turn. 3: Illustration of enlarged domain when singular point is on the boundary. Firstly consider Z @! Z u @n d, = ," ," Z = ," @ , 1 log r d, u @n 2 by definition of ! @ , 1 log r d, u @r 2 since Z u 1 = , 2 r d, Z 1 1 = , 2 " u d, ," 1 1 ! , 2 " u P " @ @ on , " @n @r ," since r = " on ," by the mean value theorem for a surface with a unique tangent at P . Thus 1 u P u P Z @!