By Ken Hayami
In 3 dimensional boundary aspect research, computation of integrals is a vital point because it governs the accuracy of the research and likewise since it often takes the foremost a part of the CPU time. The integrals which make sure the impact matrices, the interior box and its gradients comprise (nearly) singular kernels of order lIr a (0:= 1,2,3,4,.··) the place r is the space among the resource aspect and the combination element at the boundary aspect. For planar components, analytical integration can be attainable 1,2,6. in spite of the fact that, it's changing into more and more vital in functional boundary point codes to take advantage of curved components, equivalent to the isoparametric parts, to version basic curved surfaces. seeing that analytical integration isn't attainable for basic isoparametric curved parts, one has to depend on numerical integration. whilst the space d among the resource element and the point over which the combination is played is adequately huge in comparison to the aspect measurement (d> 1), the normal Gauss-Legendre quadrature formulation 1,3 works successfully. besides the fact that, while the resource is absolutely at the point (d=O), the kernel 1I~ turns into singular and the easy program of the Gauss-Legendre quadrature formulation breaks down. those integrals should be known as singular integrals. Singular integrals happen whilst calculating the diagonals of the effect matrices.
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Additional info for A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals
59) . This means that the Cauchy principal value for a~ 2. 59) . Alternatively, the physical concept equivalent to rigid body motion in elasticity may be used to calculate the diagonal coefficients of the influence matrices. 48) instead of interpolating on the boundary element. 46) does have a Cauchy principal value and can be calculated directly using a method similar to that of Gray 28 and Rudolphi et al. 51, or as a limit of a nearly singular integral by the method which will be proposed in Chapter II.
65) 41f I and the calculation of the singular integral H .. 66) = x. 67) (r,n) 41f r3 become unnecessary. This technique is equivalent to what is known as the use of rigid body motion in elastostatics. On the other hand, it is a good check to calculate Hii directly from Ci and Hij. For discontinuous elements 1, Ci = 1 / 2 for i = 1-N, and calculating the singular integrals Hii directly are reported to give more accurate results 15. 68) where u* = 1/(47l"r) and G ii = L k x = x. 71) which contribute to the non-diagonal element Hij and Gij.
43) where au = ( r , l' r, 1) , a 12 = (r , 1 . 56) --0(1), r Hence, the kernel of the integrals in gee kl , hee kl (k=t= l) are of the order 0(1) or O(r) , which means that they have a even weaker singularity compared to O( 1/ r) for integrals in gee ll and heell , and will not cause any substantial difficulty when integrating them numerically. 58) so that the Jacobian r in dS = r dr d8 will cancell the singularity in
A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals by Ken Hayami