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A Projection Transformation Method for Nearly Singular by Ken Hayami

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 be certain the impression matrices, the inner box and its gradients comprise (nearly) singular kernels of order lIr a (0:= 1,2,3,4,.··) the place r is the gap among the resource aspect and the combination element at the boundary aspect. For planar parts, analytical integration could be attainable 1,2,6. notwithstanding, it really is turning into more and more very important in sensible boundary point codes to take advantage of curved components, akin to the isoparametric components, to version normal curved surfaces. due to the fact analytical integration isn't really attainable for basic isoparametric curved parts, one has to depend upon numerical integration. whilst the gap d among the resource aspect and the point over which the combination is played is satisfactorily huge in comparison to the aspect measurement (d> 1), the normal Gauss-Legendre quadrature formulation 1,3 works successfully. even though, while the resource is admittedly at the aspect (d=O), the kernel 1I~ turns into singular and the trouble-free program of the Gauss-Legendre quadrature formulation breaks down. those integrals could be known as singular integrals. Singular integrals happen whilst calculating the diagonals of the impact matrices.

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Extra info for A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals

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E. r - 0, r - t -+ r where t is the unit tangent vector in the direction of r (as r - 0) , as shown in Fig. 5. e. 118) n -+ 1 as shown in Fig. 6 .

DB q ax. 61). This is reasonable since, physically, one would expect finite values for au/axs from the interpolation of u, au/an, au/at in the boundary S , where d = 0 . 55) have finite values. But it is difficult to calculate them accurately and efficiently using the standard Gauss-Legendre product formula, since the value of the kernels vary very rapidly near the source point Xs. In fact the nearly singular integrals 54 turn out to be more difficult to calculate than the singular integrals (O

46) is of order O(lIr2) and O(lIr3), suggesting that the integral does not have a finite value, which is contrary to the fact that q == au/an and au/at 43 (alat: tangential derivative at Xs on the boundary S) usually have finite values on the boundary. However, it will be shown in the following that the integrals Iau* and Iaq* do have Cauchy principal values. Since only the neighborhood of Xs is relevant, so long as the singularity is concerned, let us assume that the boundary S is smooth at tangential planar disc Sa of radius a centered at Xs Xs and take a local as shown in Fig.

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