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BOUNDARY ELEMENTS 41 that is, the displacement u( x) at any point is expressed in terms of the displacement u and the derivative a~ at the boundary. Generally, the boundary is a line or surface that has to be discretized by boundary elements. For this one-dimensional problem, the boundary consists only of the point x = a and no discretization is necessary. To reduce the whole problem to the boundary, the source point x is moved to a. 56) = ikEAu as before. 55). 2 Time-domain boundary elements Now consider the same problem directly in the time domain.

1 shows the reflection coefficient for different orders N and distances L of the artificial boundary. All variables are given in dimensionless form. The left figure shows the undamped case. It is evident that for frequencies below the cut-off frequency only the distance L is important whereas for frequencies above the cut-off frequency the result depends only on the order N. At the cut-off frequency itself, the reflection coefficient is always 1. It should be noted that in a practical case both Nand L may only be chosen up to a certain limit.

High-order transmitting boundaries use high-order derivatives, which are usually not available and a large distance L increases the size of the model, making the original purpose of the transmitting boundary, to reduce the model, useless. 05) reduces the reflection coefficient, especially at the cut-off frequency in the case of larger distances, as shown in the right figure. 2. The figure compares (iw/c)PN for different orders N to the exact solution ik. This comparison is made at the transmitting boundary itself (L = 0).

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Analysis of WWII Air Combat Records [CONFIDENTIAL]

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