Method for calculating multiwave scattering by layered anisotropic media. (English) Zbl 1524.74464
Summary: The method of transfer matrices of orders \(n=2\) and \(n=4\) is known from the theory of the propagation of light and elastic waves in layered media. The development of this matrix method is presented for cases \(n>4\). The transfer matrix \(\boldsymbol{T}\) of waves through a homogeneous layer of thickness \(d\) is defined as the matrix exponential \(\boldsymbol{T} = \exp (\boldsymbol{W}d)\). The elements of the matrix \(\boldsymbol{W}\) depend on the physical properties of the layer and are determined from the differential equations describing the waves. The matrix \(\boldsymbol{T}\) is calculated using the scaling and squaring method: \(\boldsymbol{T} = [\exp (\boldsymbol{W} d/(2^j))]^{2^j}\). A new way is proposed for choosing the scaling parameter \(m=2^j\), which provides the required accuracy in computing the elements of the matrix \(\boldsymbol{T}\). Two procedures for calculating the exponential of the scaled matrix \(\boldsymbol{A} =\boldsymbol{W} d /(2^j)\) are considered: 1) truncation of the Taylor series, 2) polynomial representation of the matrix exponential. Both of these methods do not require finding the eigenvalues of the matrix \(\boldsymbol{A}\). The effectiveness of these methods is compared. It is shown that when calculating the matrix exponential with double precision, the polynomial method is preferable for matrices of order \(n<10\). An alternative to multiple squaring for the cases \(j>n\) is proposed. The application of the scaling method is shown by the example of calculations of six-beam scattering of elastic waves by an anisotropic layer.
MSC:
| 74S99 | Numerical and other methods in solid mechanics |
| 65F60 | Numerical computation of matrix exponential and similar matrix functions |
| 15A16 | Matrix exponential and similar functions of matrices |
| 74J20 | Wave scattering in solid mechanics |
| 78A45 | Diffraction, scattering |
Keywords:
matrix exponential; relative truncation error; diffraction of elastic waves; acoustic stressesReferences:
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