Federal Institute for Materials Research and Testing, 12200 Berlin, Germany.
School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia.
Ultrasonics. 2014 Jul;54(5):1373-85. doi: 10.1016/j.ultras.2014.02.004. Epub 2014 Feb 14.
This paper addresses the computation of dispersion curves and mode shapes of elastic guided waves in axisymmetric waveguides. The approach is based on a Scaled Boundary Finite Element formulation, that has previously been presented for plate structures and general three-dimensional waveguides with complex cross-section. The formulation leads to a Hamiltonian eigenvalue problem for the computation of wavenumbers and displacement amplitudes, that can be solved very efficiently. In the axisymmetric representation, only the radial direction in a cylindrical coordinate system has to be discretized, while the circumferential direction as well as the direction of propagation are described analytically. It is demonstrated, how the computational costs can drastically be reduced by employing spectral elements of extremely high order. Additionally, an alternative formulation is presented, that leads to real coefficient matrices. It is discussed, how these two approaches affect the computational efficiency, depending on the elasticity matrix. In the case of solid cylinders, the singularity of the governing equations that occurs in the center of the cross-section is avoided by changing the quadrature scheme. Numerical examples show the applicability of the approach to homogeneous as well as layered structures with isotropic or anisotropic material behavior.
本文针对轴对称波导中弹性导波的频散曲线和模态形状的计算问题进行了研究。该方法基于比例边界有限元公式，该公式之前已经被提出用于板结构和具有复杂横截面的一般三维波导。该公式提出了一个哈密顿特征值问题，用于计算波数和位移幅度，这个问题可以非常有效地求解。在轴对称表示中，仅需对圆柱坐标系中的径向方向进行离散化，而圆周方向以及传播方向则通过解析方式进行描述。本文展示了如何通过采用极高阶的谱元来显著降低计算成本。此外，还提出了一种导致实系数矩阵的替代公式。讨论了这两种方法如何根据弹性矩阵影响计算效率。在实心圆柱体的情况下，通过改变求积方案，可以避免横截面中心处控制方程的奇点。数值示例展示了该方法在各向同性或各向异性材料行为的同质和分层结构中的适用性。
