基于α有限元法的二维水下声散射计算

Alpha finite element method for two-dimensional underwater acoustic scattering numerical computation

众所周知,由于存在插值误差和污染误差的原因,高波数声学问题的有限元解是不可靠的。为了提高有限元法的求解精度,本文提出了一种新型的α有限元法来求解水下声散射问题。在α有限元法中,首先运用基于点的梯度光滑技术得到点光滑的梯度场,然后对点光滑有限元法进行推导。因此,α有限元模型既包含来自点光滑有限元模型的梯度成分又包含来自标准光滑有限元模型的梯度成分,充分利用了点光滑有限元模型的"过软"特性和标准有限元模型的"过刚"特性。为了处理外声场问题,本文运用了Dt N无反射边界条件。数值计算结果表明:与标准有限元法相比,α有限元法在水下声散射计算中具备更高的计算精度和计算效率。

It is well known that the finite element method（ FEM） is unreliable to solve acoustic problems at high wavenumbers due to the interpolation and pollution error. In order to improve the performance of the standard FEM solutions for acoustic problems,a novel alpha finite element method（ α-FEM） is presented to analyze the underwater acoustic scattering problems. In the α-FEM,the node-based gradient smoothing technique is used to obtain the smoothed node-based gradient field and then the node-based smoothed finite element（ NS-FEM） is formulated. The α-FEM contains both the gradient components from the NS-FEM and the gradient components from the standard FEM,and takes advantage of the properties from the＂overly-soft＂NS-FEM and the＂overly-stiff＂FEM.In order to handle the exterior acoustic problems,the Dirichle-to-Neumann map is introduced. Several typical numerical examples are presented and it is verified that the α-FEM possesses higher computational efficiency and can provide more accurate solutions than standard FEM for underwater acoustic scattering problems.