A numerical iteration method for the Cauchy problem in linear elastic dynamics
Applicable Nonlinear Analysis, Volume 1, Issue 2, December 2024, Pages: 170–186
YAO SUN
College of Science, Civil Aviation University of China, Tianjin, China
MIN JIANG
College of Science, Civil Aviation University of China, Tianjin, China
Abstract
This paper investigates numerical solution methods for the Cauchy problem of the Navier equation on a connected domain. We utilize the measured Cauchy data on a portion of the domain boundary to invert the unknown Cauchy data on another portion of the boundary. First, we present the corresponding mathematical model, i.e., the Navier equation, and combine it with the known Cauchy data. Then, we discretize this mathematical model to obtain a corresponding ill-conditioned linear system of equations. We employ the pseudoinverse method, conjugate gradient method, and Tikhonov regularization method based on the Morozov discrepancy principle to solve this system of equations. The effectiveness of the algorithms is evaluated using spectral analysis. Next, we propose an improved Landweber iteration method. We first demonstrate the effectiveness of the algorithm using a filtering function and then analyze the errors of the algorithm to prove its stability and accuracy. Finally, we verify the stability and accuracy of the method through numerical experiments. Therefore, the proposed method is feasible and the numerical solutions obtained by this method can better approximate the true solution compared to several previous methods.
Cite this Article as
Yao Sun and Min Jiang, A numerical iteration method for the Cauchy problem in linear elastic dynamics, Applicable Nonlinear Analysis, 1(2), 170–186, 2024