Highly accurate space-time coupled least-squares finite element framework in studying wave propagation

Abstract

Simulation of stress wave propagation through solid medium is commonly carried using Galerkin weak-form cast over decoupled space and time domains. In this paper, accuracy of this commonly utilized framework is compared to that of the variationally-consistent least-squares form of the wave equation cast over space-time domain. The two formulations are tested for numerical dispersion and numerical diffusion, through two test cases. The first case studies the dispersion in harmonic shear wave propagation through a soil column over a wide range of forcing frequencies. The second test case investigates numerical diffusion in an axial wave propagation generated by constant force; which is removed after a certain time to allow free vibration to take place. Low numerical dispersion and numerical diffusion as well as high rates of convergence are the main advantages of the coupled least-squares (CLS) computational framework; when compared to the decoupled Galerkin (DG) framework. Based on studies presented here, CLS has low dispersion; yielding errors with one to two orders of magnitude less than that of DG. Also, the numerical diffusion present in DG framework causes a %40 error in DG's prediction of the stress-wave intensity. Furthermore, accumulative error during evolution is virtually nonexistent for CLS, whereas, the error steadily increases as the solution evolves in DG framework. It is also demonstrated that CLS feature of temporal meshing allows for faster computations.

Convergence Study of Variational Space‐Time Coupled Least‐Squares Frameworks in Simulation of Wave Propagation in Viscoelastic Medium

“…e first is the space-time coupled least squares (cLs) formulation, which was recently employed for solving the undamped wave equation [31]. e second is the widely utilized space-time decoupled Galerkin (dGa) formulation along with the unconditionally stable Newmark-β method.…”

“…In general, numerical errors have dispersive and diffusive (dissipative) characteristics [31]. Each of the two test models employed here exhibits dominance of one type of numerical error.…”

“…e result is an oscillating rectangular wave with sharp temporal gradient, the sharpness of which depends on the amount physical damping present in the system. Numerical simulation of undamped impact case [31] shows that the dominant error is dispersive in nature and it is most significant at the fixed boundary, as shown in Figure 1(a). Fictitious oscillations observed here decrease naturally with increase in system damping; i.e., depending on amount of damping present in the system, the square profile of the stress wave will assume smoother corners, thereby reducing the oscillatory overshoots.…”

“…In a recent paper [31], it was shown that numerical errors can be significantly reduced if the problem is cast in a spacetime coupled least squares finite element framework. As an extension to that work, in this paper, convergence characteristics of damped wave equation are studied for the decoupled Galerkin and coupled least squares formulations.…”