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.
There are many novel applications of space-time decoupled least squares and Galerkin formulations that simulate wave propagation through different types of media. Numerical simulation of stress wave propagation through viscoelastic medium is commonly carried out using the space-time decoupled Galerkin weak form in site response problem, etc. In a recent investigation into accuracy of this formulation in simulating elastic wave propagation, it was shown that the diffusive and dispersive errors are greatly reduced when space-time coupled least squares formulation is used instead in variational form. This paper investigates convergence characteristics of both formulations. To this end, two test cases, which are site response and impact models for viscoelastic medium, are employed, one dominated by dispersive and the other by diffusive numerical error. Convergence studies reveal that, compared to the commonly used space-time decoupled Galerkin and the coupled least squares formulation has much lower numerical errors, higher rates of convergence, and ability to take far larger time increments in the evolution process. In solving such models, coefficient matrices would require updating after each time step, a process that can be very costly. However large time steps allowed by cLs are expected to be a significant feature in reducing the overall computational cost.
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