There are two categories of dynamic problems: those under low-frequency loads like waves and those under high-frequency loads like explosions. The two dynamic problems can be solved by partial differential equations (PDEs). However, due to the complexity of the governing partial differential equations, they are usually reduced to ordinary differential equations (ODEs) in the time domain using space discretized by finite elements. Therefore, these dynamic problems have been attributed to ODE solution.
The traditional time-continuous Galerkin finite element method (CGFEM), comprising of time integration methods like Newmark methods, has been widely used to solve dynamic problems, producing good performances, especially for solving low-frequency dominated dynamic problems. One major disadvantage of CGFEM that hinders its application in solving high-frequency dominated dynamic problems is its inability to capture sharp gradients and discontinuities of the solution in space. Additionally, CGFEM fails to reduce the spurious numerical oscillations of the dynamic problem solution under high-frequency loading.
Recently, a time-discontinuous Galerkin finite element method (TDGFEM) has emerged as a potential approach for overcoming the setbacks of CGFEM thanks to its favorable characteristics, such as simultaneous use of finite element discretization in time and space. Importantly, TDGFEM can automatically filter spurious high-order numerical oscillations through a numerical dissipation mechanism. However, TDGFEM is limited to solving dynamic problems in the classical continuum theory framework and not the classical theory framework because it fails to describe the wave modes for materials with microstructures. As such, it is imperative to use suitable continuum theory, specifically Cosserat theory, to describe the physical chrematistics of materials with microstructures. Cosserat theory can overcome the limitations of TDGFEM, a potential approach for analyzing many dynamic problems.
In their study, Dr. Chenxi Xiu, Professor Xihua Chu and Dr. Ji Wan from Wuhan University in collaboration with Dr. Jiao Wang from Southwest Jiaotong University numerically investigated the propagation of impulse waves in a Cosserat media using TDGFEM. The Cosserat continuum theory comprised additional rotational DoF resulting in Cosserat rotational (R), shear (S) and compressive (P) waves, whose propagations were simulated under impulse loads. For the Cosserat nodal unknown vectors, three different orders of the temporal shape functions: linear (P1), quadratic (P2) and Hermite (P3) were presented, leading to different TDGFEM matric equations. The capacity and efficiency of the TDGFEM method were validated by comparing the results to those of one TDGFEM and two TCGFEMs with artificial damping. The work is published in the International Journal for Numerical Methods in Engineering.
The authors findings demonstrated the advantages and efficiency of the TDGFEM method for solving dynamic problems under impulse loading. TDGFEM effectively reduced the spurious numerical oscillations, captured the discontinuities of the solution and automatically filtered out the effects of spurious high order modes. As a result, the obtained mesh density and time step had small spurious oscillations and small errors. P1 exhibited lower-order temporal discretization than P3, which had larger order of accuracy and smaller relative errors. While Cosserat P wave was similar to classical P wave, classical S wave and Cosserat S were different and the Cosserat S was coupled with Cosserat R wave.
The three numerical methods with artificial damping showed good performance in reducing the effects of spurious oscillations. However, they exhibited weaker capacities and lower efficiencies of capturing discontinuities than TDGFEM. Furthermore, the Cosserat shear modulus and the characteristic length had a minor influence on the Cosserat P wave propagation, but it significantly influenced the Cosserat S and R wave propagations subjected to impulse loads.
In summary, TDGFEM method is extended to Cosserat media and subsequently used to simulate three different Cosserat P, S and R wave propagations in Cosserat media under impulse loads. Both one-dimensional and two-dimensional numerical analysis results showed the accuracy, efficiency and higher capacity of TDGFEM, demonstrating that it is an effective method for simulating Cosserat impulse wave propagations. In a statement to Advances in Engineering, Professor Xihua Chu stated the study will advance the application of TDGFEM to solve different dynamic problems.
Xiu, C., Chu, X., Wan, J., & Wang, J. (2021). Numerical simulation of impulse waves in Cosserat media based on a time‐discontinuous Galerkin finite element method. International Journal for Numerical Methods in Engineering, 122(17), 4507-4540.