A pixel-based finite element implementation to estimate effective wave velocity in heterogeneous media

In the present work, we present a 2D pixel-based Finite Element strategy to simulate the elastic wave propagation in heterogeneous media. An assembly-free approach is employed for the stiffness matrix, leveraging a pixel-based structured mesh to reduce the memory required to store computations. Additionally, a diagonal Lumped-Mass matrix technique is utilized to address challenges associated with the inversion and storage of the mass matrix. The Leap-frog integration method, known for its amalgamation of stability, precision, and efficiency, is adopted. The combination of these features is aimed at facilitating massive parallel computations for very large systems with 10⁸ to 10⁹ degrees of freedom. In that sense, the present work can be understood as a first step toward a very efficient massive parallel GPU-based voxel-based Finite Element implementation to treat very large digital images with personal computers. The implementation presented here has been validated against theoretical predictions and analytical results derived from classical wave propagation theory. Finally, the transmission test is simulated in two digital models, one representing a layered medium and another representing a medium with complex microstructue obtained via micro-tomography. For the first model, the results are compared with the so called Bakus average, while, for the second model, the results are compared with the corresponding outcomes acquired through an in-house developed static finite element homogenization implementation.