Performance of continuous mass-lumped tetrahedral elements for elastic wave propagation with and without global assembly

Abstract

SUMMARYWe consider isotropic elastic wave propagation with continuous mass-lumped finite elements on tetrahedra with explicit time stepping. These elements require higher-order polynomials in their interior to preserve accuracy after mass lumping and are only known up to degree 3. Global assembly of the symmetric stiffness matrix is a natural approach but requires large memory.Local assembly on the fly, in the form of matrix-vector products per element at each time step, has a much smaller memory footprint. With dedicated expressions for local assembly, our code ran about 1.3 times faster for degree 2 and 1.9 times for degree 3 on a simple homogeneous test problem, using 24 cores. This is similar to the acoustic case. For a more realistic problem, the gain in efficiency was a factor 2.5 for degree 2 and 3 for degree 3. For the lowest degree, the linear element, the expressions for both the global and local assembly can be further simplified.In that case, global assembly is more efficient than local assembly. Among the three degrees, the element of degree 3 is the most efficient in terms of accuracy at a given cost.

New Higher-Order Mass-Lumped Tetrahedral Elements for Wave Propagation Modelling

“…The older elements, apart from the one with degree 1, have degree 2 with 23 nodes and degree 3 with 50 nodes and two variants called a and b. For the implementation of the mass-lumped methods we used the algorithm described in [18]. The time step size is based on the estimates of [19] multiplied by a factor 0.9.…”

Efficient Quadrature Rules for Computing the Stiffness Matrices of Mass-Lumped Tetrahedral Elements for Linear Wave Problems

“…In particular, we show how we efficiently compute the element stiffness matrix-vector products on the fly. We do not store the matrices, since this requires storing and fetching significantly more data, and since it was shown in [19] that an on-the-fly approach is more efficient for higher-degree elements.…”

“…whereũ (e) := u (e) • φ e ,∇ is the gradient operator in reference coordinates, andc (e) := (c • φ e ) |e| |ẽ| J −t e · J −1 e is a tensor field, with J e := ∇φ e the Jacobian of the element mapping and J −t e the transposed of J −1 e . When c is constant within each element, thenc (e) is also constant and we can compute (26) using the algorithm of [19]:…”

Research Note: On the error behaviour of force and moment sources in simplicial spectral finite elements

“…Up till now, triangular spectral elements of degree 2 and 3 (Cohen et al ., 1995, 2001; Tordjman, 1995), 4 (Mulder, 1996), 5 (Chin‐Joe‐Kong et al ., 1999), infinitely many of degree 6 (Mulder, 2013) and 7 to 9 (Cui et al ., 2017; Liu et al ., 2017) have been found. Spectral elements for tetrahedra have been available since 1996 (Mulder, 1996; Chin‐Joe‐Kong et al ., 1999) but were rarely used because the requirement to maintain spatial accuracy after mass lumping comes at a high computational expense (Lesage et al ., 2010; Mulder and Shamasundar, 2016). This changed fairly recently, when a sharper accuracy criterion led to a significant cost reduction (Geevers et al ., 2018, 2019), making them more appealing for large‐scale applications.…”