New Triangular Mass-Lumped Finite Elements of Degree Six for Wave Propagation

Abstract

Abstract-Mass-lumped continuous finite elements allow for explicit time stepping with the second-order wave equation if the resulting integration weights are positive and provide sufficient accuracy. To meet these requirements on triangular and tetrahedral meshes, the construction of continuous finite elements for a given polynomial degree on the edges involves polynomials of higher degree in the interior. The parameters describing the supporting nodes of the Lagrange interpolating polynomials and the integration weights are the unknowns of a polynomial system of equations, which is linear in the integration weights. To find candidate sets for the nodes, it is usually required that the number of equations equals the number of unknowns, although this may be neither necessary nor sufficient. Here, this condition is relaxed by requiring that the number of equations does not exceed the number of unknowns. This resulted in two new types elements of degree 6 for symmetrically placed nodes. Unfortunately, the first type is not unisolvent. There are many different elements of the second type with a large range in their associated time-stepping stability limit. To assess the efficiency of the elements of various degrees, numerical tests on a simple problem with an exact solution were performed. Efficiency was measured by the computational time required to obtain a solution at a given accuracy. For the chosen example, elements of degree 4 with fourth-order time stepping appear to be the most efficient.

A Comparison of Continuous Mass-lumped Finite Elements and Finite Differences for 3D

“…On triangles in 2D, they have to be enriched with polynomials of higher degree in the interior to avoid loss of accuracy after lumping (Fried and Malkus, 1975). At present, elements are known for degrees 1 to 6 (Fried and Malkus, 1975;Tordjman, 1995;Cohen et al, 1995Cohen et al, , 2001Mulder, 1996;ChinJoe-Kong et al, 1999;Mulder, 2011). The 3-D extension to the tetrahedron (Mulder, 1996) required higher-degree polynomials in the interior of the faces and of the tetrahedron and resulted in an element of degree 2 on the edges and 4 in the interior of faces and element.…”

Should We Use the First- or Second-order Formulation with Spectral Elements for Seismic Modelling?

“…) as well as for simplex-based elements on triangles (Mulder, 1996(Mulder, , 2013 or tetrahedra (Zhebel et al, 2014). For some applications, however, a first-order formulation may be desirable.…”

An Improved Source Term for Finite-element Modelling with the Stress-velocity Formulation of the Wave Equation

“…The second-order formulation of the wave equation is often used for modelling seismic wave propagation with spectral methods, both for box-like elements on quadrilaterals and hexahedra (Komatitsch et al, 1999, e.g. ) as well as for simplex-based elements on triangles (Mulder, 1996(Mulder, , 2013 or tetrahedra (Zhebel et al, 2014;Mulder and Shamasundar, 2016). For some applications, a first-order formulation may be desirable.…”