Solving the 3D Acoustic Wave Equation with Higher-order Mass-lumped Tetrahedral Finite Elements

Zhebel, Elena; Minisini, Sara; Kononov, A. D.; Mulder, W.A.

doi:10.3997/2214-4609.20148952

Cited by 7 publications

(11 citation statements)

References 6 publications

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“…The source was represented as a tapered sinc, a variant of the one proposed by Hicks (2002). For the mass-lumped finite elements, we only considered the augmented element of degree 3, type 2 (Chin-Joe-Kong et al, 1999), which has a more favourable time-stepping stability limit than type 1 (Zhebel et al, 2011). The element is enriched with polynomials of degree 5 on the interior of the faces and of degree 6 in the interior of the tetrahedron, leading to a total of 50 nodes for each element compared to 20 for the standard element.…”

Section: Methodsmentioning

confidence: 99%

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

Zhebel¹,

Minisini²,

Kononov³

et al. 2012

Proceedings

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SUMMARYThe finite-difference method is widely used for time-domain modelling of the wave equation because of its ease of implementation of high-order spatial discretization schemes, parallelization and computational efficiency. However, finite elements on tetrahedral meshes are more accurate in complex geometries near sharp interfaces. We compared the fourth-order finite-difference method to fourth-order continuous masslumped finite elements in terms of accuracy and computational cost. The results show that for simple models like a cube with constant density and velocity, the finite-difference method outperforms the finite-element method by at least an order of magnitude. For a model with interior complexity and topography, however, the finite elements are about two orders of magnitude faster than finite differences.

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Section: Methodsmentioning

confidence: 99%

“…Chin-Joe- Kong et al (1999) found 2 types of elements with degree 3 on the edges. Lesage et al (2010) and Zhebel et al (2011) presented 3-D applications of these elements.…”

Section: Methodsmentioning

confidence: 99%

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

Zhebel¹,

Minisini²,

Kononov³

et al. 2012

Proceedings

Self Cite

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show abstract

“…For degree 3, the scheme runs slower than with local assembly on the fly and requires a lot more memory. For that reason, Zhebel et al (2011Zhebel et al ( , 2014 only mentioned local assembly.…”

Section: The Acoustic Wave Equationmentioning

confidence: 99%

“…Mulder (1996) made the generalization to tetrahedral elements with an element of degree 2 on the edges, 4 on the faces as product of a cubic bubble function and polynomial of degree 1, and degree 4 in the interior as a product of a quartic bubble function and constant polynomial. Lesage et al (2010) and Zhebel et al (2011) applied that element to acoustic wave propagation modelling.…”

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confidence: 99%

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

Mulder

Shamasundar

2016

Geophys. J. Int.

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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.

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A comparison of continuous mass‐lumped finite elements with finite differences for 3‐D wave propagation

Zhebel

Minisini

Kononov

et al. 2014

Geophysical Prospecting

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The finite‐difference method on rectangular meshes is widely used for time‐domain modelling of the wave equation. It is relatively easy to implement high‐order spatial discretization schemes and parallelization. Also, the method is computationally efficient. However, the use of finite elements on tetrahedral unstructured meshes is more accurate in complex geometries near sharp interfaces. We compared the standard eighth‐order finite‐difference method to fourth‐order continuous mass‐lumped finite elements in terms of accuracy and computational cost. The results show that, for simple models like a cube with constant density and velocity, the finite‐difference method outperforms the finite‐element method by at least an order of magnitude. Outside the application area of rectangular meshes, i.e., for a model with interior complexity and topography well described by tetrahedra, however, finite‐element methods are about two orders of magnitude faster than finite‐difference methods, for a given accuracy.

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Solving the 3D Acoustic Wave Equation with Higher-order Mass-lumped Tetrahedral Finite Elements

Cited by 7 publications

References 6 publications

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

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

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

A comparison of continuous mass‐lumped finite elements with finite differences for 3‐D wave propagation

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