ADAPTIVE FINITE ELEMENT TECHNIQUES FOR THE ACOUSTIC WAVE EQUATION

We present an adaptive finite element method for solving the acoustic wave equation. Using a global duality argument and Galerkin orthogonality, we derive an identity for the error with respect to an arbitrary functional output of the solution. The error identity is evaluated by solving the dual problem numerically. The resulting local cell-wise error indicators are used in the grid adaptation process. In this way, the space-time mesh can be tailored for the efficient computation of the quantity of interest. We give an overview of the implementation of the proposed method and illustrate its performance by several numerical examples.

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A Priori Error Estimates for the Finite Element Approximation of Westervelt's Quasi-linear Acoustic Wave Equation

SIAM Journal on Numerical Analysis ◽

10.1137/19m1240873 ◽

2019 ◽

Vol 57 (4) ◽

pp. 1897-1918 ◽

Cited By ~ 1

Author(s):

Vanja Nikolić ◽

Barbara Wohlmuth

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Error Estimates ◽

A Priori ◽

Finite Element Approximation ◽

A Priori Error Estimates ◽

Element Approximation ◽

Acoustic Wave Equation ◽

Priori Error Estimates

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Explicit Discrete Dispersion Relations for the Acoustic Wave Equation in d-Dimensions Using Finite Element, Spectral Element and Optimally Blended Schemes

Advanced Structured Materials - Computer Methods in Mechanics ◽

10.1007/978-3-642-05241-5_1 ◽

2010 ◽

pp. 3-17 ◽

Cited By ~ 4

Author(s):

Mark Ainsworth ◽

Hafiz Abdul Wajid

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Spectral Element ◽

Dispersion Relations ◽

Acoustic Wave Equation

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Theory of a 2.5-D acoustic wave equation for constant density media

Geophysics ◽

10.1190/1.1443025 ◽

1991 ◽

Vol 56 (12) ◽

pp. 2114-2117 ◽

Cited By ~ 10

Author(s):

Christopher L. Liner

Keyword(s):

Wave Propagation ◽

Wave Equation ◽

Acoustic Wave ◽

Symmetry Axis ◽

Constant Density ◽

Efficient Computation ◽

Acoustic Wave Equation ◽

Out Of Plane ◽

Acoustic Media ◽

D Wave

The theory of 2.5-dimensional (2.5-D) wave propagation (Bleistein, 1986) allows efficient computation of 3-D wavefields in c(x, z) acoustic media when the source and receivers lie in a common y-plane (assumed to be y = 0 in this paper). It is really a method of efficiently computing an inplane 3-D wavefield in media with one symmetry axis. The idea is to raytrace the wavefield in the (x, z)-plane while allowing for out‐of‐plane spreading. In this way 3-D amplitude decay is honored without 3-D ray tracing. This theory has its conceptual origin in work by Ursin (1978) and Hubral (1978). Bleistein (1986) gives an excellent overview and detailed reference to earlier work.

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Finite Element θ-Schemes for the Acoustic Wave Equation

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.10-m1018 ◽

2011 ◽

Vol 3 (1) ◽

pp. 181-203 ◽

Cited By ~ 6

Author(s):

Samir Karaa

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Stability And Convergence ◽

Optimal Error Estimates ◽

Finite Difference Schemes ◽

Time Interval ◽

Acoustic Wave Equation ◽

Time Step ◽

Polynomial Degree

AbstractIn this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1 + ∆ts), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and ∆t the time step.

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Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

Journal of Applied Mathematics and Computing ◽

10.1007/s12190-012-0558-8 ◽

2012 ◽

Vol 40 (1-2) ◽

pp. 659-682 ◽

Cited By ~ 7

Author(s):

Samir Karaa

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Stability And Convergence ◽

Acoustic Wave Equation ◽

Fully Discrete ◽

Discrete Finite Element ◽

Finite Element Schemes

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On Multiple Modes of Propagation of High-Order Finite Element Methods for the Acoustic Wave Equation

Lecture Notes in Computational Science and Engineering - Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016 ◽

10.1007/978-3-319-65870-4_36 ◽

2017 ◽

pp. 509-518 ◽

Cited By ~ 1

Author(s):

S. P. Oliveira

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Finite Element Methods ◽

High Order ◽

Acoustic Wave Equation ◽

Multiple Modes

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AN ADAPTIVE FINITE ELEMENT METHOD FOR MODELING SALT DIAPIRISM

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202506001273 ◽

2006 ◽

Vol 16 (04) ◽

pp. 587-614 ◽

Cited By ~ 6

Author(s):

P. MASSIMI ◽

A. QUARTERONI ◽

G. SCROFANI

Keyword(s):

Finite Element ◽

Adaptive Finite Element Method ◽

Adaptive Finite Element ◽

Geological Time ◽

Element Space ◽

Geological Time Scale ◽

Salt Domes ◽

Subsurface Geology ◽

Grid Adaptation ◽

Space Discretization

Various types of oil traps have been found to be associated with salt domes in subsurface geology. In this paper the diapiric rise of light salt layers through a denser overburden — the surrounding rocks — is modeled assuming that, in a geological time scale, salt and rocks layers behave like Newtonian fluids. A Lagrangian approach is adopted to track the interface between layers, within the framework of a finite element space discretization. An accurate description of large deformations due to salt movement is achieved using a grid adaptation technique based on geometrical refinement. Different geological cases have been simulated in order to describe the behavior of rocks and estimate the effect on diapiric growth of buoyancy force, differential loading, gravitational gliding and thin-skinned regional extension. Our computational model accounts also for sedimentation and compaction of the overburden.

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