Wave Equation In Heterogeneous Media Research Articles

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media

AbstractIn this paper, we consider the development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well.

Dual-pairing summation by parts finite difference methods for large scale elastic wave simulations in 3D complex geometries

High-order accurate summation-by-parts (SBP) finite difference (FD) methods constitute efficient numerical methods for simulating large-scale hyperbolic wave propagation problems. Traditional SBP FD operators that approximate first-order spatial derivatives with central-difference stencils often have spurious unresolved numerical wave-modes in their computed solutions. Recently derived high order accurate dual pair SBP operators based non-central (upwind) FD stencils have the potential to suppress these poisonous spurious wave-modes on marginally resolved computational grids. In this paper, we demonstrate that not all high order dual pair SBP FD operators are applicable. Numerical dispersion relation analysis shows that odd-order dual pair SBP FD operators also support spurious unresolved high-frequency wave-modes on marginally resolved meshes. Meanwhile, even-order dual pair SBP FD operators (of order 2,4,6) do not support spurious unresolved high frequency wave-modes and also have better numerical dispersion properties.For all the dual pair SBP FD operators we discretise the three space dimensional (3D) elastic wave equation on boundary-conforming curvilinear meshes. Using the energy method we prove that the semi-discrete approximation is stable and energy-conserving. We derive a priori error estimate and prove the convergence of the numerical error for smooth solutions. Numerical experiments for the 3D elastic wave equation in complex geometries corroborate the theoretical analysis. Numerical simulations of the 3D elastic wave equation in heterogeneous media with complex non-planar free surface topography are given, including numerical simulations of community developed seismological benchmark problems. Computational results show that even-order dual pair SBP FD operators are more efficient, robust and less prone to numerical dispersion errors on marginally resolved meshes when compared to the odd-order dual pair and traditional SBP FD operators. Finally, scaling tests demonstrate nearly perfect strong scaling and verify the efficiency of our parallel implementation of the high order upwind SBP methods.

Carleman estimates for the wave equation in heterogeneous media with non-convex interface

A wave equation whose main coefficient is discontinuous models the evolution of waves amplitude in a media composed of at least two different materials, in which the propagation speed is different. In our mathematical setting, the spatial domain where the partial differential equation evolves is an open bounded subset of R2 and the wave speed is assumed to be constant in each one of two sub-domains, separated by a smooth and possibly non-convex interface. This article is concerned with the construction of Carleman weights for this wave operator, allowing generalizations of previous results to the case of an interface that is not necessarily the boundary of a convex set. Indeed, using the orthogonal projection onto this interface, we define convex functions satisfying the transmission conditions imposed by the equation, such that, under usual hypothesis on the sign of the jump of the wave speed, can be used as Carleman weights.

Stability of High-Order Staggered-Grid Schemes for 3D Elastic Wave Equation in Heterogeneous Media

In this paper, we firstly derive the stability conditions of high-order staggered-grid schemes for the three-dimensional (3D) elastic wave equation in heterogeneous media based on the energy method. Moreover, the plane wave analysis yields a sufficient and necessary stability condition by the von Neumann criterion in homogeneous case. Numerical computations for 3D wave simulation with point source excitation are given.

Error Analysis of Chebyshev Spectral Element Methods for the Acoustic Wave Equation in Heterogeneous Media

This work concerns the error analysis of the spectral element method with Gauss–Lobatto–Chebyshev collocation points with the implicit Newmark average acceleration scheme for the two-dimensional acoustic wave equation. The analysis is restricted to homogeneous Dirichlet boundary conditions, constant compressibility and variable density. The proposed error estimates are optimal with respect to the mesh parameter although suboptimal on the polynomial degree. Numerical examples illustrate the theoretical results.

Error analysis of the spectral element method with Gauss–Lobatto–Legendre points for the acoustic wave equation in heterogeneous media

We present the error analysis of a high-order method for the two-dimensional acoustic wave equation in the particular case of constant compressibility and variable density. The domain discretization is based on the spectral element method with Gauss–Lobatto–Legendre (GLL) collocation points, whereas the time discretization is based on the explicit leapfrog scheme. As usual, GLL points are also employed in the numerical quadrature, so that the mass matrix is diagonal and the resulting algebraic scheme is explicit in time. The analysis provides an a priori estimate which depends on the time step, the element length, and the polynomial degree, generalizing several known results for the wave equation in homogeneous media. Numerical examples illustrate the validity of the estimate under certain regularity assumptions and provide expected error estimates when the medium is discontinuous.

Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain [Formula: see text] is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length [Formula: see text], we prove that the solution of any member of our family of effective equations is [Formula: see text]-close to the true oscillatory wave over a time interval of length [Formula: see text] in a norm equivalent to the [Formula: see text] norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488–513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

An improved optimal nearly analytical discretized method for 2D scalar wave equation in heterogeneous media based on the modified nearly analytical discrete operator

Conventional finite-difference methods will encounter strong numerical dispersion on coarse grids or when higher frequencies are used. The nearly analytic discrete method (NADM) and its improved version, developed recently, can suppress numerical dispersion efficiently. However, the NADM is imperfect, especially in long-time seismic wave propagation simulation. To overcome this limitation, by minimizing the energy-error function, we have developed a modified optimal nearly analytical discretized (MNAD) method. The stencil of MNAD for the 2D wave equation is a novel diamond stencil. MNAD has two major advantages: The first one is that MNAD can suppress numerical dispersion effectively on coarse grids, which significantly improved computational efficiency and reduced memory demands. The second advantage is that the energy error of MNAD is much smaller after long-time simulation. The numerical dispersion analysis shows that the maximum phase velocity error was 5.92% even if only two sampling points were adopted in each minimum wavelength. To simulate a wavefield without visible numerical dispersion, the computational speed of MNAD, measured by CPU time, was approximately 4.32 times and 1.43 times comparing with the fourth-order Lax-Wendroff correction (LWC) method and the optimal nearly analytical discretized method (ONADM), respectively. MNAD also shows good numerical stability. Its CFL condition increased 24.3% comparing with that of ONADM, from 0.523 to 0.650. The total energy error was less than 1.5% after 300-s simulation whereas the error of other numerical schemes, such as the fourth-order and eighth-order LWC, etc., was up to more than 10% under the same computational parameters. Numerical results showed that our MNAD method performed well in computational efficiency, simulation accuracy, and numerical stability, as well as provided a useful tool for large-scale, long-time seismic wave propagation simulation.

Efficient algorithm for simultaneous reduction to the $$m$$ m -Hessenberg-triangular-triangular form

This paper proposes an efficient algorithm for simultaneous reduction of three matrices by using orthogonal transformations, where $$A$$ is reduced to $$m$$ -Hessenberg form, and $$B$$ and $$E$$ to triangular form. The algorithm is a blocked version of the algorithm described by Miminis and Paige (Int J Control 35:341–354, 1982). The $$m$$ -Hessenberg-triangular–triangular form of matrices $$A$$ , $$B$$ and $$E$$ is specially suitable for solving multiple shifted systems $$(\sigma E-A)X=B$$ . Such shifted systems naturally occur in control theory when evaluating the transfer function of a descriptor system, or in interpolatory model reduction methods. They also arise as a result of discretizing the time-harmonic wave equation in heterogeneous media, or originate from structural dynamics engineering problems. The proposed blocked algorithm for the $$m$$ -Hessenberg-triangular-triangular reduction is based on aggregated Givens rotations, and is a generalization of the blocked algorithm for the Hessenberg-triangular reduction proposed by Kagstrom et al. (BIT 48:563–584, 2008). Numerical tests confirm that the blocked algorithm is much faster than its non-blocked version based on regular Givens rotations only. As an illustration of its efficiency, two applications of the $$m$$ -Hessenberg-triangular-triangular reduction from control theory are described: evaluation of the transfer function of a descriptor system at many complex values, and computation of the staircase form used to identify the controllable part of the system.

Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media

Solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering. The design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration. In this paper, we focus on multi-dimensional wave problems and consider linear second-order wave equation in heterogeneous media. We develop and analyze an LDG method, in which numerical fluxes are carefully designed to maintain the energy conserving property and accuracy. Compatible high order energy conserving time integrators are also proposed. The optimal error estimates and the energy conserving property are proved for the semi-discrete methods. Our numerical experiments demonstrate optimal rates of convergence, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property.

Nonlinearity parameters B/A and C/A and wave equation for heterogeneous media containing negative stiffness inclusions

This work considers nonlinear propagation in a medium consisting of a low volume fraction of metamaterial inclusions dispersed in a fluid-like material. The metamaterial inclusions of interest are assumed to possess non-monotonic stress-strain constitutive relations, which results in regimes of negative stiffness. For modeling purposes, the constitutive relation for these inclusions is approximated with an expansion to third order in volume strain with coefficients that can be tuned with the geometry of the metamaterial structure and ambient pressure. A far-reaching goal of this research is to model the hysteretic response of the heterogeneous medium resulting from metamaterial inclusion snapping events and the associated effect on acoustic disturbances that cycle through regimes of both positive and negative stiffness. As an initial step, results are presented here for small but finite-amplitude disturbances limited to local regions of the constitutive relation. For this case, the quadratic and cubic nonlinearity parameters B/A and C/A, respectively, as traditionally defined for fluids are obtained. An evolution equation with both quadratic and cubic nonlinearity is also obtained. Numerical solutions of the evolution equation illustrate nonlinear waveform distortion as a function of the volume fraction and constitutive behavior of the inclusions. [Work supported by ARL:UT McKinney Fellowship in Acoustics.]

A time-domain finite volume approach for prediction of muffler transmission loss including thermal effects

A time-domain finite volume approach is presented for predicting the transmission loss of muffler including thermal effects with non-uniform sound speed field and density field, in which the acoustic wave equation in heterogeneous media is solved by using unstructured finite volume method with the temperature field specified or solved by some commercial code. An improved time-domain impulse method based on the absorbing boundary condition is applied to predict the acoustic attenuation characteristics of mufflers. The approach is validated by numerical simulations of a simple expansion chamber muffler and a complex muffler with five chambers. The predicted results agree well with the corresponding experimental ones and numerical ones obtained by finite element method with commercial code SYSNOISE. The results of both mufflers under different thermal conditions indicate that the temperature distribution has a significant influence on transmission loss. According to the analysis of a complex muffler with ideal medium, it is shown that the variation of working conditions can obviously affect density and sound speed distributions but have little influence on transmission loss. On the other hand, the obtained transmission loss with the solved temperature field deviates much from the one with specified uniform temperature field.

A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation

A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.

Performance Analysis of a High-Order Discontinuous Galerkin Method Application to the Reverse Time Migration

AbstractThis work pertains to numerical aspects of a finite element method based discontinuous functions. Our study focuses on the Interior Penalty Discontinuous Galerkin method (IPDGM) because of its high-level of flexibility for solving the full wave equation in heterogeneous media. We assess the performance of IPDGM through a comparison study with a spectral element method (SEM). We show that IPDGM is as accurate as SEM. In addition, we illustrate the efficiency of IPDGM when employed in a seismic imaging process by considering two-dimensional problems involving the Reverse Time Migration.

An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms

SUMMARY We present a new numerical approach to solve the elastic wave equation in heterogeneous media in the presence of externally given source terms with arbitrary high-order accuracy in space and time on unstructured triangular meshes. We combine a discontinuous Galerkin (DG) method with the ideas of the ADER time integration approach using Arbitrary high-order DERivatives. The time integration is performed via the so-called Cauchy-Kovalewski procedure using repeatedly the governing partial differential equation itself. In contrast to classical finite element methods we allow for discontinuities of the piecewise polynomial approximation of the solution at element interfaces. This way, we can use the well-established theory of fluxes across element interfaces based on the solution of Riemann problems as developed in the finite volume framework. In particular, we replace time derivatives in the Taylor expansion of the time integration procedure by space derivatives to obtain a numerical scheme of the same high order in space and time using only one single explicit step to evolve the solution from one time level to another. The method is specially suited for linear hyperbolic systems such as the heterogeneous elastic wave equations and allows an efficient implementation. We consider continuous sources in space and time and point sources characterized by a Delta distribution in space and some continuous source time function. Hereby, the method is able to deal with point sources at any position in the computational domain that does not necessarily need to coincide with a mesh point. Interpolation is automatically performed by evaluation of test functions at the source locations. The convergence analysis demonstrates that very high accuracy is retained even on strongly irregular meshes and by increasing the order of the ADER‐DG schemes computational time and storage space can be reduced remarkably. Applications of the proposed method to Lamb’s Problem, a problem of strong material heterogeneities and to an example of global seismic wave propagation finally confirm its accuracy, robustness and high flexibility.

Error analysis of numerical schemes for the wave equation in heterogeneous media

A family of finite difference schemes for the acoustic wave equation in heterogeneous media is introduced. The precision and computational cost are analyzed in two cases. First, a two-layered medium is considered. The order of convergence at the interface is derived for each scheme. Given an a priori accuracy on the solution, the computational cost is studied as a function of the order of accuracy of the finite difference scheme. It is demonstrated that this function has a minimum. The previous results are extended to the case of random media by a numerical study. Similar conclusions about precision and cost are found.

The Phononic Lattice Solid With Fluids For Modelling Non-Linear Solid-Fluid Interactions

SUMMARY The phononic lattice solid has been developed recently as a possible approach for modelling compressional waves in complex solids at the microscopic scale. Rather than directly modelling the wave equation, the microdynamics of quasi-particles is simulated on a discrete lattice. It is comparable with the lattice gas approach to model idealized gas particles but differs fundamentally in that lattice solid particles carry pressure rather than mass and propagate through a heterogeneous medium. Their speed may be space and direction dependent while the speed of lattice gas particles is constant. Furthermore, they may be scattered by medium heterogeneities. Lattice sites in the phononic lattice solid approach are considered to be fixed in space for all time. Lattice site movements (i.e. deformations) induced by the passage of a macroscopic wave are particularly important for a fluid-filled porous medium considering that non-linear solid-fluid interactions are thought to play a role in attenuation mechanisms. We take lattice site movements into account in the phononic lattice solid and name the approach ‘the phononic lattice solid with fluids (PLSF)’ because it could lead to an improved understanding of the effect of solid-fluid interactions in wave propagation problems. the macroscopic limit of the Boltzmann equation for the PLSF yields the acoustic wave equation for heterogeneous media modified by shear and bulk viscosity terms as well as the second-order term in macroscopic velocity (for the PLS) and additional non-linear terms due to the lattice site movements. It is hoped that PLSF numerical simulation studies of waves through digitized rock matrices may lead to an improved understanding of attenuation mechanisms of waves in porous rocks.

Probabilistic approach to wave propagation problems IV: Reflecting boundaries

Here, we obtain probabilistic representations for the Laplace transform of the solution to the scalar wave equation in heterogeneous media with reflecting boundaries. Besides proving existence results, we obtain path integral representations of stationary solutions and consider some simple cases of stratified media.

Seismic energy partitioning and scattering in laterally heterogeneous ocean crust

We present finite difference forward models of elastic wave propagation through laterally heterogeneous upper oceanic crust. The finite difference formulation is a 2-D solution to the elastic wave equation for heterogeneous media and implicitly calculates P and SV propagation, compressional to shear conversion, interference effects and interface phenomena. Random velocity perturbations with Gaussian and self-similar autocorrelation functions and different correlation lengths (a) are presented which show different characteristics of secondary scattering. Heterogeneities scatter primary energy into secondary body waves and secondary Stoneley waves along the water-solid interface. The presence of a water-solid interface in the model allows for the existence of secondary Stoneley waves which account for much of the seafloor ‘noise’ seen in the synthetic seismograms for the laterally heterogeneous models.