Galerkin Method For Wave Propagation Research Articles

A nodal discontinuous Galerkin method for wave propagation in coupled acoustic–elastic media

AbstractThe accurate numerical solution at an acoustic–elastic interface is important for offshore exploration. The solution requires careful implementation for the acoustic–elastic boundary conditions. In this work, we leverage a nodal discontinuous Galerkin method, in which the unstructured uniform triangular meshes are used for the model meshing and an explicit upwind numerical flux derived from the Riemann problem is adopted to handle the boundary conditions at the acoustic–elastic interface. Several numerical results are provided to assess the accuracy and convergence properties of this method. The convergence analysis is carried out in the coupled model with a flat interface, and the accuracy of the proposed method is verified in the curved interface coupled model. Finally, a more complex model with a salt dome, inspired by real geophysical applications, is carried out in this study. The numerical results demonstrate that the proposed nodal discontinuous Galerkin method is effective and accurate for dealing with the coupled acoustic–elastic media with complex geometries.

An immersed discontinuous Galerkin method for wave propagation in acoustic elastic media

We present an immersed discontinuous Galerkin (IDG) method for interface problems simulating wave propagation in acoustic-elastic media. The method uses an interface independent mesh that allows elements to be cut by the material interface and thus leading to elements consisting of the union of an acoustic medium and an elastic medium. In this method, two different wave propagation models are treated in each interface element where piecewise polynomials constructed according to interface jump conditions are to be used. The proposed discontinuous Galerkin method is stable. We present several computational examples and results to illustrate the stability, efficacy and accuracy of the method.

Explicit and Energy-Conserving Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Media

In this work, we propose a local multiscale model reduction approach for the time domain scalar wave equation in a heterogenous media. A fine mesh is used to capture the heterogeneities of the coef...

A weight-adjusted discontinuous Galerkin method for wave propagation in coupled elastic-acoustic media

This paper presents a high-order discontinuous Galerkin (DG) scheme for the simulation of wave propagation through coupled elastic-acoustic media. We use a first-order stress-velocity formulation, and derive a simple upwind-like numerical flux which weakly imposes continuity of the normal velocity and traction at elastic-acoustic interfaces. When combined with easily invertible weight-adjusted mass matrices [1–3], the resulting method is efficient, consistent, and energy stable on curvilinear meshes and for arbitrary heterogeneous media, including anisotropy and sub-cell (micro) heterogeneities. We numerically verify the high order accuracy and stability of the proposed method, and investigate its performance for applications in photoacoustic tomography.

A discontinuous Galerkin method for wave propagation in orthotropic poroelastic media with memory terms

Poroelastic materials play an important role in biomechanical and geophysical research. In this paper, we investigate wave propagation in orthotropic poroelastic media by studying the time-domain poroelastic wave equations. Both the low frequency Biot's (LF-Biot) equations and the Biot-Johnson-Koplik-Dashen (Biot-JKD) model are considered. In LF-Biot equations, the dissipation terms are proportional to the relative velocity between the fluid and the solid by a constant. Contrast to this, the dissipation terms in the Biot-JKD model are in the form of time convolution (memory) as a result of the frequency-dependence of fluid-solid interaction at the underlying microscopic scale in the frequency domain. The dynamic tortuosity and permeability described by Darcy's law are two crucial factors in this problem, and highly linked to the viscous force. In the Biot model, the key difficulty is to handle the viscous term when the pore fluid is viscous. In the Biot-JKD model, the convolution operator involves order 1/2 shifted fractional derivatives in the time domain, which is challenging to discretize.In this work, a new method of the multipoint Padé (or Rational) approximation for Stieltjes function is applied to approximate the JKD dynamic tortuosity and then an augmented system of Biot-JKD model is obtained, where the kernel of the memory term is replaced by the finite auxiliary variables satisfying a local system of ordinary differential equations. The Runge-Kutta discontinuous Galerkin (RKDG) method with the un-splitting method is used to compute the numerical solution, and numerical examples are presented to demonstrate the high order accuracy and stability of the method. Compared with the existing approaches for solving the Biot-JKD equations, the augmented system presented here require neither the storage of solution history nor the computation of the flux of the auxiliary variables.

Discontinuous Galerkin method for wave propagation in elastic media with inhomogeneous inclusions

AbstractA discontinuous Galerkin method on unstructured grids is adapted and implemented for simulation of wave response of subvertical fractured systems in carbonate rocks for numerical solution of direct problems of seismic exploration. The present paper compares seismic responses for several mechanical-mathematical models of a fractured reservoir. The models of collectors differ in the presence and location of media interfaces relative to the collector and also in a way of its specification, namely, an explicit selection of fractures with the parameters of the medium in the domain of collector coinciding with the host medium, or differing from it. We indicate the ability to take into account inter-fracture interactions with the use of the model of a fractured layer presented in the paper and study wave processes formed as the result of interaction of seismic pulses with fractured reservoirs.

Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials

In this paper, we develop a nodal discontinuous Galerkin method for solving the time-dependent Maxwellʼs equations when metamaterials are involved. Both semi- and fully-discrete schemes are constructed. Numerical stability and error estimate are proved for both schemes. Numerical results are presented to demonstrate that the method is not only efficient but also very effective in solving metamaterial Maxwellʼs equations.

Local time stepping with the discontinuous Galerkin method for wave propagation in 3D heterogeneous media

Modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources. Seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings. Finite elements enable accurate simulation of time-dependent wave propagation in heterogeneous media. They are more costly than finite-difference methods, but this is compensated by their superior accuracy if the finite-element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength, hence with the local wave velocity. However, 3D complex geologic settings often contain details on a very small scale compared to the dominant wavelength, requiring the mesh to contain elements that are smaller than dictated by the wavelength. Also, limitations of the mesh generation software may produce regions where the elements are much smaller than desired. In both cases, this leads to a reduction of the time step required to solve the wave propagation and significantly increases the computational cost. Local time stepping (LTS) can improve the computational efficiency and speed up the simulation. We evaluated a local formulation of an LTS scheme with second-order accuracy for the discontinuous Galerkin finite-element discretization of the wave equation. We tested the benefits of the scheme by considering a geologic model for a North-Sea-type example.

A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials

Some electromagnetic materials exhibit, in a given frequency range, effective dielectric permittivity and/or magnetic permeability which are negative. In the literature, they are called negative index materials, left-handed materials or meta-materials. We propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta-material. The method we investigate can be considered as an alternative method compared to the method presented by the second author and co-workers. In particular, we shall use the abstract framework they developed to prove well-posedness of the exact problem. We recast this problem to fit later discretization by the staggered discontinuous Galerkin method developed by the first author and co-worker, a method which relies on introducing an auxiliary unknown. Convergence of the numerical method is proven, with the help of explicit inf–sup operators, and numerical examples are provided to show the efficiency of the method.

A linear stability analysis of the cell-wise relaxation implicit discontinuous Galerkin method for wave propagation

A linear stability analysis of the cell-wise relaxation implicit discontinuous Galerkin method proposed in our previous work was carried out for a scalar advection equation using three different generating patterns of triangular cells in two-dimensional space. A detailed mathematical formulation of the stability analysis is presented in this paper. It is shown that the present cell-wise implicit method with spatial accuracies up to fourth order is stable with very large Courant–Friedrichs–Lewy (CFL) numbers.

A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.

Element-free Galerkin method for wave propagation and dynamic fracture

Element-free Galerkin method (EFG) is extended to dynamic problems. EFG method, which is based on moving least square interpolants (MLS), requires only nodal data; no element connectivity is needed. This makes the method particularly attractive for moving dynamic crack problems, since remeshing can be avoided. In contrast to the earlier formulation for static problems by authors, the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions. With this formulation, the stiffness matrix is symmetric and positive semi-definite, and hence the consistency, conergence and stability analyses of time integration remain the same as those in finite element method. Numerical examples are presented to illustrate the performance of this method. The relationship between the element-free Galerkin method and the smooth particle hydrodynamics (SPH) method is also discussed in this paper. Results are presented for some one-dimensional problems and two-dimensional problems with static and moving cracks.