Theory of elastic waves produced by a spherical source for generalized boundary conditions

Both fundamental boundary-value problems for elastic waves produced by a spherical source are solved on the assumption that the boundary conditions on the surface of the source apart from being an arbitrary function of the coordinates are also an arbitrary function of time.     

Nonreflecting boundary conditions for discrete waves

We introduce a new class of nonreflecting boundary conditions for lattice models, which minimizes reflections at artificial boundaries. Exact integrodifferential boundary conditions for finite chains and half-spaces are obtained by means of Green’s functions for initial value problems. Truncating the resulting integrals in time, we obtain absorbing boundary conditions. Numerical tests illustrate the ability of these conditions to suppress reflections.     

Beam vibrations with time-dependent boundary conditions

A procedure is outlined for the solution of the vibration problem of a Bernoulli-Euler beam with time-dependent boundary conditions. The solution is greatly simplified if the dependent variable in the original partial differential equation can be changed to produce homogeneous boundary conditions and at the same time maintain a homogeneous differential equation. A method for making such a change is given and illustrated by solving a cantilever beam problem with a time-dependent tip displacement.     

Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the well-posedness and accuracy aspects of local ABCs designed for the transient modeling of the scalar anisotropic wave equation. The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL), a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation based ABCs. Specifically, we derive the necessary and sufficient condition for the well-posedness of the initial boundary value problem (IBVP) obtained by coupling an interior and a PMDL ABC. The derivation of the reflection coefficient presented in a companion paper (S. Savadatti, M.N. Guddati, J. Comput. Phys., 2010, doi:10.1016/j.jcp.2010.05.018) has shown that PMDL can correctly identify and accurately absorb outgoing waves with opposing signs of group and phase velocities provided the PMDL layer lengths satisfy a certain bound. Utilizing the well-posedness theory developed by Kreiss for general hyperbolic IBVPs, and the well-posedness conditions for ABCs derived by Trefethen and Halpern for isotropic acoustics, we show that this bound on layer lengths also ensures well-posedness. The time discretized form of PMDL is also shown to be theoretically stable and some instability related to finite precision arithmetic is discussed.     

Vibration of rectangular plates with time-dependent boundary conditions

A general method of solution for the vibration of rectangular plates with any type of time-dependent boundary conditions is developed by an extension of the method of Mindlin and Goodman [1]. For illustration, the problems of a plate with different time-dependent boundary conditions are solved and the closed form solutions for the transverse deflections of the plate are obtained. The non-dimensionalized transverse deflections, ($\text{w}\text{a}$) at the middle of the plate are evaluated numerically for different dimensions of the plate and different forcing functions. These are presented graphically against the non-dimensionalized time, T, for three cases and tabulated for other cases.     

Absorbing boundary conditions for scalar waves in anisotropic media. Part 1: Time harmonic modeling

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the accuracy aspects of local ABCs designed for the scalar anisotropic wave equation in the frequency domain (time harmonic case). The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL). PMDL is a simple variant of perfectly matched layers (PML) and is equivalent to rational approximation-based local ABCs. Specifically, we derive a sufficient condition for PMDL to accurately absorb wave modes with outgoing group velocities and this condition turns out to be a simple bound on the PMDL parameters. The reflection coefficient derived in this paper clearly reveals that the PMDL absorption is based on group velocities, and not phase velocities, and hence a PMDL can be designed to correctly identify and accurately absorb all outgoing wave modes (even those with opposing signs of phase and group velocities). The validity of the sufficient condition is demonstrated through a series of frequency domain simulations. In part 2 of this paper [S. Savadatti, M.N. Guddati, Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling, J. Comput. Phys. (2010), http://dx.doi.org/10.1016/j.jcp.2010.05.017], the accuracy condition presented here is shown to govern both the well-posedness and accuracy aspects of PMDL designed for transient (time-dependent) modeling of scalar waves in anisotropic media.     

Fringe waves radiated by a half-plane for the boundary conditions of Neumann

The fringe waves of the physical theory of diffraction are obtained in terms of Fresnel integrals for a half-plane satisfying the Neumann boundary condition. The approximate expressions of the radiated waves are also evaluated for sufficiently large wavenumbers. The fields are plotted and compared numerically.     

Mathematics of diffusion controlled precipitates with time-dependent boundary conditions

The boundary value problem of diffusion controlled precipitation with time-dependent boundary conditions is treated by an extension of the conventional eigenfunction expansion method. In this approach, expansion coefficients and eigenvalues become functions of time and eigenfunctions become functions of space and time. Mathematical techniques to determine the relevant quantities are set forth and the general solution describing precipitate growth under time-dependent conditions is obtained. The general solution is applied to the precipitation of plates and spheres and the growth of radii as a function of time is determined. In the latter case, comparisons are made with conventional methods and the fractional error for a particle to reach half its equilibrium is estimated.     

Modal decomposition method for modeling the interaction of Lamb waves with cracks

The interaction of the low-order antisymmetric (a0) and symmetric (s0) Lamb waves with vertical cracks in aluminum plates is studied. Two types of slots are considered: (a) internal crack symmetrical with respect to the middle plane of the plate and (b) opening crack. The modal decomposition method is used to predict the reflection and transmission coefficients and also the through-thickness displacement fields on both sides of slots of various heights. The model assumes strip plates and cracks, thus considering two-dimensional plane strain conditions. However, mode conversion (a0 into s0 and vice versa) that occurs for single opening cracks is considered. The energy balance is always calculated from the reflection and transmission coefficients, in order to check the validity of the results. These coefficients together with the through-thickness displacement fields are also compared to those predicted using a finite element code widely used in the past for modeling Lamb mode diffraction problems. Experiments are also made for measuring the reflection and transmission coefficients for incident a0 or s0 lamb modes on opening cracks, and compared to the numerical predictions.     

Propagation of elastic waves in layered composites with microdefect concentration zones and their simulation with spring boundary conditions

The possibility is studied of applying spring boundary conditions to describe propagation of elastic waves in layered composites with nonperfect contact of components or in the presence of groups of microdefects at the interface. Stiffnesses in spring boundary conditions are determined by crack density, the average size of microdefects, and the elastic properties of the materials surrounding them. In deriving the values of the effective stiffness parameters, the Baik-Thompson and Boström-Wickham approaches are applied, as well as the integral approach. The components of the stiffness matrices are derived from consideration of an incident, at a random angle to the interface, plane wave in the antiplane case, and at a normal angle in the plane case. The efficiency of this model and the possibility of using its results in the three-dimensional case are discussed.     

High-performance modeling acoustic and elastic waves using the parallel Dichotomy Algorithm

A high-performance parallel algorithm is proposed for modeling the propagation of acoustic and elastic waves in inhomogeneous media. An initial boundary-value problem is replaced by a series of boundary-value problems for a constant elliptic operator and different right-hand sides via the integral Laguerre transform. It is proposed to solve difference equations by the conjugate gradient method for acoustic equations and by the GMRES(k) method for modeling elastic waves. A preconditioning operator was the Laplace operator that is inverted using the variable separation method. The novelty of the proposed algorithm is using the Dichotomy Algorithm [26], which was designed for solving a series of tridiagonal systems of linear equations, in the context of the preconditioning operator inversion. Via considering analytical solutions, it is shown that modeling wave processes for long instants of time requires high-resolution meshes. The proposed parallel fine-mesh algorithm enabled to solve real application seismic problems in acceptable time and with high accuracy. By solving model problems, it is demonstrated that the considered parallel algorithm possesses high performance and efficiency over a wide range of the number of processors (from 2 to 8192).     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

Influence of boundary conditions on the ordering of elastic dipoles

We consider elastic dipoles in several two-dimensional geometries. Using Cauchy integral techniques and the image method, the state of elastic equilibrium under different boundary conditions can be determined. The results are used to find the ground state of systems of anisotropic dipoles via the simulated annealing method. Only in the case of fixed boundaries the ordering depends on the boundary condition.     

Poroelastic modeling of seismic boundary conditions across a fracture

Permeability of a fracture can affect how the fracture interacts with seismic waves. To examine this effect, a simple mathematical model that describes the poroelastic nature of wave-fracture interaction is useful. In this paper, a set of boundary conditions is presented which relate wave-induced particle velocity (or displacement) and stress including fluid pressure across a compliant, fluid-bearing fracture. These conditions are derived by modeling a fracture as a thin porous layer with increased compliance and finite permeability. Assuming a small layer thickness, the boundary conditions can be derived by integrating the governing equations of poroelastic wave propagation. A finite jump in the stress and velocity across a fracture is expressed as a function of the stress and velocity at the boundaries. Further simplification for a thin fracture yields a set of characteristic parameters that control the seismic response of single fractures with a wide range of mechanical and hydraulic properties. These boundary conditions have potential applications in simplifying numerical models such as finite-difference and finite-element methods to compute seismic wave scattering off nonplanar (e.g., curved and intersecting) fractures.