Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research. Also, learning the fundamentals of linear elastic wave theory does not require a quantum leap for electromagnetic practitioners. Integral equation methods have been around for several decades, and their introduction to electromagnetics has been due to the seminal works of Richmond and Harrington in the 1960s. There was a surge in the interest in this topic in the 1980s (notably the work of Wilton and his coworkers) due to increased computing power. The interest in this area was on the wane when it was demonstrated that differential equation methods, with their sparse matrices, can solve many problems more efficiently than integral equation methods. Recently, due to the advent of fast algorithms, there has been a revival in integral equation methods in electromagnetics. Much of our work in recent years has been in fast algorithms for integral equations, which prompted our interest in integral equation methods. While previously, only tens of thousands of unknowns could be solved by integral equation methods, now, tens of millions of unknowns can be solved with fast algorithms. This has prompted new enthusiasm in integral equation methods.

Integral Equation Methods for Electromagnetic and Elastic Waves

The perfectly matched layer (PML) for elastic waves in cylindrical and spherical coordinates is developed using an improved scheme of complex coordinates. As is known for electromagnetic waves, Berenger’s original PML scheme does not apply to cylindrical and spherical coordinates. The straightforward extension of the complex coordinates for elastic waves to cylindrical and spherical coordinates requires extra unknowns for time-domain solutions, wasting computer memory and computation time. The main idea of the improved scheme in this work is the use of integrated complex variables. It is shown that for three-dimensional cylindrical and spherical coordinates, this improved PML scheme requires no more unknowns than in Cartesian coordinates. The number of unknowns can be further reduced through the use of symmetry in the partial differential equations. The PML scheme allows an arbitrary inhomogeneity in the medium, and is suitable for numerical solutions of wave equations by finite-difference, finite-element, and pseudospectral methods for elastic waves in inhomogeneous media with cylindrical and spherical structures. Finite-difference time-domain (FDTD) results are shown to demonstrate the efficacy of the PML absorbing boundary condition.

Perfectly matched layers for elastic waves in cylindrical and spherical coordinates

A time-domain finite-difference solution to the equations of linear elasticity was used to model the propagation of lithotripsy waves in kidney stones. The model was used to determine the loading on the stone (principal stresses and strains and maximum shear stresses and strains) due to the impact of lithotripsy shock waves. The simulations show that the peak loading induced in kidney stones is generated by constructive interference from shear waves launched from the outer edge of the stone with other waves in the stone. Notably the shear wave induced loads were significantly larger than the loads generated by the classic Hopkinson or spall effect. For simulations where the diameter of the focal spot of the lithotripter was smaller than that of the stone the loading decreased by more than 50%. The constructive interference was also sensitive to shock rise time and it was found that the peak tensile stress reduced by 30% as rise time increased from 25 to 150 ns. These results demonstrate that shear waves likely play a critical role in stone comminution and that lithotripters with large focal widths and short rise times should be effective at generating high stresses inside kidney stones.

http://limu.msu.ru/sites/default/files/jasa_v118n4_2005_robin_stones.pdf

Modeling elastic wave propagation in kidney stones with application to shock wave lithotripsy

A particle velocity-strain, finite-difference (FD) method with a perfectly matched layer (PML) absorbing boundary condition is developed for the simulation of elastic wave propagation in multidimensional heterogeneous poroelastic media. Instead of the widely used second-order differential equations, a first-order hyperbolic leap-frog system is obtained from Biot’s equations. To achieve a high accuracy, the first-order hyperbolic system is discretized on a staggered grid both in time and space. The perfectly matched layer is used at the computational edge to absorb the outgoing waves. The performance of the PML is investigated by calculating the reflection from the boundary. The numerical method is validated by analytical solutions. This FD algorithm is used to study the interaction of elastic waves with a buried land mine. Three cases are simulated for a mine-like object buried in “sand,” in purely dry “sand” and in “mud.” The results show that the wave responses are significantly different in these cases. The target can be detected by using acoustic measurements after processing.

/pdf/a-staggered-grid-finite-difference-method-with-perfectly-39pxwvrirr.pdf

A staggered-grid finite-difference method with perfectly matched layers for poroelastic wave equations

A new 3D cylindrical perfectly matched layer (PML) formulation is developed for elastic wave propagation in a pressurized borehole surrounded by a triaxially stressed solid formation. The linear elastic formation is altered by overburden and tectonic stresses that cause significant changes in the wave propagation characteristics in a borehole. The 3D cylindrical problem with both radial and azimuthal heterogeneities is suitable for numerical solutions of the wave equations by finite-difference time-domain (FDTD) and pseudospectral time-domain (PSTD) methods. Compared to the previous 2.5D formulation with other absorbing boundary conditions, this 3D cylindrical PML formulation allows modeling of a borehole-conformal, full 3D description of borehole elastic waves in a stress-induced heterogeneous formation. We have developed an FDTD method using this PML as an absorbing boundary condition. In addition to the ability to solve full 3D problems, this method is found to be advantageous over the previously reported 2.5D finite-difference formulation because a borehole can now be adequately simulated with fewer grid points. Results from the new FDTD technique confirm the principle of superposition of the influence of various stress components on both the borehole monopole and dipole dispersions. In addition, we confirm that the increase in shear-wave velocity caused by a uniaxial stress applied in the propagation direction is the same as that applied parallel to the radial polarization direction.

A 3D cylindrical PML/FDTD method for elastic waves in fluid‐filled pressurized boreholes in triaxially stressed formations

A three‐dimensional finite‐difference (FD) method is used to simulate sonic wave propagation in a borehole with an inhomogeneous solid formation. The second‐order FD scheme solves the first‐order elastic wave equations with central differencing in both space and time via staggered grids. Liao’s boundary condition is used to reduce artificial reflections from the finite computational domain. In the staggered grids, sources have to be implemented at the discrete center in order to maintain the appropriate symmetry in an axisymmetric borehole environment. The FD scheme is validated for multipole sources in three special media: (i) a homogeneous medium; (ii) a homogeneous formation with a fluid‐filled borehole; and (iii) a horizontally layered formation. The staircase approximation of a circular borehole introduces little error in dipole wave fields, although it causes a noticeable phase velocity error in the monopole Stoneley wave. This error has been drastically reduced by using a material averaging scheme ...

A three‐dimensional finite difference simulation of sonic logging

Elastic wave propagation in realistic borehole environments is very complex because of the presence of borehole, dipping beds, and other irregular scatterers. In order to understand this complex wave propagation phenomenon for sonic logging applications, a three‐dimensional finite‐difference (FD) method was used to simulate elastic wave propagation on a parallel computing architecture. The FD scheme solves the first‐order elastic wave equations with central differencing in both space and time via staggered grids. Liao’s absorbing boundary condition is used to reduce artificial reflections from the finite computational domain. In this work, uniform grids in Cartesian coordinates are used to discretize the inhomogeneous medium. Because of the staircase approximation of the circular borehole, it is observed that the discretization requirement is quite different for monopole and dipole sources. Several methods are suggested to remedy this problem. The results from the FD method were validated by other methods...

/pdf/large-scale-3d-finite-difference-simulation-of-elastic-wave-2gduokuqhk.pdf

Large‐scale 3D finite‐difference simulation of elastic wave propagation in borehole environments

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A three‐dimensional finite difference simulation of sonic logging

Large‐scale 3D finite‐difference simulation of elastic wave propagation in borehole environments