Richard Paul Shaw

An integral equation approach to diffusion

Abstract A method of solution of transient diffusion, e.g. heat conduction, problems in homogeneous and isotropic media with internal sources and arbitrary (including nonlinear) boundary conditions and initial conditions is proposed. The method is based on the reduction of the problem to one only involving surface values of temperature and/or heat flux in the form of an integral equation through the introduction of fundamental solutions and the use of Greens theorem. The integral equation is solved numerically for a specific example.

Green's function for the vector wave equation in a mildly heterogeneous continuum

Abstract In this work, a fundamental solution is derived for the case of time-harmonic elastic waves originating from a point source and propagating in a three-dimensional, unbounded heterogeneous medium with a Poissons ratio of 0.25. The first step in the solution procedure is to transform the displacement vector in the Navier equations of dynamic equilibrium through scaling by the square root of the position-dependent shear modulus. Following imposition of certain constraints that are subsequently used to derive the depth profile of the elastic moduli and of the density, it becomes possible to employ Helmholtzs vector decomposition so as to generate two scalar wave equations for the dilational and rotational components of the wave motion, a process which again generates additional constraints. The corresponding Greens function is then synthesized in the conventional way followed for homogeneous media. Consideration of all intermediate constraints shows that the elastic moduli and the density all have quadratic variation with respect to the depth coordinate z , while the pressure and shear wavespeed profiles are constant and correspond to reference values at z = 0. The present methodology is based on earlier algebraic transformation techniques applied for the case of scalar wave propagation. The methodology is finally illustrated through a number of examples involving a commonly encountered geological medium.

Green's functions for heterogeneous media potential problems

Abstract Procedures are discussed for obtaining Greens functions for potential problems in heterogeneous materials. Such fundamental solutions are required in the boundary element method. An exact funddamental solution for a linearly varying, layered two dimensional potential problem is given that does not appear to be available elsewhere.

Green's functions for Helmholtz and Laplace equations in heterogeneous media

Fundamental Green’s functions are developed for a class of heterogeneous materials for Helmholtz and Laplace equations. Although limited to specific heterogeneities, the resulting Green’s functions are particularly simple and may be used directly in standard boundary integral equation methods.

Melt segregation in anatectic granites: a thermo-mechanical model

Abstract The generation and initial migration of magma produced by crustal anatexis involves partial melting and at least partial separation of melt from residual solid. The generation of anatectic granitic magma is thus governed by the flow of heat, melt and residual solid. A model which couples two-phase flow in porous media and heat flow was developed to simulate the evolution of anatectic melt around a mantle-derived mafic heat source. Results of our modelling suggest that emplacement of a mafic intrusion in the lower crust may result in partial melting of the crustal rocks and, if sufficient porosity is produced by melting, in migration and segregation of the anatectic melt. Segregation of the melt and its residuum was significant in models in which at least 25% melting of the host rock occurred. As an example, our calculations indicated that a 5-km-thick mafic intrusion emplaced in country rocks with an initial temperature of 800°C will create a zone of partial melting 5 km thick above and below the intrusion. In this case migration and segregation of the melt will produce a layer of magma approximately 1 km thick above the intrusion, and an even larger volume of melt below the intrusion. The anatectic magma moves upward by disaggregating the partially melted country rocks which form the roof of the evolving magma chamber. Upward migration by this process is limited to the portion of the crust which is partially melted by heat from the intrusion. The process of melt migration will modify the melt composition by both zone-refining and fractional crystallization in the initial stages of melt accumulation.

Harmonic wave propagation through viscoelastic heterogeneous media exhibiting mild stochasticity — I. Fundamental solutions

This work examines the propagation of time harmonic, horizontally polarized shear waves through a naturally occurring heterogeneous medium that exhibits viscous behaviour as well as random fluctuations of its elastic modulus about a mean value. As a first step, the governing equation, which is a heterogeneous Helmholtz equation, is solved using algebraic transformations and the relevant Greens function is obtained for two sets of boundary conditions, one corresponding to a finite depth layer and the other to an infinite layer. Viscous material behaviour is introduced by considering the depth-dependent elastic modulus to be a complex quantity. Subsequently, material stochasticity in the medium is handled through the perturbation approach by assuming that the elastic modulus has a small random fluctuation about its mean value. The final results are closed-form expressions for the mean value and covariance matrix of both the wave speed profile in the medium and the corresponding Greens function. In Part II, (Soil Dynam. Earth. Engng, 1996, 15, 129-39), two examples concerning seismic wave propagation in soft topsoil and in sandstone serve to illustrate the methodology and comparisons are made with Monte Carlo simulations.

Mechanics of emplacement of basic intrusions

Abstract The sunken nature of many basic intrusions can be explained by simple models for flexure of the lithosphere subjected to loading by a magma. The lithosphere is divided into the strata above the magma which deforms as a stack of elastic plates, and the substratum below the magma which is modelled as an elastic plate overlying a weak fluid asthenosphere. Plane-strain plate theory is used to calculate shape and size of plutons. Significant mechanical parameters controlling the shape are: (1) depth of emplacement; (2) width of intrusion; (3) total lithospheric thickness; (4) magma density; (5) effective thickness of the overburden; and (6) magmatic pressure. In areas with lithospheric thickness of 50–100 km, basic intrusions emplaced in the upper crust (

Harmonic wave propagation through viscoelastic heterogeneous media exhibiting mild stochasticity — II. Applications

The results obtained in Part I (Soil Dynam. Earth. Engng, 1996, 15, 119-27) of this work for modelling harmonic wave propagation through viscoelastic heterogeneous media that exhibit a small random fluctuation of their material properties about mean values are now used here to investigate SH wave propagation in two naturally occurring media, namely sandstone and topsoil. These results are in the form of depth dependent, deterministic mean values and non-isotropic covariances for both the wave speed profile in the medium and for the fundamental solution in terms of displacement due to a unit point source. The results are also compared against conventional Monte Carlo simulations.

A BIE formulation of a linearly layered potential problem

A boundary integral equation formulation is developed for a Poisson equation with a linearly layered conductivity, using a 3D Greens function obtained for this heterogeneous medium problem found by a technique previously developed for the corresponding 2D case.

Boundary integral formulation for 2D and 3D thermal problems exibiting a linearly varying stochastic conductivity

This work examines steady-state heat conduction in a stochastic, heterogeneous medium where the thermal conductivity varies linearly along one direction and its slope consists of a constant plus a zero-mean random part. As a first step, the governing Laplaces equation is solved using a coordinate transformation of the independent spatial variables and the exact Greens functions in both two and three dimensions are obtained for a linearly varying conductivity profile. In addition, a boundary integral equation statement in which the Greens functions appear as kernels is concurrently obtained. Next, material stochasticity is introduced and the perturbation approach is employed for deriving the mean value and covariance of the Greens functions using up to second order terms. Perturbations are also used in conjuction with the discretized boundary integral equation statement so that a mean vector and a covariance matrix for the response (temperature, heat flux) are also obtained. An example involving steady-state temperature distribution in a block along the direction where conductivity varies on the horizontal plane due to a buried heat source serves to illustrate the method. Finally, comparisons are made with Monte Carlo simulations.