D. R. Hartree

Bio: D. R. Hartree is an academic researcher. The author has contributed to research in topics: Exponential integrator & Parabolic partial differential equation. The author has an hindex of 1, co-authored 1 publications receiving 2334 citations.

A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

Abstract: This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equationwheresubject to the boundary conditionsA, k, q are known constants.Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2).

Pressure-Temperature-Time Paths of Regional Metamorphism I. Heat Transfer during the Evolution of Regions of Thickened Continental Crust

Abstract: The development of regional metamorphism in areas of thickened continental crust is investigated in terms of the major controls on regional-scale thermal regimes. These are: the total radiogenic heat supply within the thickened crust, the supply of heat from the mantle, the thermal conductivity of the medium and the length and time scales of erosion of the continental crust. The orogenic episode is regarded as consisting of a relatively rapid phase of crustal thickening, during which little temperature change occurs in individual rocks, followed by a lengthier phase of erosion, at the end of which the crust is at its original thickness. The principal features of pressure-temperature-time (PTt) paths followed by rocks in this environment are a period of thermal relaxation, during which the temperature rises towards the higher geotherm that would be supported by the thickened crust, followed by a period of cooling as the rock approaches the cold land surface. The temperature increase that occurs is governed by the degree of thickening of the crust, its conductivity and the time that elapses before the rock is exhumed sufficiently to be affected by the proximity of the cold upper boundary. For much of the parameter range considered, the heating phase encompasses a considerable portion of the exhumation (decompression) part of the PTt path. In addition to the detailed calculation of PTt paths we present an idealized model of the thickening and exhumation process, which may be used to make simple calculations of the amount of heating to be expected during a given thickening and exhumation episode and of the depth at which a rock will start to cool on its ascent path. An important feature of these PTt paths is that most of them lie within 50 °C of the maximum temperature attained for one third or more of the total duration of their burial and uplift, and for a geologically plausible range of erosion rates the rocks do not begin to cool until they have completed 20 to 40 per cent of the total uplift they experience. Considerable melting of the continental crust is a likely consequence of thickening of crust with an average continental geotherm. A companion paper discusses these results in the context of attempts to use metamorphic petrology data to give information on tectonic processes. © 1984 Oxford University Press.

Verification and Validation in Scientific Computing

Abstract: Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.

Relating extracellular potentials and their derivatives to anisotropic propagation at a microscopic level in human cardiac muscle. Evidence for electrical uncoupling of side-to-side fiber connections with increasing age.

Abstract: Elucidation of the mechanisms of cardiac conduction disturbances leading to reentry will require resolution of the details of multidimensional propagation at a microscopic size scale (less than 200 micron). In practice, this will necessitate the combined analysis of extracellular and transmembrane action potentials. The purpose of this paper is to demonstrate the relationships between the time derivatives of the extracellular waveforms and the underlying action potentials in the experimental analysis of anisotropic propagation at this small size scale, and apply these relationships to human atrial muscle at different ages. The extracellular waveforms and their derivatives changed from a smooth contour during transverse propagation in young preparations to complex polyphasic waveforms in the older preparations. The major problem was to estimate the size and location of small groups of fibers that generated the complex waveforms in the older preparations. We found dissimilarities in the derivatives that distinguished source (bundle) size from the distance of the source to the measurement site. The differences in the extracellular waveforms and their derivatives indicated that there was electrical uncoupling of the side-to-side connections between small groups of fibers with aging. These changes produced a prominent zigzag course of transverse propagation at a microscopic level which, in turn, accounted for the increased complexity of the waveforms. The waveform differences also correlated with the development of extensive collagenous septa that separated small groups of fibers. The electrophysiological consequence was an age-related decrease in the "effective" transverse conduction velocities to the range of the very slow conduction (less than 0.08 m/sec) which makes it possible for reentry to occur in small regions of cardiac muscle with normal cellular electrophysiological properties.

Formulation of the k-w Turbulence Model Revisited

Abstract: This paper presents a reformulated version of the author'sk-ω model of turbulence. Revisions include the addition of just one new closure coefficient and an adjustment to the dependence of eddy viscosity on turbulence properties. The result is a significantly improved model that applies to both boundary layers and free shear flows and that has very little sensitivity to finite freestream boundary conditions on turbulence properties. The improvements to the k-ω model facilitate a significant expansion of its range of applicability. The new model, like preceding versions, provides accurate solutions for mildly separated flows and simple geometries such as that of a backward-facing step. The model's improvement over earlier versions lies in its accuracy for even more complicated separated flows. This paper demonstrates the enhanced capability for supersonic flow into compression corners and a hypersonic shock-wave/ boundary-layer interaction. The excellent agreement is achieved without introducing any compressibility modifications to the turbulence model.