Lecture Notes in Engineering – serie
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The past decade has seen remarkable growth in research related to petroleum reseIVoir simulation. This growth reflects several developments, not the least of which is the increased interest in oil recovery technologies requiring sophisticated engineer ing. Augmenting this interest has been the broader availability of supercomputers capable of handling the tremendous computational demands of a typical reseIVoir simulator. The field of reseIVoir simulation incorporates several major facets of applied mathematics. First, in view of the varieyt and complexity of the processes encoun tered, it is imperative that the modeler adopt a systematic approach to establishing the equations governing reseIVoir flows. Second, the mathematical structure of these flow equations needs to be carefully analyzed in order to develop appropriate and efficient numerical methods for their solution. Third, since some aspects of the discretized flow equations are typically stiff, one must develop efficient schemes for solving large sparse systems of linear equations. This monograph has three parts, each devoted to one of these three aspects of reseIVoir modeling. The text grew out of a set of lectures presented by the authors in the autumn of 1986 at the IBM Scientific Center in Bergen, Norway. We feel that it is only appropriate to caution the reader that many of the ideas that we present in this monograph do not reflect standard approaches in petroleum reseIVoir simulation. In fact, our aim is to outline promising new ways of attacking reseIVoir simulation prob lems, rather than to compile another textbook for the mainstream.
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Turbulent reactive flows are of common occurrance in combustion engineering, chemical reactor technology and various types of engines producing power and thrust utilizing chemical and nuclear fuels. Pollutant formation and dispersion in the atmospheric environment and in rivers, lakes and ocean also involve interactions between turbulence, chemical reactivity and heat and mass transfer processes. Considerable advances have occurred over the past twenty years in the understanding, analysis, measurement, prediction and control of turbulent reactive flows. Two main contributors to such advances are improvements in instrumentation and spectacular growth in computation: hardware, sciences and skills and data processing software, each leading to developments in others. Turbulence presents several features that are situation-specific. Both for that reason and a number of others, it is yet difficult to visualize a so-called solution of the turbulence problem or even a generalized approach to the problem. It appears that recognition of patterns and structures in turbulent flow and their study based on considerations of stability, interactions, chaos and fractal character may be opening up an avenue of research that may be leading to a generalized approach to classification and analysis and, possibly, prediction of specific processes in the flowfield. Predictions for engineering use, on the other hand, can be foreseen for sometime to come to depend upon modeling of selected features of turbulence at various levels of sophistication dictated by perceived need and available capability.
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Any undisturbed rock mass is subject to natural stresses inclu ding gravitational stresses due to the mass of the overburden and possibly tectonic stresses due to the straining of the earth's crust and remanent stresses due to past tectonism. Knowledge of the in situ stress field must be integrated into any rock engineering design along with general rock mass characteristics such as de for mability, strength, permeability and time dependent behavior. For example, the choice of optimum orientation and shape of deep underground caverns or complex underground works will be controlled by the orientation and the magnitude of the in situ stress @ield if it is necessary to minimize stress concentration problems. Long term variation of the in situ stress field may also help to evaluate the potential hazard of earthquake occurences. The magnitude and orientation of the stress field ata point within a rock mass can be measured but there is no known method by which the state of stress at a point can be accurately determined by instruments located remotely. In general, measurements are made inside boreholes, on outcrops or on the internal surfaces of under ground cavities. Most of the measuring techniques intentionally disturb the state of stress in the rock and then measure consequent strains and displacements. Measured strains or displacements are then related to the stresses through assumptions of material behavior. A common procedure is to assume that the rock mass is linearly elastic, isotropic, continuous and homogeneous.
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Numerical techniques for solving many problems in continuum mechanics have experienced a tremendous growth in the last twenty years due to the development of large high speed computers. In particular, geomechanical stress analysis can now be modelled within a more realistic context. In spite of the fact that many applications in geomechanics are still being carried out applying linear theories, soil and rock materials have been demonstrated experimentally to be physically nonlinear. Soils do not recover their initial state after removal of temporary loads and rock does not deform in proportion to the loads applied. The search for a unified theory to model the real response of these materials is impossible due to the complexities involved in each case. Realistic solutions in geomechanical analysis must be provided by considering that material properties vary from point to point, in addition to other significant features such as non-homogeneous media, in situ stress condition, type of loading, time effects and discontinuities. A possible alternative to tackle such a problem is to inttoduce some simplified assumptions which at least can provide an approximate solution in each case. The validity or accuracy of the final solution obtained is always dependent upon the approach adopted. As a consequence, the choice of a reliable theory for each particular problem is another difficult decision which should be 2 taken by the analyst in geomechanical stress analysis.
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Del 6 - Lecture Notes in Engineering
Collocation Techniques for Modeling Compositional Flows in Oil Reservoirs
Häftad, Engelska, 1983
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This investigation is an outgrowth of my doctoral dissertation at Princeton University. I am particularly grateful to Professors George F. Pinder and William G. Gray of Princeton for their advice during both my research and my writing. I believe that finite-element collocation holds promise as a numer ical scheme for modeling complicated flows in porous media. However, there seems to be a "conventional wisdom" maintaining that collocation is hopelessly beset by oscillations and is, in some way, fundamentally inappropriate for multiphase flows. I hope to dispel these objections, realizing that others will remain for further work. The U. S. National Science Foundation funded much of this study through grant number NSF-CEE-8111240. TABLE OF CONTENTS ABSTRACT ;; FOREWORD ;; ; CHAPTER ONE. THE PHYSICAL SYSTEM. 1.1 Introduction. 1 1.2 The reservoir and its contents. 5 1.3 Reservoir mechanics. 9 1.4 Supplementary constraints. 18 1.5 Governing equations. 26 CHAPTER TWO. REPRESENTING FLUID-PHASE BEHAVIOR. 39 2.1 Thermodynamics of the fluid system. 40 2.2 Standard equation-of-state methods. 45 2.3 Maxwell-set interpolation.
Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems
Häftad, Engelska, 1984
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Harmonic and biharmonic boundary value problems (BVP) arising in physical situations in fluid mechanics are, in general, intractable by analytic techniques. In the last decade there has been a rapid increase in the application of integral equation techniques for the numerical solution of such problems [1,2,3]. One such method is the boundary integral equation method (BIE) which is based on Green's Formula [4] and enables one to reformulate certain BVP as integral equations. The reformulation has the effect of reducing the dimension of the problem by one. Because discretisation occurs only on the boundary in the BIE the system of equations generated by a BIE is considerably smaller than that generated by an equivalent finite difference (FD) or finite element (FE) approximation [5]. Application of the BIE in the field of fluid mechanics has in the past been limited almost entirely to the solution of harmonic problems concerning potential flows around selected geometries [3,6,7]. Little work seems to have been done on direct integral equation solution of viscous flow problems. Coleman [8] solves the biharmonic equation describing slow flow between two semi infinite parallel plates using a complex variable approach but does not consider the effects of singularities arising in the solution domain. Since the vorticity at any singularity becomes unbounded then the methods presented in [8] cannot achieve accurate results throughout the entire flow field.
Del 8 - Lecture Notes in Engineering
Multiphase Migration of Organic Compounds in a Porous Medium
A Mathematical Model
Häftad, Engelska, 1984
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Groundwater has long been one of the world's most important resources. It accounts for approximately 96% of all fresh water in the United States and supplies more than 50% of the population with potable water. Historically, this water source has generally been regarded as pristine. However, in recent years, contamination of ground water by industrial products has become a problem of growing concern. During the past four decades, the variety and quantity of organic chemicals produced in the U. S. has steadily increased. Currently, more than 40,000 different organic compounds are being manufactured, trans ported, used and eventually disposed of in the environment (Wilson, et !l (1981». Production and consumption of petroleum products has also risen in this same time period. Many of these industrial compounds are highly toxic and slightly water soluble. Thus, they pose a poten tial threat to large volumes of groundwater if they are somehow intro duced into the subsurface. Increased production of chemicals implies the increased risk of accidental spills or leakage to the soil, and indeed, the literature abounds with contamination case histories. 2 Incidences of petroleum contamination of groundwater have been documented by many authors. For example, see: Schwi11e (1967); Toms (1971); Guenther (1972); McKee, et!l (1912); Williams and Wilder (1971); Van100cke, et ~]-
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The Complex Variable Boundary Element Method or CVBEM is a generalization of the Cauchy integral formula into a boundary integral equation method or BIEM. This generalization allows an immediate and extremely valuable transfer of the modeling techniques used in real variable boundary integral equation methods (or boundary element methods) to the CVBEM. Consequently, modeling techniques for dissimilar materials, anisotropic materials, and time advancement, can be directly applied without modification to the CVBEM. An extremely useful feature offered by the CVBEM is that the pro duced approximation functions are analytic within the domain enclosed by the problem boundary and, therefore, exactly satisfy the two-dimensional Laplace equation throughout the problem domain. Another feature of the CVBEM is the integrations of the boundary integrals along each boundary element are solved exactly without the need for numerical integration. Additionally, the error analysis of the CVBEM approximation functions is workable by the easy-to-understand concept of relative error. A sophistication of the relative error analysis is the generation of an approximative boundary upon which the CVBEM approximation function exactly solves the boundary conditions of the boundary value problem' (of the Laplace equation), and the goodness of approximation is easily seen as a closeness-of-fit between the approximative and true problem boundaries.
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The increasing size and complexity of new structural forces in engineering have made it necessary for designers to be aware of their dynamic behaviour. Dynamics is a subject which has traditionally been poorly taught in most engineering courses. This book was conceived as a way of providing engineers with a deeper knowledge of dynamic analysis and of indicating to them how some of the new vibrations problems can be solved. The authors start from basic principles to end up with the latest random vibration applications. The book originated 1n a week course given annually by the authors at the Computational Mechanics Centre, Ashurst Lodge, Southampton, England. Special care was taken to ensure continuity in the text and notations. Southampton 1984 CONTENTS Page Foreword Chapter 1 Introduction to Vibration 1. Introductory Remarks 1 2. Single Degree of Freedom Systems: Equations of Motion and Types of Problem 2 3. Response 6 4. General Structures: Equations of Motion 11 5. Response 15 6. Dynamic Interaction Problems 20 Chapter 2 Free Vibration, Resonance and Damping l. Introduction 25 25 2. Spring-Mass System 3. Simple Pendulum 27 4. Beam with Central Load 28 5. Rolling of a Ship 28 6. Springs in Parallel 30 7. Springs in Series 30 8. Free Vibration 31 9. Energy of Vibrating System 33 10. Damped Free Vibration 34 11. Undamped Forced Response 38 12. Damped Forced Response 39 13. Undamped Transient Vibration 42 14. Damped Transient Vibration 43 15.
Water Quality Management
A Review of the Development and Application of Mathematical Models
Häftad, Engelska, 1985
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During 1978-1982 the International Institute for Applied Systems Analysis (IIASA) was responsible for a research project on Environmental Quality Control and Management. The project was begun under the direction of Professor O. F. Vasiliev (from the Institute of Hydrodynamics of the Siberian Branch of the USSR Academy of Sciences) and was subsequently led by myself. This review is very much a re'fiection of that IIASA project. The major themes of the IIASA project were: (i) research into the methodological aspects of modeling river and lake sys tems [some of the principal results of this research appear in M. B. Beck and G. van Straten (eds. ) (1983), Uncertainty and Forecasting of Water Quality (Springer, Berlin (West)), and in K. Fedra (1983), Environmental Modeling Under Uncertainty: Monte Carlo Simulation (IIASA Research Report RR-83-28)]; (ii) case studies in the application of mathematical models to lake eutrophi cation control [results of which are summarized in L. Somlyody, S. Hero dek, and J. Fischer (eds. ) (1983), Eutrophication of Shallow Lakes: Model ing and Management (The Lake Balaton Case Study) (IIASA Collaborative Proceedings CP-83-S3), and in K. Fedra (1983), A Modular Approach to Comprehensive System Simulation: A Case Study of Lakes and Watersheds (in W. K. Lauenroth, G. V. Skogerboe, and M. Flug (eds. ), Analysis of Ecological Systems: State-of-the-Art in Ecological Modelling, pp. 195-204. Elsevier, Amsterdam)]; iv (iii) a policy study of operational water qua,lity management [M. B. Beck (1981), Operational Water Quality Management: Beyond Planning and Design (IIASA Executive Report ER-7)].
Nonlinear Dynamics of Transcritical Flows
Proceedings of a DFVLR International Colloquium, Bonn, Germany, March 1984
Häftad, Engelska, 1985
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The German Aerospace Research Establishment (DFVLR) has initiated a new series of seminars concerning fundamental problems in applied engineering sciences. These seminars will be devoted to interdisciplinary topics related to the vast variety of DFVLR activities in the fields of fluid mechanics, flight mechanics, guidance and control, materials and struc tures, non-nuclear energetics, communication technology, and remote sensing. The purpose of the series is twofold, namely, to bring modern ideas and techniques to the attention of the DFVLR in order to stimulate internal activi ties, and secondly, to promulgate DFVLR achievements wi thin the international scientific/technical community. To this end, prominent speakers from Germany and other countries will be invited to join in a series of lectures and discussions on certain topics of mutual interest. The first colloquium of this series dealt with the dynamics of nonlinear systems, especially in relation'to its application to fluid mechanics, particularly in transcritical flows. Of special interest are questions concerning the formation of nonlinear three-dimensional structures in classical fluid mechanical stability problems, the physical process of transition to turbulence, and the appearance of chaotic solutions. The scope of lectures reaches from self-organization in physical systems to structural stability of three-dimensional vortex patterns, the treatment of dissipative and conservative systems, the formation of nonlinear structures in the region of laminar-turbulent transition, and numerical simulation of cumulus cloud convection in meteorology. The seminar should provide an insight into the extent to which theoretical findings in Non linear Dynamics apply to the comprehension of fluid-mechanical problems.
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DEFINITION AND NOMENCLATURE A Stirling engine is a mechanical device which operates on a closed regenerative thermodynamic cycle with cyclic compression and expansion of the working fluid at different temperature levels. The flow of working fluid is controlled only by the internal volume changes, there are no valves and, overall, there is a net conversion of heat to work or vice-versa. This generalized definition embraces a large family of machines with different functions; characteristics and configurations. It includes both rotary and reciprocating systems utilizing mechanisms of varying complexity. It covers machines capable of operating as a prime mover or power system converting heat supplied at high tempera ture to output work and waste heat at a lower temperature. It also covers work-consuming machines used as refrigerating systems and heat pumps abstracting heat from a low temperature source and delivering this plus the heat equivalent of the work consumed to a higher tem perature. Finally it covers work-consuming devices used as pressure generators compressing a fluid from a low pressure to a higher pres sure. Very similar machines exist which operate on an open regen erative cycle where the flow of working fluid is controlled by valves. For convenience these may be called Ericsson engines but unfortunate ly the distinction is not widely established and regenerative machines of both types are frequently called 'Stirling engines'.
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The Boundary Integral Equation (BIE) or the Boundary Element Method is now well established as an efficient and accurate numerical technique for engineering problems. This book presents the application of this technique to axisymmetric engineering problems, where the geometry and applied loads are symmetrical about an axis of rotation. Emphasis is placed on using isoparametric quadratic elements which exhibit excellent modelling capabilities. Efficient numerical integration schemes are also presented in detail. Unlike the Finite Element Method (FEM), the BIE adaptation to axisymmetric problems is not a straightforward modification of the two or three-dimensional formulations. Two approaches can be used; either a purely axisymmetric approach based on assuming a ring of load, or, alternatively, integrating the three-dimensional fundamental solution of a point load around the axis of rotational symmetry. Throughout this ~ook, both approaches are used and are shown to arrive at identi cal solutions. The book starts with axisymmetric potential problems and extends the formulation to elasticity, thermoelasticity, centrifugal and fracture mechanics problems. The accuracy of the formulation is demonstrated by solving several practical engineering problems and comparing the BIE solution to analytical or other numerical methods such as the FEM. This book provides a foundation for further research into axisymmetric prob lems, such as elastoplasticity, contact, time-dependent and creep prob lems.
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1. 1 AREAS OF APPLICATION FOR THE SHALLOW WATER EQUATIONS The shallow water equations describe conservation of mass and mo mentum in a fluid. They may be expressed in the primitive equation form Continuity Equation _ a, + V. (Hv) = 0 L(l;,v;h) at (1. 1) Non-Conservative Momentum Equations a M("vjt,f,g,h,A) = at(v) + (v. V)v + tv - fkxv + gV, - AIH = 0 (1. 2) 2 where is elevation above a datum (L) ~ h is bathymetry (L) H = h + C is total fluid depth (L) v is vertically averaged fluid velocity in eastward direction (x) and northward direction (y) (LIT) t is the non-linear friction coefficient (liT) f is the Coriolis parameter (liT) is acceleration due to gravity (L/T2) g A is atmospheric (wind) forcing in eastward direction (x) and northward direction (y) (L2/T2) v is the gradient operator (IlL) k is a unit vector in the vertical direction (1) x is positive eastward (L) is positive northward (L) Y t is time (T) These Non-Conservative Momentum Equations may be compared to the Conservative Momentum Equations (2. 4). The latter originate directly from a vertical integration of a momentum balance over a fluid ele ment. The former are obtained indirectly, through subtraction of the continuity equation from the latter. Equations (1. 1) and (1. 2) are valid under the following assumptions: 1. The fluid is well-mixed vertically with a hydrostatic pressure gradient. 2. The density of the fluid is constant.
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This book contains notes for a one-semester graduate course which is an introduction to the study of viscoelasticity and creep of concrete. Emphasis was set on the conceptual aspects rather than on the mathematical or computational refinements. The mathematical structure of viscoelasticity is discussed with some care because it clarifies the basic concepts and has important consequences in computa tional applications. Basic ideas are exemplified using the simplest problems and constitutive models in order to be able to show complete solutions. In the computational applications we have also chosen to present the sim plest situations with the greatest possible detail. It has been the author's experience that once the basic concepts are well understood the students are able to follow the rest of the course more easily and to accede to more advanced literature and applications. Chapters I to III furnish the foundations for the course, that may be expanded in diverse ways. If we are interested in finite elements applications we should look at Chapter IV and then go direct ly to Chapter VII. If we are interested in the simplified analysis of frame structures we should study Chapter VI in detail. Chapter V re views the viscoelastic behavior of concrete and Chapter VIII studies the problem of creep buckling. At the end of each chapter we give se lected references to works that complete and extend the subject matter.
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The purpose of this book is to j.ir€'~ 0l'l\' a new technique for the experimental investigation of the free wave model sound field of acoustics. The technique is based on the use of spherical harmonic functions of angle. Acousticians frequently encounter random sound fields whose properties may be closely modelled by use of the "free wave" field. This model field is defined by two basic statistical properties: stationarity in time, and homogeneity in space. Stationarity means that any single order statistic measured by a microphone in the field will be independent of the time at which the recording is taken, while homogeneity means that the measurement will also be independent of the mic- phone's position in the field. Furthermore, second order statistics obtained from the measurements of two microphones will depend only on the time lapse between the two recordings, and the relative spatial separation of the micro phones, and not on the microphones' absolute positions in space and time. The free wave field may also (equivalently) be pictured as a collection of plane sound waves which approach an observation position from all angles. These are the "free waves" of the title, with no correlation between waves at different angles and frequencies, although there may exist an angle-dependant plane wave density function. This is a measure of the density of sound energy arriving from different angles. The free wave field has proved to be a simple but remarkably powerful model.
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Computational fluid flow is not an easy subject. Not only is the mathematical representation of physico-chemical hydrodynamics complex, but the accurate numerical solution of the resulting equations has challenged many numerate scientists and engineers over the past two decades. The modelling of physical phenomena and testing of new numerical schemes has been aided in the last 10 years or so by a number of basic fluid flow programs (MAC, TEACH, 2-E-FIX, GENMIX, etc). However, in 1981 a program (perhaps more precisely, a software product) called PHOENICS was released that was then (and still remains) arguably, the most powerful computational tool in the whole area of endeavour surrounding fluid dynamics. The aim of PHOENICS is to provide a framework for the modelling of complex processes involving fluid flow, heat transfer and chemical reactions. PHOENICS has now been is use for four years by a wide range of users across the world. It was thus perceived as useful to provide a forum for PHOENICS users to share their experiences in trying to address a wide range of problems. So it was that the First International PHOENICS Users Conference was conceived and planned for September 1985. The location, at the Dartford Campus of Thames Polytechnic, in the event, proved to be an ideal site, encouraging substantial interaction between the participants.
Finite Rotations in Structural Mechanics
Proceedings of the Euromech Colloquium 197, Jabłonna, Poland, 1985
Häftad, Engelska, 1986
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The deformation near a material particle of the classical continuum is produced by successive superposition of a rigid-body translation, a pure stretch along principal directions of strain and a rigid-body ro tation of those directions. The rotational part of deformation is par ticularly important in the non-linear analysis of thin-walled solid structures such as ~eams, thin-walled bars, plates and shells, since in this case finite rotations may appear even if the strains are infinite simal. It seems that the research concerning the application of finite ro tations is carried out independently in different fields of structural mechanics. Theoretical and numerical methods developed and the results obtained for a particular type of the structure or for a particular ma terial behaviour not always are used to analyse similar problems for other types of structures or for another material behaviour. Since the research in this field had been growing rapidly, it was decided to organize an informal international meeting, under the auspi ces of the European Mechanics Co~mittee, entitled: Euromech Colloquium 197 "Finite Rotations in Structural Mechanics". The meeting was held on 17 - 20 September 1985 in Jablonna, a small suburbian area of Warsaw.
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The analysis of thin shells of revolution in general has always occupied an important place in the theory of structures, and recently the problem of hyperbolic cooling towers has attracted many investigators due to the wide use of such shells in industry. Until the early 1960's these towers were of moderate size, probably not exceeding 76m (250ft) height. In this range, the structural safety and stability were not of primary concern because, the minimum wall thickness and reinforcement were sufficient to provide the required safety. It was not necessary to use very rigorous methods to analyse the problem. The analysis involved the following assumptions, i) flexural stresses were ignored (membrane tneory), ii) The geometry of the shell was assumed to be perfect and to be idealised as a set of straight sided conical frusta, and iii) The boundary conditions at the base were taken to be fixed or continuously hinged with full tangential restraint.
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A. GENERAL REMARKS During the last century, probabilistic methods for design and analysis of engineering systems have assumed a prominent place as an engineering tool. No longer do engineers naively believe that all problems can be analyzed with deterministic methods; but rather, it has been recognized that, due to unc- tainties in the model and the excitation, it may only be possible to describe the state of a system in terms of some random measure. Thus, with the need to address safety and design issues adequately and simultaneously to minimize the cost of a system, much attention has been given to the development of probabilistic criteria which can be applied in a systematic manner [l]t. These techniques allow for uncertainties in the parameters of the model as well as for uncertainties in both the static and dynamic loadings to be considered and therefore give a better measure of the reliability of a system. Widespread application of probabilistic methods can be found in disciplines ranging from civil, mechanical and electrical engineering to biology, economics and political science.
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and Literature Rev iew Chapter 1 1. INTRODUCTION AND LITERATURE REVIEW The exploration for oil and gas in ever increasing water depths has given an impetus to research efforts on the behaviour of offshore structures under ocean environment. These structures are continuously subjected to environmental loading because of waves, wind and current. A response analysis is required to assess the safety of offshore structure under severe storm conditions as well as for estimation of damage caused by less severe but more frequently occuring sea states. A majority of the reported failures in the life time of offshore structures are in fact fatigue failures. The offshore structures are usually built in the form of welded tubular structures. The joints of these tubular members experience the fatigue damage mainly due to small defects in welding which act as crack initiators, high stress concentrations and the variable loads. The variable loads due to the ocean waves cause cyclic stress variation in the structural members and the accumu lated effect of these stresses results in the fatigue failure.
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Supercomputers and Fluid Dynamics
Proceedings of the First Nobeyama Workshop September 3–6, 1985
Häftad, Engelska, 1986
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In the past several years, it has become apparent that computing will soon achieve a status within science and engineering to the classical scientific methods of laboratory experiment and theoretical analysis. The foremost tools of state-of-the-art computing applications are supercomputers, which are simply the fastest and biggest computers available at any given time. Supercomputers and supercomputing go hand-in-hand in pacing the development of scientific and engineering applications of computing. Experience has shown that supercomputers improve in speed and capability by roughly a factor 1000 every 20 years. Supercomputers today include the Cray XMP and Cray-2, manufactured by Cray Research, Inc., the Cyber 205, manufactured by Control Data Corporation, the Fujitsu VP, manufactured by Fujitsu, Ltd., the Hitachi SA-810/20, manufactured by Hitachi, Ltd., and the NEC SX, manufactured by NEC, Inc. The fastest of these computers are nearly three orders-of-magnitude faster than the fastest computers available in the mid-1960s, like the Control Data CDC 6600. While the world-wide market for supercomputers today is only about 50 units per year, it is expected to grow rapidly over the next several years to about 200 units per year.
Modeling of Plume Rise and Dispersion — The University of Salford Model: U.S.P.R.
Häftad, Engelska, 1987
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Shell and Spatial Structures: Computational Aspects
Proceedings of the International Symposium July 1986, Leuven, Belgium
Häftad, Engelska, 1987
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In recent years powerful engineering workstations for a reasonable price become a valuable tool for the design of complicated constructions such as shell and spatial structures. This availability causes an increasing use of advanced numerical techniques for the static and dynamic analysis of these structures, also in the non-linear range. The I.A.S.S. Working Group nO 13 concerned with "Numerical Methods in Shell and Spatial Structures" and the Department of Civil Engineering of the Katholieke Universiteit Leuven have taken the initiative to organise an International Symposium, providing a forum for discussion and exchange of views between researchers, specialists in numerical analysis on one hand and designers, practising engineer ings on the other hand. These Proceedings contain the papers presented at the Symposium, held in Leuven, July 14-16 1986. The papers are organised in five sections 1. Shell structures 2. Spatial structures 3. Dynamic analysis 4. Non-linear analysis 5. Presentation and interpretation of results The papers covering more than one domain are classified following the main subject. We hope that researchers as well as practising engineers will find a lot of useful information in the book.
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The most commonly used numerical techniques in solving engineering and mathematical models are the Finite Element, Finite Difference, and Boundary Element Methods. As computer capabilities continue to impro':e in speed, memory size and access speed, and lower costs, the use of more accurate but computationally expensive numerical techniques will become attractive to the practicing engineer. This book presents an introduction to a new approximation method based on a generalized Fourier series expansion of a linear operator equation. Because many engineering problems such as the multi dimensional Laplace and Poisson equations, the diffusion equation, and many integral equations are linear operator equations, this new approximation technique will be of interest to practicing engineers. Because a generalized Fourier series is used to develop the approxi mator, a "best approximation" is achieved in the "least-squares" sense; hence the name, the Best Approximation Method. This book guides the reader through several mathematics topics which are pertinent to the development of the theory employed by the Best Approximation Method. Working spaces such as metric spaces and Banach spaces are explained in readable terms. Integration theory in the Lebesque sense is covered carefully. Because the generalized Fourier series utilizes Lebesque integration concepts, the integra tion theory is covered through the topic of converging sequences of functions with respect to measure, in the mean (Lp), almost uniformly IV and almost everywhere. Generalized Fourier theory and linear operator theory are treated in Chapters 3 and 4.