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This paper presents a technique for simulation of the stock preparation and Fourdrinier dynamics which is flexible, convenient, and detailed enough to allow the user to study physical process modifications and transient response dur ing upsets and "furnish" (fibers with different character istics, fillers, dyes, etc.) changes and to evaluate simplified mathematical models and control schemes. The technique uses digital simulation in a block diagram form, each block of which corresponds to one type of physical unit in the process (e.g., cleaner or tanks). Simu lation of a particular process then involves specification of the particular parameters of each physical unit together with the way in which they are interconnected. Individual components of the furnish may be traced through the whole process. A complete wet-end configuration, including the white water recovery system, is simulated using the technique described in the paper, and typical results obtained during grade changes and disturbances are presented.

The effective use of modern control theory requires suitable mathematical models of the objects to be controlled and of the disturbances. Paper machines have been controlled digitally for a long time. Digital control of basis weight and moisture content is in fact one of the few examples where a digital computer and sensors form a standard package. In spite of this there are still significant gaps in our knowledge of paper machine dynamics. This paper attempts to fill a few of these. The paper is based on measurements on full scale paper machines and on a pilot plant at the Swedish Forest Products Research Laboratory (STFI). The experiments on the industrial plants were performed using control computers with appropriate instrumentation both on craft paper and newsprint machines. The experiments on the pilot plant were carried out with different fibre, clay and water mixtures. These experiments gave insight into the effect of the pulp characteristics on paper machine dynamics. Optical gauges capable of measuring low fibre concentrations were used in the experiments. This made detailed analysis of the mixing dynamics possible. The possibility of obtaining better control through the incorporation of such gauges could also be evaluated.

The paper outlines the development of mathematical models to predict the wear of metal, polymer, and ceramic materials caused by the abrasive effects of paper. The collection and analysis of wear data, and the derivation of the predictive mathematical models are described. Included in the investigation are the effects on wear of total length of paper passed over the specimen, individual document length, document velocity, normal load, specimen geometry, and specimen material. The document mix used was representative of that often encountered for modern document handling machines. The results indicate good correlation between the wear observed and that predicted by the mathematical models.

The use of models in teaching elementary school mathematics has been extensively researched and recommended. Since number concepts and basic operations constitute a major part of the elementary mathema tics curriculum, a variety of models—extending on a continuum from Cuisenaire rods; Dienes blocks; and objects like sticks, straws, and buttons; to representations such as drawings and number sentences—have been recommended. The remainder of this paper is devoted to several uses of paperand-pencil models which are simple enough for children to construct. The descriptions of the models indicate the level of suggested use.

Prediction and simulation are dependent on the knowledge gained from mathematical models. The choice and the form of the model is determined by the specific application and the accuracy required. Model techniques, such as horizontal oscillation tests, provide the means of finding the coefficients of extensive non-linear models, though for other purposes, such as simulation, there is also a need for simpler, empirical models. To determine the coefficients of such models, free-running full-scale or model tests are necessary.

In this investigation, a pushrod operated valve train is simulated by mathematical models of two, three, four, ten, and fifteen degress of freedom. A number of important features are included in the simulation, such as tappet clearance, linkage separation, valve bounce, and valve spring surge. Solutions of the response to cam lift function under specified running conditions, are obtained by numerical means with a digital computer. These are compared with experimental records of pushrod load and valve displacement on a running engine. The use of similar programs to predict valve gear performance is justified by the excellent agreement obtained.

Manoeuvrability in waves is discussed from the point of view of the dangers of broaching-to when a ship is running before the sea. Conditions are assessed under which this may occur, illustrated by documented cases, including the Wahine disaster in 1968. Because of the problems involved in investigating broaching-to by means of free-running model tests, there is an urgent need for reliable mathematical models: however, theories published so far, based on two different simplifications, are shown to have limitations. It is argued that the theory must take account of pitching, surging, rolling and orbital motion of the water particles.

Although several methods are available for the design of flexible pavements, no existing technique explicitly considers the optimal combination of flexible pavement components to minimize the total in-place cost of the pavement system. The purpose of this systems analysis was to develop a rational method for the optimal selection of the arrangement of the various pavement components. This cost minimization must be realized within the boundary conditions that are imposed by the practical limitations of the design parameters. The design model consists of an objective function and various constraint equations. The total cost of the pavement system is quantitatively described by this objective function, and a minimum-cost solution is obtained for each combination of material costs and design conditions. The various constraining equations quantify the boundary conditions to which the design of a flexible pavement is subject. These physical limitations complete the realism of the mathematical model in describing the real-world situation of flexible pavement design. The design model was solved by a modified linear programming technique. In developing practical solutions to the design model, optimal flexible pavements are designed for cross-sections without subbase, cross-sections with subbase through shoulders, and cross-sections with subbase and subdrains. The design requirements for the various components are predicated on the design parameters of traffic conditions, soil support values, pavement material characteristics, environmental effects, and pavement performance requirements and on unit costs of pavement components. Substantial cost savings result in the selection of flexible pavement sections by this design procedure.

The purposes of this research Investigation were to develop mathematical models that measure the relative safety or hazard of urban grade crossings and to establish a priority rating system, based on these models, for determining protection improvements in urban areas. The mathematical techniques of discriminant analysis and regression analysis were utilized to develop discriminant models with linearly assigned probabilities. These models permit potential hazards to be expressed as the probability that a grade crossing is accident prone. A grade crossing where a vehicle-train accident had occurred during a two-year period was considered as a representative member of the population of accident prone crossings. A location which had not experienced a vehicle-train accident for at least five years prior to the date of the field investigation was assumed as a representative member of the population of non-accident prone grade crossings. Data were collected at 295 accident locations and 281 non-accident locations in urban portions of the State of Indiana. The best discriminant model is 74-percent successful in classifying the true group membership of the sample grade crossings. This model expresses potential hazard as a function of protective device, average daily highway traffic, average daily train traffic, degree of effective sight distance, and roadside distractions. A methodology was developed for selecting a minimum level of grade crossing protection and for establishing priorities for the improvement of protection at urban railroad-highway grade crossings.

The energy relations and the constitutive relations pertaining to a nonlinear transmission line are examined in detail. It is concluded that mathematical models which have been used in certain studies and which lead to the paradox concerning energy dissipation in the shock front are inadmissible. Correct models are free from such paradoxes. The work leads to the formulation of the hypothesis of realization: it is impossible to realize a continuous loss-free transmission medium which would be characterized by nonlinear distributed inductance L(I) and capacitance C(V), unless the medium is dispersive.

Recently, Leer and Holzer (1972) presented a comparison of kinetic and hydrodynamic solar-wind models. Since both their models produced the same solar- wind properties at Earth, and since the hydrodynamic description did not include viscosity, they concluded that viscosity is not important in the solar wind. The present authors disagree that this conclusion applies to the real solar wind. Also it is felt that they did not demonstrate any general applicability for their conclusion that the kinetic and hydrodynamic approaches are equivalent in the solar wind. (WDM)

This paper discusses the cornering characteristics during the progress of tire wear and how the stability and controllability of a vehicle varies in the various stages of tire wear. With the introduction of tire parameters appropriate to represent the characteristics of tire construction, the method of calculation of the cornering characteristics of a tire is proposed in this paper. Furthermore, this paper shows the values of tire parameters in the various stages of tire wear in both cases of bias belted and radial tires.

This paper presents and compares three mathematical programming models for the optimization of drainage networks. The three models are based on two extensions of linear programming: separable-convex and mixed- integer programming. The first produces a continuous range of diameters while the second limits the solution to discrete commercially available sizes. It is shown that while the first can be formulated with pipes flowing full, the second must allow for partial flow. A solution which combines the use of both of these techniques can produce a minimum cost system with partially-full flows and commercially available diameters. This solution requires less computer time than those based strictly on mixed-integer programming. An example seven-link drainage network is designed by the three proposed methods and the results are reported in the paper.

This paper considers the model-reference adaptive control problem which has received considerable attention in the literature in the last few years. An adaptive control scheme is proposed which has terms in the Liapunov function used in the design procedure which are not included in previously proposed schemes. The relationship of this new scheme to existing schemes is shown by considering the root-loci of the linearized error equations between plant and model. Finally, a second order example is given which illustrates the difference between the two previously proposed model-reference adaptive methods and the one proposed in this paper.

The efficient design and operation of a geothermal wellbore requires an understanding of the complex interactions of heat transfer, fluid flow, phase change, flow regime change, and steam-water slip. The equations flow and concepts presented here account for those interactions and are used in simulating two-phase steam-water in a wellbore. Introduction The current examination of our country's energy resources has resulted in new and greater interest in geothermal power. There are geothermal installations in New Zealand, Italy, California, Mexico, Japan, U.S.S.R., and elsewhere. Many of these operations have long been producing power from reservoirs that are principally vapor-dominated systems, where essentially dry steam is produced at the wellhead and used directly for power generation. Unfortunately, vapor-dominated systems are relatively rare and current geothermal projects in the U.S. propose to use aquifers that produce hot water or a steam-water mixture. New Zealand is the leader in developing the technology required to operate these two-phase steam-water systems. A typical aquifer, when first discovered, will produce undersaturated water at the wellbore sand-face. produce undersaturated water at the wellbore sand-face. As the fluid flows up the wellbore and pressure is lost, the fluid eventually flashes, producing a steam-water mixture in the well. The two-phase mixture is usually separated at the wellhead, with the steam being used directly to generate power." More advanced designs call for the fluid heat to be transferred to a secondary fluid, such as freon or isobutane, thereby using the potential of both steam and hot-water fractions. potential of both steam and hot-water fractions. The secondary fluid is used to generate power and the condensed water is reinjected into the formation. This approach reduces environmental problems and makes more efficient use of the available thermal energy. The reinjected water maintains aquifer pressure and prolongs the producing life of the field. As production continues over a period of months, the aquifer pressure will usually drop to the saturation point, and fully developed two-phase steam-water flow will be produced by the formation at the sandface. The two-phase mixture then flows towards the surface, experiencing pressure drop and heat transfer. In many applications, minerals are deposited on the wellbore casing where the mixture is flashing. The design, analysis, and operation of geothermal systems requires sophisticated applications of the principles of fluid mechanics, heat transfer, and flow principles of fluid mechanics, heat transfer, and flow through porous media. The mathematical equations that describe a geothermal wellbore will be presented in this paper. The simulation of the combined effects of heat paper. The simulation of the combined effects of heat transfer and two-phase flow in wells has not previously been fully presented. Various portions of the technology have been published, but in a piece-wise fashion. The aim of this work is to develop an integrated approach to the problem, an adequate treatment of which must account for problem, an adequate treatment of which must account for the following effects:Two-phase pressure drop,Flow regime change,Phase change (fluid miscibility),Relative steam-water velocity (slip),Heat transfer from the fluid. All these effects are interrelated. JPT P. 833

It has become increasingly apparent that mathematical models may play a significant role in anthropological methodology. Less apparent is the fact that such models have an important place in formulation of theory as well. At the same time, many errors appear in attempts to use mathematical models. This paper attempts to indicate the nature of such errors and a methodology by which they can be avoided. A case study is presented showing the relationship of mathematical models to anthropological theory. Department of Anthropology University of California, Los Angeles December, 1972 AS IS THE CASE for students in other disciplines, anthropologists are becoming increasingly aware that the use of mathematical models can be a very powerful tool in the analysis of data and the formulation of theory. Mathematics is useful if for no other reason than that the very nature of a mathematical model requires precise definitions, exact statements of relationships between variables, and quantification of data. In a mathematical model, it is not sufficient simply to state that variable A is correlated with variable B, but the precise way in which B changes as A changes-for example, linearly, exponentially, or otherwise-must be stated. Such rigor in defining relationships necessitates a scrutiny of data and a rethinking of theory that might not occur without the formulation of a mathematical model. A mathematical model thus provides a concise way to state relationships among variables. In addition, the attempt to establish a model for use in a particular instance may itself facilitate the process of understanding and explaining the phenomena under observation. This important aspect of mathematical models has, unfortunately, been little exploited. The purpose of this paper is to explore the ways in which mathematical models can be and have been used in this manner. In the anthropological literature, there are many examples that could be cited that demonstrate the usefulness of mathematical models for explication. White's An anatomy of kinship (1963), for instance, shows the power that formal mathematical models have if a set of basic (and exhaustive) definitions for a system can be stated explicitly. By first deriving such a set of definitions for an exogamous marriage system, he was able to demonstrate that the resulting structure has a particular mathematical form, known as a group, and hence can be analyzed by the concepts and theory of that branch of mathematics known as group theory. At the same time that this makes the structure of the marriage system so defined apparent, it also makes equally obvious any sources of error in the model should its conclusions be untenable, for errors can only lie in the basic set of definitions. These can then be modified and, if necessary, a new structure developed. In the absence of such explicit definitions and well-defined theory, the result can be near chaos-witness the debate in the early 1960's over the validity of Levi-Strauss' purported distinction between marriage systems built on "prescription" versus those derived from "preference." Without precise definitions of these terms, it becomes extremely difficult to resolve the validity of the distinction implied by the terms. While Ackerman (1964) tried to do this statistically, it is clear that in part his efforts could not be wholly convincing. He was obliged to test not just an explicit definition, but also his interpretation of Levi-Strauss' term, which need not have been the same as one by Needham. And Needham in turn does not appear to agree with Levi-Strauss (Harris 1968:512). Most mathematical models used in anthropology, though, are statistical in nature. These are the simplest to formulate and, generally speaking, most easily adapted to traditional ways of anthropological analysis. Such statistical models often lack precision, in the sense of formally showing the nature of the relationship between variables. However, there are implicit assumptions

The steady state response for models of a six-axle locomotive running on a sinusoidally irregular track has been investigated. Two mathematical models have been set up, a full model for the “stationary” vehicle in which creep between wheels and rails was neglected, and a full model for the “moving” vehicle in which creep forces, gravity stiffness effects and wheel tread profiles were considered. The use of the generalized method of complex algebra to obtain the steady state response of the railway vehicle components to varying input frequencies was used. The results given in this paper are for the case of sinusoidal lateral track irregularities only, but the method is general enough to allow also for vertical track irregularities. For the “stationary” vehicle the input frequency is increased from zero to 3 cycles per second. For the “moving” vehicle the input frequency is a function of the track wave length and the vehicle forward speed and is given in terms of the vehicle speed. The frequency response curves are computer plotted in each case. For the “moving” vehicle, responses for the cases of both new and worn wheels are obtained. The natural frequencies for the full model are also calculated. The results obtained show the effect of the creep forces and the condition of the wheels on the steady state response. It is recommended that slip and corresponding creep forces, wheel tread and rail profiles, and the gravity stiffness effect be included in the steady state response analysis of railway vehicles to track irregularities. The analysis may be used to check the response of any proposed design for a railway vehicle to economically attractive track irregularities. It may also be used to adjust geometry, spring rates and damping characteristics in order to maximize operating speeds while providing optimum damping for the trucks and body motions. This paper illustrates and describes the mathematical models used; gives generalized form for the differential equations of motion and the methods of solution. The equations of motion for the wheelsets are derived in detail including the creep forces and the wheel tread profiles.

This paper provides a brief review of the application of Central-Composite Designs (CCD) in human performance research. Particular mention is made of the mixed-factor CCD, which allows simultaneous consideration of within-subject and between-subjects factors. The current paper details the construction of two alternative versions of such a design. After the two versions have been compared on theoretical grounds, an empirical investigation is proposed to determine the relative predictive accuracy and validity of prediction equations derived from data collected in accordance with each design version.

Bracken and McGill have discussed the theory, computations, and an example of mathematical programming models with optimization problems in the constraints [Opns. Res. 21, 37–44 (1973)], and have presented a computer program for solving such models with nonlinear programs in the constraints [Opns. Res. 22, 1097–1101 (1974)]. Bracken, Falk, and McGill have given a procedure for transforming mathematical programs with two-sided optimization problems in the constraints into mathematical programs with nonlinear programs in the constraints [Opns. Res. 22, 1102–1104 (1974)], thus enabling their solution by the computer program. This paper formulates models of defense problems that are convex programs having the mathematical properties treated in the previous papers. The models include several strategic-force-planning models and two general-purpose-force planning models.

This paper presents a physiological math model which predicts the liminal contrast required to detect a target as a function of target size, background brightness and exposure time. The model is based on the premise that target detection occurs when the difference in response of cells across a target/background brightness contour is greater than some threshold. This premise is supported by recent physiological and psycho-physical measurements. A logical “wiring diagram” for the retina is conceived which resembles the actual anatomy of the retina. The photo-receptor cells, bipolar cells and ganglion cells in the model are given simple mathematical response functions. The target detection requirement for a minimum threshold response difference across a contour is used to develop a mathematical expression of threshold target brightness as a function of background brightness. There are two major parameters in the expression for threshold target brightness: 1) Photo-receptor sensitivity to incident luminance, 2) The threshold response difference required for target detection. The expression for threshold target brightness fits the experimental data well. The model is expanded to explain the variation of the sensitivity and threshold parameters with target size and exposure time. The resulting model is self-consistent and closely agrees with the experimental data for background luminance levels of less than one foot lambert and exposure times less than .1 sec. The results presented in this paper indicate that a coupling of psycho-physical measurements with physiological and anatomical findings leads to both a greater understanding of the physiological processes involved and a more powerful formulation of the psycho-physical results.

This paper discusses the appropriate scaling factors for the modeling of the fluid-mechanical interaction of complex structures. The possible mechanisms of flow-excited vibration are described, and scaling parameters are derived from considerations of the mechanical and fluid systems. The paper is particularly concerned with the flow-induced vibrations of banks of cylinders in single phase, high Reynolds number, turbulent, cross flow, as occur for example in light water nuclear reactors and heat exchangers. It is concluded that the modeling of the tube banks in light water nuclear reactors will involve a mismatch of Reynolds number, but that the major phenomena of fluid-solid interaction in the single-phase flow region will probably be closely replicated if the Reynolds number is high enough so that the entering flow is turbulent and if the scaling ratio is not too large.

This paper reports the development of disaggregate, behavioral models of transport mode choice along two new dimensions. The scope of disaggregate modeling is extended to an intercity context and the range of journey purposes is broadened to include social/recreational and business travel, as well as the journey to/from work. The intercity models developed should be viewed in the context of recent trends in demand modeling. Instead of arbitrarily selecting a mode as a base on the grounds that in term of speed, or cost, or comfort, the approach advocated in this paper examines the choice actually made by the traveler. The results reported lead to the conclusion that doubts as to the homogeneity of the transport mode choice decision are fully justified. This paper has investigated mode choice behavior stratified along two dimensions, journey purpose and trip length, and the analysis clearly demonstrates that the same mode choice process is not reflected in each case.

Since the early development stages of Apollo, the docking probe mechanism has confronted the structural dynamicist with a modeling and structural response challenge. Skylab is an excellent example of a structure where complexity of analytical models and subsequent dynamic analysis is fully justified. This paper describes the fouryear development of Skylab analytical programs, mathematical model complexity, and vibration and load analyses associated with the docking/latching maneuver. Examples are shown where coarse models produced loads in excess of structural capability. However, subsequent model refinement lowered the loads to the point where redesign was not required. A summary of experience gained over a four-year period is presented in the conclusions. is a 100-ton manned orbital scientific space station designed to accomplish scientific and technical experiments and medical investigations while in near-earth orbit. The primary source of power for Skylab is provided by large flexible solar array systems (SAS) which require special handling to prevent panel buckling or other structural damage under their own weight. The transient loading conditions, occurring during the atmospheric boost phase of flight, design the primary structural elements of Skylab with the exception of the docking interfaces, solar arrays, and deployment structure. The orbital vacuum environment causes no external forces on the vehicle. The orbital events considered in the design of Skylab are docking, latching, orbit maneuvers, and the several deployment sequences. Since docking and latching are the significant load-producing events, this paper will be limited to the discussion of these two events. In the early design stages of Skylab, an acceleration value of 0.2 g was used to simulate the loads due to the orbital docking and latching event. This was adequate for preliminary design, but changed considerably as the design progressed. Docking and latching forcing functions were derived and used to calculate the response loads at selected coordinates. This process became more complicated as the models grew in sophisticatio n. This paper describes the evolution of the Skylab models and the vibration analyses performed in support of the docking and latching event. In addition, the paper lists criteria derived from Skylab experience and considered applicable for future orbital space station development.

This paper consists of six sections of original contributions to the theory of cumulative fatigue damage of airframe structures and a section which consists of extensive bibliography of important papers in cumulative fatigue damage. Classical models for predicting the average number of cycles-to-failure, using basically one form or another of the so-called Miner’s rule, are known to be inadequate, in the sense that they either underestimate or overestimate, in some cases, the average number of cycles-to-failure. This is because they do not take into consideration, or even do not know how to take into consideration, the various co-occurring or concomitant factors, other than the various types of loads themselves, and incorporate them into the formula for predicting the expected number of cycles-to-failure. We believe that, in this paper we have suggested methods to circumvent the drawbacks from which the classical Miner’s rule suffers, and we have also offered models, in the more difficult and realistic situation, in which the additional increments in the propagation of the dominant crack form a dependent sequence of chance variables. It is the fond hope of the authors that they and other workers in the field will have the opportunity of applying the realistic approach to predicting the average number of cycles of a large airframe, based on actual flight test data, and to compare its performance with that of methods advanced in a follow-up effort.

This paper is concerned with the mathematical programming problems which arise in formulating a model to optimize the allocation of investment funds between different proposed improvement schemes in a road network. An account is first given of a number of different formulations which have been suggested and it is shown that the only acceptable models require the use of programming techniques which axe not sufficiently powerful to be able to optimize investment when more than a relatively small number of projects are considered. This has led to the development of heuristic algorithms. The remainder of the paper seeks to make the case for more careful consideration of the ways in which heuristic algorithms are employed. Although it is framed in terms of road networks, the argument is one generally applicable to many areas of engineering optimization where heuristics are used.

Many urban service systems have the need to estimate travel times between arbitrary points on the highway network. Some of these systems, such as Dial-a-Ride, require these estimates to be performed in real time. This paper describes and evaluates various computerized methods for performing these estimates. This paper begins by considering the characteristics on estimating techniques. Some experimental data are presented. Then various models for travel time estimation are described and evaluated using the City of Cambridge, Mass., as a test case. A framework for travel time prediction is established along with a characterization of the various errors inherent in the techniques. Tradeoffs between accuracy of the estimate, speed of the estimating procedure, and computer storage requirements are presented.

Abstract During the past decade, efforts in reservoir modeling have focused on the three areas of capability efficiency, and reliability. Capability means the ability to handle larger and more complex problems where complexity includes physical problems where complexity includes physical phenomena, such as gas percolation and variable phenomena, such as gas percolation and variable PVT properties, and severe heterogeneity due to PVT properties, and severe heterogeneity due to property variation or geometry, or both. Efficiency property variation or geometry, or both. Efficiency is increased by improving model formulations anti solution techniques to increase tolerable time-step size and reduce computer time per time step. Reliability refers to ease of use and minimum burden in selecting or experimenting with time-step size, solution technique options, iteration parameters, and closure tolerances. parameters, and closure tolerances. The single facet of a reservoir simulator that has the greatest combined influence in all three categories is the technique used to solve the large systems of equations arising from the numerical approximation of the nonlinear fluid flow equations. Available techniques include both direct solution and iterative methods such as ADIP, SOR, and SIP. Iterative methods are currently used almost to the exclusion of direct solution because of the significantly higher computer storage and time requirements of the latter. This paper describes some new ordering schemes for Gaussian elimination that reduce computing time and storage requirements by factors as large as 6 and 3, respectively, relative to more standard orderings. Computational work estimates are given for these methods, for the standard Gaussian ordering, and for several iterative methods. These work estimates are checked by comparisons of actual run times using different solution techniques. Numerical examples are given to illustrate the increased efficiency and reliability that can be achieved in many cases through use of the new direct solution methods. Introduction It is well known that the way we number or order the unknowns of a sparse system of linear algebraic equations can drastically affect the amount of computation and storage for a direct solution. However, until recently the best ordering scheme that appeared in the literature numbered the points of a three-dimensional grid first along the shortest direction - i.e., the dimension with the fewest number of grid points - then in the next shortest direction, and finally in the longest direction. This ordering, which we shall call the standard ordering for Gaussian elimination is still widely used even though it is substantially slower than many other orderings. Ogbuobiri et al. present a survey of the literature related to ordering schemes that exploit matrix sparsity. These schemes are grouped into the two classes of matrix-banding schemes and optimal or pseudo-optimal schemes. The latter schemes pseudo-optimal schemes. The latter schemes purport to yield generally greater efficiency. purport to yield generally greater efficiency. In a recent paper, Georges has shown that for five-point difference approximations on square n x n two-dimensional grids, the total work for certain orderings of the grid points is less than C1n3 and the storage is less than C2n2 log n, compared with n4 and n3, respectively, for the standard ordering. Moreover, George has shown that no ordering scheme can require less work than the order of n3. For the special case of n - 21 he shows that work W is less than 10n3 and the storage S is less than 8ln2 for symmetric matrices. For nonsymmetric matrices these results become W less than 20n3 and S less than 16ln2, respectively. In this paper we describe some specific orderings in the matrix-banding class. Analyses of work and storage requirements are given for these orderings as applied to the diffusivity-type pressure equation that arises in reservoir simulation problems. These work and storage requirements are compared with those of the standard Gaussian ordering and of some iterative methods. These comparisons are performed for problems ranging from simple performed for problems ranging from simple homogeneous squares to practical reservoir problems of typical heterogeneity and irregular problems of typical heterogeneity and irregular geometry. The work requirements of the orderings presented here are also compared experimentally with those of one of the leading pseudo-optimal schemes. SPEJ P. 295

In this paper we study the flow of traffic on roads for which passing is not permitted. Such traffic conditions exist in rural areas, hilly or mountainous regions, tunnels, etc. The traffic flow process is considered as taking place in a one directional road of infinite length, with passing not permitted. This paper studies measures of traffic flow in a road where vehicles have one of two free speeds. Measures of traffic flow are obtained for general vehicle entrance processes and derived explicitly for the homogeneous Poisson case.

This paper presents sufficient conditions for a multiattribute utility function to be either multiplicative or additive. The number of requisite assumptions to imply the main result is equal to the number of attributes. Because the assumptions involve only trade-offs between two attributes at a time or lotteries over one attribute, it is reasonable to expect that decision makers can ascertain whether these assumptions are appropriate for their specific problems. Procedures are given for verifying the assumptions and assessing the resulting utility functions. The paper concludes with a sketch of a recent application of the results to a six-attribute problem relating to the development of Mexico City's airport facilities.

Analytically tractable statistical-physical models of man-made noise environments have been constructed [1]-[3]. These permit quantitative description of the various types of electromagnetic interference appearing in typical radio receivers and, in particular here, for the communication links employed in mobile transportation systems and urban environments generally. This paper presents a summary of some of the principal analytical results obtained to date [1], [4], and includes some suggested next steps for joint theoretical and experimental study of these increasingly important phenomena. First-order probability density functions (pdf's) and probability distributions (pd's) are obtained explicitly; (higher order pdf's and pd's may also be found by similar methods) [2]. These models are based on a Poisson distribution of sources in space. The approach is canonical, in that the results are, in form, independent of particular emitted waveforms, propagation conditions, source distributions, beam patterns, and specific system parameters, as long as the interference is narrow-band following (at least) the aperture and/or the RF stages of a typical receiver. Considered here only are the cases of communication interference, where source and receiver bandwidths are comparable. The paper concludes with a short discussion of some features of suggested future interaction between theory and experiment.

This paper is intended to provide a general background to the two following papers, “A Simultaneous Destination and Mode Choice Model for Shopping Trips” and “Some Estimation Results of a Simultaneous Model of Auto Ownership and Mode Choice to Work.” Some of the deficiences of the conventional urban transport modelling system are reviewed and a case is made for the use of simultaneous models estimated with disaggregate data.

Almost all the analytical models in the literature for traffic flow on two-lane roads are concerned with steady state situations, assuming that passing is possible at any point along the road. In this paper we study the steady state situation, assuming that passing is possible only at passing points, which are located at equal distances along the road. The mathematical model, presented in this paper, shows how expected velocities, expected driver loss time and platoon length are affected by such characteristics as speeds of fast and slow vehicles, rates at which fast and show vehicles enter the road, spacing of passing opportunities, and probability of a vehicle being able to pass given a passing opportunity.

This paper summarises the program of work carried out at TRRL up to 1971 on traffic restraint treated as a policy for transport planning. The special techniques required were developed and are described here. The theoretical framework within which local traffic effects can be treated at a strategic level is developed using marginal cost road pricing as an example, and the necessarily stringent pricing establishing the convergence, stability, and repeatability of the results is described for a practical algorithm which can readily be used in other transport planning program systems. The application of these techniques to analyse the comparative effects of different traffic restraint policies, and the variations on the techniques required to handle several groups of travellers who react differently to restraint measures, are the subject of companion papers to appear later in this journal. (a). See also IRRD abstract no. 212050 for part 2. /TRRL/

The paper reviews problems encountered when using overhung boring bars fitted with single point tooling for profile boring operations. Above certain length to diameter ratios, which are affected by the diameter of the bars and the base fixing conditions, the stiffness is inadequate permitting the build-up of unstable vibrations and making it difficult to maintain accuracy of the form of the bore. Approximate design rules are given for steel boring bars, bars fitted with shanks made in materials with a high modulus of elasticity, and for bars fitted with simple dampers. The paper concludes with a brief description of possible new methods of damping boring bars and of the problems of devising simple but acceptable mathematical models representing the dynamic system.

Behavior of elastic-plastic structures under repetitive and fluctuating loads is considered in this paper. Plastic deformation either stabilizes after a finite number of cycles or continues during the cycling. In the first case the structure is said to have shaken down to the boundary at the loading program. If plastic deformation does not stabilize an elastic-plastic structure becomes unserviceable due to either alternating plasticity when yielding occurs repeatedly in the opposite senses or accumulation of plastic strains and progressive increase of permanent displacements. This paper attempts to survey the shakedown theory, including the accommodation of a structure to the prescribed loading range as well as inadaptation and unserviceability. The notion of shakedown, incremental collapse, ratchetting and alternating plastic deformation are first illustrated with examples. The fundamental theorems on shakedown and inadaptation are presented next, attention being directed to generalizations of the classical Melan and Koiter theorems. Applications of the theorems are given. Available methods for determining the shakedown range on generating self-equilibrated stress fields are discussed. Special techniques applicable to problems involving both dead and fluctuating loads are invoked. Recent studies accounting for inertia forces, geometrical changes, material hardening, variation of material properties with temperature, displacement assessment, etc. are referred to.

The first part of this paper is a review of current knowledge of the probability distribution of stress peaks for a general type of broadband response of a linear structure under stationary and Gaussian random excitations. It is shown that the random stress in a linear structure always can be decomposed into two components and that the peaks in one of the components are Rayleigh-distributed. The second part of this paper presents a solution to the distribution of stress peaks in a nonlinear structure, with the restrictions that the response of the structure is dominated by a single mode and that the nonlinear deflection (not necessarily the stress) is of a narrow band. The theoretical result for the nonlinear stress peaks shows marked differences from that of the linear stress peaks. In particular, there is a finite probability measure for the stress peaks to assume negative values in a nonlinear structure, even when the deflection peaks are always positive. For the positive stress peaks, which are smaller than about two times the standard deviation of the stresses, the nonlinear theory predicts a smaller probability measure than that of a Rayleigh distribution.

Here is an approach for solving reservoir flow problems where behavior is dominated by a rate-limiting step. Simple models are developed for gravity drainage where vertical flow occurs, for water underrunning of viscous oils, for gravity segregation of water banks in gas caps, and for control of coning by injection of oil. Introduction Forecasting the behavior of a reservoir is one of the more important but complicated tasks of engineers in the oil industry. Knowledge of reserves remaining in a reservoir is vital to planning optimum depletion of a field. Unfortunately, the engineer assigned the task of predicting reserves often faces a difficult choice. For the most accurate answer, he can use a computer program that takes into account all of the pertinent factors, but this approach is usually pertinent factors, but this approach is usually expensive and time consuming, and requires a detailed knowledge of the reservoir. On the other hand, he can use conventional one-dimensional displacement calculations that are easily applied but that in some cases do not adequately describe the reservoir flow system. The purpose of this paper is to describe a middle-ground approach that in special situations has many of the advantages of the above methods with out their more serious drawbacks. This approach uses mathematical models that describe the principal flow mechanisms and can be quickly applied by hand calculations. The technology of predicting reservoir behavior has grown steadily since the pioneering work of Muskat and of Buckleys and Leverett. Muskat's tank-type or zero-dimensional method of predicting behavior in dissolved gas drive reservoirs has been invaluable to the industry. Another milestone was reached with the Buckley-Leverett method of predicting linear displacement of oil by water or gas when flow was principally along the bedding plane. The classic work of Hurst, Muskat, and van Everdingen and Hursts laid a firm foundation for problems involving unsteady-state flow of fluids. Later, progress was made by Welge in solving one-dimensional progress was made by Welge in solving one-dimensional displacement equations more easily. The advent of digital computers led to the development of methods of solving problems of greater and greater complexity. Indicating the progress being made with computers, Douglas et al. in one paper, included the effect of capillary pressure in one-dimensional flow, and in another paper dealt with the flow of two phases in two dimensions. The utility of computers in predicting reservoir behavior has continued to grow as programs have become more user-oriented and as computers have become faster and more economical to employ. But even today, the time, effort, and money required to use computers to solve reservoir problems cannot always be justified. Thus, other tools are needed. An excellent example of another approach is given in a paper by Joslin. In analyzing a gas injection project in a large Venezuelan reservoir, Joslin realized project in a large Venezuelan reservoir, Joslin realized that gas had overridden the entire oil sand because production was above the critical rate. However, the production was above the critical rate. However, the presence of pancake-like shale members penetrated presence of pancake-like shale members penetrated by the wells prevented coning of gas into perforations located below the shale near the base of the sand. Oil recovery was predicted by assuming that gas displaced oil vertically downward throughout the producing area. Another paper demonstrating the practical use of simple mathematical models is that of Matthews and Lefkovits for predicting producing rates for wells in depletion-type reservoirs. JPT P. 1145

In this section the results of a recent study reported in Ref. 19 are briefly reviewed. The objective of the paper is a set of stability criteria for a spacecraft characterized as a rigid body having attached a flexible appendage idealized as a collection of elastically interconnected particles. The appendage is permitted in the paper to be configured in a general fashion, but the special case of the planar appendage shown in Fig. 16 is adopted in order to obtain explicit closed form stability criteria without the aid of computer simulation. The well Down stability procedures offered by Liapunov’s direct (second) method are invoked in the paper. Among the many theorems on stability and instability of the class developed by Liapunov two have direct relevance to the development. Described in detail by Pringle, [20] the theorems of interest are par aphrased below.

The past decade has given rise to a considerable amount of research in the fields of inventory control and production planning. One of the most useful mathematical models originated with the paper of K. D. Arrow, T. E. Harris, and J. Marschak, “Optimal Inventory Policy,” Econometrica 19, 250–272 (1951). This model was further improved in Richard Bellman's book, Dynamic Programming, pp. 152–182, Princeton University Press, Princeton, New Jersey, 1957. In the present paper, another formulation of the same problem is exhibited. The attentive reader will observe many similarities to the dynamic programming formulation.

The numerical results presented in this paper, in Tables I through VI, apply only to the PSEG system. First presented is a summary of the forced-outage data collected; see Table I. Table II compares the actual distribution of up-period duration with an exponential distribution. Then, Tables III and IV give estimates of average up-period duration and the probabilities of an outage being a halfoutage for the four classes of units. Table V shows the component failure rates for mature units. Table VI gives the details of the durations and postponements of down periods. Comments on the numerical results are given in the section "Results" at the beginning of the paper.

The objectives of this paper are: 1. (1) A brief discussion of where and how computer techniques which are based on mathematical models can be used in geology. 2. (2) A summary of recent developments in computer techniques for geological problems. 3. (3) A more detailed discussion of some general, statistical techniques with applications to practical examples. It is shown that methods of trend analysis for a single variable should be modified when ( a ) the residuals of the variable are autocorrelated, and ( b ) the computations are carried out using transformed data for the variable. A technique of two-dimensional power spectrum analysis is employed for the quantitative description of periodicities in the arrangement of the grains for gabbroic rocks from the Muskox Layered Intrusion, Canada. Finally, the trend of a multivariate system (chemical data for the Yellowknife Volcanic Belt, Canada) is analyzed by using the multivariate Markov scheme method. Brief explanations for a number of non-geological terms that are used in the paper are given in an appendix.

Abstract Water drive in a homogeneous porous medium with a uniform permeability distribution has been extensively studied in the past, both theoretically and experimentally. In this paper equations are derived for a two-dimensional water drive in a porous system with either a continuously or a porous system with either a continuously or a discontinuously varying permeability distribution perpendicular to the layer. The influence of perpendicular to the layer. The influence of capillary forces bas not been taken into account. A necessary condition for the validity of the equations is that the water should underrun the oil. It is shown that the permeability distribution has much influence on the oil recovery. A large difference in recovery was apparent from a comparison of three systems, in which the oil production decreased as follows: (1) permeability increasing in an upward direction perpendicular to the layer, (2) homogeneous, uniform permeability throughout, (3) permeability increasing in a downward direction perpendicular to the layer. Introduction The recovery from a homogeneous, two-dimensional system in response to a normal water drive can be calculated with the help of theories developed by Beckers. He describes segregated flow of oil and water in which the water underruns the oil due to gravity and viscous forces. In his approach it is impossible to include capillarity, which means that the system does not describe a transition zone. However, we have a great deal of scaled experimental evidence which shows that moderate initial transition zones disappear during the displacement process where gravity and/or viscous tonguing effects occur. The experiments then show a sharp interface, and a transition zone is only found in the top of the water tongue. Neglect of this small zone affects breakthrough-time calculations, but on the other hand, gives the advantage of being able to treat the problem with the end-point permeabilities only. Although not immediately recognizable, there is a form of crossflow involved in this concept. It can be found from the material balance if the layer is divided into two sections parallel to the bedding plane. The theory describes experiments with a maximum initial transition zone of about one-third of the layer thickness. There are no restrictions on the mobility ratio. The production curve predicts too early a breakthrough, but shortly thereafter very satisfactory agreement is found. For many practical cases, these theories can be reduced to the simplified formulation given by Dietz. Experimental verification shows that his theory describes horizontal scaled-model experiments satisfactorily for mobility ratios larger than 6. Although gravity forces are completely neglected in the theory but not in the experiments, only the viscous forces are now responsible for tongue forming. Neglect of gravity delays breakthrough in comparison with predictions from the Beckers theory, but generally gives a somewhat better though not correct prediction of the moment of breakthrough. These findings have encouraged us to apply the same principles to inhomogeneous systems. To delay the theoretical breakthrough, gravity forces parallel to the bedding plane have been included, parallel to the bedding plane have been included, which results in a in a better fit with the over-all production curve for tilted layers. production curve for tilted layers. The equations derived in this paper can be applied to systems with either a continuous or a discontinuous permeability distribution perpendicular to the layer They are therefore applicable to a great number of inhomogeneous systems. Areal extension can be effected with sweep efficiency factors or by applying stream-tube models.

Abstract This paper concerns the use of network principles to study displacement phenomena in porous media. The information presented is for equal-viscosity, equal-density miscible displacements. The paper explains the reasons for using an interconnected network of capillary tubes to model the interconnected network of pores in a reservoir rock. A method is presented for defining the heterogeneity of a presented for defining the heterogeneity of a network of tubes based on tube-size and tube-location distribution functions. A technique is described for constructing a network whose heterogeneity models the heterogeneity of pores in a reservoir rock. The use of networks to provide information which can be used in the solution of reservoir engineering problems is illustrated with example calculations of the effect of heterogeneity on fingering, breakthrough, and selective plugging in linear systems, and the effect of heterogeneity on areal sweep efficiency in a five-spot pattern. Introduction Oil in a reservoir is contained in an interconnected three-dimensional network of pores. Direct evidence of the nature of this network of pores comes from examination of petrographic thin sections and three dimensional Scanning Electron Microscope (SEM) pictures of the pores. The SEM pictures show that the pores in a reservoir rock are channels through which flow can occur. These channels have highly irregular configurations so irregular that it is not practical at this time to calculate flow behavior through individual channels or through the interconnected network of the channels. It is practical, however, to use a computer to calculate flow behavior in an interconnected network of capillary tubes and several investigators have studied the problem of using a network of tubes to model a network of pores. pores. Fatt pioneered the idea of using a network of cubes model for reservoir engineering studies. He demonstrated that capillary pressures, relative permeabilities, and electrical resistivities permeabilities, and electrical resistivities calculated for a network model have the same characteristics as those measured for real pores in reservoir rocks. From this, Fatt concluded that the network of tubes is a valid model of real porous media. Rose reinforced Fatt's conclusion and showed that computers can be used to study the displacement characteristics of networks and to obtain results "…which can be supposed to have a direct bearing on the mechanics of petroleum recovery…" This paper takes two steps beyond the work of Fatt and Rose. First, it describes a technique for constructing a network whose heterogeneity models the heterogeneity of natural pores. This is done by matching calculated equal-viscosity miscible displacement behavior in the network with measured behavior in a laboratory core. Second, it illustrates the use of the network model for calculating the effects of heterogeneity on fingering, breakthrough, and plugging in linear systems and areal sweep efficiency in a five-spot pattern. The networks used in the studies in this paper consist of several hundred interconnected capillary tubes of different sizes. Four different types of connections or configurations were investigated and are shown below. These configurations are discussed in detail later in the paper. SPEJ P. 99

A method is developed to generate by one nth order sensitivity model all the sensitivity functions [(@?"x"i(@?"q"j)]|"q"0, i = 1, ..., fn, j = 1 ..., r for a single input, linear, time-invariant, controllable nth order system which depends on r different parameters. This represents an improvement over known methods for generating the sensitivity functions, which generally require a composite dynamic system of order n(r+1). In the development, two useful properties of the sensitivity functions of the states of the companion form of a system, where the system A matrix is in companion form, are used. It is shown that all the sensitivity functions of the states of the companion form system can be generated by linear combinations of the signals on one sensitivity model of the system and the system states. The Leverrier algorithm has been extended so that the method of the paper is readily implemented on a digital or hybrid computer. Finally, an example is given, demonstrating how the technique of the paper can result in considerable time savings in the computer simulation and sensitivity analysis of systems.