Showing papers on "Discretization published in 1976"

Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics

Abstract: The field equations of three-dimensional elastostatics are transformed to boundary integral equations. The elastic body is divided into subregions, and the surface and interfaces are represented by quadrilateral and triangular elements with quadratic variation of geometry and linear, quadratic or cubic variation of displacement and traction with respect to intrinsic co-ordinates. The integral equation is discretized for each subregion, and a system of banded form obtained. For the integration of kernel-shape function products, Gaussian quadrature formulae are chosen according to upper bounds for error in terms of derivatives of the integrands. Use of the integral formulation is illustrated by the analysis of a prestressed concrete nuclear reactor pressure vessel.

Analysis of Microstrip-Like Transmission Lines by Nonuniform Discretization of Integral Equations

Abstract: The nonuniform discretization of the integral equation on the tangential electromagnetic (EM) field on the boundary surface is proposed as a numerically efficient method to analyze the microstrip-like transmission lines. The calculated results of the propagation constant of the microstrip line based on this method are compared with other published analytical results. Various types of planar striplines are treated by the same formulas. The dominant and higher order modes of shielded microstrip line are discussed and compared with the longitudinal-section electric (LSE) and linear synchronous motor (LSM) modes of a two-medium waveguide.

One-dimensional mathematical model for large-strain consolidation

Abstract: A mathematical model is developed to represent the one-dimensional large-strain consolidation of a fully saturated clay. The fluid limit is postulated to be that water content associated with a ‘stress-free’ condition of the soil, and it is taken as the reference state from which strains are measured. Experimental results from a series of permeability tests suggest that the relationship between the logarithm of the coefficient of permeability and the void ratio is not a straight line for the entire range of void ratio considered. In addition, the variation of the constrained modulus as consolidation progresses is taken into account. The resulting boundary value problem involves a non-linear partial differential equation with void ratio as the dependent variable, and the numerical solution is accomplished by a step-by-step procedure combined with a weighted residual technique which leads to a finite element discretization in the spatial variable and a finite difference discretization in the time variable. ...

Nonlinear feedback in simple locomotion systems

Abstract: The purpose of this short paper is to analyze postural stability and periodic motion of a simple model of a locomotion system. Nonlinear feedback is used to linearize the system. The effects of discretization, quantization and noise are studied via simulation of a compound inverted pendulum with realistic actuator models.

A finite difference method of solving anisotropic scattering problems.

Abstract: A new method of solving radiative transfer problems is described including a comparison of its speed with that of the doubling method, and a discussion of its accuracy and suitability for computations involving variable optical properties. The method uses a discretization in angle to produce a coupled set of first-order differential equations which are integrated between discrete depth points to produce a set of recursion relations for symmetric and anti-symmetric angular sums of the radiation field at alternate depth points. The formulation given here includes depth-dependent anisotropic scattering, absorption, and internal sources, and allows arbitrary combinations of specular and non-Lambertian diffuse reflection at either or both boundaries. Numerical tests of the method show that it can return accurate emergent intensities even for large optical depths. The method is also shown to conserve flux to machine accuracy in conservative atmospheres

Monte Carlo simulation of microbial population growth

Abstract: Mathematical models of cell populations which recognize them to be segregated into individual cells differing from one another in state and behavior lead to complicated integro-differential equations. Analysis of fluctuations in small populations leads to the even more complex situation of coupled integro-differential equations. An early technique by Kendall for simulating the behavior of populations which obey the birth-and-death model has been extended in this work to include the factor of variation in individual cellular state and behavior. The resulting algorithm is probabilistically exact in the sense that the random variables to be generated have exactly derived distribution functions, and involves no arbitrary discretization of the time interval, characteristic of simulation techniques in general. The simulated systems include populations distributed according to their age in a birth-and-death process and a batch culture of cells distributed according to their mass which grow and reproduce by binary fission.

The incremental perturbation method for large displacement analysis of elastic‐plastic structures

Abstract: A procedure of applying the perturbation method is presented for the incremental numerical analysis of materially and combined non-linear problems of discrete and discretized structral systems. Small but finite strain and stress increments are strictly distinguished from the strain rates and stress rates, respectively. It is shown that, by applying the perturbation procedure not only to the non-linear strain-displacement relations and equilibrium equations, but also to the constitutive equations in terms of rate quantities, all the governing equations can be satisfied to any desired accuracy at every instantaneous configuration in between the starting and terminal points of an incremental step. The proposed method provides also means of finding, to a desired accuracy, every point on an equilibrium path of a discrete system at which a new element will start yielding or unloading and possible critical points on the path. The significance of the proposed method is expected to be appreciated particularly in numerical investigations of critical behaviours and post-buckling behaviours.

Resistivity modeling for arbitrarily shaped two dimensional structures. Part I. Theoretical formulation

Abstract: A numerical technique was developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations, making point as well as area discretization, of the subsurface. Potential distribution at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct, explicit, matrix inversion techniques. An empirical mixed boundary condition is used at the ''infinitely distant'' edges of the lower half-space. Accurate solutions using area discretization method were obtained with significantly less attendant computational costs than with the relaxation, finite-element or network solution techniques, for models of comparable dimensions.

Evaluation of mean square error involved in approximating the areal average of a rainfall event by a discrete summation

Abstract: Two-dimensional areal processes are commonly evaluated in hydrology through a discretization in space over the region in which the process is being studied. Such a discretization involves an error in going from the continuous process to the discrete one. This error is studied theoretically, and graphs are presented for its evaluation as function of the size of the area, the functional form of the correlation equation in space, and the level of discretization or size of the sample. Correlation structures of the Bessel type and of the single and double exponential kind are considered, and their different implications are discussed.

General Software for Partial Differential Equations

Abstract: Several general purpose digital computer programs for numerically solving partial differential equations (PDEs) have been developed. This chapter discusses the possible advantages of PDECOL, which is a new general purpose digital computer program for numerically solving partial differential equations. The most significant feature of PDECOL is the spatial discretization technique that is implemented. The discretization technique can best be described as a finite element collocation method that uses piecewise polynomials for the trial function space. The chapter highlights the use of PDECOL to solve four different PDE problems illustrating few advantages and disadvantages of the use of higher order methods. PDECOL is an extremely versatile and unique software package. It can reliably solve a very broad class of nontrivial and nonlinear systems of partial differential equations. Its higher order methods can produce extremely accurate solutions quite efficiently when compared to lower order methods. The package is quite portable and very easy to use.

Discretization of stress fields in the finite element method

Abstract: Several aspects of the discretization of stress fields, as opposed to displacement fields, are reviewed. The most classical satisfies rigorous equilibrium, both translational and rotational, in the interior domain of each element and reciprocity of surface tractions at interelement boundaries (strong diffusivity). The difficulties associated with kinematical deformation modes are analysed and resolved by different procedures: the composite element technique; quasi-diffusivity controlled by the dual patch test; discretization of the displacement connectors, or hybridation; discretization of rotational equilibrium. This last and recent approach is discussed in some detail. It involves direct or indirect use of first-order stress functions, whose Co continuity is sufficient for strong diffusivity. One of its advantages is the possibility of curving the boundaries by a geometric isoparametric coordinate transformation. Some numerical convergence tests are presented.

Dynamic analyses of suspension bridge structures

Abstract: A method of dynamic analysis for vertical, torsional and lateral free vibrations of suspension bridges has been developed that is based on linearized theory and the finite-element approach. The method involves two distinct steps: (1) specification of the potential and kinetic energies of the vibrating members of the continuous structure, leading to derivation of the equations of motion by Hamilton's Principle, (2) use of the finite-element technique to: (a) discretize the structure into equivalent systems of finite elements, (b) select the displacement model most closely approximating the real case, (c) derive element and assemblage stiffness and inertia properties, and finally (d) form the matrix equations of motion and the resulting eigenvalue problems. The stiffness and inertia properties are evaluated by expressing the potential and kinetic energies of the element (or the assemblage) in terms of nodal displacements. Detailed numerical examples are presented to illustrate the applicability and effectiveness of the analysis and to investigate the dynamic characteristics of suspension bridges with widely different properties. This method eliminates the need to solve transcendental frequency equations, simplifies the determination of the energy stored in different members of the bridge, and represents a simple, fast and accurate tool for calculating the natural frequencies and modes of vibration by means of a digital computer. The method is illustrated by calculating the modes and frequencies of a bridge and comparing them with the measured frequencies.

On a numerical solution of the supersonic panel flutter eigenproblem

Abstract: An automated digital computer procedure is presented in this paper which enables efficient solution of the eigenvalue problem associated with the supersonic panel flutter phenomena. The step-by-step incremental solution procedure is based on an inverse iteration technique which effectively utilizes solution results from the previous step in determining such results during the current solution step. Also, the computations are limited to the determination of a few specific roots only, which are expected to contain the flutter mode, and this is achieved at each step without having to compute any other root. The structural discretization achieved by the finite-element method yields highly banded stiffness, mass, and aerodynamic matrices; the aerodynamic matrix evaluated by the linearized piston theory is real but unsymmetric in nature. The solution algorithm presented in this paper fully exploits the banded form of the associated matrices, and the resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be most efficient and economical when compared to existing procedures of such analysis. Numerical results are presented for a two-dimensional panel flutter problem.

Flexibility and economics of implementation of the finite element and difference techniques in nonlinear magnetic fields of power devices

Abstract: This paper presents a comparison between the finite element and difference methods applied to two dimensional nonlinear magnetic field problems. The comparison is based on the determination of the magnetic field distribution in a shell-type single phase transformer with substantial magnetic core saturation. The comparison between the two discretization schemes is based on an equal number of finite element and finite difference domains. The finite element mesh is superimposed on the finite difference grid, to assure an equitable basis of comparison. Aspects of flexibility of implementation and economics (computer execution time and storage requirements) for the two techniques are compared. In addition, the degree of agreement between numerical results of the two methods and corresponding experimental test data is given.

The solution of ill‐conditioned linear systems arising from Fredholm equations of the first kind by steepest descents and conjugate gradients

Abstract: It is shown that the methods of steepest descents and conjugate gradients give smooth solutions of the ill-conditioned systems of linear equations obtained by the discretization of Fredholm equations of the first kind without the use of any regularization techniques. A modification of the usual method of steepest descents is described which is equivalent to applying two cycles of the usual method with very little additional work. A number of examples are worked including one with a degenerate kernel in which the algebraic system is singular.

Positive quadratures for volterra equations

Abstract: The present paper deals with discretizations to linear Volterra equations which preserve the possible positivity of the Volterra operator. It is shown that the method must be implicit and e.g. that the repeated trapezoidal rule has this property. It is then shown how this property can be used in studying the asymptotic behaviour of the solutionsx n of the discretized equations, asn→∞ with a fixed steph.

Resistivity modelling for arbitrarily shaped two dimensional structures. part ii. user's guide to the fortran algorithm resis2d

Abstract: The development of a numerical solution technique is described to obtain the potential distribution in three-dimensional space due to a point source of charge injection in or on the surface of a half space containing any arbitrary two-dimensional conductivity distribution. Finite difference approximations are made to discretize the governing Poisson's equation with appropriate boundary conditions. The discretization of Poisson's equation by elemental area brought about a numerical formulation for a more effective matrix technique to be utilized to solve for the potential distribution at each node of a discretized half-space. A FORTRAN algorithm named RESIS2D was written to implement the generalized solution method. A brief description of the FORTRAN program in terms of its construction is given. The formal input and output parameters for the relevant subroutines are discussed. The program is designed to be implemented on a CDC 7600 machine. The language of the algorithm is FORTRAN IV; certain programming norms for the CDC 7600 machine and the RUN76 compiler are routinely used. Some variables are stored in the LCM (Large Core Memory) of this machine, and their calling sequence and usage apply to the CDC7600 alone. The resulting solution of the potential distribution can be obtained for amore » current source or sink located on the surface or at any arbitrary surface location. Any arbitrary configuration of transmitter or receiver electrode arrays, therefore, could be simulated to obtain the resistivity response over arbitrarily shaped two-dimensional geologic bodies. For brevity, in the source deck provided in this report, only two electrode arrays commonly used in geothermal reservoir delineation are illustrated. (JGB)« less

On plastic dynamics of discrete structural models

Abstract: This paper discusses the dynamic loading of rigid-perfectly plastic structures by adopting a structural model discretized with constant stress finite elements. This assumption together with the hypothesis of small displacements and of a piecewise linear yield surface leads to the formulation of a problem in linear inequalities, which is, implicitly, time discretized. It is shown how the non linearity of the associated plastic flow laws leads to an equivalent extremal formulation. A numerical algorithm for solving the problem is directly derived from the mechanical statements. Numerical examples illustrate this approach. In the appendix an approximate technique is developed, which takes into account the influence of the strain-hardening and the strain rate sensitivity of material.

A defect correction method for functional equations

Abstract: We want to solve numerically the functional equation F(y)=o. For that purpose we use a discretization method with the property that the global discretization error admits an asymptotic expansion. We combine this with Newton's method and find numerical methods which are related to Pereyra's technique [8]. The first step of these methods have been given for the special case of initial value problems for ordinary differential equations by Zadunaisky [14,15] and Stetter [12].

A submerged singularity method for calculating potential flow velocities at arbitrary near-field points

Abstract: A discrete singularity method has been developed for calculating the potential flow around two-dimensional airfoils. The objective was to calculate velocities at any arbitrary point in the flow field, including points that approach the airfoil surface. That objective was achieved and is demonstrated here on a Joukowski airfoil. The method used combined vortices and sources ''submerged'' a small distance below the airfoil surface and incorporated a near-field subvortex technique developed earlier. When a velocity calculation point approached the airfoil surface, the number of discrete singularities effectively increased (but only locally) to keep the point just outside the error region of the submerged singularity discretization. The method could be extended to three dimensions, and should improve nonlinear methods, which calculate interference effects between multiple wings, and which include the effects of force-free trailing vortex sheets. The capability demonstrated here would extend the scope of such calculations to allow the close approach of wings and vortex sheets (or vortices).

On the Origins and Numerical Solution of Some Sparse Nonlinear Systems

Abstract: Publisher Summary This chapter presents some source problems that give rise to sparse nonlinear systems of algebraic equations. It discusses some numerical methods for their solution. The chapter reviews a class of nonlinear diffusion equations and an associated discretization that takes the form of a sparse nonlinear system. This system has the agreeable property of admitting a unique solution that may be determined by a globally convergent iterative method. The chapter explains a problem involving heated networks. Its solution is seen to be equivalent to finding a Brouwer fixed point. In this connection, a new method is described in the chapter for the computation of such fixed points. Many linear and nonlinear algebraic systems arise as discrete analogs of partial differential equations.

Elastoplastic Torsion of Anisotropic Bars

Abstract: An elastoplastic analysis for the torsion of anisotropic bars has been presented on the basis of a hybrid model using the finite element method. The element formulation is based on modified Hellinger-Reissner variational principle. The analysis is applicable to nonhomogeneous and multiply connected sections and it uses isoparametric “linear” quadrilateral element for the discretization of the sections. The initial stress iterative technique has been used for obtaining the elastoplastic solutions. The results of a number of numerical examples have been presented to demonstrate the accuracy and efficiency of the proposed model.

Covariance-invariant signal processing

Abstract: When discretizing continuous-time systems or signals, one is often interested in preserving a property termed covariance-invariance. In this paper a technique is outlined for synthesizing discrete-time systems and signals which are covariance-invariant with corresponding continuous-time systems and signals. Applications of the technique to process simulation, minimum mean-squared error estimation, and digital filter synthesis are outlined, with example designs presented for covariance-invariant Butterworth and Chebychev digital filters. Based on the frequency response of these designs it is argued that the method of covariance-invariance is superior to the methods of impulse-invariance and bilinear-z as a response matching design technique for the synthesis of digital filters. This superiority is especially apparent at sampling rates that are marginal with respect to filter critical frequencies.