Showing papers on "Discretization published in 1975"
TL;DR: In this paper, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived starting from continuum mechanics principles, and a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure.
Abstract: SUMMARY Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.
789 citations
TL;DR: The software interface provides centered differencing in the spatial variable for time-dependent nonlinear PDEs, giving a semidiscrete system of nonlinear ordinary differential equations (ODEs), which are then solved using one of the recently developed robust ODE integrators.
Abstract: The numerical solution of physically realistic nonlinear partial differential equations (PDEs) is a complicated and highly problem-dependent process which usually requires the scientist to undertake the difficult and time-consuming task of developing his own computer program to solve his problem. This paper presents a software interface which can eliminate much of the expensive and time-consuming effort involved in the solution of nonlinear PDEs. The software interface provides centered differencing in the spatial variable for time-dependent nonlinear PDEs, giving a semidiscrete system of nonlinear ordinary differential equations (ODEs), which are then solved using one of the recently developed robust ODE integrators. Besides being portable, efficient, and easy to use, the software interface along with an ODE integrator will discretize the problem, select the time step and order, solve the nonlinear equations (checking for convergence, etc.), and maintain a user-specified time integration accuracy, all automatically and reliably. Physically realistic examples are given to illustrate the use and capability of the software.
218 citations
TL;DR: This short paper considers a discretization procedure often employed in practice and shows that the solution of the discretized algorithm converges to the Solution of the continuous algorithm, as theDiscretization grids become finer and finer.
Abstract: The computational solution of discrete-time stochastic optimal control problems by dynamic programming requires, in most cases, discretization of the state and control spaces whenever these spaces are infinite. In this short paper we consider a discretization procedure often employed in practice. Under certain compactness and Lipschitz continuity assumptions we show that the solution of the discretized algorithm converges to the solution of the continuous algorithm, as the discretization grids become finer and finer. Furthermore, any control law obtained from the discretized algorithm results in a value of the cost functional which converges to the optimal value of the problem.
191 citations
TL;DR: Fortran IV subroutines for the in-core solution of linear algebraic systems with a sparse, symmetric, skyline-stored coefficient matrix are presented and the application to ‘superelement’ condensation of large-scale systems is discussed.
88 citations
TL;DR: A comprehensive linear head injury model development program using the finite element technique for constructing the models is presented and a special modification of the isoparametric element is shown to be particularly suited to simulation of the dynamic response of brain matter.
Abstract: A comprehensive linear head injury model development program using the finite element technique for constructing the models is presented. A two-dimensional axisymmetrical model and a two-dimensional plane strain model are constructed and analyzed in studies that were preliminary to the development of a fully three-dimensional model. Selection of the appropriate temporal integration operator and the particular isoparametric element formulation employed are examined. A special modification of the isoparametric element is shown to be particularly suited to simulation of the dynamic response of brain matter. Three-dimensional discretizations are generated with automatic mesh generation techniques applied to a conceptualized I-J-K space discretization. Resulting models possess recognizable skull geometry and the importance of this simulation is emphasized.
49 citations
TL;DR: The initial-boundary value problem for a linear parabolic equation in an infinite cylinder under the Dirichlet boundary condition is solved by applying the finite element discretization in the space dimension and A0-stable multistep discretizations in time.
Abstract: The initial-boundary value problem for a linear parabolic equation in an infinite cylinder under the Dirichlet boundary condition is solved by applying the finite element discretization in the space dimension and A0-stable multistep discretizations in time. No restriction on the ratio of the time and space increments is imposed. The methods are analyzed and bounds for the discretization error in the L2-norm are given.
44 citations
TL;DR: In this paper, a generalized eigenvalue problem for a large, sparse complex matrix is solved by permutation of the matrix into a convenient banded form and by writing recursion relations for the determinant.
34 citations
TL;DR: A method is proposed of modeling the storage process which has been called ‘the divided interval technique’ and its results are compared, by means of examples, with those from the existing method.
Abstract: Discrete state-discrete time stochastic matrix methods of reservoir analysis may be used to estimate the relationships between reservoir capacity, yield, and probability of emptiness for given inflow distributions. To avoid excessive computation and computer storage demands, these methods most commonly use models in which the state variable is discretized into a small number of finite states. This, however, generally leads to inaccurate solutions. To improve computational efficiency and allow solutions with reasonable accuracy with small numbers of finite states, a method is proposed of modeling the storage process which has been called ‘the divided interval technique.’ This method is described, and its results compared, by means of examples, with those from the existing method.
34 citations
TL;DR: In this paper, the collinear collision of an atom with a diatomic molecule was formulated quantum mechanically using the close coupling approach, and the resulting continuously infinite set of coupled differential equations can be reformulated as a discrete set of coupling differential equations.
Abstract: The collinear collision of an atom with a diatomic molecule A + (B,C) is formulated quantum mechanically using the close coupling approach. Neglecting rearrangement collisions but including electronically adiabatic inelastic and dissociative processes, it is shown that the resulting continuously infinite set of coupled differential equations can be reformulated as a discrete set of coupled differential equations. The technique is analogous to a procedure developed to treat gas–solid energy transfer. However, the boundary conditions involving three free particles in the final state has no analog in the gas–solid formulation of the problem. These are formulated also in terms of the discretized equations, and various simplifications and symmetries of the equations are discussed.
31 citations
TL;DR: In this article, a nonlinear observer (extended linear observer) which estimates the transient state of a power system is presented, where the observer estimates the state successfully not only in the case of hunting but also in step-out transient states of a synchronous machine.
Abstract: This paper presents a new nonlinear observer (extended linear observer) which estimates the transient state of power system. The transient state of the power system is represented as a continuous time nonlinear differential equation and observation of output is obtained as a discretized value at every cycle of system frequency by using the mean value detecting circuit device. The nonlinear differential equation of state is discretized at each observation instant by Taylor expansion, and then the theory of discrete linear observers is applied to the discretized system. Digital computer simulation confirms that the observer estimates the state successfully not only in the case of hunting but also in step-out transient state of a synchronous machine.
30 citations
TL;DR: In this article, the frequency and mode of vibration for suspension cables are determined using a discretized system composed of a linkage of straight bars connected by frictionless pins with concentrated masses at the connection points.
Abstract: Natural frequencies and modes of vibration for suspension cables are determined using a discretized system composed of a linkage of straight bars connected by frictionless pins with concentrated masses at the connection points. The governing equations are linearized which limits the applicability to small oscillations. The frequencies are determined by a generalized Holzer method coupled with the solution of the associated boundary value problem as a set of initial-value problems. The method is applied to sample numerical problems and the results are compared with the results obtained by alternate approaches. The sensitivity of the results to the nature of the discretization is studied. Also, several parameter studies are conducted to determine how the natural vibrations are altered in response to dimensional variations in the suspension cable. The results of the parameter studies are summarized in a nondimensional form.
TL;DR: The truncation cancellation procedure (TCP) as mentioned in this paper was proposed to cancel a portion of the truncation error in the convection term with that in the accumulation term to reduce the hyperbolic character of the Peclet number.
Abstract: A finite-difference method is presented for treating the convection-diffusion (C-D) equation which is fourth order correct when the Peclet number approaches infinity. In this limit, when the dimensionless time step equals the dimensionless spatial increment, the discretization is exact. For very small time steps, the method reduces to a difference form which is identical to one proposed by Stone and Brian and obtained by Price, Cavendish, and Varga by applying the Galerkin method to the C-D equation using chapeau basis functions. For larger time steps, it yields significantly better accuracy. Since the difficulty in representing the C-D equation numerically arises due to the hyperbolic character assumed as the Peclet number becomes large, the method aims at reducing the truncation error in the convection term. The rationale underlying the treatment is to cancel a portion of the error in the convection term with that in the accumulation term. Thus, the technique presented is referred to as the truncation cancellation procedure (TCP). Results are presented in both Cartesian and radial coordinates. Comparisons are made with exact analytic solutions, conventional numerical techniques and other high order accuracy numerical methods. (12 refs.)
TL;DR: In this paper, a method for solving structural design problems that allows a continuous distribution of material along structural elements is presented, which is an extension of the generalized steepest descent method presented in Reference 1.
Abstract: A method for solving structural design problems that allows a continuous distribution of material along structural elements is presented. The method is an extension of the generalized steepest descent method presented in Reference 1. Inequality constraints on design variables, displacement, natural frequency, and buckling are explicitly treated and a minimum weight cost function is employed. A steepest descent method for boundary-value state equations is developed and a computational algorithm is given. Several example problems in minimum weight structural design are solved and compared with results obtained by discretization techniques.
TL;DR: Simulation results are described that verify the qualitative carry-over of known results for the linear-Gaussian problem: the greater the lag, the greaterThe improvement over filtering obtained through the use of smoothing is greater than the signal-to-noise ratio (SNR).
Abstract: The fixed-lag smoothing of random telegraph type signals is studied. The smoothers are derived by first obtaining fixed-point smoothing equations and then using a time discretization. Simulation results are described that verify the qualitative carry-over of known results for the linear-Gaussian problem: the greater the lag, the greater the improvement; beyond a certain lag, no further improvement is obtained by the increase of lag; and the higher the signal-to-noise ratio (SNR), the greater is the improvement over filtering obtained through the use of smoothing. Smoothing errors of one-half the corresponding filtering error are demonstrated.
TL;DR: In this paper, the authors used the finite element techniques for the forced vibration analysis of horizontally curved box girder bridges and applied the mass condensation technique to reduce the number of coupled differential equations obtained from finite element method.
Abstract: The finite element techniques is used for the forced vibration analysis of horizontally curved box girder bridges Annular plates and cylindrical shell elements are used to discretize the slab, bottom flanges, and webs Rectangular plate elements and pin-jointed bar elements are used for discretization of diaphragms A vehicle, as the applied time varying forcing function, is simulated by two sets of concentrated forces, having components in the radial and transverse directions and moving with constant angular velocities on circumferential paths of the bridge The effect of centrifugal forces is considered and the effect of damping of the bridge is neglected in the analysis The mass condensation technique is used to reduce the number of coupled differential equations obtained from the finite element method The resulting differential equations are solved by the linear acceleration method A number of bridges with practical geometries are analyzed and impact factors are calcualted /Author/
TL;DR: The application of spatial discretization (discrete ordinate method) to a class of integro-differential equations is discussed in this paper, where it is shown that consistency in the approximation of the operators implies convergence of the approximate solution to the true solution.
Abstract: The application of spatial discretization (discrete ordinate method) to a class of integro-differential equations is discussed. It is shown that consistency in the approximation of the operators implies convergence of the approximate solution to the true solution.
TL;DR: In this article, the authors studied the problem of uniqueness in determining a certain kind of stratified conductivity structure of a medium from its magnetotelluric surface impedance, where the model is assumed to belong to the class con- sisting of homogeneous, isotropic parallel layers lying on top of a homo- geneous half-space substratum.
Abstract: Summary The problem of uniqueness in determining a certain kind of stratified conductivity structure of a medium from its magnetotelluric surface impedance is studied. The model is assumed to belong to the class con- sisting of homogeneous, isotropic parallel layers lying on top of a homo- geneous half-space substratum. An equal penetration discretization is further assumed. It is shown that for this class of models the impedance can be repre- sented in a rational polynomial form of a-penetration operator u. From the algebraic properties of these polynomials it is proved that no two such conductivity models can have the same surface impedance. The magnetotelluric method is aimed at exploring the electric structure of the Earth's crust. This method is based on simultaneous measurements of the natural magnetic and the induced electric fields at the Earth's surface. Using these observa- tions, the magnetotelluric impedance, which relates the electrical and magnetic fields at the surface, is determined as function of frequency. The interpretation of the obtained data in the frequency domain is usually done by assuming a model, computing its theoretical impedance and fitting the results with those obtained from observations. The derivation of the theoretical impedance for a given model is called the ' direct problem ' while the determination of the model's parameters from the surface impedance is known as the ' inverse problem '. A crucial question in any geophysical inverse problem is the question of theoretical uniqueness of the solution. Bailey (1970) gives a uniqueness proof of the inverse geomagnetic induction problem for a sphere. Another elegant treatment of the geomagnetic induction inverse problem for a fiat Earth is given by Wiedelt (1972). In that paper the similarities between the magnetotelluric and the geomagnetic induction problems is shown. In both works the models considered have con- ductivity distributions which are analytic functions of depth. It is the purpose of this work to study the question of uniqueness within a different class of models in which the conductivity profile contain finite number of discon- tinuities rather than being analytic function of depth. The simplest models of this kind are those consisting of homogeneous, isotropic, parallel layers above a half space (Cagniard 1953). In such a model the electrical conductivities are functions of depth only and the problem turns out to be one
01 Jan 1975
TL;DR: In this paper, the optimal design of continuous or pieeewise continuous structural systems can be approached along two avenues: discretizing the system at the outset and a sequence of problems involving refined domains may be studied in an effort to extract an approximate optimal solution.
Abstract: General Remarks. The optimal design of continuous or pieeewise continuous structural systems can be approached along two avenues. On one hand the system can be discretized at the outset and a sequence of problems involving refined domains may be studied in an effort to extract an approximate optimal solution. On the other hand, the continuous nature of the problem may be recognized and optimality criteria for the continua sought directly.
01 Jan 1975
TL;DR: The optimization of basic continuous structural members—such as a bar, a beam, a plate or a shell—under various conditions has not been fully explored yet and is still of utmost interest.
Abstract: In the recent years a considerable amount of effort has been devoted to the field of synthesis of complex discretized structures with the use of mathematical programming. Nevertheless, optimization of basic continuous structural members—such as a bar, a beam, a plate or a shell—under various conditions has not been fully explored yet and is still of utmost interest. In addition to providing well-known solutions with which the validity of various discrete optimization schemes may be tested, it also permits an insight into the non-trivial and most often overlooked problem of existence and uniqueness of optimal solutions.
TL;DR: A unified approach for deriving Various classes of such algorithms using the “positivity” concept and an equivalent feedback representation of the M.R.A.S.S.'s adaptive laws for continuous time is presented.
IBM1
TL;DR: In this paper, a numerical procedure for the solution of the Vlasov-Poisson system of equations in two and three phase-space variables is described, where time integration is done by advancing the distribution in real phase space as in finite difference methods.
TL;DR: In this article, the structural response to a single finite loading step is assumed to involve regularly progressive yielding (no local unloading), and an extremum property of this structural response is established, by recognizing that the relations governing the configuration change coincide with the Kuhn-Tucker conditions of a particular nonlinear constrained optimization problem, subject to sign constraints alone.
Abstract: Discrete or discretized structures are considered in the range of large displacements. Elastic plastic behavior is assumed, under the hypothesis that both yield functions and hardening rules are piecewise linear. The structural response to a single finite loading step is assumed to involve regularly progressive yielding (no local unloading). An extremum property of this structural response is established, by recognizing that the relations governing the configuration change coincide with the Kuhn-Tucker conditions of a particular nonlinear constrained optimization problem, subject to sign constraints alone. This extremum property can be regarded as an extension of the theorem of minimum potential energy. Other properties, even if computationally less attractive, broaden the theory developed, so that some results previously obtained are derived as special cases.
TL;DR: In this paper, the smoothing property of parabolic equations is used to show that for times uniformly bounded away from the initial time, the continuous-in-time Galerkin approximation and the GAs approximation based on the Crank-Nicolson time discretization give optimal order errors in the maximum norm.
Abstract: The smoothing property of parabolic equations is used to show that for times uniformly bounded away from the initial time the continuous-in-time Galerkin approximation and the Galerkin approximation based on the Crank–Nicolson time discretization give optimal order errors in the maximum norm under the mild restriction that the initial data for the Galerkin processes are optimally close to initial data for the parabolic boundary value problem in the mean square sense.
TL;DR: A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two-dimensional partial differential equations with discrete form of a complicated operator by performing simple numerical operations on elementary matrix factors.
Abstract: A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two-dimensional partial differential equations. The process is based on a novel approach to finite difference analysis in which differential operators are approximated by rectangular matrices. In this approach, the discrete form of a complicated operator is obtained by performing simple numerical operations on elementary matrix factors. The analysis is augmented by a listing of a computer program based on the method for the automatic generation of finite difference formulas.
TL;DR: The multiple exchange algorithm for restricted range approximation is discussed and efficient formulas are derived for the numerical implementation of the method.
Abstract: The multiple exchange algorithm for restricted range approximation is discussed. Efficient formulas are derived for the numerical implementation of the method. Discretization effects are analyzed mathematically. The method is applied to a certain problem arising in digital filter design.
TL;DR: A Galerkin procedure is used to obtain a semi-discretization for parabolic equations such as the heat equationut=txx, and it is proved that the operation count is 0 (h−2) as compared to 0(h−3) with the classical Crank-Nicolson.
Abstract: A Galerkin procedure is used to obtain a semi-discretization for parabolic equations such as the heat equationut=txx. The time variable being left continuous, the higher order approximation thus obtained for the space variable is then matched by a higher order discretization of the system of ordinary differential equations that results. Specifically we choose the Pade (2,2), and show how complex factorization it can be practically used. Moreover we prove that the operation count is 0 (h−2) as compared to 0(h−3) with the classical Crank-Nicolson. Numerical calculations are available.
TL;DR: In this article, the relative expectation and variance biases for several common distributions from a point of view somewhat different from that taken by the well-known Sheppard's corrections are studied.
Abstract: A continuous r.v. can only hypothetically assume any value on a continuum. When only an approximating discrete r.v. (“discretization”) is observable, estimation procedures employing the hypothetical continuous variate are sometimes biased. This paper studies the relative expectation and variance biases for several common distributions from a point of view somewhat different from that taken by the well-known Sheppard's Corrections. For each distribution a table is provided to enable the statistician to determine whether a serious bias problem exists. It is noted that while these biases are often of little consequence to the applied statistician, estimators that are a function of several discretized r.v.'s may be seriously biased.
TL;DR: In this paper, the primal and dual finite element methods were compared, and the authors concluded that linear elements and the dual method provided the most efficient combination for solving the problem of control theory.
Abstract: Two problems in control theory, one with state constraints and the other with control constraints, have been approximated by the finite element method. This discretization has been applied to both the primal and the dual formulation, in order to make a number of observations and comparisons: 1. The rate of convergence as the grid interval h is decreased, for polynomial elements of different degrees. 2. The presence or absence of a boundary layer in the error, concentrated at the "contact points" where the constraints change between binding and nonbinding. 3. The advantages of simpler constraints in the dual formulation, and the disadvantages of replacing strict convexity by ordinary convexity. 4. The numerical efficiency of each possible variation in achieving an approximate solution of reasonable accuracy. We concluded that in our model problems, linear elements and the dual method provide the most efficient combination.
TL;DR: In this paper, an extension of the method of integral coordinates to the stability analysis of hybrid dynamical systems containing two-and three-dimensional elastic members is presented, which yields stability criteria with relative ease and in a form that permits ready physical interpretation.
Abstract: This paper presents an extension of the method of integral coordinates to the stability analysis of hybrid dynamical systems containing two- and three-dimensional elastic members. A common approach to such problems consists of system discretization by means of series representation in terms of admissible functions, an approach involving lengthy computations. By contrast, the method of integral coordinates yields stability criteria with relative ease and in a form that permits ready physical interpretation. As an application, the attitude stability of a force-free satellite containing a membrane is investigated and closed-form stability criteria are derived.