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Showing papers on "Discretization published in 1969"


Journal ArticleDOI
TL;DR: In this article, the essential shakedown theory and the basis of relevant solution procedures are presented in compact form. And for systems with associated flow-laws, the second shakedown theorem (Koiter's) is extended in order to allow variable dislocations (e.g. temperature cycles).
Abstract: The matrix description of the mechanical behaviour resting on finite element discretization and piecewise linearization of yield surfaces, is adopted instead of the traditional continuous field description. By the use of linear programming concepts, the essential of a general shakedown theory and the basis of relevant solution procedures are presented in compact form. For systems with associated flow-laws, the second shakedown theorem (Koiter's) is extended in order to allow for variable dislocations (e. g. temperature cycles). For systems with nonassociated flow-laws two theorems are given which supply lower and upper bounds to the safety factor.

185 citations


Journal ArticleDOI
TL;DR: In this paper, an integral equation on the area of the crack for the relative displacement across the crack is presented. But the kernel of this integral equation is non-integrable and a method for discretizing it and a numerical method of solution is carried out.
Abstract: An infinite elastic medium is initially at rest in a prestressed state of plane- or anti-plane strain. At time t = 0 a plane crack comes into existence which occupies a strip parallel to the y axis and whose width varies in time. Assuming that the components of the traction are known on the crack surface it is possible to set up an integral equation on the area of the crack for the relative displacement across the crack. Although the kernel of this integral equation is non-integrable a method is found for discretizing it and a numerical method of solution is carried out. The results, which in some cases are the solutions of diffraction problems, are presented graphically.

104 citations



Journal ArticleDOI
TL;DR: In this paper, an alternating direction iteration method is formulated, and con- vergence is proved, for the solution of certain systems of nonlinear equations, such as a heat conduction problem with a nonlinear boundary con- dition.
Abstract: An alternating direction iteration method is formulated, and con- vergence is proved, for the solution of certain systems of nonlinear equations. The method is applied to a heat conduction problem with a nonlinear boundary con- dition. e 1. Alternating direction methods are often used for solving the sets of linear equations arising from the discretization of elliptic boundary value problems (11, (2), (3). In this paper, an alternating direction method is formulated for a certain nonlinear system of equations. Convergence of the method is established in the case of a single iteration parameter. Finally, the method is applied to a set of equa- tions arising from a steady-state heat conduction problem with nonlinear boundary conditions. Such boundary conditions occur when energy is transmitted from the boundary of the region by means of radiation or by means of natural convection

38 citations


Journal ArticleDOI
TL;DR: In this article, a series of experiments executed with water are described and the results compared with theoretical predictions for power law and sinusoidal variations in surface temperature in a semi-infinite domain.

34 citations


Journal ArticleDOI
TL;DR: An algorithm is presented for solving a system of linear equations Bu = k where B is tridiagonal and of a special form, and it is shown that this algorithm is almost twice as fast as the Gaussian elimination method usually suggested for solving such systems.
Abstract: An algorithm is presented for solving a system of linear equations Bu = k where B is tridiagonal and of a special form. This form arises when discretizing the equation - d/dx (p(x) du/dx) = k(x) (with appropriate boundary conditions) using central differences. It is shown that this algorithm is almost twice as fast as the Gaussian elimination method usually suggested for solving such systems. In addition, explicit formulas for the inverse and determinant of the matrix B are given.

21 citations


Journal ArticleDOI
TL;DR: This paper presents a method which solves the identification problem of linear dynamical systems with transport lags and is digitally oriented and shows how a continuous-time system can be identified by discrete techniques.
Abstract: Linear dynamical systems with transport lags are characterized by linear differential-difference equations. The task of identifying unknown parameters in such systems from the input-output data is difficult due to mathematical complications associated with differentialdifference equations. This paper presents a method which solves the identification problem. The method is digitally oriented and shows how a continuous-time system can be identified by discrete techniques. The solution is based on Kalman's least square method. The identification procedure essentially involves two steps: 1) discretizing the continuous system via finite difference approximation, and 2) estimating the parameters through the identification of the resulting discrete model. Experimental results have verified the validity of the proposed method.

18 citations


Journal ArticleDOI
TL;DR: In this paper, the convergence to the solutions of a class of optimal control problems is proved for a method by Rosen in which the differential equations, constraints, and cost functional are discretized, and the resulting mathematical programming problem is solved approximately by a penalty-function approach.
Abstract: Under suitable restrictions, convergence to the solutions of a class of optimal control problems is proved for a method by Rosen in which the differential equations, constraints, and cost functional are discretized, and the resulting mathematical programming problem is solved approximately by a penalty-function approach.

16 citations


Journal ArticleDOI
TL;DR: In this paper, a matrix method for the stationary creep analysis of structures is presented, which implies discretization of the system, piecewise linearization of stress-strain rate laws, some linear elastic calculations which are useful also for other purposes, and the solution of a quadratic program.
Abstract: The paper presents a matrix method for the stationary creep analysis of structures. The method implies discretization of the system, piecewise linearization of the (generalized) stress-strain rate laws, some linear elastic calculations which are useful also for other purposes, and the solution of a quadratic program. The procedure adapts to various degrees of accuracy, and is highly suitable for computers.

7 citations


Journal ArticleDOI
TL;DR: In this paper, a set of requirements such as conservation and correct asymptotic behavior, which, although quite reasonable, cannot be satisfied by a finite difference method on a fixed word length machine are presented.

5 citations


Journal ArticleDOI
TL;DR: The solution of the difference approximation to the heat conduction equation is shown to reflect accurately the pattern of behaviour of the differential equation, and this result is applied to the phenomenon of 'persistent discretisation error' in the solution to the difference equation.
Abstract: The stability of the Crank Nicolson scheme for the numerical solution of the heat conduction equation subject to separated boundary conditions is demonstrated. This result is extended to separable equations with variable coefficients and to the heat conduction equation in cylindrical geometry which has a singular coefficient. The solution of the difference approximation to the heat conduction equation is shown to reflect accurately the pattern of behaviour of the differential equation, and this result is applied to the phenomenon of 'persistent discretisation error' in the solution to the difference equation.

Journal ArticleDOI
TL;DR: In this article, a hybrid computer simulation method to solve the three-dimensional nonuniform diffusion equation is described, where the partial differential equation is transformed into a set of algebraic equations and the entire process is performed in the hybrid computer system using the digital computer as the control.
Abstract: Evaporation from finite areas with constant sources is determined by simulating the mass transfer equation. The equation of mass transfer in a turbulent atmosphere describes the movement of mass, water vapor, and momentum. The process of transferring water vapor by the evaporation phenomena from a constant source can be analyzed by the solution of this diffusion equation. A hybrid computer simulation method to solve the three-dimensional nonuniform diffusion equation is described. The proposed technique can handle any geometric configuration of lakes and reservoirs. Using finite difference method, the partial differential equation is transformed into a set of algebraic equations. The analogy between these algebraic equations and the node equations of a passive resistance network plane makes the hybrid computer simulation technique applicable to solve problems of this type. A vertical plane, perpendicular to the direction of the wind, is simulated by a passive resistance network. The movement in the direction of the wind is discretized with resistances simulating the step size. The problem is solved by moving the vertical plane in the direction of the wind. Obtaining the solution using a simulation technique involves a repetitive process of observing voltages on the simulation network. The entire process is performed in the hybrid computer system using the digital computer as the control. This then is a general method available to analyze the evaporation phenomena using a mathematical solution of the diffusion equation.