Linear element

Abstract: The object of this paper is to outline a stability theory based on functional methods. Part I of the paper was devoted to a general feedback configuration. Part II is devoted to a feedback system consisting of two elements, one of which is linear time-invariant, and the other nonlinear. An attempt is made to unify several stability conditions, including Popov's condition, into a single principle. This principle is based on the concepts of conicity and positivity, and provides a link with the notions of gain and phase shift of the linear theory. Part II draws on the (generalized) notion of a "sector non-linearity." A nonlinearity N is said to be INSIDE THE SECTOR {\alpha,\beta} if it satisfies an inequality of the type \langle(Nx-\alphax)_{t}, (Nx-\betax)_{t}\rangle\leq0 . If N is memoryless and is characterized by a graph in the plane, then this simply means that the graph lies inside a sector of the plane. However, the preceding definition extends the concept to include nonlinearities with memory. There are two main results. The first result, the CIRCLE THEOREM, asserts in part that: If the nonlinearity is inside a sector {\alpha, \beta} , and if the frequency response of the linear element avoids a "critical region" in the complex plane, then the closed loop is bounded; if \alpha > 0 then the critical region is a disk whose center is halfway between the points -1/\alpha and -1/\beta , and whose diameter is greater than the distance between these points. The second result is a method for taking into account the detailed properties of the nonlinearity to get improved stability conditions. This method involves the removal of a "multiplier" from the linear element. The frequency response of the linear element is modified by the removal, and, in effect, the size of the critical region is reduced. Several conditions, including Popov's condition, are derived by this method, under various restrictions on the nonlinearity N ; the following cases are treated: (i) N is instantaneously inside a sector {\alpha, \beta} . (ii) N satisfies (i) and is memoryless and time-invariant. (iii) N satisfies (ii) and has a restricted slope.

Abstract: This paper examines the theoretical bases for the smoothed finite element method (SFEM), which was formulated by incorporating cell-wise strain smoothing operation into standard compatible finite element method (FEM) The weak form of SFEM can be derived from the Hu–Washizu three-field variational principle For elastic problems, it is proved that 1D linear element and 2D linear triangle element in SFEM are identical to their counterparts in FEM, while 2D bilinear quadrilateral elements in SFEM are different from that of FEM: when the number of smoothing cells (SCs) of the elements equals 1, the SFEM solution is proved to be ‘variationally consistent’ and has the same properties with those of FEM using reduced integration; when SC approaches infinity, the SFEM solution will approach the solution of the standard displacement compatible FEM model; when SC is a finite number larger than 1, the SFEM solutions are not ‘variationally consistent’ but ‘energy consistent’, and will change monotonously from the solution of SFEM (SC = 1) to that of SFEM (SC → ∞) It is suggested that there exists an optimal number of SC such that the SFEM solution is closest to the exact solution The properties of SFEM are confirmed by numerical examples Copyright © 2006 John Wiley & Sons, Ltd

Abstract: This is the first in a series of papers in which a new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz--Zhu patch recovery method. In addition, for uniform triangular meshes, the method is superconvergent for the linear element under the chevron pattern, and ultraconvergent at element edge centers for the quadratic element under the regular pattern. Applications of this new gradient recovery technique will be discussed in forthcoming papers.

Abstract: In a light-emitting element drive circuit in an active matrix display device, at least one current-control transistor controls a current flowing through a light-emitting element. The current-control transistor and the light-emitting element are connected in parallel to each other. A constant current source is connected to a junction between one electrode of the light-emitting element and one electrode of the transistor through which the current 8s controlled to flow. The other electrodes of the light-emitting element and the transistor are connected to a common electrode which may be grounded via a resistor. In other configuration, it may be arranged that the light-emitting element and a capacitance are connected in parallel to each other. In this case, the current-control transistor is connected to a function between the light-emitting element and the capacitance so as to use charging and discharging operations of the capacitance for driving the light-emitting element.