Discrete approximations

As mentioned before, the Laplace transform is a convenient way of representing a process model. However, in today s world of computers discrete time representations of systems is preferred. The simplest way is a first-order discretization  

Note X is a deviation variable, A r is the discretization interval. For a first-order system of Eqn. (5.10) one obtains  

The input variable u is given a time index -1 rather than k, since we know that it is physically impossible for u to have an impact ony at time k. Equation (5.44) can be written as  

The expression for Kiag is valid as long as Af is relatively small compared with r. If this is not the case, a better expression for Kiag is  

The difference between the determinations of Ku,g is illustrated in Table 5.3 where Af=5. 

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In this section, we discuss the role of numerical simulations in studying the response of materials and structures to large deformation or shock loading. The methods we consider here are based on solving discrete approximations to the continuum equations of mass, momentum, and energy balance. Such computational techniques have found widespread use for research and engineering applications in government, industry, and academia. 

In all of these alkali-metal and alkaline earth-metal orthophosphates there are discrete, approximately regular tetrahedral PO4 units in... 

In the cellular automata (CA) algorithm, many small cells replace the differential equations by discrete approximations in suitable applications the approximate description that the cellular automata method provides can be every bit as effective as a more abstract equation-based approach. In addition, while differential equations rarely help a novice user to develop an intuitive understanding of a process, the visualization provided by CA can be very informative. 

Further Analysis of Solutions to the Time-Independent Wave Packet Equations of Quantum Dynamics II. Scattering as a Continuous Function of Energy Using Finite, Discrete Approximate Hamiltonians. 

To determine A we need to implement the path integration as a numerical path summation. The path integral in Eqs. 23-24 is in fact isomorphic to the configuration integral of a flexible polymer which interacts with the external potential V(r(P)). This analogy can be made more explicit by considering a discrete approximation to the path integral [71]. If the path is cut up into P... 

The basic strategy to accomplish this goal is to develop discrete approximations for integration (like rectangular or trapezoidal) in terms of functions of z. These are called z forms, and they are different than z transforms. Then we substitute the z form for 1/s in G(j,. So the first thing we must do is develop discrete approximations for integration (1/s). 

Example 18.11. Suppose we want to find a discrete approximation for a first-order lag. This would be called a first-order digital filter. Let x, be the output of the filter and m ) be the input. 

The stochastic stress o in Grassia and Hinch s algorithm is given by a discrete approximation to the time average of Eq. (2.378) over one timestep, as... 

In Section II we developed the concept of the convolution integral and its discrete approximation. No conceptual difficulty is therefore encountered in computing A from B and G ... 

Continuous functions and signals in the time domain are denoted by lower case letters with the argument in parentheses, e.g. x(t). Sampling at constant intervals A t produces a discrete approximation x[n] to the continuous signal, defined at times f = n A t, n = 0,1,2. Square brackets are used for the arguments of discrete functions. The Fourier transform establishes the connection between the time and frequency domains [76] ... 

The PID control law considered here contains the P, PI, PD, and PID control laws as special cases. The velocity form of the discrete approximation of an ideal PID controller is given by [2]... 

The discrete approximation for the second order spatial derivative (d1c/dx2)i n at x = x results in a similar manner, namely by expressing the concentrations at the neighboring gridpoints x i and Xj, i from the Taylor series in x at constant t = tn ... 

In this chapter, all the discrete approximations required for simulation are established, that is, for first and second derivatives, both central and asymmetric forms, and for a range of numbers of points used. 

Most discrete approximations that have been mentioned in this book are consistent, with the exception of one. This is the DuFort-Frankel method [216], described on page 153 in Chap. 9. It is stable for all A, yet it has a consistency problem. Giving (9.19) the same treatment as above, one ends with... 

For some years the method in widespread use for the electron—hydrogen problem was to solve the coupled integrodifferential equations of the coordinate representation with the appropriate boundary and orthogonality conditions (Seaton, 1973). This was so laborious that the use of Stur-mians as a discrete approximation to the target spectrum could not be implemented to convergence. 

Huang. Y.. Iyengar. S.S.. Kouri, D.J. and Hoffman. D.K. (1996) Further analysis of solutions to the time-independent wnve packet equations of quantum dynamics. 2. Scattering as a continuous function of energt- using finite, discrete approximate Hamiltonians, J. Chem. Phys. 105, 927-939. 

It is convenient now to pass to the consideration of one cluster and change the discrete approximation to the continual one. The transformation means the passage from the summation over discrete functions in expression (566) to the integration of continual functions by the rule... 

The numerical solution of both the fractional Fokker-Planck equation in terms of the Griinwald-Letnikov scheme used to find a discretized approximation to the fractional Riesz operator exhibits reliable convergence, as corroborated by direct solution of the corresponding Langevin equation. 

One method to solve partial differential equations using the numerical schemes developed for solving time dependent ordinary differential methods is the method of lines. In this method, the spatial derivatives at time t are replaced by discrete approximations such as finite differences or finite element methods such as collocation or Galerkin. The reason for this approach is the advanced stage of development of schemes to solve ordinary differential equations. The resulting numerical schemes are frequently similar to those developed directly for partial differential equations. 

The expression we derive below leads to an action and to a stationary (minimum) condition on the classical path. The optimal path is a discrete approximation to a classical trajectory. Interestingly, in the integral limit (an infinitesimal time step), the action below was used already by Gauss ( ) to compute classical trajectories [14]. At variance with Gauss we keep a finite At. 

Instead of starting from the Newton s equation, we use the action formalisms. In the time formulation, we obtain the equation of motion by requiring that the action is stationary that is, S/ X(t) = 0 or in the discrete approximation to the path 95/9X,- = 0 j [see Eq. (4)]. The Ssdet action can be written in that... 

The LS models are apparently more accurate than the computationally cheaper VOF methods, but the LS models still suffer from not having the possibility to prescribe a physical coalescence criterion instead of relying on the numerical mode outcome of the discretization approximations. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

Let be a discrete approximation to in the i-th GCV at time step n. The total variation (TV) at time step n is defined by ... 

Fig. 2,3.6 The saddle coil is an approximation of a cylinder with a sinusoidal current distribution, (a) Sinusoidal current distribution on a cylinder, (b) Discrete approximation of the cylinder by six parallel wires carrying current /. (c) Saddle coil obtained from (b) by connecting the current bearing wires. Adapted from [Krel] with permission from publicis MCD.

Finally, let us suppose that 0S (x) is an arbitrary but smooth function ofx and that we can approximate this function by breaking the surface region into a large number of discrete intervals in which 0S (x) can be approximated as a constant, corresponding, for example, to the average value over the interval. Then the solution, in terms of — (39/3T)y=0, for each individual interval is of the type shown in Step (3). Thus the discrete approximation to the whole solution is a sum of solutions of the type from Step (3) for each subinterval, namely,... 

Now, in the limit Ax 0, the discrete approximation to the surface temperature distribution passes smoothly to the continuous function 9S (x), and the summation in (11-106) becomes an integral overx. Thus,... 

Cellular automata (CA) are an alternative method to solve differential equations. They can be considered as discrete approximations to partial differentia equations. Cellular automata are mathematical systems consisting of many identical components or cells. Each cell is a kind of virtual robot that responds to signals according... 

The cell temperatures may be updated by solving equation (20.62) numerically. Replacing the space differentials by discrete approximations gives the time differential for cell temperature for every cell except the last as ... 

FIGURE 25.8 Example of a discrete approximation of the continuous concentration function c(x). 

An openloop-unstable, first-order process has the transfer function A discrete approximation of a PI controller is used. 

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