Discretisation

In this approach, continuously varying quantities are computed, generally as a function of time as some process, such as casting or mechanical working, proceeds, by discretising them in small regions, the finite elements of the title. The more... 

The method for estimating parameters from Monte Carlo simulation, described in mathematical detail by Reilly and Duever (in preparation), uses a Bayesian approach to establish the posterior distribution for the parameters based on a Monte Carlo model. The numerical nature of the solution requires that the posterior distribution be handled in discretised form as an array in computer storage using the method of Reilly 2). The stochastic nature of Monte Carlo methods implies that output responses are predicted by the model with some amount of uncertainty for which the term "shimmer" as suggested by Andres (D.B. Chambers, SENES Consultants Limited, personal communication, 1985) has been adopted. The model for the uth of n experiments can be expressed by... 

Whilst finite-element modelling of gap junctions occurs at a sub-cellular level, these models do not consider the operation of intact organs. Conversely, in models of the complete heart the discretisation is usually on a milhmetre scale. However, the cochlea (see Figure 9.3) is already being simulated on a 0.01 mm, or cellular, scale. Although cochlear malfunction is not hfe threatening, damage to it does adversely affect the ability of almost 1000000000 people to communicate. 

Irrespective of the type of extended system we are interested in we impose periodic boundary conditions in position space - "the large period" BK. Such conditions imply a discretisation of momentum and reciprocal space 27] which means that integrations are replaced by summations ... 

Generation of quant network the product flow is discretisized... 

For computational reasons, it is necessary to discretise the space in such a way that one considers only a finite number of z-coordinates for the segments of the molecules to take positions. Again, the coarse graining is justified as long as the relevant phenomena do not take place on a smaller length scale. Here we consider lattices with sites that fit united atoms, e.g. CH2 units. The... 

Boriskina, S.V., Sewell, P. Benson, T.M., and Nosich A.I., 2004, Accurate simulation of 2D optical microcavities with uniquely solvable BIEs and trigonometric-Galerkin discretisation, J. Opt. Soc. Am. A 21(3) 393-402. 

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

Due to their higher flexibility and accuracy, Finite Elements Methods (FEMs) [5] are often preferred to numerical methods their basic concept consists first of all in establishing a weak variational formulation of the mathematical problem the second step is to introduce the concept of shape functions that are defined into small sub-regions of the domain (see also Chapter 3). Finally, the variational equations are discretised and form a linear system where the unknowns are the coefficients in the linear combination. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

Figure 4.3 Two examples of a 27t-spectrometer collecting electrons within a plane, (a) Suggested set-up with full rotational symmetry around the symmetry axis z. The electric field is supplied by conical bodies, and the detector collects simultaneously all electrons emitted from the source Q into the plane perpendicular to the z-axis (for directional information of electron emission using a discretised detector, see below). From [Kuy68]. (b) Spatial view of a toroidal analyser. The outer and inner toroids which again possess axial symmetry provide the electric field. The detector is a position-sensitive detector (see Section 4.3.2) which records directional information about the electron emission. Hence, the angle dependence of electron emission from the source Q into the plane perpendicular to the symmetry axis is preserved. See also [EBM81, Hue93]. Part (b) reprinted from Nucl. Instr. Meth. B12, Toffoletto et al, 282 (1985) with kind permission of Elsevier Science - NL, Sara Burgerhartstraat 25, 1085 KV Amsterdam, The Netherlands.

In this approach, the ODE or DAE process models are discretised into a set of algebraic equations (AEs) using collocation or other suitable methods and are solved simultaneously with the optimisation problem. Application of the collocation techniques to ODEs or DAEs results in a large system of algebraic equations which appear as constraints in the optimisation problem. This approach results in a large sparse optimisation problem. 

In this approach the set of DAEs presented in Equation 5.4 is discretised into a set of Algebraic Equations (AEs) by applying collocation method. 

The reflux ratio is discretised into two time intervals for task 1 and one time interval for task 2. Thus a total of 3 reflux ratio levels and 3 switching times are optimised for the whole multiperiod operation. Three cases are considered, corresponding to different values of the main-cut 1 product. For each case the... 

The reflux ratio is discretised into two control intervals for each operation task. Three cases were considered with different sales values for the two main-cuts. For all cases, the optimal recovery of component 1 in task 1 and that of component 2 in task 2, the optimal amounts of main-cut 1, off-cut 1 and main-cut 2, the optimal... 

In digital simulation, when discretising the diffusion equation, we have a first derivative with respect to time, and one or more second derivatives with respect to the space coordinates sometimes also spatial first derivatives. Efficient simuiation methods will always strive to maximise the orders. 

The coefficients are used to compute the derivatives, but can also be useful in the discretisation of derivative boundary conditions or in the setting up of discretisation matrices in some problems. 

All the techniques described above can also be applied to the numerical solution of systems of odes, and here we are getting closer to what happens when we solve pdes, because in effect, one reduces them to ode systems when discretising them. 

For brevity, the Euler method will be treated as a special case of RK, considered as RK1. The method is then to start by calculating a vector of k values, one for each y element. Discretising directly from (4.51), this is... 

In some cases, for example electrochemical pdes with derivative boundary conditions, the discretisation process for both the pde and the boundary conditions leads to a mix of a differential equation system and one or more plain algebraic equations. They might be, for example, equations of the form... 

We might wish to solve it using an implicit method, for example, BI (Sect. 4.6). Discretising (4.61) then gives... 

At any point with index i, that is at X = iH, the diffusion (1.1) is discretised on the left-hand side in the Euler manner (4.4, or in other words the forward difference formula 3.1) and on the right-hand side with the central three-point approximation (3.41), giving for the iteration going from time T to the next time T + 8T,... 

If several species are involved (in this case there is the product prod, but we are not interested in it), the equations are extended in an obvious manner, apart from some tricks to be seen in a later chapter in connection with implicit methods. This is one of the attractive aspects of method EX. If the her is second order, there will be a term in C in the discrete equation, and it will present no problem in the discretisation step [146]. 

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