Node equations

Differentiation of locally defined shape functions appearing in Equation (2.34) is a trivial matter, in addition, in isoparametric elements members of the Jacobian matrix are given in terms of locally defined derivatives and known global coordinates of the nodes (Equation 2.27). Consequently, computation of the inverse of the Jacobian matrix shown in Equation (2.34) is usually straightforward. 

This relationship is exactly the same as the one that was found in the TMB case. Similar conditions apply to the two compoimds in the other three columns. As a consequence, Eqs. 17.10a and 17.11a apply as well. Combining the node equations and the propagation equation (Eq. 17.15), we can derive the concentration profiles of both components in each column. This calculation must be done at the end of each period, as is explained in the next few subsections. These concentration profiles are complex functions of the experimental conditions that depend on the rank of the period considered. One interesting result is that these functions have asymptotic limits that are easily derived from simple theorems on the infinite limits of suites, series, and products. This allows the calculation of the steady-state profiles. 

Using the propagation velocity associated with a concentration and the node equations, it is easy to calculate the concentration plateau of each component in the columns 111 and IV and the location of the concentration shocks at both ends... 

These network equations differ from the Kirchhoff equations used in electrical circuit theory where the sum in the node Equation 10.5 is zero. O Keeffe (Struct. Bonding 1989, 71, 161-190) has shown that a correct mathematical correspondence requires that Kirchhoff s loop law be equivalenced with Equation 10.5 and the junction law with Equation 10.6. This requires replacing the nodes of the bond network with the loops of the equivalent Kirchhoff network and vice versa. For practical purposes it is simpler to stay with Equations 10.5 and 10.6... 

For the solution of the PDE models of the columns, a Galerkin method on finite elements is used for the liquid phase and orthogonal collocation for the solid phase. The switching of the node equations is considered explicitly, that is, a full hybrid plant model is used. The objective function F is the sum of the costs incurred for each cycle (e.g., the desorbent consumption) and a regularizing term that is added in order to smooth the input sequence in order to avoid high fluctuations of the inputs from cycle to cyde. The first equality constraint represents the plant model... 

The jump length distribution for a walker moving along the backbone of the network, segment AB in Fig. 6.6, is w(x) = Pout (- — 0 + Pm ix. + 1), where / is the distance between two consecutive nodes. Equations (5.30) and (4.46) yield the following expression for the front velocity ... 

The memory elements are identified in the equations by characteristics common to all memory elements. The first criterion is that the loading of a value into a memory element must be controlled by the clock the second, that the element must retain its value when not clocked. Whether a node is controlled by the clock is easily determined in the node equation. The second criterion is true if the node equation contains a reference to the node itself, and this self reference is enabled by the clock. Having identified clocked memory elements, a partition of the net into memory elements and strictly combinational parts can be found. Prom this information, a description in PTL can easily be derived. 

Upon solving the discretized P-field, the local constants, A and B, can be calculated according to Equations (7) at all internal mesh nodes. At a botmdary node. Equation (7) can only be used to find B along the boundary. However, Equations (1) and (8) can then be used to find B across. Afterwards, Equation (5) is applied toward the interior to obtain an equation, which is used for the calculation of A across. The set of values of A at all mesh nodes is the discretized (pjj-field. 

The objective equation [18.1] indicates that the total cost of all routes, which is the weighted sum of the arcs safety and travel time, should be minimized. Equation [18.2] specifies that the customer node i may be visited only once. Equation [18.3] requires that the vehicle should enter and leave a customer node an equal number of times in the same route. Equations [18.4] and [18.5] define that each route starts and ends at the school (depot). Equation [18.6] guarantees that the vehicle capacity is not exceeded in any route. Equation [18.7] requires conservation of flow of each route at each customer node. Equation [18.8] limits the total time of a vehicle s route to the time limit. Equation [18.9] puts a lower bound in the number of students of each route. Equations [18.10] and [18.11] are the integrality and non-negativity requirements on the variables. 

The conservation equations are numerically discretized along the axial direction by the forward finite difference method. The discretization or nodalization is done by dividing the coolant channel and the water rods into nodes of equal length and by approximating the time-dependent thermal-hydraulic variables as spatially uniform within these nodes. Equations (5.36)-(5.39) are expressed as follows for each axial node i. 

Using the fundamental relationship for current through a diode as expressed back in Chapter 3 (see Eq. (3.27)), two node equations can be written as ... 

The procedure for formulating the boundary node equation set is then to sum over the interior triangles for all the boundary nodes in exactly the same manner as for the interior nodes. Then additional terms are added to the boundary node equations to implement the results of Eq. (13.36) if the boundary condition is that of a mixed type of if there is a non-zero normal derivative term. If the boundary condition is of the fixed value type, then the matrix equation diagonal element is simply set to the fixed value as given by the last line of Eq. (13.35). 

A second more extensive extension would be the development of programs to solve a number of coupled second order PDEs - for example 3 PDEs in three physical variables. These could also be time dependent. Such coupled PDEs occur frequently in physical problems. While extending the previous code for this case is relatively straightforward in principle, this is not a trivial task. Only some considerations for coupled equations will be considered here for N coupled equations. If one allows in the most general case, all ranges of derivatives to be expressed in the coupled equations, then one has N variables at each node and each defining PDE has N variables. Thus the number of node equations is increased by the factor N and the number of possible non-zero elements per row is increased by the factor N, giving a value of NXN = as the increased factor for the possible number of non-zero matrix elements. For the case of 3 coupled variables this is a factor of 9. Thus the computational time for eoupled systems of equations can increase very fast. 

The FE solvers developed in this chapter have made use of some of the approximate matrix solution techniques developed in the previous chapter. However, most of the code is new because of the different basic formulation of the finite element approach. In this work the method of weighted residuals has been used to formulate sets of FE node equations. This is one of the two basic methods typically used for this task. In addition, the development has been based upon flie use of basic triangular spatial elements used to cover a two dimensional space. Other more general spatial elements have been sometimes used in the FE method. Finally the development has been restricted to two spatial dimensions and with possible an additional time dimension. The code has been developed in modular form so it can be easily applied to a variety of physical problems. In keeping with the nonlinear theme of this work, the FE analysis can be applied to either linear or nonlinear PDEs. 

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