Nodes boundary

A quadrilateral mesh may be logically rectangular or arbitrarily connected. A two-dimensional logieally reetangular mesh has four elements eonneeted to eaeh interior node. Boundary nodes have less than four eonneeted elements. The reetangular and quadrilateral meshes shown in Fig. 9.1 are logieally reetangular. An arbitrary conneetivity mesh may have an arbitrary number of elements eonneeted to a node. Examples of arbitrary eonneetivity meshes are shown in Fig. 9.2. 

Fig. 12.6. (a) Hopf bifurcation loci for the Takoudis-Schmidt Aris model with k, = 10-3 and k2 = 2x 10-3. Also shown (broken curves) are the saddle-node boundaries from Fig. 12.6. (b)-(i) The eight qualitative arrangements of Hopf and saddle-node bifurcation points. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A415,... 

This problem is described mathematically as an ordinary-differential-equation boundary-value problem. After discretization (Eq. 4.27) a system of algebraic equations must be solved with the unknowns being the velocities at each of the nodes. Boundary conditions are also needed to complete the system of equations. The most straightforward boundary-condition imposition is to simply specify the values of velocity at both walls. However, other conditions may be appropriate, depending on the particular problem at hand. In some cases a balance equation may be required to describe the behavior at the boundary. 

The effort to develop a neutron transport code based on an improved nodal method has been continued in order to treat the Hex-Z geometry of FBR cores more accurately. In order to reduce truncation error of the code, a new method to treat the radial leakage has been developed, in which the distribution of node boundary fluxes is obtained from local two-dimensional flux distribution. The local flux distribution is evaluated from average fluxes at surrounding nodes and node boundaries by second-order polynomials. An FBR core model with extremely large-sized assemblies was calculated by the new method and the results were compared with those of a reference Monte Carlo calculation. While the previous method overestimated the criticality by 0.3% dk for a control rod-insertion case, the new method agreed well with the Monte Carlo result. 

In order to treat the Hex-Z geometry of FBR cores more accurately, effort to develop a neutron transport code based on an improved nodal method was continued. In the previous version of the nodal code, the radial transverse leakage on node boundaries was assumed to be distributed uniformly, which generates some truncation errors. This year, a new treatment for the radial transverse leakage was introduced to the code by adopting a second order polynomial expansion of the flux at the node vertex point. A benchmark test of an FBR core showed the new nodal method can predict the keff within errors of 0.02%dk/k, on the other hand, the previous treatment has errors of 0.1%dk/k. 

Segmentation Inner node Adjoining stmt Enclosed curve boundary Inner node boundary lines... 

CMT level did not drain until the top node liquid temperature had reached saturation. Draining stagnated to the level of the node boundary and continued after the liquid in the node next below was in saturation. During the stagnation the injection flow decreased dramatically and started to oscillate. 

Therefore the second-order derivative of/ appearing in the original form of / is replaced by a term involving first-order derivatives of w and/plus a boundary term. The boundary terms are, normally, cancelled out through the assembly of the elemental stiffness equations over the common nodes on the shared interior element sides and only appear on the outside boundaries of the solution domain. However, as is shown later in this chapter, the appropriate treatment of these integrals along the outside boundaries of the flow domain depends on the prescribed boundary conditions. 

Papanastasiou et al. (1992) suggested that in order to generate realistic solutions for Navier-Stokes equations the exit conditions should be kept free (i.e. no outflow conditions should be imposed). In this approach application of Green s theorem to the equations corresponding to the exit boundary nodes is avoided. This is eqvrivalent to imposing no exit conditions if elements with... 

Location arrays showing the numbers of elements that contain each given node in the fixed mesh and its boundary are prepared and stored in a file. 

After identification of the elements that contain feet of particle trajectories the old time step values of F at the feet are found by interpolating (or extrapolating for boundary nodes) its old time step nodal values. In the example shown in Figure 3.6 the old time value of Fat the foot of the trajectory passing through A is found by interpolating its old nodal values within element (e). 

In some applications it may be necessary to prescribe a pressure datum at a node at the domain boundary. Although pressure has been eliminated from the working equations in the penalty scheme it can be reintroduced through the penalty terms appearing in the boundary line integrals. 

Step 2 an initial configuration representing the partially filled discretized domain is considered and an array consisting of the appropriate values of F - 1, 0.5 and 0 for nodes containing fluid, free surface boundary and air, respectively, is prepared. The sets of initial values for the nodal velocity, pressure and temperature fields in the solution domain are assumed and stored as input arrays. An array containing the boundary conditions along the external boundaries of the solution domain is prepared and stored. 

TOTAL NUMBER OF BOUNDARY CONDITIONS DIMENSIONS OF THE SOLUTION DOMAIN DEGREE OF FREEDOM PER NODE... 

MAXDF) ARRAY FOR SORTING BOUNDARY CONDITIONS TOTAL I UMBER OF F.I.EMENTS IN THE MESH (MAXBC) ARRAY FOR BOUNDARY NODES... 

NCOD ARRAY FOR IDETJTIFTCATION OF BOUNDARY NODES BC ARRAY FOR STORING BOUNDARY CONDITION VALUES... 

As an example, consider the residue curve map for the nonazeotropic mixture shown in Eigure 2. It has no distillation boundary so the mixture can be separated into pure components by either the dkect or indkect sequence (Eig. 4). In the dkect sequence the unstable node (light component, L) is taken overhead in the first column and the bottom stream is essentially a binary mixture of the intermediate, I, and heavy, H, components. In the binary I—H mixture, I has the lowest boiling temperature (an unstable node) so it is recovered as the distillate in the second column and the stable node, H, is the corresponding bottoms stream. The indkect sequence removes the stable node (heavy component) from the bottom of the first column and the overhead stream is an essentially binary L—I mixture. Then in the second column the unstable node, L, is taken overhead and I is recovered in the bottoms. 

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