Spatial Meshes

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. 

It is easier to work with logieally reetangular meshes beeause they map 

Data may be located at the center, face, edge, or node of the element as shown in Fig. 9.3. All variables are typically element-centered except for the velocity. The velocity is typically positioned on the element face, or node. If the velocity is positioned on the side, then the direction is commonly assumed to be normal to the face. 

Computational accuracy can be dramatically improved by dynamically adding elements where they minimize the error. For example, more elements ean be added in the neighborhood of a strong gradient in the velocity to help resolve shocks and vortex sheets. Elements may be removed from regions of smooth flow to minimize the computational cost without degrading the overall accuracy. The concept is shown in Fig. 9.6 where a finer mesh overlays the original mesh. This mesh refinement can be carried out to as many levels as necessary [15], [16], [17]. 

Accuracy can also be improved by moving the mesh. The original number 

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Solutions to models with different length scales may contain regions such as shocks, steep fronts and other near discontinuities. Adaptive meshing strategies, in which a spatial mesh network is adjusted dynamically so as to capture the local behavior accurately, will be described. The algorithm will be tested on an example of filtration combustion. 

Here M represents the number of spatial gridpoints and the spatial mesh constant... 

To alleviate further numerical problems due to sharp changes in some of the physical variables across the front and also because of the numerical difficulties presented by the Dirac delta function in the surface tension term, the interface is defined having a fixed thickness that is proportional to the spatial mesh. This allows us to smooth the functions across the interface and replace the Dirac delta function with a smoothed delta function S. In other words, we maintain a fixed thickness of the interface within the LS approach. 

Fig. 9.31. Spatial propagation of a Ce tide resembling the waves seen in hepa-tocytes, oocytes, or endothelial cells (type 2 wave). The transient pattern is obtained as in fig. 9.30 for parameter values yielding oscillations of a period of the order of 1 min Vq =-1.68 jiM/min, = 93 xM/min, - 500 p,M/min, Kf = 0.66 pM, 2 = 11 -M, k -16.8 min, A , = 1 min other parameter values are as in fig. 9.10. The spatial mesh contains 30 x 30 points (similar results are obtained with a mesh of 60 x 60 points). The black bar in the upper, left part denotes the initial, transient stimulation, which consists in raising locally the level of cytosoUc Ca to 1.5 pM at the left extremity while the rest of the cell is in the resting level of 0.1 pM the scale of Ca concentration extends from 0 (white) to 1.5 pM (black) (Dupont Goldbeter, 1992b, 1994).

C.R. Johnson and R.S. MacLeod. Nonuniform spatial mesh adaption using a posteriori error estimates applications to forward and inverse problems. Appl. Numer. Math., 14 331-326, 1994. This is a paper by the author which describes the apphcation of the h-method of mesh refinement for large scale two- and three-dimensional bioelectric field problems. 

Therefore, powerful time integrators such as LIMFX, RADAU5 or DASSL [4, 6, 10] should be used to solve these complex systems. The advantage of the whole strategy is obvious the user only provides a suitable spatial mesh, the demanding time integration of the stiff system is automatically and adaptively done by a black-box solver. This first element of adaptivity is already quite popular. 

In the second calculation set, the Su approximation is used and the spatial mesh is determined according to the criteria specified by Hunt and Coonfield." The results of these calculations are compared with experimental results in Table 1." ... 

Direct Numerical Simulations (DNS) (Fig. 12.3-1 A) The Navier-Stokes equations are solved as such, yielding the full details of micro- and macro-mixing. The reaction rates in (12.3-la), (12.3-lb), and (12.3-5b) are point values, as defined in Chapter 1. DNS requires a time-accurate calculation of the statistically stationary behavior and extremely fine temporal and spatial meshes. 

Write a general implicit program for transient compressible liquids and gases, taking constant spatial meshes for simplicity. For both flows, assume an initially hydrostatic reservoir, with the sandface suddenly exposed to a prescribed pressure level different from hydrostatic. Run the simulations to steady-state and monitor the flow rate history at the well. Show that the asymptotic results agree with the three complementary steady flow formulations amd solutions given in this chapter. 

Both the spatial meshe and the temporal step need to be analysed so that a convergence solution can be guaranteed. Selections of a spatial mesh and a temporal step... 

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