Three-dimensions numerical methods

Gresho, P. M., Lee, R. L. and Sani, R. L., 1980. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in fluids, Ch. 2, Pineridge Press, Swansea, pp. 27-75. 

Pleshanov (P4) extends the integral heat balance method to bodies symmetric in one, two, or three dimensions, using a quadratic polynomial for the approximate temperature function. Solutions are obtained in terms of modified Bessel functions which agree well with numerical finite-difference calculations. 

The principal numerical problem associated with the solution of (7) is that lengthy calculations are required to integrate several coupled nonlinear equations in three dimensions. However, models based on a fixed coordinate approach may be used to predict pollutant concentrations at all points of interest in the airshed at any time. This is in contrast to moving cell methods, wherein predictions are confined to the paths along which concentration histories are computed. 

Each of the function-minimization procedures involves some assessment of the objective function contour in parameter space. This contour is easily visualized in three dimensions, as shown in Figure 19.1(a), but the large number of adjustable parameters used in typical regressions makes graphical visualization cumbersome if not impossible. Numerical methods, such as those described in this chapter, are therefore required. 

Van kelen, H.A.M. 1980. Isotropic yield surfaces in three dimensions for use in soil mechanics. International Journal for Numerical and Analytical Methods in Geomechanics 4 98-101. 

This is Laplace s equation, which describes potential flow. It is widely used in heat flow and electrostatic field problems an enormous number of solutions to Laplace s equation are known for various geometries. These can be used to predict the two-dimensional flow in oil fields, underground water flow, etc. The same method can be used in three dimensions, but solutions are more difficult. The solutions to the two-dimensional Laplace equation for common problems in petroleum reservoir engineering are summarized by Muskat [3]. The analogous solutions for groundwater flow are shown in the numerous texts on hydrology, e.g., Todd [4]. See Chap. 10 for more on potential flow. 

The spectrum of CFD is as broad as the applications of the Navier-Stokes equations are. At the one end one can purchase design packages for pipe systems that solve problems in a few seconds on personal computers, on the other hand there are codes - and physical problems - that may require days on large computers. We shall be concerned with three methods designed to solve a large range of fluid motion in two or three dimensions. Two of them are commonly used in commercial CFD-codes. Before we specify these methods, we will shortly summarize the request for an approximate numerical solution. 

The MAC method, which allows arbitrary free surface flows to be simulated, is widely used and can be readily extended to three dimensions. Its drawback lies in the fact that it is computationally demanding to trace a large number of particles, especially in 3D simulation. In addition, it may result in some regions void of particles because the density of particles is finite. The impact of the MAC method is much beyond its interface capmring scheme. The staggered mesh layout and other features of MAC have become a standard model for many other Eulerian codes (even numerical techniques involving mono-phase flows). 

At present there are several methods of filler structure (distribution) determination in polymer matrix, both experimental [10, 35] and theoretical [4]. All the indicated methods describe this distribution by fractal dimension of filler particles network. However, correct determination of any object fractal (Hausdorff) dimension includes three obligatory conditions. The first from them is the indicated above determination of fiiactal dimension numerical magnitude, which should not be equal to object topological dimension. As it is known [36], any real (physical) fractal possesses fiiactal properties within a certain scales range. Therefore, the second condition is the evidence of object self-similarity in this scales range [37]. And at last, the third condition is the correct choice of measurement scales range itself As it has been shown in Refs. [38, 39], the minimum range should exceed at any rate one self-similarity iteration. 

Greengard L, Rokhlin V (1997) A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer 6 229-269... 

Accounting for more active variables expands the feasible set in additional dimensions. For example, expanding the dimensionality of the analysis to include an additional active variable, X34, pre-exponential factor of reaction 0H + CH3-> CH2 + H20, reveals more. X34 is the second highest impact parameter for 55 and the fourth highest for 57. The three-dimensional feasible set is shown in Fig. 7. A cross-section of this feasible set by the plane X34 = 0 forms the darker shaded area in Fig. 6. With the addition of experiments and active variables, the geometry of the feasible set grows in complexity and is difficult to visualize. This necessitates developing theoretical and numerical methods to accurately account for known constraints (usually implied by data) in nonlinear and hybrid models. This is precisely the capability of the numerical methods described next. 

A, R. Ingraffea and C. Manu, "Stress-intensity factor computation In three dimensions with quarter-point eiements," Int. J. Numer. Methods Engrg. 15(10), 1427-1445... 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

Summary. Optimization of the three-dimensional pore structure of a hydrodemetallation catalyst will be described. A random network model with different connectivities has been used. The influence of connectivity, diffusion coefficient, outer dimension of pellet and operating time on optimcd pore structure has been investigated. Numerical methods employed will be discussed. 

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