Discrete Formulations for Solution

The various attempts in the literature to cast the above expression entirely in terms of and Nj belong to either of the following two categories. 

17 This discussion originates from a paper of Kumar and Ramkrishna (1996a). 

Substitution of (4.5.8) into (4.5.6) will yield the right-hand side of (4.5.7). The pivots and Xj must depend on the frequency function a v, v ) as well as the number densities in the two intervals so that they must strictly be regarded as time dependent in a dynamic problem. Furthermore, the pivots would not remain the same for all the terms in the functional v, t]. These observations provide further 

Some latitude exists here in the choice of the average value withdrawn from the integral so that slightly different forms of (4.5.9) can also be envisaged. In this approach, one is then required to calculate the integral of the aggregation frequency at each step. 

In either of the preceding categories, since the integrand contains the unknown number density, the mathematically rigorous choice for the pivot, which is consistent with the mean value theorem is of course not accessible. The finer the interval, the less crucial would be the location of the pivot in /f. The fineness required would depend on the extent to which the phenomenological functions of the population balance model such as the aggregation and breakage functions vary in the interval. 

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On a lattice, so-called crankshaft moves are trivial implementations of concerted rotations [77]. They have been generalized to the off-lattice case [78] for a simplified protein model. For concerted rotation algorithms that allow conformational changes in the entire stretch, a discrete space of solutions arises when the number of constraints is exactly matched to the available degrees of freedom. The much-cited work by Go and Scheraga [79] formulates the loop-closure problem as a set of algebraic equations for six unknowns reducible... 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

In the present paper the Boundary Finite Element Method is presented as a boundary discretization method for the numerical investigation of interfacial stress concentrations in composite laminates. In contrast to the classical boundary element method, the element formulation is finite element based, which avoids the necessity of a fundamental solution. Comparative results from finite element calculations show good agreement both for the laminate free-edge effect and for the example of the stress concentrations near cracks in composite laminates. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

There are a number of approaches to discrete formulation some are illustrated here in terms of a steady one-dimensional fin problem (recall Ex. 2.9). For an infinitely long fin, with a specified base temperature To, transferring heat with a coefficient h to an ambient at temperature T00, an exact solution for temperature is... 

S2 and S4 Models for Cylindrical Geometries. For the solution of the RTE in cylindrical media, the formulations for the S2 and S4 discrete ordinates (DO) approximations (based on Ref. 69) will be presented here. Note that, in a more recent study, Jendoubi et al. [75] used a similar DO approximation in cylindrical geometry and evaluated the effect of anisotropic phase function on the accuracy of the model. 

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

When a discrete optimization problem is in the polynomially solvable complexity category, it is usually clear how to proceed with its analysis. A clever and efficient algorithm is at hand. Often an exact Unear programming formulation is also known, and a strong duality theory is available for sensitivity studies. Very complete analysis should be possible, unless limits on the time available for solution (e.g., in a real-time setting) mandate quicker methods. 

Almost all contributions to MIRSP use a time-continuous formulation and consider constant consumption and/or production rates for the products at the sites. One exception is Persson and Gothe-Lundgren (2005) using a time-discrete formulation. Moreover, production planning decisions are incorporated besides routing and inventory management. The second exception propose Christiansen et al. (2011) where a solution procedure for a MIRSP with time-varying consumption rates is described (but no mathematical model). 

Expressions (173) and (174) provide efficient formulations for the solution of deep-penetration problems whose geometrical irregularity is in the vicinity of the detector. The unperturbed flux distribution is calculated with a low-order, low-dimensional calculational method (such as a one-dimensional discrete-ordinates method). The irregularity in the geometry is treated as an alteration to the unperturbed system. The 5 or distribution is then obtained from the solution of Eq. (44a) or Eq. (163), respectively, for the local region of the detector and the alteration. High-order and multidimensional calculational methods (such as Monte Carlo) can be used for this local solution. Equation (173) provides an efficient formulation for the calculation of the response of many different detectors (such as different reaction rates and the spatial distribution of a given reaction rate) in a... 

Curran DAS, Cross MM, Lewis BA (1980) Solution of parabolic differential equations by the boundary element method using discretization in time. Appl Math Model 4 398-400 Dai SC, Qi F, Tanner RI (2006) Strain and strain-rate formulation for flow-induced crystallization. Polym Eng Sci 46 659-669... 

A typical sequence of steps is presented below. For each step, the current objective value (in discrete space) and a measure of its infeasibility (shortfall in demands met in m /s) is obtained and these are collated in Table 1. Due to the highly constrained nature of the discrete formulation, an exactly feasible solution is unlikely to be achieved. However, the aim is not so much to solve the problem directly with the visualization tool but to provide good initial solutions for the rigorous optimization procedure. 

This paper describes a method for solving the problem of elastohydrodynamic lubrication in elliptical contacts which Includes thermal effects. The technique is based on discrete formulations and computer solutions of the fluid flow, the solid elastic deflections and the heat transfer within both the fluid and the bounding solids. Detailed fluid pressure, film thickness and temperature solutions are reported for several conditions. The results clearly Indicate the departure from isothermal conditions which exists as the entrainment velocity, and particularly the sliding velocity, are Increased. The inclusion of thermal effects decreases the film thickness and the magnitude of the pressure spike by significant amounts... 

The differential equations presented in Section 5-2 describe the continuous movement of a fluid in space and time. To be able to solve those equations numerically, all aspects of the process need to be discretized, or changed from a continuous to a discontinuous formulation. For example, the region where the fluid flows needs to be described by a series of connected control volumes, or computational cells. The equations themselves need to be written in an algebraic form. Advancement in time and space needs to be described by small, finite steps rather than the infinitesimal steps that are so familiar to students of calculus. All of these processes are collectively referred to as discretization. In this section, disaetiza-tion of the domain, or grid generation, and discretization of the equations are described. A section on solution methods and one on parallel processing are also included. 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

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