Explicit Finite-Difference Formulation

As a simple illustration, we wish to develop the unsteady one-dimensional finite-difference formulation for a flat plate of thickness i having an initial temperature difference AT = 7i — r2 between its surfaces. Assume the surface of the plate with temperature T is suddenly insulated. 

Applying the first law of thermodynamics to the system shown in Fig. 4.20 (a uniform grid is assumed), and relating the result to temperatures by means of the Fourier law, we obtain 

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Note that the left side of this equation is simply the fiiiile difference formulation of the problem for the steady case. This is not surprising since the formulation must reduce to the steady case for = Tj,. Also, we are still not committed to explicit or implicit formulation since we did not indicate the time step on the left side of the equation. Wc now obtain the explicit finite difference formulation by expressing the left side at time step i as... 

Schematic for the explicit finite difference formulation of the convection condition at the left boundary of a plane wall.

Noie that in the case of no heal generation and t = 0.5, the explicit finite difference formulation for a general interior node reduces to T , = (T/,-1 +, )/2, which has the interesting interpretation that the temperature... 

To gain a better understanding of the stability criterion, consider the explicit finite difference formulation for an interior node of a plane wall (Eq. 5 47) for the case of no heat generation,... 

The exterior surface of the Trombe v/ail is subjected to convection as well as to heat flux. The explicit finite difference formulation at that boundary is obtained hy writing an energy balance on the volume element represented by node 5,... 

C 1 he explicit finite difference formulation of a general interior node for transient two-dimensional heat conduction is given by... 

Consider transient heat conduction in a plane wall with variable heal generation and constant thermal conductivity. The nodal network of (he medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of A.r. The wall is initially at a specified temperaWre. The temperature at the right bound ary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary... 

The finite difference formulation of transieiii heat conduction problems is based on an energy balance that also accounts for tire variation of the energy content of the volume element during a time interval At. The heat transfer and heat generation terms are expressed at the previous time. step fin the explicit method, and at the new time step i I 1 in the implicit method. For a general node III, the finite difference formulations are expressed as... 

Glass et al. [7] reported that MacCormack s method [22], which is a second-order accurate explicit scheme, can handle these moving discontinuities quite well and is valid for the HHC problems. Since the hyperbolic problems considered here have step discontinuities at the thermal wave front, MacCormack s prediction-correction scheme is used in the present study. When MacCormack s method is applied to Eqs. (2.7) and (2.8), the following finite difference formulation results ... 

So far, we have considered the explicit scheme of finite-difference formulations and its stability criterion for an illustrative example. The use of the explicit scheme becomes somewhat cumbersome when a rather small Ax is selected to eliminate the truncation error for accuracy. The Ai allowed then by the stability criterion may be so small that an enormous amount of calculations may be required. We now intend to eliminate this difficulty by giving different forms to the equations resulting from the finite-difference formulation. Let us take the case of one-dimensional conduction in unsteady problems, for which we obtained the difference equation given by Eq. (4.50). Consider a formulation of the problem in terms of backward rather than forward differences in time. That is, decrease the time from f +i = (n + l)Ai to tn = nAt. Thus we obtain... 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

In this section, an explicit time advance scheme for unsteady flow problems is outlined [30]. The momentum equation is discretized by an explicit scheme, and a Poisson equation is solved for the pressure to enforce continuity. The continuity is discretized in an implicit manner. In the original formulation, the spatial derivatives were approximated by finite difference schemes. 

The GC distribution has been used here avoiding the concept of temperature but explicitly based on the electronic information. This description has been recognized adequate to introduce the S-R interactions on the descriptors and permits by the means of a charge-dependent interaction potential to overcome the problem of the discontinuities in the derivatives of the energy. This treatment recovers the piecewise dependence when the interaction vanishes, i.e., Uv 0, as expected. This formulation, as has been pointed out, is more realistic than evaluating the descriptors in isolated systems by finite difference methods, hi conclusion, the GC distribution for open molecular domains enables to introduce statistical concepts to describe electron distributions in the molecular structure even they are few body systems. 

In many books, radial flow theory is studied superficially and dismissed after cursory derivation of the log r pressure solution. Here we will consider single-phase radial flow in detail. We will examine analytical formulations that are possible in various physical limits, for different types of liquids and gases, and develop efficient models for time and cost-effective solutions. Steady-state flows of constant density liquids and compressible gases can be solved analytically, and these are considered first. In Examples 6-1 to 6-3, different formulations are presented, solved, and discussed the results are useful in formation evaluation and drilling applications. Then, we introduce finite difference methods for steady and transient flows in a natural, informal, hands-on way, and combine the resulting algorithms with analytical results to provide the foundation for a powerful write it yourself radial flow simulator. Concepts such as explicit versus implicit schemes, von Neumann stability, and truncation error are discussed in a self-contained exposition. 

Other works used, instead of spectral, lower order accuracy finite difference approximations, but they have to be noted here since they employed more suitable numerical formulations for the constitutive equations that explicitly avoided the introduction of artificial diffusivity in the numerical solution [61, 63-66]. Those works also employed the FENE-P model to simulate dilute polymer solutions [61, 63, 64] or the Giesekus model for surfactant turbulent fiow [19, 65, 66[. 

The large-dimension limit has recently resolved at least some of the difficulties of the molecular model. The molecule-like structure falls out quite naturally from the rigid bent triatomic Lewis configuration obtained in the limit D — oo [5], and the Langmuir vibrations at finite D can be analyzed in terms of normal modes, which provide a set of approximate quantum numbers [6,7]. These results are obtained directly from the Schrodinger equation, in contrast to the phenomenological basis of some of the earlier studies. When coupled with an analysis of the rotations of the Lewis structure, this approach provides an excellent alternative classification scheme for the doubly-excited spectrum [8]. Furthermore, an analysis [7] of the normal modes offers a simple explanation of the connection between the explicitly molecular approaches of Herrick and of Briggs on the one hand, and the hyperspherical approach, which is rather different in its formulation and basic philosophy. 

An explicit mathematical formulation to the finite-parameter approach to the local bifurcations was given by Arnold [19], based on the notion of versal families. Roughly speaking, versality is a kind of structural stability of the family in the space of families of dynamical systems. Different versions of such stability are discussed in detail in [97]. 

The procedure to calculate fiber orientation is the same as explained above, but their implementation into explicit solvers and non-linear material models is more complex than it is for quasi-static load-cases and purely elastic material models. The fiber orientation is characterized by a so called orientation distribution function (ODE) that describes the chance of a fiber being oriented into a certain direction. For isotropic, elastic matrix materials an integral of the individual stiffness in every possible direction weighted with the ODE provides the complete information about the anisotropic stiffness of the compound. However, this integral can not be solved in case of plastic deformation as needed for crash-simulation. Therefore it is necessary to approximate and reconstruct the full information of the ODE by a sum of finite, discrete directions with their stiffness, so called grains [10]. Currently these grains are implemented into a material description and different methods of formulation are tested. 

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