Solution algorithms

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

Nichols, B. D., Hirt, C. W. and Hitchkiss, R. S., 1980. SOLA-VOF a solution algorithm for transient fluid flow with multiple free surface boundaries. Los Alamos Scientific Laboratories Report No. La-8355, Los Alamos, NM. 

The derived working equations are solved using the following solution algorithm ... 

The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

SOLUTION ALGORITHMS BASED ON THE GAUSSIAN ELIMINATION METHOD... 

SOLUTION ALGORITHMS GAUSSIAN ELIMINATION METHOD 205 6.4.2 Frontal solution technique... 

Validation and Application. VaUdated CFD examples are emerging (30) as are examples of limitations and misappHcations (31). ReaUsm depends on the adequacy of the physical and chemical representations, the scale of resolution for the appHcation, numerical accuracy of the solution algorithms, and skills appHed in execution. Data are available on performance characteristics of industrial furnaces and gas turbines systems operating with turbulent diffusion flames have been studied for simple two-dimensional geometries and selected conditions (32). Turbulent diffusion flames are produced when fuel and air are injected separately into the reactor. Second-order and infinitely fast reactions coupled with mixing have been analyzed with the k—Z model to describe the macromixing process. 

P. Colella and H.M. Glaz, Efficient Solution Algorithms for the Riemann Problem for Real Gasses, J. Comput. Phys. 59 (1985). 

Hunt and Kulmala have solved the full turbulent fluid flow for the Aaberg system using the k-e turbulent model or a variation of it as described in Chapter 13— the solution algorithm SIMPLE, the QUICK scheme, etc. Both commercial software and in-house-developed codes have been employed, and all the investigators have produced very similar findings. 

Cloutman, L. D., C. W. Hirt, and N. C. Romero. 1976. SOLA-ICE a numerical solution algorithm for transient compressible fluid flows. Los Alamos Scientific Laboratory report LA-6236. 

The analysis of NNs has shown that, in order to assure both accuracy and smoothness for the approximating function, the solution algorithm will have to allow the model (essentially its size) to evolve dynamically with the... 

Schilling, G., Pantehdes, C.C., 1996. A simple continuous-time process scheduling formulation and a novel solution algorithm. Comput. Chem. Eng., 20(Suppl.) S1221-1226 Umeda, T., Harada, T., Shiroko, K., 1979. A thermodynamic approach to the synthesis of heat integration systems in chemical processes. Comput. Chem. Eng., 3 273-282 Wang, Y.P., Smith, R., 1994. Wastewater minimization. Chem. Eng. Sci., 49(7) 981-1002... 

Schilling, G., Pantelides, C.C., 1996. A simple continuous-time process scheduling formulation and a novel solution algorithm. Comput. Chem. Eng., 20(Suppl.) S1221-S1226. 

The models developed to take the PIS operational philosophy into account are detailed in this chapter. The models are based on the SSN and continuous time model developed by Majozi and Zhu (2001), as such their model is presented in full. Following this the additional constraints required to take the PIS operational philosophy into account are presented, after which, the necessary changes to constraints developed by Majozi and Zhu (2001) are presented. In order to test the scheduling implications of the developed model, two solution algorithms are developed and applied to an illustrative example. The final subsection of the chapter details the use of the PIS operational philosophy as the basis of operation to design batch facilities. This model is then applied to an illustrative example. All models were solved on an Intel Core 2 CPU, T7200 2 GHz processor with 1 GB of RAM, unless specifically stated. 

The two sets of constraints presented in Sections 3.1 and 3.2 constitute an overall mathematical model, which is used in the proposed two-stage solution algorithm. 

As in the previous case the solution procedure described in Sect. 4.4 was used to solve the example. The resulting models were formulated in GAMS 22.0, as with the previous case. CPLEX 9.1.2 was used to solve the MILP and the DICOPT2 solution algorithm was used to solve the exact MINLP. In the DICOPT solution algorithm, CLPEX 9.1.2 was the MIP solver and CONOPT3 the NLP solver. The same processor as the previous example was used to find a solution. 

The example was formulated in GAMS 22.0 and solved using the DICOPT2 solution algorithm, with CPLEX 9.1.2 as the MIP solver and CONOPT3 as the NLP solver. The model was solved using a Pentium 4 3.2 GHz processor and required 16.8 CPU seconds to find a solution. DICOPT did 4 major iterations to find the final solution. The optimal number of time points was 8, which resulted in 192 binary variables for the model. 

The model for the second illustrative example was formulated in GAMS 22.0 and solved using the DICOPT2 solution algorithm. The MIP solver used was CPLEX... 

As mentioned previously the objective was the minimisation of the required storage vessel while still achieving the minimum wastewater target. This was done using the solution algorithm discussed earlier. 

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

As there is no simple algorithm which can generate a solution just in one step several algorithms have to be combined [7-10]. Those algorithms which may improve the solution are called operators. The overall combinatorial solution algorithm consists of many operators. Any operator may work in any combination on the solution. Some examples are ... 

The mathematical model of two-stage stochastic mixed-integer linear optimization problems was discussed as well as state-of-the-art solution algorithms. A new hybrid evolutionary algorithm for solving this class of optimization problems was presented. The new algorithm exploits the specific problem structure by stage decomposition. 

The differences between the gas-phase and solution algorithms appear from this point on. To derive equation 3.3, the perfect gas mixture was assumed, and A related to an equilibrium constant given in terms of the partial pressures of the reactants and the activated complex [1], This Kp is then easily connected with A H° and A .S ". As stated, the perfect gas model is a good assumption for handling the results of the large majority of gas-phase kinetic experiments. 

The extension of the simple ideal binary system considered in the preceding section to a nonideal multicomponent column is not diflicult. The only changes that have to be made to the basic structure of the solution algorithm arc ... 

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