Homotopy method

Wayburn, T. L., and J. D. Seader, in A. W. Westerberg and H. H. Chien (eda.), Foundations of Computer-Aided Process Design, 765, CACHE, 1983. 

Seader, J. D., Class notes from MIT course Modeling, Simulation and Optimization of Chemical Processes,1988. 

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Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer, in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate/given x. One can estimate the value of the first derivative at Xq using... 

AIChE monograph Senes, AIChE, New York, 81, No. 15 (1985)]. Homotopy methods begin from a known solution of a companion set of equations and follow a path to the desired solution of the set of equations to be solved. In most cases, the path exists and can be followed. In one implementation, the set of equations to be solved, call tf x), and the companion set of equations, call it g x), are connected together by a set of mathematical homotopy equations ... 

Christiansen et al. (54) applied the Naphtali-Sandholm method to natural gas mixtures. They replaced the equilibrium relationships and component vapor rates with the bubble-point equation and total liquid rate to get practically half the number of functions and variables [to iV(C + 2)]. By exclusively using the Soave-Redlich-Kwong equation of state, they were able to use analytical derivatives of revalues and enthalpies with respect to composition and temperature. To improve stability in the calculation, they limited the changes in the independent variables between trials to where each change did not exceed a preset maximum. There is a Naphtali-Sandholm method in the FraChem program of OLI Systems, Florham Park, New Jersey CHEMCAD of Coade Inc, of Houston, Texas PRO/II of Simulation Sciences of Fullerton, California and Distil-R of TECS Software, Houston, Texas. Variations of the Naphtali-Sandholm method are used in other methods such as the homotopy methods (Sec. 4,2.12) and the nonequilibrium methods (Sec. 4.2.13). 

The homotopy methods can be divided into two general classes, mathematical homotopies and physical or parametric homotopies. The mathematical homotopies are conventions without a physical relationship to the MESH equations and this occasionally causes problems. The physical homotopies have a basis in the MESH equations and these will be emphasized. Taylor, Wayburn, and Vickery (80) state that the physical homotopies should outperform the mathematical homotopies and are easier to implement. 

The global Newton methods, such as the Naphtali-Sandholm method (Sec. 4.2.9), are often used to solve highly nonideal systems. These are frequently prone to failure. Good explanations of the theory of homotopy methods are provided by Seader (86) and Wayburn (83). A homotopy method can greatly expand the global Newton method ability to solve difficult nonideal systems. Homotopy methods have been associated with the Naphtali-Sandholm method, where the derivatives of the if-values and enthalpies with respect to all compositions directly appear within the Jacobian. Using a thermodynamic homotopy for another method such as a Tomich has not been presented in the literature. 

The number of equations, M5C + 1), for a large number of trays and components, can be excessive. The global Newton method will suffer from the same problem of requiring initial values near the answer. This problem is aggravated with nonequilibrium models because of difficulties due to nonideal if-values and enthalpies then compounded by the addition of mass transfer coefficients to the thermodynamic properties and by the large number of equations. Taylor et al. (80) found that the number of sections of packing does not have to be great to properly model the column, and so the number of equations can be reduced. Also, since a system is seldom mass-transfer-limited in the vapor phase, the rate equations for the vapor can be eliminated. To force a solution, a combination of this technique with a homotopy method may be required. 

It is common to design and operate reasonably close to the minimum reflux or minimum boilup (Sec. 3.1.4). A computer solution at such low reflux ratios can be unstable and fail. A solution may only be reached if very good initial values are available. The technique of "sneaking up on an answer" is powerful in these cases. Initially, the column is solved at a higher reflux ratio. This solution is used as the initial value for the subsequent calculation, in which the reflux ratio is slightly lowered. This process is continued until the desired reflux ratio is reached. Other examples of how to use the solution of one simulation to initialize another simulation are described by Brierley and Smith (106). The "sneaking-up technique is part of the basis of the homotopy methods (Sec. 4.2.12) and these and other forcing techniques may also be used. 

F(x) Vector representing the difficult, desired solution in a homotopy method, Sec. 4,2.12. 

For detailed descriptions of homotopy methods in chemical engineering, see Seader [Computer Modeling of Chemical Processes, AIChE Monograph Series 15, 81, (1985)]. 

Figure 7.1 Illustration of the principle of the Fused Spheres Guided Homotopy Method (FSGH), applied for the generation of dot representations of density scalable MIDCO surfaces for the water molecule. Three families of atomic spheres (thin lines) and their envelope surfaces (heavy lines) are shown in the upper part of the figure. In the lower part of the figure, the selected point sets on the innermost family of spheres are connected by interpolating lines to the exposed points (black dots) on the envelope surfaces of two enlarged families of spheres. Linear interpolation along the lines for two selected density values leads to two families of white dots, generating approximations of two MIDCO s (heavy lines in the lower figure).

The FSGH method (Fused Sphere Guided Homotopy method) [43]. This method has been designed for the construction of approximate, density scalable ("inflatable") isodensity contour surfaces and their dot representations (i.e., for continuous transformations between different isodensity surfaces of a given molecule). 

Density Scalable Atomic Sphere (DSAS) surfaces [255]. This technique generates radii for atomic spheres for any desired electron density at the surface. The method is used for inexpensive representations of MIDCO s of large molecules, in combination with the Fused Sphere Guided Homotopy method (FSGH) [43]. 

Homotopy methods can be viewed as methods that widen the domain of conver-... 

This often gives an excellent guess of the solution for the next value of a. Sometimes even these methods do not solve the very hardest problems. In that case, a homotopy method may be necessary (Finlayson, 1990). 

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