Jacobian method

GENERAL SOLUTION FOR TWO DEGREES OF FREEDOM AND RELATIONSHIP TO JACOBIAN METHODS... 

Another possibility would be to use a multirate code which integrates each differential equation individually using different step sizes. Orailoglu (16) and Gear (17) discuss this approach but their procedure uses multistep Jacobian methods which are not efficient for our system of equations. What we need is a fixed order single step multirate method. 

A very useful method for determining thermodynamic relations is the Jacobian method (cf. Callen, 1960). Let us apply this method to show that... 

The Jacobian method for determining the identities is presented in Appendix 10.B. j is the Laplace variable and should not be confused with the entropy density. 

Pinkerton, R.C. A Jacobian method for the rapid evaluation of thermodynamic derivatives, without the use of tables. 1. Phys. Chem. 56, 799-800 (1952)... 

Note that the transformed representation is a product of Maxwellian distributions of the center-of-mass and relative kinetic energy. To complete the transformation of (2.14) requires expressing dci dc in terms of the new coordinates. The result, obtained by the Jacobian method outlined in the Appendix, is dc rfcg = d M dc so that the integral in (2.14) is the product... 

Use the Jacobian method to show that the differential volume element in spherical polar coordinates is sin 6 dc dd d(p. 

Use the Jacobian method to prove the statement made in Section 2.4, dcidcz = dCMdc the coordinate transformation is given by (2.15). 

Newton-Raphson method (or any or several variants to it) is used to solve the equations, the jacobian matrix and its LU fac tors are... 

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

Partial derivatives, as introduced in Section 2.12 are of particular importance in thermodynamics. The various state functions, whose differentials are exact (see Section 3.5), are related via approximately 1010 expressions involving 720 first partial derivatives Although some of these relations are not of practical interest, many are. It is therefore useful to develop a systematic method of deriving them. Hie method of Jacobians is certainly the most widely applied to the solution of this problem. It will be only briefly described here. For a more advanced treatment of the subject and its application to thermodynamics, die reader is referred to specialized texts. 

Unlike the other alternative methods, analytical expressions of partial derivatives are required and the Jacobian must be evaluated in the Newton-Raphson method. These requirements sometimes prove to be the undoing when the method is applied to complicated equations. Brown (B12) has developed a modification to the Newton-Raphson method, which requires only some of the partial derivatives to be calculated. We have tested Brown s method on our sample problems but have found that it actually required more computing time than the unmodified Newton-Raphson method. 

In this method the inverse of the Jacobian is approximated by — H and the successive approximations are generated according to the following formula ... 

The advantage of this method is that it avoids both the evaluation of partial derivatives and the inversion of the Jacobian. To start the iterations, an initial estimate H0 is also required an identity matrix is frequently used for this purpose. 

Unlike the previous method Wolfe s method requires the inversion of an (n + 1) x (n + 1) matrix at each iteration, but a short-cut procedure may be used to take advantage of the fact that at each iteration only one column of this matrix is modified. Another disadvantage of this method is that it cannot use the information from a previous case (e.g., the Jacobian) to obtain a better starting point. To the best of our knowledge there is no demonstrable advantage to recommend the use of this method in pipeline network problems. 

If the Newton-Raphson method is used to solve Eq. (1), the Jacobian matrix (df/3x)u is already available. The computation of the sensitivity matrix amounts to solving the same Eq. (59) with m different right-hand side vectors which form the columns — (3f/<5u)x. Notice that only the partial derivatives with respect to those external variables subject to actual changes in values need be included in the m right-hand sides. 

Algorithmic Details for NLP Methods All the above NLP methods incorporate concepts from the Newton-Raphson method for equation solving. Essential features of these methods are that they rovide (1) accurate derivative information to solve for the KKT con-itions, (2) stabilization strategies to promote convergence of the Newton-like method from poor starting points, and (3) regularization of the Jacobian matrix in Newton s method (the so-called KKT matrix) if it becomes singular or ill-conditioned. 

A more efficient way of solving the DFT equations is via a Newton-Raphson (NR) procedure as outlined here for a fluid between two surfaces. In this case one starts with an initial guess for the density profile. The self-consistent fields are then calculated and the next guess for density profile is obtained through a single-chain simulation. The difference from the Picard iteration method is that an NR procedure is used to estimate the new guess from the density profile from the old one and the one monitored in the single-chain simulation. This requires the computation of a Jacobian matrix in the course of the simulation, as described below. 

Like Newton s method, the Newton-Raphson procedure has just a few steps. Given an estimate of the root to a system of equations, we calculate the residual for each equation. We check to see if each residual is negligibly small. If not, we calculate the Jacobian matrix and solve the linear Equation 4.19 for the correction vector. We update the estimated root with the correction vector,... 

In this section we consider how Newton-Raphson iteration can be applied to solve the governing equations listed in Section 4.1. There are three steps to setting up the iteration (1) reducing the complexity of the problem by reserving the equations that can be solved linearly, (2) computing the residuals, and (3) calculating the Jacobian matrix. Because reserving the equations with linear solutions reduces the number of basis entries carried in the iteration, the solution technique described here is known as the reduced basis method. ... 

Fig. 4.4. Comparison of the computing effort, expressed in thousands of floating point operations (Aflop), required to factor the Jacobian matrix for a 20-component system (Nc = 20) during a Newton-Raphson iteration. For a technique that carries a nonlinear variable for each chemical component and each mineral in the system (top line), the computing effort increases as the number of minerals increases. For the reduced basis method (bottom line), however, less computing effort is required as the number of minerals increases.

Such a scheme is sometimes called a soft Newton-Raphson formulation because the partial derivatives in the Jacobian matrix are incomplete. We could, in principle, use a hard formulation in which the Jacobian accounts for the devia-tives dy/dm,i and daw/dm,i. The hard formulation sometimes converges in fewer iterations, but in tests, the advantage was more than offset by the extra effort in computing the Jacobian. The soft method also allows us to keep the method for calculating activity coefficients (see Chapter 8) separate from the Newton-Raphson formulation, which simplifies programming. 

Another way to calculate the partial derivatives is possible. Figure 15.12 represents a typical module. If a module is simulated individually rather than in sequence after each unknown input variable is perturbed by a small amount, to calculate the Jacobian matrix, (C + 2)nci + ndi + 1 simulations will be required for the ith module, where nci = number of interconnecting streams to module i and ndi = number of unspecified equipment parameters for module /. This method of calculation of the Jacobian matrix is usually referred to as full-block perturbation. 

Note that the Jacobian matrix dh/dx on the left-hand side of Equation (A.26) is analogous to A in Equation (A.20), and Ax is analogous to x. To compute the correction vector Ax, dh/dx must be nonsingular. However, there is no guarantee even then that Newton s method will converge to an x that satisfies h(x) = 0. 

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