Nonlinear equations Newton-Raphson iteration

The NONLIN module is responsible for intializing the concentration vector, C(t), for l i NRCT. Here NRCT is the number of reactants. If there are no equilibrium reactions, then C i) is set to IC i), the initial concentration vector, for 1 < f < NRCT. If equilibrium reactions do exist, then the type (2) equations (with derivatives set to zero) and the Type (1) and Type (3) equations are all solved simultaneously for the equilibrium concentrations of all reactants. Because the equilibrium equations are generally nonlinear, the Newton-Raphson iteration method is used to solve these equations. Also, since there is no symbol manipulation capability in the current version of CRAMS, numerical differentiation is used to calculate the required partial derivatives. That is, the rate expressions cannot at this time be automatically differentiated by analytical methods. A three point differentiation formula is used 27) ... 

Geochemists, however, seem to have reached a consensus (e.g., Karpov and Kaz min, 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 1983) that Newton-Raphson iteration is the most powerful and reliable approach, especially in systems where mass is distributed over minerals as well as dissolved species. In this chapter, we consider the special difficulties posed by the nonlinear forms of the governing equations and discuss how the Newton-Raphson method can be used in geochemical modeling to solve the equations rapidly and reliably. 

Fig. 4.2. Newton-Raphson iteration for solving two nonlinear equations containing the unknown variables x and y. Planes are drawn tangent to the residual functions R and R2 at an initial estimate (r, > (o)) to the value of the root. The improved guess (v(l y(l)) is the point at which the tangent planes intersect each other and the plane R = 0.

The multidimensional counterpart to Newton s method is Newton-Raphson iteration. A mathematics professor once complained to me, with apparent sincerity, that he could visualize surfaces in no more than twelve dimensions. My perspective on hyperspace is less incisive, as perhaps is the reader s, so we will consider first a system of two nonlinear equations / = a and g = b with unknowns, v and y. 

For the simulation of the reactor behaviour the system of ordinary differential equations was integrated by means of a Runge-Kutta-Merson method with variable step length, whereas the nonlinear algebraic equations were solved by a Newton-Raphson iteration. 

Eqn (23) is a second order nonlinear difference equation the Jacobian of which is easily established as a regular tridiagonal matrix with a dominating diagonal, similar to system matrices found in the simulation of distillation columns. The analytical derivation of the Jacobian and the Newton-Raphson iteration is trivial. In figure 3 is shown an example where the intermediate pressures are plotted as functions of the total pressure drop across the line segment. The example is artificially chosen such that all e-parameters are the same, i.e. ... 

The program LSV4IRC is a simulation of a reversible reaction with input values of p (dimensionless uncompensated resistance and qc (dimensionless double layer capacity). Unequal intervals are used, with asymmetric 4-point second spatial derivatives, and second order extrapolation in the time direction. The nonlinear set of 6 equations for the boundary values is solved by Newton-Raphson iteration. Some results are seen in Chap. 11. 

Mass balance of solid Mass balance of water Mass balance of air Momentum balance for the medium Internal energy balance for the medium The resulting system of Partial Differential Equations is solved numerically. Finite element method is used for the spatial discretization while finite differences are used for the temporal discretization. The discretization in time is linear and the implicit scheme uses two intermediate points, t and t between the initial 1 and final t limes. Finally, since the problems are nonlinear, the Newton-Raphson method has been adopted following an iterative scheme. 

Computational method and estimation of parameters. The system of three differential equations which pre-sents the design model is nonlinear and subject to boundary conditions. For solving numerically the model equations the method of orthogonal collocation was used (90, 91). As collocation functions the so-called shifted Legendre polymonials were applied. As a rule the collocation was done for 5 inner points. The lumped equations were solved by means of the Newton-Raphson iteration method. 

The main advantage of the implicit techniques is their stability for any given value of the step size. This advantage, however, requires the solution of a set of nonlinear equations via an iterative approach. For this purpose, methods such as successive substitution or Newton-Raphson can be used [1]. 

The boundary conditions that may be simulated with the flow-compaction module are autoclave pressure, impermeability or permeability with prescribed bag pressure, no displacement or no normal displacement (tangent sliding condition). The governing equations (Eqs [13.3] and [13.4]) are coupled during individual time-steps of the transient solution. A Newton-Raphson iterative procedure is used to solve the resulting nonlinear system of equations. Details in the solution of the flow-compaction for autoclave processing can be found in reference 17. 

Equation (49) contains two possible sources of nonlinearities material nonlinearity due to Schapery s law, and geometric nonlinearity arising from the large displacement (and small strain) formulation. In order to obtain a solution to this nonlinear equation at any time step, the Newton-Raphson iterative technique is used. The incremental displacement Aw obtained at the end of the iteration is used to update the total displacement for the time step,... 

The mathematical structure of the problem is thus a set of partial differential equations which are coupled with a set of nonlinear algebraic equations at each axial position. The algebraic equations are solved using a Newton-Raphson iteration with the derivatives calculated numerically. The same reaction rate expression (P2)was used. 

The values of the functions Uf, V ,and Wf have to be calculated at discrete points and 10 elements with 6 collocation points per element were found to suffice. Rather than solve (15) and (16) separately it was found best to use collocation on these also, using four radial collocation points. This gives 459 simultaneous nonlinear algebraic equations which were solved by a modified Newton-Raphson Iteration. The modifications were the use of under-relaxation and less frequent evaluation of the Jacobian. 

The kinetic equations such as Equation 7.37, Equation 7.38, Equation 7.40, and Equation 7.41 are known as transcendental equations, whose direct solution cannot be obtained. Such a kinetic equation is generally solved by the use of approximation techniques such as Newton-Raphson iterative method and nonlinear least-squares method. But, these methods have limitations of a different nature. For instance, the nonlinear least-squares method, which is most commonly used in such kinetic studies, tends to provide less reliable values of calculated kinetic parameters with increase in the number of such parameters. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

This represents a set of nonlinear algebraic equations that can he solved with the Newton-Raphson method. However, in this case, a viable iterative strategy is to evaluate the transport coefficients at the last value and then solve... 

The unknowns in Eq. (38) are p and 2n. These are found by demanding hm to obey the image constraint equations (18) and normalization (3). Because the unknowns enter in a nonlinear way, the resulting M + 1 equations were solved by an iterative technique—Newton-Raphson relaxation (see, e.g., Hildebrand, 1956). Empirical cases are studied in Section XI. 

Of course, the roots of Eq. (46a) may easily be found. The unknown quantities fi and 2m may be found by substituting hm from Eq. (46a) into the imaging equations (18) and normalization equation (3). Because of the nonlinear nature of these equations, a Newton-Raphson (or other iterative) method of solution would be necessary. 

SC (simultaneous correction) method. The MESH equations are reduced to a set of N(2C +1) nonlinear equations in the mass flow rates of liquid components ltJ and vapor components and the temperatures 2J. The enthalpies and equilibrium constants Kg are determined by the primary variables lijt vtj, and Tf. The nonlinear equations are solved by the Newton-Raphson method. A convergence criterion is made up of deviations from material, equilibrium, and enthalpy balances simultaneously, and corrections for the next iterations are made automatically. The method is applicable to distillation, absorption and stripping in single and multiple columns. The calculation flowsketch is in Figure 13.19. A brief description of the method also will be given. The availability of computer programs in the open literature was cited earlier in this section. 

The resulting C + E equations are nonlinear in unknowns nj, nj, and tt but In nj are iteration variables since nj occur in logarithmic terms. These equations are linearized using first-order Taylor Series (Newton-Raphson method), in the variables An , A (In nj), and ir, and with n nj are reduced to S + 1 + E linear equations in unknowns AN, A (In N), and tt. When extended to include P mixed phases, we nave shown that they are nearly identical to the equations of the RAND Method and have the same coefficient matrix. 

The equilibrium configuration of the surface region comprising n layers is determined by solving simultaneously the 4n equations obtained by equating to zero the partial derivatives of AU with respect to each of the variables. The equations so obtained are nonlinear and are solved by an iterative Newton-Raphson procedure (12), which necessitates calculating the second partial derivatives of AU with respect to all possible pairs of variables. A Bendix G15D computer was used for all numerical computations—i.e., evaluation of the various lattice sums, calculation of the derivatives of AU, and solution of the linearized forms in the Newton-Raphson treatment. 

Often the melting point and the heat of fusion at the melting point are used as estimates of T and A Hi. It should be noted that the latter equation is nonlinear, since y- on the right-hand side is a function of x . Hence the determination of x calls for an iterative numerical procedure, such as the Newton-Raphson or the secant methods. 

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