SOLVER, nonlinear analysis

Contact conditions add even more difficulty and complexity to an already very complex and difficult analysis of rubber products and tires. Contact conditions are unilateral and need to be constantly checked during the incremental nonlinear analysis. In addition, they are not smooth, thus degrading the performance of nonlinear solvers. A number of numerical regularization parameters need to be introduced to prevent chattering and ensure robustness of a finite element analysis (FEA) with frictional contact. 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

XPPAUT (http //www.math.pitt.edu/ bard/xpp/xpp.html) offers deterministic simulations with a set of very good stiff solvers. It also offers fitting, stability analysis, nonlinear systems analysis, and time-series analysis, like histograms. The GUI is simple. It is mainly available under Linux, but also mns on Windows. 

An Excel spreadsheet illustrating the use of the Solver tool for nonlinear least-squares analysis of a fluorescent decay curve of a ruby crystal. The sum of the squares of residuals is calculated in cell C14 and is minimized in Solver by iterative variation of the parameters in cells CIO, Cll, and C12. 

The set of n equations of the type shown in Equation (B.4.3) needs to be solved. This set of equations is nonlinear if /(x,, a) is nonlinear. Thus, the solution of this set of equations requires a nonlinear algebraic equation solver. These are readily available. For information on the type of solution, consult any text on numerical analysis. Since the solution involves a set nonlinear algebraic equation, it is performed by an iterative process. That is, initial guesses for the parameters a are required. Often, the solution will terminate at local minimum rather than the global minimum. Thus, numerous initial guesses should be used to assure that the final solution is independent of the initial guess. 

SolvStat.xls is a command macro that returns the standard deviations for nonlinear regression analysis performed by the Solver. See "Instructions for Using SolvStat" at the end of this appendix. 

Effort to entire the capability of the general purpose nonlinear stmctural analysis program FINAS is continued, particularly with respect to adoptive mesh generation based on r-method and h-method. FINAS was mounted as the solver in CAE systems such as CADAS, ATLAS, FEMAP and so on. FINAS is currently used by many engineers at about 40 sites including fabricators and universities. The latest version, VI 2.0, was translated into English. 

Nonlinear least-squares analysis can be used to obtain best fit values of the unknown parameters in a nonlinear rate model. Elementary nonlinear regressions can be performed using the SOLVER function in EXCEL. 

The FE solvers developed in this chapter have made use of some of the approximate matrix solution techniques developed in the previous chapter. However, most of the code is new because of the different basic formulation of the finite element approach. In this work the method of weighted residuals has been used to formulate sets of FE node equations. This is one of the two basic methods typically used for this task. In addition, the development has been based upon flie use of basic triangular spatial elements used to cover a two dimensional space. Other more general spatial elements have been sometimes used in the FE method. Finally the development has been restricted to two spatial dimensions and with possible an additional time dimension. The code has been developed in modular form so it can be easily applied to a variety of physical problems. In keeping with the nonlinear theme of this work, the FE analysis can be applied to either linear or nonlinear PDEs. 

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