Large Sets Of Nonlinear Equations Research Articles

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.

Implicit Temporal Discretization and Exact Energy Conservation for Particle Methods Applied to the Poisson–Boltzmann Equation

We report on a new multiscale method approach for the study of systems with wide separation of short-range forces acting on short time scales and long-range forces acting on much slower scales. We consider the case of the Poisson–Boltzmann equation that describes the long-range forces using the Boltzmann formula (i.e., we assume the medium to be in quasi local thermal equilibrium). We develop a new approach where fields and particle information (mediated by the equations for their moments) are solved self-consistently. The new approach is implicit and numerically stable, providing exact energy conservation. We test different implementations that all lead to exact energy conservation. The new method requires the solution of a large set of non-linear equations. We consider three solution strategies: Jacobian Free Newton Krylov, an alternative, called field hiding which is based on hiding part of the residual calculation and replacing them with direct solutions and a Direct Newton Schwarz solver that considers a simplified, single, particle-based Jacobian. The field hiding strategy proves to be the most efficient approach.

Advanced Computational Techniques for Solving Desalination Plant Models Using Neural and Genetic Based Methods

Simulation of multi-stage flash (MSF) desalination processes that have a production capacity range between 50,000 to 75,000 m3/d is an intensive computational problem that requires high computer processing speed despite the availability of advanced processing computer power in hand nowadays. In this work, a comparative study is conducted to explore the performance of different numerical techniques to solve large set of nonlinear equations generated by large scale MSF models. These algorithms can be categorized into three groups, namely: conventional numerical approximation methods, multi-objective optimization based methods, and the last group comprises artificial neural networks (ANN) based models and genetic algorithms (GA) based methods. The problem of solving large sets of nonlinear equations with upper and lower constraints is accomplished successfully using all algorithms with different prediction efficiency and speed. The idea of using GA and ANN based algorithms in simulating the MSF model is basically used to generate feasible initial solution estimates that were used as starting guesses for other numerical methods in the former case and to eliminate the step of providing these initial guesses in the later case. Significant reduction of computation effort was attained using ANN-based techniques. The outcome of this work can be utilized to develop new generations of process simulators that are based on well-trained ANNs in order to achieve speedup of computations and to generate more reliable predictions without detracting from accuracy.

Transient response of finned-tube condenser in household refrigerators

A distributed parameter model for prediction of the transient performance of a condenser is presented. The model is capable of predicting the refrigerant temperature distribution, tube wall temperature, quality of refrigerant, inventory mass of refrigerant as a function of position and time. An efficient two-level iteration method is proposed to obtain the numerical solution of the model without solving a large set of non-linear equations simultaneously. A finned tube condenser of 12 ft 3 refrigerator with R12 as working fluid was chosen as a sample and some tests were carried out to determine its transient response. The examination of results indicates that the theoretical model provides a reasonable prediction of dynamic response compared to the experimental data. Transient behavior of temperature, pressure, mass flow rate, mass of liquid and vapor of refrigerant, quality, heat transfer in household refrigerators have been presented. Also time-dependent displacement of interface between saturated and superheated regions has been shown. Extensive examinations of theoretical and experimental results show that with utilization of a controllable compressor, power consumption can be reduced.

Transient response of dry expansion evaporator in household refrigerators

A distributed parameter model for prediction of the transient performance of an evaporator is presented. The model is capable of predicting the refrigerant temperature distribution, tube wall temperature, quality of refrigerant, inventory mass of refrigerant as a function of position and time. An efficient two-level iteration method is proposed to obtain the numerical solution of the model without solving a large set of non-linear equations simultaneously. A round bound evaporator of 12 ft3 refrigerator with R12 as working fluid were chosen as a sample and some tests were carried out to determine its transient response. The results indicate that the theoretical model provides a reasonable prediction of dynamic response compared with the experimental data. Transient behaviour of temperature, pressure, mass flow rate, mass of liquid and vapour of refrigerant, quality, heat transfer in household refrigerators have been presented. Also time dependent displacement of interface between saturated and superheated regions has been shown. Extensive investigation of theoretical and experimental results shows that with a controllable compressor, power consumption can be reduced.

Simulation of failure of structures using dynamics and optimization techniques

Simulation of failure processes requires solution of large sets of non-linear equations, which is an inherently difficult task. Many attempts have been made to stabilize the solution procedures but with limited success. This paper presents a broad review of techniques to stabilize the solution procedures both from the field of structural mechanics (both static and dynamic) and the related field of optimization. In particular two new methods for structural mechanics applications are presented: the arc-length strain constraint and the trust region method (TRM). The TRM uses a step size constraint to ensure the validity of the Taylor expansion. The review finds interesting similarities between arc-length methods, time integration procedures and TRM, providing new insights. The new techniques have been implemented in a FE-code and some numerical examples are presented. The examples show that the results from static arc-length procedures are somewhat artificial and difficult to interpret since they follow a displacement controlled failure. The dynamics must be included to follow the more realistic load controlled failure. The dynamics also to some extent stabilizes the solution process. The convergence test of the TRM shows that it improves robustness allowing larger time increments to be used, but the rate of convergence is decreased why efficiency needs to be further improved.

Transient response of dry expansion evaporator in household refrigerators

A distributed parameter model for prediction of the transient performance of an evaporator is presented. The model is capable of predicting the refrigerant temperature distribution, tube wall temperature, quality of refrigerant, inventory mass of refrigerant as a function of position and time. An efficient two-level iteration method is proposed to obtain the numerical solution of the model without solving a large set of non-linear equations simultaneously. A round bound evaporator of 12 ft 3 refrigerator with R12 as working fluid were chosen as a sample and some tests were carried out to determine its transient response. The results indicate that the theoretical model provides a reasonable prediction of dynamic response compared with the experimental data. Transient behaviour of temperature, pressure, mass flow rate, mass of liquid and vapour of refrigerant, quality, heat transfer in household refrigerators have been presented. Also time dependent displacement of interface between saturated and superheated regions has been shown. Extensive investigation of theoretical and experimental results shows that with a controllable compressor, power consumption can be reduced.

A distributed model for prediction of the transient response of an evaporator

A distributed parameter model for predicting the transient performance of an evaporator is presented. The model is capable of predicting distributions of the refrigerant velocity, void fraction, temperature, tube wall temperature, air temperature and humidity, in both position and time domains. An efficient and stable two-level iteration method is proposed to obtain the numerical solutions of the model without solving a large set of non-linear equations simultaneously. The dynamic behaviour of he evaporator is investigated with a step change in the infrigerant flowrate. Comparisons are made with experimental measurements on a commercial evaporator with refrigerant R134a as the working fluid. The results indicate that the model provides a reasonable estimation of the dynamic response compared with the experimental data. It is also observed that the evaporator thermal response is faster for a step decrease than for a step increase in refrigerant flowrate.

An inverse eigenvalue problem: ComputingB-stable Runge-Kutta methods having real poles

The implementation of implicit Runge-Kutta methods requires the solution of large sets of nonlinear equations. It is known that on serial machines these costs can be reduced if the stability function of ans-stage method has only ans-fold real pole. Here these so-called singly-implicit Runge-Kutta methods (SIRKs) are constructed utilizing a recent result on eigenvalue assignment by state feedback and a new tridiagonalization, which preserves the entries required by theW-transformation. These two algorithms in conjunction with an unconstrained minimization allow the numerical treatment of a difficult inverse eigenvalue problem. In particular we compute an 8-stage SIRK which is of order 8 andB-stable. This solves a problem posed by Hairer and Wanner a decade ago. Furthermore, we finds-stageB-stable SIRKs (s=6,8) of orders, which are evenL-stable.

A scheme for the implementation of implicit Runge-Kutta methods

The computational work required to implement implicit Runge-Kutta methods is often dominated by the cost of solving large sets of nonlinear equations. As an alternative to modified Newton methods, iteration schemes, which sacrifice superlinear convergence for reduced linear algebra costs, have been proposed. A new scheme of this type is considered here. This scheme avoids expensive vector transformations, is computationally more efficient, and gives improved performance.

Phase transitions in lattice fluids. I. Short-range interactions

The phase transitions of lattice fluids with molecules defined by first-neighbour exclusion, and interacting at short range, are considered using extended Kikuchi approximations. On the square lattice, a second-neighbour interaction does not produce a disordered low-density transition, but does make the packing transition first order below a critical temperature epsilon beta =-1.05. The resulting phase diagram therefore has only generalized fluid and solid regions. In a Kikuchi double- square calculation with third-neighbour interaction a disordered transition is produced, but this overlaps with the ordered state and again there is no stable intermediate liquid phase. Short-range interactions produce qualitatively the same effect on the triangular and simple cubic lattices. Particularly for the ordered state, the large set of non-linear equations produced by an analytical treatment of extended Kikuchi approximations have limited the range of possible calculations.