Nonlinear Solution Techniques Research Articles

Non-linear dynamic modelling using Component Mode Synthesis

This paper investigates the coupling of Component Mode Synthesis (CMS) and non-linear solution techniques to simulate the non-linear dynamic response of structures. CMS techniques can be used to model flexibility of substructures in a primarily rigid body dynamic solution through the use of modal superposition. The limitation of this technique, as implemented in commercial rigid body dynamics codes, is that only the linear elastic response of the substructure can be modelled. An implicit method is presented, which couples a fixed interface CMS technique with classical plasticity theory to approximate non-linear material effects during a dynamic simulation. An example problem is provided to indicate the accuracy and limitations of this type of solution technique.

Phase stability analysis of liquid-liquid equilibrium with stochastic methods

Minimization of Gibbs free energy using activity coefficient models and nonlinear equation solution techniques is commonly applied to phase stability problems. However, when conventional techniques, such as the Newton-Raphson method, are employed, serious convergence problems may arise. Due to the existence of multiple solutions, several problems can be found in modeling liquid-liquid equilibrium of multicomponent systems, which are highly dependent on the initial guess. In this work phase stability analysis of liquid-liquid equilibrium is investigated using the NRTL model. For this purpose, two distinct stochastic numerical algorithms are employed to minimize the tangent plane distance of Gibbs free energy: a subdivision algorithm that can find all roots of nonlinear equations for liquid-liquid stability analysis and the Simulated Annealing method. Results obtained in this work for the two stochastic algorithms are compared with those of the Interval Newton method from the literature. Several different binary and multicomponent systems from the literature were successfully investigated.

Phase stability analysis using interval Newton method with NRTL model

The Gibbs energy minimization using activity coefficient models and nonlinear equation solution techniques are commonly applied for phase stability problems. However, dependence on the initial estimates and multiple solutions for these highly nonlinear equations are common drawbacks for some of the conventional approaches. We have used interval Newton method with the local composition model of NRTL for the phase stability analysis of 10 binary systems and 2 ternary systems at various feed compositions to locate all the stationary points. Results indicate that the interval Newton method is reliable and efficient.

Numerical Conservation Properties of H(div)-Conforming Least-Squares Finite Element Methods for the Burgers Equation

Least-squares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation highlights the smoothness of the flux vector for weak solutions, namely, $\bff(u) \in H({\rm div},\Omega)$. The standard least-squares (LS) finite element (FE) procedure is applied to the reformulated equations using H(div)-conforming FE spaces and a Gauss--Newton nonlinear solution technique. Numerical results are presented for the one-dimensional Burgers equation on adaptively refined space-time domains, indicating that the H(div)-conforming FE methods converge to the entropy weak solution of the conservation law. The H(div)-conforming LSFEMs do not satisfy a discrete exact conservation property in the sense of Lax and Wendroff. However, weak conservation theorems that are analogous to the Lax--Wendroff theorem for conservative finite difference methods are proved for the H(div)-conforming LSFEMs. These results illustrate that discrete exact conservation in the sense of Lax and Wendroff is not a necessary condition for numerical conservation but can be replaced by minimization in a suitable continuous norm.

Elasto-plastic element-free Galerkin method

In this paper the element free Galerkin method (EFGM) has been extended to be used in the elastoplastic stress analysis. The developed method has been examined in planar stress analysis around the tip of a crack and in its opening mode of loading. To do this, at the first step by using the incremental relations of plastic deformation a system of elastoplastic EFGM equations has been derived. Since the obtained relations are nonlinear, a nonlinear solution technique has been chosen. To examine the validity of this technique, stress fields in two different plates with and without a crack have been calculated and the results have been compared with other similar analytical works in the literature. In doing so the power law work hardening behavior has been employed and the value of J-integral has been used as a base for comparison of the results.

Japanese Forest Sector Modeling through a Partial Equilibrium Market Model

The structure of Japanese timber markets has changed drastically during recent decades. After the introduction of a large amount of imported softwood products. Japanese timber producers have faced global competition with foreign timber suppliers such as Canada, the US, and recently Nordic countries. In this paper, we present a forest sector model for lumber markets with a focus on eight aggregate regions (Tohoku, Kanto, Hokuriku, Chubu, Kinki, Chugoku, Shikoku, and Kyushu) in Japan. The proposed model is based on the Samuelson-partial equilibrium formulation, which searches for an optimal solution by maximizing the net social payoff subject to demand and supply constraints. A nonlinear programming solution technique is incorporated into the proposed model. Three types of lumber are considered, i.e., domestic lumber, the lumber processed in Japan from imported logs, and imported lumber from the US and Canada. Using data for 1998, our analysis indicates that the derived equilibrium solution has a higher price for the imported lumber supply in all regions, and a lower price for the other two products in most regions than the actual current price in 1998. The derived net social payoff gains 1.6% compared with the one derived with the current set of prices and quantities.

Flux-limiting and non-linear solution techniques for simulation of transport in porous media

The conservation laws governing the flow of liquids in porous media are often non-linear and have steep fronts that require resolution in time. It is one of the aims of this work to analyse the use of higher order spatial weighting schemes and temporal methods for reducing numerical dispersion. Another important ingredient in the development of an efficient simulator is the treatment of the non-linear system that results from the discrete analogue of the conservation law. In this work a vertex-centered finite volume method is used for discretising a representative conservation law in one-dimension and two non-linear iterative methods, an inexact full Newton method and the modified Shamanskii method, are scrutinised. Two case studies are chosen to highlight the performance of the chosen numerical techniques. At first, the focus is on the accuracy and efficiency of the spatial weighting methods for a linear advection-dispersion equation and then, a two-phase flow problem is analysed to gauge the performance of the non-linear solvers. In both cases, comparisons with exact solutions are presented.

An improved single-parameter tip-timing method for turbomachinery blade vibration measurements using optical laser probes

This paper presents an improved analysis method for the interpretation of the vibration data measured at turbomachinery blade tips using optical laser probes. A multi-degree-of-freedom numerical simulator, which includes the structural and geometric properties of the bladed-disk assembly, the external forcing terms and the characteristics of the optical probe, has been developed to assess the reliability of the various data processing techniques to identify the natural frequencies and mode shapes of bladed-disk assemblies. It has been demonstrated that the Zablotsky—Korostelev single parameter technique, which is a de-facto standard in the aerospace industries, has inherent limitations associated with it. An improved and more rigorous method is presented for deriving the blade arrival times and a non-linear solution technique is suggested for their numerical determination. Finally, the effect of blade mistuning on the accuracy of the proposed method is also investigated.

Unified nonlinear elastic frame analysis

This paper develops a unified nonlinear analysis technique for two-dimensional structural frames. Any level of nonlinear effects can be rationally treated, and results automatically obtained to any specified accuracy. The procedure only requires minimum discretisation of the structure as a self-adaptive procedure is used to solve the member equations. The nonlinear solution techniques used enable either discrete equilibrium positions or the complete load-deflection response to be calculated.

THE MEAN-VARIANCE APPROACH TO PORTFOLIO OPTIMIZATION SUBJECT TO TRANSACTION COSTS

Tra,nsact>ion costss are a. source of concern for port,folio managers. Due to nonlinearity of the cost function, the ordinary quadratic programming solution technique cannot be applied. This paper addresses the portfolio optinlization problem subject to transaction costs. The transaction cost is assumed to be a V-sha,ped function of difference between an existing and new portfolio. A nonlinear programming solution technique is used to solve t,he proposed problem. The port,folio optimiza,t,ion syst,em ca,lled POSTRAC (Portfolio Optirniza,tion System with TRAnsaction Costs) is proposed. The experimental analysis indicates that ignoring the transa,ction costss results in inefficient portfolios. It is also shown tlmt there does not exist statistica,lly significant difference in portfolio performance with different methods to estimate the expected return of se~urit~ies, when considering the tra,nsact,ion costs int,o the p~rt~folio return.

Algorithms for constrained molecular dynamics

AbstractIn molecular dynamics simulations, the fastest components of the potential field impose severe restrictions on the stability and hence the speed of computational methods. One possibility for treating this problem is to replace the fastest components with algebraic length constraints. In this article the resulting systems of mixed differential and algebraic equations are studied. Commonly used discretization schemes for constrained Hamiltonian systems are discussed. The form of the nonlinear equations is examined in detail and used to give convergence results for the traditional nonlinear solution technique SHAKE iteration and for a modification based on successive overrelaxation (SOR). A simple adaptive algorithm for finding the optimal relaxation parameter is presented. Alternative direct methods using sparse matrix techniques are discussed. Numerical results are given for the new techniques, which have been implemented in the molecular modeling software package CHARMM and show as much as twofold improvement over SHAKE iteration. © 1995 John Wiley & Sons, Inc.

Vision-assisted lead-to-pad alignment technique for placement of surface mount components

In this paper, a lead-to-pad alignment technique for surface mount component placement on printed circuit boards is developed. The problem is expressed as a nonlinear optimization problem with a linear objective function and quadratic constraints. A solution procedure is developed and compared with a known nonlinear optimization solution technique. Results show substantial improvements in execution time of the optimization and yield of the placement operation when implemented in a robotic cell.

Large deformation finite element analysis of laminated circular composite plates

A 48 d.o.f., four-noded quadrilateral laminated composite shell finite element is particularised to a sector finite element and is used for the large deformation analysis of circular composite laminated plates. The strain-displacement relationships for the sector element are obtained by reducing those of the quadrilateral shell finite element by substituting proper values for the geometric parameters. Subsequently, the linear and tangent stiffness matrices are formulated using conventional methods. The Newton-Raphson method is employed as the nonlinear solution technique. The computer code developed is validated by solving an isotropic case for which results are available in the literature. The method is then applied to solve problems of cylindrically orthotropic circular plates. Some of the results of cylindrically orthotropic case are compared with those available in the literature. Subsequently, application is made to the case of laminated composite circular plates having different lay-up schemes. The computer code can handle symmetric/unsymmetric lay-up schemes. The large displacement analysis is useful in estimating the damage in composite plates caused by low-velocity impact.

Superconducting axisymmetric finite elements based on a gauged potential variational principle—II. Solution and numerical results

Part II of a two part paper discusses an incremental-iterative nonlinear solution technique for solving the nonlinear finite element equations of the superconducting state of a superconductor. The untreated equations are highly ill-conditioned and are impossible to solve within the typical 16-place double precision supplied by most computers. A combination of matrix scaling and mesh grading techniques is used to reduce the condition number of the tangent stiffness matrix and increase the accuracy of the current carrying boundary layer representation. Numerical results for a one-dimensional model of a time-independent superconductor treated by the Ginzburg-Landau model are presented and discussed. The computed solutions clearly display the Meissner effect of magnetic field expulsion from the central region of the superconductor. These results are compared to the physics of a low-viscosity fluid problem. From this analogy, a physical argument is advanced about the macroscopic behavior of superconductors.

Optimization of Process Systems with Discontinuities

The method for optimizing problems that comprise differential and algebraic equations with profile discontinuities is developed. With the proposed approach polynomial approximations are constructed on continuous profiles and orthogonal collocation on finite elements is used to discretize the differential equations. The discontinuities caused by inequality constraints or bounds on profiles are handled efficiently by the smooth approximation procedure. The proposed procedure is based on a hyperbolic approximation and converts the discontinuity problem into a continuous nonlinear program. The discretized problem is solved with Minos, a nonlinear programming solution technique. The proposed procedure is an attempt to develop a general and effective method to handle complexities on profiles and thus make it possible to simultaneously solve problems that include differential and algebraic equations.

An error estimate and step size control method for nonlinear solution techniques

The condition number of the tangent stiffness matrix is shown to reflect the error encountered in the solution of nonlinear structural systems. Due to the linearization process in the incremental solution, two sources of error exist: (1) the error associated with the solution of the linearized set of equations; and (2) the error associated with the unbalanced force residual. While the condition number itself reflects the first error, its change from step to step represents the second error. Being an expensive quantity to compute, only an estimate of the condition number that is based on the L 1-norm of the tangent stiffness matrix has been effectively used in guiding the solution algorithm in adjusting the step size according to the degree of nonlinearity of the system. It is also used to identify the stable from the unstable paths of the response curve. Several examples are solved to show the applicability of the proposed step size control with different solution techniques.

Finite element procedures for strain estimations of sheet metal forming parts

AbstractA finite element algorithm is developed to predict the strain distribution of sheet forming parts and to evaluate their formability at the design stage. The algorithm is simply called the inverse approach since we know the positions of the material points on the final product and we determine the positions of the corresponding material points in the initial blank. This is also a direct algorithm which avoids the path dependent incremental procedure of plasticity and contact. This paper presents some theoretical aspects related to the large logarithmic strains with membrane triangular elements, the deformation theory of plasticity, the isotropic and anisotropic material properties of metal sheets and the non‐linear solution techniques. Some particular problems are also studied, such as the influence of friction forces under the punch and blankholders, the ‘optimization’ of the initial blank's contour and the deep drawing in several steps. The results are compared with those obtained by experiments and by the more classical incremental approach.

An Algorithm for Solving the General Bilevel Programming Problem

The conflict that naturally arises in a hierarchical system can often be modeled as a multistage optimization problem. This paper considers the sequential uncooperative problem in which two decision makers wish to maximize their own objective functions over a feasible region defined by interactive strategy sets. The resultant problem is known as the bilevel program. Such programs are inherently nonconvex and resistant to standard nonlinear programming solution techniques such as piecewise linearization and convex underestimating envelopes. Alternatively, a grid search algorithm is offered which exhibits the desirable property of monotonicity. The algorithm is based on two sets of necessary conditions previously developed and combined here to provide an operational check for stationarity and local optimality. Potential solutions are obtained from a parameterized master program whose feasible region approximates that of the original problem. As the one-dimensional parameter is varied over the unit interval and the master problem solved, a nondecreasing sequence of lower bounds to the first decision maker’s objective function is produced. If the solution to the original problem is stable under small perturbations, the sequence is shown to converge to the global optimum. A discussion of the computational requirements follows the presentation of the algorithm.