Finite element methods for nonlinear shell analysis

Summary Finite element methods for nonlinear shell analysis are analyzed using both the minimum potential energy and the mixed formulations. Existence and local uniqueness of both the exact solutions and the corresponding finite element solutions are proved. Error bounds, which are of the same order as for the corresponding linear problems, are established.     

Sensitivity analysis for nonlinear programming using penalty methods

In this paper we establish a theoretical basis for utilizing a penalty-function method to estimate sensitivity information (i.e., the partial derivatives) of a localsolution and its associated Lagrange multipliers of a large class of nonlinear programming problems with respect to a general parametric variation in the problem functions. The local solution is assumed to satisfy the second order sufficient conditions for a strict minimum. Although theoretically valid for higher order derivatives, the analysis concentrates on the estimation of the first order (first partial derivative) sensitivity information, which can be explicitly expressed in terms of the problem functions. For greater clarity, the results are given in terms of the mixed logarithmic-barrier quadratic-loss function. However, the approach is clearly applicable toany algorithm that generates a once differentiable solution trajectory.Supported by the U.S. Army Research Office, Durham.     

STABILITY ANALYSIS OF NONLINEAR GALERKIN ＆ GALERKIN METHODS FOR NONLINEAR EVOLUTION EQUATIONS

Abstract. Ogr object in this artlcle is to describe tbe Galerkln scheme and nonlin-eax Galerkin scheme for the approximation of nonlinear evolution equations, and tostudy the stability of these schemes. Spatial discretizatlon can be pedormed by eitherGalerkln spectral method or nonlinear Galerldn spectral method; time discretizatlort isdone hy Euler sin.heine wklch is explicit or implicit in the nonlinear terms. According tothe stability analysis of the above schemes, the stability of nonllneex Galerkln methodis better than that of Galexkln method.     

Local convergence analysis of tensor methods for nonlinear equations

Tensor methods for nonlinear equations base each iteration upon a standard linear model, augmented by a low rank quadratic term that is selected in such a way that the mode is efficient to form, store, and solve. These methods have been shown to be very efficient and robust computationally, especially on problems where the Jacobian matrix at the root has a small rank deficiency. This paper analyzes the local convergence properties of two versions of tensor methods, on problems where the Jacobian matrix at the root has a null space of rank one. Both methods augment the standard linear model by a rank one quadratic term. We show under mild conditions that the sequence of iterates generated by the tensor method based upon an ideal tensor model converges locally and two-step Q-superlinearly to the solution with Q-order 3/2, and that the sequence of iterates generated by the tensor method based upon a practial tensor model converges locally and three-step Q-superlinearly to the solution with Q-order 3/2. In the same situation, it is known that standard methods converge linearly with constant converging to 1/2. Hence, tensor methods have theoretical advantages over standard methods. Our analysis also confirms that tensor methods converge at least quadratically on problems where the Jacobian matrix at the root is nonsingular.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Research supported by AFOSR grant AFOSR-90-0109, ARO grant DAAL 03-91-G-0151, NSF grants CCR-8920519 CCR-9101795.     

Convergence analysis of the nonlinear block scaled ABS methods

In this paper, by a further investigation of the algorithm structure of the nonlinear block scaled ABS methods, we convert it into an inexact Newton method. Based on this equivalent version, we establish the semilocal convergence theorem of the nonlinear block scaled ABS methods and obtain convergence conditions that mainly depend on the behavior of the mapping at the initial point. This complements the convergence theory of the nonlinear block scaled ABS methods.     

Quasi-Newton methods for constrained nonlinear systems: complexity analysis and applications

We address the solution of constrained nonlinear systems by new linesearch quasi-Newton methods. These methods are based on a proper use of the projection map onto the convex constraint set and on a derivative-free and nonmonotone linesearch strategy. The convergence properties of the proposed methods are presented along with a worst-case iteration complexity bound. Several implementations of the proposed scheme are discussed and validated on bound-constrained problems including gas distribution network models. The results reported show that the new methods are very efficient and competitive with an existing affine-scaling procedure.     

Local analysis of Newton-type methods for variational inequalities and nonlinear programming

This paper presents some new results in the theory of Newton-type methods for variational inequalities, and their application to nonlinear programming. A condition of semistability is shown to ensure the quadratic convergence of Newton's method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth functions is given. The second part of the paper considers some particular variational inequalities with unknowns (x, ), generalizing optimality systems. Here only the question of superlinear convergence of {x ^k } is considered. Some necessary or sufficient conditions are given. Applied to some quasi-Newton algorithms they allow us to obtain the superlinear convergence of {x ^k }. Application of the previous results to nonlinear programming allows us to strengthen the known results, the main point being a characterization of the superlinear convergence of {x ^k } assuming a weak second-order condition without strict complementarity.     

OnM-functions and nonlinear relaxation methods

Globally convergent nonlinear relaxation methods are considered for a class of nonlinear boundary value problems (BVPs), where the discretizations are continuousM-functions.It is shown that the equations with one variable occurring in the nonlinear relaxation methods can always be solved by Newton's method combined with the Bisection method. The nonlinear relaxation methods are used to get an initial approximation in the domain of attraction of Newton's method. Numerical examples are given.     

The analysis study on nonlinear iterative methods for inverse problems

In this paper, the nonlinear iterative methods, which are different from the classical algorithms, to solve inverse problems are presented. Our methods by denoting some parameters and some properties of the algorithm in both noise and noiseless cases are studied. Finally, the convergence of the sequence generated by the algorithm without noise is discussed.     

Stability analysis of one-leg methods for nonlinear functional differential and functional equations

This paper is concerned with the numerical solution of nonlinear functional differential and functional equations. The adaptation of one-leg methods is considered. It is proved that an A-stable one-leg method is globally stable and a strongly A-stable one-leg method is asymptotically stable under suitable conditions. A numerical test is given to confirm the theoretical results.     

Stability and error analysis of one-leg methods for nonlinear delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability behaviour and error analysis of one-leg methods with respect to nonlinear DDEs. The new concepts of GR-stability, GAR-stability and weak GAR-stability are introduced. It is proved that a strongly A-stable one-leg method with linear interpolation is GAR-stable, and that an A-stable one-leg method with linear interpolation is GR-stable, weakly GAR-stable and D-convergent of order s, if it is consistent of order s in the classical sense.     

Parallel nonlinear multisplitting methods

Summary Linear multisplitting methods are known as parallel iterative methods for solving a linear systemAx=b. We extend the idea of multisplittings to the problem of solving a nonlinear system of equationsF(x)=0. Our nonlinear multisplittings are based on several nonlinear splittings of the functionF. In a parallel computing environment, each processor would have to calculate the exact solution of an individual nonlinear system belonging to his nonlinear multisplitting and these solutions are combined to yield the next iterate. Although the individual systems are usually much less involved than the original system, the exact solutions will in general not be available. Therefore, we consider important variants where the exact solutions of the individual systems are approximated by some standard method such as Newton's method. Several methods proposed in literature may be regarded as special nonlinear multisplitting methods. As an application of our systematic approach we present a local convergence analysis of the nonlinear multisplitting methods and their variants. One result is that the local convergence of these methods is determined by an induced linear multisplitting of the Jacobian ofF.Dedicated to the memory of Peter Henrici     

Error analysis of finite element methods for mildly nonlinear variational inequalities

The finite element method is applied through the use of a variational inequality to an obstacle problem involving nonhomogeneous boundary data. For piecewise linear conforming trial functions energy norm error bounds are derived.     

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.     

Formulation and analysis of two energy-consistent methods for nonlinear elastodynamic frictional contact problems

Energy-conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability. Furthermore, some physical phenomena (such as friction) can generate dissipation; then in this work, we present and analyse two energy-consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon. The first approach permits one to enforce, respectively, the Kuhn–Tucker and persistency conditions during each time step by combining an adapted continuation of the Newton method and a Lagrangean formulation. In addition the second method which is based on the work in [P. Hauret, P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916] represents a specific penalisation of the unilateral contact conditions. Some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods.     

Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations

Summary An existence and uniqueness result is given for nonlinear Volterra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows twosided error bounds to be derived.     

The convergence analysis of inexact Gauss–Newton methods for nonlinear problems

In this paper, inexact Gauss–Newton methods for nonlinear least squares problems are studied. Under the hypothesis that derivative satisfies some kinds of weak Lipschitz conditions, the local convergence properties of inexact Gauss–Newton and inexact Gauss–Newton like methods for nonlinear problems are established with the modified relative residual control. The obtained results can provide an estimate of convergence ball for inexact Gauss–Newton methods.     

Tangent methods for nonlinear equations

Summary Equation-solving methods that utilize alternate function and derivative values are developed. The procedures are similar to the secant or to Muller's method, and are especially competitive when the derivative is simpler to obtain than the function itself. Related hybrid methods are also found to be attractive.Work supported by the British Science Research Council at the University of Dundee, and by the U.S. Atomic Energy Commission.