Approximate Solutions to Nonlinear Problems of Solid Mechanics by Quasilinearization Method

A new approach to parallelization of materially non‐linear problems in solid mechanics is developed. It is based on approximating generalized models of subdomains. The procedure does not retain the same substructuring technique used in a linear version. The convergence proof of the single‐ and multilevel‐domain decomposition algorithms uses the principle of minimum potential energy dissipation and investigated properties of the substructural models. The high efficiency of the approach introduced is shown through the study of several examples. The method developed in this paper for steady creep can be used without modification to solve non‐linear elasticity problems and, at active loading, plasticity problems for bodies of the power‐law strain–stress diagrams. Copyright © 2001 John Wiley & Sons, Ltd.

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, Lp-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.

Analytical computation of amplification of coupling in relativistic equations with Yukawa potential

Alternative stress-integration schemes for large-deformation problems of solid mechanics

This study is aimed to relate nonlinear infinite-horizon optimal control problems (NLIHOCPs) with open-loop information. The difficulties of solving the two-point boundary value problems (TPBVPs) arising from NLIHOCPs can be assigned to the nonlinearity of differential equations, the combination of split boundary conditions, and how the transversality conditions in infinite-horizon are treated. In this paper, we propose a combined quasilinearization method (QLM) with the exponential Bernstein functions (EBFs) for solving nonlinear TPBVPs on the semi-infinite domain. First, the QLM is used to reduce the nonlinear TPBVP to a sequence of linear differential equations. Then, a collocation method based on the EBFs is utilized to find the approximate solution of the resulting linear differential equations. By applying the EBFs, the transversality conditions for TPBVP on the semi-infinite domain are satisfied. The convergence of the QLM + EBFs is proved. Some numerical experiments are performed to confirm the validity of the proposed computational scheme.

This paper presents the solution of geometrically nonlinear problems in solid mechanics by the Domain-Boundary Element Method. Because of the Total-Lagrange approach, the arising domain and boundary integrals are evaluated in the undeformed configuration. Therefore, the system matrices remain unchanged during the solution procedure, and their time-consuming computation needs to be performed only once. While the integral equations for axisymmetric finite deformation problems will be derived in detail, the basic ideas of the formulation in two and three dimensions can be found in [1]. The present formulation includes torsional problems with finite deformations, where additional terms arise due to the curvilinear coordinate system. A Newton–Raphson scheme is used to solve the nonlinear set of equations. This involves the solution of a large system of linear equations, which has been a very time-consuming task in former implementations, [1, 2]. In this work, an iterative solver, i.e. the generalized minimum residual method, is used within the Newton–Raphson algorithm, which leads to a significant reduction of the computation time. Finally, numerical examples will be given for axisymmetric and two/three-dimensional problems.

Solution bound and nearly exact solution to nonlinear solid mechanics problems based on the smoothed FEM concept

Chapter 2 - Galerkin Method of Approximation: Irreducible and Mixed Forms

An algorithm, based on quasi-linearization and state variable parametrization by using Chebyshev polynomials, is proposed to solve the nonlinear optimal control problem subject to terminal state constraints and control saturation constraints. By the virtue of the second method of quasi-linearization and state parametrization, the difficult nonlinear optimal control problem is converted into a sequence of standard quadratic programming problems. An example is given to show the effectiveness of the proposed algorithm.

If a naturally straight thin rod, such as a plastic or metal ruler, is subjected to a small compressive thrust applied to its ends, it remains straight. If the thrust is slowly increased beyond a certain critical value, called the buckling load, the rod assumes a configuration, called a buckled state, that is not straight. See Fig. 1.1. This process is called buckling. Depending on the precise mode of loading and the nature of the rod, the transition to a buckled state can be very rapid. If the thrust is further increased, the deflection of the rod from its straight state is likewise increased. If this entire process is repeated, the rod may well buckle into another configuration such as the reflection of the first through a plane of symmetry. The performance of a whole series of such experiments on different rods would lead to the observation that the buckling loads and the nature of buckled states depend upon the material and shape of the rod and upon the manner in which it is supported at its ends. It can also be observed that the results of experiments are highly sensitive to slight deviations of the rod from perfect straightness or of the thrust from perfect symmetry. The study of buckling for different bodies is one of the richest sources of important problems in nonlinear solid mechanics.

Principles of Construction of Adjoint Operators in Non-Linear Problems Dual Spaces and Adjoint Operators Construction of Adjoint Operators Based on Using the Lagrange Identity Definition of Adjoint Operators Based on Using Taylor's Formula Operators of the Class D and their Adjoint Operators Properties of Adjoint Operators Constructed on the Basis of Various Principles General Properties of Main and Adjoint Operators Corresponding to Non-Linear Operators Properties of Operators of the Class D Properties of Adjoint Operators Constructed with the Use of the Taylor Formula Solvability of Main and Adjoint Equations in Non-Linear Problems Main and Adjoint Equations. Problems Solvability of the Equation F(u) = y Solvability of the Equation A(u)v = y Solvability of the Equation A(u)v = y Solvability of the Equation A*(u)w = p Solvability of the Equation A*(u)w = p Transformation Groups, Conservation Laws and Construction of the Adjoint Operators in Non-Linear Problems Definitions. Non-Linear Equations and Operators. Conservation Laws Transformation of Equations Adjoint Equations Relationship between Different Adjoint Operators General Remarks on Constructing the Adjoint Equations with the Use of the Lie Groups and Conservation Laws Construction of Adjoint Operators with Prescribed Properties The Noether Theorem, Conservation Laws and Adjoint Operators On Some Applications of Adjoint Equations Perturbation Algorithms in Non-Linear Problems Perturbation Algorithms for Original Non-Linear Equations and Equations Involving Adjoint Operators Perturbation Algorithms for Non-Linear Functionals Based on Using Main and Adjoint Equations Spectral Method in Perturbation Algorithms Justification of the N-th Order Perturbation Algorithms Convergence Rate Estimates for Perturbation Algorithms. Comparison with the Successive Approximation Method Justification of Perturbation Algorithms in Quasi-Linear Elliptic Problems Adjoint Equations and the N-th Order Perturbation Algorithms in Non-Linear Problems of Transport Theory Some Problems of Transport Theory The N-th Order Perturbation Algorithms for an Eigenvalue Problem A Problem of Control and its Approximate Solution with the Use of Perturbation Algorithms Investigation and Approximate Solution of a Non-Linear Problem for the Transport Equation Adjoint Equations and Perturbation Algorithms for a Quasilinear Equation of Motion Statement of the Problem. Basic Assumptions. Operator Formulation Transformation of the Problem. Properties of the Non-Linear Operator Adjoint Equation An Algorithm for Computing the Functional The Problem on Chemical Exchange Processes Adjoint Equations and Perturbation Algorithms for a Non-Linear Mathematical Model of Mass Transfer in Soil Mathematical Models of Mass Transfer in Soil Formulation of a Non-Linear Mathematical Model Transformation of the Problem. Properties of the Non-Linear Operator Perturbation Algorithm. Adjoint Equation Approximate Solution of the Problem on Finding an Effective Dispersion Coefficient An Algorithm for Solving the Problem Applications of Adjoint Equations in Science and Technology Adjoint Equations in Data Assimilation Problems Application of Adjoint Equations for Solving the Problem of Liquid Boundary Conditions in Hydrodynamics Shape Optimization Using Adjoint Equation Approaches Global Transport of Pollutants Problems of Climate Change Sensitivity in Various Regions of the World

In connection with the problem of stress distribution in a deep beam with one hole(1)it is of interest to study also the more practical case where two circular holes are perforated for the purpose of the weight saving, etc. In this paper, the problem is considered as that of a deep beam having two circular holes, the depth of which is great enough compared with the size of holes. The solution is given for bipolar co-ordinates, for which the co-ordinate curves are co-axial circles.Since the problem is considered as that of plane stress, the stresses can be expressed in terms of a single functionX, say stress function, as given in (3) in this particular coordinates.(2) We assume this functionXto be (X1+X0) and suppose thatX1gives no stresses at great distance from two holes and the stresses in terms ofX0only satisfy the conditions at that point, though each function satisfies independently the linear partial differential equation of the same type as given in (4) or (5) in our particular co-ordinates.(2) Two solutions are obtained, (1) for the case of a deep beam containing two circular holes of equal radii on each side of the neutral axis and (2) for the case of a deep beam containing two circular holes on the neutral axis.The results of mathematical investigation may be summarised as follow;For the case (I). In this case, thex-axis is the neutral axis and they-axis is the vertical axis, on which the centers of two holes lie. These two circular holes lie at equal distance on either side of the neutral axis. The stress function which satisfies the conditions above mentioned is given by (8). The stresses in beam are given in (9), (10), (11) and (12). The constants in (12) which are determined by the conditions on the periphery of holes (13), namely, (14) are given in (15). The stress distribution round each hole is as shown in Fig. 3 by plotting the values of stresses on the normals to the periphery of hole. The stress round each hole has two maximum values at the symmetrical points A and B on the vertical axis through centers of holes. These two maximum values are some multiples of the stresses at the corresponding points in a non-perforated beam. These multiples are 2.85 atAand 3.22 atBfor P/D= 3.75, Pbeing the center distance and Dthe dia. cf two holes. For larger values of P/D, these multiples are nearly constant as shown in Fig. 1. The stress distribution on the neutral axis and on the vertical axis through centers of holes are given in Fig. 4 by plotting the values of stresses on the normals to the neutral and vertical axis respectively.For the case (2). In this case, the y-axis is the neutral axis of beam, on which two equal circular holes lie. The stress function for this case is given in (17) and the stresses in beam are given by (18), (19), (20) and (21). The constants in (21) are determined by the conditions round each circular hole (22), namely, (23) and are given in (24). The stress distribution round each circular hole in this case is not symmetry about the vertical axis through its center as shown in Fig. 5, since the existence of the another Hole on the same neutral axis has great influence upon the stress distribution in the neighbourhood of the other hole. But this effect due to the another hole will be decreased as the center distance of two holes is increased and, for larger value of P/D=10, the stress round of each hole may be approximately given by that for the case of a deep beam having one circular hole as shown in Fig. 2. The stress for the latter case is given byββ=MR/I (sin θ-sin 3θ) M=uniform bending moment.R=radius of circular hole.

High precision approximate analytic expressions for the ground state energies and wavefunctions of the generalized anharmonic oscillator with the potential U(x)=g2x2/2+λ|x|p are obtained by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The suitable initial guess for the first iteration ensures the high accuracy of the resulting expressions in a wide range of the parameters g⩾0, λ⩾0 and p⩾2, the latter need not be an integer. It is found that the precision of the obtained wavefunctions is close to 0.1% that is more accurate than Wentzel–Kramers–Brillouin (WKB) solutions by two to three orders of magnitude. In addition, the SWKB excited state energies have been calculated using the obtained ground state wavefunctions. The latter results are in good agreement with the exact ones. All of the above mentioned enables us to make accurate analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the generalized anharmonic oscillators.

A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics

Application of the method of integral relations and the method of quasi-linearization to Goursat's problem