Construction of Basis Functions and Their Application to Boundary-Value Problems of Mechanics of Continuous Media

A unified method for constructing basis (eigen) functions is proposed to solve problems of mechanics of continuous media, problems of cubature and quadrature, and problems of approximation of hypersurfaces. Numerical-analytical methods are described, which allow obtaining approximate solutions of internal and external boundary-value problems of mechanics of continuous media of a certain class (both linear and nonlinear). The method is based on decomposition of the sought solutions of the considered partial differential equations into series in basis functions. An algorithm is presented for linearization of partial differential equations and reduction of nonlinear boundary-value problems, which are reduced to systems of linear algebraic equations with respect to unknown coefficients without using traditional methods of linearization.     

Evaluation of a simple nonlinear elastic constitutive form

A nonlinear, two constant stress-deformation form is deduced for elastic materials. At very large stretch ratios of greater than about 3 or 4, the model exhibits the strain stiffening behavior common to many elastomers. The constitutive form is very simple since the two material constants enter it as multiplying constants times certain nonlinear deformation terms. The model is evaluated with respect to data upon natural rubber under both uniaxial and bi-axial stress conditions. The model is also used to evaluate data obtained from a nonlinear membrane inflation experiment. The latter experimental capability and corresponding data are new.Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.     

A UNIVERSAL APPROACH FOR CONTINUOUS OR DISCRETE NONLINEAR PROGRAMMINGS WITH MULTIPLE VARIABLES AND CONSTRAINTS

IntroductionExtensive research works have been published for solving nonlinear mathematicprogramming problems.Nonetheless,it is still difficult to find an effective and universalapproach for general programming problems with multiple design variables and …     

非线性杆单元荷载矩阵积分算子直接集成算法

在文献[1]的基础上，将积分算子直接集成单元刚度矩阵的方法扩展应用于非线性杆单元的荷载矩阵集成．该方法将分布荷载产生的内力函数分解为由未知参数表示的线形项，和与分布荷载特性相关的已知函数项的组合，计算积分算子，由积分算子的线形组合得到分布荷戟产生的杆件内力函数，从而得到荷载矩阵．该方法没有采用任何假定，具有广泛的普适性，能用于求解任意刚度分布杆件上作用任意分布荷载或集中荷载的单元荷载矩阵．对于刚度非连续、荷载分布非连续的情况，该方法避免了高斯数值积分的误差，具有更高的计算精度。     

Multiple scale fourier transformation: An application to nonlinear dispersive waves

A Fourier transformation involving multiple scales is applied to describe the far-field asymptotic behaviour of nonlinear dispersive waves. It is shown that a nonlinear asymptotic perturbation can be carried out in terms of simple calculations with respect to Dirac delta functions involving a multiple scale wave number and frequency space. Fourier transformed versions of the nonlinear Schrödinger and Korteweg-de Vries equations are derived explicitly.     

On Saint-Venant's Principle in three-dimensional nonlinear elasticity

Consider a long elastic isotropic beam with a convex cross-section and a sufficiently smooth boundary. Suppose that a self-equilibrated load is applied at each end but the sides are stress-free and there are no internal body forces. It is proved in the context of three-dimensional, nonlinear elastostatics that if the first four derivatives of the displacement vector are a priori assumed to be everywhere sufficiently small with respect to the physical constants and the geometry of the cross-section, then the strains at any point decay exponentially with the distance of the point from the nearest end.This result is an extension of known results on Saint-Venant's Principle in linear and two-dimensional nonlinear elasticity.     

Application of stochastic finite element method in estimation of elastic constants for NCF composites

The main aim of this paper is to evaluate the effect of micro-structure variations on elastic constants of NCF (non-crimp fabric) composites. To reach this aim, a three-dimensional unit cell model is developed by the finite element code ABAQUS and mean value of elastic constants in NCF composites are determined by analytical and unit cell based methods with various boundary conditions. In addition, first- and second-order stochastic finite element formulations are derived and variances of elastic constants are computed with respect to the dispersion in length and width of the voids. Results show up to 9.1% scattering in the elastic constants.     

Influence of Nonlinear Local Properties on Effective Transport

The assumption of constant local coefficients is one of the first restrictions in most of the smoothing theories for transport in porous media. In this paper we present a formal analysis of the effects produced by nonconstant local transport coefficients on the nonlinear behavior of the effective transport properties. In particular, we use the volume averaging method to study heat transport in a two-component system considering the local thermal conductivities as analytical functions of the temperature. Within this approach we obtain a general expression for the effective nonlinear thermal conductivity dependence on the averaged temperature gradient. The important result is that the effective conductivity is obtained by a linearly bounded problem (the closure problem), just as if the conductivities were constants, by replacing the constant conductivities by the actual temperature dependent ones. As an example, we model the porous medium as cylindrical inclusions in a periodic array and solve the closure problem for the case of the one-equation model. We analyze the values of the second derivative of the thermal conductivity with respect to the temperature to establish the range where the nonlinear corrections must be considered to correctly describe the effective transport.     

Modified projective and modified function projective synchronization of a class of real nonlinear systems and a class of complex nonlinear systems

The modified projective and modified function projective synchronization of a class of chaotic real nonlinear systems, or a class of chaotic complex nonlinear systems, have been widely reported in the previous studies, respectively. In the paper, the modified projective and modified function projective synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems are first investigated. Based on the Lyapunov stability theory, the drive real system and response complex system can be synchronized up to the desired scaling constants and functions, respectively. The corresponding numerical simulations are performed to verify and demonstrate the validity and feasibility of the presented idea.     

Bending instabilities of elastic tubes

The present paper examines instabilities of long thin elastic tubes. Both initially straight and initially bent tubes are analyzed under in-plane bending. Tube response, a combination of ovalization instability and bifurcation instability (buckling), is investigated using a nonlinear finite element (FE) technique, which employs polynomial functions in the longitudinal tube direction and trigonometric functions to describe cross-sectional deformation. It is demonstrated that the interaction between the two instability modes depends on the value and the sign of the initial tube curvature. The ovalization of initially bent tubes is examined in detail and, in particular, the case of opening moments. Furthermore, the paper emphasizes on bifurcation instability. It is shown that buckling may occur prior to or beyond the ovalization limit point, depending on the value of the initial curvature. Using the nonlinear FE formulation, the location of bifurcation on the primary path is detected, post-buckling equilibrium paths are traced, and the corresponding wavelengths of the buckled configurations are calculated. Moreover, results over a wide range of initial curvature values are presented, extending the findings of previous works. Finally, several analytical approaches, introduced in previous research works, are also employed to estimate the moments causing ovalization and bifurcation instability. These approaches are based on nonlinear flexible shell theory or simplified ring analysis. The efficiency and accuracy of those analytical methods with respect to the nonlinear FE formulation are examined.     

Stability of solutions of one class of nonlinear dynamic equations

We study the stability of the zero solution of a nonlinear dynamic equation on a time scale under certain assumptions on the right-hand side of this equation. In addition to conditions for the existence and uniqueness of a solution of the Cauchy problem, we also assume that the exponential function of the linear approximation is bounded, and the norms of the nonlinear part and its derivatives with respect to the components of the space variable are majorized by power functions of the norm of the space variable. Using the generalized method of Lyapunov functions, we obtain sufficient conditions for the stability of the zero solution of the nonlinear equation under consideration.     

随机桁架结构几何非线性问题的混合摄动-伽辽金法求解

提出应用混合摄动$\!$-$\!$-$\!$伽辽金法求解随机桁架结构的几何非线性问题.将含位移项的随机割线弹性模量以及随机响应表示为幂多项式展开,利用高阶摄动方法确定随机结构几何非线性响应的幂多项式展开的各项系数.将随机响应的各阶摄动项假定为伽辽金试函数,运用伽辽金投影对试函数系数进行求解,从而得到随机桁架结构几何非线性响应的显式表达式.同已有的随机伽辽金法相比,本文所给的试函数由摄动解的线性组合而成,在求解非线性问题时,试函数的获取具有自适应性.数值算例结果表明,对于具有不同概率分布的多随机变量问题,本文方法无需对随机变量的概率分布形式进行转换,避免了转换误差,因而比同阶的广义正交多项式方法(generalizedpolynomial chaos, GPC)计算精度高.同时,在结果精度相当时,和GPC方法相比,本文方法得到的试函数系数的非线性方程维度不大,方程的求解工作量小且更易求解.当随机量涨落较大时,混合摄动$\!$-$\!$-$\!$伽辽金法计算所得的结构响应的各阶统计矩比高阶摄动法所得结果更逼近于蒙特卡洛模拟结果,显示了该方法对几何非线性随机问题求解的有效性.      

Adaptive tracking control for a class of nonlinear non-strict-feedback systems

This paper focuses on the adaptive tracking control problem for a class of nonlinear non-strict-feedback systems. By introducing a compact set, the restrictive assumption that the lower bounds of the control gain functions must be positive constants is canceled in the proposed method, and the compact set is proved to be invariant set eventually. The functions in non-strict-feedback system are no longer required to be differentiable, and the neural networks are constructively used to deal with the unknown system functions, which contain the whole state variables of the non-strict-feedback system. Furthermore, it is rigorously proved that all the closed-loop signals are bounded and the tracking error converges to a small residual set asymptotically. Finally, simulation examples are provided to demonstrate the effectiveness of the designed method.     

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.     

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.     

Finite element model updating using variable separation

In the field of structural dynamics, reliable finite element response predictions are becoming increasingly important to industry and there is a genuine interest to improve these in the light of measured frequency response functions. Unlike modal-based model updating formulations, response-based methods have been applied only with limited success due to incomplete measurements and numerical ill-conditioning problems. The least squares approximation method is one of the methods used but often poses a problem of pseudo inverse due to the number of incomplete measurements. The proposed algorithm is a modification and extension of a previously-developed nonlinear least squares method for damage detection and finite element model updating. The paper derives explicit expressions for the first and second order partial derivatives with respect to the correction parameters and for the Jacobian matrix used in the Newton–Raphson solution of the nonlinear set of equations in order to avoid the pseudo inverse and to build a symmetrical system. The proposed method, assigned to a frequency parameterization which considers the minimum distance to be minimized, shows a good numerical stability. The performance of the method in localizing structural damage and updating model is examined using simulated measurements.     

Implicit Difference Methods for First-Order Partial Differential Functional Equations

We present a new class of numerical methods for quasilinear first-order partial differential functional equations. The numerical methods are difference schemes implicit with respect to time variable. We give a complete convergence analysis for the methods and show by an example that the new methods are considerably better than explicit schemes. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators with respect to the functional variable. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 201–215, April–June, 2005.     

A comparative analysis of methods for solving equations in the nonlinear elasticity theory

An approach to solving a variational equation used in geometrically and physically nonlinear problems of deformable body mechanics is considered. This approach is based on the continuation of a solution with respect to the loading parameter. Large systems of nonlinear ordinary differential equations arise in such problems. Usually, these systems are solved by the Euler methods. It is proposed to use the Runge-Kutta and multistep methods and to estimate the total computational cost. A dependence of numerical errors on the number of integration steps is obtained. An optimal method for solving nonlinear problems is chosen on the basis of this dependence.     

Orbit trap rendering methods for generating colorful symmetric images in three-dimensional space

Automatic generation of colorful symmetric images is considered by using orbit trap rendering methods. Orbit traps with appropriate symmetries are constructed to determine the density functions for the creation of colorful images. Furthermore, complete proofs of the orbit trap methods compatible with equivariant functions with respect to the tetrahedral and cubic symmetries are given.