A geometrically nonlinear (3,2) reduced degree-of-freedom unified zigzag laminated beam element for large deformation analysis

A geometrically nonlinear (3,2) unified zigzag beam element is developed with a reduced number of degree-of-freedom for the large deformation analysis. The main merit of the beam element model is the Kirchhoff and Cauchy shear stress solution for large deformation and large strain analysis is more accurate. The geometrically nonlinearity is considered in the calculation of the zigzag coefficients. Thus, the results of shear Cauchy stress are matching well with solid element analysis in case of the beam with aspect ratio greater than 20 under large deformation. The zigzag coefficients are derived explicitly. The Green strain and the second Piola Kirchhoff stress are used. The second Piola Kirchhoff shear stress is continuous at the interface between adjacent layers priori. The bottom surface second Piola Kirchhoff shear stress condition is used to determine the zigzag coefficient and the top surface second Piola Kirchhoff shear stress condition is used to reduce one degree-of-freedom. The nonlinear finite element equations are derived. In the numerical tests, several benchmark problems with large deformation are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for beam with aspect ratio greater than 20. The second Piola Kirchhoff and Cauchy shear stress accuracy is also good. A convergence study is also presented.     

The radial basis reproducing kernel particle method for geometrically nonlinear problem of functionally graded materials

In this paper, the radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is proposed for solving geometrically nonlinear problem of functionally graded materials (FGM). Compared with the RKPM, the advantages of the proposed method are that it can eliminate the negative effect of different kernel functions on the computational accuracy, and has higher computational accuracy and stability. Using the Total Lagrange (T.L.) formulation and the weak form of Galerkin integration, the corresponding formulae for geometrically nonlinear problem of FGM are derived. The penalty factor, shaped parameter of the RBF, the control parameter of influence domain radius, loading step number and node distribution are discussed. Furthermore, the effects of different gradient functions and exponents on displacement and stress are analyzed. Newton-Raphson (N-R) iterative method is utilized for numerical solution. The proposed method is correct and effective for solving geometrically nonlinear problem of FGM, which can be demonstrated by several numerical examples.     

Stabilization of the dynamic elastica only by damping torque

We consider a system of partial differential equations called the dynamic elastica, which describes the motion of a geometrically nonlinear (i.e., largely deflecting) elastic beam and is derived based on Euler–Bernoulli’s assumption. The aim of the paper is to examine the effect of damping torque (i.e., external torque generated as a negative feedback of the angular velocity of the beam’s centerline) on the stability of the elastica.     

Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods

This article introduces a coupled methodology for the numerical solution of geometrically nonlinear static and dynamic problem of thin rectangular plates resting on elastic foundation. Winkler–Pasternak two-parameter foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the discrete singular convolution (DSC) and harmonic differential quadrature (HDQ) methods, respectively. Two different realizations of singular kernels such as the regularized Shannon’s kernel (RSK) and Lagrange delta (LD) kernel are selected as singular convolution to illustrate the present DSC algorithm. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse, an N-wave pulse, and a triangular load are investigated and the results are presented graphically. The effects of Winkler and Pasternak foundation parameters, influence of mass of foundation on the response have been investigated. In addition, the influence of damping on the dynamic analysis has been studied. The accuracy of the proposed DSC–HDQ coupled methodology is demonstrated by the numerical examples.     

Design and analysis of two discrete-time ZD algorithms for time-varying nonlinear minimization

Nonlinear minimization, as a subcase of nonlinear optimization, is an important issue in the research of various intelligent systems. Recently, Zhang et al. developed the continuous-time and discrete-time forms of Zhang dynamics (ZD) for time-varying nonlinear minimization. Based on this previous work, another two discrete-time ZD (DTZD) algorithms are proposed and investigated in this paper. Specifically, the resultant DTZD algorithms are developed for time-varying nonlinear minimization by utilizing two different types of Taylor-type difference rules. Theoretically, each steady-state residual error in the DTZD algorithm changes in an O(τ ³) manner with τ being the sampling gap. Comparative numerical results are presented to further substantiate the efficacy and superiority of the proposed DTZD algorithms for time-varying nonlinear minimization.     

Flexure Hinge Mechanisms Modeled by Nonlinear Euler-Bernoulli-Beams

A flexure hinge is an innovative engineering solution for providing relative motion between two adjacent stiff members by the elastic deformation of an arbitrary shaped flexible connector. In the literature, modeling of compliant mechanisms incorporating flexure hinges is mainly focused on linear methods. However, geometrically nonlinear effects cannot be ignored generally. This study presents a nonlinear modeling technique for flexure hinges based on the Euler-Bernoulli beam theory, in contrast to the predominant linear modeling approaches. Higher order beam elements of variable cross-section are employed to model the flexure hinge region. A Newton-Raphson scheme is applied to solve the resulting nonlinear system equations. The proposed approach reduces the overall degrees of freedom and is computationally efficient compared to commonly applied 3D finite element methods. A compliant displacement amplification mechanism is studied by means of the proposed method, where an excellent agreement with results of a reference solution is achieved. The modeling approach is suitable for the structural optimization of compliant mechanisms towards a less intuitive design process. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bending of a Composite Material Strip with Curved Structure in the Geometrically Nonlinear Statement

To date, bending problems for strips or plates made of composite materials with curved structure have been investigated only in the linear statement. However, in many cases, the necessity arises to investigate the corresponding bending problems in the geometrically nonlinear statement. Therefore, in the present paper, some bending problems for a composite strip with a periodically curved structure is investigated in such a statement using the exact nonlinear equations of elasticity theory in Lagrangian coordinates. The numerical results are obtained by employing the FEM with the use of the Newton-Raphson and the Modified Newton-Raphson algorithms.     

A spatial shear deformable beam finite element based on the absolute nodal coordinate formulation

In the present paper a three-dimensional beam finite element undergoing large deformations is proposed. Since the definition of the proposed finite element is based on the absolute nodal coordinate formulation (ANCF), no rotational coordinates occur in the formulation. In the current approach, the orientation of the cross section is parameterized by means of slope vectors. Since those are no unit vectors, the cross-section can deform, similar to existing thick beam and shell elements. The nodal displacements and the directional derivatives of the displacements are chosen as nodal coordinates, but in contrast to standard ANCF elements, the proposed formulation is based on the two transversal slope vectors per node only. Different approaches for the virtual work of elastic forces are presented: a continuum mechanics based formulation, as well as a structural mechanics based formulation, which is in accordance with classical nonlinear beam finite elements. Since different interpolation functions as in standard ANCF elements are used, a much better convergence rate (up to order four) can be obtained. Therefore, the present element has high potential for application in geometrically nonlinear problems. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularization methods for optimization problems with probabilistic constraints

We analyze nonlinear stochastic optimization problems with probabilistic constraints on nonlinear inequalities with random right hand sides. We develop two numerical methods with regularization for their numerical solution. The methods are based on first order optimality conditions and successive inner approximations of the feasible set by progressive generation of p-efficient points. The algorithms yield an optimal solution for problems involving α-concave probability distributions. For arbitrary distributions, the algorithms solve the convex hull problem and provide upper and lower bounds for the optimal value and nearly optimal solutions. The methods are compared numerically to two cutting plane methods.     

A Variational Integrator for the Chaplygin–Timoshenko Sleigh

The paper introduces a mechanically inspired nonholonomic integrator for numerical simulation of the dynamics of a constrained geometrically exact beam that is a field-theoretic analogue of the Chaplygin sleigh. The integrator features an exact constraint preservation, an excellent numerical energy conservation throughout a large number of iterations, while avoiding the use of unnecessary Lagrange multipliers. Simulations of the dynamics of the constrained beam reveal typical for nonholonomic system’s behavior, such as motion reversals and locomotion generation.

     

Conservative numerical methods for the Full von Kármán plate equations

This article is concerned with the numerical solution of the full dynamical von Kármán plate equations for geometrically nonlinear (large‐amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von Kármán system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semidiscrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretizations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1948–1970, 2015     

Mixed algorithm in searches of mechanical system steady-state conditions for low precision of the state estimation

This paper focuses on algorithms of numerical solution of nonlinear system of equations. Analytical formulas of their nonlinear functions my not be calculated with the requested precision. Additionally, analytical formulas of the partial derivatives are unknown. They are evaluated numerically by finite differences method. It effects in erroneous estimations. Described situation is critical when steady state conditions are searched for mechanical systems. According to the precision of the numerical procedures, their dynamic equations are known with limited precision. This same stands for the system's final conditions (obtained by a numerical integration). It the actual case, the classical Newton-Raphson algorithm can be ineffective. As an alternative, a mixed search algorithm is proposed in the paper. By contrast to the classical algorithm, within the search direction defined by the Newton-Raphson indication, (potentially erroneous) a local minimum is searched. Both the algorithms are tested on analytical functions; on randomized functions and on a model of a mechanical system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of different least-squares mixed finite element formulations for hyperelasticity

The main goal of this contribution is the solution of geometrically nonlinear problems using the mixed least-squares finite element method (LSFEM). An investigation of a hyperelastic material law based on logarithmic deformation measures is performed. The basis for the proposed LSFEM is a div-grad first-order system consisting of the equilibrium condition and the constitutive equation, see e.g. Cai and Starke [1]. For the interpolation of the solution variables vector-valued Raviart-Thomas functions for the approximation of the stresses and standard Lagrange polynomials for the displacements are used. In order to show the performance of the presented formulations a numerical example is investigated, where we compare the different interpolation combinations used. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Highly accurate algorithms endowing with boundary functions for solving a nonlinear beam equation involving an integral term

In this paper, the problem of a nonlinear beam equation involving an integral term of the deformation energy, which is unknown before the solution, under different boundary conditions with simply supported, 2‐end fixed, and cantilevered is investigated. We transform the governing equation into an integral equation and then solve it by using the sinusoidal functions, which are chosen both as the test functions and the bases of numerical solution. Because of the orthogonality of the sinusoidal functions, we can find the expansion coefficients of the numerical solution that are given in closed form by using the Drazin inversion formula. Furthermore, we introduce the concept of fourth‐order and fifth‐order boundary functions in the solution bases, which can greatly raise the accuracy over 4 orders than that using the partial boundary functions. The iterative algorithms converge very fast to find the highly accurate numerical solutions of the nonlinear beam equation, which are confirmed by 6 numerical examples.     

Numerical methods for the Lotka–McKendrick's equation

The Lotka–McKendrick's model is a well-known model which describes the evolution in time of the age structure of a population. In this paper we consider this linear model and discuss a range of methods for its numerical solution. We take advantage of different analytical approaches to the system, to design different numerical methods and compare them with already existing algorithms. In particular we set up some algorithms inspired by the approach based on Volterra integral equations and we also consider a direct approach based on the nonlinear system that describes the evolution of the age profile of the population.     

Advanced combinations of splitting–shooting–integrating methods for digital image transformations

An image consists of many discrete pixels with greyness of different levels, which can be quantified by greyness values. The greyness values at a pixel can also be represented by an integral as the mean of continuous greyness functions over a small pixel region. Based on such an idea, the discrete images can be produced by numerical integration; several efficient algorithms are developed to convert images under transformations. Among these algorithms, the combination of splitting–shooting–integrating methods (CSIM) is most promising because no solutions of nonlinear equations are required for the inverse transformation. The CSIM is proposed in [6] to facilitate images and patterns under a cycle transformations T⁻¹T, where T is a nonlinear transformation. When a pixel region in two dimensions is split into N² subpixels, convergence rates of pixel greyness by CSIM are proven in [8] to be only O(1/N). In [10], the convergence rates O_p(1/N^1.5) in probability and O_p(1/N²) in probability using a local partition are discovered. The CSIM is well suited to binary images and the images with a few greyness levels due to its simplicity. However, for images with large (e.g., 256) multi-greyness levels, the CSIM still needs more CPU time since a rather large division number is needed.In this paper, a partition technique for numerical integration is proposed to evaluate carefully any overlaps between the transformed subpixel regions and the standard square pixel regions. This technique is employed to evolve the CSIM such that the convergence rate O(1/N²) of greyness solutions can be achieved. The new combinations are simple to carry out for image transformations because no solutions of nonlinear equations are involved in, either. The computational figures for real images of 256×256 with 256 greyness levels display that N=4 is good enough for real applications. This clearly shows validity and effectiveness of the new algorithms in this paper.     

Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories

This study investigates the nonlinear free vibration of functionally graded material (FGM) beams by different shear deformation theories. The volume fractions of the material constituents and effective material properties are assumed to be changing in the thickness direction according to the power-law form. The von Kármán geometric nonlinearity has been considered in the formulation. The Ritz method and Lagrange equation are adopted to yield the discrete formulations. A direct numerical integration method for the motion equation in matrix form is developed to solve the nonlinear frequencies of FGM beams. Comparing with the global concordant deformation assumption (GCDA), a new deformation assumption named as local concordant deformation assumption (LCDA) is proposed in this study. The LCDA fits with the real deformation of the vibrating beam better, thus more accurate results of the nonlinear frequency can be expected. In numerical results, the comparison study of the GCDA and LCDA is carried out. In addition, the effects of power-law index, slenderness ratio and maximum deflection for different shear deformation theories and boundary conditions on the nonlinear frequency of the beam are discussed.     

Constrained euler buckling

Summary We consider elastic buckling of an inextensible beam confined to the plane and subject to fixed end displacements, in the presence of rigid, frictionless side-walls which constrain overall lateral displacements. We formulate the geometrically nonlinear (Euler) problem, derive some analytical results for special cases, and develop a numerical shooting scheme for solution. We compare these theoretical and numerical results with experiments on slender steel beams. In contrast to the simple behavior of the unconstrained problem, we find a rich bifurcation structure, with multiple branches and concomitant hysteresis in the overall load-displacement curves. Dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.