薄壳屈曲问题的有限质点法求解

有限质点法是一种新型的数值方法,它从牛顿力学的角度出发,以质点为研究对象,将求解域离散为质点系统,同时着眼于该系统内各个质点所受的力,进而追踪到整个质点系统的运动状态,在处理结构或机构的大变位、大变形等非线性问题时具有独特的优势。该文将有限质点法应用于薄壳的屈曲问题研究,为追踪其完整的屈曲路径,将显式弧长法的加载策略与其相结合;针对屈曲或大变形后出现的接触问题,改进了一种适用于显式求解的接触算法。最后,通过自编程序,分别选取薄壳屈曲问题的静力、动力等经典算例,并将该方法的计算结果和相关文献及试验结果进行了对比。结果表明,该文的方法用于薄壳的屈曲问题求解是可行的,能有效捕捉薄壳屈曲的完整过程。     

A modified Newton‐type Koiter‐Newton method for tracing the geometrically nonlinear response of structures

The Koiter‐Newton (KN) method is a combination of local multimode polynomial approximations inspired by Koiter's initial postbuckling theory and global corrections using the standard Newton method. In the original formulation, the local polynomial approximation, called a reduced‐order model, is used to make significantly more accurate predictions compared to the standard linear prediction used in conjunction with Newton method. The correction to the exact equilibrium path relied exclusively on Newton‐Raphson method using the full model. In this paper, we proposed a modified Newton‐type KN method to trace the geometrically nonlinear response of structures. The developed predictor‐corrector strategy is applied to each predicted solution of the reduced‐order model. The reduced‐order model can be used also in the correction phase, and the exact full nonlinear model is applied only to calculate force residuals. Remainder terms in both the displacement expansion and the reduced‐order model are well considered and constantly updated during correction. The same augmented finite element model system is used for both the construction of the reduced‐order model and the iterations for correction. Hence, the developed method can be seen as a particular modified Newton method with a constant iteration matrix over the single KN step. This significantly reduces the computational cost of the method. As a side product, the method has better error control, leading to more robust step size adaptation strategies. Numerical results demonstrate the effectiveness of the method in treating nonlinear buckling problems.     

Improved nonlinear buckling analysis of structures

The theory of structural stability is both an important and a difficult subject, whose main field of application is found in the design of thin wall lightweight structures and shells. It is unavoidable to employ nonlinear model and to use finite element numerical method in order to obtain the critical load under the action of which the instability of structures will occur. Thus, it becomes very important how to simplify such a nonlinear problem that appears in instability analysis. In this paper, an improvement on previous nonlinear buckling analysis is proposed, in which the emphasis is on the shortcut of calculation of pre-buckling fundamental path. The nonlinear equation used to solve the displacement vector is translated into the linear one to solve the load factor by means of the concept of energy conservation. The detailed procedure to calculate the nonlinear buckling load is presented and two simple examples are also shown as the application of suggested method. The theoretical analysing and numerical examples show that the suggested method is valid for predicting the nonlinear critical loads of structures.     

Material and Geometric Nonlinear Analysis of Functionally Graded Plate-Shell Type Structures

A nonlinear formulation for general Functionally Graded Material plate-shell type structures is presented. The formulation accounts for geometric and material nonlinear behaviour of these structures. Using the Newton–Raphson incremental-iterative method, the incremental equilibrium path is obtained, and in case of snap-through occurrence the automatic arc-length method is used. This simple and fast element model is a non-conforming triangular flat plate/shell element with 24 degrees of freedom for the generalized displacements. It is benchmarked in the solution of some illustrative plate- shell examples and the results are presented and discussed with numerical alternative models. Benchmark tests with material and geometrically nonlinear behaviour are also proposed.     

Topology optimization of energy absorbing structures with maximum damage constraint

A novel density‐based topology optimization framework for plastic energy absorbing structural designs with maximum damage constraint is proposed. This framework enables topologies to absorb large amount of energy via plastic work before failure occurs. To account for the plasticity and damage during the energy absorption, a coupled elastoplastic ductile damage model is incorporated with topology optimization. Appropriate material interpolation schemes are proposed to relax the damage in the low‐density regions while still ensuring the convergence of Newton‐Raphson solution process in the nonlinear finite element analyses. An effective method for obtaining path‐dependent sensitivities of the plastic work and maximum damage via adjoint method is presented, and the sensitivities are verified by the central difference method. The effectiveness of the proposed methodology is demonstrated through a series of numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

POD‐based model reduction with empirical interpolation applied to nonlinear elasticity

The constantly rising demands on finite element simulations yield numerical models with increasing number of degrees‐of‐freedom. Due to nonlinearity, be it in the material model or of geometrical nature, the computational effort increases even further. For these reasons, it is today still not possible to run such complex simulations in real time parallel to, for example, an experiment or an application. Model reduction techniques such as the proper orthogonal decomposition method have been developed to reduce the computational effort while maintaining high accuracy. Nonetheless, this approach shows a limited reduction in computational time for nonlinear problems. Therefore, the aim of this paper is to overcome this limitation by using an additional empirical interpolation. The concept of the so‐called discrete empirical interpolation method is translated to problems of solid mechanics with soft nonlinear elasticity and large deformations. The key point of the presented method is a further reduction of the nonlinear term by an empirical interpolation based on a small number of interpolation indices. The method is implemented into the finite element method in two different ways, and it is extended by using different solution strategies including a numerical as well as a quasi‐Newton tangent. The new method is successfully applied to two numerical examples concerning hyperelastic as well as viscoelastic material behavior. Using the extended discrete empirical interpolation method combined with a quasi‐Newton tangent enables reductions in computational time of factor 10 with respect to the proper orthogonal decomposition method without empirical interpolation. Negligibly, orders of error can be reached. Copyright © 2015 John Wiley & Sons, Ltd.     

LATIN: A new view and an extension to wave propagation in nonlinear media

The LATIN (acronym of LArge Time INcrement) method was originally devised as a non‐incremental procedure for the solution of quasi‐static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time‐dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. Copyright © 2017 John Wiley & Sons, Ltd.     

含界面脱粘及表板基体开裂损伤的复合材料夹层板非线性稳定性的研究

本文作者基于"zig-zag"模型和Mindlin一阶剪切变形板理论,推导了复合材料夹层板屈曲分析的有限元列式,在该列式中考虑了面板的横向剪切变形和芯体的面内刚度对夹层板力学性能的影响。针对具有面板和芯体间界面脱粘和纤维增强树脂基体微裂纹损伤的夹层板损伤特征,分别提出了分层模型和多标量损伤模型,并推导了多标量形式的损伤本构关系。采用修正的 Newton-Raphson迭代格式求解含损复合材料夹层板的非线性稳定性性状。通过算例研究了脱粘面积、基体的损伤演化、表板的铺设方式及载荷形式对复合材料夹层板屈曲性态的影响。本文作者给出的有限元模型和结论,对复合材料夹层板结构设计的损伤容限的制定具有一定的参考价值。     

Combined interior‐point method and semismooth Newton method for frictionless contact problems

In the present paper, a solution scheme is proposed for frictionless contact problems of linear elastic bodies, which are discretized using the finite element method with lower order elements. An approach combining the interior‐point method and the semismooth Newton method is proposed. In this method, an initial active set for the semismooth Newton method is obtained from the approximate optimal solution by the interior‐point method. The simplest node‐to‐node contact model is considered in the present paper, that is, pairs of matching nodes exist on the contact surfaces. However, the discussions can be easily extended to a node‐to‐segment or segment‐to‐segment contact model. In order to evaluate the proposed method, a number of illustrative examples of the frictionless contact problem are shown. The proposed combined method is compared with the interior‐point method and the semismooth Newton method. Two numerical examples that are difficult to solve using the semismooth Newton method are solved effectively using the proposed combined method. It is shown that the proposed method converges within far fewer iterations than the semismooth Newton methods or the interior‐point method. Copyright © 2009 John Wiley & Sons, Ltd.     

Cyclic steady states of nonlinear electro‐mechanical devices excited at resonance

We present an efficient numerical method to solve for cyclic steady states of nonlinear electro‐mechanical devices excited at resonance. Many electro‐mechanical systems are designed to operate at resonance, where the ramp‐up simulation to steady state is computationally very expensive – especially when low damping is present. The proposed method relies on a Newton–Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time‐stepping from zero initial conditions. We use a recently developed high‐order Eulerian–Lagrangian finite element method in combination with an energy‐preserving dynamic contact algorithm in order to solve the coupled electro‐mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro‐electro‐mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed‐ups of the form S  = 0.6·ξ  ^? 0.8, where ξ is the linear damping ratio of the corresponding resonance frequency. Copyright © 2016 John Wiley & Sons, Ltd.     

Accurate and efficient a posteriori account of geometrical imperfections in Koiter finite element analysis

The Koiter method recovers the equilibrium path of an elastic structure using a reduced model, obtained by means of a quadratic asymptotic expansion of the finite element model. Its main feature is the possibility of efficiently performing sensitivity analysis by including a posteriori the effects of the imperfections in the reduced nonlinear equations. The state‐of‐art treatment of geometrical imperfections is accurate only for small imperfection amplitudes and linear pre‐critical behaviour. This work enlarges the validity of the method to a wider range of practical problems through a new approach, which accurately takes into account the imperfection without losing the benefits of the a posteriori treatment. A mixed solid‐shell finite element is used to build the discrete model. A large number of numerical tests, regarding nonlinear buckling problems, modal interaction, unstable post‐critical and imperfection sensitive structures, validates the proposal. Copyright © 2017 John Wiley & Sons, Ltd.     

Stabilization of bi‐linear mixed finite elements for tetrahedra with enhanced interpolation using volume and area bubble functions

In order to overcome the oscillatory effects of the mixed bi‐linear Galerkin formulation for tetrahedral elements, a stabilization approach is presented. To this end the mixed method of incompatible modes and the mixed method of enhanced strains are reformulated, thus giving both the interpretation of a mixed finite element method with stabilization terms. For non‐linear problems, these are non‐linearly dependent on the current deformation state and therefore are replaced by linearly dependent stabilization terms. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step, typically arising for mixed‐enhanced elements, is completely avoided. The stabilization matrices for the mixed method of incompatible modes and the mixed method of enhanced strains are obtained with volume and area bubble functions. Various numerical examples are presented, which illustrate successfully the stabilization effect for bi‐linear tetrahedral elements. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability criteria for neutral systems with time‐varying delays and nonlinear uncertainties

Abstract

The asymptotic stability problem for a class of neutral systems with time‐varying delays and nonlinear uncertainties is investigated in this paper. LMI‐based delay‐dependent criteria are proposed to guarantee the asymptotic stability of the considered systems. New Lyapunov‐Krasovskii functional and Leibniz‐Newton formulae are used to find the delay‐dependent stability results. Finally, some numerical examples are illustrated to show the improved results from using this method.     

拱结构的弹性二次屈曲性能

本文详细阐述了拱的非线性屈曲的两种形式,即极值点屈曲和二次分叉屈曲;建立了一种简捷的计算二次分叉屈曲的方法,并且通过例题证实了其可靠性;通过对拱屈曲前后荷载--位移曲线的全过程分析,研究了矢跨比下拱的各种屈曲性能;提出的索--拱杂交结构能显著地提高拱的承载力并能有效地防止二次分叉屈曲的发生,计算结果对拱的设计有参考价值。     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Buckling Study of Thin‐walled Channel Beams with Double‐box Flanges in Pure Bending

Abstract: This paper provides the results of numerical and experimental investigations of buckling problems of cold‐formed, thin‐walled channel beams with double‐box flanges under pure bending. A local and global buckling analysis is realised numerically with the use of the finite strip method. A local buckling has been experimentally studied and also numerically with the use of the finite element method. Experimental tests of beams subjected to pure bending are conducted. The results of numerical and experimental investigations are presented and compared. A fundamental influence of double‐box flanges on the critical load is shown.     

Time integration of mechanical systems with elastohydrodynamic lubricated joints using Quasi‐Newton method and projection formulations

This work deals with the efficient time integration of mechanical systems with elastohydrodynamic (EHD) lubricated joints. Two novel approaches are presented. First, a projection function is used to formulate the well‐known Swift–Stieber cavitation condition and the mass‐conservative cavitation condition of Elrod as an unconstrained problem. Based on this formulation, the pressure variable from the EHD problem is added to the dynamic equations of a multi‐body system in a monolithic manner so that cavitation is solved within a global iteration. Compared with a partitioned state‐of‐the‐art formulation, where the pressure is solved locally in a nonlinear force element, this global search reduces simulation time. Second, a Quasi‐Newton method of DeGroote is applied during time integration to solve the nonlinear relation between pressure and deformation. Compared with a simplified Newton method, the calculation of the time‐consuming parts of the Jacobian are avoided, and therefore, simulation time is reduced significantly, when the Jacobian is calculated numerically. Solution strategies with the Quasi‐Newton method are presented for the partitioned formulation as well as for the new DAE formulations with projection function. Results are given for a simulation example of a rigid shaft in a flexible bearing. Copyright © 2016 John Wiley & Sons, Ltd.     

Discontinuous Galerkin methods for non‐linear elasticity

This paper presents the formulation and a partial analysis of a class of discontinuous Galerkin methods for quasistatic non‐linear elasticity problems. These methods are endowed with several salient features. The equations that define the numerical scheme are the Euler–Lagrange equations of a one‐field variational principle, a trait that provides an elegant and simple derivation of the method. In consonance with general discontinuous Galerkin formulations, it is possible within this framework to choose different numerical fluxes. Numerical evidence suggests the absence of locking at near‐incompressible conditions in the finite deformations regime when piecewise linear elements are adopted. Finally, a conceivable surprising characteristic is that, as demonstrated with numerical examples, these methods provide a given accuracy level for a comparable, and often lower, computational cost than conforming formulations. Stabilization is occasionally needed for discontinuous Galerkin methods in linear elliptic problems. In this paper we propose a sufficient condition for the stability of each linearized non‐linear elastic problem that naturally includes material and geometric parameters; the latter needed to account for buckling. We then prove that when a similar condition is satisfied by the discrete problem, the method provides stable linearized deformed configurations upon the addition of a standard stabilization term. We conclude by discussing the complexity of the implementation, and propose a computationally efficient approach that avoids looping over both elements and element faces. Several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous Galerkin method within the class. Copyright © 2006 John Wiley & Sons, Ltd.     

Non‐linear analysis of thin‐walled members of open cross‐section

The paper presents a means of determining the non‐linear stiffness matrices from expressions for the first and second variation of the Total Potential of a thin‐walled open section finite element that lead to non‐linear stiffness equations. These non‐linear equations can be solved for moderate to large displacements. The variations of the Total Potential have been developed elsewhere by the authors, and their contribution to the various non‐linear matrices is stated herein. It is shown that the method of solution of the non‐linear stiffness matrices is problem dependent. The finite element procedure is used to study non‐linear torsion that illustrates torsional hardening, and the Newton–Raphson method is deployed for this study. However, it is shown that this solution strategy is unsuitable for the second example, namely that of the post‐buckling response of a cantilever, and a direct iteration method is described. The good agreement for both of these problems with the work of independent researchers validates the non‐linear finite element method of analysis. Copyright © 1999 John Wiley & Sons, Ltd.