A Koiter‐Newton approach for nonlinear structural analysis

A new approach termed the Koiter‐Newton is presented for the numerical solution of a class of elastic nonlinear structural response problems. It is a combination of a reduction method inspired by Koiter's post‐buckling analysis and Newton arc‐length method so that it is accurate over the entire equilibrium path and also computationally efficient in the presence of buckling. Finite element implementation based on element independent co‐rotational formulation is used. Various numerical examples of buckling sensitive structures are presented to evaluate the performance of the method. The examples demonstrate that the method is robust and completely automatic and that it outperforms traditional path‐following techniques. This improved efficiency will open the door for the direct use of detailed nonlinear finite element models in the design optimization of next generation flight and launch vehicles. Copyright © 2013 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.     

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.     

Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver

Linear buckling constraints are important in structural topology optimization for obtaining designs that can support the required loads without failure. During the optimization process, the critical buckling eigenmode can change; this poses a challenge to gradient‐based optimization and can require the computation of a large number of linear buckling eigenmodes. This is potentially both computationally difficult to achieve and prohibitively expensive. In this paper, we motivate the need for a large number of linear buckling modes and show how several features of the block Jacobi conjugate gradient (BJCG) eigenvalue method, including optimal shift estimates, the reuse of eigenvectors, adaptive eigenvector tolerances and multiple shifts, can be used to efficiently and robustly compute a large number of buckling eigenmodes. This paper also introduces linear buckling constraints for level‐set topology optimization. In our approach, the velocity function is defined as a weighted sum of the shape sensitivities for the objective and constraint functions. The weights are found by solving an optimization sub‐problem to reduce the mass while maintaining feasibility of the buckling constraints. The effectiveness of this approach in combination with the BJCG method is demonstrated using a 3D optimization problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Geometric Imperfection Sensitivity of Curved Panels Under Combined Compression and In‐Plane Bending – A Study Using Adaptive Meshing and DIC

Abstract: The existence of geometric imperfections, either resulting from the use of manufacturing tolerances or because of damage in situ is known to have a detrimental effect on the buckling and post‐buckling behaviour of thin‐walled structures, which depending upon factors such as geometry and loading, can be significant. This paper presents an automated technique, based on the use of topography data obtained from optical measurement, for creating finite meshes representative of the geometry of real structures, which can be analysed to obtain accurate predictions of unstable behaviour. The technique incorporates an algorithm to allow mesh density to be varied across the specimen according to level of curvature thus ensuring the meshes generated are not only accurate but also computationally efficient by reducing the number of degrees of freedom where appropriate. The results of applying this technique to a series of curved panels subject to combined shear and in‐plane bending (an example of a component of an aero engine blade) are reported.     

Optimization of composite structures under multiple load cases using a discrete approach based on lamination parameters

This paper presents a new optimization technique applicable to optimization of composite structures subjected to multiple objectives. The composite structures may be composed of an arbitrary number of laminates. The technique is especially suited for the case where the layers of the laminates may assume a discrete number of orientations. However, given the efficiency of the technique, it is readily extendable to situations where the ply orientations vary quasi‐continuously, for instance, by one degree in one degree. The high efficiency is obtained through application of lamination parameters, which, in the case of symmetric laminates, consist of only 10 parameters per laminate. Three traditional structures, a rectangular composite plate, a cantilever composite beam, and a stiffened composite panel, are optimized against buckling when subjected to multiple load cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Topology optimization of planar linkage mechanisms

In spite of increasing interest in gradient‐based topology optimization of linkage mechanisms, it is still difficult to solve practical, realistic problems. Besides the apparent difficulty resulting from high nonlinearity, the optimization problem faces other major difficulties: difficulty to satisfy the discrete DOF condition with continuous design variables and lack of intrinsic mechanisms to generate distinct black‐and‐white layouts. To deal with the DOF issue, we propose a new formulation, which maximizes a single objective function, the energy transmittance efficiency. It is shown that the efficiency function maximization handles DOF redundancy and deficiency simultaneously. To obtain distinct linkage layouts, a common practice is to introduce an artificial mass constraint and/or to remove unnecessary links during optimization. However, we do not use any artificial mass constraint but post‐process the optimized result to obtain the final layout by a special post‐processing algorithm. In this study, the linkage design model consists of nonlinear ground bars and zero‐length springs. The springs are used to fix bar‐connecting nodes to the ground, generating pinned joints. After verifying the effectiveness of the proposed approach for four‐bar linkage synthesis, we synthesize an automobile steering mechanism satisfying the Ackermann condition. The steering mechanism problem is solved here for the first time. Copyright © 2014 John Wiley & Sons, Ltd.     

Using the sequential linear integer programming method as a post‐processor for stress‐constrained topology optimization problems

This paper deals with topology optimization of load‐carrying structures defined on discretized continuum design domains. In particular, the minimum compliance problem with stress constraints is considered. The finite element method is used to discretize the design domain into n finite elements and the design of a certain structure is represented by an n‐dimensional binary design variable vector. In order to solve the problems, the binary constraints on the design variables are initially relaxed and the problems are solved with both the method of moving asymptotes and the sparse non‐linear optimizer solvers for continuous optimization in order to compare the two solvers. By solving a sequence of problems with a sequentially lower limit on the amount of grey allowed, designs that are close to ‘black‐and‐white’ are obtained. In order to get locally optimal solutions that are purely {0, 1}ⁿ, a sequential linear integer programming method is applied as a post‐processor. Numerical results are presented for some different test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

不同截面梁构件的刚度和稳定性优化设计

本文运用有限元分析与优化设计软件JIFEX,对五种常用截面梁结构的尺寸和形状进行了抗剪、抗弯、抗扭的刚度优化设计和在轴力、剪力作用下的结构稳定性优化设计.通过对优化设计的计算结果分析,得到了对工程设计有意义的若干结论.然后通过飞机结构中一种波形梁构件的优化,进一步讨论了波形梁的波数对结构稳定性和刚度的影响.     

Topology optimization of finite strain viscoplastic systems under transient loads

A transient finite strain viscoplastic model is implemented in a gradient‐based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark‐beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capability of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.     

An optimization of composite lattice cylinder using the approximate method

Recently, the approximate methods based on continuous models have been developed to perform structural analysis of composite lattice structures due to their relative simplicity and computational efficiency. This paper defines the modified effective stiffness considering the directionally dependent mechanical properties to an intersection of ribs and mode shape function of a composite lattice cylinder. It subsequently presents an approximate method based on the continuous model of conducting a buckling analysis of the composite lattice cylinder with various boundary conditions under uniform compression. This method considers the coupled buckling mode as well as the global and local buckling modes. The validity of the present method is verified by comparing the results of the finite element analysis. In addition, a parametric analysis is performed to investigate the effects of the design parameters on the critical load and buckling mode shape of the composite lattice cylinder based on the present method. Finally, we apply the present method to perform the optimization of a composite lattice cylinder for a high-speed vehicle to minimize the mass. Consequently, it is concluded that the present method is very suitable to optimization of composite lattice cylinders due to their relative simplicity and computational efficiency.     

Robust stability analysis of structures with uncertain parameters using mathematical programming approach

This paper presents a novel mathematical programming approach for the static stability analysis of structures with uncertainties within the framework of FEM. The considered uncertain parameters are material properties, geometry of element cross section, and loading conditions, all of which are described by an interval model. The proposed method formulates the two cases of interest, namely, worst and best buckling load calculation, into a pair of mathematical programming problems. Two straightforward advantages are exhibited by such formulations. The first advantage is that the proposed formulation can overcome the interference on the sharpness of bounds of the buckling load due to the interval dependence issue. The second benefit is that the information of uncertain parameters causing the extremities of buckling load can always be retrieved as by‐products of the uncertain stability analysis. Some numerical examples are presented to illustrate the capability of the proposed method on various structures and the sharpness of the bounds of the buckling load factors. The efficiency and effectiveness of the proposed method are also demonstrated through comparison with the classical Monte Carlo simulation method. Copyright © 2014 John Wiley & Sons, Ltd.     

Experimental and Numerical Post‐buckling Analysis of Thin Aluminium Aeronautical Panels under Shear Load

In this work, a novel multi‐hinged test fixture, which can apply a pure shear load to thin panels, is presented. The new device has been used to perform several experimental tests on the shear behaviour of aluminium alloy panels. Four different test configurations have been considered: stiffened and not stiffened panels, with and without a central rectangular cut‐out. The specimens, made of Al 6082 T6, are tested under shear load boundary conditions up to buckling and post‐buckling occurrence. Furthermore, finite element analyses have been performed in order to simulate the four panel configurations tests. The new test fixture has shown the capability to buckle a panel attaining very large out‐of‐plane displacements, if compared to the outcomes of a traditional picture frame test fixture, under the same external load. Numerical and experimental results are finally compared, showing a satisfactory agreement.     

复合材料层合板屈曲稳定性优化设计及其灵敏度计算方法

笔者在有限元分析基础上研究了以屈曲稳定性作为约束条件或优化目标的复合材料层合板结构优化设计及其灵敏度分析方法,重点讨论了屈曲临界荷载灵敏度对内力场和载荷的依赖关系及其在铺层优化、尺寸优化和形状优化问题中的不同计算方法,并在JIFEX软件中实现了复杂结构复合材料层合板优化设计方法。数值算例验证了本文算法和程序的有效性。     

Post‐buckling analysis of imperfect laminates using finite strips based on a higher‐order plate theory

A finite strip procedure has been developed for the post‐buckling analysis of composite laminates when subjected to progressive end shortening. The finite strips are developed based on a higher‐order shear deformation plate theory and there are nine variables at each nodal line. Initial imperfection expressed in the form of suitable trigonometric function is allowed. Examples including isotropic plates and laminates with arbitrary lay‐up arrangement are presented. Numerical results for laminates with and without initial imperfection are used to illustrate the effect of imperfection. Copyright © 2003 John Wiley & Sons, Ltd.     

Tracing the equilibrium path beyond compound critical points

Procedures to handle critical points with coincident or nearly coincident buckling loads in a geometrically non‐linear continuation process are presented. The proposed branch switching algorithm is based on a Liapunov–Schmidt–Koiter‐type asymptotic reduction which is used for the prediction step into the post‐buckling range. A critical review of the existing methodology for the numerical treatment of post‐bifurcation branches at a multiple bifurcation point is also given. In the review the main emphasis is given to robustness, but also implementational issues and computational requirements are addressed. Some modifications to the existing algorithms are proposed in order to improve their robustness or efficiency. Numerical examples of a flat plate and a thin‐walled structure are shown. Copyright © 1999 John Wiley & Sons, Ltd.     

考虑制造因素的变刚度层合板的抗屈曲铺层优化设计

与常规层合板相比,变刚度层合板的制造、有限元建模分析和铺层设计有其特殊之处。首先对设计时需考虑的制造因素进行了归纳,提出了变刚度层合板的铺层设计要求。然后给出了变刚度层合板的理想模型和考虑丝束宽度模型的建模方法。基于理想模型对ABAQUS的前处理模块进行二次开发,利用编制的参数化建模程序分析了不同铺放角的变刚度层合板的屈曲性能,并讨论了最小曲率半径对铺层的限制和变刚度设计提高屈曲载荷的机制。基于变刚度层合板的抗屈曲机制建立了一种铺层优化设计方法,使用遗传算法经两步优化得到最优铺层。对最优铺层建立考虑丝束宽度的模型以研究丝束宽度和铺层偏移对变刚度层合板抗屈曲铺层优化结果的影响。研究表明,在变刚度层合板的抗屈曲铺层优化中使用简化的理想模型通常来说是可行的。在考虑制造因素的情况下,优化后的变刚度层合板较常规层合板屈曲载荷有显著提高。     

带脱层的复合材料层板的屈曲分析

用基于Mindlin 板理论的有限元方法进行了带脱层损伤的复合材料层板的屈曲载荷分析。为了获得物理上可能的屈曲模态, 即避免上下脱层的相互贯穿, 在接触区域引入一些假想弹簧, 并给出了这些假想弹簧刚度系数的计算公式和接触计算的迭代格式, 通过这些弹簧对原始刚度矩阵进行修正可以有效地求解屈曲载荷特征值分析中的接触问题。数值计算结果表明了本算法的有效性和引入接触分析对这类屈曲分析的重要性。同时, 还对脱层的大小、形状、位置和脱层的纤维铺层方向对屈曲载荷的影响进行了研究。      

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.