Sensitivity computation and shape optimization for a non-linear arch model with limit-points instabilities

A shape optimization method for geometrically non-linear structural mechanics based on a sensitivity gradient is proposed. This gradient is computed by means of an adjoint state equation and the structure is analysed with a total Lagrangian formulation. This classical method is well understood for regular cases, but standard equations have to be modified for limit points and simple bifurcation points. These modifications introduce numerical problems which occur at limit points. Numerical systems are very stiff and the quadratic convergence of Newton–Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables. A geometrically non-linear curved arch is implemented with a finite element method via a formal calculus approach. Thickness and/or shape for differentiable costs under linear and non-linear constraints are optimized. Numerical results are given for linear and non-linear examples and are compared with analytic solutions. © 1998 John Wiley & Sons, Ltd.     

A method for modal reanalysis of topological modifications of structures

A method for structural modal reanalysis for three cases of topological modifications, the number of degrees of freedom (DOFs) is unchanged, decreased, and increased, is presented. In this method, the newly added DOFs are linked to the original DOFs of the modified structure by means of the dynamic reduction so as to obtain the condensed equation. The methods for forming the stiffness and mass increments, Δ K and Δ M , resulting from the three cases of topological modifications of structures are discussed. The extended Kirsch method is used to improve the accuracy of the starting solutions of the initial structure. And then, the eigenvectors of newly added DOFs resulting from topological modification can be recovered. At last, the Rayleigh–Ritz analysis is used to evaluate the eigenvalues and eigenvectors for the modified structure. Three numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural modal reanalysis of three cases of the topological modifications and it is easy to implement on a computer. By comparing with previous method, it can be seen that the present method can give good approximate eigenvalues and eigenvectors, even if the topological modifications are very large. Copyright © 2005 John Wiley & Sons, Ltd.     

平衡矩阵理论的探讨及一索杆梁杂交空间结构的静力和稳定性分析

首先对索杆杂交结构采用平衡矩阵理论进行了探讨。对于索杆杂交结构而言,可以根据平衡矩阵确定体系在初状态几何下的整体预应力分布和大小,但是必须考虑变形相容条件和本构关系即各个单元的柔度分布。只有满足体系柔度分布的杆件截面尺寸的结构才能最终施工张成给定的预应力分布和设计几何。由相容方程和平衡方程提出了包含局部超静定结构的索杆杂交结构的柔度分布确定方法和独立柔度分布模态这一概念。算例分析结果验证了方法的正确性。其次,对一索杆梁杂交空间结构的计算分析方法及其初步的静力及其稳定性进行了研究。基于局部分析法对索杆梁体系的预应力分布和大小进行了优化分析。在优化目标下,最优的预应力分布是外环预应力较大,中环、内环预应力要小很多。并以预应力作为整体自平衡的初始内力情况下对其进行了初步静力和稳定性分析。结果表明柔性体系和刚性体系的杂交实际上只能为刚性体系提供大变形的安全储备,而并不能有效的降低体系的挠度,但是预应力的存在却可以有效的改变梁单元轴力的性质和水平。相比单层网壳,索杆梁体系的稳定性能有了较大提高。     

一种基于冲击振动响应分析的桥梁橡胶支座病害诊断方法

提出了一种利用桥梁在冲击荷载作用下动力响应诊断既有桥梁橡胶支座脱空或者受力不均病害的理论方法。以支座刚度的相对下降量为病害指数,计算桥梁冲击响应及其对病害指数的灵敏度,建立灵敏度方程,采用有限元更新技术和优化算法识别支座刚度。一座铁路简支单箱梁桥和一座公路装配式简支箱梁桥的分析结果表明,利用所提病害诊断方法可以准确识别单个或多个支座的刚度变化量,从而判断桥梁支座是否脱空或受力不均。对于铁路简支单箱梁桥,宜将冲击点和响应点布置在扭转振动较大的梁翼缘上;对于公路装配式简支箱梁桥,宜将冲击点和响应点设置在每片主梁的轴线上。     

A new discretization method and its application to solve incompressible Navier–Stokes equations

 A new numerical method is presented in this paper. This method directly solves partial differential equations in the Cartesian coordinate system. It can be easily applied to solve irregular domain problems without introducing the coordinate transformation technique. The concept of the present method is different from the conventional discretization methods. Unlike the conventional numerical methods where the discrete form of the differential equation only involves mesh points inside the solution domain, the new discretization method reduces the differential equation into a discrete form which may involve some points outside the solution domain. The functional values at these points are computed by the approximate form of the solution along a vertical or horizontal line. This process is called extrapolation. The form of the solution along a line can be approximated by Lagrange interpolated polynomial using all the points on the line or by low order polynomial using 3 local points. In this paper, the proposed new discretization method is first validated by its application to solve sample linear and nonlinear differential equations. It is demonstrated that the present method can easily treat different solution domains without any additional programming work. Then the method is applied to simulate incompressible flows in a smooth expansion channel by solving Navier–Stokes equations. The numerical results obtained by the new discretization method agree very well with available data in the literature. All the numerical examples showed that the present method is very efficient, which is suitable for solving irregular domain problems. Received 19 July 2000     

A local boundary integral equation method for potential problems

Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two-dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques.     

Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads

The governing differential equation for buckling of a multi-step non-uniform column subjected to combined concentrated and distributed axial loads, each step of which has an arbitrary number of cracks with or without spring supports, is expressed in the form of bending moment. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the column. In this paper, the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six important cases. The exact buckling solutions of one-step non-uniform columns are thus found. Then a new approach that combines the exact buckling solution of a one-step column and the transfer matrix method is presented to establish the eigenvalue equation for buckling of a multi-step non-uniform column with spring supports. The main advantage of the proposed method is that the eigenvalue equation for buckling of a non-uniform column with an arbitrary number of cracks, any kind of two end supports and various spring supports at intermediate points can be conveniently determined from a second order determinant. Due to the decrease in the determinant’s order as compared with previously developed procedures the computational time required by the present method can be reduced significantly. A numerical example is given to examine the accuracy of the proposed method and to investigate the effect of cracks on buckling of a multi-step non-uniform column.     

局部修改对结构模态参数的影响范围

本文讨论局部修改对于具有结构阻尼线性系统的模态参数的影响范围.所提出的数值方法和近似作图法可以给出在局部修改刚度或阻尼时系统固有频率和阻尼损耗因子变化范围的上下限.它们可以用来估计局部修改在改进结构动态响应方面所具有的能力,为选择合理的修改部位及修改量提供必要的数据.     

A full tangent stiffness field-boundary-element formulation for geometric and material non-linear problems of solid mechanics

The field-boundary-element method naturally admits the solution algorithm in the incompressible regimes of fully developed plastic flow. This is not the case with the generally popular finite-element method, without further modifications to the method such as reduced integration or a mixed method for treating the dilatational deformation. The analyses by the field-boundary-element method for geometric and material non-linear problems are generally carried out by an incremental algorithm, where the velocities (or displacement increments) on the boundary are treated as the primary variables and an initial strain iteration method is commonly used to obtain the state of equilibrium. For problems such as buckling and diffused tensile necking, involving very large strains, such a solution scheme may not be able to capture the bifurcation phenomena, or the convergence will be unacceptably slow when the post-bifurcation behaviour needs to be analysed. To avoid this predicament, a full tangent stiffness field-boundary-element formulation which takes the initial stress–velocity gradient (displacement gradient) coupling terms accurately into account is presented in this paper. Here, the velocity field both inside and on the boundary are treated as primary variables. The large strain plasticity constitutive equation employed is based on an endochronic model of combined isotropic/kinematic hardening plasticity using the concepts of material director triad and the associated plastic spin. A generalized mid-point radial return algorithm is presented for determining the objective increments of stress from the computed velocity gradients. Numerical results are presented for problems of diffuse necking, involving very large strains and plastic instability, in initially perfect elastic–plastic plates under tension. These results demonstrate the clear superiority of the full tangent stiffness algorithm over the initial strain algorithm, in the context of the integral equation formulations for large strain plasticity.     

Crack-like formation of failure-decided angle points on middle planes of the layers resistant to bending in multi-layer structure - a continuum model

A theoretical model is considered that describes, in a continuum approximation, formation of a segment of angle points on the middle planes of thin layers forming a multi-layer structure. These points are associated with the jumps of the slope of the middle planes on the segment. A 2-D case is dealt with. The structure is assumed to be a half-plane with its boundary parallel to the layers and acted upon by a symmetric distribution of the displacements normal to the boundary. The layers forming the structure are assumed capable of mutually gliding with respect to each other and of revealing their flexure rigidity under the above loading. The continuum approximation to describe the above multi-layer structure has been applied. Physically the above mathematical angle points may (depending on the layer material properties) emerge either as a result of transverse fracture of the layers or as a result of intensive local plastic deformation (formation of the plastic `hinges'). As a result, the bending moment drops drastically, so that it is assumed dropping down to zero. This condition is employed to determine the distribution of the above slope jumps. The segment length is determined by equating the bending moment at the remote (from the boundary) end of the segment to a critical (specified) value of the bending moment. Thus, the problem of determining the slope jumps on the segment is reduced to a Fredholm integral equation of the first kind with the kernel having an integrable singularity. This equation has been solved numerically. The results of the calculations are presented.     

3D structural characterisation, deformation measurements and assessment of low-density wood fibreboard under compression: The use of X-ray microtomography

A low-density wood fibreboard has been compressed along its transversal direction. The experiment was carried out in ESRF synchrotron (ID19) and X-ray microtomographic images were recorded for each state. Stipulating that the fibreboard is a discontinuous material essentially made of air, that the compression simply re-organises the spatial distribution of the fibres and does not involve their intrinsic mechanical properties, we are able to deduce the material points density variations along the thickness of the panel. Good agreement is achieved between the macroscopic deformation of the sample and the microscopic compression rate evaluation. Then, the modifications of structural parameters are investigated by 3D image analysis. The relationship between the local structure and the behaviour of the wood fibreboard are deduced. Finally, a modelling approach allows the local densification to be predicted and confirms the initial hypotheses about the local behaviour of the material. In particular, polyester bonds are not involved.     

Crack-like formation of failure-decided angle points on middle planes of the layers resistant to bending in multi-layer structure - a continuum model

A theoretical model is considered that describes, in a continuum approximation, formation of a segment of angle points on the middle planes of thin layers forming a multi-layer structure. These points are associated with the jumps of the slope of the middle planes on the segment. A 2-D case is dealt with. The structure is assumed to be a half-plane with its boundary parallel to the layers and acted upon by a symmetric distribution of the displacements normal to the boundary. The layers forming the structure are assumed capable of mutually gliding with respect to each other and of revealing their flexure rigidity under the above loading. The continuum approximation to describe the above multi-layer structure has been applied. Physically the above mathematical angle points may (depending on the layer material properties) emerge either as a result of transverse fracture of the layers or as a result of intensive local plastic deformation (formation of the plastic `hinges'). As a result, the bending moment drops drastically, so that it is assumed dropping down to zero. This condition is employed to determine the distribution of the above slope jumps. The segment length is determined by equating the bending moment at the remote (from the boundary) end of the segment to a critical (specified) value of the bending moment. Thus, the problem of determining the slope jumps on the segment is reduced to a Fredholm integral equation of the first kind with the kernel having an integrable singularity. This equation has been solved numerically. The results of the calculations are presented.     

Structural modifications simulated by virtual distortions

A numerical strategy for the simulation of structural modifications by virtual distortions is proposed. Two cases of structural modification are considered: the first concerns modifications of material distribution, and the second modifications of local constitutive relations (e.g. unilateral constraints for stresses or deformations). A formaulation of the fundamental equations of the simulation method is presented. These equations are applicable to the general structural modification problem of a truss-like structure. Then numerical algorithms which refer to particular applications, such as progressive collapse analysis or the analysis of structures with gaps, are discussed. The versatility of the method is illustrated with a number of examples, and the computational advantages of structural modification by the virtual distortion method are discussed.     

Enhanced solution control for physically and geometrically non‐linear problems. Part I—the subplane control approach

Geometrically or physically non‐linear problems are often characterized by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to inefficiency of standard numerical solution techniques. Special numerical procedures are therefore required to pass critical points. This paper presents a solution technique which is based on a constraint equation that is defined on a subplane of the degrees‐of‐freedom (dof's) hyperspace or a hyperspace constructed from specific functions of the degrees‐of‐freedom. This unified approach includes many existing methods which have been proposed by various authors. The entire computational process is driven from only one control function which is either a function of a number of degrees‐of‐freedom (local subplane method) or a single automatically weighted function that incorporates all dof's directly or indirectly (weighted subplane method). The control function is generally computed in many points of the structure, which can be related to the finite element discretization. Each point corresponds to one subplane. In the local subplane method, the subplane with the control function that drives the load adaptation is selected automatically during the deformation process. Part I of this two‐part series of papers fully elaborates the proposed solution strategy, including a fully automatic load control, i.e. load estimation, adaptation and correction. Part II presents a comparative analysis in which several choices for the control function in the subplane method are confronted with classical update algorithms. The comparison is carried out by means of a number of geometrically and physically non‐linear examples. General conclusions are drawn with respect to the efficiency and applicability of the subplane solution control method for the numerical analysis of engineering problems. Copyright © 1999 John Wiley & Sons Ltd.     

A universal method for structural static reanalysis of topological modifications

This paper presents a universal method, iterative combined approximation (iterative CA) approach, for structural static reanalysis of all types of topological modifications. The proposed procedure is basically an approximate two-step method. First, the newly added degrees of freedom (DOFs) are assumed to be linked to the original DOFs of the modified structure by means of the Guyan reduction so as to obtain the condensed equation. Second, the displacements of the original DOFs of the modified structure are solved by using the iterative CA approach. And the displacements of the newly added DOFs resulting from topological modification can be recovered. Four numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural static reanalysis of all types of the topological modifications and it is easy to implement on a computer. Copyright © 2004 John Wiley & Sons, Ltd.     

Material metric, connectivity and dislocations in continua

Summary. In this paper, we present a framework within which the role of the initial material structure of a body, or its developing structure in the course of deformation, is accounted for in the constitutive equation of the material. The problem is dealt with at the local level, i.e., at the material neighborhood. If a neighborhood has structure then its geometry cannot be represented by a Euclidean metric. The entire body may thus be non–Euclidean. We formulate the constitutive equation for large deformation in the case where either a neighborhood is non–Euclidean because of its initial structure, or it becomes so by virtue of irreversible internal motion and/or induced dislocation fields, which we discuss at some length. In the formulation of the theory, questions of connectivity arise and are dealt with in the text.     

带有加强筋的Mindlin板动态刚度阵法

以加筋中厚矩形板为研究对象，推导了加筋板的动态刚度阵，为动态刚度阵法提供一种新单元。板的运动微分方程由Mindlin厚板理论给出，同时还考虑了板平面内的振动。对于板上加强筋的处理，则通过Hamilton原理对板的运动方程作相应的修正，最终得到加筋板的运动微分方程。而方程的解析解直接用于单元刚度阵的推导，所得加筋板单元的动态刚度阵结合传统有限元方法的单元组装和求解方法即可用于计算整个结构的动力响应。此外，还给出了加筋板单元的均方响应计算公式，可用来计算结构的平均振动能量。最后通过数值算例验证本文方法，计算结果与传统有限元方法进行分析比较。     

Inverse heat conduction problems in three-dimensional anisotropic functionally graded solids

A meshless method based on the local Petrov–Galerkin approach is applied to inverse transient heat conduction problems in three-dimensional solids with continuously inhomogeneous and anisotropic material properties. The Heaviside step function is used as a test function in the local weak form, leading to the derivation of local integral equations. Nodal points are randomly distributed in the domain analyzed, and each node is surrounded by a spherical subdomain in which a local integral equation is applied. A meshless approximation based on the moving least-squares method is employed in the implementation. After performing spatial integrations, we obtain a system of ordinary differential equations for certain nodal unknowns. A backward finite-difference method is used for the approximation of the diffusive term in the heat conduction equation. A truncated singular-value decomposition is used to solve the ill-conditioned linear system of algebraic equations at each time step. The effectiveness of the meshless local Petrov–Galerkin (MLPG) method for this inverse problem is demonstrated by numerical examples.     

A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks

A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process. This revised version was published online in July 2006 with corrections to the Cover Date.