Singular perturbations for sensitivity analysis in symmetric bifurcation buckling

A direct procedure for the evaluation of imperfection‐sensitivity in bifurcation problems is presented. The problems arise in the context of the general theory of elastic stability (Koiter's theory) for discrete structural systems, in which the total potential energy is employed together with a stability criterion based on energy derivatives. The imperfection sensitivity of critical states, such as bifurcations and trifurcations, is usually represented as a plot of the critical load versus the amplitude ε of the imperfection considered. However, such plots have a singularity at the point with ε=0, so that a regular perturbation expansion of the solution is not possible. In this work, we describe a direct procedure to obtain the sensitivity of the critical load (eigenvalue of the bifurcation problem) and the sensitivity of the critical direction (eigenvector of the bifurcation problem) using singular perturbation analysis. The perturbation expansions are constructed as a power series in terms of the imperfection amplitude, in which the exponents and the coefficients are the unknowns of the problem. The solution of the exponents is obtained by means of trial and error using a least degenerate criterion, or by geometrical considerations. To compute the coefficients a detailed formulation is presented, which employs the conditions of equilibrium and stability at the critical state and their contracted forms. The formulation is applied to symmetric bifurcations, and the coefficients are solved up to third‐order terms in the expansion. The algorithmis illustrated by means of a simple example (a beam on an elastic foundation under axial load) for which the coefficients are computed and the imperfection‐sensitivity is plotted. Copyright © 2001 John Wiley & Sons, Ltd.     

OPTIMIZATION OF SPACE TRUSSES AGAINST INSTABILITY USING DESIGN SENSITIVITY DERIVATIVES

This paper addresses the problem of the maximization of the critical load of shallow space trusses of given configuration and volume. Through implicit differentiation of the nonlinear equilibrium equations and the stability criterion, the sensitivity derivatives of the critical load parameter with respect to the design variables are developed. Optimum designs are generated using a projected Lagrangian technique known as the Variable Metric method for constrained optimization (VMCON).     

Fractional Order Thermoelasticity Theory for a Half-Space Subjected to an Axisymmetric Heat Distribution

The present work is concerned with a traction-free thermoelastic half-space subjected to a known axisymmetric temperature distribution. The thermoelastic interactions inside the medium are investigated by employing the fractional order theory of thermoelasticity. The problem is solved by using Laplace's and Hankel's transforms. The inverse transforms are computed numerically. The variations of temperature, displacements, and stresses inside the half-space are investigated. The field variables for a particular material are graphically presented. Comparisons are made within the theory in the presence and absence of fractional order parameter.     

An approximation technique for design sensitivity analysis of the critical load in non‐linear structures

A new technique of approximating design sensitivities of the critical load is presented in this paper. The technique results in stable and reliable estimations of design sensitivities at prebuckling points. Since taking derivatives of an approximated eigenvalue problem gives unstable sensitivities as the point approaches the critical load, the sensitivities are approximated directly from the exact sensitivity expressions. The sensitivities are approximated by applying two common approaches that are used in the critical load estimation and are called ‘one‐ and two‐point approximation’. The reliability and applicability of the proposed technique are demonstrated through several numerical examples of truss and beam structures. Two‐point approximation of design sensitivities gives better results than one‐point approximation. Copyright © 1999 John Wiley & Sons, Ltd.     

测量生物组织等效杨氏模量的毫米压痕仪

生物组织的弹性是诊断其是否发生病变的重要依据,杨氏模量是反映组织弹性的重要参数。以Labview为软件平台结合数据采集和运动控制等硬件设备,研制了一套测量弹性模量的毫米压痕弹性仪。以市购冷鲜牛肉、猪肉、猪肝和猪肾为试样,测量了球形压头向试样施加的负载力和对应的试样压痕深度,在对测量数据进行校准的基础上,利用基于赫兹接触力学模型的压头半径R、作用力F、压痕深度δ与试样等效杨氏模量E~*间的解析关系,得到各试样的等效杨氏模量。实验结果表明,测得的组织试样等效杨氏模量数值与文献基本相符,所设计的毫米级压痕法测试装置可用于生物组织等效杨氏模量的检测。     

Eigenvalue analysis for acoustic problem in 3D by boundary element method with the block Sakurai–Sugiura method

This paper presents accurate numerical solutions for nonlinear eigenvalue analysis of three-dimensional acoustic cavities by boundary element method (BEM). To solve the nonlinear eigenvalue problem (NEP) formulated by BEM, we employ a contour integral method, called block Sakurai–Sugiura (SS) method, by which the NEP is converted to a standard linear eigenvalue problem and the dimension of eigenspace is reduced. The block version adopted in present work can also extract eigenvalues whose multiplicity is larger than one, but for the complex connected region which includes a internal closed boundary, the methodology yields fictitious eigenvalues. The application of the technique is demonstrated through the eigenvalue calculation of sphere with unique homogenous boundary conditions, cube with mixed boundary conditions and a complex connected region formed by cubic boundary and spherical boundary, however, the fictitious eigenvalues can be identified by Burton–Miller's method. These numerical results are supported by appropriate convergence study and comparisons with close form.     

Elasticity Solution of a Laminated Clamped Cylindrical Shell with Piezoelectric Layer under Dynamic Load

Dynamic elasticity solution for a clamped, laminated cylindrical shell with two orthotropic layers bounded with a piezoelectric layer and subjected to impulse load distributed on inner surface is presented. The piezoelectric layer serves as sensor/actuator. The governing elasticity PDE equations are reduced to ordinary differential equations by means of Legendre polynomial expansion for displacement and electric potential in the axial direction. The resulting equations are transferred into state space form and reduced to an eigenvalue problem by using Galerkin's finite element in radial direction. The static and dynamic results are presented for [0/90/Piezo] lamination. The radius to thickness ratio effect on dynamic behavior is studied. The results are compared for different thickness ratios and applied electric loads with simply-supported shell results. Time responses for sensor and actuated shell are presented and natural frequencies are compared with simply-supported shell results.     

A direct application of higher-order parametric programming techniques to structural optimization

Computational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems (COP) without any parameter. An auxiliary parametric programming problem (APPP) is formulated in order to solve COP. The procedure is started with an arbitrary initial solution which is the trivial solution of APPP corresponding to the initial value of the parameter. Then the optimal solution for the final value of the parameter, which is the optimal solution of COP, is estimated by Taylor's expansion with respect to the parameter where higher-order terms are incorporated. It is shown that the incorporation of the higher-order terms indeed leads to a faster convergence of the solution. As an extension of the method, a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables. Logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously. The derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables. Finally, a method of simultaneous analysis and optimization is developed for trusses with geometrical non-linearity.     

Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

When exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally efficient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is first reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identified by the Wittrick–Williams algorithm. Then the generalized linear eigenvalue problem is effectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick–Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large‐scale structures. Copyright © 2003 John Wiley & Sons, Ltd.     

Minimum weight design of non‐linear elastic structures with multimodal buckling constraints

It well known that multimodal instability is an event particularly relevant in structural optimization. Here, in the context of non‐linear stability theory, an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints. Given an initial design, the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight. Multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem, and the difficulties related to the non‐differentiability in the common sense of state variables in multimodal critical states, are overcome by means of the Lagrange multiplier method. Further constraints impose that only the first critical equilibrium states (local maxima or bifurcation points) on the initial equilibrium path of the actual designs are taken into account. Copyright © 2002 John Wiley & Sons, Ltd.     

Buckling analysis of composite plates using moving least squares differential quadrature method

This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.     

Highly efficient general method for sensitivity analysis of eigenvectors with repeated eigenvalues without passing through adjacent eigenvectors

It is well known that the sensitivity analysis of the eigenvectors corresponding to multiple eigenvalues is a difficult problem. The main difficulty is that for given multiple eigenvalues, the eigenvector derivatives can be computed for a specific eigenvector basis, the so-called adjacent eigenvector basis. These adjacent eigenvectors depend on individual variables, which makes the eigenvector derivative calculation elaborate and expensive from a computational perspective. This research presents a method that avoids passing through adjacent eigenvectors in the calculation of the partial derivatives of any prescribed eigenvector basis. As our method fits into the adjoint sensitivity analysis , it is efficient for computing the complete Jacobian matrix because the adjoint variables are independent of each variable. Thus our method clarifies and unifies existing theories on eigenvector sensitivity analysis. Moreover, it provides a highly efficient computational method with a significant saving of the computational cost. Additional benefits of our approach are that one does not have to solve a deficient linear system and that the method is independent of the existence of repeated eigenvalue derivatives of the multiple eigenvalues. Our method covers the case of eigenvectors associated to a single eigenvalue. Some examples are provided to validate the present approach.     

A novel computational method for solving Troesch's problem with high-sensitivity parameter

Troesch's problem is a nonlinear boundary value problem arising in the confinement of a plasma column by radiation pressure, and also in the theory of gas porous electrodes. It is well known that finding numerical solutions to this problem is challenging, especially when the sensitivity parameter is large. In this article, we present an efficient and accurate numerical method for solving Troesch's problem. The method presented in this work is capable of computing the solution, even for extremely high-sensitivity parameter. The method is based on the the Newton–Raphson–Kantorovich approximation method in function space combined with the standard finite difference method. Although, available numerical solvers fail to provide accurate numerical solutions when the sensitivity parameter λ becomes large (λ exceeds 100) [1–5], the method proposed here is able to provide accurate numerical solutions for extremely large values of this sensitivity parameter, up to λ = 500. Numerical experiments are provided to show the accuracy of the method compared to existing solvers, as well as its capability to compute the solution for high values of the sensitivity parameter λ.     

Thermal post-buckling of slender composite and FGM columns through a simple and novel FE formulation

A simple and novel finite element (FE) formulation is proposed to study the thermal post-buckling of composite and FGM columns with axially immovable ends and operating in severe thermal environment. A linear eigenvalue analysis gives the critical buckling temperature but practically the buckled columns can withstand additional thermal load beyond critical temperature, which can be obtained using von-Karman geometric nonlinearity, applicable for moderately large deflections. In the present study, the solution of the non-linear post-buckling problem is obtained by treating it as a linear eigenvalue problem using the concept of effective stiffness. Here, the total degrees of freedom (dof) of the discretized column are reduced and the post-buckling load is obtained without the need for iterative analysis. Comparison of the numerical results obtained from this FE formulation is in very good agreement with those obtained from the earlier FE formulations.     

Prediction methods of fatigue critical point for notched components under multiaxial fatigue loading

Two methods based on local stress responses are proposed to locate fatigue critical point of metallic notched components under non‐proportional loading. The points on the notch edge maintain a state of uniaxial stress even when the far‐field fatigue loading is multiaxial. The point bearing the maximum stress amplitude is recognized as fatigue critical point under the condition of non‐mean stress; otherwise, the Goodman's empirical formula is adopted to amend mean stress effect prior to the determination of fatigue critical point. Furthermore, the uniaxial stress state can be treated as a special multiaxial stress state. The Susmel's fatigue damage parameter is employed to evaluate the fatigue damage of these points on the notch edge. Multiaxial fatigue tests on thin‐walled round tube notched specimens made of GH4169 nickel‐base alloy and 2297 aluminium‐lithium alloy are carried out to verify the two methods. The prediction results show that both the stress amplitude method and the Susmel's parameter method can accurately locate the fatigue critical point of metallic notched components under multiaxial fatigue loading.     

On special points on load-displacement paths in the prebuckling domain of thin shells

Points on load-displacement paths of thin shells subjected to proportional loading, at which the second and/or the third derivative of the displacement components with respect to the load parameter vanishes, are termed ‘special points’. Such points represent global characteristics of the state of deformation of the shell in the sense that for the respective values of the load parameter the mentioned rate(s) of all displacement components at all points of the shell must vanish. It will be shown that special points on load-displacement paths correspond to special points of one order lower on the Det K_T–λ diagram, where Det K_T is the determinant of the tangent stiffness matrix within the framework of the finite element method and λ is a dimensionless load parameter. Points of inflection on load-displacement diagrams, for example, correspond to extreme values on the Det K_T–λ diagram. The main reason for the occupation with special points on load-displacement paths is that points of inflection and flat points on these paths correspond to special points on eigenvalue curves in the context of accompanying linear stability analyses of geometrically non-linear prebuckling analyses of thin shells by the finite element method, investigated in a companion paper.     

Second-order sensitivity analysis in vibration and buckling problems

A perturbation approach for the first- and second-order sensitivity analysis of eigenvalue problems, as they arise in buckling and vibration of structural systems, is presented. A particular feature introduced in the formulation is the dependence of all matrices that model the problem not only on the design parameter, but also on a pre-critical state. Thus, the sensitivity of the response of the fundamental path in buckling problems, or a pre-stressed state, in vibration problems, are introduced in the analysis. Two techniques are considered in this work: the direct and the adjoint methods. First- and second-order sensitivity equations for eigenvalues and eigenvectors are obtained, and it is shown that second-order sensitivity of an eigenvalue can be accomplished by using only one adjoint problem.     

Asymmetric buckling of shell-stiffened annular plates

The present study is devoted to some stability problems of annular plates with shell-stiffening. The plate is simply supported, or clamped, or elastic restrains are applied on the boundaries of the plate. Though the load is axisymmetric we shall assume that the deformations are not necessarily axisymmetric. The displacement fields in the plate and the shell are expanded into Fourier series. For the shell all physical quantities are derived from an appropriately chosen Galerkin function. The main goal is to clarify the effect of a stiffening shell on the buckling load. Hence the field equations together with the boundary- and continuity conditions are clarified. These provide the eigenvalue problem from which the critical load can be calculated. In order to solve the eigenvalue problem we have established, appropriate numerical methods are used.     

Buckling analysis of shear deformable plates by boundary element method

In this paper, the derivation and numerical implementation of boundary integral equations for the buckling analysis of shear deformable plates are presented. Plate buckling equations are derived as a standard eigenvalue problem. The formulation is formed by coupling boundary element formulations of shear deformable plate and two dimensional plane stress elasticity. The eigenvalue problem of plate buckling yields the critical load factor and buckling modes. The domain integrals which appear in this formulation are treated in two different ways: initially the integrals are evaluated using constant cells, and next, they are transformed into equivalent boundary integrals using the dual reciprocity method (DRM). Several examples with different geometry, loading and boundary conditions are presented to demonstrate the accuracy of the formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical solution of thermal instability of a rotating micropolar fluid layer

In this paper, we have considered the thermal instability of a rotating, heat conducting, micropolar fluid layer heated from below and confined between two rigid boundaries. The onset of thermal instability is governed by a linear eigenvalue problem. The solution of the eigenvalue problem is obtained by using finite difference method and Wilkinson's iteration technique. The effects of rotation and micropolar parameters on the critical Rayleigh number and the wave number at the threshold of instability are discussed in detail.