Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties

This paper presents the local boundary integral formulation for an elastic body with nonhomogeneous material properties. All nodal points are surrounded by a simple surface centered at the collocation point. Only one nodal point is included in each the sub-domain. On the surface of the sub-domain, both displacements and traction vectors are unknown generally. If a modified fundamental solution, for governing equation, which vanishes on the local boundary is chosen, the traction value is eliminated from the local boundary integral equations for all interior points. For every sub-domain, the material constants correspond to those at the collocation point at the center of sub-domain. Meshless and polynomial element approximations of displacements on the local boundaries are considered in the numerical analysis. Received 16 August 1999     

STRIP DISTRIBUTED TRANSFER FUNCTION METHOD FOR ANALYSIS OF PLATES

A semi-analytical method, called the strip distributed transfer function method, is developed for analysis of plate structures that are composed of rectangular plates. In the method, a rectangular plate (substructure) is divided into a number of strips; the response of each strip is interpolated in the unknown nodal line displacements, which are functions of the strip longitudinal co-ordinate and time. The nodal line displacements are determined in an exact and closed form by the distributed transfer functions that are defined along the strips. Synthesis of the substructures using the strip distributed transfer functions yields accurate prediction of the static and dynamic response, natural frequencies and buckling loads of the structure. The proposed method is compared with some existing techniques in numerical examples.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Quadrature rules for triangular and tetrahedral elements with generalized functions

Quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains. These quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries. Specifically, the integrands considered here consist of a quadratic polynomial multiplied by a Heaviside or Dirac delta function operating on a linear polynomial. This form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function. Exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values. Compared with methods involving explicit integration over subdomains, the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical Jacobian information describing the dependence of the integrals on the nodal level set values. The accuracy of the method is demonstrated for a simple conduction problem with the Neumann and Robin‐type boundary conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

Henyey-Greenstein and Mie phase functions in Monte Carlo radiative transfer computations

Monte Carlo radiative transfer simulation of light scattering in planetary atmospheres is not a simple problem, especially the study of angular distribution of light intensity. Approximate phase functions such as Henyey-Greenstein, modified Henyey-Greenstein, or Legendre polynomial decomposition are often used to simulate the Mie phase function. An alternative solution using an exact calculation alleviates these approximations.     

Exact geometry four-node solid-shell element for stress analysis of functionally graded shell structures via advanced SaS formulation

Abstract

The exact geometry four-node solid-shell element formulation using the sampling surfaces (SaS) method is developed. The SaS formulation is based on choosing inside the shell N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the use of Lagrange basis polynomials of degree N???1 in the through-thickness interpolations of displacements, strains, stresses and material properties leads to a very compact form of the SaS shell formulation. The SaS are located at Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. To implement efficient 3D analytical integration, the extended assumed natural strain method is employed. As a result, the proposed hybrid-mixed solid-shell element exhibits a superior performance in the case of coarse meshes. To circumvent shear and membrane locking, the assumed stress and strain approximations are utilized in the framework of the mixed Hu-Washizu variational formulation. It can be recommended for the 3D stress analysis of thick and thin doubly-curved functionally graded shells because the SaS formulation with only Chebyshev polynomial nodes allows the obtaining of numerical solutions, which asymptotically approach the 3D solutions of elasticity as the number of SaS tends to infinity.     

受约束的圆柱壳的振动分析

本文提出的解析法可对具有任意经典边界条件和在两端间具有若干约束的圆柱壳的自振特性进行计算。文中将亮体轴向模态位移用完备正交级数表示,然后将壳体振型函数分解为一个一致收敛级数迭加上另一线性多项式,从而解决了振型级数的收敛性和可微性。这样处理比stokes变换方法更具有一般性,不但可以得到受约束的圆柱壳固有频率的精确值,而且可以求得相应的振型。     

Multiplier methods for engineering optimization

Multiplier methods used to solve the constrained engineering optimization problem are described. These methods solve the problem by minimizing a sequence of unconstrained problems defined using the cost and constraint functions. The methods, proposed in 1969, have been determined to be quite robust, although not as efficient as other algorithms. They can be more effective for some engineering applications, such as optimum design and control of large scale dynamic systems. Since 1969 several modifications and extensions of the methods have been developed. Therefore, it is important to review the theory and computational procedures of these methods so that more efficient and effective ones can be developed for engineering applications. Recent methods that are similar to the multiplier methods are also discussed. These are continuous multiplier update, exact penalty and exponential penalty methods.     

Exact reconstruction of multiple concentrated damages on beams

In this paper, an exact procedure for the reconstruction of multiple concentrated damages on a straight beam is proposed. The concentrated damages are modelled as Dirac’s delta distributions capturing the effect of concentrated stiffness reduction. The presented procedure requires the knowledge of vibration mode shape displacements together with the relevant natural frequency, for the reconstruction of each damage position and intensity. The exact solution of the inverse problem at hand has been pursued by exploiting the analytical structure of the explicit closed form expressions provided for the vibration mode shapes of beams in the presence of an arbitrary number of cracks. The proposed procedure is first presented under the hypothesis that the displacements of a vibration mode shape are known at the cracked cross-sections. In this case, explicit closed form expressions of the crack severities are formulated. A further simple reconstruction approach allows the evaluation of the exact positions and intensity of the concentrated damages, if displacements of two vibration mode shapes are known at a single cross-section between two consecutive cracks. The proposed reconstruction procedure is applied for the identification of multiple cracks on a free–free beam where measurements have been simulated by means of a finite element analysis.     

APPROXIMATE DISCRETE VARIABLE OPTIMIZATION OF FRAME STRUCTURES WITH DUAL METHODS

The purpose of this work is to present an efficient method for optimum design of frame structures, using approximation concepts. A dual strategy in which the design variables can be considered as discrete variables is used. A two-level approximation concept is used. In the first level, all the structural response quantities such as forces and displacements are approximated as functions of some intermediate variables. Then the second level approximation is employed to convert the first-level approximation problem into a series of problems of separable forms, which can be solved easily by dual methods with discrete variables. In the second-level approximation, the objective function and the approximate constraints are linearized. The objective of the first-level approximation is to reduce the number of structural analyses required in the optimization problem and the second level approximation reduces the computational cost of the optimization technique. A portal frame and a single layer grid are used as design examples to demonstrate the efficiency of the proposed method.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

Single-Stage Procedures for Ranking Multiply-Classified Variances of Normal Populations

The author and Sobel have proposed single-stage procedures for ranking the variances of normal populations which are classified according to a single qualitative factor. Using the multiplicative model for variances introduced by the author in an earlier paper, it is shown how to generalize these procedures so that they can deal with situations in which the variances are classified according to two or more qualitative factors. The rationale of such experiments is discussed. Goals and associated probability requirements and procedures are described. Explicit expressions for the exact probability of a correct selection, using the proposed procedures, are obtained for certain special cases. The determination of sample size to guarantee the probability requirement is considered, “large-sample” approximations to these sample sizes are given, and the accuracy of the approximations is assessed.     

一维有限元后处理超收敛解答计算的EEP法

提出一维有限元法后处理中超收敛解答的一种自然合理的算法,称为单元能量投影法(EEP)。理论分析和数值算例表明,提出的方法简便易行、行之有效、效果显著;此外,还有一些颇合人意的优点,如:任一点的应力和位移的误差与结点位移的误差具有相同的收敛阶(m次单元可达mh2阶)、结点两边单元各自算出的应力自动平衡、自由端点的应力自动为精确值等。     

Multipoint approximation development: thermal structural optimization case study

Function approximation is one of the most important fields of research in design optimization. Accurate function approximation reduces the repetitive cost of finite element analysis. Many local approximations, such as one‐ and two‐point local approximations, are already available. These are, however, only valid in a small domain and require stringent move‐limits on design variables. The objective of this research is to achieve an efficient and accurate multipoint approximation to constraints by integrating the most accurate segment of all local approximations previously constructed. The proposed multipoint approximation is constructed by combining weighting functions with local function approximations. With function and gradient information at a series of points, local approximations are established at those points. Once established, all local approximations are blended into a multipoint approximation by use of a weighting function. Function and gradient values of this multipoint approximation correspond directly with exact counterparts at the points where the local approximations were generated. Finally, the multipoint approximation is applied to a plate and wing box thermal‐structural optimization. Published in 2000 by John Wiley & Sons, Ltd.     

A simple approach towards fully stressed design in hyperelasticity exploiting the isoparametric finite element concept

A novel approach towards fully stressed designs in hyperelasticity is discussed leading to closed‐form expressions for the sensitivities of the objective and displacements with respect to design variations. The key idea is the modification of the classical approach coupled with a so‐called design element method offering a lot of parallelism to standard finite element methods. We bypass implicit constraints on dependent quantities and derive an explicit linearly constrained optimization problem solved by means of first‐order procedures. The results obtained with the proposed method are adequate from an engineering point of view though being computed with a simple method. Copyright © 2000 John Wiley & Sons, Ltd.     

Linear equality constraints in finite element approximation

The typical numerical problem associated with finite element approximations is a quadratic programming problem with linear equality constraints. When nodal variables are employed, the coefficient matrix of the constraint equations, [ A ], acquires a block-diagonal structure. The transformation from polynomial coefficients to nodal variables involves finding a basis for [ A ] and computing its inverse. Simultaneous satisfaction of completeness and C¹ (or higher) continuity requirements establishes linear relationships among the nodal variables and precludes inversion of the basis by exclusively element-level operations. Linear dependencies among the constraint equations and among the nodal variables can be evaluated by the simplex method. The computational procedure is outlined.     

Comparison of finite element reliability methods

The spectral stochastic finite element method (SSFEM) aims at constructing a probabilistic representation of the response of a mechanical system, whose material properties are random fields. The response quantities, e.g. the nodal displacements, are represented by a polynomial series expansion in terms of standard normal random variables. This expansion is usually post-processed to obtain the second-order statistical moments of the response quantities. However, in the literature, the SSFEM has also been suggested as a method for reliability analysis. No careful examination of this potential has been made yet. In this paper, the SSFEM is considered in conjunction with the first-order reliability method (FORM) and with importance sampling for finite element reliability analysis. This approach is compared with the direct coupling of a FORM reliability code and a finite element code. The two procedures are applied to the reliability analysis of the settlement of a foundation lying on a randomly heterogeneous soil layer. The results are used to make a comprehensive comparison of the two methods in terms of their relative accuracies and efficiencies.     

基于EEP技术的一维有限元结点位移误差计算

利用单元能量投影(Element Energy Projection,简称EEP)法所计算的EEP超收敛解,在不改变有限元网格及其整体刚度矩阵的情况下,导出残差的等效结点荷载向量,只经回代过程即可得到具有更高阶精度的结点位移的误差估计,使结点位移精度得到极大提高。该文以一般的二阶常微分方程边值和初值问题为例,给出算法和相应的数值算例。从中可以看出,本法十分简单而高效:对于m≥1次单元,采用EEP简约格式和凝聚格式修正后的结点位移,分别具有O(h^2m+2)和O(h⁽3m+mod (m, 2)))的超常规的超收敛阶。该文给出了典型算例,并对该法的进一步拓展和应用作了讨论。     

A variational formulation in fracture mechanics

A variational approach to linear elasticity problems is considered. The family of variational principles is proposed based on the linear theory of elasticity and the method of integrodifferential relations. The idea of this approach is that the constitutive relation is specified by an integral equality instead of the local Hooke’s law and the modified boundary value problem is reduced to the minimization of a nonnegative functional over all admissible displacements and equilibrium stresses. The conditions of decomposition on two separated problems with respect to displacements and stresses are found for the variational problems formulated and the relation between the approach under consideration and the minimum principles for potential and complementary energies is shown. The effective local and integral criteria of solution quality are proposed. A numerical algorithm based on the piecewise polynomial approximations of displacement and stress fields over an arbitrary domain triangulation are worked out to obtained numerical solutions and estimate their convergence rates. Numerical results for 2D linear elasticity problems with cracks are presented and discussed.     

The interface crack under combined loading: an eigenvalue problem for the gap

To avoid the difficulty of overlapping material near the crack tips, the tips of the crack are assumed to be closed (the Comninou model). The exact series solution to the problem of the interface crack under combined loading is complicated, and it is difficult to take advantage of the smallness of the left contact zone to obtain simple asymptotic approximations to the quantities of physical interest. Here, the gap is shown to satisfy an eigenvalue problem. The method of matched inner and outer expansions is used to obtain a simple uniform approximation to this quantity. Similar results are obtained for the tangential shift, which satisfies the same differential equation as the gap. The tractions at the interface also satisfy a second order differential equation and simple uniform approximations are obtained for these quantities.