OPTIMUM SHAPE DESIGN AND POSITIONING OF FEATURES USING THE BOUNDARY INTEGRAL EQUATION METHOD

A design optimization procedure is developed using the boundary integral equation (BIE) method for linear elastostatic two-dimensional domains. Optimal shape design problems are treated where design variables are geometric parameters such as the positions and sizing dimensions of entire features on a component or structure. A fully analytical approach is adopted for the design sensitivity analysis where the BIE is implicitly differentiated. The ability to evaluate response sensitivity derivatives with respect to design variables such as feature positions is achieved through the definition of appropriate design velocity fields for these variables. How the advantages of the BIE method are amplified when extended to sensitivity analysis for this category of shape design problems is also highlighted. A mathematical programming approach with the penalty function method is used for solving the overall optimization problem. The procedure is applied to three example problems to demonstrate the optimum positioning of holes and optimization of radial dimensions of circular arcs on structures.     

ANALYSIS OF THREE-DIMENSIONAL TRANSIENT ACOUSTIC WAVE PROPAGATION USING THE BOUNDARY INTEGRAL EQUATION METHOD

This paper describes a boundary integral equation (boundary element) method for the solution of a variety of transient acoustic problems. The spatial and temporal discretization employs quadratic isoparametric elements with high-order Gauss quadrature, and the ensuing equations are implicit. The implicit formulation both eliminates the instabilities reported in explicit treatments, and permits a freedom of choice of timestep which can reduce costs dramatically. The accuracy of the approach is demonstrated by comparison with the analytical solution for a sphere. Results for more demanding sphere–cone–sphere geometries extending to seven wavelengths long are presented, and compared to those obtained from a related frequency domain approach.     

SENSITIVITY ANALYSIS FORMULATION FOR THREE-DIMENSIONAL CONDUCTION HEAT TRANSFER WITH COMPLEX GEOMETRIES USING A BOUNDARY ELEMENT METHOD

A special boundary integral formulation had been proposed to analyse many engineering problems of conduction heat transfer in complex three-dimensional geometries (closely spaced surface and circular hole in infinite domain or simple modification of it) by Rezayat and Burton. One example of such geometries is the mold sets in the injection molding process. In this paper, an efficient and accurate approach for the design sensitivity analysis (DSA) is presented for these kinds of problems in the similar complex geometries using the direct differentiation approach (DDA) based on the above special boundary integral formulation. The present approach utilizes the implicit differentiation of the boundary integral equations with respect to the design variables (radii and locations of circular holes) to yield the sensitivity equations. A sample problem (heat transfer of injection molding cooling system) is solved to demonstrate the accuracy of the present sensitivity analysis formulation. Although the techniques introduced here are applied to a particular problem in heat transfer of injection molding cooling system, their potential application is quite broad.     

拱坝体形优化中的三维解析敏度

本文针对基于有限元分析的拱坝体形优化，采用改进的半解析法进行三维敏度分析。推导了敏度分析有关公式，并研制了相应的程序。敏度验证和拱坝优化算例表明，本文方法精度高、数值稳定，对基于三维有限元分析的拱坝体形优化是有效的。     

SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION FOR STRUCTURAL DURABILITY

In this paper, a design sensitivity analysis (DSA) method for fatigue life of 3-D solid structural components of mechanical systems with respect to shape design parameters is presented. The DSA method uses dynamic stress DSA obtained using an analytical approach to predict dynamic stress increment due to design changes; computes fatigue life of the component, including crack initiation and crack propagation, using the predicted dynamic stress; and uses the difference of the new life and the original life at the same critical point to approximate the sensitivity of fatigue life. A tracked vehicle roadarm is presented in this paper to demonstrate accuracy and efficiency of the DSA method. Also, this method is applied to support design optimization of the tracked vehicle roadarm considering crack initiation lives as design constraints. © 1997 by John Wiley & Sons, Ltd.     

OVERHAUSER TRIANGULAR ELEMENTS FOR THREE-DIMENSIONAL POTENTIAL PROBLEMS USING BOUNDARY ELEMENT METHODS

Interpolation to boundary data and one-dimensional Overhauser parabola blending methods are used to derive Overhauser triangular elements. The elements are C¹-continuous at inter-element nodes and no functional derivatives are required as nodal parameters. These efficient parametric representation elements are used to solve three-dimensional potential problems using the Boundary Element Method (BEM). Results obtained are generally as accurate as those obtained using Overhauser quadrilateral elements.     

结构优化设计中的灵敏度分析

阐述了灵敏度分析在带有频率约束的结构优化设计中的重要性，并且指出主模态局部化与非经典系统的灵敏性是结构设计师需要研究的课题。     

正交各向异性厚板振动分析的边界积分方程法

本文采用正交各向异性厚板静力问题的基本解作为边界积分方程的核函数,利用加权残数法建立了正交各向异性厚板振动分析的边界积分方程。文中详细地讨论了边界积分方程的数值处理过程并给出了若干数值算例以论证本文方法的正确性。      

ITERATIVE SOLUTION OF LARGE THREE-DIMENSIONAL BEM ELASTOSTATIC ANALYSES USING THE GMRES TECHNIQUE

The GMRES (Generalized Minimal RESidual) iterative method is receiving increased attention as a solver for the large dense and unstructured matrices generated by boundary element elastostatic analyses. Existing published results are predominantly for two-dimensional problems, of only medium size. When these methods are applied to large three-dimensional problems, which actually do require efficient iterative methods for practical solution, they fail. This failure is exacerbated by the use of the pre-conditioning otherwise desirable in such problems. The cause of the failure is identified as being in the orthogonalization process, and is demonstrated by the divergence of the ‘true’ residual, and the residual calculated during the GMRES algorithm. It is shown that double precision arithmetic is required for only the small fraction of the work comprising the orthogonalization process, and exploitation of this largely removes the penalties associated with the use of double precision. Additionally, it is shown that full re-orthogonalization can be employed to overcome the lack of convergence, extending the applicability of the GMRES to significantly larger problems. The approach is demonstrated by solving three-dimensional problems comprising ∼4000 and ∼5000 equations. © 1997 John Wiley & Sons, Ltd.     

SENSITIVITY ANALYSIS FOR BOUNDARY ELEMENT ERROR ESTIMATION AND MESH REFINEMENT

The subject of this paper is the sensitivity analysis of approximate boundary element solutions with respect to the positions of the collocation points. The direct differentiation approach is considered here and the analysis is performed analytically. Since only the collocation points are perturbed, the shape of the body and the corresponding discretization remain unaltered. This aspect makes the present work quite different in spirit with respect to earlier analyses on shape sensitivities. Sensitivities of approximate BEM solutions with respect to the positions of collocation points are shown to be related to the residual of hypersingular integral equations. Numerical results confirm that the present approach can be seen as the analytical counterpart of an adaptive scheme for mesh refinement presented by the same author in some recent papers. Some other advantages of the present approach over the former one are also outlined.     

DESIGN SENSITIVITY ANALYSIS AND OPTIMAL DESIGN OF COMPOSITE STRUCTURES USING HIGHER ORDER DISCRETE MODELS

This paper deals with the structural optimization of multilaminated composite plate structures of arbitrary geometry and layup, using single layer higher order shear deformation theory discrete models. The structural and sensitivity analysis formulation is developed for a family of C° Lagrangian elements. The design sensitivities of static response for objective and/or constraint functions, such as maximum displacements, stress failure criterion and elastic strain energy, with respect to ply angles and ply thickness are presented. The objectives of the design are the minimization of the structural elastic strain energy, minimization of maximum deflection and/or the minimization of the structure volume. The accuracy and relative performance of the proposed discrete models are compared and discussed among developed elements and alternative models. Several test designs are optimized to show the applicability of the proposed refined discrete models.     

基于哈密顿解法的厚板边界效应典型算例分析

在平板弯曲问题中存在边界效应现象。这个现象用Reissner厚板理论可以论证,而经典薄板理论则无法解释。边界效应现象在板的自由边界附近表现最为典型。该文首先选取均载作用下的三边简支,一边自由的矩形厚板作为典型问题,采用哈密顿解法求出解析解。然后结合这个典型问题,对边界效应现象进行分析,对其影响因素和影响范围进行讨论。     

MIXED ELEMENTS IN THE SENSITIVITY ANALYSIS OF STRUCTURES

In this paper is developed the sensitivity analysis of structures based on mixed elements. It examines the sensitivity of static, dynamic and stability behaviour of structures. The results are compared with analytical and displacement finite element solutions. The advantages and disadvantages of mixed elements in sensitivity analysis of structures are discussed with reference to applications.     

A SINGLE-DOMAIN BOUNDARY ELEMENT METHOD FOR 3-D ELASTOSTATIC CRACK ANALYSIS USING CONTINUOUS ELEMENTS

A boundary element method is presented for single-domain analysis of cracked three-dimensional isotropic elastostatic solids. A numerical treatment for the hypersingular Boundary Integro-Differential Equation (BIDE) for displacement derivatives is described, in which continuous boundary elements may be used. Hadamard principal values of the hypersingular integrals arising in the formulation are evaluated using polar co-ordinates defined on the tangent planes at the source point, and the free term coefficients are calculated directly using a numerical technique. The forms of the Boundary Integral Equation (BIE) and the BIDE are considered for a source point on the coincident surfaces of a crack, and a scheme is given for defining the Traction Boundary Integral Equation TBIE so that it optimally incorporates the traction information deficient in its complementary partner, the BIE. Numerical results for some example mixed-mode crack problems are presented.     

ON SENSITIVITY ANALYSIS AND OPTIMAL DESIGN OF SPECIALLY ORTHOTROPIC LAMINATES

Sensitivity analysis with respect to fibre orientation as well as to ply thickness is performed for specially orthotropic laminates. Eigenfrequencies, buckling load and displacement are all related to the same non-dimensional functional, which is therefore taken as the objective for our optimization.

If the laminates are assumed to be made of plies of the same material, then closed form results are obtainable, and it is essential that these results are well understood. For more advanced models the main parameters will be same and so will the main aspects of the sensitivity analysis.

The principal conclusions of this unconstrained optimization are that the optimal fibre orientation depends mainly on the displacement mode and that the problem of optimal thickness distribution is not well-defined.     

求解夹杂问题的有限部积分──边界元法

利用Kelvin解及有限部积分的概念和方法,导出求解含夹杂二维有限弹性体的超奇异积分方程,继而使用有限部积分与边界元结合的方法,为其建立了数值求解方法,即有限部积分与边界元法.最后计算了若干典型数值例子夹杂端部的应力强度因子.      

SENSITIVITY ANALYSIS AND A MIXED APPROACH TO THE OPTIMIZATION OF SYMMETRIC LAYERED COMPOSITE PLATES

This paper studies the sensitivity analysis of the static structural compliance of symmetric layered composite plates with respect to the fiber orientation in terms of continuous and finite element discretized formulations. From the optimality criterion obtained based on the sensitivity analysis, it is found that the number of different angles of the fibers in an optimal plate is no more than 4 under rather general conditions. Making use of the sensitivity information and optimality criterion approach, we construct a mixed method for adjusting the fiber orientations to minimize the structural compliance of composite plates. Furthermore, the non-convcxity of compliances of composite plates is shown by numerical experiment.     

板弯曲问题三维虚边界元分析

本文抛弃以往解板弯曲问题的假设,直接从三维弹性力学微分方程出发,依据三维弹性力学问题的Kelvin解,应用最小二乘法建立了三维虚边界元法解板弯曲问题的一般方法。文中给出了具有各种约束的矩形板的数值算例,以作为本方法的应用。本文方法与边界元直接法相比,优点在于无需处理奇异积分,且系数阵是对称的：再者,本文方法思想简单,且程序实现容易,易于被工程界接受。     

STABILITY ANALYSIS AND DESIGN OF TIME-STEPPING SCHEMES FOR GENERAL ELASTODYNAMIC BOUNDARY ELEMENT MODELS

In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models. However, there has been no theory to support scientists and engineers in assessing the stability of their boundary element algorithms or to help them with the design of new, more stable algorithms. In this paper we present a general framework for the analysis of the stability of any time-domain boundary element model. We illustrate how the stability theory can be used to assess the stability of existing boundary element models and how the insight gained from this analysis can be used to design more stable time-stepping schemes. In particular, we describe a new time-stepping procedure that we have developed, which has substantially enhanced stability characteristics and greater accuracy for the same computational effort. The new scheme, which we have called ‘the half-step scheme’, is shown to have substantially improved performance for the displacement discontinuity boundary element method commonly used to model dynamic fracture interaction and propagation. © 1997 by John Wiley & Sons, Ltd.     

ON THE CHOICE OF A DERIVATIVE BOUNDARY ELEMENT FORMULATION USING HERMITE INTERPOLATION

This paper reports on some problems that can arise with the use of regularized derivative boundary integral equations. It concentrates on developing a formulation for the simple Laplace equation using a cubic Hermite interpolation and shows how certain combinations of derivative and conventional boundary integral equations can result in a solution scheme severely lacking in stability. With some simple two- and three-dimensional geometries, the derivative equations on their own do not provide enough information to solve a Dirichlet problem. Even combinations of the conventional and derivative equations fail for some simple geometries. We conclude that the only consistently successful combination is that of the conventional equation with the tangential derivative equation, which showed cubic convergence of results with mesh refinement. Numerical results are presented for this scheme in both two and three dimensions.