Approximation of derivatives in semi-analytical structural optimization
Approximation of derivatives in semi-analytical structural optimization
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DOI：
10.1016/j.compstruc.2007.04.014
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发表时间：
2008-07-01
影响因子：
4.7
通讯作者：
Daoud, Fernass
中科院分区：
文献类型：
--
作者：
Bletzinger, Kai-Uwe;Firl, Matthias;Daoud, Fernass
本文提出了一种简单而普遍适用的方法，用于检测和消除任何一种fe配方的半分析设计灵敏度误差。半解析方法的基本性质是刚度矩阵和载荷矢量的导数用有限差分近似。显然，这种方法会产生截断误差，这取决于所选择的步长和力学模型的运动学假设。这些近似误差导致了半解析灵敏度分析中的精度问题[Barthelemy B， Haftka RT.]。形状灵敏度计算半解析方法的精度分析。[j].机械工程学报，1998,18(6):444 - 444。在这个贡献中，两个梁单元（Euler-Bernoulli运动学，Timoshenko运动学）被用来通过对误差项的解析计算来强调近似误差的后果。这两种元素在灵敏度误差上表现出严重的差异。通过这些简单的1-d元素所获得的思想可以通过Reissner-Mindlin和Kirchhoff运动学进一步扩展到3-d元素。由于导数的有限差分近似误差较大，需要对其进行修正以获得精确的灵敏度。文献中存在各种各样的方法试图消除设计灵敏度的误差。Haftka和Adelmann， Mlejnek， Cheng和Olhoff， V. Keulen和De Boer等人发表了重要的贡献。本文提出了一种基于刚体旋转矢量积空间的修正系数计算方法。对于刚度矩阵的导数，一种直接的推导产生了刚体条件。由于受扰单元的基础发生了变化，该导数的近似违反了刚体条件。通过提出的方法，可以得到一组与特定有限元的刚体旋转矢量相关的修正因子。由于这组因子对近似刚度矩阵导数的修正，最终得到了“精确”的灵敏度。改进的刚度矩阵导数近似满足上述刚体条件。该方法的基本优点是效率高，不依赖于有限元公式。与目前发表的许多其他修正方法相比，该方法适用于所有类型的有限元，无需进行大的修改。这就产生了适用于大量有限元的通用形状优化算法，而无需对每个单个元素进行解析推导。(c) 2007 Elsevier Ltd.版权所有。
This paper presents a straightforward and generally applicable method for detection and elimination of errors in semi-analytical design sensitivities for any kind of FE-formulation. The basic property of the semi-analytical approach is that derivatives of the stiffness matrix and the load vector are approximated by finite differences. Obviously, truncation errors occur by this method which depend on the chosen step size and the kinematic assumptions of the mechanical model. The accuracy problems in the semi-analytical sensitivity analysis result from these approximation errors [Barthelemy B, Haftka RT. Accuracy analysis of the semi-analytical method for shape sensitivity calculation. Mech Struct Mach 1988;18:407-32]. In this contribution two beam elements (Euler-Bernoulli kinematics, Timoshenko kinematics) are utilized to emphasize the consequences of the approximation errors by an analytical computation of the error terms. These two elements show serious differences in the errors of the sensitivities. The ideas gained by these simple 1-d elements are extended further to 3-d elements with Reissner-Mindlin and Kirchhoff kinematics.The errors of the finite difference approximation of the derivatives may become serious, so it is necessary to correct them to obtain exact sensitivities. There exists a great variety of methods in the literature which try to eliminate the errors in the design sensitivities. Important contributions are published by Haftka and Adelmann, Mlejnek, Cheng and Olhoff, V. Keulen and De Boer among many others.In this paper, a method for the computation of correction factors based on product spaces of rigid body rotation vectors is presented. A straightforward derivation yields to a rigid body condition for the stiffness matrix derivative. The approximation of this derivative violates this rigid body condition due to the changed basis of the perturbed element. By the proposed method one obtains a set of correction factors related to the rigid body rotation vectors of the specific finite element. Due to the modification of the approximated stiffness matrix derivative by this set of factors one finally gets 'exact' sensitivities. The improved approximation of the stiffness matrix derivative satisfies the above mentioned rigid body condition.The basic advantage of the proposed method is the efficiency and the independence on the Finite Element formulation. In contrast to many other correction methods published so far, this approach is applicable to all kind of Finite Elements without major modifications.This gives rise to general shape optimization algorithms for a huge amount of finite elements without the necessity to derive each single element analytically. (c) 2007 Elsevier Ltd. All rights reserved.