This paper presents a straightforward and generally applicablemethod fordetection and elimination of errors in semi-analytical design sensitivitiesfor any kind of F.E. formulation. The basic property of the semi-analyticalapproach is that derivatives of the stiffness matrix and the load vectorare approximated by finite differences. Obviously truncation errors occurby this method which depend on the chosen step size and the kinematic assumptionsof the mechanical model. The accuracy problems in the semi-analytical sensitivityanalysis result from these approximation errors [1]. In this contributiontwo beam elements (Euler-Bernoulli kinematics, Timoshenko kinematics) andtwo cantilever models are used to emphasize the consequences of the approximationerrors by an analytical computation of the error terms. These two elementsshow serious differences in the errors of the sensitivities. The ideasgained by these simple 1-d elements are extended further to 3-d elementswith Reissner-Mindlin kinematics. The errors of the finite difference approximationof the derivatives may become serious, so it is necessary to correct themto obtain exact sensitivities. There exists a great variety of methodsin the literature which try to eliminate the errors in the design sensitivities.Important contributions are published by Haftka and Adelmann, Mlejnek,Cheng and Olhoff and v. Keulen among many others. In this paper a methodfor the computation of correction factors based on product spaces of rigidbody rotation vectors is presented. A straightforward derivation yieldsto a rigid body condition for the stiffness matrix derivative. The approximationof this derivative violates this rigid body condition due to the changedbasis of the perturbed element. By the proposed method one obtains a setof correction factors related to the rigid body rotation vectors of thespecific finite element. Due to the modification of the approximated stiffnessmatrix derivative by this set of factors one finally gets ’exact’ sensitivities.The improved approximation of the stiffness matrix derivative satisfiesthe above mentioned rigid body condition. The basic advantage of the proposedmethod is the efficiency and the independence on the Finite Element formulation.In contrast to many other correction methods published so far, this approachis applicable to all kind of Finite Elements without major modifications.This gives rise to general shape optimization algorithms for a huge amountof finite elements without the necessity to derive each single elementanalytically.
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