• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.12
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• pp.2631-2635
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• 2002

The semi-analytic sensitivity analysis using Pade approximation is presented for linear elastic structures. Although the semi-analytic method has several advantages, accuracy of the method prevents it from practical application. One of promising remedies is the use of geometric series for the matrix inversion. Though series expansion of order three has been successfully applied to the calculation of the structural sensitivity in the most range of the design perturbation, it is prone to have a slow convergence for large perturbation. To overcome this shortage, Pade approximation is introduced so that it can broaden the trust region of the perturbation without adding expansion terms. Numerical results show that the confident sensitivity can be obtained with tiny expenses of computation effort.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.4
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• pp.593-600
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• 2003

Structural optimization often requires the evaluation of design sensitivities. The Semi Analytic Method(SAM) fur computing sensitivity is popular in shape optimization because this method has several advantages. But when relatively large rigid body motions are identified for individual elements. the SAM shows severe inaccuracy. In this study, the improvement of design sensitivities corresponding to the rigid body mode is evaluated by exact differentiation of the rigid body modes. Moreover. the error of the SAM caused by numerical difference scheme is alleviated by using a series approximation for the sensitivity derivatives and considering the higher order terms. Finally the present study shows that the refined SAM including the iterative method improves the results of sensitivity analysis in dynamic problems.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.2
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• pp.262-267
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• 2003

Error of the geometric series expansion method for the structural sensitivity analysis is estimated. Although the semi-analytic method has several advantages, accuracy of the method prevents it from practical application. One of the promising remedies is the use of geometric series formula for the matrix inversion. Its result of the sensitivity analysis converges that of the global difference method which is known as reliable one. To reduce computational efforts and to obtain reliable results, it is important to know how many terms need to expand. In this paper, the error formula is presented and Its usefulness is illustrated through numerical experiments.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.4
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• pp.585-592
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• 2003

Among various sensitivity evaluation techniques, semi-analytical method(SAM) is quite popular since this method is more advantageous than analytical method(AM) and global finite difference method(FDM). However, SAM reveals severe inaccuracy problem when relatively large rigid body motions are identified fur individual elements. Such errors result from the numerical differentiation of the pseudo load vector calculated by the finite difference scheme. In the present study, an iterative method combined with mode decomposition technique is proposed to compute reliable semi-analytical design sensitivities. The improvement of design sensitivities corresponding to the rigid body mode is evaluated by exact differentiation of the rigid body modes and the error of SAM caused by numerical difference scheme is alleviated by using a Von Neumann series approximation considering the higher order terms for the sensitivity derivatives.

• Proceedings of the KSME Conference
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• 2001.06a
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• pp.686-691
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• 2001

Structural optimization often require the evaluation of design sensitivities. The Semi Analytic method(SAM) is popular for shape optimization because this method has several advantages. But when relatively large rigid body motions are identified for individual elements, the SA method shows severe inaccuracy. In this paper, the improvement of design sensitivities corresponding to the rigid body mode is evaluated by exact differentiation of the rigid body modes. Moreover, the error of the SA method caused by numerical difference scheme is alleviated by using a series approximation for the sensitivity derivatives and considering the higher order terms.

• Proceedings of the KSME Conference
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• 2003.04a
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• pp.491-496
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• 2003

In sensitivity analysis, semi-analytical method(SAM) reveals severe inaccuracy problem when relatively large rigid body motions are identified for individual elements. Recently such errors of SAM resulted by the finite difference scheme have been improved by the separation of rigid body mode. But the eigenvalue should be obtained first before the sensitivity analysis is performed and it takes much time in the case that large system is considered. In the present study, by constructing a reduced one from the original system, iterative method combined with mode decomposition technique is proposed to compute reliable semi-analytical design sensitivities. The sensitivity analysis is performed by the eigenvector acquired from the reduced system. The error of SAM caused by difference scheme is alleviated by Von Neumann series approximation.

• Transactions of the Korean Society of Mechanical Engineers
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• v.17 no.1
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• pp.11-20
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• 1993

Sensitivity information is required in the optimal design process. In structural optimization, sensitivity calculation is a bottleneck due to its complexities and expensiveness. Various schemes have been proposed for the calculation. Analytic and finite difference methods are the most popular at the present time. However, they have advantages and disadvantages in different ways. Semi-anayltic method has been suggested to overcome the difficulties. In spite of the excellency, the semi-analytic method has been found to possess numerical error quite much with respect to shape variables. In this research, the error from each method is evaluated and compared using a shape variable. A two-dimensional beam is selected for an example since it has mathematical solution. An efficient method is suggested for the structural optimization which utilizes finite element method.

• Proceedings of the KSME Conference
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• 2008.11a
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• pp.418-423
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• 2008

Among various sensitivity evaluation techniques, semi-analytical method is quite popular since this method is more advantageous than analytical method and global finite difference method. However, SAM reveals severe inaccuracy problem when relatively large rigid body motions are identified for individual elements. Such errors result from the numerical differentiation of the pseudo load vector calculated by the finite difference scheme. In the present study, the adjoint variable method combined with complex variable is proposed to obtain the shape and size sensitivity for structural optimization. The complex variable can present accurate results regardless of the perturbation size as well as easy to be implemented. Through a few numerical examples of the static problem for the structural sensitivity, the efficiency and reliability of the adjoint variable method combined with complex variable is demonstrated.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.3
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• pp.243-250
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• 2009

The adjoint variable method can reduce computation time and save computer resources because it can selectively provide the sensitivity information for the positions that designers wish to measure. However, the adjoint variable method commonly employs exact analytical differentiation with respect to the design variables. It can be cumbersome to precisely differentiate every given type of finite element. This trouble can be overcome only if the numerical differentiation scheme can replace this exact manner of differentiation. But, the numerical differentiation scheme causes of severe inaccuracy due to the perturbation size dilemma. For assuring the accurate sensitivity without any dependency of perturbation size, this paper employs a complex variable that has been mainly used for computational fluid dynamics problems. The adjoint variable method combined with complex variables is applied to obtain the shape and size sensitivity for structural optimization. Numerical examples demonstrate that the proposed method can predict stable sensitivity results and that its accuracy is remarkably superior to traditional sensitivity evaluation methods.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.8
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• pp.745-752
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• 2009

In robust design, the mean and variance of design performance are frequently used to measure the design performance and its robustness under uncertainties. In this paper, we present the Gauss-type quadrature formula as a rigorous method for mean and variance estimation involving arbitrary input distributions and further extend its use to robust design optimization. One dimensional Gauss-type quadrature formula are constructed from the input probability distributions and utilized in the construction of multidimensional quadrature formula such as the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed. The proposed approach is applied to a simple bench mark problems and robust topology optimization of structures considering various types of uncertainty.