• Transactions of the Korean Society of Mechanical Engineers A
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• v.28 no.10
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• pp.1590-1597
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• 2004

Three methods of design sensitivity analysis for structures such as numerical method, analytical method and semi-analytical method have been developed for the last three decades. Although analytical design sensitivity analysis can provide very exact result, it is difficult to implement into practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable fur most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate in nonlinear design sensitivity analysis because its computational cost depends on the number of design variables and large numerical errors can be included. Thus the semi-analytical method is more suitable for complicated design problems. Moreover, semi-analytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure fur the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and the computational technique is proposed for evaluating the partial differentiation of internal nodal force, so called pseudo-load. Numerical examples coupled with commercial finite element package are shown to verify usefulness of proposed semi-analytical sensitivity analysis procedure and computational technique for pseudo-load.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.25 no.2
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• pp.182-191
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• 2001

The paper describes the improvement on concurrent subspace optimization(CSSO) via automatic differentiation. CSSO is an efficient strategy to coupled multidisciplinary design optimization(MDO), wherein the original design problem is non-hierarchically decomposed into a set of smaller, more tractable subspaces. Key elements in CSSO are consisted of global sensitivity equation, subspace optimization, optimum sensitivity analysis, and coordination optimization problem that require frequent use of 1st order derivatives to obtain design sensitivity information. The current version of CSSO adopts automatic differentiation scheme to provide a robust sensitivity solution. Automatic differentiation has numerical effectiveness over finite difference schemes tat require the perturbed finite step size in design variable. ADIFOR(Automatic Differentiation In FORtran) is employed to evaluate sensitivities in the present work. The use of exact function derivatives facilitates to enhance the numerical accuracy during the iterative design process. The paper discusses how much the automatic differentiation based approach contributes design performance, compared with traditional all-in-one(non-decomposed) and finite difference based approaches.

• 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.

• Journal of the Korean Mathematical Society
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• v.49 no.1
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• pp.99-112
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• 2012

This paper attempts to present a numerical method for solving optimal control problems. The method is based upon constructing the n-th degree Jacobi polynomials to approximate the control vector and use differentiation matrix to approximate derivative term in the state system. The system dynamics are then converted into system of algebraic equations and hence the optimal control problem is reduced to constrained optimization problem. Numerical examples illustrate the robustness, accuracy and efficiency of the proposed method.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.29 no.9 s.240
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• pp.1209-1216
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• 2005

Three methods for design sensitivity such as numerical differentiation, analytical method and semi-analytical method have been developed for the last three decades. Although analytical design sensitivity analysis is exact, it is hard to implement for practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable for most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate because its computational cost depends on the number of design variables and large numerical errors can be included especially in nonlinear design sensitivity analysis. Thus semi-analytical method is more suitable for complicated design problems. Moreover semi-analytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure for the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and computational technique is proposed, which evaluates the pseudo-load for design sensitivity analysis easily by using the design variation of corresponding internal nodal forces. Errors in semi-analytical design sensitivity analysis are examined and numerical examples are illustrated to confirm the reduction of numerical error considerably.

• Transactions of the Korean Society of Automotive Engineers
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• v.6 no.5
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• pp.128-137
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• 1998

A numerical approach for performing kinematic design sensitivity analysis of vehicle suspension systems is presented. Compared with the conventional analytical methods, which require explicit derivation of sensitivity equations, the proposed numerical method can be applied to any type of suspension systems without obtaining sensitivity equations, once any kinematic analysis procedure is established. To obtain sensitivity equations, a numerical differentiation algorithm that uses the third order Lagrange polynomial is developed. The algorithm efficiently and accurately computes the sensitivity of various vehicle static design factors with respect to kinematic design variables. Through a suspension design problem, the validity and usefulness of the method is demonstrated.

• Geophysics and Geophysical Exploration
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• v.25 no.4
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• pp.242-251
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• 2022

Automatic differentiation automatically calculates the derivatives of a function using the chain rule once the forward operation of a function is defined. Given the recent development of computing libraries that support automatic differentiation, many researchers have adopted automatic differentiation techniques to solve geophysical inverse problems. We analyzed the advantages, disadvantages, and performances of automatic differentiation techniques using the gradient calculations of seismic full waveform inversion objective functions. The gradients of objective functions can be expressed as multiplications of the derivatives of the model parameters, wavefields, and objective functions using the chain rule. Using numerical examples, we demonstrated the speed of analytic differentiation and the convenience of complex gradient calculations for automatic differentiation. We calculated derivatives of model parameters and objective functions using automatic differentiation and derivatives of wavefields using analytic differentiation.

• Journal of Industrial Technology
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• v.18
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• pp.233-240
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• 1998

In this study, ride sensitivity analysis of train with non-linear suspension elements is performed. Non-linear characteristics of springs and dampers for primary and secondary suspensions of a train is parameterized. Equation of motion of the train model is derived, and using the direct differentiation method, sensitivity equations are obtained. For a nominal ride quality performance index, sensitivity analysis with respect to various design parameters regarding non-linear suspension parameters is carried out.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1996.04a
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• pp.378-382
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• 1996

This paper presents a 'mixed' method for performing the sensitivity analysis for multibody dynamics. The mixed method uses both the analytical derivation and the numerical evaluation, in which premitive derivations rely on the analytical process and their associated individual terms are evaluated by the numerical precess. Therefore, this method can eliminate difficulty in dervation of the direct differentiation. Furthermore, by using the joint coordinate formulation for the equations of motion, compulational efficiencyand numerical accuracy are achieved.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2000.04b
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• pp.177-186
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• 2000

An improved multi-level（IML) optimization algorithm using automatic differentiation (AD) for multi-objective optimum design of framed structures is proposed in this paper. In order to optimize the steel frames under seismic load, two main objective functions need to be considered for minimizing the structural weight and maximizing the strain energy. For the efficiency of the proposed algorithm, multi-level optimization techniques using decomposition method that separately utilizes both system-level and element-level optimizations and an artificial constraint deletion technique are incorporated in the algorithm. And also to save the numerical efforts, an efficient reanalysis technique through approximated structural responses such as moments, frequencies, and strain energy with respect to intermediate variables is proposed in the paper. Sensitivity analysis of dynamic structural response is executed by AD that is a powerful technique for computing complex or implicit derivatives accurately and efficiently with minimal human effort. The efficiency and robustness of the IML algorithm, compared with a plain multi-level (PML) algorithm, is successfully demonstrated in the numerical examples.