• Proceedings of the KSME Conference
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• 2001.06c
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• pp.398-401
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• 2001

There are two methods to calculate design sensitivity such as direct differentiation method and adjoint method. A sort of direct differentiation method for design sensitivity analysis costs too much when number of design variables is much larger than the number of response functions whose design sensitivity analyses are required. Therefore, an adjoint method is suggested for the case that the dimension of design variables is lager than the number of response function. An adjoint method is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation, requiring only the eigenvalue and its associated eigenvectors for mode being differentiated. This method has been extended to the repeated eigenvalue problem. In this paper, we propose an adjoint method for deign sensitivity analysis of damped vibratory systems with distinct eigenvalues.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.29 no.11 s.242
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• pp.1527-1533
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• 2005

Three methods for design sensitivity analysis such as finite difference method(FDM), direct differentiation method(DDM) and adjoint variable method(AVM) are well known. FDM and DDM for design sensitivity analysis cost too much when the number of design variables is too large. An AVM is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation. Because the adjoint equation is independent of the number of design variables, an AVM is efficient for when number of design variables is too large. In this study, AVM has been extended to the eigenproblem of damped systems whose eigenvlaues and eigenvectors are complex numbers. Moreover, this method is implemented into a commercial finite element analysis program by means of the semi-analytical method to show applicability of the developed method into practical structural problems. The proposed_method is compared with FDM and verified its accuracy for analytical and practical cases.

• 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 Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.161-171
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• 1999

Aerodynamic sensitivity analysis codes are developed via the hand-differentiation using a direct differentiation method and an adjoint method respectively from discrete two-dimensional compressible Navier-Stokes equations. Unlike previous other researches, Baldwin-Lomax algebraic turbulence model is also differentiated by hand to obtain design sensitivities with respect to design variables of interest in turbulent flows. Discrete direct sensitivity equations and adjoint equations are efficiently solved by the same time integration scheme adopted in the flow solver routine. The required memory for the adjoint sensitivity code is greatly reduced at the cost of the computational time by allowing the large banded flux jacobian matrix unassembled. Direct sensitivity code results are found to be exactly coincident with sensitivity derivatives obtained by the finite difference. Adjoint code results of a turbulent flow case show slight deviations from the exact results due to the limitation of the algebraic turbulence model in implementing the adjoint formulation. However, current adjoint sensitivity code yields much more accurate sensitivity derivatives than the adjoint code with the turbulence eddy viscosity being kept constant, which is a usual assumption for the prior researches.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.23 no.6
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• pp.683-691
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• 2010

In this paper, using an adjoint variable method, we develop a design sensitivity analysis(DSA) method applicable to heat conduction problems in steady state. Also, a topology design optimization method is developed using the developed DSA method. Design sensitivity expressions with respect to the thermal conductivity are derived. Since the already factorized system matrix is utilized to obtain the adjoint solution, the cost for the sensitivity computation is trivial. For the topology design optimization, the design variables are parameterized into normalized bulk material densities. The objective function and constraint are the thermal compliance of structures and allowable material volume respectively. Through several numerical examples, the developed DSA method is verified to yield very accurate sensitivity results compared with finite difference ones, requiring less than 0.25% of CPU time for the finite differencing. Also, the topology optimization yields physical meaningful results.

• 한국전산유체공학회:학술대회논문집
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• 2009.04a
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• pp.57-65
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• 2009

An efficient and high-fidelity design approach for wing-body shape optimization is presented. Depending on the size of design space and the number of design of variable, aerodynamic shape optimization process is carried out via different optimization strategies at each design stage. In the first stage, global optimization techniques are applied to planform design with a few geometric design variables. In the second stage, local optimization techniques are used for wing surface design with a lot of design variables to maintain a sufficient design space with a high DOF (Degree of Freedom) geometric change. For global optimization, Kriging method in conjunction with Genetic Algorithm (GA) is used. Asearching algorithm of EI (Expected Improvement) points is introduced to enhance the quality of global optimization for the wing-planform design. For local optimization, a discrete adjoint method is adopted. By the successive combination of global and local optimization techniques, drag minimization is performed for a multi-body aircraft configuration while maintaining the baseline lift and the wing weight at the same time. Through the design process, performances of the test models are remarkably improved in comparison with the single stage design approach. The performance of the proposed design framework including wing planform design variables can be efficiently evaluated by the drag decomposition method, which can examine the improvement of various drag components, such as induced drag, wave drag, viscous drag and profile drag.

• 한국전산유체공학회:학술대회논문집
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• 2003.10a
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• pp.240-242
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• 2003

A method for performing two-dimensional lift-constraint drag minimization in inviscid compressible flows on unstructured meshes is developed. Sensitivities of objective function with respect to the design variables are efficiently obtained by using a continuous adjoint method. In addition, parallel algorithm is used in multi-point design optimization to enhance the computational efficiency. The characteristics of single-point and multi-point optimization are examined, and the comparison of these two method is presented.

• Journal of the Korean Society of Fisheries and Ocean Technology
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• v.35 no.4
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• pp.435-447
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• 1999

It is main objective of this approach to present a method to analyse stochastic design sensitivity for problems of structural dynamics with randomness in design parameters. A combination of the adjoint variable approach and the second order perturbation method is used in the finite element approach. An alternative form of the constant functional that holds for all times is introduced to consider the time response of dynamic sensitivity. The terminal problem of the adjoint system is solved using equivalent homogeneous equations excited by initial velocities. The numerical procedures are shown to be much more efficient when based on the fold superposition method: the generalized co-ordinates are normalized and the correlated random variables are transformed to uncorrelated variables, whereas the secularities are eliminated by the fast Fourier transform of complex valued sequences. Numerical algorithms have been worked out and proved to be accurate and efficient : they can be readily adapted to fit into the existing finite element codes whose element derivative matrices can be explicitly generated. The numerical results of two cases -2 dimensional portal frame for the comparison with reference and 3-dimensional frame structure - for the deterministic sensitivity analysis are presented.

• 한국전산유체공학회:학술대회논문집
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• 2000.05a
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• pp.66-72
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• 2000

Aerodynamic sensitivity analysis is performed for the Navier-Stokes equations coupled with two-equation turbulence models using a discrete adjoint method and a direct differentiation method respectively. Like the mean flow equations, the turbulence model equations are also hand-differentiated to accurately calculate the sensitivity derivatives of flow quantities with respect to design variables in turbulent viscous flows. Both the direct differentiation code and the adjoint variable code adopt the same time integration scheme with the flow solver to efficiently solve the differentiated equations. The sensitivity codes are then compared with the flow solver in terms of solution accuracy, computing time and computer memory requirements. The sensitivity derivatives obtained from the sensitivity codes with different turbulence models are compared with each other. Using two-equation turbulence models, it is observed that a usual assumption of constant turbulent eddy viscosity in adjoint methods may lead to seriously inaccurate results in highly turbulent flows.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.43 no.6
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• pp.1020-1026
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• 1994

This paper presents a technique to optimize the shape of waveguide components in H-plane. The technique utilizes the numerical optimization process which employs the vector finite element method. In the optimization process, the sensitivity of an objective function with respect to design variables is computed by introducting adjoint variables, which makes the computation easy. The steepest descent method is then employed to update design variables. As a numerical example, an H-plane waveguide teejunction was considered to obtain optimized shape. Comparison between the initial and optimized shape was made.