• Transactions of the Korean Society of Automotive Engineers
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• v.7 no.5
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• pp.141-153
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• 1999

A finite element procedure for the analysis of rubber-like hyperelastic material is developed. The volumetric incompressiblity conditions of the rubber deformation is included in the formulation by using penalty method. In this paper, the behavior of the rubber deformation is represented by hyperelastic constitutive relations based on a generalized Mooney-Rivlin model. The principle of virtual work is used to derive nonlinear finite element equation for the large displacement problem and presented in total-Lagrangian description. The finite element procedure using analytic differentiation resulted in very close solution to the result of the well known commercial packages NISAII AND ABAQUS. Numerical tests show that the results from the numerical differentiation method coincide very well with those from the analytic method and the well known commercial packages in static analysis. The convergency of rubber usingν iteration method is also discussed.

• Proceedings of the KSME Conference
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• 2004.11a
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• pp.492-497
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• 2004

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.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2001.10a
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• pp.53-58
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• 2001

This paper deals with the application of numerical differentiation in the structural analysis. In the structural analysis, the derivative values of the given function are sometimes used in calculation of structural behaviors. For calculating the derivative values, both the time and labor are needed when the structures consist of non-linear geometries such as arches or curved beams. From this viewpoint the numerical differentiation scheme is applied into the structural analysis. The numerical results obtained from the numerical differentiation are agreed very well with those obtained from the exact derivatives by analytical method. It is expected that the numerical differentiation can be utilized practically in the structural analysis.

• 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.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.19 no.4 s.74
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• pp.441-447
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• 2006

This paper deals with the application of the numerical differentiation in the structural analyses. Derivative values of the geometry of structure are definitely needed for analysing the structural behavior. In this study, free vibration problems of arches are chosen for verifying the numerical differential technique in the structural analyses. The curvature parameters composed with the derivatives of arch geometry obtained herein are quite agreed with those of analytical method. Also, natural frequencies with curvature parameters obtained by using the forward fifth polynomial method are quite agreed with those in the literature. The numerical differentiation technique can be practically utilized in the structural analyses.

• 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.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2003.11a
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• pp.814-818
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• 2003

This paper deals with the application of numerical differentiation in free vibration analysis. In the free vibration analysis, the derivative values of the given function are certainly used in calculation of structural parameters. For deriving the derivative values, both the time and labor are needed when the structures consist of non-linear geometries such as arches or curved beams. From this viewpoint, the numerical differentiation scheme is applied into the free vibration analysis. The numerical results obtained from the numerical differentiations are agreed very well with those obtained from the exact derivatives by analytical method. It is expected that the numerical differentiations can be utilized practically in the free vibration analysis.

• 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.

• 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.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1999.10a
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• pp.359-369
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• 1999

The paper describes the study of concurrent subspace optimization(CSSO) for coupled multidisciplinary design optimization (MDO) techniques in mechanical systems. This method is a solution to large scale coupled multidisciplinary system, wherein the original problem is decomposed into a set of smaller, more tractable subproblems. Key elements in CSSO are consisted of global sensitivity equation(GSE), subspace optimization (SSO), optimum sensitivity analysis(OSA), and coordination optimization problem(COP) so as to inquiry valanced design solutions finally, Automatic differentiation has an ability to provide a robust sensitivity solution, and have shown the numerical numerical effectiveness over finite difference schemes wherein the perturbed step size in design variable is required. The present paper will develop the automatic differentiation based concurrent subspace optimization(AD-CSSO) in MDO. An automatic differentiation tool in FORTRAN(ADIFOR) will be employed to evaluate sensitivities. The use of exact function derivatives in GSE, OSA and COP makes Possible to enhance the numerical accuracy during the iterative design process. The paper discusses how much influence on final optimal design compared with traditional all-in-one approach, finite difference based CSSO and AD-CSSO applying coupled design variables.