Krylov 부공간 모델차수축소법을 이용한 주파수응답함수의 직접 설계민감도 해석

한정삼,Han, Jeong-Sam 한국전산구조공학회 2010 한국전산구조공학회논문집 Vol.23 No.2

Krylov 부공간 모델차수축소법은 초기 유한요소모델과 축소모델의 전달함수의 계수인 모멘트를 일치시키는 방법을 이용하는 축소기법으로 이미 대형 유한요소모델의 주파수응답함수의 효율적인 계산에 많이 사용되고 있는 방법 중의 하나이다. 본 논문에서는 Krylov 부공간 축소기법을 이용한 관심 주파수영역에 대한 주파수응답 해석과 이를 통하여 계산된 주파수응답의 여러 가지 설계변수에 대한 설계민감도 해석 방법을 제안하였다. 일반적으로 기계시스템의 주파수응답을 고려한 최적설계를 위해서는 설계변수에 대한 관심 주파수영역에서의 주파수응답 및 그의 민감도 정보가 요구되므로, 고려하는 유한요소모델이 대형일 경우에는 관심 주파수영역에서의 반복적인 해석으로 인한 계산비용의 문제가 심각하게 대두된다. 본 논문에서는 축소모델을 이용하여 주파수응답과 주파수응답의 설계민감도 해석을 수행하여 계산의 효율성을 극대화하였다. 계산상 시스템행렬의 민감도 계산에는 시간측면과 구현의 용이성 측면에서 장점이 있는 준해석적 방법을 이용하였다. 수치 예제를 통하여 축소기법을 이용한 주파수응답의 설계민감도 해석 결과를 초기 유한요소모델의 민감도 결과와 비교하여 우수한 정확성 및 효율성을 확인하였다. 본 논문에서 제안된 방법을 주파수응답을 고려하는 최적설계에 이용하는 경우, 결과의 정확성 및 계산비용 측면에서 매우 효과적인 방법이 될 수 있을 것으로 판단된다. In this paper a frequency response analysis using Krylov subspace-based model reduction and its design sensitivity analysis with respect to design variables are presented. Since the frequency response and its design sensitivity information are necessary for a gradient-based optimization, problems of high computational cost and resource may occur in the case that frequency response of a large sized finite element model is involved in the optimization iterations. In the suggested method model order reduction of finite element models are used to calculate both frequency response and frequency response sensitivity, therefore one can maximize the speed of numerical computation for the frequency response and its design sensitivity. As numerical examples, a semi-monocoque shell and an array-type $4{\times}4$ MEMS resonator are adopted to show the accuracy and efficiency of the suggested approach in calculating the FRF and its design sensitivity. The frequency response sensitivity through the model reduction shows a great time reduction in numerical computation and a good agreement with that from the initial full finite element model.

CAD 기반 최적설계 시스템인 DS / FDM의 개발과 공학 구조물에 대한 적용 ( Development of DS / FDM - a Robust CAD-based Optimal Design System and Its Application to Engineering Structures )

한정삼,프랭크 웁하우스(Frank Uphaus),김영렬,곽병만 대한기계학회 2000 대한기계학회 춘추학술대회 Vol.- No.-

다중 하중 과도응답해석 과정에 대한 크리로프 부공간 모델차수축소법의 자동화

한정삼,김승현 한국전산구조공학회 2021 한국전산구조공학회논문집 Vol.34 No.2

다중 하중 과도응답해석은 시간에 따른 작용 하중에 대한 과도응답을 확인하므로 정교한 시스템 모델링 및 조밀한 시간 간격을 가 질수록 해당 시스템에 대한 동특성은 정확하게 나타내지만 이에 따른 계산 시간은 크게 증가하게 된다. 크리로프 부공간 기반 모델차 수축소법은 기계 시스템이 가지는 동적 특성과 거의 동일한 결과를 나타내면서 계산 시간을 줄일 수 있기 때문에 효율적인 과도응답 해석 방법이다. 본 연구에서는 다중 하중 및 이동 하중을 가지는 수치 예제를 통하여 크리로프 부공간 모델차수축소법 기반 과도응답 해석을 수행하고, 이를 통해 초기 시스템 및 축소차수 모델의 정확성 및 효율성을 비교하였다. 또한, 시스템 행렬 추출, 크리로프 부공 간의 기저 벡터로 구성되는 변환행렬 생성 및 축소차수모델 생성 그리고 이를 바탕으로 과도응답해석을 하는 절차를 수립하여 상용 유한요소 프로그램인 ANSYS Workbench ACT를 통해 과도응답해석 과정 자동화를 구현하여 그 효용성과 효율성을 보였다. In general, several computational resources are required to perform multiple-loading transient response analyses. In this paper, we present the procedure for multiple-loading transient response analysis using the Krylov subspace model order reduction and Newmark’s time integration scheme. We utilized ANSYS MAPDL, Python, and ANSYS ACT to automate the transient response analysis procedure in the ANSYS Workbench environment and studied several engineering numerical examples to demonstrate the feasibility and efficiency of the proposed approach.

Krylov 벡터를 이용한 모델차수축소법

한정삼,Han, Jeong-Sam 한국전산구조공학회 2012 전산구조공학 Vol.25 No.4

축소된 유한요소모델을 이용한 하드디스크 구동부의 고유치 및 주파수응답 해석

한정삼 대한기계학회 2007 大韓機械學會論文集A Vol.31 No.5

Efficient frequency response and its direct sensitivity analyses for large-size finite element models using Krylov subspace-based model order reduction

한정삼 대한기계학회 2012 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.26 No.4

In this paper, we examine an efficient calculation of the approximate frequency response (FR) for large-size finite element (FE) models using the Krylov subspace-based model order reduction (MOR) and its direct design sensitivity analysis with respect to design variables for sizing. Information about both the FR and its design sensitivity is necessary for typical gradient-based optimization iterations;therefore, the problem of high computational cost may occur when FRs of a large-size FE models are involved in the optimization problem. In the method suggested in this paper, reduced order models, generated from the original full-order FE models through the Arnoldi process, are used to calculate both the FR and FR sensitivity. This maximizes the speed of numerical computation of the FR and its design sensitivity. Assuming that the Krylov basis vectors remain constant with respect to the perturbation of a design variable, the FR sensitivity analysis is performed in a more efficient manner. As numerical examples, a car body with 535,992 degrees of freedom (DOF)and a 6 × 6 micro-resonator array with 368,424 DOF are adopted to demonstrate the numerical accuracy and efficiency of the suggested approach. Using the reduced-order models, we found that the FR and FR sensitivity are in a good agreement with those using the fullorder FE model. The reduction in computation time is also found to be significant because of the use of Krylov subspace-based reduced models.

Krylov subspace-based model order reduction for Campbell diagram analysis of large-scale rotordynamic systems

한정삼 국제구조공학회 2014 Structural Engineering and Mechanics, An Int'l Jou Vol.50 No.1

This paper focuses on a model order reduction (MOR) for large-scale rotordynamic systems by using finite element discretization. Typical rotor-bearing systems consist of a rotor, built-on parts, and a support system. These systems require careful consideration in their dynamic analysis modeling because they include unsymmetrical stiffness, localized nonproportional damping, and frequency-dependent gyroscopic effects. Because of this complex geometry, the finite element model under consideration may have a very large number of degrees of freedom. Thus, the repeated dynamic analyses used to investigate the critical speeds, stability, and unbalanced response are computationally very expensive to complete within a practical design cycle. In this study, we demonstrate that a Krylov subspace-based MOR via moment matching significantly speeds up the rotordynamic analyses needed to check the whirling frequencies and critical speeds of large rotor systems. This approach is very efficient, because it is possible to repeat thedynamic simulation with the help of a reduced system by changing the operating rotational speed, which canbe preserved as a parameter in the process of model reduction. Two examples of rotordynamic systems show that the suggested MOR provides a significant reduction in computational cost for a Campbell diagram analysis, while maintaining accuracy comparable to that of the original systems.

4중 안정성 일체형 메커니즘의 설계, 해석 및 제작

한정삼 대한기계학회 2007 大韓機械學會論文集A Vol.31 No.5

Comparison of model order reductions using Krylov and modal vectors for transient analysis under seismic loading

한정삼 국제구조공학회 2020 Structural Engineering and Mechanics, An Int'l Jou Vol.76 No.5

Generally, it is necessary to perform transient structural analysis in order to verify and improve the seismic performance of high-rise buildings and bridges against earthquake loads. In this paper, we propose the model order reduction (MOR) method using the Krylov vectors to perform seismic analysis for linear and elastic systems in an efficient way. We then compared the proposed method with the mode superposition method (MSM) by using the limited numbers of modal vectors (or eigenvectors) calculated from the modal analysis. In the calculation, the data of the El Centro earthquake in 1940 were adopted for the seismic loading in the transient analysis. The numerical accuracy and efficiency of the two methods were compared in detail in the case of a simplified high-rise building.

Calculation of design sensitivity for large-size transient dynamic problems using Krylov subspace-based model order reduction

한정삼 대한기계학회 2013 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.27 No.9

Nowadays, transient dynamic responses of a large-size finite element (FE) model can be solved within a reasonable computation time owing to rapid improvement in both numerical schemes and computing resources. However, increasing demands for accurate simulation and complicated modeling have led to larger and more complex finite element models, which consequently result in considerably high computational cost. In addition, when structural optimizations include transient responses such as displacement, velocity, and acceleration,the optimizations often do not end within a reasonable process time because the large-size simulation must be repeated many times. In order to reduce the computational cost in this respect, model order reduction (MOR) for the original full-order model (FOM) can be used for the transient response simulation. In this paper, a transient dynamic response analysis using Krylov subspace-based MOR and its design sensitivity analysis with respect to sizing design variables is suggested as an approach to the handling of large-size finite element models. Large-size finite element models can incur the problem of a long computation time in gradient-based optimization iterations because of the need for repeated simulation of transient responses. In the suggested method, the reduced order models (ROMs) generated from the original FOMs using implicit moment-matching via the Arnoldi process are used to calculate the transient response and its design sensitivity. As a result, the speed of numerical computation for the transient response and its design sensitivity is maximized. Newmark’s time integration method is employed to calculate transient responses and their design sensitivities. In the case of the transient sensitivity analysis, we apply a temporal discretization scheme to the design sensitivity equation derived by directly differentiating the governing equation with respect to design variables. This methodology has been programmed on the MATLAB with the FE information extracted from the FE package ANSYS. Two application examples are provided to demonstrate the numerical accuracy and efficiency of the suggested approach. The relative errors of transient response and design sensitivity between the FOMs and ROMs are also compared according to the orders of the reduced model. Calculation of transient dynamic responses and their sensitivities using Krylov subspacebased MOR shows a sizeable reduction in computation time and a good agreement with those provided by the FOM.