• Title, Summary, Keyword: Shape design sensitivity

### Shape Optimization for Improving Fatigue Life of a Lower Control Arm Using the Experimental Design (실험계획법을 적용한 Lower Control Arm의 피로수명 형상 최적설계)

• 김민수;이창욱;손성효;임홍재;허승진
• Transactions of the Korean Society of Automotive Engineers
• /
• v.11 no.3
• /
• pp.161-166
• /
• 2003
• In order to improve the fatigue lift of a lower control arm in the vehicle suspension, a new shape optimization procedure is presented. In this approach, the shape control point concept is introduced to reduce the numbers of shape design variables. Also, the two-level orthogonal way is employed to evaluate the design sensitivity of fatigue life with respect to those shape design variables, because the analytical design sensitivity information is not directly supplied from the commercial CAE softwares. In this approach, only the six design variables are used to approximate the shape of lower control arm. Then, performed are only 10 fatigue life analyses including the baseline design, 8 DOE models and the final design. The final design, the best combination obtained from the sensitivity information, can maximize the fatigue lift nearly two times as that of the baseline design, while reducing the 12 percentage of weight than it.

### Computer Simulation and Shape Design Sensitivity Analysis of the Valve inside the Reciprocal Compressor using Finite Element Model (유한 요소 모델을 이용한 왕복동식 압축기 밸브의 거동 해석 및 형상 설계 민감도 해석)

• 이제원;왕세명;주재만;박승일;이성태
• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
• /
• /
• pp.796-801
• /
• 2002
• The goal of this research is the shape design of the valve using a computer simulation. For an analysis a basic mathematical model describing compression cycle is considered as consisting of five sets of coupled equations. These are the volume equation (kinematics), valve dynamic equation (dynamics), ideal gas equation (thermodynamics), Bernoulli equation (fluid dynamics), and dynamic equation of fluid particle based on Helmholtz equation (acoustics). Valve motion is made by the superposition of free vibration modes obtained by the finite element method. That is, the eigenvalues and eigenvectors are the sufficient modeling factors fur the valve in the simulation program. Thus, to design a shape of the valve, shape design sensitivity through chain-ruled derivatives is considered from two sensitivity coefficients, one is the design sensitivity of the capability of compressor with respect to the eigenvalues of the valve, and the other is the design sensitivity of the eigenvalue with respect to the shape change of the valve. In this research, the continuum design sensitivity analysis concepts are used for the latter.

### Direct Differentiation Method for Shape Design Sensitivity Analysis of Axisymmetric Elastic Solids by the BEM and Shape Optimization of Turbin Disc (경계요소법에 의한 축대칭 탄성체의 형상설계 민감도해석을 위한 직접미분법과 터빈 디스크의 형상최적설계)

• Lee, Bu-Yun
• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.20 no.5
• /
• pp.1458-1467
• /
• 1996
• A direct differentiationmethod is presented for the shape design sensitivity analysis of axisymmeetric elastic solids. Based on the exisymmetric boundary integralequaiton formulation, a new boundary ntegral equatio for sensitivity analysis is derived by taking meterial derivative to the same integral identity that was used in the adjoint variable melthod. Numerical implementation is performed to show the applicaiton of the theoretical formulation. For a simple example with analytic solution, the sensitivities by present method are compared with analytic sensitivities. As an application to the shape optimization, an optimal shape of a gas turbine disc toinimize the weight under stress constraints is found by incorporating the sensitivity analysis algorithm in an optimizatio program.

### Shape Design Sensitivity Analysis of Supercavitating Flow Problem (초공동(超空洞) 유동 문제의 형상 설계민감도 해석)

• Choi, J.H.;Gwak, H.G.;Grandhi, R.
• Proceedings of the KSME Conference
• /
• /
• pp.1047-1052
• /
• 2004
• An efficient boundary-based technique is developed for addressing shape design sensitivity analysis in supercavitating flow problem. An analytical sensitivity formula in the form of a boundary integral is derived based on the continuum formulation for a general functional defined in potential flow problems. The formula, which is expressed in terms of the boundary solutions and shape variation vectors, can be conveniently used for gradient computation in a variety of shape design in potential flow problems. While the sensitivity can be calculated independent of the analysis means, such as the finite element method (FEM) or the boundary element method (BEM), the FEM is used for the analysis in this study because of its popularity and easy-touse features. The advantage of using a boundary-based method is that the shape variation vectors are needed only on the boundary, not over the whole domain. The boundary shape variation vectors are conveniently computed by using finite perturbations of the shape geometry instead of complex analytical differentiation of the geometry functions. The supercavitating flow problem is chosen to illustrate the efficiency of the proposed methodology. Implementation issues for and optimization procedure are addressed in this flow problem.

### Shape Design Sensitivity Analysis using Isogeometric Approach (CAD 형상을 활용한 설계 민감도 해석)

• Ha, Seung-Hyun;Cho, Seon-Ho
• Proceedings of the Computational Structural Engineering Institute Conference
• /
• /
• pp.577-582
• /
• 2007
• A variational formulation for plane elasticity problems is derived based on an isogeometric approach. The isogeometric analysis is an emerging methodology such that the basis functions in analysis domain arc generated directly from NURBS (Non-Uniform Rational B-Splines) geometry. Thus. the solution space can be represented in terms of the same functions to represent the geometry. The coefficients of basis functions or the control variables play the role of degrees-of-freedom. Furthermore, due to h-. p-, and k-refinement schemes, the high order geometric features can be described exactly and easily without tedious re-meshing process. The isogeometric sensitivity analysis method enables us to analyze arbitrarily shaped structures without re-meshing. Also, it provides a precise construction method of finite element model to exactly represent geometry using B-spline base functions in CAD geometric modeling. To obtain precise shape sensitivity, the normal and curvature of boundary should be taken into account in the shape sensitivity expressions. However, in conventional finite element methods, the normal information is inaccurate and the curvature is generally missing due to the use of linear interpolation functions. A continuum-based adjoint sensitivity analysis method using the isogeometric approach is derived for the plane elasticity problems. The conventional shape optimization using the finite element method has some difficulties in the parameterization of boundary. In isogeometric analysis, however, the geometric properties arc already embedded in the B-spline shape functions and control points. The perturbation of control points in isogeometric analysis automatically results in shape changes. Using the conventional finite clement method, the inter-element continuity of the design space is not guaranteed so that the normal vector and curvature arc not accurate enough. On tile other hand, in isogeometric analysis, these values arc continuous over the whole design space so that accurate shape sensitivity can be obtained. Through numerical examples, the developed isogeometric sensitivity analysis method is verified to show excellent agreement with finite difference sensitivity.

### Shape Design Sensitivity Analysis and Optimization of General Plane Arch Structures (일반 평면 아치 구조물의 형상설계민감도 해석 및 최적설계)

• 최주호
• Proceedings of the Computational Structural Engineering Institute Conference
• /
• /
• pp.238-245
• /
• 2000
• A general formulation for shape design sensitivity analysis over a plane arch structure is developed based on a variational formulation of curved beam in linear elasticity. Sensitivity formula is derived using the material derivative concept and adjoint variable method for the stress defined at a local segment. Obtained sensitivity expression, which can be computed by simple algebraic manipulation of the solution variables, is well suited for numerical implementation since it does not involve numerical differentiation. Due to the complete description for the shape and its variation of the arch, the formulation can manage more complex design problems with ease and gives better optimum design than before. Several examples are taken to show the advantage of the method, in which the accuracy of the sensitivity is evaluated. Shape optimization is also conducted with two design problems to illustrate the excellent applicability.

### Boundary-Based Shape Design Sensitivity Analysis of Elastostatics Problems (정탄성학 문제에서 경계 기반 형상설계 민감도 해석)

• Won Jun-Ho;Choi Joo-Ho
• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.30 no.2
• /
• pp.149-156
• /
• 2006
• A boundary-based design sensitivity analysis(DSA) technique is proposed for addressing shape optimization issues in the elastostatics problems. Sensitivity formula is derived based on the continuum formulation in a boundary integral form, which consists of the boundary solutions and shape variation vectors. Though the boundary element method(BEM) has been mainly used to obtain the boundary solution, the FEM is used in this paper because this is much more popular, and has greatly improved meshing and computing power recently. The advantage of the boundary DSA is that the shape variation vectors, which are also known as design velocity fields, are needed only on the boundary. Then, the step for determining the design velocity field over the whole domain, which was necessary in the domain-based DSA, is eliminated, making the process easy to implement and efficient. Problem of fillet design is chosen to illustrate the efficiency of the proposed method. Accuracy of the sensitivity is good with this method even by employing the free mesh for the FE analysis.

### 3D Optimal Design of Transformer Tank Shields using Design Sensitivity Analysis

• Yingying Yao;Ryu, Jae-Seop;Koh, Chang-Seop;Dexin Xie
• KIEE International Transaction on Electrical Machinery and Energy Conversion Systems
• /
• v.3B no.1
• /
• pp.23-31
• /
• 2003
• A novel 3D shape optimization algorithm is presented for electromagnetic devices carry-ing eddy current. The algorithm integrates the 3D finite element performance analysis and the steepest descent method with design sensitivity and mesh relocation method. For the design sensitivity formula, the adjoint variable vector is defined in complex form based on the 3D finite element method for eddy current problems. A new 3D mesh relocation method is also proposed using the deformation theory of the elastic body under stress to renew the mesh as the shape changes. The design sensitivity f3r the sur-face nodal points is also systematically converted into that for the design variables for the parameterized optimization application. The proposed algorithm is applied to the optimum design of the tank shield model of the transformer and the effectiveness is proved.

### 3D Shape Optimization of Nonlinear Electromagnetic Device Using Parameterized Sensitivity Analysis (매개화된 민감도 해석에 의한 비선형 전자소자의 3차원 형상최적화)

• ;Yingying Yao
• The Transactions of the Korean Institute of Electrical Engineers B
• /
• v.53 no.8
• /
• pp.469-476
• /
• 2004
• In this paper, a 3D shape optimization algorithm which guarantees a smooth optimal shape is presented using parameterized sensitivity analysis. The design surface is parameterized using Bezier spline and B-spline, and the control points of the spline are taken as the design variables. The parameterized sensitivity for the control points are found from that for nodal points. The design sensitivity and adjoint variable formulae are also derived for the 3D non-linear problems. Through an application to the shape optimization of 3D electromagnet to get a uniform magnetic field, the effectiveness of the proposed algorithm is shown.

### 3D Shape Optimization of Electromagnetic Device Using Design Sensitivity Analysis and Mesh Relocation Method (설계민감도해석과 요소망 변형법을 이용한 전자소자의 3차원 형상최적화)

• ;Yao Yingying
• The Transactions of the Korean Institute of Electrical Engineers B
• /
• v.52 no.7
• /
• pp.307-314
• /
• 2003
• This paper presents a 3D shape optimization algorithm for electromagnetic devices using the design sensitivity analysis with finite element method. The structural deformation analysis based on the deformation theory of the elastic body under stress is used for mesh renewing. The design sensitivity and adjoint variable formulae are derived for the 3D finite element method with edge element. The results of sensitivity analysis are used as the input data of the structural analysis to calculate the relocation of the nodal points. This method makes it possible that the new mesh of analysis region can be obtained from the initial mesh without regeneration. The proposed algorithm is applied to the shape optimization of 3D electromagnet pole to net a uniform flux density at the target region.