• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.27 no.1
• /
• pp.1-8
• /
• 2014

Using the level set and the meshfree methods, we develop a topological shape optimization method applied to linear elasticity problems. Design gradients are computed using an efficient adjoint design sensitivity analysis(DSA) method. The boundaries are represented by an implicit moving boundary(IMB) embedded in the level set function obtainable from the "Hamilton-Jacobi type" equation with the "Up-wind scheme". Then, using the implicit function, explicit boundaries are generated to obtain the response and sensitivity of the structures. Global nodal shape function derived on a basis of the reproducing kernel(RK) method is employed to discretize the displacement field in the governing continuum equation. Thus, the material points can be located everywhere in the continuum domain, which enables to generate the explicit boundaries and leads to a precise design result. The developed method defines a Lagrangian functional for the constrained optimization. It minimizes the compliance, satisfying the constraint of allowable volume through the variations of boundary. During the optimization, the velocity to integrate the Hamilton-Jacobi equation is obtained from the optimality condition for the Lagrangian functional. Compared with the conventional shape optimization method, the developed one can easily represent the topological shape variations.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.24 no.1
• /
• pp.61-67
• /
• 2011

This study is concerned with the shape optimization of structural members frequently found in critical area in a structure system, that is, highly stressed zone. Isogeometry analysis is well known to be the very efficient way to integrate the geometric modeling(CAD) and computational analysis(CAE). This can be accomplished by directly using the geometric modeling by NURBS(Non-Uniform Rational Basis Spline). In this study, an efficient computer code adopting the isogeometry concept has been developed for the structural analysis, in which CAD information can be directly used in the finite element modeling. In order to show the validity of the present code, the present results are compared with those by using the commercial package, that is, MSC/NASTRAN. The present isogeometric analysis procedure has been integrated with the optimization procedure to deal with the optimization problem found in the context of structural mechanics. The present system has been successfully applied to the shape optimization of cantilever structure having bracket. From the present study, it can be seen the validity of the present approach and computer codes developed in this study. This paper ends with some discussions about the practical usefulness of the present approach which is based on isogeometry analysis, and extension of the present study.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.25 no.6
• /
• pp.497-504
• /
• 2012

Finite element analysis is to approximate a geometry model developed in computer-aided design(CAD) to a finite element model, thus the conventional shape design sensitivity analysis and optimization using the finite element method have some difficulties in the parameterization of geometry. However, isogeometric analysis is to build a geometry model and directly use the functions describing the geometry in analysis. Therefore, the geometric properties can be embedded in the NURBS basis functions and control points so that it has potential capability to overcome the aforementioned difficulties. In this study, the isogeometric structural analysis and shape design sensitivity analysis in the generalized curvilinear coordinate(GCC) systems are discussed for the curved geometry. Representing the higher order geometric information, such as normal, tangent and curvature, yields the isogeometric approach to be the best way for generating exact GCC systems from a given CAD geometry. The developed GCC isogeometric structural analysis and shape design sensitivity analysis are verified to show better accuracy and faster convergency by comparing with the results obtained from the conventional isogeometric method.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.28 no.4
• /
• pp.425-431
• /
• 2015

Based on a bond-based peridynamics theory for dynamic crack propagation problems, this paper presents a design sensitivity analysis and optimization method. Peridynamics has a peculiar advantage over the existing continuum theory in the mathematical modelling of problems where discontinuities arise. For the design optimization of the crack propagation problems, a non-shape design sensitivity is derived using the adjoint variable method. The obtained adjoint sensitivity of displacement and strain energy turns out to be very accurate and efficient compared to the finite different sensitivity. The obtained design sensitivities are futher utilized to optimally control the position of bifurcation point in the design optimization of crack propagation in a plate under tension. A numerical experiment demonstrates that the optimal distribution of material density could delay the position of bifurcation.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.27 no.1
• /
• pp.9-16
• /
• 2014

Using a level set method and topological derivatives, a topological shape optimization method that is independent of an initial design is developed for linearly elastic structures. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in the level set function, which facilitates to handle complicated topological shape changes. The "Hamilton-Jacobi(H-J)" equation and computationally robust numerical technique of "up-wind scheme" lead the initial implicit boundary to an optimal one according to the normal velocity field while minimizing the objective function of compliance and satisfying the constraint of allowable volume. Based on the asymptotic regularization concept, the topological derivative is considered as the limit of shape derivative as the radius of hole approaches to zero. The required velocity field to update the H-J equation is determined from the descent direction of Lagrangian derived from optimality conditions. It turns out that the initial holes are not required to get the optimal result since the developed method can create holes whenever and wherever necessary using indicators obtained from the topological derivatives. It is demonstrated that the proper choice of control parameters for nucleation is crucial for efficient optimization process.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.27 no.5
• /
• pp.337-343
• /
• 2014

This paper presents a continuum-based shape design sensitivity analysis(DSA) method for crack propagation problems using a reproducing kernel method(RKM), which facilitates the remeshing problem required for finite element analysis(FEA) and provides the higher order shape functions by increasing the continuity of the kernel functions. A linear elasticity is considered to obtain the required stress field around the crack tip for the evaluation of J-integral. The sensitivity of displacement field and stress intensity factor(SIF) with respect to shape design variables are derived using a material derivative approach. For efficient computation of design sensitivity, an adjoint variable method is employed tather than the direct differentiation method. Through numerical examples, The mesh-free and the DSA methods show excellent agreement with finite difference results. The DSA results are further extended to a shape optimization of crack propagation problems to control the propagation path.

• Proceedings of the Computational Structural Engineering Institute Conference
• /
• 2009.04a
• /
• pp.408-411
• /
• 2009

본 논문에서는 아이소-지오메트릭 해석 방법을 사용하여 응력 제한 조건이 있는 형상 최적설계 문제를 다룬다. 아이소-지오메트릭 해석 방법은 해석에 사용되는 기저 함수와 기하 모델을 구성하는 함수가 일치하여 기하학적으로 정확하기 때문에 설계민감도 해석 및 형상 최적설계에 있어서 강점이 있다. 많은 최적화 문제에서 최대 강성을 확보하는 방향으로 최적화가 진행되고 있는데 이때 응력 조건을 고려하지 않는 경우가 대부분이다. 응력 제한조건이 있는 구조물에서 아이소-지오메트릭 형상 최적설계를 적용시켜 봄으로써 그 효용성을 확인하였다.

• Proceedings of the Computational Structural Engineering Institute Conference
• /
• 2009.04a
• /
• pp.412-415
• /
• 2009

본 논문에서는 등기하 해석법을 이용하여 설계 의존형 하중조건을 갖는 구조물에 대한 형상 최적설계 를 수행하였다. 유한요소 기반 형상 최적설계는 설계영역 매개화에 어려움이 있으나 등기하 해석법은 NURBS 기저 함수와 조정점을 이용함으로써 기하학적 표현이 용이하다는 장점을 가지고 있다. 기하학적으로 정확한 모델은 응답 및 설계민감도 해석에 사용되며, 설계구배 기반의 최적화에 있어서 중요한 역할을 한다. 하중조건이 설계영역의 변화에 따라 변하는 최적설계 문제에서 경계에서 설계민감도가 부정확한 경우, 설계공간에서 최적설계가 균일한 수렴성을 갖기 어렵다. 즉 유한요소법을 이용한 형상 최적설계에서 설계 의존형 하중조건을 갖는 문제를 푸는 경우, 최적설계를 진행할 때 변하는 경계의 부정확성 때문에 정확한 설계민감도를 얻기가 어려운 점이 있다. 본 논문에서는, 엄밀한 기하형상을 표현하는 등기하 설계민감도를 활용한 형상 최적설계 기법이 설계 의존형 하중조건을 갖는 문제에서 좋은 결과를 제시함을 확인하였다.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.27 no.2
• /
• pp.63-70
• /
• 2014

Design sensitivity analysis and topology design optimization for a piezoelectric crystal resonator are developed. The piezoelectric crystal resonator is deformed mechanically when subjected to electric charge on the electrodes, or vice versa. The Mindlin plate theory with higher-order interpolations along thickness direction is employed for analyzing the thickness-shear vibrations of the crystal resonator. Thin electrode plates are masked on the top and bottom layers of the crystal plate in order to enforce to vibrate it or detect electric signals. Although the electrode is very thin, its weight and shape could change the performance of the resonators. Thus, the design variables are the bulk material densities corresponding to the mass of masking electrode plates. An optimization problem is formulated to find the optimal topology of electrodes, maximizing the thickness-shear contribution of strain energy at the desired motion and restricting the allowable volume and area of masking plates. The necessary design gradients for the thickness-shear frequency(eigenvalue) and the corresponding mode shape(eigenvector) are computed very efficiently and accurately using the analytical design sensitivity analysis method using the eigenvector expansion concept. Through some demonstrative numerical examples, the design sensitivity analysis method is verified to be very efficient and accurate by comparing with the finite difference method. It is also observed that the optimal electrode design yields an improved mode shape and thickness-shear energy.

• Proceedings of the Computational Structural Engineering Institute Conference
• /
• 2010.04a
• /
• pp.512-515
• /
• 2010

본 논문에서는 등기하 해석법을 이용하여 설계의존형 하중조건을 갖는 구조물에 대한 형상 최적설계를 수행하였다. 유한요소법 기반 형상 최적설계는 CAD 모델과 해석 모델의 차이로 인해, 설계영역 매개변수화에 어려움이 있다. 등기하 해석법은 CAD 모델과 동일한 NURBS 기저 함수와 조정점을 해석에 이용함으로써 설계의 기하학적 변화를 해석모델에 직접적으로 표현할 수 있는 장점을 가진다. 하중조건이 설계 영역이 변화함에 따라 변하는 최적설계 문제의 경우, 정확한 설계 영역 표현은 법선 벡터, 즉 변화하는 하중의 방향과 곡률과 같은 고차항의 정보를 정확하게 표현할 수 있고, 따라서 목적함수를 최소 또는 최대화시키는 최적의 해로 이끌어 낸다. 유한요소법 또는 밀도법을 이용한 형상 최적설계에서 설계의존형 하중조건을 갖는 구조물의 문제를 푸는 경우, 최적설계가 진행됨에 있어 변화하는 경계의 부정확성 때문에 정확한 설계민감도를 얻기가 어려운 점이 있다. 본 논문에서는, 수치 예제를 통해 등기하 해석 기반의 형상 최적설계 방법론이 설계의존형 하중조건을 갖는 구조물 문제에서 수월성을 가짐을 확인하였다.