• Title/Summary/Keyword: Variational formulation

### Design Sensitivity Analysis and Optimization of Plane Arch Structures Using Variational Formulation (변분공식화를 이용한 2차원 아치 구조물의 설계민감도 해석 및 최적설계)

• 최주호
• Journal of the Computational Structural Engineering Institute of Korea
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• v.14 no.2
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• pp.159-171
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• 2001
• 평면 아치 구조물에 대해 선형 탄성 변분방정식에 기반을 둔 설계민감도 해석을 위한 일반적 이론을 개발하였다. 아치 구조물내의 임의 마디에 정의된 응력범함수를 고려하였고 이에 대한 설계민감도 공식을 유도하기 위해 전미분(material derivative) 개념과 보조(adjoint) 변수 방법을 도입하였다. 얻어진 민감도 공식은 구조해석 결과를 얻고 나면 이들로부터 단순 대수연산을 통해 계산이 되므로 적용이 간편할 뿐 아니라 해의 정확도가 높은 잇점이 있다. 본 방법은 아치의 형상을 매개변수를 통해 표현하므로 얕은 아치에 국한하지 않고 어떠한 형상도 고려가 가능하며, 나아가서 아치의 형상변화를 형상에 대해 수직뿐 아니라 접선방향도 포함하여 일반적으로 고려하므로 다양한 형상설계가 가능하다. 몇 가지 예제에서 민감도 계산을 수행함으로써 본 방법의 정확도와 효율성을 입증하였으며, 두 가지의 설계최적화 문제를 대상으로 실제로 두께 및 형상최적설계를 수행하였다.

### IMAGE SEGMENTATION BASED ON THE STATISTICAL VARIATIONAL FORMULATION USING THE LOCAL REGION INFORMATION

• Park, Sung Ha;Lee, Chang-Ock;Hahn, Jooyoung
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.2
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• pp.129-142
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• 2014
• We propose a variational segmentation model based on statistical information of intensities in an image. The model consists of both a local region-based energy and a global region-based energy in order to handle misclassification which happens in a typical statistical variational model with an assumption that an image is a mixture of two Gaussian distributions. We find local ambiguous regions where misclassification might happen due to a small difference between two Gaussian distributions. Based on statistical information restricted to the local ambiguous regions, we design a local region-based energy in order to reduce the misclassification. We suggest an algorithm to avoid the difficulty of the Euler-Lagrange equations of the proposed variational model.

### The Development of Incompatible Finite Elements for Plane Stress/Strain Using Multivariable Variational formulation (다변수 변분해법에 의한 비적합 4절점 사각형 평면응력 및 평면변형률 요소의 개발)

• 주상백;신효철
• Transactions of the Korean Society of Mechanical Engineers
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• v.18 no.11
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• pp.2871-2882
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• 1994
• Two kinds of 4-node plane stress/strain finite elements are presented in this work. They are derived from the modified Hellinger-Reissner variational principle so as to employ the internal incompatible displacement and independent stress fields, or the incompatible displacement and strain fields. The introduced incompatible functions are selected to satisfy the constant strain condition. The elements are evaluated on several problems of bending and material incompressibility with regular and distorted elements. The results show that the new elements perform excellently in the calculation of deformation and stresses.

### Transient Linear Viscoelastic Stress Analysis Based on the Equations of Motion in Time Integral (시간적분형 운동방정식에 근거한 동점탄성 문제의 응력해석)

• Lee, Sung-Hee;Sim, Woo-Jin
• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.9
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• pp.1579-1588
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• 2003
• In this paper, the finite element equations for the transient linear viscoelastic stress analysis are presented in time domain, whose variational formulation is derived by using the Galerkin's method based on the equations of motion in time integral. Since the inertia terms are not included in the variational formulation, the time integration schemes such as the Newmark's method widely used in the classical dynamic analysis based on the equations of motion in time differential are not required in the development of that formulation, resulting in a computationally simple and stable numerical algorithm. The viscoelastic material is assumed to behave as a standard linear solid in shear and an elastic solid in dilatation. To show the validity of the presented method, two numerical examples are solved nuder plane strain and plane stress conditions and good results are obtained.

### Study on the Generalization of the Extended Framework of Hamilton's Principle in Transient Continua Problems (확장 해밀턴 이론의 일반화에 대한 고찰)

• Kim, Jinkyu;Shin, Jinwon
• Journal of the Computational Structural Engineering Institute of Korea
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• v.29 no.5
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• pp.421-428
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• 2016
• The present work extends the recent variational formulation to more general time-dependent problems. Thus, based upon recent works of variational formulation in dynamics and pure heat diffusion in the context of the extended framework of Hamilton's principle, formulation for fully coupled thermoelasticity is developed first, then, with thermoelasticity-poroelasticity analogy, poroelasticity formulation is provided. For each case, energy conservation and energy dissipation properties are discussed in Fourier transform domain.

### Variational Formulation of Hybrid-Trefftz Plate Elements and Evaluation of Their Static Performance (하이브리드 트레프츠 평판 요소의 변분 수식화와 성능 평가)

• Choo, Yeon-Seok;Lee, Byung-Chai
• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.2
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• pp.302-309
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• 2003
• Hybrid-Trefftz plate bending elements are known to be robust and free of shear locking in the thin limit because of Internal displacements fields and linked boundary displacements. Also, their finite element approximation is very simple regardless to boundary shape since all element matrices can be calculated using only boundary integrals. In this study, new hybrid-Trefftz variational formulation based on the total potential energy principle of internal displacements and displacement consistency conditions at the boundary is derived. And flat shell elements are derived by combining hybrid-Trefftz bending stiffness and plane stress stiffness with drilling dofs.

### Variational Formulation for Shape Optimization of Spatial Beam Structures (정식화를 이용한 3차원 구조물의 형상 최적설계)

• 최주호;김종수
• Proceedings of the Computational Structural Engineering Institute Conference
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• pp.123-130
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• 2002
• A general formulation for shape design sensitivity analysis over three dimensional beam structure is developed based on a variational formulation of the beam in linear elasticity. Sensitivity formula is derived based on variational equations in cartesian coordinates using the material derivative concept and adjoint variable method for the displacement and Von-Mises stress functionals. Shape variation is considered for the beam shape in general 3-dimensional direction as well as for the orientation angle of the beam cross section. In the sensitivity expression, the end points evaluation at each beam segment is added to the integral formula, which are summed over the entire structure. The sensitivity formula can be evaluated with generality and ease even by employing piecewise linear design velocity field despite the bending model is fourth order differential equation. For the numerical implementation, commercial software ANSYS is used as analysis tool for the primal and adjoint analysis. Once the design variable set is defined using ANSYS language, shape and orientation variation vector at each node is generated by making finite difference to the shape with respect to each design parameter, and is used for the computation of sensitivity formula. Several numerical examples are taken to show the advantage of the method, in which the accuracy of the sensitivity is evaluated. The results are found excellent even by employing a simple linear function for the design velocity evaluation. Shape optimization is carried out for the geometric design of an archgrid and tilted bridge, which is to minimize maximum stress over the structure while maintaining constant weight. In conclusion, the proposed formulation is a useful and easy tool in finding optimum shape in a variety of the spatial frame structures.

### THE SOLUTIONS OF A PERTURBED ELLIPTIC EQUATION WITH EXPONENTIAL GROWTH

• Jung, Tacksun;Choi, Q-Heung
• Korean Journal of Mathematics
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• v.17 no.1
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• pp.103-112
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• 2009
• We show the existence of the solutions for the following nonlinear elliptic problem under the Dirichlet boundary condition. To show the existence of the solutions we use the variational formulation.

### A method to evaluate the frequencies of free transversal vibrations in self-anchored cable-stayed bridges

• Monaco, Pietro;Fiore, Alessandra
• Computers and Concrete
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• v.2 no.2
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• pp.125-146
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• 2005
• The objective of this paper is setting out, for a cable-stayed bridge with a curtain suspension, a method to determine the modes of vibration of the structure. The system of differential equations governing the vibrations of the bridge, derived by means of a variational formulation in a nonlinear field, is reported in Appendix C. The whole analysis results from the application of Hamilton's principle, using the expressions of potential and kinetic energies and of the virtual work made by viscous damping forces of the various parts of the bridge (Monaco and Fiore 2003). This paper focuses on the equation concerning the transversal motion of the girder of the cable-stayed bridge and in particular on its final form obtained, restrictedly to the linear case, neglecting some quantities affecting the solution in a non-remarkable way. In the hypotheses of normal mode of vibration and of steady-state, we propose the resolution of this equation by a particular method based on a numerical approach. Respecting the boundary conditions, we derive, for each mode of vibration, the corresponding frequency, both natural and damped, the shape-function of the girder axis and the exponential function governing the variability of motion amplitude in time. Finally the results so obtained are compared with those deriving from the dynamic analysis performed by a finite elements calculation program.

### WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY

• Hwang, Jin-Soo;Nakagiri, Shin-Ichi
• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.443-453
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• 2007
• We study the equation of a membrane with strong viscosity. Based on the variational formulation corresponding to the suitable function space setting, we have proved the fundamental results on existence, uniqueness and continuous dependence on data of weak solutions.