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P1 비순응 유한요소를 이용한 Navier-Stokes 유동 문제의 위상최적화
장강원(Gang-Won Jang),장세명(Se-Myong Chang) 대한기계학회 2010 대한기계학회 춘추학술대회 Vol.2010 No.11
An alternative approach for topology optimization of Navier-Stokes flow problems is presented by using P1 nonconforming finite elements. Due to the discrete divergence-free property of the P1 nonconforming element, the continuity equation of flow can be imposed by using the penalty method into the momentum equation. So, instead of using the mixed formulation, the velocity-only formulation can be built and thus numerical expense for optimization can be efficiently reduced. Moreover, while nodes of other quadrilateral nonconforming elements are mostly located at the midpoints of element edges, the present P1 nonconforming elements have vertex-wisely defined degrees of freedom, so its implentation is as simple as the standard bilinear conforming elements. The effectiveness of the proposed formulation is verified by showing examples with various Reynolds numbers.
P1 비순응 요소를 이용한 비압축성 유동 문제의 위상최적화
장강원(Gang-Won Jang),장세명(Se-Myong Chang) 대한기계학회 2012 大韓機械學會論文集A Vol.36 No.10
P1 비순응 요소를 이용하여 정상 비압축성 Navier-Stokes 유동의 위상최적화 문제를 푸는 방법을 제시한다. 본 연구는 Stokes 유동의 위상최적화 문제에 P1 비순응 요소를 적용하여 그 수치적 효용성을 보인 바 있는 이전 연구에 대한 후속 연구이다. 비압축성 물질 해석에서 잠김현상이 발생하지 않으며 선형 형상함수를 가지는 P1 비순응 요소의 장점이 관성항을 가지는 유체 문제의 해석과 설계에도 유효한 지를 파악하고자 한다. 일반적으로 사용되는 혼합정식화법과 비교하여 P1 비순응 요소의 사용은 벌칙 함수를 이용하여 연속 방정식을 따로 사용하지 않고 운동방정식에 부과할 수 있기 때문에 자유도의 개수를 감소시킬 수 있다. 벌칙 파라미터가 해의 정확도에 주는 영향과 적정 범위는 수치적으로 검토하도록 한다. 또한 보통의 사각 비순응 요소들이 요소면의 중앙에 절점을 가지고 고차의 형상함수를 지니는데 비하여, 본 연구에서 제시하는 P1 비순응 요소는 요소의 꼭지점에 절점을 가지고 {1, x, y}의 P1 형상함수로 구성됨으로써 수치적인 구현의 용이함이 일반 선형 사절점 요소와 동일하다. 제안한 방법의 효용성을 다양한 레이놀즈수에 따른 유동최적화 문제들을 살펴봄으로써 검증하도록 한다. An alternative approach for topology optimization of steady incompressible Navier-Stokes flow problems is presented by using P1 nonconforming finite elements. This study is the extended research of the earlier application of P1 nonconforming elements to topology optimization of Stokes problems. The advantages of the P1 nonconforming elements for topology optimization of incompressible materials based on locking-free property and linear shape functions are investigated if they are also valid in fluid equations with the inertia term. Compared with a mixed finite element formulation, the number of degrees of freedom of P1 nonconforming elements is reduced by using the discrete divergence-free property; the continuity equation of incompressible flow can be imposed by using the penalty method into the momentum equation. The effect of penalty parameters on the solution accuracy and proper bounds will be investigated. While nodes of most quadrilateral nonconforming elements are located at the midpoints of element edges and higher order shape functions are used, the present P1 nonconforming elements have P1, {1, x, y}, shape functions and vertex-wisely defined degrees of freedom. So its implentation is as simple as in the standard bilinear conforming elements. The effectiveness of the proposed formulation is verified by showing examples with various Reynolds numbers.
고차 박판보 해석을 이용한 직사각단면 박판보 조인트 구조물 해석
장강원(Gang-Won Jang),최수민(Soomin Choi),김윤영(Yoon Young Kim) 대한기계학회 2011 대한기계학회 춘추학술대회 Vol.2011 No.4
세 개의 직사각 단면 박판보가 하나의 조인트에서 맞물리는 구조물을 고차 박판보 이론에 기반하여 해석을 수행한다. 워핑, 디스토선을 포함하는 일차원 변위들 간의 연성효과를 조인트의 공유 모서리들에서 삼차원 변위의 연속성을 고려함으로써 이끌어 낼 수 있다. 제안한 고차 박판보 이론 기반의 조인트 변위 관계식을 사용할 경우 예측한 조인트 유연성 효과는 조인트 구조물을 삼차원 쉘 요소로 상세 모델링할 경우와 거의 비슷한 결과를 얻을 수 있다. 세 개의 박판보가 등각으로 맞물리는 조인트 구조물이 평면의 정적 굽힘을 받는 예제를 살펴봄으로써 제안한 이론의 정확성을 검증하도록 한다. A three-beam-jo int structure which consists of thin-walled beams with a rectangular cross-section is analyzed based on a higher order thin-walled beam theory. Coupling effect among one-dimensional deformation measures including warping and distortion is found by considering continuity condition on shared edges of joints. Joint flexibility effect which is observed when the joint structure is modeled by using full three-dimensional shell elements can be accurately calculated by applying the proposed joint match condition for a higher order beam theory. Static analysis results for a three-beam-joint structure with equal joint angles show the effectiveness of the proposed approach.
장강원(Gang-Won Jang),김윤영(Yoon Young Kim) 한국자동차공학회 2005 한국자동차공학회 Symposium Vol.- No.-
If thin-walled closed beams are analyzed by the standard Timoshenko beam elements, their structural behavior, especially near boundaries, cannot be accurately predicted because of the incapability of the Timoskenko theory to predict the sectional warping and distortional deformations. If a higher-order thin-walled box beam theory is used, on the other hand, accurate results comparable to those obtained by plate finite elements can be obtained. However, currently available two-node displacement based higher-order beam elements are not efficient in capturing exponential solution behavior near boundaries. Based on this motivation, we consider developing higher-order mixed finite elements. Instead of using the standard mixed formulation, we propose to develop the mixed formulation based on the state-vector form so that only the field variables that can be prescribed on the boundary are interpolated for finite element analysis. By this formulation, less field variables are used than by the standard mixed formulation, and the interpolated field variables have the physical meaning as the boundary work conjugates. To facilitate the discretization, two-node elements are considered. The effects of interpolation orders for the generalized stresses and displacements on the solution behavior are investigated along with a numerical example.
Mixed State-Vector Finite Element Analysis for a Higher-Order Box Beam Theory
장강원(Gang-Won Jang),김윤영(Yoon Young Kim) 대한기계학회 2005 대한기계학회 춘추학술대회 Vol.2005 No.11
If thin-walled closed beams are analyzed by the standard Timoshenko beam elements, their structural behavior, especially near boundaries, cannot be accurately predicted because of the incapability of the Timoskenko theory to predict the sectional warping and distortional deformations. If a higher-order thin-walled box beam theory is used, on the other hand, accurate results comparable to those obtained by plate finite elements can be obtained. However, currently available two-node displacement based higher-order beam elements are not efficient in capturing exponential solution behavior near boundaries. Based on this motivation, we consider developing higher-order mixed finite elements. Instead of using the standard mixed formulation, we propose to develop the mixed formulation based on the state-vector form so that only the field variables that can be prescribed on the boundary are interpolated for finite element analysis. By this formulation, less field variables are used than by the standard mixed formulation, and the interpolated field variables have the physical meaning as the boundary work conjugates. To facilitate the discretization, two-node elements are considered. The effects of interpolation orders for the generalized stresses and displacements on the solution behavior are investigated along with a numerical example.