• 제목/요약/키워드: Gauss-type quadrature

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가우스구적법을 이용한 구조물의 강건최적설계 (Robust Structural Optimization Using Gauss-type Quadrature Formula)

  • 이상훈;서기석
    • 대한기계학회논문집A
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    • 제33권8호
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    • pp.745-752
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    • 2009
  • In robust design, the mean and variance of design performance are frequently used to measure the design performance and its robustness under uncertainties. In this paper, we present the Gauss-type quadrature formula as a rigorous method for mean and variance estimation involving arbitrary input distributions and further extend its use to robust design optimization. One dimensional Gauss-type quadrature formula are constructed from the input probability distributions and utilized in the construction of multidimensional quadrature formula such as the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed. The proposed approach is applied to a simple bench mark problems and robust topology optimization of structures considering various types of uncertainty.

ERROR BOUNDS FOR GAUSS-RADAU AND GAUSS-LOBATTO RULES OF ANALYTIC FUNCTIONS

  • Ko, Kwan-Pyo
    • 대한수학회논문집
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    • 제12권3호
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    • pp.797-812
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    • 1997
  • For analytic functions we give an expression for the kernel $K_n$ of the remainder terms for the Gauss-Radau and the Gauss-Lobatto rules with end points of multiplicity r and prove the convergence of the kernel we obtained. The error bound are obtained for the type $$\mid$R_n(f)$\mid$ \leq \frac{1}{\pi}l(\Gamma) max_{z \in \Gamma} $\mid$K_n(z)$\mid$ max_{z \in \Gamma} $\mid$f(z)$\mid$$, where $l(\Gamma)$ denotes the length of contour $\Gamma$.

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Robust concurrent topology optimization of multiscale structure under load position uncertainty

  • Cai, Jinhu;Wang, Chunjie
    • Structural Engineering and Mechanics
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    • 제76권4호
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    • pp.529-540
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    • 2020
  • Concurrent topology optimization of macrostructure and microstructure has attracted significant interest due to its high structural performance. However, most of the existing works are carried out under deterministic conditions, the obtained design may be vulnerable or even cause catastrophic failure when the load position exists uncertainty. Therefore, it is necessary to take load position uncertainty into consideration in structural design. This paper presents a computational method for robust concurrent topology optimization with consideration of load position uncertainty. The weighted sum of the mean and standard deviation of the structural compliance is defined as the objective function with constraints are imposed to both macro- and micro-scale structure volume fractions. The Bivariate Dimension Reduction method and Gauss-type quadrature (BDRGQ) are used to quantify and propagate load uncertainty to calculate the objective function. The effective properties of microstructure are evaluated by the numerical homogenization method. To release the computation burden, the decoupled sensitivity analysis method is proposed for microscale design variables. The bi-directional evolutionary structural optimization (BESO) method is used to obtain the black-and-white designs. Several 2D and 3D examples are presented to validate the effectiveness of the proposed robust concurrent topology optimization method.

2차 요소를 이용한 2차원 상향가중 유한요소모형 (2-D SU/PG Finite Element Model Using Quadratic Elements)

  • 최승용;김병현;김상호;한건연
    • 한국수자원학회논문집
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    • 제42권12호
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    • pp.1053-1067
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    • 2009
  • 본 연구의 목적은 하도의 형상이 불규칙한 자연하천에서 2차원 흐름 특성을 해석하고 예측하기 위해 2차 요소를 이용한 정확하고 효과적인 상향가중 유한요소모형의 개발에 있다. 모형의 개발을 위해 선형 삼각형 요소, 선형 사각형 요소와 혼합요소를 적용하였고 2차 삼각형, 사각형 요소와 혼합요소를 적용하여 모형을 개발하였으며, 지배방정식의 수치적분식으로 Gauss Quadrature 방법을 사용하였다. 개발된 모형의 적용성 검증을 위해 하상융기가 있는 수로, U자형 수로 등에 모의를 실시하여 해석해 및 실측치와 비교 검토하였다. 모의 결과 2차 요소가 선형 요소에 비해 보다 정확한 해를 제공하는 것으로 판단되었으며 2차요소를 적용한 상용모형인 RMA-2 모형과 비교한 결과 본 연구 개발 모형이 보다 정확한 해를 나타내는 것을 확인할 수 있었다. 개발된 모형을 향후 자연하천에 적용할 경우 기존의 모형에 비해 향상된 결과를 얻을 수 있을 것으로 판단된다.

On the receding contact between a two-layer inhomogeneous laminate and a half-plane

  • Liu, Zhixin;Yan, Jie;Mi, Changwen
    • Structural Engineering and Mechanics
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    • 제66권3호
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    • pp.329-341
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    • 2018
  • This paper considers the smooth receding contact problem between a homogeneous half-plane and a composite laminate composed of an inhomogeneously coated elastic layer. The inhomogeneity of the elastic modulus of the coating is approximated by an exponential function along the thickness dimension. The three-component structure is pressed together by either a concentrated force or uniform pressures applied at the top surface of the composite laminate. Both semianalytical and finite element analysis are performed to solve for the extent of contact and the contact pressure. In the semianalytical formulation, Fourier integral transformation of governing equations and boundary conditions leads to a singular integral equation of Cauchy-type, which can be numerically integrated by Gauss-Chebyshev quadrature to a desired degree of accuracy. In the finite element modeling, the functionally graded coating is divided into homogeneous sublayers and the shear modulus of each sublayer is assigned at its lower boundary following the predefined exponential variation. In postprocessing, the stresses of any node belonging to sublayer interfaces are averaged over its surrounding elements. The results obtained from the semianalytical analysis are successfully validated against literature results and those of the finite element modeling. Extensive parametric studies suggest the practicability of optimizing the receding contact peak stress and the extent of contact in multilayered structures by the introduction of functionally graded coatings.

유한요소법을 이용한 축대칭 구조물의 비선형 거동해석 (Analyses of Non-linear Behavior of Axisymmetric Structure by Finite Element Method)

  • 구영덕;민경탁
    • 전산구조공학
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    • 제10권2호
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    • pp.139-148
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    • 1997
  • A finite element method is programmed to analyse the nonlinear behavior of axisymmetric structures. The lst order Mindlin shell theory which takes into account the transversal shear deformation is used to formulate a conical two node element with six degrees of freedom. To evade the shear locking phenomenon which arises in Mindlin type element when the effect of shear deformation tends to zero, the reduced integration of one point Gauss Quadrature at the center of element is employed. This method is the Updated Lagrangian formulation which refers the variables to the state of the most recent iteration. The solution is searched by Newton-Raphson iteration method. The tangent matrix of this method is obtained by a finite difference method by perturbating the degrees of freedom with small values. For the moment this program is limited to the analyses of non-linear elastic problems. For structures which could have elastic stability problem, the calculation is controled by displacement.

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Concrete fragmentation modeling using coupled finite element - meshfree formulations

  • Wu, Youcai;Choi, Hyung-Jin;Crawford, John E.
    • Interaction and multiscale mechanics
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    • 제6권2호
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    • pp.173-195
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    • 2013
  • Meshfree methods are known to have the capability to overcome the strict regularization requirements and numerical instabilities that encumber the finite element method (FEM) in large deformation problems. They are also more naturally suited for problems involving material perforation and fragmentation. To take advantage of the high efficiency of FEM and high accuracy of meshfree methods, a coupled finite element (FE) and reproducing kernel (RK, one of the meshfree approximations) formulation is described in this paper. The coupling of FE and RK approximation is implemented in an evolutionary fashion, where the extent and location of the evolution is dependent on a triggering criteria provided by the material constitutive laws. To enhance computational efficiency, Gauss quadrature is applied to integrate both FE and RK domains so that no state variable transfer is required when mesh conversion is performed. To control the hourglassing that might occur with 1-point integrated hexahedral grids, viscous type hourglass control is implemented. Meanwhile, the FEM version of the K&C concrete (KCC) model was modified to make it applicable in both FE and RK formulations. Results using this code and the KCC model are shown for the modeling of concrete responses under quasi-static, blast and impact loadings. These analyses demonstrate that fragmentation phenomena of the sort commonly observed under blast and impact loadings of concrete structures was able to be realistically captured by the coupled formulation.

전단변형(剪斷變形)과 회전관성(回轉慣性)을 고려(考慮)한 Timoshenko 보의 자유진동(自由振動) 해석(解析) (Free Vibration Analysis of a Degenerated Timoshenko Beam Including the Effect of Shear Deformation and Rotatory Inertia)

  • 변동균;신영식;장종탁
    • 대한토목학회논문집
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    • 제3권4호
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    • pp.109-122
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    • 1983
  • 본(本) 연구(硏究)에서는 전단변형(剪斷變形)(Shear deformation)과 회전관성(回轉慣性)(Rotatory inertia)의 영향(影響)을 고려(考慮)한 4절점(節點) 8자유도(自由度)를 갖는 Timoshenko 보 요소(要素)(TB4)를 3차원(次元) 연속체(連續體)로부터 유도(誘導)하고 있다. TB4보 요소(要素)는 3차(次) 보간함수(補間凾數)(Interpolation function)로 표시(表示)되는 연직(鉛直)처짐(Transverse deflection) W와 평면회전(平面回轉)(Plane rotation) ${\theta}$를 절점(節點)의 자유도(自由度)로 취(取)하고 있다. TB4요소(要素)의 강도(剛度)매트릭스와 질량(質量)매트릭스는 보의 운동방정식(運動方程式)을 Galerkin 가중잔차법(加重殘差法)(Weighted residual method)으로 Discretization하여 3개(個)의 Gauss점(點)을 이용(利用)한 RSI(Reduced shear integration)기법(技法)에 의한 수치적분(數値積分)으로 구해진다. TB4보 요소(要素)의 정확성(正確性)과 수감상태(收歛狀態)를 고찰(考察)하기 위하여 여러 가지 예제(例題)를 해석(解析)한 결과(結果), 보의 L/h 비(比)에 관계없이 보의 정적해석(靜的解析)(Static analysis)이라 자유진동해석(自由振動解析)(Free vibration analysis)에 있어서 TB4보 요소(要素)는 다른 Timoshenko보 요소(要素)들 보다 월등(越等)히 우수(優秀)한 정확도(正確度)와 수감현상(收歛現象)을 보여 주고 있다.

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