• Title/Summary/Keyword: Gauss Quadrature

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FAMILIES OF NONLINEAR TRANSFORMATIONS FOR ACCURATE EVALUATION OF WEAKLY SINGULAR INTEGRALS

  • BEONG IN YUN
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.27 no.3
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    • pp.194-206
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    • 2023
  • We present families of nonlinear transformations useful for numerical evaluation of weakly singular integrals. First, for end-point singular integrals, we define a prototype function with some appropriate features and then suggest a family of transformations. In addition, for interior-point singular integrals, we develop a family of nonlinear transformations based on the aforementioned prototype function. We take some examples to explore the efficiency of the proposed nonlinear transformations in using the Gauss-Legendre quadrature rule. From the numerical results, we can find the superiority of the proposed transformations compared to some existing transformations, especially for the integrals with high singularity strength.

Free Vibration Analysis of Compressive Tapered Members Resting on Elastic Foundation Using Differential Quadrature Method (미분구적법(DQM)을 이용한 탄성지반 위에 놓인 변단면 압축부재의 자유진동 해석)

  • 이병구;최규문;이태은;김무영
    • Journal of the Computational Structural Engineering Institute of Korea
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    • v.15 no.4
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    • pp.629-638
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    • 2002
  • This paper deals with the free vibration analysis of compressive tapered members resting on elastic foundation using the Differential Quadrature Method. Based on the differential equation subjected to the boundary conditions, adopted from the open literature, which governs the free vibrations of such member, this equation is applied to the Differential Quadrature Method. For computing natural frequencies, the numerical procedures are developed by QR Algorithm, in which the Chebyshev-Gauss-Lobatto method is used for choosing the grid points. The numerical methods developed herein for computing natural frequencies are programmed in FORTRAN code, and all solutions obtained in this study are quite agreed with those in the open literature.

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

  • Choi, Seung-Yong;Kim, Byung-Hyun;Kim, Sang-Ho;Han, Kun-Yeun
    • Journal of Korea Water Resources Association
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    • v.42 no.12
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    • pp.1053-1067
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    • 2009
  • The objective of this study is to develop an efficient and accurate quadratic finite element model based on Streamline Upwind/Petrov Galerkin (SU/PG) scheme for analyzing and predicting two dimensional flow features in complex natural rivers. For a development of model, quadratic tin, quadrilateral and mixed elements as well as linear tin, quadrilateral and mixed elements were used in the model. Also, this model was developed through reinforcement of Gauss Quadrature which was necessary to integral of governing equation. Several tests for bottom-rising channel and U-type channel were performed for the purpose of validation and verification of the developed model. Such results showed that solutions of second order elements are better accurate and improved than those of linear elements. Results obtained by the developed model and RMA-2 model are compared, and the results for the developed model were better accurate than those of RMA-2 model. In the future if the developed model is applied in natural rivers, it can provide better accurate results than those of existing model.

Free Vibration Analysis of Beam-columns Resting on Pasternak Foundation by Differential Quadrature Method (미분구적법에 의한 Pasternak지반 위에 놓인 보-기둥의 자유진동 해석)

  • 이태은;이병구;강희종
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • 2004.05a
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    • pp.957-962
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    • 2004
  • This paper deals with the free vibration analysis of beam-columns resting on Pasternak foundation by the Differential Quadrature Method. Based on the differential equation subjected to the boundary conditions, adopted from the open literature, which governs the free vibrations of such member, this equation is applied to the Differential Quadrature Method. For computing natural frequencies, the numerical procedures are developed by QR Algorithm, in which the Chebyshev-Gauss-Lobatto method is used for choosing the grid points. The numerical methods developed herein for computing natural frequencies are programmed in FORTRAN code, and all solutions obtained in this study are quite agreed with those in the open literature.

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The Free Vibration Analyses by Using Two Dimensional 6-Node Element and Three Dimensional 16-Node element with Modification of Gauss Sampling Point (가우스 적분점을 수정한 2차원 6-절점 요소 및 3차원 16-절점 요소에 의한 자유진동해석)

  • 김정운;경진호;권영두
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.18 no.11
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    • pp.2922-2931
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    • 1994
  • We propose a modified 6-node element, where the sampling point of Gauss quadrature moved in the thickness direction. The modified 6-node element has been applied to static problems and forced motion analyses. In this study, this method is extended to the finite element analysis of the natural frequencies of two dimensional problems. We also propose a modified 16-node element for three dimensional problems, which behaves much like a 20-node element with smaller degree of freedom. The modified 6-node and 16-node elements have been applied to the modal analyses of beams and plates, respectively. The results agree well with the results of the 8-node or 20-node element models.

Maximum likelihood estimation of Logistic random effects model (로지스틱 임의선형 혼합모형의 최대우도 추정법)

  • Kim, Minah;Kyung, Minjung
    • The Korean Journal of Applied Statistics
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    • v.30 no.6
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    • pp.957-981
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    • 2017
  • A generalized linear mixed model is an extension of a generalized linear model that allows random effect as well as provides flexibility in developing a suitable model when observations are correlated or when there are other underlying phenomena that contribute to resulting variability. We describe maximum likelihood estimation methods for logistic regression models that include random effects - the Laplace approximation, Gauss-Hermite quadrature, adaptive Gauss-Hermite quadrature, and pseudo-likelihood. Applications are provided with social science problems by analyzing the effect of mental health and life satisfaction on volunteer activities from Korean welfare panel data; in addition, we observe that the inclusion of random effects in the model leads to improved analyses with more reasonable inferences.

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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    • v.66 no.3
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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.

Robust concurrent topology optimization of multiscale structure under load position uncertainty

  • Cai, Jinhu;Wang, Chunjie
    • Structural Engineering and Mechanics
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    • v.76 no.4
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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.

Free vibration analysis of functionally graded beams with variable cross-section by the differential quadrature method based on the nonlocal theory

  • Elmeiche, Noureddine;Abbad, Hichem;Mechab, Ismail;Bernard, Fabrice
    • Structural Engineering and Mechanics
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    • v.75 no.6
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    • pp.737-746
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    • 2020
  • This paper attempts to investigate the free vibration of functionally graded material beams with nonuniform width based on the nonlocal elasticity theory. The theoretical formulations are established following the Euler-Bernoulli beam theory, and the governing equations of motion of the system are derived from the minimum total potential energy principle using the nonlocal elasticity theory. In addition, the Differential Quadrature Method (DQM) is applied, along with the Chebyshev-Gauss-Lobatto polynomials, in order to determine the weighting coefficient matrices. Furthermore, the effects of the nonlocal parameter, cross-section area of the functionally graded material (FGM) beam and various boundary conditions on the natural frequencies are examined. It is observed that the nonlocal parameter and boundary conditions significantly influence the natural frequencies of the functionally graded material beam cross-section. The results obtained, using the Differential Quadrature Method (DQM) under various boundary conditions, are found in good agreement with analytical and numerical results available in the literature.

Ultimate Load of RC Structures Bonded with the Soffit Plate by p-Version Nonlinear Analysis (p-Version 비선형 해석에 의한 팻취보강된 RC구조물의 극한강도 산정)

  • 안재석;박진환;홍종현;우광성
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • 2004.04a
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    • pp.365-372
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    • 2004
  • A new finite element model will be presented to analyze the nonlinear behavior of not only RC beams and slabs, but also RC beams strengthened by a patch repair. The numerical approach is based on the p-version degenerate shell element including theory of anisotropic laminated composites, theory of materially and geometrically nonlinear plates. In the nonlinear formulation of this model, the total Lagrangian formulation is adopted with large deflections and moderate rotations being accounted for in the sense of von Karman hypothesis. The material model is based on hardening rule, crushing condition, plate-end debonding strength model and so on. The Gauss-Lobatto numerical quadrature is applied to calculate the stresses at the nodal points instead of Gauss points. The validity of the proposed p-version finite element model is demonstrated through several numerical examples for the load-deflection curves, the ultimate loads, and the failure modes of reinforced connote slabs and RC beams bonded with steel plates or FRP plates compared with available experimental and numerical results.

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