• Title, Summary, Keyword: point collocation method

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A comparative study of three collocation point methods for odd order stochastic response surface method

  • Li, Dian-Qing;Jiang, Shui-Hua;Cheng, Yong-Gang;Zhou, Chuang-Bing
    • Structural Engineering and Mechanics
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    • v.45 no.5
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    • pp.595-611
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    • 2013
  • This paper aims to compare three collocation point methods associated with the odd order stochastic response surface method (SRSM) in a systematical and quantitative way. The SRSM with the Hermite polynomial chaos is briefly introduced first. Then, three collocation point methods, namely the point method, the root method and the without origin method underlying the odd order SRSMs are highlighted. Three examples are presented to demonstrate the accuracy and efficiency of the three methods. The results indicate that the condition that the Hermite polynomial information matrix evaluated at the collocation points has a full rank should be satisfied to yield reliability results with a sufficient accuracy. The point method and the without origin method are much more efficient than the root method, especially for the reliability problems involving a large number of random variables or requiring complex finite element analysis. The without origin method can also produce sufficiently accurate reliability results in comparison with the point and root methods. Therefore, the origin often used as a collocation point is not absolutely necessary. The odd order SRSMs with the point method and the without origin method are recommended for the reliability analysis due to their computational accuracy and efficiency. The order of SRSM has a significant influence on the results associated with the three collocation point methods. For normal random variables, the SRSM with an order equaling or exceeding the order of a performance function can produce reliability results with a sufficient accuracy. The order of SRSM should significantly exceed the order of the performance function involving strongly non-normal random variables.

A POINT COLLOCATION SCHEME FOR THE STATIONARY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

  • Kim, Yongsik
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1737-1751
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    • 2013
  • An efficient and stable point collocation scheme based on a meshfree method is studied for the stationary incompressible Navier-Stokes equations. We describe the diffuse derivatives associated with the moving least square method. Using these diffuse derivatives, we propose a point collocation method to fit in solving the Navier-Stokes equations which improves the stability of the direct point collocation scheme. The convergence of the numerical solution is investigated from numerical examples. The driven cavity ow and the backward facing step ow are implemented for the reliability of the scheme. Also, the viscous ow on complicated geometry is successfully calculated such as the ow past a circular cylinder in duct.

FCM for the Multi-Scale Problems (고속 최소자승 점별계산법을 이용한 멀티 스케일 문제의 해석)

  • 김도완;김용식
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • pp.599-603
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    • 2002
  • We propose a new meshfree method to be called the fast moving least square reproducing kernel collocation method(FCM). This methodology is composed of the fast moving least square reproducing kernel(FMLSRK) approximation and the point collocation scheme. Using point collocation makes the meshfree method really come true. In this paper, FCM Is shown to be a good method at least to calculate the numerical solutions governed by second order elliptic partial differential equations with geometric singularity or geometric multi-scales. To treat such problems, we use the concept of variable dilation parameter.

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Development of a meshless finite mixture (MFM) method

  • Cheng, J.Q.;Lee, H.P.;Li, Hua
    • Structural Engineering and Mechanics
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    • v.17 no.5
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    • pp.671-690
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    • 2004
  • A meshless method with novel variation of point collocation by finite mixture approximation is developed in this paper, termed the meshless finite mixture (MFM) method. It is based on the finite mixture theorem and consists of two or more existing meshless techniques for exploitation of their respective merits for the numerical solution of partial differential boundary value (PDBV) problems. In this representation, the classical reproducing kernel particle and differential quadrature techniques are mixed in a point collocation framework. The least-square method is used to optimize the value of the weight coefficient to construct the final finite mixture approximation with higher accuracy and numerical stability. In order to validate the developed MFM method, several one- and two-dimensional PDBV problems are studied with different mixed boundary conditions. From the numerical results, it is observed that the optimized MFM weight coefficient can improve significantly the numerical stability and accuracy of the newly developed MFM method for the various PDBV problems.

Electromagnetic Field Analysis Using the Point Collocation Method Based on the FMLSRK Approximation

  • Kim, Hong-Kyu;Chong, Jin-Kyo;Park, Kyong-Yop;Kim, Do-Wan
    • KIEE International Transaction on Electrical Machinery and Energy Conversion Systems
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    • v.4B no.4
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    • pp.180-183
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    • 2004
  • This paper presents a description of the point collocation method and its application to the electromagnetic field computation. The interpolation scheme is based on the fast moving least square reproducing kernel approximation. In the method, the integration cell is not required and the essential boundary conditions can be enforced directly. Numerical simulations on 1-D and 2-D problems are carried out to validate the method. It is found that computational efficiency is higher than the general mesh-free methods.

An Upwind Meshfree Method for the Supersonic Flow

  • Ahn, Mu-Young;Chang, Keun-Shik
    • 한국전산유체공학회:학술대회논문집
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    • pp.74-75
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    • 2006
  • Recently much attention has been drawn to meshfree method since conventional methods such as FDM, FVM and FEM have suffered from difficulty with mesh generation for complex geometry and deformable bodies. In this paper, an upwind point collocation meshfree method developed by the authors is applied to two shock wave diffraction problems. One is the shock diffraction over a 90-degree corner and the other is the single Mach reflection on a ramp. The scheme showed stability and the results showed accuracy.

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Numerical Techniques in Calculation of Hydrodynamic Stability for Vertical Natural Convection Flows (수직(垂直) 자연대류(自然對流)의 수동력학적(水動力學的) 안정성(安定性) 계산에 관한 수치해석(數値解析) 방법(方法))

  • Hwang, Young-Kyu
    • Solar Energy
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    • v.8 no.1
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    • pp.82-94
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    • 1988
  • The hydrodynamic stability equations for natural convection flows adjacent to a vertical isothermal surface in cold or warm water (Boussinesq or non-Boussinesq situation for density relation), constitute a two-point-boundary-value (eigenvalue) problem, which was solved numerically using the simple shooting and the orthogonal collocation method. This is the first instance in which these stability equations have been solved using a computer code COLSYS, that is based on the orthogonal collocation method, designed to solve accurately two-point-boundary-value problem. Use of the orthogonal collocation method significantly reduces the error propagation which occurs in solving the initial value problem and avoids the inaccuracy of superposition of asymptotic solutions using the conventional technique of simple shooting.

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Stress Analysis of Linear Elastic Solid Problems by using Enhanced Meshfree Method based on Fast Derivatives Approximation (고속 도함수 근사화에 의해 개선된 무요소법을 이용한 선형탄성 고체문제의 응력해석)

  • 이상호;김효진;윤영철
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • pp.583-590
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    • 2002
  • Point collocation method based on the fast derivatives approximation of meshfree shape function is applied to solid mechanics in this study. Enhanced meshfree approximation with approximated derivative of shape function is reviewed, and formulation of linear elastic solid mechanics by point collocation method is presented. It implies that governing equation of solid mechanics with strong form is directly formulated without no numerical integration cells or grid. The regularity of weight function is not required due to a use of approximated derivative, so we propose the exponential type weight function that is discontinuous in first derivative. The convergence and stability of the proposed method is verified by passing the generalized patch test. Also, the efficiency and applicability of the proposed method in solid mechanics is verified by solving types of solid problems. Numerical results show that not only a use of proposed weight function leads lower error and higher convergence rate than that of the conventional weight functions, but also the improved collocation method with derivative approximation enables to compute the derivatives of shape function very fast and accurately enough to replace the classical direct derivative calculation.

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Analysis of 2-D Potential Problem with L-shape Domain by p-Convergent Boundary Element Method (p-수렴 경계요소법에 의한 L-형 영역을 갖는 2차원 포텐셜 문제 해석)

  • Woo, Kwang-Sung;Jo, Jun-Hyung
    • Journal of the Computational Structural Engineering Institute of Korea
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    • v.22 no.1
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    • pp.117-124
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    • 2009
  • The p-convergent boundary element method has been proposed to analyze two-dimensional potential problem on the basis of high order Legendre shape functions that have different property comparing with the shape functions in conventional boundary element method. The location of nodes corresponding to high order shape function are not defined along the boundary, called by nodeless node, similar to the p-convergent finite element method. As the order of shape function increases, the collocation point method is used to solve linear simultaneous equations. The collocation patterns of p-convergent boundary element method consist of non-symmetric hierarchial or symmetric non-hierarchical. As the order of shape function increases, the number of collocation point increases. The singular integral that appears in p-convergent boundary element has been calculated by special numeric quadrature technique and semi-analytical integration technique. The L-shape domain problem including singularity in the vicinity of reentrant comer is analyzed and the numerical results show that the relative error is smaller than $10^{-2}%$ range as compared with other results in literatures. In case of same condition, the symmetric p-collocation point pattern shows high accuracy of solution.

A Comparative Study of Transcription Techniques for Nonlinear Optimal Control Problems Using a Pseudo-Spectral Method

  • Kim, Chang-Joo;Sung, Sangkyung
    • International Journal of Aeronautical and Space Sciences
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    • v.16 no.2
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    • pp.264-277
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    • 2015
  • This article investigates various transcription techniques for the Legendre pseudospectral (PS) method to compare the pros and cons of each approach. Eight combinations from four different types of collocation points and two discretization methods for dynamic constraints, which differentiate Legendre PS transcription techniques, are implemented to solve a carefully selected test set of nonlinear optimal control problems (NOCPs). The convergence property and prediction accuracy are compared to provide a useful guideline for selecting the best combination. The tested NOCPs consist of the minimum time, minimum energy, and problems with state and control constraints. Therefore, the results drawn from this comparative study apply to the solution of similar types of NOCPs and can mitigate much debate about the best combinations. Additionally, important findings from this study can be used to improve the numerical efficiency of the Legendre PS methods. Three PS applications to the aerospace engineering problems are demonstrated to prove this point.