• East Asian mathematical journal
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• 제15권1호
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• pp.121-133
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• 1999

• 대한수학회논문집
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• 제15권1호
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• pp.143-153
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• 2000

In [8], Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we propose the block diagonal preconditioners. The preconditioned conjugate residual method is robust in that the convergence is uniform as the parameter, v, goes to $\sfrac{1}{2}$. Computational experiments are included.

• Interaction and multiscale mechanics
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• 제3권2호
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• pp.123-143
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• 2010

The essential idea of the element-free Galerkin method (EFG) is that moving least-squares (MLS) approximation are used for the trial and test functions with the variational principle (weak form). By using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae for an improved element-free Galerkin (IEFG) method for solving three-dimensional problems in linear elasticity. There are fewer coefficients in improved moving least-squares (IMLS) approximation than in MLS approximation. Also fewer nodes are selected in the entire domain with the IEFG method than is the case with the conventional EFG method. In this paper, we selected a few example problems to demonstrate the applicability of the method.

• 대한기계학회논문집A
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• 제25권10호
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• pp.1605-1612
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• 2001

Least-squares meshfree method is presented. Conventional meshfree methods based on the Galerkin formulation suffer from inaccurate numerical integration. Least-squares formulation exhibits rather different integration-related characteristics. It is demonstrated through numerical examples that least-squares formulation is much more robust to integration errors than the Galerkin's. Therefore efficient meshfree methods can be devised by combining very simple integration algorithms and least-squares formulation.

• East Asian mathematical journal
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• 제14권2호
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• pp.399-410
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• 1998

In [3], Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated-linear systems. In this work, we present computational experiments of W-cycle multigrid method. Computational experiments show that the convergence is uniform as the parameter, $\nu$, goes to 1/2.

• Structural Engineering and Mechanics
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• 제21권3호
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• pp.315-332
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• 2005

The Element-free Galerkin Method has become a very popular tool for the simulation of mechanical problems with moving boundaries. The internally applied Moving Least Squares interpolation uses in general Gaussian or cubic weighting functions and has compact support. Due to the approximative character of this interpolation the obtained shape functions do not fulfill the interpolation conditions, which causes additional numerical effort for the application of the boundary conditions. In this paper a new weighting function is presented, which was designed for meshless shape functions to fulfill these essential conditions with very high accuracy without any additional effort. Furthermore this interpolation gives much more stable results for varying size of the influence radius and for strongly distorted nodal arrangements than existing weighting function types.

• 대한기계학회논문집A
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• 제26권11호
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• pp.2312-2321
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• 2002

The first-order least-squares meshfree method for linear elasticity is presented. The conventional and the compatibility-imposed least-squares formulations are studied on the convergence behavior of the solution and the robustness to integration error. Since the least-squares formulation is a type of mixed formulation and induces positive-definite system matrix, by using shape functions of same order for both primal and dual variables, higher rate of convergence is obtained for dual variables than Galerkin formulation. Numerical examples also show that the presented formulations do not exhibit any volumetric locking for the incompressible materials.

• 호남수학학술지
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• 제38권1호
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• pp.47-57
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• 2016

In the discrete formulation of the bubble stabilized Legendre Galerkin methods, the system of equations includes the artificial viscosity term as the parameter. We investigate the estimation of this parameter to get the least-squares solution which minimizes the sum of the squares of errors at each node points. Some numerical results are reported.

• East Asian mathematical journal
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• 제31권5호
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• pp.685-693
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• 2015

In this paper, we introduce fully discrete mixed discontinuous Galerkin approximations for parabolic problems. And we analyze the error estimates in $l^{\infty}(L^2)$ norm for the primary variable and the error estimates in the energy norm for the primary variable and the flux variable.

• 한국전산구조공학회논문집
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• 제27권1호
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• pp.17-26
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• 2014

본 논문은 MLS(Moving Least Squares) 차분법을 바탕으로 동적균열전파 해석을 수행하기 위한 알고리즘을 제시한다. MLS 차분법은 절점만으로 이루어진 수치모델을 사용하며, 이동최소제곱법을 이용하여 전개한 Taylor 다항식을 기초로 미분근사식을 유도하기 때문에, 요소망의 제약에서 완벽하게 벗어난 절점해석이 가능하다. 시간항을 포함하는 동적 평형방정식은 Newmark 방법으로 시간적분 하였다. 동적하중을 받는 균열이 전파할 때, 매 시간단계마다 절점모델을 재구성하지 않고 균열선단 주변에서 국부적인 수정을 통해 해석이 가능하다. 동적균열을 묘사하기 위해 가시한계법(visibility criterion)을 적용하였고, 동적 에너지해방률을 산정하여 균열의 진전유무와 그에 상응하는 진전방향을 결정하였다. 모드 I 상태와 혼합모드 상태에서 균열이 진전하는 현상을 모사하였고, 이론해와 Element-Free Galerkin법으로 계산한 결과와의 비교를 통해 개발된 알고리즘의 정확성과 안정성을 검증하였다.