점성균열 모델을 위한 국부단위분할이 적용된 무요소법

지광습(Zi Goangseup),정진규(Jung Jin-kyu),김병민(Kim Byeong Min) 대한토목학회 2006 대한토목학회논문집 A Vol.26 No.5A

본 연구에서는 이차원 연속체에 존재하는 점성균열을 무요소법에서 국부 단위분할 원리에 근거하여 정식화하였다. 균열이 한 절점의 영향영역(domain of influence)을 완전히 통과하는 경우 그 절점의 형상함수는 계단함수로 확장되고, 균열 끝이 영향영역 내에 위치하는 경우 특이성이 제거된 가지함수(branch function).로 확장된다. 이러한 해의 영역의 확장은 국부 단위분할 원리를 만족하는 변위계에서만 이루어지므로, 약형 정식화는 표준 Galerkin방법에 의해서 얻어진다. 균열과 상호작용하는 영향영역만 확장되기 때문에, 성긴 형태의 시스템의 행렬을 유지하게 된다. 그러므로 확장에 의해 발생하는 계산비용의 증가는 최소화된다. 동적인 문제에서 균열성장에 관한 조건은 재료안정론으로부터 얻어졌다. 즉, 재료 한 점에서 어느 방향으로든 변형열화가 집중하게 되면, 그 방향에 점성균열을 삽입하여 연속체가 비연속체로 되도록 하였다. 균열의 성장속도도 같은 조건으로부터 자연스럽게 얻어졌다. 전통적인 무요소법보다 더 나은 정확도와 빠른 수렴성을 보이는 것이 확인되었으며, 이 기법의 적용성을 보이기 위해 잘 알려진, 정적 및 동적문제에 적용하였다. The element free Galerkin method is extended by the local partition of unity method to model the cohesive cracks in two dimensional continuum. The shape function of a particle whose domain of influence is completely cut by a crack is enriched by the step enrichment function. If the domain of influence contains a crack tip inside, it is enriched by a branch enrichment function which does not have the LEFM stress singularity. The discrete equations are obtained directly from the standard Galerkin method since the enrichment is only for the displacement field, which satisfies the local partition of unity. Because only particles whose domains of influence are influenced by a crack are enriched, the system matrix is still sparse so that the increase of the computational cost is minimized. The condition for crack growth in dynamic problems is obtained from the material instability; when the acoustic tensor loses the positive definiteness, a cohesive crack is inserted to the point so as to change the continuum to a discontiuum. The crack speed is naturally obtained from the criterion. It is found that this method is more accurate and converges faster than the classical mesh less methods which are based on the visibility concept. In this paper, several wellknown static and dynamic problems were solved to verify the method.

파티션 함수의 크기가 무요소해법에 미치는 효과에 대한 고찰

홍원택 ( Won Tak Hong ) 아시아.유럽미래학회 2015 유라시아연구 Vol.12 No.4

기존의 유한 요소 연구에서는 근사하고자하는 함수의 변화가 크거나 불연속에 가까운 경우 이러한 변화를 해석하기 위하여 필요한 요소의 숫자를 단계적으로 증가시켜 가며 근사의 적정성을 살펴보게 된다.본 연구에서는 기존방법과는 차별된 무요소 해석에 대한 가능성을 살펴보기 위하여 간단한 포아송 모델문제를 활용하여 최적 파티션 함수의 크기에 대해 알아보았다. 본 연구는 무요소해법을 이용하여 함수를 근사 할 때 파티션 함수의 최적 크기가 존재함을 시사한다. 파티션 함수의 최적크기는 무요소법을 이용하여 유로피안이나 아메리칸 옵션의 그릭을 살펴볼 때 유용하다. 무요소 해석을 위한 파티션 함수의 크기에 대한 최근의 연구 결과들을 살펴보면 파티션 함수가 완전히 겹치지 않고 일부 포개질 경우 더 정확히 문제를 풀 수 있다고 알려져 있다. 하지만 파티션함수의 크기가 극단적으로 커지는 경우에는 연구결과를 찾을 수 없었다. 새로운 모델링 기법인 무요소법의 경우 파티션 함수의 크기를 키울수록 방정식의해의 변화를 더 효율적으로 기술할 수 있으리라 기대하였으나 특별한 경우에는 기존연구 결과와 일부상반된 결과가 얻어짐을 확인 하였다. 즉, 기존 연구에서는 파티션 함수의 겹치는 영역이 작아져 파티션함수의 크기가 커지면 커질수록 좋은 결과를 얻었으나 파티션 함수의 겹치는 영역이 극단적으로 작아지면 오히려 그 반대의 현상이 나타날 수도 있음을 확인하였다. There is a tight link between the size of flat-top region and the gradient of the partition of unity function. Recent studies show that the condition number of the system can grow arbitrarily large with linear finite element shape functions as a partition of unity even though local approximation functions are linearly independent. As a result, a flat-top partition of unity function is introduced to avoid linear independence. The wide flat-top region in the partition of unity function has been known to ensure the linear independence of mesh-free basis functions. The wider the flat-top is, the smaller so better in the condition number. However, the wider flat-top partition of unity function could result in a large gradient of the partition of unity function as well as large condition number while having a strongly linear independent local basis functions. We demonstrate this phenomenon in a simple Poisson model problem when the flat-top region expands and starts to fill up the support of the partition of unity functions. For the particular problems we have considered, we find that there is an optimal size for the flat-top region in the support of partition of unity function.

New higher-order triangular shell finite elements based on the partition of unity

전형민 국제구조공학회 2020 Structural Engineering and Mechanics, An Int'l Jou Vol.73 No.1

Finite elements based on the partition of unity (PU) approximation have powerful capabilities for p-adaptivity and solutions with high smoothness without remeshing of the domain. Recently, the PU approximation was successfully applied to the three-node shell finite element, properly eliminating transverse shear locking and showing excellent convergence properties and solution accuracy. However, the enrichment with the PU approximation results in a significant increase in the number of degrees of freedom; therefore, it requires greater computational cost, thus making it less suitable for practical engineering. To circumvent this disadvantage, we propose a new strategy to decrease the total number of degrees of freedom in the existing PU-based shell element, without loss of optimal convergence and accuracy. To alleviate the locking phenomenon, we use the method of mixed interpolation of tensorial components and perform convergence studies to show the accuracy and capability of the proposed shell element. The excellent performances of the new shell elements are illustrated in three benchmark problems.

PATCHWISE REPRODUCING POLYNOMIAL PARTICLE METHOD FOR THICK PLATES

HYUNJU KIM,BONGSOO JANG 한국산업응용수학회 2013 Journal of the Korean Society for Industrial and A Vol.17 No.2

Reproducing Polynomial Particle Method (RPPM) is one of meshless methods that use meshes minimally or do not use meshes at all. In this paper, the RPPM is employed for free vibration analysis of shear-deformable plates of the first order shear deformation model (FSDT), called Reissner-Mindlin plate. For numerical implementation, we use flat-top partition of unity functions, introduced by Oh et al, and patchwise RPPM in which approximation functions have high order polynomial reproducing property and the Kronecker delta property. Also, we demonstrate that our method is highly effective than other existing results for various aspect ratios and boundary conditions.

hp-adaptive finite element method for linear elasticity using higher-order virtual node method

오현철,이병채 대한기계학회 2015 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.29 No.10

Higher-order polygonal finite elements are developed for adaptive analyses of linear elastic problem. These elements are constructedusing virtual node method based on partition of unity coupled with polynomial enrichment functions. Because the element shape functionsare polynomials, the stiffness matrix is computed precisely with standard Gauss quadrature rules. Several numerical examples oflinear elasticity are presented to validate the accuracy and convergence of the proposed elements. One of the advantages of the proposedelements is that they can be used as transition elements with hanging nodes on higher-order approximation meshes. Building on this advantage,h- and hp-adaptive finite element analyses of numerical examples with local singularities are performed on triangular quadtreemeshes in order to demonstrate the performance of the adaptive strategies using the proposed elements.

Investigation of allowable time-step sizes for generalized finite element analysis of the transient heat equation

O'Hara, P.,Duarte, C.A.,Eason, T. Techno-Press 2010 Interaction and multiscale mechanics Vol.3 No.3

This paper investigates the heat equation for domains subjected to an internal source with a sharp spatial gradient. The solution is first approximated using linear finite elements, and sufficiently small time-step sizes to yield stable simulations. The main area of interest is then in the ability to approximate the solution using Generalized Finite Elements, and again explore the time-step limitations required for stable simulations. Both high order elements, as well as elements with special enrichments are used to generate solutions. When compared to linear finite elements, the high order elements deliver better accuracy at a given level of mesh refinement, but do not offer an increase in critical time-step size. When special enrichment functions are used, the solution can be approximated accurately on very coarse meshes, while yielding solutions which are both accurate and computationally efficient. The major conclusion of interest is that the significantly larger element size yields larger allowable time-step sizes while still maintaining stability of the time-stepping algorithm.

Benchmark tests of MITC triangular shell elements

전형민,Paul Mukai,김산 국제구조공학회 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.68 No.1

In this paper, we compare and assess the performance of the standard 3- and 6-node MITC shell elements (Lee and Bathe 2004) with the recently developed MITC triangular elements (Lee et al. 2014, Jeon et al. 2014, Jun et al. 2018) which were based on the partitions of unity approximation, bubble node, or both. The convergence behavior of the shell elements are measured in well-known benchmark tests; four plane stress tests (mesh distortion test, cantilever beam, Cook’s skew beam, and MacNeal beam), two plate tests (Morley’s skew plate and circular plate), and six shell tests (curved beam, twisted beam, pinched cylinder, hemispherical shells with or without hole, and Scordelis-Lo roof). To precisely compare and evaluate the solution accuracy of the shell elements, different triangular mesh patterns and distorted element mesh are adopted in the benchmark problems. All shell finite elements considered pass the basic tests; namely, the isotropy, the patch, and the zero energy mode tests.

Benchmark tests of MITC triangular shell elements

Jun, Hyungmin,Mukai, Paul,Kim, San Techno-Press 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.68 No.1

In this paper, we compare and assess the performance of the standard 3- and 6-node MITC shell elements (Lee and Bathe 2004) with the recently developed MITC triangular elements (Lee et al. 2014, Jeon et al. 2014, Jun et al. 2018) which were based on the partitions of unity approximation, bubble node, or both. The convergence behavior of the shell elements are measured in well-known benchmark tests; four plane stress tests (mesh distortion test, cantilever beam, Cook's skew beam, and MacNeal beam), two plate tests (Morley's skew plate and circular plate), and six shell tests (curved beam, twisted beam, pinched cylinder, hemispherical shells with or without hole, and Scordelis-Lo roof). To precisely compare and evaluate the solution accuracy of the shell elements, different triangular mesh patterns and distorted element mesh are adopted in the benchmark problems. All shell finite elements considered pass the basic tests; namely, the isotropy, the patch, and the zero energy mode tests.