Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes I. Volume integration

You, Hojun,Kim, Chongam Elsevier 2019 Journal of computational physics Vol.395 No.-

<P><B>Abstract</B></P> <P>This work deals with the development of the direct reconstruction method (DRM) and its application to the volume integration of the discontinuous Galerkin (DG) method on multi-dimensional high-order mixed-curved meshes. The conventional quadrature-based DG methods require the humongous computational cost on high-order curved elements due to their non-linear shape functions. To overcome this issue, the flux function is directly reconstructed in the physical domain using nodal polynomials on a target space in a quadrature-free manner. Regarding the target space and distribution of the nodal points, DRM has two variations: the brute force points (BFP) and shape function points (SFP) methods. In both methods, one nodal point corresponds to one nodal basis function of the target space. The DRM-BFP method uses a set of points that empirically minimizes a condition number of the generalized Vandermonde matrix. In the DRM-SFP method, the conventional nodal points are used to span an enlarged target space of the flux function. It requires a larger number of reconstruction points than DRM-BFP but offers easy extendability to the higher-degree polynomial space and a better de-aliasing effect. A robust way to compute orthonormal polynomials is provided to achieve lower round-off errors. The proposed methods are validated by the 2-D/3-D Navier-Stokes equations on high-order mixed-curved meshes. The numerical results confirm that the DRM volume integration greatly reduces the computational cost and memory overhead of the conventional quadrature-based DG methods on high-order curved meshes while maintaining an optimal order-of-accuracy as well as resolving the flow physics accurately.</P> <P><B>Highlights</B></P> <P> <UL> <LI> As a quadrature-free method, the DRM framework is proposed and applied to the volume integration of DG methods for the Navier-Stokes equations. </LI> <LI> According to the target space and interpolation nodes, DRM-BFP and DRM-SFP are developed on multi-dimensional higher-order mixed-curved mesh. </LI> <LI> Through numerical analyses and computations, efficiency and accuracy of DRM are extensively validated on higher-order mixed curved mesh. </LI> </UL> </P>

Improved three-variable element-free Galerkin method for vibration analysis of beam-column models

Chen Wu,Hong Xiang,Xipeng Du 대한기계학회 2016 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.30 No.9

A frame structure is simulated using a beam-column model, and an improved three-variable Element-free Galerkin (EFG) method is presented to compute the natural frequencies and dynamic responses. The moving least squares method in combination with the generalized moving least squares method is used to construct a shape function, after which three variables including the horizontal displacement, vertical displacement and rotational angle of every node are approximated. The Galerkin weak form was employed to derive the discretized system equations, and the Newmark time integration method was used for the time history analyses. In the modelling process, the penalty method was used to impose the essential boundary conditions to obtain the corresponding formulae of the improved threevariable EFG for elastodynamic analyses. An algorithm procedure for the improved three-variable EFG was designed. Applicability of the improved method was demonstrated by solving numerical examples for different values of number of nodes, scaling factor and order of basis function. The numerical examples illustrated that the improved method is computationally efficient and easier to pre-process than the finite element method.

전자파 수치 해석을 위해 갤러킨 기법과 보간법을 혼용하여 개선시킨 모멘트법

황지환(Ji-Hwan Hwang),권순구(Soon-Gu Kwon),오이석(Yisok Oh) 한국전자파학회 2012 한국전자파학회논문지 Vol.23 No.4

본 논문에서는 3차원 공간의 전자파 수치 해석을 위한 모멘트법(method of moments)의 개선된 해석 기법을 선보인다. 전자파 산란 특성을 해석하기 위해 기본적으로 EFIE(Electric Field Integral Equation)와 RWG(Rao-Wilton- Glisson) 기저 함수를 이용하였으며, 계산 효율을 높이기 위해 기존의 갤러킨(Galerkin) 기법과 중심점 보간(interpolation)법을 혼용하여 해석 시간을 단축시켰다. 이때, 계산 정확도 유지를 위해 임피던스 행렬의 각 원소간 거리를 상대 거리 지수로 정의하여 보간법 적용이 가능한 먼 거리 원소를 구분하였다. 제안된 해석 기법의 성능 검증은 금속구의 Mie-series 해법을 이용한 이론적 RCS(Radar Cross Section)를 비교/분석하였다. 또한, 본 연구 결과를 삼면-/전방향- 전파반사기와 같은 산란체에 적용하여 레이더 후방 산란 특성을 분석하였다. An improved method of moments using a hybrid Galerkin-interpolation technique for numerical analysis of electromagnetic wave scattering in the 3-dimensional space is presented in this paper. Basically, the EFIE(electric field integral equation) and RWG(Rao-Wilton-Glisson) basis function are used to compute a property of electromagnetic wave scattering. We propose a hybrid technique combining the existing Galerkin"s method with the interpolation method to improve the efficiency of the numerical computation. Then, an index of relative distance of each cells was defined to distinguish the relatively far elements, which interpolation method can be applied. To verify the performance of the proposed technique, the analytical Mie-series solution was used to compute the theoretical RCS of a conducting sphere for the purpose of comparison. We also applied this hybrid technique to various scatterers such as trihedral/omni-directional corner-reflectors to analyze the radar backscattering properties.

A discontinuous Galerkin method for elliptic interface problems with application to electroporation

Guyomarc'h, Gré,gory,Lee, Chang-Ock,Jeon, Kiwan John Wiley Sons, Ltd. 2009 Communications in numerical methods in engineering Vol.25 No.10

<P>We solve elliptic interface problems using a discontinuous Galerkin (DG) method, for which discontinuities in the solution and in its normal derivatives are prescribed on an interface inside the domain. Standard ways to solve interface problems with finite element methods consist in enforcing the prescribed discontinuity of the solution in the finite element space. Here, we show that the DG method provides a natural framework to enforce both discontinuities weakly in the DG formulation, provided the triangulation of the domain is fitted to the interface. The resulting discretization leads to a symmetric system that can be efficiently solved with standard algorithms. The method is shown to be optimally convergent in the L<SUP>2</SUP>-norm. We apply our method to the numerical study of electroporation, a widely used medical technique with applications to gene therapy and cancer treatment. Mathematical models of electroporation involve elliptic problems with dynamic interface conditions. We discretize such problems into a sequence of elliptic interface problems that can be solved by our method. We obtain numerical results that agree with known exact solutions. Copyright © 2008 John Wiley & Sons, Ltd.</P>

유한요소 교호법을 이용한 삼차원 균열의 탄소성 J 적분 해석

박재학(Jai Hak Park) 대한기계학회 2009 大韓機械學會論文集A Vol.33 No.2

SGBEM(Symmetric Galerkin Boundary Element Method)-FEM alternating method has been proposed by Nikishkov, Park and Atluri. In the proposed method, arbitrarily shaped three-dimensional crack problems can be solved by alternating between the crack solution in an infinite body and the finite element solution without a crack. In the previous study, the SGBEM-FEM alternating method was extended further in order to solve elastic-plastic crack problems and to obtain elastic-plastic stress fields. For the elastic-plastic analysis the algorithm developed by Nikishkov et al. is used after modification. In the algorithm, the initial stress method is used to obtain elasticplastic stress and strain fields. In this paper, elastic-plastic J integrals for three-dimensional cracks are obtained using the method. For that purpose, accurate values of displacement gradients and stresses are necessary on an integration path. In order to improve the accuracy of stress near crack surfaces, coordinate transformation and partitioning of integration domain are used. The coordinate transformation produces a transformation Jacobian, which cancels the singularity of the integrand. Using the developed program, simple three-dimensional crack problems are solved and elastic and elastic-plastic J integrals are obtained. The obtained J integrals are compared with the values obtained using a handbook solution. It is noted that J integrals obtained from the alternating method are close to the values from the handbook.

A discontinuous Galerkin method with Lagrange multiplier for hyperbolic conservation laws with boundary conditions

Pergamon Press ; Elsevier Science Ltd 2015 COMPUTERS & MATHEMATICS WITH APPLICATIONS - Vol.70 No.4

We introduce a discontinuous Galerkin method with Lagrange multiplier (DGLM) to approximate the solution to the hyperbolic conservation laws with boundary conditions. Lagrange multipliers are introduced on the edge/face of the element via weak divergence (Wang and Ye, 2014). The final global system has reduced numbers of unknowns of the standard DG methods. Numerical fluxes from finite volume/difference method are not considered. For the time discretization, backward Euler difference method is used. Stability of the approximate solution is proved in energy norm. Discontinuity of the solution is allowed in the error analysis. Local error estimates of O(h<SUP>r+12</SUP>+Δt) with P<SUB>r</SUB>(E) elements (r≥d+12) are derived, where h and Δt are the maximum diameter of the elements and time steps, respectively, and d is the dimension of the spatial domain. The high order approximation is obtained under an appropriate condition on the stabilizing parameter. It is shown that the method preserves the property of the local mass conservation. An explanation on algorithmic aspects is given.

Special Transmission Gear Invalidation Analysis Coupled with Finite Element Method Based on Meshless Local Petrov-Galerkin Method

Bing Dai,J. P Shao,Xuemei Wu,Guangbin Yu,Ye Song,Ge Jianghua 보안공학연구지원센터 2015 International Journal of u- and e- Service, Scienc Vol.8 No.10

Aiming at high nonlinear problem of special transmission gear invalidation analysis, provided a method which based on Meshless local Petrov-Galerkin method coupled with finite element method and method to solve fracture problem of special transmission gear. Simulation calculation has been done to non-involute beveloid gear developed by project team. Calculation result verified efficiency of the simulation method. The method has important meaning to novel gear development and research.

Improved Element-Free Galerkin method (IEFG) for solving three-dimensional elasticity problems

Zhang, Zan,Liew, K.M. Techno-Press 2010 Interaction and multiscale mechanics Vol.3 No.2

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.

Modeling Fracture of Concrete with a Simplified Meshless Discrete Crack Method

N. Sagaresan 대한토목학회 2012 KSCE JOURNAL OF CIVIL ENGINEERING Vol.16 No.3

Fracture in quasi-brittle materials such as concrete is accompanied by excessive cracking. Numerical analysis of concrete fracture is either based on smeared crack method or discrete crack method. Smeared crack methods are computationally less challenging than the discrete crack method. However, this simplicity brings loss of accuracy. We propose a novel simplified and highly efficient meshless method for discrete cracks and study fracture of concrete. The method exploits the advantages of smeared crack method and maintains the accuracy of discrete crack method. The discrete crack is modeled by set of discrete crack segments placed through the entire domain of influence of a node. We use Neo-Hooke material in the bulk material and a cohesive zone model once discrete cracks occur. We demonstrate the accuracy of the proposed meshless discrete crack method for complex problems involving mode-I and mixed mode failure.

유한요소 교호법으로 구한 삼차원 균열 탄성해의 정확성 향상 및 검토

박재학(Jai Hak Park),G.P. Nikishkov 대한기계학회 2010 大韓機械學會論文集A Vol.34 No.5

SGBEM-FEM 교호법이 Nikishkov, Park 및 Atluri 에 의하여 제안되었었다. 제안된 방법을 사용하면 임의 형태의 평면 혹은 비평면 삼차원 균열에 대하여 복합 모드의 응력강도계수를 구할 수 있다. 그러나 현장에서의 적용을 위해서는 이 방법의 정확성 및 신뢰성에 대한 검토가 더욱 필요하다. 따라서 본 논문에서는 응력강도계수에 영향을 주는 주요한 몇 가지 인자를 검토하였다. 그리고 원통의 내부 및 외부에 존재하는 원주방향 표면균열에 대한 응력강도계수를 구하여 기존의 해와 비교하였다. 그 결과 SGBEM-FEM 교호법은 이들 균열에 대하여 정확한 해를 주고 있음을 확인하였다. An SGBEM (symmetric Galerkin boundary element method)-FEM alternating method has been proposed by Nikishkov, Park and Atluri. This method can be used to obtain mixed-mode stress intensity factors for planar and nonplanar three-dimensional cracks having an arbitrary shape. For field applications, however, it is necessary to verify the accuracy and consistency of this method. Therefore, in this study, we investigate the effects of several factors on the accuracy of the stress intensity factors obtained using the abovementioned alternating method. The obtained stress intensity factors are compared with the known values provided in handbooks, especially in the case of internal and external circumferential semi-elliptical surface cracks. The results show that the SGBEM-FEM alternating method yields accurate stress intensity factors for three-dimensional cracks, including internal and external circumferential surface cracks and that the method can be used as a robust crack analysis tool for solving field problems.