The continuous-discontinuous Galerkin method applied to crack propagation

Tiago L.D. Forti,Nadia C.S. Forti,Fábio L.G. Santos,Marco A. Carnio 사단법인 한국계산역학회 2019 Computers and Concrete, An International Journal Vol.23 No.4

The discontinuous Galerkin method (DGM) has become widely used as it possesses several qualities, such as a natural ability to dealing with discontinuities. DGM has its major success related to fluid mechanics. Its major importance is the ability to deal with discontinuities and still provide high order of approximation. That is an important advantage when simulating cracking propagation. No remeshing is necessary during the propagation, since the crack path follows the interface of elements. However, DGM comes with the drawback of an increased number of degrees of freedom when compared to the classical continuous finite element method. Thus, it seems a natural approach to combine them in the same simulation obtaining the advantages of both methods. This paper proposes the application of the combined continuous-discontinuous Galerkin method (CDGM) to crack propagation. An important engineering problem is the simulation of crack propagation in concrete structures. The problem is characterized by discontinuities that evolve throughout the domain. Crack propagation is simulated using CDGM. Discontinuous elements are placed in regions with discontinuities and continuous elements elsewhere. The cohesive zone model describes the fracture process zone where softening effects are expressed by cohesive zones in the interface of elements. Two numerical examples demonstrate the capacities of CDGM. In the first example, a plain concrete beam is submitted to a three-point bending test. Numerical results are compared to experimental data from the literature. The second example deals with a full-scale ground slab, comparing the CDGM results to numerical and experimental data from the literature.

Application of the Runge Kutta Discontinuous Galerkin-Direct Ghost Fluid Method to internal explosion inside a water-filled tube

Jinwon Park 대한조선학회 2019 International Journal of Naval Architecture and Oc Vol.11 No.1

This paper aims to assess the applicability of the Runge Kutta Discontinuous Galerkin-Direct Ghost Fluid Method to the internal explosion inside a water-filled tube, which previously was studied by many researchers in separate works. Once the explosive charge located at the inner center of the water-filled tube explodes, the tube wall is subjected to an extremely high intensity fluid loading and deformed. The deformation causes a modification of the field of fluid flow in the region near the water-structure interface so that has substantial influence on the response of the structure. To connect the structure and the fluid, valid data exchanges along the interface are essential. Classical fluid structure interaction simulations usually employ a matched meshing scheme which discretizes the fluid and structure domains using a single mesh density. The computational cost of fluid structure interaction simulations is usually governed by the structure because the size of time step may be determined by the density of structure mesh. The finer mesh density, the better solution, but more expensive computational cost. To reduce such computational cost, a non-matched meshing scheme which allows for different mesh densities is employed. The coupled numerical approach of this paper has fewer difficulties in the implementation and computation, compared to gas dynamics based approach which requires complicated analytical manipulations. It can also be applied to wider compressible, inviscid fluid flow analyses often found in underwater explosion events.

Application of the Runge Kutta Discontinuous Galerkin-Direct Ghost Fluid Method to internal explosion inside a water-filled tube

Park, Jinwon The Society of Naval Architects of Korea 2019 International Journal of Naval Architecture and Oc Vol.11 No.1

This paper aims to assess the applicability of the Runge Kutta Discontinuous Galerkin-Direct Ghost Fluid Method to the internal explosion inside a water-filled tube, which previously was studied by many researchers in separate works. Once the explosive charge located at the inner center of the water-filled tube explodes, the tube wall is subjected to an extremely high intensity fluid loading and deformed. The deformation causes a modification of the field of fluid flow in the region near the water-structure interface so that has substantial influence on the response of the structure. To connect the structure and the fluid, valid data exchanges along the interface are essential. Classical fluid structure interaction simulations usually employ a matched meshing scheme which discretizes the fluid and structure domains using a single mesh density. The computational cost of fluid structure interaction simulations is usually governed by the structure because the size of time step may be determined by the density of structure mesh. The finer mesh density, the better solution, but more expensive computational cost. To reduce such computational cost, a non-matched meshing scheme which allows for different mesh densities is employed. The coupled numerical approach of this paper has fewer difficulties in the implementation and computation, compared to gas dynamics based approach which requires complicated analytical manipulations. It can also be applied to wider compressible, inviscid fluid flow analyses often found in underwater explosion events.

Implicit Discontinuous Galerkin Scheme for Discontinuous Bathymetry in Shallow Water Equations

이해균 대한토목학회 2020 KSCE JOURNAL OF CIVIL ENGINEERING Vol.24 No.9

One important issue in the approach with shallow water equations, which is not restricted to discontinuous Galerkin formulation, is the limitation to step geometries (discontinuous bathymetry), due to the hydrostatic assumption employed for the derivation of shallow water equations from the Navier-Stokes equations. In addition, the explicit Runge-Kutta time-stepping schemes do not come without any drawbacks even though the majority of discontinuous Galerkin applications have employed explicit ones due to simplicity. In this study, the recently developed implicit discontinuous Galerkin scheme is combined with the surface gradient method for steps (SGMS). The developed scheme is verified with flows over discontinuous bathymetry, i.e., vertical steps and weirs. For one-dimensional problems, the flows over a step and over a rectangular weir are simulated. As for two-dimensional problems, the flow over a weir and the dam-break flow over a step followed by the L-shaped and 45°-bend channels are simulated. The numerical solutions are compared with the experimental data. In all cases, good agreements were observed and the effectiveness of the developed scheme was verified.

Error estimates of fully discrete discontinuous Galerkin approximations for linear Sobolev equations

M. R. Ohm,신준용,Hyun Yong Lee 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5

In this paper, we construct fully discrete discontinuous Galerkin approximations to the solution of linear Sobolev equations. We apply a symmetric interior penalty method which has an interior penalty term to compensate the continuity on the edges of interelements. The optimal convergence of the fully discrete discontinuous Galerkin approximations in ℓ∞(L2) norm is proved. In this paper, we construct fully discrete discontinuous Galerkin approximations to the solution of linear Sobolev equations. We apply a symmetric interior penalty method which has an interior penalty term to compensate the continuity on the edges of interelements. The optimal convergence of the fully discrete discontinuous Galerkin approximations in ℓ∞(L2) norm is proved.

A PRIORI ERROR ESTIMATES OF A DISCONTINUOUS GALERKIN METHOD FOR LINEAR SOBOLEV EQUATIONS

MI RAY OHM,JUN YONG SHIN,HYUN YOUNG LEE 한국산업응용수학회 2009 Journal of the Korean Society for Industrial and A Vol.13 No.3

A discontinuous Galerkin method with interior penalty terms is presented for linear Sobolev equation. On appropriate finite element spaces, we apply a symmetric interior penalty Galerkin method to formulate semidiscrete approximate solutions. To deal with a damping term ∇ㆍ(∇ut) included in Sobolev equations, which is the distinct character compared to parabolic differential equations, we choose special test functions. A priori error estimate for the semidiscrete time scheme is analyzed and an optimal L<SUP>∞</SUP>(L²) error estimation is derived.

고정밀 압축성 유동 해석을 위한 임의의 고차정확도를 갖는 비정렬 불연속 갤러킨-다차원 공간 제한 기법 연구

박진석,김종암 한국항공우주학회 2011 한국항공우주학회 학술발표회 논문집 Vol.2011 No.11

본 연구에서는 압축성 유동장을 정밀하게 해석하기 위해서 불연속 갤러킨 기법을 위한 강건하고 효율적인 다차원 충격파 포착 기법을 개발하고자 한다. 그동안 본 연구 그룹에서는 유한 체적법을 바탕으로 다차원 물리 유동 특성을 반영한 다차원 공간 제한 기법을 성공적으로 개발하였으며, 이를 바탕으로 고차 내삽에 용이한 불연속 갤러킨 기법으로 확장하고자 한다. 이를 위해서 본 연구에서는 기존에 2 차 정확도의 유한 체적법에서 제안한 MLP 기울기 제한자와 더불어서, 고차 내삽기법에 적합한 MLP 기반 Troubled-cell 표시자를 추가적으로 개발하였다. 이를 결합하여 개발한 DG-MLP 기법의 경우 연속적인 구간에서 높은 정확도를 유지하면서도 매우 강건하고 정확하게 물리적 비선형파를 포착함을 확인할 수 있었다. The present paper deals with the robust and efficient shock capturing strategy for arbitrary higher-order discontinuous Galerkin methods to resolve high speed compressible flow accurately. This approach is a continuous work of extending multi-dimensional limiting process (MLP), which has been successfully developed in finite volume method (FVM), into discontinuous Galerkin Method on unstructured grids. Based on successful analyses and implementations of the MLP slope limiting in FVM, MLP is extendable into DG framework with the MLP-based troubled-cell marker and the hierarchical MLP slope limiting. It is observed that the proposed approach yields outstanding performances in resolving non-compressive as well as compressive flow features.

Time-discontinuous Galerkin quadrature element methods for structural dynamics

Minmao Liao,Yupeng Wang 국제구조공학회 2023 Structural Engineering and Mechanics, An Int'l Jou Vol.85 No.2

Three time-discontinuous Galerkin quadrature element methods (TDGQEMs) are developed for structural dynamic problems. The weak-form time-discontinuous Galerkin (TDG) statements, which are capable of capturing possible displacement and/or velocity discontinuities, are employed to formulate the three types of quadrature elements, i.e., single-field, singlefield/least-squares and two-field. Gauss-Lobatto quadrature rule and the differential quadrature analog are used to turn the weakform TDG statements into a system of algebraic equations. The stability, accuracy and numerical dissipation and dispersion properties of the formulated elements are examined. It is found that all the elements are unconditionally stable, the order of accuracy is equal to two times the element order minus one or two times the element order, and the high-order elements possess desired high numerical dissipation in the high-frequency domain and low numerical dissipation and dispersion in the lowfrequency domain. Three fundamental numerical examples are investigated to demonstrate the effectiveness and high accuracy of the elements, as compared with the commonly used time integration schemes.

hp-DISCONTINUOUS GALERKIN METHODS FOR THE LOTKA-MCKENDRICK EQUATION: A NUMERICAL STUDY

Jeong, Shin-Ja,Kim, Mi-Young,Selenge, Tsendanysh Korean Mathematical Society 2007 대한수학회논문집 Vol.22 No.4

The Lotka-McKendrick model which describes the evolution of a single population is developed from the well known Malthus model. In this paper, we introduce the Lotka-McKendrick model. We approximate the solution to the model using hp-discontinuous Galerkin finite element method. The numerical results show that the presented hp-discontinuous Galerkin method is very efficient in case that the solution has a sharp decay.

ERROR ESTIMATES OF FULLY DISCRETE DISCONTINUOUS GALERKIN APPROXIMATIONS FOR LINEAR SOBOLEV EQUATIONS

Ohm, M.R.,Shin, J.Y.,Lee, H.Y. The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5

In this paper, we construct fully discrete discontinuous Galerkin approximations to the solution of linear Sobolev equations. We apply a symmetric interior penalty method which has an interior penalty term to compensate the continuity on the edges of interelements. The optimal convergence of the fully discrete discontinuous Galerkin approximations in ${\ell}^{\infty}(L^2)$ norm is proved.