Discontinuous Galerkin isogeometric analysis with peridynamic model for crack simulation of shell structure
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Keywords:
Isogeometric analysis
Peridynamics
Abstract Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.
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Among the time-integration methods used in solid mechanics, Newmark family of algorithms (along with Galerkin finite element) has been most popular and has gathered much attention over the years. Recently, the time discontinuous Galerkin method (TDG) and discontinuous Galerkin method (DGM) are being investigated by the researchers due to their advantages in wave propagation problems. In this study, TDG and DGM are combined to have space-time discontinuous Galerkin formulation for the first time for solid mechanics problems. The basic formulations are presented and some examples are studied.
Newmark-beta method
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Polyhedron
Volume mesh
Penalty Method
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Peridynamics
Isogeometric analysis
Meshfree methods
Basis function
Representation
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Isogeometric analysis
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In this paper, we generalize the compact subcell weighted essentially non oscillatory (CSWENO) limiting strategy for Runge-Kutta discontinuous Galerkin method developed recently by us in 2021 for structured meshes to unstructured triangular meshes. The main idea of the limiting strategy is to divide the immediate neighbors of a given cell into the required stencil and to use a WENO reconstruction for limiting. This strategy can be applied for any type of WENO reconstruction. We have used the WENO reconstruction proposed by Zhu and Shu in 2019 and provided accuracy tests and results for two-dimensional Burgers' equation and two dimensional Euler equations to illustrate the performance of this limiting strategy.
Stencil
Limiting
Runge–Kutta methods
Limiter
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Abstract To solve multiphysics problems, weak coupling of finite element calculations can be carried out: the subproblems of which the physical nature differs are solved separately on their own meshes. In this case, Galerkin projection provides useful tool to ensure the transfer of physical fields between different meshes. In terms of implementation, the Galerkin projection system can be either accurately assembled over the intersection of two meshes or approximately integrated over the target mesh. This paper describes and compares these two implementation techniques for the Galerkin projection. Copyright © 2013 John Wiley & Sons, Ltd.
Multiphysics
Projection method
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In this paper, we develop discontinuous Galerkin methods with penalty terms, namaly symmetric interior penalty Galerkin methods to solve nonlinear parabolic equations. By introducing an appropriate projection of u onto finite element spaces, we prove the optimal convergence of the fully discrete discontinuous Galerkin approximations in normed space.
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13th International Symposium on Multiscale, Multifunctional and Functionally Graded Materials (2014)
Peridynamics is a nonlocal reformulation of classical continuum mechanics. In contrast to classical models, governing equations in peridynamics are based on spatial integration, rather than spatial differentiation, of displacement fields. Therefore, peridynamics has been applied to the description of material failure and damage. As a nonlocal model, peridynamics is computationally more expensive than classical models; this motivates the development of concurrent multiscale methods, for which peridynamics is applied in regions where discontinuities appear or may be generated, whereas classical models are used elsewhere. A main challenge in concurrent multiscale modeling is how to couple different models without introducing spurious effects. We derive blending schemes to concurrently couple peridynamics and classical continuum mechanics, avoiding common artifacts present in these types of methods. We demonstrate the performance of the coupling schemes analytically and numerically.
Peridynamics
Classification of discontinuities
Spurious relationship
Multiscale Modeling
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This paper focuses on interior penalty discontinuous Galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra which satisfies certain shape regularity conditions characterized in a recent paper by two of the authors in [17]. Such general meshes have important application in computational sciences. The usual $H^1$ conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation. However, the interior penalty discontinuous Galerkin method provides a simple and effective alternative approach which is efficient and robust. This article provides a mathematical foundation for the use of interior penalty discontinuous Galerkin methods in general meshes.
Polyhedron
Volume mesh
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