Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework

AbstractLiouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of $$10^{-6}$$ 10 - 6 .

Download Full-text

A local discontinuous Galerkin gradient discretization method for linear and quasilinear elliptic equations

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019022 ◽

2019 ◽

Vol 53 (4) ◽

pp. 1269-1303

Author(s):

Assyr Abdulle ◽

Giacomo Rosilho de Souza

Keyword(s):

Discontinuous Galerkin ◽

Elliptic Equations ◽

Quasilinear Elliptic Equations ◽

Discretization Method ◽

Local Approach ◽

Artificial Boundary Conditions ◽

Non Local ◽

Quasilinear Elliptic ◽

Local Scheme ◽

Convergence Of The Scheme

A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic problems in high gradient regions. Using the gradient discretization framework we prove convergence of the scheme for linear and quasilinear equations under minimal regularity assumptions. The error due to artificial boundary conditions is also analyzed, shown to be of higher order and shown to depend only locally on the regularity of the solution. Numerical experiments illustrate our theoretical findings and the local method’s accuracy is compared against the non local approach.

Download Full-text

Quasi-static crush modelling of carbon/epoxy composites with discontinuous Galerkin/anisotropic extrinsic cohesive law method

Composite Structures ◽

10.1016/j.compstruct.2019.111480 ◽

2019 ◽

Vol 230 ◽

pp. 111480

Author(s):

Xing Liu ◽

Ling Wu ◽

Danny Van Hemelrijck ◽

Lincy Pyl

Keyword(s):

Discontinuous Galerkin ◽

Epoxy Composites ◽

Cohesive Law

Download Full-text

Discontinuous Galerkin methods for a kinetic model of self-organized dynamics

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202518500318 ◽

2018 ◽

Vol 28 (06) ◽

pp. 1171-1197

Author(s):

Francis Filbet ◽

Chi-Wang Shu

Keyword(s):

Kinetic Model ◽

Discontinuous Galerkin ◽

Existence Theory ◽

Galerkin Methods ◽

Hydrodynamic Models ◽

Numerical Resolution ◽

Self Organized ◽

Orientation Interaction ◽

Non Local ◽

Free Transport

This paper deals with the numerical resolution of kinetic models for systems of self-propelled particles subject to alignment interaction and attraction–repulsion. We focus on the kinetic model considered in [P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008) 1193–1215; P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal. 20 (2013) 89–114] where alignment is taken into account in addition to an attraction–repulsion interaction potential. We apply a discontinuous Galerkin method for the free transport and non-local drift velocity together with a spectral method for the velocity variable. Then, we analyze consistency and stability of the semi-discrete scheme. We propose several numerical experiments which provide a solid validation of the method and illustrate its underlying concepts.

Download Full-text