The application of a high-order discontinuous Galerkin time-domain method for the computation of electromagnetic resonant modes

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).

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“…for all v ∈ L 2 (Γ), where u e and q e are obtained from the local problems defined in Equation (2).…”

Section: The Hdg Rationalementioning

confidence: 99%

“…Second, an element-by-element problem is solved to compute the value of the electric potential u and its gradient, i.e. the electric field, in the elements, according to Equation (2). Finally, an element-by-element postprocess is performed to obtain a superconvergent solution u e by solving (5)- (6).…”

Section: Hdg-nefem: Exact Geometry and Degree Adaptivitymentioning

confidence: 99%

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Giacomini

Sevilla

2019

SN Appl. Sci.

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“…However, it requires the generation of curvilinear high‐order meshes, which is still challenging for complex geometries in three dimensions 22 . The use of exact boundary representation has shown advantages in this context, 23,24 but the automatic generation of meshes for NEFEM is even more challenging and, nowadays, it is only available for 2D geometries 25 …”

Section: Introductionmentioning

confidence: 99%

High‐order discontinuous Galerkin method for time‐domain electromagnetics on geometry‐independent Cartesian meshes

Navarro-García

Sevilla

Nadal

et al. 2021

Numerical Meth Engineering

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In this work we present the Cartesian grid discontinuous Galerkin (cgDG) finite element method, a novel numerical technique that combines the high accuracy and efficiency of a high‐order discontinuous Galerkin discretization with the simplicity and hierarchical structure of a geometry‐independent Cartesian mesh. The elements that intersect the boundary of the physical domain require special treatment in order to minimize their effect on the performance of the algorithm. We considered the exact representation of the geometry for the boundary of the domain avoiding any nonphysical artifacts. We also define a stabilization procedure that eliminates the step size restriction of the time marching scheme due to extreme cut patterns. The unstable degrees of freedom are eliminated and the supporting regions of their shape functions are reassigned to neighboring elements. A subdomain matching algorithm and an a posterior enrichment strategy are presented. Combining these techniques we obtain a final spatial discretization that preserves stability and accuracy of the standard body‐fitted discretization. The method is validated through a series of numerical tests and it is successfully applied to the solution of problems of interest in the context of electromagnetic scattering with increasing complexity.

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“…The numerical simulation of wave propagation phenomena is of great interest in many areas of science and engineering, including seismic, acoustic, and electromagnetic applications. In this context, the use of time‐domain solvers enables to significantly reduce the memory requirements and the need to devise effective preconditioners, when compared to frequency‐domain solvers.…”

Section: Introductionmentioning

confidence: 99%

Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

Cottereau

Sevilla

2018

Numerical Meth Engineering

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Summary This paper considers the stability of an explicit leapfrog time marching scheme for the simulation of acoustic wave propagation in heterogeneous media with high‐order spectral elements. The global stability criterion is taken as a minimum over local element stability criteria, obtained through the solution of element‐borne eigenvalue problems. First, an explicit stability criterion is obtained for the particular case of a strongly heterogeneous and/or rapidly fluctuating medium using asymptotic analysis. This criterion is only dependent upon the maximum velocity at the vertices of the mesh elements, and not on the velocity at the interior nodes of the high‐order elements. Second, in a more general setting, bounds are derived using statistics of the coefficients of the elemental dispersion matrices. Different bounds are presented, discussed, and compared. Several numerical experiments show the accuracy of the proposed criteria in one‐dimensional test cases as well as in more realistic large‐scale three‐dimensional problems.

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The application of a high-order discontinuous Galerkin time-domain method for the computation of electromagnetic resonant modes

Abstract: K. (2017). The application of a high-order discontinuous Galerkin time-domain method for the computation of electromagnetic resonant modes. Applied Mathematical Modelling

Cited by 9 publications

References 42 publications

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

High‐order discontinuous Galerkin method for time‐domain electromagnetics on geometry‐independent Cartesian meshes

Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

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