Foundations of the blended isogeometric discontinuous Galerkin (BIDG) method

Michoski, Craig; Chan, Jesse; Engvall, Luke; Evans, John A.

doi:10.1016/j.cma.2016.02.015

Cited by 20 publications

(25 citation statements)

References 70 publications

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“…Moreover, adaptive mesh methods for shock problems have long been known to be highly effective [45,46], particularly when shock fronts can be precisely tracked. However, AMR and hp-adaptive methods are constrained by their meshing/griding geometries, even as isoparametric and isogeometric methods have been developed to, at least in part, reduce these dependencies [47].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Solving differential equations using deep neural networks

et al. 2020

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Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental features in numerical PDEs and nonlinear analysis: irregular solutions. First, the Sod shock tube solution to compressible Euler equations is discussed, analyzed, and then compared to conventional finite element and finite volume methods. These methods are extended to consider performance improvements and simultaneous parameter space exploration. Next, a shock solution to compressible magnetohydrodynamics (MHD) is solved for, and used in a scenario where experimental data is utilized to enhance a PDE system that is a priori insufficient to validate against the observed/experimental data. This is accomplished by enriching the model PDE system with source terms and using supervised training on synthetic experimental data. The resulting DNN framework for PDEs seems to demonstrate almost fantastical ease of system prototyping, natural integration of large data sets (be they synthetic or experimental), all while simultaneously enabling single-pass exploration of the entire parameter space.

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Section: Numerical Experimentsmentioning

confidence: 99%

Solving differential equations using deep neural networks

et al. 2020

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“…This is especially important for discretizations on curvilinear meshes, where the exact integration of spatially varying geometric factors and Jacobians can be either prohibitively expensive for high‐order curvilinear mappings or impossible for rational mappings. ()…”

Section: An Energy‐stable Wadg Formulation For Elastic Wave Propagationmentioning

confidence: 99%

Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media

Chan

2017

Numerical Meth Engineering

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Weight-adjusted inner products are easily invertible approximations to weighted L 2 inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time-domain method for wave propagation which is low storage, energy stable, and high-order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight-adjusted DG methods to the case of matrix-valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind-like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high-order accuracy of weight-adjusted DG for several problems in elastic wave propagation.

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“…. The former shows up in estimates for L 2 projection on curvilinear elements [30,12], while the latter term illustrates the additional effect of the smoothness of J on the WADG pseudo-projection. Since (2) requires approximating uJ by polynomials, the approximation power of P N is split between u and J.…”

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Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes

Chan¹,

Hewett²,

Warburton³

2017

SIAM J. Sci. Comput.

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Traditional time-domain discontinuous Galerkin (DG) methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices. In this work, we propose a weight-adjusted DG (WADG) methods for curvilinear meshes which reduce storage costs while retaining energy stability. A priori error estimates show that high order accuracy is preserved under sufficient conditions on the mesh, which are illustrated through convergence tests with different sequences of meshes. Numerical and computational experiments verify the accuracy and performance of WADG for a model problem on curved domains. arXiv:1608.03836v1 [math.NA] 12 Aug 2016 topologically (vertex, edges, faces, interior); however, the number of nodes exceeds the cardinality of natural approximation spaces on simplices. Additionally, such nodal sets have only been constructed up to degree 4 for tetrahedra.A similar approach is taken for flux reconstruction schemes on simplices, which are closely related to filtered nodal DG methods [23,24] where nodes are taken to be unisolvent quadrature points [25,26]. Unlike nodal sets for mass-lumped simplices, these quadrature points do not contain nodes which lie on the boundary, necessitating an additional interpolation step in the computation of numerical fluxes. However, numerical evidence indicates that co-locating nodes and quadrature points reduces instabilities resulting from the aliasing of spatially varying Jacobians [27], though an analysis of high order convergence and energy stability for curvilinear simplices are open problems.Krivodonova and Berger introduced an inexpensive treatment of curved boundaries for two-dimensional flow problems by modifying the DG formulation on affine triangles [28]. This was extended to wave propagation problems by Zhang in [8], and by Zhang and Tan for elements with non-boundary curved faces in [7]. A theoretical stability and convergence analysis remains to be shown, though numerical results suggest that each of these approaches preserves stability and high order accuracy on curvilinear meshes under the condition that curved triangles are well-approximated by planar triangles. However, sufficiently large differences between curved and planar triangles still result in unstable schemes [8].An alternative treatment addressing increased storage costs of curvilinear DG was addressed by Warburton using the Low-Storage Curvilinear DG (LSC-DG) method [29,30]. Under LSC-DG, the spatial variation of the Jacobian is incorporated into the physical basis functions over each element, resulting in identical mass matrices over each element. Work in [30] also includes a priori estimates for projection errors under the LSC-DG basis, and gives sufficient conditions under which convergence is guaranteed. Furthermore, the DG variational formulation is constructed to be a priori stable for surface quadratures with positive weights, allowing for stable under-integration of high order integrands present for curvilinear elements.In [31], the weight-adjusted DG (WADG)...

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Foundations of the blended isogeometric discontinuous Galerkin (BIDG) method

Cited by 20 publications

References 70 publications

Solving differential equations using deep neural networks

Solving differential equations using deep neural networks

Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media

Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes

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