Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion

Chan, Jesse; Evans, John A.

doi:10.1016/j.cma.2018.01.022

Cited by 26 publications

(23 citation statements)

References 45 publications

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“…A low storage 4th order Runge-Kutta scheme [49] is used for all numerical experiments. The time-step is estimated based on formulas derived for linear advection in [29,50]…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

Chan

2019

J Sci Comput

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High order entropy stable discontinuous Galerkin (DG) methods for nonlinear conservation laws satisfy an inherent discrete entropy inequality. The construction of such schemes has relied on the use of carefully chosen nodal points [1][2][3][4][5] or volume and surface quadrature rules [6,7] to produce operators which satisfy a summation-by-parts (SBP) property. In this work, we show how to construct "modal" DG formulations which are entropy stable for volume and surface quadratures under which the SBP property in [6] does not hold. These formulations rely on an alternative skew-symmetric construction of operators which automatically satisfy the SBP property. Entropy stability then follows for choices of volume and surface quadrature which satisfy sufficient accuracy conditions. The accuracy of these new SBP operators depends on a separate set of conditions on quadrature accuracy, with design order accuracy recovered under the usual assumptions of degree 2N −1 volume quadratures and degree 2N surface quadratures. We conclude with numerical experiments verifying the accuracy and stability of the proposed formulations, and discuss an application of these formulations for entropy stable DG schemes on mixed quadrilateral-triangle meshes. IntroductionHigh order methods for the simulation of time-dependent compressible flow have the potential to achieve higher levels of accuracy at lower costs compared to current low order schemes [8]. In addition to superior accuracy, the low numerical dispersion and dissipation of high order methods [9] enables the accurate propagation of waves over long distances and time scales. The same properties also make high order methods attractive for unsteady phenomena such as vorticular and turbulent flows, which are sensitive to numerical dissipation [8,10].

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“…A low storage 4th order Runge-Kutta scheme [49] is used for all numerical experiments. The time-step is estimated based on formulas derived for linear advection in [29,50]…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…This discrepancy can be understood in terms of constants in finite element inverse and trace inequalities. It was shown in [29,50] that, for linear problems, the maximum stable time-step scales inversely with the order-dependent constants C I , C T , where…”

Section: A Dependence Of Inverse and Trace Constants On Quadraturementioning

confidence: 99%

Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

Chan

2019

J Sci Comput

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“…Namely, we want to quantify the error made in the approximation of the L 2 -projection of a function f (x, y) into the curved-GD approximation space. Following [15], we consider the coordinate transform…”

Section: Wadg-gd Projection Accuracymentioning

confidence: 99%

Robust approaches to handling complex geometries with Galerkin difference methods

Kozdon

Wilcox

Hagstrom

et al. 2019

Journal of Computational Physics

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The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using the tensor product constructions to quadrilateral elements for discretizing partial differential equations. Here we propose two approaches to handling complex geometries using the GD basis within a discontinuous Galerkin finite element setting: (1) using non-conforming, curvilinear GD elements and (2) coupling affine GD elements with curvilinear simplicial elements. In both cases the (semidiscrete) discontinuous Galerkin method is provably energy stable even when variational crimes are committed and in both cases a weight-adjusted mass matrix is used, which ensures that only the reference mass matrix must be inverted. Additionally, we give sufficient conditions on the treatment of metric terms for the curvilinear, nonconforming GD elements to ensure that the scheme is both constant preserving and conservative. Numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes.which means that for GD these calculations can be performed along the grid lines. We then define the metric terms needed at the quadrature nodes as the interpolation of these metric derivatives. Namely, we definewhere the operator diag(·) turns a vector into a diagonal matrix. When needed, the Jacobian determinant can be computed from these interpolated values as:An important property of our approach to computing the metric terms is that we preserve discretely the divergence theorem, which will enable us to show that the scheme can be made both conservative and constant preserving.

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“…Another example is the inverse mass matrix approximation proposed by Chan and Evans,20 in the context of discontinuous Galerkin IGA for wave propagation, where an approximated weight-adjusted RMM is derived empirically. It is interesting to note that the same inverse mass matrix is obtained if we apply the proposed variational framework to the LC ansatz basis…”

Section: Convenient Dual Basesmentioning

confidence: 99%

“…These element matrices are then used to assemble the discrete EOMs, obtained from momentum (20) and equilibrium (19) equations, particularized for the infinite lattice in the form…”

Section: Dispersion Analysismentioning

confidence: 99%

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

González

Kopačka

Kolman

et al. 2018

Numerical Meth Engineering

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A variational framework is employed to generate inverse mass matrices for isogeometric analysis (IGA). As different dual bases impact not only accuracy but also computational overhead, several dual bases are extensively investigated.Specifically, locally discontinuous biorthogonal basis functions are evaluated in detail for B-splines of high continuity and Bézier elements with a standard C 0 continuous finite element structure. The boundary conditions are enforced by the method of localized Lagrangian multipliers after generating the inverse mass matrix for completely free body. Thus, unlike inverse mass matrix methods without employing the method of Lagrange multipliers, no modifications in the reciprocal basis functions are needed to account for the boundary conditions. Hence, the present method does not require internal modifications of existing IGA software structures. Numerical examples show that globally continuous dual basis functions yield better accuracy than locally discontinuous biorthogonal functions, but with much higher computational overhead. Locally discontinuous dual basis functions are found to be an economical alternative to lumped mass matrices when combined with mass parameterization. The resulting inverse mass matrices are tested in several vibration problems and applied to explicit transient analysis of structures. KEYWORDSBézier extraction, explicit transient analysis, free vibration, inverse mass matrix, isogeometric analysis, localized Lagrange multipliers Int J Numer Methods Eng. 2019;117:939-966. wileyonlinelibrary.com/journal/nme

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Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion

Cited by 26 publications

References 45 publications

Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

Robust approaches to handling complex geometries with Galerkin difference methods

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

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