An immersed discontinuous Galerkin method for compressible Navier‐Stokes equations on unstructured meshes

Hong, Xia; Febrianto, Eky; Zhang, Qiaoling; Cirak, Fehmi

doi:10.1002/fld.4765

Cited by 8 publications

(7 citation statements)

References 63 publications

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“…The EB approaches mentioned above use Cartesian background grids. Fidkowski and Darmofal [48] presented a 2D dG method using triangular cut cells for compressible flow, whereas, to the best of our knowledge, the only EB method involving both two-and three-dimensional computations of compressible fluids was recently proposed by Xiao et al [49], who used triangular and tetrahedral meshes as their background grid.…”

Section: Introductionmentioning

confidence: 99%

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

Gulizzi,

Almgren,

Bell

2021

Preprint

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We present a computational framework for solving the equations of inviscid gas dynamics using structured grids with embedded geometries. The novelty of the proposed approach is the use of high-order discontinuous Galerkin (dG) schemes and a shock-capturing Finite Volume (FV) scheme coupled via an hp adaptive mesh refinement (hp-AMR) strategy that offers high-order accurate resolution of the embedded geometries. The hp-AMR strategy is based on a multi-level block-structured domain partition in which each level is represented by block-structured Cartesian grids and the embedded geometry is represented implicitly by a level set function. The intersection of the embedded geometry with the grids produces the implicitly-defined mesh that consists of a collection of regular rectangular cells plus a relatively small number of irregular curved elements in the vicinity of the embedded boundaries. High-order quadrature rules for implicitly-defined domains enable high-order accuracy resolution of the curved elements with a cell-merging strategy to address the small-cell problem. The hp-AMR algorithm treats the system with a second-order finite volume scheme at the finest level to dynamically track the evolution of solution discontinuities while using dG schemes at coarser levels to provide high-order accuracy in smooth regions of the flow. On the dG levels, the methodology supports different orders of basis functions on different levels. The space-discretized governing equations are then advanced explicitly in time using high-order Runge-Kutta algorithms. Numerical tests are presented for two-dimensional and three-dimensional problems involving an ideal gas. The results are compared with both analytical solutions and experimental observations and demonstrate that the framework provides high-order accuracy for smooth flows and accurately captures solution discontinuities.

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Section: Introductionmentioning

confidence: 99%

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

Gulizzi,

Almgren,

Bell

2021

Preprint

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“…As in immersed, or embedded domain, methods the faces of the tetrahedra can be projected to the curved domain boundaries if higherorder boundary approximation is desired, see e.g. [46][47][48].…”

Section: Finite Element Discretisation With Mollified Basis Functionsmentioning

confidence: 99%

Mollified finite element approximants of arbitrary order and smoothness

Febrianto

Ortiz

Cirak

2021

Computer Methods in Applied Mechanics and Engineering

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The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily shaped polytopes. We propose the mollified basis functions of arbitrary order and smoothness for partitions consisting of convex polytopes. On each polytope an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolutions of the local approximants with a mollifier. The mollifier is chosen to be smooth, to have a compact support and a unit volume. The approximation properties of the obtained basis functions are governed by the local polynomial approximation order and mollifier smoothness. The convolution integrals are evaluated numerically first by computing the boolean intersection between the mollifier and the polytope and then applying the divergence theorem to reduce the dimension of the integrals. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, are not necessarily aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method as in immersed/embedded finite elements. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems.

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“…Performance of simulations using this type of EBM is yet to be validated by experiments featuring large motion and deformation of structural objects. Xiao et al [21] used an embedded boundary method with a cut-cell approach coupled with both finite volume and discontinuous-Galerkin fluid algorithms. In their method, multiple approaches are proposed to overcome the excessive stable time step restrictions imposed by the cut-cells.…”

Section: Introductionmentioning

confidence: 99%

Coupling with the Embedded Boundary Method in a Runge-Kutta Discontinuous-Galerkin Direct Ghost-Fluid Method (RKDG-DGFM) Framework for Fluid-Structure Interaction Simulations of Underwater Explosions

Brown

2021

JMSE

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Solution of near-field underwater explosion (UNDEX) problems frequently require the modeling of two-way coupled fluid-structure interaction (FSI). This paper describes the addition of an embedded boundary method to an UNDEX modeling framework for multiphase, compressible and inviscid fluid using the combined algorithms of Runge-Kutta, discontinuous-Galerkin, level-set and direct ghost-fluid methods. A computational fluid dynamics (CFD) solver based on these algorithms has been developed as described in previous work. A fluid-structure coupling approach was required to perform FSI simulation interfacing with an external structural mechanics solver. Large structural deformation and possible rupture and cracking characterize the FSI phenomenon in an UNDEX, so the embedded boundary method (EBM) is more appealing for this application in comparison to dynamic mesh methods such as the arbitrary Lagrangian-Eulerian (ALE) method to enable the fluid-structure coupling algorithm in the fluid. Its limitation requiring a closed interface that is fully submerged in the fluid domain is relaxed by an adjustment described in this paper so that its applicability is extended. Two methods of implementing the fluid-structure wall boundary condition are also compared. The first solves a local 1D fluid-structure Riemann problem at each intersecting point between the wetted elements and fluid mesh. In this method, iterations are required when the Tait equation of state is utilized. A second method that does not require the Riemann solution and iterations is also implemented and the results are compared.

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An immersed discontinuous Galerkin method for compressible Navier‐Stokes equations on unstructured meshes

Cited by 8 publications

References 63 publications

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

Mollified finite element approximants of arbitrary order and smoothness

Coupling with the Embedded Boundary Method in a Runge-Kutta Discontinuous-Galerkin Direct Ghost-Fluid Method (RKDG-DGFM) Framework for Fluid-Structure Interaction Simulations of Underwater Explosions

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