A two-fluid spectral-element method

Helenbrook, Brian T.

doi:10.1016/s0045-7825(01)00275-4

Cited by 43 publications

(31 citation statements)

References 21 publications

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“…However, in many cases this may be o set by the fact that the cost of the mesh-movement is smaller than the ow solver. Such is the case in spectral=hp simulations [2,12] where the ow resolution is much greater than the number of elements in the mesh. For implicit ÿnite-volume solvers, this cost can also be negligible because it is more di cult to solve the implicit Navier-Stokes equations [4].…”

Section: Resultsmentioning

confidence: 99%

Mesh deformation using the biharmonic operator

Helenbrook

2003

Numerical Meth Engineering

164

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SUMMARYThe use of the biharmonic operator for deforming a mesh in an arbitrary-Lagrangian-Eulerian simulation is investigated. The biharmonic operator has the advantage that two conditions can be speciÿed on each boundary of the mesh. This allows both the position and the normal mesh spacing along a boundary to be controlled, which is important for two-uid interfaces and periodic boundaries. At these boundaries, we can simultaneously ÿx the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary. In addition, results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using secondorder partial di erential equations. A ÿnal advantage is that with the biharmonic operator, the integrity of a grid in a moving boundary layer can be preserved as the boundary moves. The main disadvantage of the approach is its increased computational expense.

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Section: Resultsmentioning

confidence: 99%

Mesh deformation using the biharmonic operator

Helenbrook

2003

Numerical Meth Engineering

164

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“…The p component of this algorithm was proposed by R/ onquist and Patera [13] and analyzed by Maday and Munoz [12] for a Galerkin spectral element discretization of the Laplace equation. Helenbrook [8] combined p-multigrid with geometric multigrid and applied it to an unstructured streamwise-upwind-Petrov-Galerkin (SUPG) discretization of the incompressible Navier-Stokes equations. Recently there has also been work combining overlapping Schwarz relaxation methods with multigrid for spectral element discretizations [7].…”

Section: Introductionmentioning

confidence: 99%

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

Helenbrook

Atkins

2006

AIAA Journal

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We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

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“…In 2001, Helenbrook [24] applied the spectral multigrid method to a 2D incompressible NavierStokes equation using triangular elements. For the first time, a geometric multigrid method was used for the solution at p = 1.…”

Section: Historymentioning

confidence: 99%

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

Mitchell

2010

Numerical Linear Algebra App

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SUMMARYRecently the hp version of the finite element method, in which adaptivity occurs in both the size, h, of the elements and in the order, p, of the approximating piecewise polynomials, has received increasing attention. It is desirable to combine this optimal order discretization method with an optimal order algebraic solution method like multigrid. An intriguing notion is to use the values of p as the levels of a multilevel method. In this paper we present such a method, known as hp-multigrid, for high-order finite elements and hp-adaptive grids. We present a survey of the development of p-multigrid and hp-multigrid, define an hp-multigrid algorithm based on the p-hierarchical basis for the p levels and h-hierarchical basis for an h-multigrid solution of the p = 1 'coarse grid' equations, and present numerical convergence results using hp-adaptive grids. The numerical results suggest that the method has a convergence rate of

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A two-fluid spectral-element method

Cited by 43 publications

References 21 publications

Mesh deformation using the biharmonic operator

Mesh deformation using the biharmonic operator

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

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