An adaptive least-squares method for the compressible Euler equations

Taghaddosi, Farzad; Habashi, Wagdi G.; Guevremont, G.; Ait‐Ali‐Yahia, D.

doi:10.2514/6.1997-2097

Cited by 11 publications

(18 citation statements)

References 20 publications

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“…More recently, impressive results using anisotropic elements have been reported by Habashi et al 12,13,14,15,16,17 With their approach, all grid adaptions are local operations, as opposed to global remeshings. Highly-stretched elements are obtained, achieved by equating the interpolation error along each edge, again using a spring analogy minimization.…”

Section: Introductionmentioning

confidence: 85%

On multi-dimensional unstructured mesh adaption

Wood

Kleb

1999

14th Computational Fluid Dynamics Conference

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Anisotropic unstructured mesh adaption is developed for a truly multi-dimensional upwind fluctuation splitting scheme, as applied to scalar advection-diffusion. The adaption is performed locally using edge swapping, point insertion/deletion, and nodal displacements. Comparisons are made versus the current state of the art for aggressive anisotropic unstructured adaption, which is based on a posteriori error estimates. Demonstration of both schemes to model problems, with features representative of compressible gas dynamics, show the present method to be superior to the a posteriori adaption for linear advection. The performance of the two methods is more similar when applied to nonlinear advection, with a difference in the treatment of shocks. The a posteriori adaption can excessively cluster points to a shock, while the present multi-dimensional scheme tends to merely align with a shock, using fewer nodes. As a consequence of this alignment tendency, an implementation of eigenvalue limiting for the suppression of expansion shocks is developed for the multi-dimensional distribution scheme. The differences in the treatment of shocks by the adaption schemes, along with the inherently low levels of artificial dissipation in the fluctuation splitting solver, suggest the present method is a strong candidate for applications to compressible gas dynamics.

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Section: Introductionmentioning

confidence: 85%

On multi-dimensional unstructured mesh adaption

Wood

Kleb

1999

14th Computational Fluid Dynamics Conference

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“…In Gnoffo's work, structured grids are replaced by tension springs, and the spring stiffness is defined as a function of the gradient of certain flow property so that more nodes could be "pulled" to the regions with higher gradient. Subsequently, Nakahashi and Deiwert [1985;1987a;1987b] introduced torsion springs into Gnoffo's spring model to increase the orthogonality of the structured grids, and Ait-Ali-Yahia et al [1996] and Taghaddosi et al [1997] combined the vertex spring analogy method with an edge-based error estimator and developed a directionally adaptive finite element method for quadrilateral grids. From these work, it was found that the vertex spring analogy method is very effective for both 2D and 3D structured grids.…”

Section: Introductionmentioning

confidence: 99%

An r-Adaptive Technique for Unstructured Grids Based on the Segment Spring Analogy Method

Sun

Zhang

2015

Int. J. Comput. Methods

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A simple and effective r-adaptive technique for unstructured grids based on the segment spring analogy method is proposed. The finite element method and a corresponding error estimate method using second derivatives are used for computation. The traditional segment spring analogy method is modified, based on an idea of controlling the equilibrium length of the fictitious springs, and used for mesh adjustment. The principle of making numerical errors distributed uniformly over all elements is applied. Three numerical examples involving the one-dimensional (1D) convection-diffusion equation, the two-dimensional (2D) linear parabolic equation and the 2D Euler equations are presented and the effectiveness of the proposed r-type grid adaptive technique is examined.

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“…Also in LSFEM, non-self-adjoint equation systems, such as the ones that govern fluid dynamics, result in symmetric positive-definite global system matrices, which can be solved efficiently by iterative solvers. In literature it is possible to find many uses of LSFEM for the solution of compressible flows (Fletcher, 1979;Jiang and Carey, 1990;Taghaddosi et al, 1999;Moussaoui, 2003;Reddy et al, 2004;Gerritsma et al, 2008).…”

Section: Introductionmentioning

confidence: 99%

“…Jiang and Carey (1987) used LSFEM with h-type adaptive 2D, non-conforming quadrilateral meshes. Taghaddosi et al (1999) and Guaily and Megahed (2010) used LSFEM on r-refined quadrilateral meshes. Chang and Chen (2011) coupled LSFEM with a re-triangulation algorithm to perform optimal adaptive solutions of advection-diffusion equation on 2D triangular meshes.…”

Section: Introductionmentioning

confidence: 99%

Least-squares finite element solution of Euler equations with H-type mesh refinement and coarsening on triangular elements

Akargun

Sert²

2014

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose -The purpose of this paper is to demonstrate successful use of least-squares finite element method (LSFEM) with h-type mesh refinement and coarsening for the solution of two-dimensional, inviscid, compressible flows. Design/methodology/approach -Unsteady Euler equations are discretized on meshes of linear and quadratic triangular and quadrilateral elements using LSFEM. Backward Euler scheme is used for time discretization. For the refinement of linear triangular elements, a modified version of the simple bisection algorithm is used. Mesh coarsening is performed with the edge collapsing technique. Pressure gradient-based error estimation is used for refinement and coarsening decision. The developed solver is tested with flow over a circular bump, flow over a ramp and flow through a scramjet inlet problems. Findings -Pressure difference based error estimator, modified simple bisection method for mesh refinement and edge collapsing method for mesh coarsening are shown to work properly with the LSFEM formulation. With the proper use of mesh adaptation, time and effort necessary to prepare a good initial mesh reduces and mesh independency control of the final solution is automatically taken care of. Originality/value -LSFEM is used for the first time for the solution of inviscid compressible flows with h-type mesh refinement and coarsening on triangular elements. It is shown that, when coupled with mesh adaptation, inherent viscous dissipation of LSFEM technique is no longer an issue for accurate shock capturing without unphysical oscillations.

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An adaptive least-squares method for the compressible Euler equations

Cited by 11 publications

References 20 publications

On multi-dimensional unstructured mesh adaption

On multi-dimensional unstructured mesh adaption

An r-Adaptive Technique for Unstructured Grids Based on the Segment Spring Analogy Method

Least-squares finite element solution of Euler equations with H-type mesh refinement and coarsening on triangular elements

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