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An adaptive finite element method for the compressible Euler equations
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.ORCID iD: 0000-0003-4256-0463
2010 (English)In: INT J NUMER METHOD FLUID, 2010, Vol. 64, no 10-12, 1102-1128 p.Conference paper, Published paper (Refereed)
Abstract [en]

We present an adaptive finite element method for the compressible Euler equations, based on a posteriori error estimation of a quantity of interest in terms of a dual problem for the linearized equations. Continuous piecewise linear approximation is used in space and time, with componentwise weighted least-squares stabilization of convection terms and residual-based shock-capturing. The adaptive algorithm is demonstrated numerically for the quantity of interest being the drag force on a body.

Place, publisher, year, edition, pages
2010. Vol. 64, no 10-12, 1102-1128 p.
Keyword [en]
adaptive finite element method, a posteriori error estimation, dual problem, compressible euler equations, circular cylinder, wedge, sphere
National Category
Computer and Information Science
Identifiers
URN: urn:nbn:se:kth:diva-25412OAI: oai:DiVA.org:kth-25412DiVA: diva2:358023
Conference
15th International Conference on Finite Elements in Flow Problems Tokyo, JAPAN, APR 01-03, 2009
Note
QC 20101020Available from: 2010-10-20 Created: 2010-10-20 Last updated: 2012-01-16Bibliographically approved
In thesis
1. An adaptive finite element method for the compressible Euler equations
Open this publication in new window or tab >>An adaptive finite element method for the compressible Euler equations
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This work develops a stabilized finite element method for the compressible Euler equations and proves an a posteriori error estimate for the approximated solution. The equations are approximated by the cG(1)cG(1) finite element method with continuous piecewise linear functions in space and time. cG(1)cG(1) gives a second order accuracy in space, and corresponds to a Crank-Nicholson type of discretization in time, resulting in second order accuracy in space, without a stabilization term.

The method is stabilized by componentwise weighted least squares stabilization of the convection terms, and residual based shock capturing. This choice of stabilization gives a symmetric stabilization matrix in the discrete system. The method is successfully implemented for a number of benchmark problems in 1D, 2D and 3D. We observe that cG(1)cG(1) with the above choice of stabilization is robust and converges to an accurate solution with residual based adaptive mesh refinement.

We then extend the General Galerkin framework from incompressible to compressible flow, with duality based a posteriori error estimation of some quantity of interest. The quantities of interest can be stresses, strains, drag and lift forces, surface forces or a mean value of some quantity. In this work we prove a duality based a posteriori error estimate for the compressible equations, as an extension of the earlier work for incompressible flow [25].

The implementation and analysis are validated in computational tests both with respect to the stabilization and the duality based adaptation

 

 

 

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. xii, 39 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2009:13
Identifiers
urn:nbn:se:kth:diva-10582 (URN)978-91-7415-365-1 (ISBN)
Presentation
2009-06-10, D42, KTH, Lindstedtsvägen 5, Plan 4, Stockholm, 14:15 (English)
Opponent
Supervisors
Available from: 2009-05-28 Created: 2009-05-28 Last updated: 2010-10-20Bibliographically approved
2. Adaptive Algorithms and High Order Stabilization for Finite Element Computation of Turbulent Compressible Flow
Open this publication in new window or tab >>Adaptive Algorithms and High Order Stabilization for Finite Element Computation of Turbulent Compressible Flow
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This work develops finite element methods with high order stabilization, and robust and efficient adaptive algorithms for Large Eddy Simulation of turbulent compressible flows.

The equations are approximated by continuous piecewise linear functions in space, and the time discretization is done in implicit/explicit fashion: the second order Crank-Nicholson method and third/fourth order explicit Runge-Kutta methods. The full residual of the system and the entropy residual, are used in the construction of the stabilization terms. These methods are consistent for the exact solution, conserves all the quantities, such as mass, momentum and energy, is accurate and very simple to implement. We prove convergence of the method for scalar conservation laws in the case of an implicit scheme. The convergence analysis is based on showing that the approximation is uniformly bounded, weakly consistent with all entropy inequalities, and strongly consistent with the initial data. The convergence of the explicit schemes is tested in numerical examples in 1D, 2D and 3D.

To resolve the small scales of the flow, such as turbulence fluctuations, shocks, discontinuities and acoustic waves, the simulation needs very fine meshes. In this thesis, a robust adjoint based adaptive algorithm is developed for the time-dependent compressible Euler/Navier-Stokes equations. The adaptation is driven by the minimization of the error in quantities of interest such as stresses, drag and lift forces, or the mean value of some quantity.

The implementation and analysis are validated in computational tests, both with respect to the stabilization and the duality based adaptation.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. xii, 54 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:13
Keyword
Compressible flow, adaptivity, finite element method, a posteriori error estimates, Implicit LES
National Category
Computational Mathematics Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-34532 (URN)978-91-7501-053-3 (ISBN)
Public defence
2011-09-01, F3, Entre plan,, Lindstedtsvägen 26, KTH, Stockholm, 17:58 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note
QC 20110627Available from: 2011-06-27 Created: 2011-06-09 Last updated: 2011-07-21Bibliographically approved

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