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Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
Uppsala University.ORCID-id: 0000-0003-1216-1672
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

In numerical simulations, problems stemming from aerodynamics pose many challenges for the method used. Some of these are addressed in this thesis, such as the fluid interacting with objects, the presence of shocks, and various types of boundary conditions.

Scenarios of the kind mentioned above are described mathematically by initial boundary value problems (IBVPs). We discretize the IBVPs using high order accurate finite difference schemes on summation by parts form (SBP), combined with weakly imposed boundary conditions, a technique called simultaneous approximation term (SAT). By using the energy method, stability can be shown.

The weak implementation is compared to the more commonly used strong implementation, and it is shown that the weak technique enhances the rate of convergence to steady state for problems with solid wall boundary conditions. The analysis is carried out for a linear problem and supported numerically by simulations of the fully non-linear Navier–Stokes equations.

Another aspect of the boundary treatment is observed for fluid structure interaction problems. When exposed to eigenfrequencies, the coupled system starts oscillating, a phenomenon called flutter. We show that the strong implementation sometimes cause instabilities that can be mistaken for flutter.

Most numerical schemes dealing with flows including shocks are first order accurate to avoid spurious oscillations in the solution. By modifying the SBP-SAT technique, a conservative and energy stable scheme is derived where the order of accuracy can be lowered locally. The new scheme is coupled to a shock-capturing scheme and it retains the high accuracy in smooth regions.

For problems with complicated geometry, one strategy is to couple the finite difference method to the finite volume method. We analyze the accuracy of the latter on unstructured grids. For grids of bad quality the truncation error can be of zeroth order, indicating that the method is inconsistent, but we show that some of the accuracy is recovered.

We also consider artificial boundary closures on unbounded domains. Non-reflecting boundary conditions for an incompletely parabolic problem are derived, and it is shown that they yield well-posedness. The SBP-SAT methodology is employed, and we prove that the discretized problem is stable.

Ort, förlag, år, upplaga, sidor
Uppsala: Uppsala University, 2012. , s. 26
Serie
Acta Universitatis Upsaliensis : Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 985
Nyckelord [en]
summation by parts, simultaneous approximation term, accuracy, stability, finite difference methods
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
URN: urn:nbn:se:lnu:diva-77769ISBN: 978-91-554-8509-2 (tryckt)OAI: oai:DiVA.org:lnu-77769DiVA, id: diva2:1248194
Disputation
2012-12-07, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, 10:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2018-09-18 Skapad: 2018-09-14 Senast uppdaterad: 2018-10-23Bibliografiskt granskad
Delarbeten
1. Analysis of the order of accuracy for node-centered finite volume schemes
Öppna denna publikation i ny flik eller fönster >>Analysis of the order of accuracy for node-centered finite volume schemes
2009 (Engelska)Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 59, nr 10, s. 2659-2676Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

The order of accuracy of the node-centered finite volume methods is analyzed, and the analysis is based on an exact derivation of the numerical errors in one dimension. The accuracy for various types of grids are considered. Numerical simulations and analysis are performed for both a hyperbolic and a elliptic case, and the results agree. The impact of weakly imposed boundary conditions is analyzed and verified numerically. We show that the error contribution from the primal and dual grid can be treated separately.

Nationell ämneskategori
Beräkningsmatematik Datavetenskap (datalogi)
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
urn:nbn:se:lnu:diva-77773 (URN)10.1016/j.apnum.2009.06.001 (DOI)000268513700023 ()
Tillgänglig från: 2018-09-14 Skapad: 2018-09-14 Senast uppdaterad: 2018-09-25Bibliografiskt granskad
2. Fluid structure interaction problems: the necessity of a well posed, stable and accurate formulation
Öppna denna publikation i ny flik eller fönster >>Fluid structure interaction problems: the necessity of a well posed, stable and accurate formulation
2010 (Engelska)Ingår i: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 8, nr 5, s. 1111-1138Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We investigate problems of fluid structure interaction type and aim for a formulation that leads to a well posed problem and a stable numerical procedure. Our first objective is to investigate if the generally accepted formulations of the fluid structure interaction problem are the only possible ones. Our second objective is to derive a stable numerical coupling. To accomplish that we will use a weak coupling procedure and employ summation-by-parts operators and penalty terms. We compare the weak coupling with other common procedures. We also study the effect of high order accurate schemes. In multiple dimensions this is a formidable task and we start by investigating the simplest possible model problem available. As a flow model we use the linearized Euler equations in one dimension and as the structure model we consider a spring.

Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
urn:nbn:se:lnu:diva-77772 (URN)10.4208/cicp.260409.120210a (DOI)000284672100007 ()
Tillgänglig från: 2018-09-14 Skapad: 2018-09-14 Senast uppdaterad: 2018-10-03Bibliografiskt granskad
3. A stable and conservative method for locally adapting the design order of finite difference schemes
Öppna denna publikation i ny flik eller fönster >>A stable and conservative method for locally adapting the design order of finite difference schemes
2011 (Engelska)Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, nr 11, s. 4216-4231Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.

Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

Nationell ämneskategori
Beräkningsmatematik Datavetenskap (datalogi)
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
urn:nbn:se:lnu:diva-77771 (URN)10.1016/j.jcp.2010.11.020 (DOI)000290185000007 ()
Tillgänglig från: 2018-09-14 Skapad: 2018-09-14 Senast uppdaterad: 2018-09-25Bibliografiskt granskad
4. Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations
Öppna denna publikation i ny flik eller fönster >>Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations
2012 (Engelska)Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, nr 14, s. 4867-4884Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier–Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.

Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
urn:nbn:se:lnu:diva-77770 (URN)10.1016/j.jcp.2012.04.007 (DOI)000304257600021 ()
Tillgänglig från: 2018-09-14 Skapad: 2018-09-14 Senast uppdaterad: 2018-10-03Bibliografiskt granskad
5. Exact non-reflecting boundary conditions revisited: well-posedness and stability
Öppna denna publikation i ny flik eller fönster >>Exact non-reflecting boundary conditions revisited: well-posedness and stability
2012 (Engelska)Rapport (Övrigt vetenskapligt)
Abstract [en]

Exact non-reflecting boundary conditions for an incompletely parabolic system have been studied. It is shown that well-posedness is a fundamental property of the non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, energy stability follows automatically. The stability in combination with the high order accuracy results in a reliable, efficient and accurate method. The theory is supported by numerical simulations.

Ort, förlag, år, upplaga, sidor
Uppsala University, 2012. s. 31
Serie
Technical Report ; 2012-032
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
urn:nbn:se:lnu:diva-77780 (URN)
Tillgänglig från: 2018-09-14 Skapad: 2018-09-14 Senast uppdaterad: 2019-08-30Bibliografiskt granskad

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