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Fluid structure interaction problems: the necessity of a well posed, stable and accurate formulation
Uppsala university ; University Witwatersrand, South Africa ; Swedish Def Res Agcy.
Uppsala university.ORCID iD: 0000-0003-1216-1672
2010 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 8, no 5, p. 1111-1138Article in journal (Refereed) 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.

Place, publisher, year, edition, pages
2010. Vol. 8, no 5, p. 1111-1138
National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-77772DOI: 10.4208/cicp.260409.120210aISI: 000284672100007OAI: oai:DiVA.org:lnu-77772DiVA, id: diva2:1248198
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-10-03Bibliographically approved
In thesis
1. Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
Open this publication in new window or tab >>Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
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.

Place, publisher, year, edition, pages
Uppsala: Uppsala University, 2012. p. 26
Series
Acta Universitatis Upsaliensis : Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 985
Keywords
summation by parts, simultaneous approximation term, accuracy, stability, finite difference methods
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:lnu:diva-77769 (URN)978-91-554-8509-2 (ISBN)
Public defence
2012-12-07, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, 10:15 (English)
Opponent
Supervisors
Available from: 2018-09-18 Created: 2018-09-14 Last updated: 2018-10-23Bibliographically approved

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