Analysis of a FEM/BEM coupling method for transonic flow computations

Berger, Harald; Warnecke, Gerald; Wendland, Wolfgang L.

doi:10.1090/s0025-5718-97-00878-8

Cited by 16 publications

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“…Conditions of the uniqueness and existence of the solution to the problem were obtained. The established solvability of the problem permits one to analyze convergence of finite element approximations for the transonic flow using a technique developed in [2], [3], [20].…”

Section: Resultsmentioning

confidence: 99%

“…According to (3), where v out and β ± are given functions, the flow velocity is subsonic at the inlet and outlet of the channel. However, its value is not prescribed at the inlet.…”

Section: Formulation Of the Problem Uniqueness Of The Solutionmentioning

confidence: 99%

“…Let ϕ ∈ W 6,2 (G) be a smooth solution of problem(2),(3) such that the sonic line and points A, B, C, E, N are located as in Theorem 1, and w = (γ + 1)(ϕ x − 1) satisfies the inequalities w y < 0 and w 2 y − ww 2 x > 0 inḠ. In addition, assume that w x = 0 at a point P of the sonic line, w x > 0 on the upstream portion AP of the sonic line, while w x < 0 on the downstream portion P B (with the exception of point P ).…”

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confidence: 99%

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Solvability of a problem for transonic flow with a local supersonic region

Kuzmin

2001

NoDEA, Nonlinear differ. equ. appl.

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A nonlinear perturbation problem for steady two-dimensional inviscid transonic flow with a local supersonic region is considered. A damping condition is prescribed on a portion of the boundary in order to prevent the arising of shock waves in the flow. The existence of a smooth solution to the problem is established. A proof is based on the Fredholm alternative and on a technique for uniqueness analysis worked out by and Cook (1978).2000 Mathematics Subject Classification: 35M10, 76H05.

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Section: Resultsmentioning

confidence: 99%

“…According to (3), where v out and β ± are given functions, the flow velocity is subsonic at the inlet and outlet of the channel. However, its value is not prescribed at the inlet.…”

Section: Formulation Of the Problem Uniqueness Of The Solutionmentioning

confidence: 99%

mentioning

confidence: 99%

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Solvability of a problem for transonic flow with a local supersonic region

Kuzmin

2001

NoDEA, Nonlinear differ. equ. appl.

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“…Bei vielen Aufgabenstellungen kann die Randelementmethode nur in Verbindung mit finiten Elementen, die auf Teilgebieten mit tnhomogenit/iten, Nichtlinearit/iten und/oder nichtverschwindender rechter Seite eingesetzt sind, verwendet werden. Praktische Beispiele gibt es nicht nur in der Festk6rpermechanik [44], sondern auch in anderen Gebieten der Mathematischen Physik [8,56,66].…”

Section: Alerkin-schemaunclassified

Einführung in moderne Galerkin-Randeiementmethoden mit einer Anwendung aus dem Maschinenbau

Andrä

1999

Forsch Ing-Wes

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58Zusammenfassung Die Galerkin-Randelementmethode ist ein N~iherungsverfahren ffir Integralgleichungen, stellt jedoch gegenfiber klassischen Integralgleichungsverfahren ein universelles Werkzeug zur L6sung praktischer Ingenieurprobleme dar und kann sehr gut mit Finite-ElementSubstrukturen gekoppelt werden. Die Randelementmethode, bei der als Hauptvorteil nur ein Oberfl~ichennetz generiert werden muff, ist bei spezieUen Anwendungen beispielsweise ffir Kerb-und Ri~probleme der FEM fiberlegen. Die einzelnen Schritte zur L6sung eines elliptischen Randwertproblems fiber ein System yon Randintegralgleichungen wird am Beispiel des dreidimensionalen linearen Elastizit~itsproblems erl~iutert. Zur mathematischen Untersuchung yon elliptischen Differentialgleichungen und/~quivalenten Integralgleichungen hat sich die Theorie der Sobolev-R~iume als besonders geeignet herausgestellt. Grundbegriffe zu Sobolev-R~iumen werden eingeffihrt, so da/~ der Leser nicht in Lehrbfichern nachschlagen muff. Die 0berffihrung der elliptischen Randwertprobleme auf Systeme yon stark singul~iren und hypersingul/iren Integralgleichungen wird mit dem Calder6n-Projektor durchgeffihrt, zu dessen Definition Fundamentall6sungen verwendet werden. Die Diskretisierung des vorher abgeleiteten Systems yon Randintegralgleichungen mit der Galerkin-Randelementmethode wird dargestellt. Schliefllich wird die n~iherungsweise L6sung yon nichtlinearen Problemen unter Verwendung der Galerkin-Randelementmethode am Beispiel einer elasto-plastischen Randwertaufgabe erl/iutert. Eine numerische Testrechnung flit ein Festigkeitsproblem aus dem Maschinenbau wird kurz diskutiert.Introduction to modern 6alerkin-type boundary element methods with an application from mechanical engineering Abstract The Galerkin-type boundary element method (BEM) is an discretization procedure for integral equations, represents itself however compared with classical integral equation methods as an universal tool for the solution of practical engineering problems and can be coupled very easily with finite element substructures. The BEM, whose main advantage lies in the fact that only a surface mesh must be generated, is superior to FEM in special applications, i.e. in elastostatics (notch problems) and fracture mechanics. In this paper the individual steps to solving an elliptical boundary value problem of 3-D linear elasticity theory by way of an equivalent system of boundary integral equations will be explained. For the mathematical investigation of elliptical differential equations and integral equations, the theory of Sobolev spaces has proved to be especially suitable. Basic terms to Sobolev spaces will be introduced so that the reader does not have to refer to textbooks for new terms. The transformation of elliptical boundary value problems to systems of singular and hypersingular integral equations will be explained with help of a Calder6n projector, which is defined by using fundamental solutions. The discretization of the obtained integral equations with the Galerkin-type BEM will be pr...

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“…A more detailed account can be found in Berger, Warnecke and Wendland [5], [6]. We consider the 2-dimensional steady, inviscid, isoenergetic, isentropic (irrotational), flow of an ideal gas.…”

Section: Transonic Potential Flowmentioning

confidence: 99%

A second order finite difference error indicator for adaptive transonic flow computations

Göhner¹,

Warnecke²

1995

Numerische Mathematik

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An adaptive finite element method for the calculation of transonic potential flows was developed. An error indicator based on first order finite differences of gradients is introduced as a local error estimator. It measures second order distributional derivatives. Estimates involving this error estimator, a residual and the error are given. The error estimator can be used as a criterion for mesh refinement. We also give some computational results. (1991): 65N15, 65N30, 35M10 Mathematics Subject Classification

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Analysis of a FEM/BEM coupling method for transonic flow computations

Cited by 16 publications

References 50 publications

Solvability of a problem for transonic flow with a local supersonic region

Solvability of a problem for transonic flow with a local supersonic region

Einführung in moderne Galerkin-Randeiementmethoden mit einer Anwendung aus dem Maschinenbau

A second order finite difference error indicator for adaptive transonic flow computations

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