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

Introduction to modern galerkin-type boundary element methods with an application from mechanical engineering

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Zusammenfassung

Die Galerkin-Randelementmethode ist ein Näherungsverfahren für Integralgleichungen, stellt jedoch gegenüber klassischen Integralgleichungsverfahren ein universelles Werkzeug zur Lösung praktischer Ingenieurprobleme dar und kann sehr gut mit Finite-Element-Substrukturen gekoppelt werden. Die Randelementmethode, bei der als Hauptvorteil nur ein Oberflächennetz generiert werden muß, ist bei speziellen Anwendungen beispielsweise für Kerb- und Rißprobleme der FEM überlegen. Die einzelnen Schritte zur Lösung eines elliptischen Randwertproblems über ein System von Randintegralgleichungen wird am Beispiel des dreidimensionalen linearen Elastizitätsproblems erläutert. Zur mathematischen Untersuchung von elliptischen Differentialgleichungen und äquivalenten Integralgleichungen hat sich die Theorie der Sobolev-Räume als besonders geeignet herausgestellt. Grundbegriffe zu Sobolev-Räumen werden eingeführt, so daß der Leser nicht in Lehrbüchern nachschlagen muß. Die Überführung der elliptischen Randwertprobleme auf Systeme von stark singulären und hypersingulären Integralgleichungen wird mit dem Calderon-Projektor durchgeführt, zu dessen Definition Fun-damentallösungen verwendet werden. Die Diskretisierung des vorher abgeleiteten Systems von Randintegralgleichungen mit der Galerkin-Randelementmethode wird dargestellt. Schließlich wird die näherungsweise Lösung von nichtlinearen Problemen unter Verwendung der Galerkin-Randelementmethode am Beispiel einer elastopla-stischen Randwertaufgabe erläutert. Eine numerische Testrechnung für ein Festigkeitsproblem aus dem Maschinenbau wird kurz diskutiert.

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 Calderón projector, which is defined by using fundamental solutions. The discretization of the obtained integral equations with the Galerkin-type BEM will be presented. Finally the approximation of non-linear problems by using the Galerkin-type BEM will be shown. A numerical test for a strength problem will be discussed shortly.

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Abbreviations

ℕ:

Menge der natürlichen Zahlen

ℝ:

Menge der reellen Zahlen

ℝ+ :

Menge der nichtnegativen reellen Zahlen

ℂ:

Menge der komplexen Zahlen

C l :

Raum der l-mal stetig differenzierbaren, beschränkten Funktionen (Auch alle Ableitungen seien beschränkt.)

C l,λ :

Raum der l-mal stetig differenzierbaren, beschränkten Funktionen, deren l-te Ableitung λ-hölderstetig ist.

D(Ω)=C ∞0 (Ω):

L. Schwartzsche Grundfunktionen: Menge aller fmiten, unendlich oft differenzierbaren Funktionen mit einem Träger in Ω (Ω offen)

D′(Ω):

Raum der L. Schwartzschen Distributionen

ℰl(Ω):

l-mal stetig differenzierbare Funktionen auf ω

ℰ′(Ω):

Raum der stetigen linearen Funktionale auf ℰ(Ω)

L p :

Raum der lebesguemeßbaren Funktionen

L loc p :

Menge der lokal (auf jedem Kompaktum) lebesguemeßbaren Funktionen

W l2 :

Sobolev-Slobodeckij-Raum

H l :

mit Fourier-Transformation definierter Sobolev-Raum(Hilbert-Raum)

DC m :

Raum der Cauchy-Daten m-ter Ordnung

N k,κ :

Regularitätseigenschaft eines Gebietes

D s :

partielle Ableitungen im verallgemeinerten Sinne, wobei s=(s 1, s 2,..., s n) ein Multiindex ist

γ:

Spuroperator; \(\gamma _0 \phi = \phi = \phi |_{\partial \Omega } ,\gamma _m \phi = \left( {\phi |_{\partial \Omega } ,\frac{{\partial \phi }}{{\partial n}}|_{\partial \Omega } ,...\frac{{\partial ^m \phi }}{{\partial n^m }}|_{\partial \Omega } } \right)\)

Z:

Fortsetzungsoperator

ℱ :

Fourier-Transformation

L :

Fréchet-Raum

δ:

Laplace-Operator

δ*:

Navier-Operator

δ 0 :

Diracsche Deltadistribution

δ ij :

Kronecker-Symbol

t=(t 1, t 2, t 3)T :

Randspannungsvektor

u=(u 1, u 2, u 3)T :

Verschiebungsvektor

ε ij :

Verzerrungstensor

σ ij :

Spannungstensor

ε mnl :

Levi-Civita-Tensor

\(\hat D = (D_{ijkl} )\) :

Tensor 4. Stufe der Elastizitätskonstanten

λ, μ :

Lamé-Konstanten

g ij :

Metriktensor im Riemann-Raum

J(.):

Jacobi-Determinante

x=(x 1,x 2,x 3)T :

Ortsvektor

n:

äußerer Einheitsnormalenvektor

Ω:

Gebiet

Ωc :

Komplementärmenge zu Ω

Γ=∂Ω:

Rand des Gebietes

K :

Kompaktum (beschränkte und abgeschlossene Menge)

K n :

n-dimensionaler verallgemeinerter Einheitswürfel

M :

Mannigfaltigkeit

U :

Umgebung

B ɛ (x):

Kugel mit dem Mittelpunkt ξ ∈ ℝ3 und dem Radius ɛ

C 1 :

Calderon-Projektor für das Innengebiet

h :

Gitterparameter

H :

Grobgitterparameter

I:

Einheitsoperator

P:

Differentialoperator des Randwertproblems

R1, R2 :

Matrizen von (tangentialen) Differentialoperatoren

S(x, y − x):

hypersinguläre Fundamentallösung

T(x, y − x):

singuläre Fundamentallösung

U(x, y − x):

schwach singuläre Fundamentallösung

D:

hypersingulärer Integraloperator

K:

singulärer Integraloperator

V:

schwach singulärer Integraloperator

℘h :

orthogonale Projektion auf V h

S l :

Oberflächenstück mit der Nummer l

V l :

Parametergebiet mit der Nummer l

\(\hat \pi \) :

Referenzelement: \(\hat \pi = \left\{ {z \in \mathbb{R}^2 :0 < z_1 < 1,0 < z_2 < z_1 } \right\}\)

π hi :

geometrisches Randelement mit der Nummer i

ξ(i) :

bijektive Abbildungsfunktion des Refe renzelements \(\hat \pi \) auf das Originalelement π hi

N (n) t , N (n) r :

Basisfunktionen bzgl. des Knotens n

A pq,ij :

verallgemeinerte Elementsteifigkeitsmatrix

N1, N2 :

Formfunktionsmatrizen

M ij :

Massematrix

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  1. Institut für Technische Mechanik, Universität Karlsruhe, Postfach 6980, D-76128, Karlsruhe, Germany

    H. Andrä

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Ich möchte besonders den Herren Prof. Dr.-Ing. A. Albers, Prof. Dr. U. Langer, Prof. Dr.-Ing. E. Schnack, Prof. Dr.-Ing. W.L. Wendland und Prof. Dr. J.R. Whiteman für die zahlreichen wertvollen Hinweise und Anregungen herzlich danken, die in der vorliegenden Arbeit eingeflossen sind.

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Andrä, H. Einführung in moderne Galerkin-Randeiementmethoden mit einer Anwendung aus dem Maschinenbau. Forsch Ing-Wes 65, 58–90 (1999). https://doi.org/10.1007/PL00010775

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