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Integration der Finite-Elemente-Methode in die Ausgleichsrechnung zur Parameteridentifikation
(2014)
Integration der Finite-Elemente-Methode in die Ausgleichsrechnung zur Parameteridentifikation
(2014)
Die Strukturüberwachung von Ingenieurbauwerken beruht heutzutage auf einer Auswertung räumlich und zeitlich verteilter hybrider Messungen, die z. B. mittels Tachymeter, Neigungssensoren, faseroptischen Sensoren (FOS), Dehnmessstreifen (DMS), GPS etc. erfasst werden. Für eine gemeinsame Auswertung müssen neue Methoden adaptiert werden, da diese, wie Lienhart (2012) aufzeigt, nur unter Verwendung eines mechanischen ‘Bauwerkmodells erfolgen kann.
In vielen Ingenieurwissenschaften, wie z. B. dem Bauingenieurwesen, findet die Modellierung physikalisch-mechanischer Eigenschaften von Strukturen mithilfe der Finite-Elemente-Methode (FEM) statt. Die Verifizierung eines derartigen Modells erfolgt vorwiegend lediglich durch stellenweise Messung von z. B. Durchbiegungen und einer anschließenden Gegenüberstellung mit den berechneten Modellwerten. Dies ist meist der Tatsache geschuldet, dass für die FE-Modellierung in der Regel kommerzielle Programme verwendet werden, und somit auf viele Teilprozesse des Auswertealgorithmus nicht zugegriffen werden kann. Aus diesem Grund erfolgt in vielen akademischen Fragestellungen die FE-’Modellierung mit Open-Source-Software, wie z. B. FEniCS (2013) oder OpenSees (2013), wodurch auch eine kombinierte Auswertung von Messungen und Modell nach der Methode 'der kleinsten Quadrate ermöglicht wird.
In diesem Beitrag wird eine messungs- und modellbasierte Strukturanalyse (MeMoS) durch (die Integration der Finite-Elemente-Methode in die Ausgleichungsrechnung am Beispiel eines Vier-Punkt-Biegeversuchs vorgestellt. In numerischen Untersuchungen wird gezeigt, wie diese integrierte Analyse für eine Parameteridentifikation angewendet werden kann. Für diese Untersuchungen wird ein Finite-Elemente-Modell mit bekannten Randbedingungen und Materialeigenschaften aufgestellt. Die Durchbiegungen, die als Beobachtungen in die Ausgleichung eingehen, werden mithilfe von Simulationsrechnungen erzeugt; der zu fidentifizierende Parameter ist der Elastizitätsmodul eines Balkens.
Es wird untersucht, mit welcher Genauigkeit Durchbiegungsmessungen durchgeführt werden müssen und an welcher Stelle des Bauwerks diese Messungen erfolgen sollen, um den Elastizitätsmodul möglichst genau zu bestimmen. Des Weiteren wird der Einfluss der Anzahl der Messstellen auf den zu identifizierenden Parameter untersucht.
The finite volume method (FVM), like the finite element method (FEM), is a numerical method for determining an approximate solution for partial differential equations. The derivation of the two methods is based on very different considerations, as they have historically evolved from two distinct engineering disciplines, namely solid mechanics and fluid mechanics. This makes FVM difficult to learn for someone familiar with FEM. In this paper we want to show that a slight modification of the FEM procedure leads to an alternative derivation of the FVM. Both numerical methods are starting from the same strong formulation of the problem represented by differential equations, which are only satisfied by their exact solution. For an approximation of the exact solution, the strong formulation must be converted to a so-called weak form. From here on, the two numerical methods differ. By appropriate choice of the trial function and the test function, we can obtain different numerical methods for solving the weak formulation of the problem. While typically in FEM the basis functions of the trial function and test function are identical, in FVM they are chosen differently. In this paper, we show which trial and test function must be chosen to derive the FVM alternatively: The trial function of the FVM is a “shifted” trial function of the FEM, where the nodal points are now located in the middle of an integration interval rather than at the ends. Moreover, the basis functions of the test function are no longer the same as those of the trial function as in the FEM, but are shown to be a constant equal to 1. This is demonstrated by the example of a 1D Poisson equation.
The best-known discretization methods for solving engineering problems formulated as partial differential equations are finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). While the finite volume method is used in fluid mechanics, the finite element method is predominant in solid state mechanics. At first glance, FVM and FEM are two highly specialized methods. However, both methods can solve problems of both solid mechanics and fluid mechanics well. Since experimental mechanics deals not only with solid state physics but also with fluid mechanics problems, we want to understand FVM in the sense of FEM in this work. In the long term, we want to use the variational calculus to unify many important numerical methods in engineering science into a common framework. In this way, we expect that experiences can be better exchanged between different engineering sciences and thus innovations in the field of experimental mechanics can be advanced. But in this work, we limit ourselves to the understanding of the FVM with the help of the variational calculus already known in FEM. We use a simple 1D Poisson equation to clarify the point. First, we briefly summarize the FVM and FEM. Then we will deal with the actual topic of this paper, as we establish the FEM and the FVM on a common basis by variation formulation. It is shown here that the FVM can be understood in terms of the finite element method with the so-called Galerkin-Petrov approach.
The best-known discretization methods for solving engineering problems formulated as partial differential equations are finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). While the finite volume method is used in fluid mechanics, the finite element method is predominant in solid state mechanics. At first glance, FVM and FEM are two highly specialized methods. However, both methods can solve problems of both solid mechanics and fluid mechanics well. Since experimental mechanics deals not only with solid state physics but also with fluid mechanics problems, we want to understand FVM in the sense of FEM in this work. In the long term, we want to use the variational calculus to unify many important numerical methods in engineering science into a common framework. In this way, we expect that experiences can be better exchanged between different engineering sciences and thus innovations in the field of experimental mechanics can be advanced. But in this work, we limit ourselves to the understanding of the FVM with the help of the variational calculus already known in FEM. We use a simple 1D Poisson equation to clarify the point. First, we briefly summarize the FVM and FEM. Then we will deal with the actual topic of this paper, as we establish the FEM and the FVM on a common basis by variation formulation. It is shown here that the FVM can be understood in terms of the finite element method with the so-called Galerkin-Petrov approach.
Many engineering structures are made of composite materials or metal foam. To simulate the deformational behaviour of these structures often requires a high number of discretisation elements. This in turn yields a very large system of linear equations that are extremely time and memory consuming or practically impossible to solve. It is therefore desirable to find an approach to overcome this obstacle.
Many engineering structures are made of composite materials or metal foam. To simulate the deformational behaviour of these structures often requires a high number of discretisation elements. This in turn yields a very large system of linear equations that are extremely time and memory consuming or practically impossible to solve. It is therefore desirable to find an approach to overcome this obstacle.
Der Grad der Finite-Elemente-Diskretisierung wird vom Verhältnis der Details zur Objektgröße bestimmt. Die Diskretisierung eines großen Objektes mit vielen kleinen Details führt zu einer hohen Anzahl an Elementen bzw. Knotenpunkten. Die Berechnung solcher Körper erfordern nicht nur sehr hohe Rechenzeit, sondern was die Berechnung unmöglich macht, ist der sehr hohe Speicherbedarf. Mit Hilfe eines Ersatzkörpers wird dieses Problem umgangen.