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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.
By means of a small-scale truss bridge, the ability of the Measurement- and Model-based Structural Analysis to detect and localise damage was examined in. Although there was no noteworthy difficulty in detecting damage, it turned out that damage localisation responds sensitively to systematic influences, i.e. non-modelled properties of the mechanical model. Therefore, another experiment is being conducted to re-examine the Measurement- and Model-based Structural Analysis. For this purpose, the bending test is carried out as it has been already theoretically respectively numerically discussed in. In this attempt, the systematic influences such as residual stress are kept as low as possible.
By means of a small-scale truss bridge, the ability of the Measurement- and Model-based Structural Analysis to detect and localize damage was examined. Although there was no noteworthy difficulty in detecting damage, it turned out that damage localization responds sensitively to systematic influences, i.e. non-modelled properties of the mechanical model. Therefore, another experiment is being conducted to re-examine the Measurement- and Model-based Structural Analysis. For this purpose, the bending test is carried out as it has been already theoretically respectively numerically discussed. In this attempt, the systematic influences such as residual stress are kept as low as possible.
The Measurement- and Model-based Structural Analysis (MeMoS) integrates a finite element model into least squares adjustment and thus allows to evaluate a mechanical model and measurements in a combined analysis. To examine the capability to detect and localise damage using this integrated analysis MeMoS, a small-scale truss bridge made of aluminium profiles is built as a test specimen for this purpose.
To examine the capability to detect and localise damage using the Measurement- and Model-based Structural Analysis (MeMoS), a small-scale truss bridge (1520 mm × 720 mm × 720 mm) made of aluminium profiles is built as a test specimen for this purpose. The truss frame of the test bridge is made of aluminium profiles with a sophisticated design of the cross-sectional area. In comparison, with solid profiles, only a fraction of the material is needed to produce the profiles, while their bending resistance decreases slightly. The profiles are built into a truss frame by connecting them by means of fastening sets made of steel. The bridge model is mounted on four steel bearings which each of them consist of a cylinder arranged between two plates. Fixed bearings are made by holding onto one end of the bridge. The bridge is subjected by an external load by placing a heavy object beneath it. At the same time, measurements can be conducted below the bridge. Therefore, the bridge specimen is elevated by attaching it on a pedestal with four columns. Damages can be induced by loosening the fastening pieces.
Many engineering structures are nowadays made of composite materials or metal foam. These modern engineering materials contain very complex inner geometry. 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. In this paper a numerical method is proposed to find an approximate substitute model for geometrical complex structures.
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.
The determination of material parameters from displacement field measurement is being examined for linear elastic solid. A frequently used approach to compute material constants can be found in many studies. Even though they presented the approach in many different variations, but in the end they are essentially based on the same algorithm: Parameters are iteratively tuned until the computed results are in accordance with the measurements. The main drawback of this approach is that mainly commercial software is used that hinders us to investigate its inner evaluation process. This leads to the question, how the results from this commercial software can be trusted. On the contrary to these debatable approaches, we present a method that inverts the procedure of finite element method by using the most general model for a least-squares adjustment – the GAUSS-HELMERT Model.