Here, 4-0H and 4-0L modes are defined as follows. When a bell has formal asymmetrical factors on its circumference, each vibration mode splits into two modes whose vibration frequencies are slightly different from each other. Here, using 4-0 mode vibration as an example, a little bit higher vibration frequency is defined as the 4-0H mode, as well as the 4-0L mode for a little bit lower frequency. The reason for the split can be explained as follows.
the different phases involved in the process, namely the Black-Oil model (see ). Many of the simulation tools used to obtain the solution of the mathematical mod- els use finite differences, see , or finiteelement methods. Other formulations, based on finite volume methods and combinations offiniteelement and finite volume methods, are also available in the literature . In this work, we follow a strategy, see [14, 8], where the parabolic system of PDEs is transformed into a system formed by an elliptic equation for the pressure and a hyperbolic equation for the satura- tion. The latter equation is similar to the well-known Buckley-Leverett equation (first introduced in ). Theoretical analysisof this kind of formulation have been developed in [9, 17]. An interesting work in this study corresponds to the treatment of the two phase model based on a conversion into an elliptic-hyperbolic system, developing both analytical study and numerical solution is . Also in  a numer- ical resolution of a simplified model is performed. In  the authors prove local existence and uniqueness of a classical solution for the original hyperbolic-elliptic system arising in the modeling of oil-recovery processes, using the Arzela-Ascoli theorem. Another relevant contribution concerning existence and uniqueness of the solution in models of filtration of immiscible fluids in a porous a media is . The elliptic (pressure) equation and the hyperbolic (saturation) equation are coupled by means of the saturation which appears in both of them, more precisely, in the coefficients of mobility. The numerical scheme developed in this work consists of solving the pressure equation (the elliptic part of the model) by means of a finiteelementmethod and solving the saturation equation (the hyperbolic part) via a finite volume scheme. The reason behind the use of these two different techniques in the same problem is that finiteelement methods have been devised
This new family of methods provides powerful tools for dealing with problems not easily solvable by the finiteelement methods. Particularly, in aero- nautical engineering problems, MMs are used, e.g., in solving simple analysis, such as loaded beams , thin plates , or thin shells . On the other hand, more specific problems are analyzed, problems dealing with the boundaries offinite ele- ment method, where difficulties appear: crack growth in panels , large deformations, composite plates analysis, etc.
ABSTRACT: When an automobile passes over a bridge dynamic effects are produced in vehicle and structure. In addition, the bridge itself moves when exposed to the wind inducing dynamic effects on the vehicle that have to be considered. The main objective of this work is to understand the influence of the different parameters concerning the vehicle, the bridge, the road roughness or the wind in the comfort and safety of the vehicles when crossing bridges. Non linear finiteelement models are used for structures and multibody dynamic models are employed for vehicles. The interaction between the vehicle and the bridge is considered by contact methods. Road roughness is described by the power spectral density (PSD) proposed by the ISO 8608. To consider that the profiles under right and left wheels are different but not independent, the hypotheses of homogeneity and isotropy are assumed. To generate the wind velocity history along the road the Sandia method is employed. The global problem is solved by means of the finiteelementmethod. First the methodology for modelling the interaction is verified in a benchmark. Following, the case of a vehicle running along a rigid road and subjected to the action of the turbulent wind is analyzed and the road roughness is incorporated in a following step. Finally the flexibility of the bridge is added to the model by making the vehicle run over the structure. The application of this methodology will allow to understand the influence of the different parameters in the comfort and safety of road vehicles crossing wind exposed bridges. Those results will help to recommend measures to make the traffic over bridges more reliable without affecting the structural integrity of the viaduct.
The finiteelementanalysis is made by a plane stress linear static study that allows the consideration of the thickness of the pieces. Wood is considered as an orthotropic material and the values of the elastic properties perpendicular to the grain are achieved by the arithmetic average in the radial and tangential directions. In order to perform the numerical simulation of the joint, each piece is modelled in the ANSYS finiteelement software taking the elementof its internal library called PLANE42. This element is used for two dimensions modelling of solid structures and can be used either as a plane element (plane stress or plane strain) or as an axisymmetric element. The element is defined by four nodes having two degrees of freedom at each node (translations in the nodal x and y directions) and it has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities (Figure 4) .
We present a mixed ﬁnite elementmethod for a class of non-linear Stokes models arising in quasi-Newtonian ﬂuids. Our results include, as a by-product, a new mixed scheme for the linear Stokes equation. The approach is based on the introduction of both the ﬂux and the tensor gradient of the velocity as further unknowns, which yields a twofold saddle point operator equation as the resulting variational formulation. We prove that the continuous and discrete formu- lations are well posed, and derive the associated a priori error analysis. The corresponding Galerkin scheme is deﬁned by using piecewise constant functions and Raviart–Thomas spaces of lowest order.
 Hübner, G.:Eine Betrachtung zur Physik der Schallabstrahlung. Acustica Vol. 75 (1991), S. 130-144  Hübner, G., Messner, J. und Rieger, W.:Schalleistungsbestimmung mit der Direkten Finiten Elemente Methode Schriftenreihe Forschung der Bundesanstalt für Arbeitsschutz, Fb 660, Dortmund 1992, Verlag für neue Wissenschaft, Bremerhaven  Hübner, G.:Erweiterung der DFEM auf allgemein gestaltete Strahler - die Beugung in ihrer Rückwirkung auf abgestrahlte Schalleistungen, Fortschritte der Akustik, Referate der DAGA ‘91, Bochum, 1991, S. 237-240  Hübner, G. ; Gerlach, A.: Determination of the airborne sound power radiated by structure -borne sound sources of arbitrary shape using the Direct FiniteElementMethod - further developments. Conference Proceedings on CD-ROM, 137 th Meeting of the Acoustical Society of America and the 2 nd Convention of the European Acoustics Association: Forum Acusticum integrating the 25 th German Acoustics DAGA Conference, Berlin, March 14-19, 1999  Hübner, G.; Gerlach A.:Schallleistungs- bestimmung mit der DFEM. Forschungsbericht BAU. Dortmund/Berlin 1999.  Gerlach A.:Ein Beitrag zur Erweiterung und Anwendung der Direkten Finiten Elemente Methode zur Bestimmung der abgestrahlten Luftschalleistung dreidimensional ausgedehnter Körperschallquellen. Dissertation. Universität Stuttgart, 2000.  Hübner, G. ; Gerlach, A.:Determination of the airborne sound power radiated by structure-borne sound sources of arbitrary shape - using non-contacting vibration measurements. Proceedings of Inter-Noise, Christchurch, New Zealand, 1998  Hübner, G. ; Gerlach, A.:Zusammenhang der DFEM-Schalleistungs- beschreibung mit der Rayleighschen Schallfelddarstellung ebener Strahler. 24. Jahrestagung für Akustik DAGA '98, Fortschritte der Akustik (DAGA '98), Zürich, 1998, S. 682 – 683  Hübner G.: Script of lectures
In this paper we adapt to the vibration problem the mixed finiteelementmethod proposed and analyzed by Arunakirinathar and Reddy in  for the load problem for elastic curved rods. With this purpose, we settle the corresponding spectral problem by including the mass terms arising from displacement and rotational inertia in the model, as proposed in . Our assumptions on the rods are slightly weaker than those in . On the one hand we allow for non-constant geometric and physical coefficients varying smoothly along the rod. On the other hand, we do not assume that the Frenet basis associated with the line of cross-section centroids is a set of principal axes. We prove that the resulting method yield optimal order approximation of displacements and rotations of the vibration modes, as well as a double order of convergence for the vibration frequencies. Under mild assumptions, we also prove that the error estimates do not degenerate as the thickness becomes small, which allow us to conclude that the method is locking free.
This is the second part of a work dealing with a low-order mixed ﬁnite elementmethod for a class of nonlinear Stokes models arising in quasi-Newtonian ﬂuids. In the ﬁrst part we showed that the resulting variational formulation is given by a twofold saddle point operator equation, and that the corresponding Galerkin scheme becomes well posed with piecewise constant functions and Raviart–Thomas spaces of lowest order as the associated ﬁnite element sub- spaces. In this paper we develop a Bank–Weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed ﬁnite element solutions. Several numerical results illus- trating the eﬃciency of the method are also provided.
The same set of results obtained in the first model is also presented. Figure 3.20 depicts the spatially distributed transfer functions, Figure 3.21 the Fourier spectral amplitude and Figure 3.22 the synthetic seismograms from the three considered methods. The agreement between the results from the full model and the reduced domain method is almost exact, while the results from the classical model exhibit significant differences. The classical model captures only the natural frequency of the microzone, which is in part dominated by the mechanical effect. The classical model however underpredicts the amplitude associated to the natural mode. From the analysis conducted in section 4.1 it is known that near the edges of the canyon large amplifications due to the diffraction effect should be generated. This diffraction effect, which is effectively being captured by the regional model, and subsequently applied to the reduced model seems to be the most relevant aspect of the excitation and the mechanical effect appears to play a minor role in the local response. In a more detailed study a separation of the geometric effects contained in the regional model and the mechanical effects, contained in the micro-zone, should be conducted. That study should provide correction factors to be applied to the standard one-dimensional models to account for the geometric effect.
The dynamic behaviour of short simply-supported railway bridges under convoy cir- culations and, especially, the effect of soil structure interaction (SSI) in the maximum expectable deck transverse response is the aim of this study. These structures due to their usually light weight may experience excessively high acceleration levels under resonant conditions. In order to approach this wave propagation problem, a coupled three-dimensional Boundary Element-FiniteElement model formulated in the time domain is used to reproduce the soil and structural behaviour, respectively. As the resonant phenomenon in this application is highly influenced by the free vibration re- sponse of the deck, a sensitivity analysis is designed in order to first analyse how SSI affects the free vibration response of beams under the circulation of a single moving load in a wide range of velocities. A subset of beam bridges is defined considering span lengths ranging from 12.5 to 25 m, and fundamental frequencies covering asso- ciated typologies. A single soil layer is considered with shear wave velocities ranging from 150 to 365 m/s . From the single load free vibration parametric analysis conclu- sions are derived regarding the conditions of maximum free vibration and cancellation of the response. These conclusions are used afterwards to justify how resonant am- plitudes of the bridge under the circulation of railway convoys are affected by the soil properties, leading to substantially amplified responses or to almost cancelled ones, and numerical examples are included to show the aforementioned situations.
In this work, a new methodology to analyze Fiber Specklegram Sensors by using the FiniteElementMethod (FEM) was presented. In particular, the proposed methodology allowed reconstructing the intensity profile of a SMMF specklegram under controlled conditions of stress and, in turn, the evaluation of Fiber Specklegram Sensors interrogated by optical power variation (PFSS). All the propagation modes supported by a SMMF, for each stress condition, were calculated and superposed for reconstructing perturbed fiber speckle patterns. Then, the performance of the PFSS was evaluated for different radius of filtering fiber and force- gauges. It evidences that, in these types of sensors, metrological characteristics as linearity, sensitivity and dynamic range, can be tuned mechanically, being this an important result for the implementation of any FSS. The analysis allowed for the first time, under a deterministic scheme, the formal identification of design criteria for this kind of measuring systems. Results are in agreement with experimental ones previously reported.
The problem of Darcy flow is of great importance in civil, geotechnical and petroleum engineering. It describes the flow of a fluid through a porous medium. The natural unknowns are the fluid pressure and the fluid velocity, being the latter the unknown of primary interest in many applications. The problem can be reduced to an elliptic equation for the pressure with a Neumann boundary condition. Although this reduced problem can be solved with appropriate accuracy by a classical Galerkin finiteelementmethod, typically there is a loss of accuracy in the approximation of the velocity through the pressure gradients. Moreover, with this reduced formulation, local mass conservation is not guaranteed. For this reason, the primal formulation for the pressure is not considered adequate for practical engineering applications.
On the other hand, the Methodof Images (MOI) is very suitable to solve unbounded problems, which is the reason why it has been used in the analysisof transmission lines (Bracken, 1976), or grounding systems (Dawalibi, 1994). But the method has not been developed for bounded problems, which is a great disadvantage in regards with the bounded methods seen before. This has made the method to be disregarded even though it has good features like speed of solution or implementation simplicity.
As a recent example of the above, we recall here that in  and  we introduce and analyse a dual-mixed formulation for a class of quasi-Newtonian Stokes flows whose kinematic viscosities are nonlinear monotone functions of the gradient of the velocity. The mixed finiteelementmethod proposed there simply relies on the introduction of the flux and the tensor gradient of the velocity as auxiliary unknowns, which yields a two-fold saddle point operator equation as the resulting variational formulation. Therefore, the abstract theory developed in , which is a slight generalization of the well known Babuˇ ska-Brezzi theory, is applied to prove that the continuous and discrete schemes are well posed. In particular, it is shown that the stability of the Galerkin scheme only requires low-order finiteelement subspaces: it suffices to use Raviart–Thomas spaces of order zero to approximate the flux and piecewise constant functions to approximate the other unknowns. In addition, since the monotonicity certainly includes the linear case, we also obtain as a by-product a new mixed finiteelementmethod for the linear Stokes equation (problem (1.1) with α = 0).
The numerical technique used for the evaluation of the hydraulic head loss caused by the WSBs layer is the one presented in section 2.2.1 for the analysisof free surface and seepage flow. This analogy between rock- fill and blocks is explained in Fig. 6: in the above part of the figure, the homogenization procedure used for the flow through rockfill is presented and the analogous consideration for the blocks is shown in the lower part. The blocks layer and their orifices are considered as a continuum porous layer and the hydraulic head loss (i) is quantified considering a Darcy type resistance law i = A v, being v the velocity and A the Darcy’s coefficient only depending of the size and shape of the block and defining the permeability of the layer.
calculated sequentially in a time step of 0.01 µ s . As the boundary condition ofanalysis areas, the 2nd-order Higdon boundary operator was accommodated on the boundaries to make them non reflective. The sound source was assumed to be a group of point sources within a space of 10 mm, to be perfectly reflective and to have the piston action. The values used in the calculation are shown in Table 1. As the initial condition, a single sinusoidal wave of 1 MHz is set to the stress tensors ( σ xx and σ yy ).
Since the fluid is supposed to be inviscid, only the normal component of displacements vanishes on Γ W , namely, U F ⋅ = ν 0 on Γ ∩ ∂Ω W F , whereas for boundary displacement of porous medium we suppose U A = 0 on . Γ ∩ ∂Ω W A Similarly, on interface Γ I between fluid and porous medium we consider the usual interface conditions of continuity of forces and normal displacements, that is,