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.

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the different phases involved in the process, namely the Black-Oil model (see [21]). Many **of** the simulation tools used to obtain the solution **of** the mathematical mod- els use **finite** differences, see [15], or **finite** **element** methods. Other formulations, based on **finite** volume methods and combinations **of** **finite** **element** and **finite** volume methods, are also available in the literature [12]. 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 [6]). Theoretical **analysis** **of** 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 [9]. Also in [9] a numer- ical resolution **of** a simplified model is performed. In [17] 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 [4]. 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 **finite** **element** **method** 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 **finite** **element** methods have been devised

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This new family **of** methods provides powerful tools for dealing with problems not easily solvable **by** the **finite** **element** methods. Particularly, in aero- nautical engineering problems, MMs are used, e.g., in solving simple **analysis**, such as loaded beams [11], thin plates [12], or thin shells [13]. On the other hand, more specific problems are analyzed, problems dealing with the boundaries **of** **finite** ele- ment **method**, where difficulties appear: crack growth in panels [14], large deformations, composite plates **analysis**, etc.

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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 **finite** **element** 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 **finite** **element** **method**. 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.

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The **finite** **element** **analysis** 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 **finite** **element** software taking the **element** **of** 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) [4].

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We present a mixed ﬁnite **element** **method** 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.

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[1] Hübner, G.:Eine Betrachtung zur Physik der Schallabstrahlung. Acustica Vol. 75 (1991), S. 130-144 [2] 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 [3] 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 [4] Hübner, G. ; Gerlach, A.: Determination **of** the airborne sound power radiated **by** structure -borne sound sources **of** arbitrary shape using the Direct **Finite** **Element** **Method** - 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 [5] Hübner, G.; Gerlach A.:Schallleistungs- bestimmung mit der DFEM. Forschungsbericht BAU. Dortmund/Berlin 1999. [6] 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. [7] 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 [8] 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 [9] Hübner G.: Script **of** lectures

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Fault activation is gener- ally caused by stress concentration at the fault tip, so in this study, a computational model of a typical underground stope with a hidden [r]

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In this paper we adapt to the **vibration** problem the mixed **finite** **element** **method** proposed and analyzed **by** Arunakirinathar and Reddy in [2] 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 [10]. Our assumptions on the rods are slightly weaker than those in [2]. 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.

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This is the second part **of** a work dealing with a low-order mixed ﬁnite **element** **method** 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.

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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.

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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**-**Finite** **Element** 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.

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In this work, a new methodology to analyze Fiber Specklegram Sensors **by** using the **Finite** **Element** **Method** (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.

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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 **finite** **element** **method**, 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.

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On the other hand, the **Method** **of** Images (MOI) is very suitable to solve unbounded problems, which is the reason why it has been used in the **analysis** **of** 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.

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As a recent example **of** the above, we recall here that in [12] and [13] 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 **finite** **element** **method** 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 [11], 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 **finite** **element** 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 **finite** **element** **method** for the linear Stokes equation (problem (1.1) with α = 0).

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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 **analysis** **of** 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.

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calculated sequentially in a time step **of** 0.01 µ s . As the boundary condition **of** **analysis** 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 ).

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Esto explica como a pesar de que el método de elementos finitos FEM (Finite Element Method) - considerado una de las herramientas más poderosas - no resulta muy útil en esto[r]

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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,

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