Finite element method (FEM)

An application of the Finite Element Method (FEM) to the solution of a geometric problem is shown. The problem is related to curve fitting i.e. pass a curve trough a set of given points even if they are irregularly spaced. Situations where cur ves with cusps can be encountered in the practice and therefore smooth interpolatting curves may be unsuitable. In this paper the possibilities of the FEM to deal with this type of problems are shown. A particular example of application to road planning is discussed. In this case the funcional to be minimized should express the unpleasent effects of the road traveller. Some comparative numerical examples are also given.

The most popular approach in applications is based on the mixed formulation, with pressure and velocity as unknowns. It is well-known that the Galerkin scheme associated to this formulation is not always well-posed and stability is ensured only for certain combinations of finite element subspaces. In this framework, several stabilization methods have been proposed in the literature. We consider a stabilized mixed finite element method introduced by Masud and Hughes in [15] for isotropic porous media. This method is based on the addition of suitable residual type terms to the standard dual-mixed approach. The resulting scheme is stable for any combination of continuous velocity and pressure interpolations, and has the singularity that the stabilization parameters can be chosen independently of the mesh size. This property was already present in the modified mixed formulation introduced in [3] for second order elliptic problems. The stabilization introduced in [15] was also applied in [4] to analyze a mixed discontinuous Galerkin method for Darcy flow. A similar idea is used in [7] to derive several unconditionally stable mixed finite element methods for Darcy flow. Finally, concerning the a posteriori error analysis of the method proposed in [15], a residual based a posteriori error estimate of the velocity in L 2 -norm was derived in [13].

We present and analyse a new mixed finite element method for the generalized Stokes prob- lem. The approach, which is a natural extension of a previous procedure applied to quasi- Newtonian Stokes flows, is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two-fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babuˇ ska-Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. In particular, the finite element subspaces providing stability coincide with those employed for the usual Stokes flows except for one of them that needs to be suitably enriched. We also develop an a-posteriori error estimate (based on local problems) and propose the associated adaptive algorithm to com- pute the finite element solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities.

In general, any semi-discrete dynamics generates spurious high-frequency oscillations that do not exist at the continuous level. Moreover, a dispersion phenomenon appears and the velocity of propagation of these high frequency numerical waves may converge to zero when the mesh size tends to zero. Note that these spurious oscillations correspond to the high frequencies of the discrete model and therefore, they weakly converge to zero when the discretization parameter h does. Consequently, their existence is compatible with the convergence of the numerical scheme. However, when we are dealing with the exact controllability problem, an uniform time for the control of all numerical waves is needed. Since the velocity of propagation of some high frequency numerical waves may tend to zero with the mesh size, the uniform controllability properties of the semi-discrete model may eventually disappear for a fixed time T > 0. This is the case when the semi-discrete model is obtained by finite differences or the classical finite element method (see [8],[19] for a detailed analysis of the 1-D case and [18] for the 2-D case, in the context of the dual observability problem).

ABSTRACT: This communication deals with the numerical solution of the acoustical behavior of elastic porous materials. Assuming a periodic structure, we use new poroelastic models obtained by homogenization techniques. In order to compute the coefficients in these new models, we solve boundary-value problems in the unitary cell. Finally, we focus our attention on non-dissipative poroelastic materials with open pore and propose a finite element method in order to compute the response to a harmonic excitation of a three-dimensional enclosure containing a free fluid and a poroelastic material. The finite element used for the fluid is the lowest order face element introduced by Raviart and Thomas that avoids the spurious modes whereas, for displacements in porous medium, the “mini element” is used in order to achieve stability of the method.

As a first approximation for 2D case, Eq. (17) can be solved as an infinite waveguide where the refraction index is supposed constant along the propagation axis ( z -direction). Here, the electric field can be computed by FEM (See more details in [32-34]). To calculate the field by FEM, the fiber cross-section is discretized into small elements and Eq. (17) is computed for each element (See Fig. 1(b) and (c)). After calculating the electric field E for each node, and for the M -modes supported by a SMMF, M  2  a NA   2 /  2 with a

The FEM analysis model of the bell was structured as follows. First, its two dimensional FEM cross sectional shape was made. The full model was configured so that its cross sectional shape was rotated by 360 degrees. The actual bell shape of the Hojobo Temple (in Isehara, Kanagawa Prefecture, Japan) was utilized as the analysis model [1][2] .

makes use of an Integral Equation (IE) representation of the electromagnetic field in the exterior región (out of the truncated domain). However, in contrast to conventional FEM-IE approaches, this is done in such a way that the original sparse structure of the FEM matrices is retained. FE-IIEE has demonstrated to be suitable for hp-adaptivity [7], as it provides an adjustable, and arbitrarily exact, radiation boundary condition for the wave propagation problem. For further reading the interested reader is referred to [8] in which a comparison study between FE-IIEE and other truncation techniques suitable for hp-adaptive methods such as the use of infinite elements [9] or the use of Perfect matched Layers (PML) [10], is presented. FE- IIEE key advantages are achieved at the expense of performing a few iterations in which the exterior field is calculated using the IE expressions and the Green's function of the exterior problem. As it will be clear later, the convolutional character (double loop) of these computations leads to a computational complexity of 0(N 2 ), being N the number of unknowns

time, many systems of ordinary differential equations (ODEs); and (ii) the calculation of some integrals, which come from the Galerkin projection, whose integrands are the product of functions defined in two different meshes. The first shortcoming is in some way related to the second because the integrals have to be computed exactly, but in general it cannot be done this way and they have to be numerically calculated with high accuracy to keep the method stable; see, in this respect, [2] where a study on the behavior of the method with different quadrature rules is performed. The use of high order quadrature rules means that many quadrature points per element should be employed to evaluate the integrals, and, therefore, since each quadrature point has an associated departure point, many systems of ODEs have to be solved numerically at every time step; hence, the whole procedure may become less efficient than it looks at first, in particular when working in unstructured meshes, because the numerical calculation of the feet of the characteristic curves requires locating and identifying of the elements containing such points, and this task is not easy to do in such meshes.

A new stabilized mixed finite element method for plane linear elasticity was presented and analyzed recently in [10]. The approach there is based on the introduction of suitable Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement. The resulting augmented method, which is easily generalized to 3D, can be viewed as an extension to the elasticity problem of the non-symmetric procedures utilized in [8] and [11]. It is shown in [10] that the continuous and discrete augmented formulations are well-posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions, respectively. Moreover, the augmented variational formulation introduced in [10], being strongly coercive in the case of Dirichlet boundary conditions, allows the utilization of arbitrary finite element subspaces for the corresponding discrete scheme, which constitutes one of its main advantages. In particular, Raviart-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the rotation can be used. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which yields the introduction of

The finite element method (FEM) has developed rapidly as a topic ofresearch in electrical engineering. It is a numerical method that can be used to calculate the electromagnetic fields, electromagnetic forces, energy storage in electromagnetic fields, power dissipation, eddy current, thermal conduction etc. Calculation ofthe transmission line impedance using the FEM requires that an appropriate boundary be defined first. Ali bodies (including the air and the ground) included in the boundary are modeled as areas. Each area has its own geo­ metric shape and physical properties, such as conductivity, permeability, nonlinearity ofmagnetization etc. Ali areas are subdivided into first order triangular finite elements. The boundary conditions must then be given to allow Maxwell's equations to be solved correctly according to the nature ofthe problem.

A Fixed Grid Finite Element Method (FGFEM) in addition to a Mooney-Rivlin hyper- elastic model is introduced. This method uses the mixed finite element formulation to treat the elements, but classifying these in a fixed cartesian grid that is superimposed over the model geometry. In order to do this, the boundary tracking is achieved by solving the level set equation. A numerical extrapolation of the displacement field from the solid domain to the entire fixed grid domain is done. The system of equations are solved by the use of an incremental Newton-Raphson scheme. Finally, some numerical examples are implemented and good convergence results are obtained for the displacement field, showing that FGFEM for the hyperelastic model is suitable for mechanical problems undergoing large strains and large displacements.

[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

High-order accurate and unconditionally stable time-discontinuous methods are implemented with nonreflecting boundary conditions in an adaptive space-time finite element method for acoustic radiation and scattering problems in exterior domains. An h-adaptive space-time procedure based on the Z 2 error estimate and the superconvergent patch recovery (SPR) technique, together with a temporal error estimate arising from the discontinuous jump in solution between time steps is used to maintain accuracy within a prescribed tolerance and drive dynamic mesh distributions. Error estimates of the nonreflecting boundaries are also monitored in the solution process. A new superconvergent interpolation method is developed for projection between adaptive meshes. Numerical studies of time-dependent scattering from an ellipse demonstrate the efficiency and reliability gained from the adaptive solution.

The Method of Moments is a numerical analysis technique that is used to solve the Maxwell Equations, like other numerical methods such as the Finite Element Method but, unlike this last one, which determines the electric field in volumetric elements solving the Electric Field Differential Equation, the Method of Moments solves the Electric Field Integral Equation obtaining the surface current in triangular elements, and when the current distribution in the object is known, then the total electric field in any point of the space can be obtained. In this TFG it will be explained in detail how to get to the Electric Field Integral Equation, starting from Maxwell’s Electromagnetic Equations in the frequency domain, as well as how to numerically solve the problem to sufficiently well approximated results. Since a direct approach to the topic might be of high complexity, a brief introduction of the Method of Moments is given, and an electrostatic problem is also solved, as a demonstration.

Analytical solutions of the Helmholtz equation can only be obtained for very simple enclosure geometries. So, to be able to accurately calculate the sound field in arbitrary shaped enclosures, one has to resort to numerical methods. One of the most widely known methods for this purpose is the Finite Element Method (FEM) [4, 6].

ABSTRACT: The aim of this study was to analyse the stress distribution in maxillary canines restored with different post systems and definitive crowns. The models of restored teeth with glass fiber, quartz fiber, titanium posts and crowns were developed with the Finite Element Method (FEM) in order to analyse their stress distribution when subjected to external compressive loads. Von Mises stress distribution values, which are considered potential fracture indicator, showed that natural tooth and glass fiber post-restored tooth, under a load of 550 N, presented similar stress values. The behaviour of a glass fiber post-restored tooth is similar to that of a natural tooth, since it produces an appropriate stress distribution, and in this investigation, they have the best biomechanical performance.

Electrostatics is an important subject of Electromagnetic Theory that contributes to the understanding of complex phenomena and industrial applications such as high voltage breakdown (Abdel, 2018), aerosol particles (Kawada, 2002), or the analysis of molecular surfaces (Bulat, 2010). For this reason, some numerical techniques to solve electrostatics problems have been developed and are still under investigation. See, for example (Tausch, 2001) for the Method of Moments, (Karkkainen, 2001) for the Finite Difference Method or (Hamou, 2015) for the Finite Element Method. The Finite Element Method (FEM) is the most used method in electromagnetics research as it may be seen in the number of publications found in the scientific community. It solves the Laplace equation, taken as a boundary-value problem (Jin, 2017). However, this very characteristic of being able to solve bounded problems is, sometimes, its weakest feature, because

As in many scientific and technological fields, piezoelectric studies started from the necessity of develop practical devices: the initial emphasis in the decade of 1960 was in applicability [?]. While the electrical and mechanical basics were rapidly understood, more advanced issues (e.g. numerical solutions, nonlinearities) were dealt with in the 1990s [?]. From a mechanical point of view piezoelectric devices have been extensively studied for both small and large strains. For the former, the authors of the present work have published several articles using finite element methods (FE): [?], [?], [?], [?] and [?]. For the latter, polyconvex approaches are reported in [?]. Other numerical techniques such as the boundary element method have been applied to model small strain piezoelectrics, see [?].

As it was said in the previous sections, a mixed formulation usually rep- resents a better and a more efficient way to predict the behaviour of shell elements. Therefore, a 4-node mixed shell element has been chosen to be im- plemented, see Figure 7.6. The element has been proposed by K.J. Bathe and E.N. Devorkin in reference (DB84), and exhibits the following charac- teristics: “(i) The element is able to represent the six rigid body modes, (ii) it also can approximate the Kirchhoff-Love hypothesis of negligible shear de- formation effects and can be used for thin shells, and (iii) the element does not contain spurious energy modes”(DB84). When talking about the six rigid