Finite element method

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TítuloA posteriori error analysis of an augmented mixed finite element method for Darcy flow

TítuloA posteriori error analysis of an augmented mixed finite element method for Darcy flow

We develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for Darcy flow. The stabilized formulation is obtained by adding to the standard dual- mixed approach suitable residual type terms arising from Darcy’s law and the mass conservation equation. We derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding Galerkin scheme are well-posed. Then, we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient. Finally, we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice.
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20 Lee mas

TítuloA mixed finite element method for the generalized Stokes problem

TítuloA mixed finite element method for the generalized Stokes problem

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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27 Lee mas

Self-adaptive hp finite element method with iterative mesh truncation technique accelerated with Adaptive Cross Approximation

Self-adaptive hp finite element method with iterative mesh truncation technique accelerated with Adaptive Cross Approximation

To alleviate the computational bottleneck of a powerful two-dimensional self-adaptive hp finite element method (FEM) for the analysis of open región problems, which uses an iterative computation of the Integral Equation over a fictitious boundary for truncating the FEM domain, we propose the use of Adaptive Cross Approximation (ACÁ) to effectively accelerate the computation of the Integral Equation. It will be shown that in this context ACÁ exhibits a robust behavior, yields good accuracy and compression levéis up to 90%, and provides a good fair control of the approximants, which is a crucial advantage for hp adaptivity. Theoretical and empirical results of performance (computational complexity) comparing the accelerated and non-accelerated versions of the method are presented. Several canonical scenarios are addressed to resemble the behavior of ACÁ with h, p and hp adaptive strategies, and higher order methods in general.
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22 Lee mas

Solving New Poroelastic Models by Finite Element Method

Solving New Poroelastic Models by Finite Element Method

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.
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8 Lee mas

BOUNDARY CONTROLLABILITY OF A LINEAR SEMI-DISCRETE 1-D WAVE EQUATION DERIVED FROM A MIXED FINITE ELEMENT METHOD

BOUNDARY CONTROLLABILITY OF A LINEAR SEMI-DISCRETE 1-D WAVE EQUATION DERIVED FROM A MIXED FINITE ELEMENT METHOD

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).
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46 Lee mas

An application of the finite element method to curve fitting

An application of the finite element method to curve fitting

The finite element method was first used more than twenty five years ago [l] in the solving of a structural problem. Since then it has been developed to a considerable degree to the point that now it constitutes an important tool for the studying of problems in the field of mathematics, enabling a large variety of physical situations to be dealt with. The mathematical foundations of the method are well established [2] , and numerous variations and formu- lations are possible: weighting function techniques (Galer kin, collocation, etc.), semianalytical procedures (finite strips, layers and finite prisms, etc.), as the well known boundary element method, are just some of the examples of the numerous possibilities which exist. See [3] for a sum mary of such examples. In the present work, a concrete application is described, namely applying the method in the planning of roads.
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7 Lee mas

A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations

A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations

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.
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26 Lee mas

TítuloA residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity

TítuloA residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity

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
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27 Lee mas

Numerical modeling of fiber specklegram sensors by using finite element method (FEM)

Numerical modeling of fiber specklegram sensors by using finite element method (FEM)

Abstract: Although experimental advances in the implementation and characterization of fiber speckle sensor have been reported, a suitable model to interpret the speckle-pattern variation under perturbation is desirable but very challenging to be developed due to the various factors influencing the speckle pattern. In this work, a new methodology based on the Finite Element Method (FEM) for modeling and optimizing Fiber Specklegram Sensors (FSSs) is proposed. The numerical method allows computational visualization and quantification, in near field, of changes of a Step Multi-Mode Fiber (SMMF) specklegram, due to the application of a Uniformly Distributed Force Line (UDFL). In turn, the local modifications of the fiber speckle produce changes in the optical power captured by a Step Single-Mode Fiber (SSMF) located just at the output end of the SMMF, causing a filtering effect that explains the operation of the FSSs. For each external force, the stress distribution and the propagations modes supported by the SMMF are calculated numerically by means of FEM. Then, those modes are vectorially superposed to reconstruct each perturbed fiber specklegram. Finally, the performance of the sensing mechanism is evaluated for different radius of the filtering SSMF and force-gauges, what evidences design criteria for these kinds of measuring systems. Results are in agreement with those theoretical and experimental ones previously reported.
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14 Lee mas

Vibration analysis of a temple bell by Finite Element Method

Vibration analysis of a temple bell by Finite Element Method

Figure 7 shows the sectional shape and dimensions of Hojobo’s Doza. Though the real Doza shape is circular, the FEM-simulated shape was configured with quadrilateral factors (See Fig. 1) on condition that the area of each Doza is the same. This is due to the limit of the shape of the minimum finite element for FEM analysis. Although Hojobo’s bell has two Dozas placed 180 degrees opposite each other on the circumference, we investigated three cases in which the numbers of Dozas are 1, 2 and 4.

7 Lee mas

THE ONE-SIDED DRIVING INTERACTION BETWEEN TWO MONOPOLES - EFFECTIVE AS THE CORE OF THE 3-DIMENSIONAL DFEM SOUND POWER AND SCATTERING DESCRIPTION

THE ONE-SIDED DRIVING INTERACTION BETWEEN TWO MONOPOLES - EFFECTIVE AS THE CORE OF THE 3-DIMENSIONAL DFEM SOUND POWER AND SCATTERING DESCRIPTION

The second method determines the sound power without any airborne sound field quantity cal- culations. The so called Direct Finite Element Method ( DFEM ) determines the air borne sound power directly in one step from the source vibration quantities only together with geometrical source data. This method was issued several years ago both for baffled plane sources [1], [2] and later on was developed, tested and approved for true 3-dimensional sources [2-6].

7 Lee mas

Método de las Imágenes para una Carga en un Medio de Tres Capas e Implementación de la Condición de Dirichlet

Método de las Imágenes para una Carga en un Medio de Tres Capas e Implementación de la Condición de Dirichlet

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
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8 Lee mas

ADAPTIVE SPACE-TIME FINITE ELEMENT METHODS FOR ACOUSTICS SIMULATIONS IN UNBOUNDED DOMAINS

ADAPTIVE SPACE-TIME FINITE ELEMENT METHODS FOR ACOUSTICS SIMULATIONS IN UNBOUNDED DOMAINS

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.
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6 Lee mas

Software alternativo para el cálculo de los parámetros matriciales eléctricos de líneas de transmisión

Software alternativo para el cálculo de los parámetros matriciales eléctricos de líneas de transmisión

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.
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99 Lee mas

A English

A English

A study of stresses, displacements and strain, based on a classic solid mechanic’s model, specifically a plate with hole, around which a stresses concentration, is presented. For this purpose, the stresses analysis was carried out in a hole concentrator subjected to tensile. The model’s material was ASTM A36. The stresses were analytically calculated, through von Mises’ theory. In addition, the analysis of the stresses using the finite element method (FEM) was carried out. Subsequently, displacements and unitary deformation were determined in the part. The results obtained report an error of 8.7% between the stresses of von Mises through analytical calculus and using the FEM.
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8 Lee mas

Development of a System for the Computation of Electromagnetic Wave Scattering from Non-Penetrable Objects by Solving the Electric Field Integral Equation

Development of a System for the Computation of Electromagnetic Wave Scattering from Non-Penetrable Objects by Solving the Electric Field Integral Equation

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.
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56 Lee mas

A Fixed Grid method for hyperelastic models in large strain analysis using the level set method

A Fixed Grid method for hyperelastic models in large strain analysis using the level set method

Numerical methods as the eXtended Finite Element Method (X-FEM) may be compared with FGFEM since both treat discontinuities in a similar fashion. In [6] the Level Set Method is coupled to X-FEM in order to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries. The disadvantage in this work is that numerical integration of the elements at the interfaces are treated via re-meshing process, which implies more computation costs.

12 Lee mas

OPTIMIZATION METHODS FOR THE ITERATIVE SOLUTION OF COUPLED FLUID-STRUCTURAL PROBLEMS

OPTIMIZATION METHODS FOR THE ITERATIVE SOLUTION OF COUPLED FLUID-STRUCTURAL PROBLEMS

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].

6 Lee mas

Dissipative Finite Element Formulation Applied to Piezoelectric Materials With the Debye Memory

Dissipative Finite Element Formulation Applied to Piezoelectric Materials With the Debye Memory

This work presents a finite element study of the Debye memory in piezoelectric devices. This memory dependency is due to the spontaneous polarization of the electric dipoles and it can be understood as a transient viscosity-like effect. The formulation assumes a small strain and rotation hypothesis and the main contribution is the inclusion of time-dependent constitutive behavior. For this purpose, a unique numerical formulation that uses convolution integrals is developed to solve the time-dependent electric constitutive equation. A consistent and monolithic finite element formulation is then obtained and implemented. Finally, a commercial piezoelectric device is simulated for two operational modes: an actuator and a sensor. Several important conclusions on the coupled mechanical and electric fields are reported and the stability of the time integration scheme is tested by representing the time evolution of the electro-mechanic energy.
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8 Lee mas

On Microstructure-Based Mechanical Behaviour of a Ductile Iron Component

On Microstructure-Based Mechanical Behaviour of a Ductile Iron Component

Castings are produced by a manufacturing method which gives the components local properties that are dependent on design, metallurgy and casting method. E.g. the wall thickness influences the resulting coarseness and type of microstructure, and the material will have local material properties which depend on the local metallurgical and thermal history. The mechanical behaviour of a cast iron component can vary significantly in the casting volume, which makes it difficult to optimize the castings with good accuracy. Structural analyses of cast products in service, e.g. using Finite Element Method (FEM) simulations, are typically based on the assumption of constant material properties throughout the product. This is not an optimal representation of the variations that are actually found in the casting. By predicting the distribution of microstructural features and establishing quantitative relationships between microstructure and mechanical behaviour, it will be possible to calculate the local material properties and the deformation behaviour of cast products with higher precision.
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7 Lee mas

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