Three-dimensional finite element analysis of MHD duct flow by the penalty function formulation

A numerical scheme based on the Finite Element Method (FEM) is presented to calculate the full solution of a three-dimensional steady magnetohydrodynamic (MHD) flow with moderately high Hartmann numbers and interaction parameters. An incompressible, viscous and electrically conducting liquid-metal is considered. Assuming a low magnetic Reynolds number, the solution method solves the coupled Navier-Stokes and Maxwell's equations through the use of a penalty function method. Results are presented for Hartmann numbers in the range 10(2)-10(3).

Electroosmotic pumping in rectangular microchannels: A numerical treatment by the finite element method

In this work, the analysis of electroosmotic pumping mechanisms in microchannels is performed through the solution of Poisson-Boltzmann and Navier Stokes equations by the Finite Element Method. This approach is combined with a Newton-Raphson iterative scheme, allowing a full treatment of the non-linear Poisson-Boltzmann source term which is normally approximated by linearizations in other methods.

Application of the finite element method in Dentistry

The finite element method (FEM) involves a series of computational procedures to calculate the stress in each element, which performs a model solution. Such a structural analysis allows the determination of stress resulting from external force, pressure, thermal change, and other factors. This method is extremely useful for indicating mechanical aspects of biomaterials and human tissues that can hardly be measured in vivo. The results obtained can then be studied using visualization software within the FEM environment to view a variety of parameters, and to fully identify implications of the analysis. Objective: An overview to show application of FEM in dentistry was undertaken. Literature review: This paper shows the basic concept, advances, advantages, limitations and applications of finite element method (FEM) in dentistry. Conclusion: It is extremely important to verify what the purpose of the study is in order to correctly apply FEM.

Application of hte finite element method in Dentistry

Introduction: The finite element method (FEM) involves a series of computational procedures to calculate the stress in each element, which performs a model solution. Such a structural analysis allows the determination of stress resulting from external force, pressure, thermal change, and other factors. This method is extremely useful for indicating mechanical aspects of biomaterials and human tissues that can hardly be measured in vivo. The results obtained can then be studied using visualization software within the FEM environment to view a variety of parameters, and to fully identify implications of the analysis. Objective: An overview to show application of FEM in dentistry was undertaken. Literature review: This paper shows the basic concept, advances, advantages, limitations and applications of finite element method (FEM) in dentistry. Conclusion: It is extremely important to verify what the purpose of the study is in order to correctly apply FEM.

A generalized finite element method with global-local enrichment functions for confined plasticity problems

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J (2) plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.

Numerical assessment of stability of interface discontinuous finite element pressure spaces

The stability of two recently developed pressure spaces has been assessed numerically: The space proposed by Ausas et al. [R.F. Ausas, F.S. Sousa, G.C. Buscaglia, An improved finite element space for discontinuous pressures, Comput. Methods Appl. Mech. Engrg. 199 (2010) 1019-1031], which is capable of representing discontinuous pressures, and the space proposed by Coppola-Owen and Codina [A.H. Coppola-Owen, R. Codina, Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions, Int. J. Numer. Methods Fluids, 49 (2005) 1287-1304], which can represent discontinuities in pressure gradients. We assess the stability of these spaces by numerically computing the inf-sup constants of several meshes. The inf-sup constant results as the solution of a generalized eigenvalue problems. Both spaces are in this way confirmed to be stable in their original form. An application of the same numerical assessment tool to the stabilized equal-order P-1/P-1 formulation is then reported. An interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation's stability. An analogous result is also reported for the mini-element P-1(+)/P-1 when the velocity bubbles are removed in an arbitrary band of elements. (C) 2012 Elsevier B.V. All rights reserved.

Numeric reconstruction of cytoskeleton with finite element method and topology optimization method

The importance of mechanical aspects related to cell activity and its environment is becoming more evident due to their influence in stem cell differentiation and in the development of diseases such as atherosclerosis. The mechanical tension homeostasis is related to normal tissue behavior and its lack may be related to the formation of cancer, which shows a higher mechanical tension. Due to the complexity of cellular activity, the application of simplified models may elucidate which factors are really essential and which have a marginal effect. The development of a systematic method to reconstruct the elements involved in the perception of mechanical aspects by the cell may accelerate substantially the validation of these models. This work proposes the development of a routine capable of reconstructing the topology of focal adhesions and the actomyosin portion of the cytoskeleton from the displacement field generated by the cell on a flexible substrate. Another way to think of this problem is to develop an algorithm to reconstruct the forces applied by the cell from the measurements of the substrate displacement, which would be characterized as an inverse problem. For these kind of problems, the Topology Optimization Method (TOM) is suitable to find a solution. TOM is consisted of an iterative application of an optimization method and an analysis method to obtain an optimal distribution of material in a fixed domain. One way to experimentally obtain the substrate displacement is through Traction Force Microscopy (TFM), which also provides the forces applied by the cell. Along with systematically generating the distributions of focal adhesion and actin-myosin for the validation of simplified models, the algorithm also represents a complementary and more phenomenological approach to TFM. As a first approximation, actin fibers and flexible substrate are represented through two-dimensional linear Finite Element Method. Actin contraction is modeled as an initial stress of the FEM elements. Focal adhesions connecting actin and substrate are represented by springs. The algorithm was applied to data obtained from experiments regarding cytoskeletal prestress and micropatterning, comparing the numerical results to the experimental ones

Finite element methods for the Stokes system based on a Zienkiewicz type N-simplex

Hermite interpolation is increasingly showing to be a powerful numerical solution tool, as applied to different kinds of second order boundary value problems. In this work we present two Hermite finite element methods to solve viscous incompressible flows problems, in both two- and three-dimension space. In the two-dimensional case we use the Zienkiewicz triangle to represent the velocity field, and in the three-dimensional case an extension of this element to tetrahedra, still called a Zienkiewicz element. Taking as a model the Stokes system, the pressure is approximated with continuous functions, either piecewise linear or piecewise quadratic, according to the version of the Zienkiewicz element in use, that is, with either incomplete or complete cubics. The methods employ both the standard Galerkin or the Petrov–Galerkin formulation first proposed in Hughes et al. (1986) [18], based on the addition of a balance of force term. A priori error analyses point to optimal convergence rates for the PG approach, and for the Galerkin formulation too, at least in some particular cases. From the point of view of both accuracy and the global number of degrees of freedom, the new methods are shown to have a favorable cost-benefit ratio, as compared to velocity Lagrange finite elements of the same order, especially if the Galerkin approach is employed.

The meccano method for finite element and isogeometric analysis

Congresos y conferencias

Adaptive Finite Element Method with a local element parametrization nested meses

[EN]This work presents a novel approach to solve a two dimensional problem by using an adaptive finite element approach. The most common strategy to deal with nested adaptivity is to generate a mesh that represents the geometry and the input parameters correctly, and to refine this mesh locally to obtain the most accurate solution. As opposed to this approach, the authors propose a technique using independent meshes : geometry, input data and the unknowns. Each particular mesh is obtained by a local nested refinement of the same coarse mesh at the parametric space…

Mixed finite-element methods for elliptic convection-dominated problems arising in semiconductor physics

In this work we develop and analyze an adaptive numerical scheme for simulating a class of macroscopic semiconductor models. At first the numerical modelling of semiconductors is reviewed in order to classify the Energy-Transport models for semiconductors that are later simulated in 2D. In this class of models the flow of charged particles, that are negatively charged electrons and so-called holes, which are quasi-particles of positive charge, as well as their energy distributions are described by a coupled system of nonlinear partial differential equations. A considerable difficulty in simulating these convection-dominated equations is posed by the nonlinear coupling as well as due to the fact that the local phenomena such as "hot electron effects" are only partially assessable through the given data. The primary variables that are used in the simulations are the particle density and the particle energy density. The user of these simulations is mostly interested in the current flow through parts of the domain boundary - the contacts. The numerical method considered here utilizes mixed finite-elements as trial functions for the discrete solution. The continuous discretization of the normal fluxes is the most important property of this discretization from the users perspective. It will be proven that under certain assumptions on the triangulation the particle density remains positive in the iterative solution algorithm. Connected to this result an a priori error estimate for the discrete solution of linear convection-diffusion equations is derived. The local charge transport phenomena will be resolved by an adaptive algorithm, which is based on a posteriori error estimators. At that stage a comparison of different estimations is performed. Additionally a method to effectively estimate the error in local quantities derived from the solution, so-called "functional outputs", is developed by transferring the dual weighted residual method to mixed finite elements. For a model problem we present how this method can deliver promising results even when standard error estimator fail completely to reduce the error in an iterative mesh refinement process.

Analysis of optimal control problems for the incompressible MHD equations and implementation in a finite element multiphysics code

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics

Over the years the Differential Quadrature (DQ) method has distinguished because of its high accuracy, straightforward implementation and general ap- plication to a variety of problems. There has been an increase in this topic by several researchers who experienced significant development in the last years. DQ is essentially a generalization of the popular Gaussian Quadrature (GQ) used for numerical integration functions. GQ approximates a finite in- tegral as a weighted sum of integrand values at selected points in a problem domain whereas DQ approximate the derivatives of a smooth function at a point as a weighted sum of function values at selected nodes. A direct appli- cation of this elegant methodology is to solve ordinary and partial differential equations. Furthermore in recent years the DQ formulation has been gener- alized in the weighting coefficients computations to let the approach to be more flexible and accurate. As a result it has been indicated as Generalized Differential Quadrature (GDQ) method. However the applicability of GDQ in its original form is still limited. It has been proven to fail for problems with strong material discontinuities as well as problems involving singularities and irregularities. On the other hand the very well-known Finite Element (FE) method could overcome these issues because it subdivides the computational domain into a certain number of elements in which the solution is calculated. Recently, some researchers have been studying a numerical technique which could use the advantages of the GDQ method and the advantages of FE method. This methodology has got different names among each research group, it will be indicated here as Generalized Differential Quadrature Finite Element Method (GDQFEM).

Fluid-structure interaction by a coupled lattice Boltzmann-finite element approach

In this thesis, a strategy to model the behavior of fluids and their interaction with deformable bodies is proposed. The fluid domain is modeled by using the lattice Boltzmann method, thus analyzing the fluid dynamics by a mesoscopic point of view. It has been proved that the solution provided by this method is equivalent to solve the Navier-Stokes equations for an incompressible flow with a second-order accuracy. Slender elastic structures idealized through beam finite elements are used. Large displacements are accounted for by using the corotational formulation. Structural dynamics is computed by using the Time Discontinuous Galerkin method. Therefore, two different solution procedures are used, one for the fluid domain and the other for the structural part, respectively. These two solvers need to communicate and to transfer each other several information, i.e. stresses, velocities, displacements. In order to guarantee a continuous, effective, and mutual exchange of information, a coupling strategy, consisting of three different algorithms, has been developed and numerically tested. In particular, the effectiveness of the three algorithms is shown in terms of interface energy artificially produced by the approximate fulfilling of compatibility and equilibrium conditions at the fluid-structure interface. The proposed coupled approach is used in order to solve different fluid-structure interaction problems, i.e. cantilever beams immersed in a viscous fluid, the impact of the hull of the ship on the marine free-surface, blood flow in a deformable vessels, and even flapping wings simulating the take-off of a butterfly. The good results achieved in each application highlight the effectiveness of the proposed methodology and of the C++ developed software to successfully approach several two-dimensional fluid-structure interaction problems.

Patient-specific finite-element simulation of the human cornea: A clinical validation study on cataract surgery

The planning of refractive surgical interventions is a challenging task. Numerical modeling has been proposed as a solution to support surgical intervention and predict the visual acuity, but validation on patient specific intervention is missing. The purpose of this study was to validate the numerical predictions of the post-operative corneal topography induced by the incisions required for cataract surgery. The corneal topography of 13 patients was assessed preoperatively and postoperatively (1-day and 30-day follow-up) with a Pentacam tomography device. The preoperatively acquired geometric corneal topography – anterior, posterior and pachymetry data – was used to build patient-specific finite element models. For each patient, the effects of the cataract incisions were simulated numerically and the resulting corneal surfaces were compared to the clinical postoperative measurements at one day and at 30-days follow up. Results showed that the model was able to reproduce experimental measurements with an error on the surgically induced sphere of 0.38D one day postoperatively and 0.19D 30 days postoperatively. The standard deviation of the surgically induced cylinder was 0.54D at the first postoperative day and 0.38D 30 days postoperatively. The prediction errors in surface elevation and curvature were below the topography measurement device accuracy of ±5μm and ±0.25D after the 30-day follow-up. The results showed that finite element simulations of corneal biomechanics are able to predict post cataract surgery within topography measurement device accuracy. We can conclude that the numerical simulation can become a valuable tool to plan corneal incisions in cataract surgery and other ophthalmosurgical procedures in order to optimize patients' refractive outcome and visual function.