EFFECT – Efficient finite element code

The theme of this dissertation is the finite element method applied to mechanical structures. A new finite element program is developed that, besides executing different types of structural analysis, also allows the calculation of the derivatives of structural performances using the continuum method of design sensitivities analysis, with the purpose of allowing, in combination with the mathematical programming algorithms found in the commercial software MATLAB, to solve structural optimization problems. The program is called EFFECT – Efficient Finite Element Code. The object-oriented programming paradigm and specifically the C ++ programming language are used for program development. The main objective of this dissertation is to design EFFECT so that it can constitute, in this stage of development, the foundation for a program with analysis capacities similar to other open source finite element programs. In this first stage, 6 elements are implemented for linear analysis: 2-dimensional truss (Truss2D), 3-dimensional truss (Truss3D), 2-dimensional beam (Beam2D), 3-dimensional beam (Beam3D), triangular shell element (Shell3Node) and quadrilateral shell element (Shell4Node). The shell elements combine two distinct elements, one for simulating the membrane behavior and the other to simulate the plate bending behavior. The non-linear analysis capability is also developed, combining the corotational formulation with the Newton-Raphson iterative method, but at this stage is only avaiable to solve problems modeled with Beam2D elements subject to large displacements and rotations, called nonlinear geometric problems. The design sensitivity analysis capability is implemented in two elements, Truss2D and Beam2D, where are included the procedures and the analytic expressions for calculating derivatives of displacements, stress and volume performances with respect to 5 different design variables types. Finally, a set of test examples were created to validate the accuracy and consistency of the result obtained from EFFECT, by comparing them with results published in the literature or obtained with the ANSYS commercial finite element code.

Predicting trabecular bone microdamage initiation and accumulation using a non-linear perfect damage model

Studies evaluating the mechanical behavior of the trabecular microstructure play an important role in our understanding of pathologies such as osteoporosis, and in increasing our understanding of bone fracture and bone adaptation. Understanding of such behavior in bone is important for predicting and providing early treatment of fractures. The objective of this study is to present a numerical model for studying the initiation and accumulation of trabecular bone microdamage in both the pre- and post-yield regions. A sub-region of human vertebral trabecular bone was analyzed using a uniformly loaded anatomically accurate microstructural three-dimensional finite element model. The evolution of trabecular bone microdamage was governed using a non-linear, modulus reduction, perfect damage approach derived from a generalized plasticity stress-strain law. The model introduced in this paper establishes a history of microdamage evolution in both the pre- and post-yield regions

A mixed finite element method for nonlinear diffusion equations

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.

Effect of expiratory flow limitation on respiratory mechanical impedance: a model study

Large phasic variations of respiratory mechanical impedance (Zrs) have been observed during induced expiratory flow limitation (EFL) (M. Vassiliou, R. Peslin, C. Saunier, and C. Duvivier. Eur. Respir. J. 9: 779-786, 1996). To clarify the meaning of Zrs during EFL, we have measured from 5 to 30 Hz the input impedance (Zin) of mechanical analogues of the respiratory system, including flow-limiting elements (FLE) made of easily collapsible rubber tubing. The pressures upstream (Pus) and downstream (Pds) from the FLE were controlled and systematically varied. Maximal flow (Vmax) increased linearly with Pus, was close to the value predicted from wave-speed theory, and was obtained for Pus-Pds of 4-6 hPa. The real part of Zin started increasing abruptly with flow (V) >85%Vmax and either further increased or suddenly decreased in the vicinity of V¿max. The imaginary part of Zin decreased markedly and suddenly above 95%Vmax. Similar variations of Zin during EFL were seen with an analogue that mimicked the changes of airway transmural pressure during breathing. After pressure andV measurements upstream and downstream from the FLE were combined, the latter was analyzed in terms of a serial (Zs) and a shunt (Zp) compartment. Zs was consistent with a large resistance and inertance, and Zp with a mainly elastic element having an elastance close to that of the tube walls. We conclude that Zrs data during EFL mainly reflect the properties of the FLE.

An alternative finite element formulation for determination of streamlines in two-dimensional problems

It is well known that the numerical solutions of incompressible viscous flows are of great importance in Fluid Dynamics. The graphics output capabilities of their computational codes have revolutionized the communication of ideas to the non-specialist public. In general those codes include, in their hydrodynamic features, the visualization of flow streamlines - essentially a form of contour plot showing the line patterns of the flow - and the magnitudes and orientations of their velocity vectors. However, the standard finite element formulation to compute streamlines suffers from the disadvantage of requiring the determination of boundary integrals, leading to cumbersome implementations at the construction of the finite element code. In this article, we introduce an efficient way - via an alternative variational formulation - to determine the streamlines for fluid flows, which does not need the computation of contour integrals. In order to illustrate the good performance of the alternative formulation proposed, we capture the streamlines of three viscous models: Stokes, Navier-Stokes and Viscoelastic flows.

Effect of particle morphology on the mechanical and thermo-mechanical behavior of polymer composites

Fiber reinforced polymer composites have been used in many applications, such as in automobile, aerospace and naval industries, due basically to their high strength-to-weight and modulus-to-weight, among other properties. Even though particles are usually not able to lead to the level of reinforcement of fibers, particle reinforced polymer composites have been proposed for many new applications due to their low cost, easy fabrication and isotropic properties. In this work, polymer composites were prepared by incorporating glass particles of different morphologies on poly(aryl sulfones) matrices. Particles with aspect ratios equal to 1, 2.5 and 10 were used. The prepared composites were characterized using electron microscopy and thermal analysis. Mechanical properties of the composites were evaluated using a four-point bending test. The thermo-mechanical behavior of the obtained composites was also investigated. The results showed that the morphology of the particles alter significantly the mechanical properties of composites. Particles with larger values of aspect ratio led to large elastic modulus but low levels of strain at failure. This result was explained by modeling the thermo-mechanical behavior of the composites using a viscoelastic model. Parameters of the model, obtained from a Cole-Cole type of plot, demonstrated that interactions at the polymer-reinforcing agent interface were higher for composites with large aspect ratio particles. Higher levels of interactions at interfaces can lead to higher degrees of stress transfer and, consequently, to composites with large elastic modulus, as experimentally observed.

Multicomponent diffusion during Prato cheese ripening: mathematical modeling using the finite element method

The partial replacement of NaCl by KCl is a promising alternative to produce a cheese with lower sodium content since KCl does not change the final quality of the cheese product. In order to assure proper salt proportions, mathematical models are employed to control the product process and simulate the multicomponent diffusion during the reduced salt cheese ripening period. The generalized Fick's Second Law is widely accepted as the primary mass transfer model within solid foods. The Finite Element Method (FEM) was used to solve the system of differential equations formed. Therefore, a NaCl and KCl multicomponent diffusion was simulated using a 20% (w/w) static brine with 70% NaCl and 30% KCl during Prato cheese (a Brazilian semi-hard cheese) salting and ripening. The theoretical results were compared with experimental data, and indicated that the deviation was 4.43% for NaCl and 4.72% for KCl validating the proposed model for the production of good quality, reduced-sodium cheeses.

Mechanical simulation of the endovascular repair of abdominal aortic aneurysms

Ce travail de thèse porte sur la simulation du déploiement des prothèses vasculaires de type stent-graft (SG) lors de la réparation endovasculaire (EVAR) des anévrismes de l’aorte abdominale (AAA). Cette étude se présente en trois parties: (i) tests mécaniques en flexion et compression de SG couramment utilisés (corps et jambage de marque Cook) ainsi que la simulation numérique desdits tests, (ii) développement d’un modèle numérique d’anévrisme, (iii) stratégie de simulation du déploiement des SG. La méthode numérique employée est celle des éléments finis. Dans un premier temps, une vérification du modèle éléments finis (MEF) des SG est realisée par comparaison des différents cas de charge avec leur pendant expérimental. Ensuite, le MEF vasculaire (AAA) est lui aussi vérifié lors d’une comparaison des niveaux de contraintes maximales principales dans la paroi avec des valeurs de la littérature. Enfin, le déploiement est abordé tout en intégrant les cathéters. Les tests mécaniques menés sur les SG ont été simulés avec une différence maximale de 5,93%, tout en tenant compte de la pré-charge des stents. Le MEF de la structure vasculaire a montré des contraintes maximales principales éloignées de 4,41% par rapport à un modèle similaire précédemment publié. Quant à la simulation du déploiement, un jeu complet de SG a pu être déployé avec un bon contrôle de la position relative et globale, dans un AAA spécifique pré-déformé, sans toutefois inclure de thrombus intra-luminal (TIL). La paroi du AAA a été modélisée avec une loi de comportement isotropique hyperélastique. Étant donné que la différence maximale tolérée en milieu clinique entre réalité et simulation est de 5%, notre approche semble acceptable et pourrait donner suite à de futurs développements. Cela dit, le petit nombre de SG testés justifie pleinement une vaste campagne de tests mécaniques et simulations supplémentaires à des fins de validation.

Finite element method for the accurate solution of diatomic molecules

We present the Finite-Element-Method (FEM) in its application to quantum mechanical problems solving for diatomic molecules. Results for Hartree-Fock calculations of H_2 and Hartree-Fock-Slater calculations of molecules like N_2 and C0 have been obtained. The accuracy achieved with less then 5000 grid points for the total energies of these systems is 10_-8 a.u., which is demonstrated for N_2.

Solution of the Hartree-Fock-Slater equations for diatomic molecules by the finite-element method

We present the finite-element method in its application to solving quantum-mechanical problems for diatomic molecules. Results for Hartree-Fock calculations of H_2 and Hartree-Fock-Slater calculations for molecules like N_2 and CO are presented. The accuracy achieved with fewer than 5000 grid points for the total energies of these systems is 10^-8 a.u., which is about two orders of magnitude better than the accuracy of any other available method.

A high-order arbitrarily unstructured finite-volume model of the global atmosphere: tests solving the shallow-water equations

Simulations of the global atmosphere for weather and climate forecasting require fast and accurate solutions and so operational models use high-order finite differences on regular structured grids. This precludes the use of local refinement; techniques allowing local refinement are either expensive (eg. high-order finite element techniques) or have reduced accuracy at changes in resolution (eg. unstructured finite-volume with linear differencing). We present solutions of the shallow-water equations for westerly flow over a mid-latitude mountain from a finite-volume model written using OpenFOAM. A second/third-order accurate differencing scheme is applied on arbitrarily unstructured meshes made up of various shapes and refinement patterns. The results are as accurate as equivalent resolution spectral methods. Using lower order differencing reduces accuracy at a refinement pattern which allows errors from refinement of the mountain to accumulate and reduces the global accuracy over a 15 day simulation. We have therefore introduced a scheme which fits a 2D cubic polynomial approximately on a stencil around each cell. Using this scheme means that refinement of the mountain improves the accuracy after a 15 day simulation. This is a more severe test of local mesh refinement for global simulations than has been presented but a realistic test if these techniques are to be used operationally. These efficient, high-order schemes may make it possible for local mesh refinement to be used by weather and climate forecast models.

Finite element aided design evolution of composite leaf spring

This paper shows the process of the virtual production development of the mechanical connection between the top leaf of a dual composite leaf spring system to a shackle using finite element methods. The commercial FEA package MSC/MARC has been used for the analysis. In the original design the joint was based on a closed eye-end. Full scale testing results showed that this configuration achieved the vertical proof load of 150 kN and 1 million cycles of fatigue load. However, a problem with delamination occurred at the interface between the fibres going around the eye and the main leaf body. To overcome this problem, a second design was tried using transverse bandages of woven glass fibre reinforced tape to wrap the section that is prone to delaminate. In this case, the maximum interlaminar shear stress was reduced by a certain amount but it was still higher than the material’s shear strength. Based on the fact that, even with delamination, the top leaf spring still sustained the maximum static and fatigue loads required, the third design was proposed with an open eye-end, eliminating altogether the interface where the maximum shear stress occurs. The maximum shear stress predicted by FEA is reduced significantly and a safety factor of around 2 has been obtained. Thus, a successful and safe design has been achieved.

Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

Reappraisal of the effective elastic thickness for the sub-Andes using 3-D finite element flexural modelling, gravity and geological constraints

P>Estimates of effective elastic thickness (T(e)) for the western portion of the South American Plate using, independently, forward flexural modelling and coherence analysis, suggest different thermomechanical properties for the same continental lithosphere. We present a review of these T(e) estimates and carry out a critical reappraisal using a common methodology of 3-D finite element method to solve a differential equation for the bending of a thin elastic plate. The finite element flexural model incorporates lateral variations of T(e) and the Andes topography as the load. Three T(e) maps for the entire Andes were analysed: Stewart & Watts (1997), Tassara et al. (2007) and Perez-Gussinye et al. (2007). The predicted flexural deformation obtained for each T(e) map was compared with the depth to the base of the foreland basin sequence. Likewise, the gravity effect of flexurally induced crust-mantle deformation was compared with the observed Bouguer gravity. T(e) estimates using forward flexural modelling by Stewart & Watts (1997) better predict the geological and gravity data for most of the Andean system, particularly in the Central Andes, where T(e) ranges from greater than 70 km in the sub-Andes to less than 15 km under the Andes Cordillera. The misfit between the calculated and observed foreland basin subsidence and the gravity anomaly for the Maranon basin in Peru and the Bermejo basin in Argentina, regardless of the assumed T(e) map, may be due to a dynamic topography component associated with the shallow subduction of the Nazca Plate beneath the Andes at these latitudes.