La descripción del flujo de fluidos involucra la solución de las ecuaciones de Navier-Stokes compresible, un problema muy complejo cuya estructura matemática no es del todo comprendida. Por lo tanto, mediante análisis asintótico, se pueden derivar modelos simplificados bajo ciertas hipótesis sobre el problema hechas en términos de parámetros adimensionales que miden la importancia relativa de los diferentes procesos físicos. Los flujos a baja velocidad se pueden describir por diferentes modelos que incluyen las ecuaciones de Navier Stokes incompresible cuya matemática es mucho mas conocida. Sin embargo, algunos flujos importantes no se pueden considerar incompresibles debido a la presencia de efectos térmicos. En esta clase de problemas se pueden derivar otra clase de ecuaciones simplificadas: las ecuaciones de Boussinesq y las ecuaciones de bajo numero de Mach.La complejidad de estos problemas matemáticos hace que su solución numérica sea muy difícil. En estos problemas el método de los elementos finitos es inestable, lo que en la práctica implica soluciones numéricas que presentan oscilaciones nodo a nodo de naturaleza no física. En las ecuaciones de Navier Stokes incompresible, dos fuentes bien conocidas de inestabilidad son la condición de incompresibilidad y la presencia del término convectivo. Muchas técnicas de estabilización utilizadas hoy en día se basan en la separación de escalas, descomponiendo la incógnita en una parte gruesa inducida por la discretización del domino y una parte fina de subescala. Modelar la subescala y su influencia conduce a un problema modificado para la escala gruesa que resulta estable.Aunque las técnicas de estabilización son ampliamente utilizadas hoy en día, importantes problemas permanecen abiertos. Contribuyendo a su comprensión, en este trabajo se analizan varios aspectos del modelado de las subescalas. Para problemas escalares de segundo orden, se encuentra la dependencia de la subescala con el tamaño de la malla en el caso general de mallas anisótropas. Estas ideas son extendidas a sistemas de ecuaciones para considerar el problema de Oseen. También se analiza el modelado de las subescalas en problemas transitorios, obteniendo un mejor esquema de integración temporal para el problema de escala gruesa. Para considerar flujos a baja velocidad, se presenta la extensión de estas técnicas a problemas no lineales acoplados, lo que esta íntimamente relacionado con el problema del modelado de la turbulencia, que es un tema en si mismo.Los flujos acoplados térmicamente, aparte del interés intrínseco que merecen, son importantes desde un punto de vista ingenieril. Una solución precisa del problema de flujo es necesaria para definir las cargas térmicas sobre las estructuras, que en muchos casos responden fuertemente, haciendo el problema acoplado. Esta clase de problemas, que motivaron este trabajo, incluyen la respuesta estructural en el caso de un incendio. / A general description of a fluid flow involves the solution of the compressible Navier-Stokes equations, a very complex problem whose mathematical structure is not well understood. Therefore, simplified models can be derived by asymptotic analysis under some assumptions on the problem, made in terms of dimensionless parameters that measure the relative importance of different physical processes. Low speed flows can be described by several models including the incompressible Navier Stokes equations whose mathematical structure is much better understood. However many important flows cannot be considered as incompressible, even at low speed, due to the presence of thermal effects. In such kind of problems another class of simplified equations can be derived: the Boussinesq equations and the Low Mach number equations.The complexity of these mathematical problems makes their numerical solution very difficult. For these problems the standard finite element method is unstable, what in practice means that node to node oscillations of non physical nature may appear in the numerical solution. In the incompressible Navier Stokes equations, two well known sources of numerical instabilities are the incompressibility constraint and the presence of the convective terms. Many stabilization techniques used nowadays are based on scale separation, splitting the unknown into a coarse part induced by the discretization of the domain and a fine subgrid part. The modelling of the subgrid scale and its influence leads to a modified coarse scale problem that now can be shown to be stable. Although stabilization techniques are nowadays widely used, important problems remain open. Contributing to their understanding, several aspects of the subgrid scale modelling are analyzed in this work. For second order scalar problems, the dependence of the subgrid scale on the mesh size, in the general anisotropic case, is clarified. These ideas are extended to systems of equations to consider the Oseen problem. The modelling of the subgrid scales in transient problems is also analyzed, leading to an improved time discretization scheme for the coarse scale problem. To consider low speed flow models, the extension of these techniques to nonlinear and coupled problems is presented, something that is intimately related to the problem of turbulence modelling, which a entire subject on its own right. Thermally coupled flow problems, despite the intrinsic interest they deserve, are important from an engineering point of view. An accurate solution of a flow problem is needed to define thermal loads on structures which, in many cases have a strong response, making the problem coupled. This kind of problems, that motivated this work, include the problem of a structural response in the case of fires.

There exists considerable motivation to reduce vehicle weight through the adoption of lightweight materials, such as aluminum alloys, while maintaining energy absorption and component integrity under crash conditions. To this end, it is of particular interest to study the crash behaviour of lightweight tubular hydroformed structures to determine how the forming behaviour affects the axial crush response. Thus, the current research has studied the dynamic crush response of both non-hydroformed and hydroformed EN-AW 5018 and AA5754 aluminum alloy tubes using both experimental and numerical methods. Experiments were performed in which hydroforming process parameters were varied in a parametric fashion after which the crash response was measured. Experimental parameters included the tube thickness and the hydroformed corner radii of the tubes. Explicit dynamic finite element simulations of the hydroforming and crash events were carried out with particular attention to the transfer of forming history from the hydroforming simulations to the crash models. The results showed that increases in the strength of the material due to work hardening during hydroforming were beneficial in increasing energy absorption during crash. However, it was shown that thinning in the corners of the tube during hydroforming decreased the energy absorption capabilities during axial crush. Residual stresses resulting from hydroforming had little effect on the energy absorption characteristics during axial crush. The current research has shown that, in addition to capturing the forming history in the crash models, it is also important to account for effects of material non-linearity such as kinematic hardening, anisotropy, and strain-rate effects in the finite element models. A model combining a non-linear kinematic hardening model, the Johnson-Cook rate sensitive model, and the Yld2000-2d anisotropic model was developed and implemented in the finite element simulations. This combined model did not account for the effect of rotational hardening (plastic spin) due to plastic deformation. It is recommended that a combined constitutive model, such as the one described in this research, be utilized for the finite element study of materials that show sensitivity to the Bauschinger effect, strain-rate effects, and anisotropy.

There exists considerable motivation to reduce vehicle weight through the adoption of lightweight materials, such as aluminum alloys, while maintaining energy absorption and component integrity under crash conditions. To this end, it is of particular interest to study the crash behaviour of lightweight tubular hydroformed structures to determine how the forming behaviour affects the axial crush response. Thus, the current research has studied the dynamic crush response of both non-hydroformed and hydroformed EN-AW 5018 and AA5754 aluminum alloy tubes using both experimental and numerical methods. Experiments were performed in which hydroforming process parameters were varied in a parametric fashion after which the crash response was measured. Experimental parameters included the tube thickness and the hydroformed corner radii of the tubes. Explicit dynamic finite element simulations of the hydroforming and crash events were carried out with particular attention to the transfer of forming history from the hydroforming simulations to the crash models. The results showed that increases in the strength of the material due to work hardening during hydroforming were beneficial in increasing energy absorption during crash. However, it was shown that thinning in the corners of the tube during hydroforming decreased the energy absorption capabilities during axial crush. Residual stresses resulting from hydroforming had little effect on the energy absorption characteristics during axial crush. The current research has shown that, in addition to capturing the forming history in the crash models, it is also important to account for effects of material non-linearity such as kinematic hardening, anisotropy, and strain-rate effects in the finite element models. A model combining a non-linear kinematic hardening model, the Johnson-Cook rate sensitive model, and the Yld2000-2d anisotropic model was developed and implemented in the finite element simulations. This combined model did not account for the effect of rotational hardening (plastic spin) due to plastic deformation. It is recommended that a combined constitutive model, such as the one described in this research, be utilized for the finite element study of materials that show sensitivity to the Bauschinger effect, strain-rate effects, and anisotropy.

The overall goal of the present research is to provide a computationally based methodology to realize the projected extraordinary properties of Carbon Nanotube (CNT)- reinforced composites and polymeric nanofibers for engineering applications. The discovery of carbon nanotubes (CNT) and its derivatives has led to considerable study both experimentally and computationally as carbon based materials are ideally suited for molecular level building blocks for nanoscale systems. Research in nanomechanics is currently focused on the utilization of CNTs as reinforcements in polymer matrices as CNTs have a very high modulus and are extremely light weight. The nanometer dimension of a CNT and its interaction with a polymer chain requires a study involving the coupling of the length scales. This length scale coupling requires analysis in the molecular and higher order levels. The atomistic interactions of the nanotube are studied using molecular dynamic simulations. The elastic properties of neat nanotube as well as doped nanotube are estimated first. The stability of the nanotube under various conditions is also dealt with in this dissertation. The changes in the elastic stiffness of a nanotube when it is embedded in a composite system are also considered. This type of a study is very unique as it gives information on the effect of surrounding materials on the core nanotube. Various configurations of nanotubes and nanocomposites are analyzed in this dissertation. Polymeric nanofibers are an important component in tissue engineering; however, these nanofibers are found to have a complex internal structure. A computational strategy is developed for the first time in this work, where a combined multiscale approach for the estimation of the elastic properties of nanofibers was carried out. This was achieved by using information from the molecular simulations, micromechanical analysis, and subsequently the continuum chain model, which was developed for rope systems. The continuum chain model is modified using properties of the constituent materials in the mesoscale. The results are found to show excellent correlation with experimental measurements. Finally, the entire atomistic to mesoscale analysis was coupled into the macroscale by mathematical homogenization techniques. Two-scale mathematical homogenization, called asymptotic expansion homogenization (AEH), was used for the estimation of the overall effective properties of the systems being analyzed. This work is unique for the formulation of spectral/hp based higher-order finite element methods with AEH. Various nanocomposite and nanofibrous structures are analyzed using this formulation. In summary, in this dissertation the mechanical characteristics of nanotube based composite systems and polymeric nanofibrous systems are analyzed by a seamless integration of processes at different scales.

Granular materials exist in many forms in nature ranging from space debris to sand dunes and from breakfast cereals to pharmaceutical tablets. They can behave like a solid or a viscous fluid or a gas. The gas-like nature of granular materials in rapid flows allows the use of models based on kinetic theory thus revealing in depth complex physics and phenomena. However unlike conventional fluids here the energy balance requires additional dissipation terms as a consequence of inelasticity. The complexity of their interaction and diversity in application has led to numerous studies using experimental methods and numerical simulations in order to determine the most appropriate constitutive relationships for granular gases. With large dissipation the form of the constitutive relationship becomes particularly important, especially in the presence of non-equipartition and anisotropy. This thesis is focused on constitutive models of simple granular flows. A vibrated bed is often used as an idealisation of granular flows, providing a convenient approximation to the simplest type of flow: binary and instantaneous collisions with no rotations. Using finite element method (FE) based COMSOL modules we solve conservation of mass, momentum and energy resulting from granular kinetic theory in axi-symmetric form to generate time and spatial resolved solutions of packing fraction, velocity and granular temperature and compare the predictions to numerical simulation and experiment. At first we show the comparison for two closure sets, one based on a simple near elastic approach while the second based on revised Enskog theory for dense inelastic flows. The results for the second approach show good agreement with the results of previously validated near elastic models and experimental results. The observed differences between the two closure sets are small except for the observation of temperature upturn in a dilute region of the cell away from base. One cause of this is the presence of additional constitutive terms in the balance equations and are a consequence of inelasticity. The models also consider time varying effects at low frequency of excitation. These solutions show existence of wave-like effects in the cell with associated temperature upturn within the hydrodynamic applicability region. Presence of instantaneous cyclic rolling is also seen in both approaches. Evidence from MD simulations and experiments qualitatively support the findings of hydrodynamic models in phase resolved as well as time average behaviour. Subsequently, the frequency of vibration was varied to unlink the wave motion from the bulk temperature. Lack of agreement between experiment and the model predictions are shown to be due to lack of separation of time scale between the grain-base interaction and the base frequency. A sharp decrease of heat flux is measured showing that the energy input is frequency dependent. Analysis of the bulk behaviour shows that at high frequency, hard sphere based models are able to capture the steady state behaviour reasonably well. Further investigations that modulate the driving with a low frequency amplitude change revealed the dynamic nature of flow with the low frequency component. No significant influence of high frequency signal is noted except the reduction of base heat flux. Independent analysis of bulk behaviour for modulated wave excitation using MD simulations and hydrodynamic models showed wave motion in a pattern similar to non-modulated low frequency vibration. A one-dimensional inviscid model was used to determine the underlying scaling relationships for near elastic granular flows. A form of non-dimensionalisation predicts scaling behaviour for the granular flow. The predictions show good results for the dilute flows using hard sphere MD simulations. Results from MD simulations confirm dilute limit scaling of base temperature, packing fractions and heat flux coefficients. At higher inelasticity and loading condition the model fails to capture the real physics suggesting the need for a more accurate model. This simplified model does, however, set the basis for describing the main scalings for vibrofluidized granular beds, and in the future we anticipate that effects of further inelasticity and enhanced density could be incorporated.

The imaging of the interior of the Earth using ground motion data, or seismic tomography, has been a subject of great interest for over a century. The full elastic wave equations are not typically used in standard tomography codes. Instead, the elastic waves are idealized as rays and only phase velocity and travel times are considered as input data. This results in the inability to resolve features which are on the order of one wavelength in scale. To overcome this problem, models which use the full elastic wave equation and consider total seismograms as input data have recently been developed. Unfortunately, those methods are much more computationally expensive and are only in their infancy. While the finite element method is very popular in many applications in solid mechanics, it is still not the method of choice in many seismic applications due to high pollution error. The pollution effect creates an increasing ratio of discretization to best approximation error for problems with increasing wave numbers. It has been shown that standard finite element methods cannot overcome this issue. To compensate, the meshes for solving high wave number problems in seismology must be increasingly refined, and are computationally infeasible due to the large scale requirements. A new generalized least squares method was recently introduced. The main idea is to select test spaces such that the discrete problem inherits the stability of the continuous problem. In this dissertation, a discontinuous Petrov-Galerkin method with optimal test functions for 2D time-harmonic seismic tomography problems is developed. First, the abstract DPG framework and key results are reviewed. 2D DPG methods for both static and time-harmonic elasticity problems are then introduced and results indicating the low-pollution property are shown. Finally, a matrix-free inexact-Newton method for the seismic inverse problem is developed. To conclude, results obtained from both DPG and standard continuous Galerkin discretization schemes are compared and the potential effectiveness of DPG as a practical seismic inversion tool is discussed. / text

No description available.

In intraplate seismic zones (e.g. the New Madrid Seismic Zone, NMSZ, in the southcentral United States), the source of stress that drives earthquake is very complex. Data from the NMSZ indicate 3 earthquake of magnitude M~7, occurring at an approximate interval of 500 years during the last 2000 years. One hypothesis that satisfies these conditions proposes that short-lived bursts of earthquakes may result from perturbations in the local or regional stress field. This causes relaxation of a lower crustal weak zone which drive repeated earthquakes. The number of earthquakes is dependent on the geometry and rheology of the weak zone. Using finite element techniques which employ contact surfaces to model discrete faulting events and a maximum shear stress criteria evaluated at each node. We investigate the relevant parameter space, as it affects the concentration of stress at the base of the seismogenic fault and the number of earthquakes generated over a given time interval. Parameters that can be varied include earthquake stress drop, background tectonic stress, and maximum shear stress at failure. Results show that solutions are non-unique. With the addition of existing observational evidence, however, we can place bounds on the range of parameters which satisfy above observations.

The most commonly used method for the time-domain Maxwell equations is the Finite-Difference Time-Domain method (FDTD). This is an explicit, second-order accurate method, which is used on a staggered Cartesian grid. The main drawback with the FDTD method is its inability to accurately model curved objects and small geometrical features. This is due to the Cartesian grid, which leads to a staircase approximation of the geometry and small details are not resolved at all. This thesis presents different ways to circumvent this drawback, but still take advantage of the benefits of the FDTD method. An approach to avoid staircasing errors but still retain the efficiency of the FDTD method is to use a hybrid grid. A few layers of unstructured cells are used close to curved objects and a Cartesian grid is used for the rest of the domain. For the choice of solver on the unstructured grid two different alternatives are compared: an explicit Finite-Volume Time-Domain (FVTD) solver and an implicit Finite-Element Time-Domain (FETD) solver. The hybrid solvers calculate the scattering from complex objects much more efficiently compared to using FDTD on highly resolved Cartesian grids. For the same accuracy in the solution roughly a factor of 10 in memory requirements and a factor of 20 in execution time are gained. The ability to model features that are small relative to the cell size is often important in electromagnetic simulations. In this thesis a technique to generalize a well-known subcell model for thin wires, in order to take arbitrarily oriented wires in FETD and FDTD into account, is proposed. The method gives considerable modeling flexibility compared to earlier methods and is proven stable. The results show excellent consistency and very good accuracy on different antenna configurations. The recursive convolution method is often used to model frequency dispersive materials in FDTD. This method is used to enable modeling of such materials in the unstructured FVTD and FETD solvers. The stability of both solvers is analyzed and their accuracy is demonstrated by computing the radar cross section for homogeneous as well as layered spheres with frequency dependent permittivity.

Este trabalho aborda a solução numérica das equações lineares de águas rasas. O método dos elementos finitos e utilizado para a discretização espacial das equações que modelam o problema, e para a discretização temporal, o esquema semi-implícito de Crank-Nicolson é empregado. Além de alguns conceitos comuns quando se trabalha com escoamentos geofísicos, são descritas também a formulação das equações de águas rasas, sua linearização e uma solução analítica para um caso onde o parâmetro de Coriolis é nulo. A escolha adequada de pares de elementos finitos é a principal dificuldade quando se trabalha com esse método para a resolução da equação de águas rasas. Assim, é discutido o uso de quatro pares de elementos finitos e técnicas de estabilização para contornar o surgimento de modos espúrios na solução discreta. Os resultados numéricos são realizados com auxílio do software FreeFem++, onde se pode notar a capacidade dos pares de elementos de reproduzirem o escoamento, através da solução discreta, além das propriedades de conservação de massa e energia de cada discretização. / This work is about the numerical solution of the linear shallow water equations. The finite element method is used for spatial discretization of the equations that model the problem and for the time discretization the semi-implicit Crank-Nicolson scheme is used. Besides the concepts related to geophysical flows, the formulation of the shallow water equations, their linearization and an analytical solution for a case where the Coriolis parameter is zero are also described. The appropriate choice of a pair of finite elements is the main difficulty when working with this method for solving the shallow water equations. The use of four pairs of finite elements and stabilization techniques to circumvent the appearance of spurious modes in the discrete solution are discussed. The numerical results are performed using the software FreeFem++, where one can notice the ability of the elements to represent the discrete solution and mass and energy conservation of each discretization.