The Fourier finite element method for the corner singularity expansion of the Heat equation

Near the non-convex vertex the solution of the Heat equation is of the form u = (c star epsilon) chi r(pi/omega) sin(pi theta/omega) + omega, omega is an element of L-2(R+; H-2), where c is the stress intensity function of the time variable t,* the convolution, epsilon (x, t) = re(-r2/4t)/2 root pi t(3), chi a cutoff function and omega the opening angle of the vertex. In this paper we use the Fourier finite element method for approximating the stress intensity function c and the regular part omega, and derive the error estimates depending on the regularities of c and omega. We give some numerical examples, confirming the derived convergence rates. (C) 2014 Elsevier Ltd. All rights reserved.X111Ysciescopu

Nonreflecting boundary conditions for time-dependent wave propagation

Many problems in computational science arise in unbounded domains and thus require an artificial boundary B, which truncates the unbounded exterior domain and restricts the region of interest to a finite computational domain, . It then becomes necessary to impose a boundary condition at B, which ensures that the solution in coincides with the restriction to of the solution in the unbounded region. If we exhibit a boundary condition, such that the fictitious boundary appears perfectly transparent, we shall call it exact. Otherwise it will correspond to an approximate boundary condition and generate some spurious reflection, which travels back and spoils the solution everywhere in the computational domain. In addition to the transparency property, we require the computational effort involved with such a boundary condition to be comparable to that of the numerical method used in the interior. Otherwise the boundary condition will quickly be dismissed as prohibitively expensive and impractical. The constant demand for increasingly accurate, efficient, and robust numerical methods, which can handle a wide variety of physical phenomena, spurs the search for improvements in artificial boundary conditions. In the last decade, the perfectly matched layer (PML) approach [16] has proved a flexible and accurate method for the simulation of waves in unbounded media. Standard PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as first-order systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step we propose instead a simple PML formulation directly for the wave equation in its second-order form. Our formulation requires fewer auxiliary unknowns than previous formulations [23, 94]. Starting from a high-order local nonreflecting boundary condition (NRBC) for single scattering [55], we derive a local NRBC for time-dependent multiple scattering problems, which is completely local both in space and time. To do so, we first develop a high order exterior evaluation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the local NRBC on the artificial boundary. By combining that evaluation formula with the decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NRBC for time-dependent multiple scattering. Remarkably, the information transfer (of time retarded values) between sub-domains will only occur across those parts of the artificial boundary, where outgoing rays intersect neighboring sub-domains, i.e. typically only across a fraction of the artificial boundary. The accuracy, stability and efficiency of this new local NRBC is evaluated by coupling it to standard finite element or finite difference methods

Spectral elements for guided waves. Formulation, Dispersion Analysis and Simulation Results

Résumé : La surveillance de l’intégrité des structures (Structural Health Monitoring - SHM) est une nouvelle technologie, et comme toute nouvelle avancée technologique, elle n’a pas encore réalisé son plein potentiel. Le SHM s’appuie sur des avancées dans plusieurs disciplines, dont l’évaluation non-desctructive, les matériaux intelligents, et les capteurs et actionneurs intégrés. Une des disciplines qui permet son déploiement est la simulation numérique. Le SHM englobe une variété de techniques basées sur la génération d’ondes vibratoires et d’ondes ultrasonores guidées. L’utilisation d’ondes guidées offre en particulier une vaste gamme d’avantages. Le défi majeur associé à la pleine utilisation de la simulation numérique dans la conception d’un système SHM basé sur l’utilisation d’ondes guidées réside dans les ressources de calcul requises pour une simulation précise. La principale raison pour ces exigences est la dispersion induite par la discrétisation numérique, tel qu’indiqué dans la littérature. La méthodes des éléments spectraux (SEM) est une variante de la p-version de la méthode des éléments finis (FEM) qui offre certains outils pour solutionner le problème des erreurs de dispersion, mais la littérature souffre toujours d’une lacune dans l’étude systématique des erreurs de dispersion numérique et de sa dépendance sur les paramètres de simulation. Le présent ouvrage tente de combler cette lacune pour les théories d’ingénierie en vibrations. Il présente d’abord le développement de la formulation des éléments spectraux pour différentes théories d’ingénierie pertinentes pour la propagation des ondes vibratoires dans différents types de structures, comme des tiges et des plaques. Puis, une nouvelle technique pour le calcul des erreurs de dispersion numériques est présentée et appliquée systématiquement dans le but d’évaluer la dispersion numérique induite en termes d’erreurs dans les vitesses de propagation. Cette technique est utilisable pour les différentes formes de propagation des ondes vibratoires dans les éléments structuraux visés dans la présente thèse afin d’évaluer quantitativement les exigences de précision en termes de paramètres de maillage. Les ondes de Lamb constituent un cas particulier de la déformation plane des ondes élastiques, en raison de la présence des doubles frontières à traction libre qui couplent les ondes longitudinales et de cisaillement et qui conduisent à une infinité de modes propagatifs qui sont dispersifs par nature. La simulation des ondes de Lamb n’a pas fait l’objet d’analyse systématique de la dispersion numérique dans la littérature autant pour la SEM que la FEM. Nous rapportons ici pour la première fois les résultats de l’analyse de dispersion numérique pour la propagation des ondes Lamb. Pour toutes les analyses de dispersion numérique présentées ici, l’analyse a été effectuée à˘ala fois dans le domaine fréquentiel et dans le domaine temporel. En se basant sur la nouvelle compréhension des effets de discrétisation numérique de la propagation des ondes guidées, nous étudions l’application de la SEM à la simulation numérique pour des applications de conception en SHM. Pour ce faire, l’excitation piézoélectrique est développée, et une nouvelle technique de condensation statique est développée et mise en œuvre pour les équations de la matrice semi-discrète, qui élimine le besoin de solution itérative, ainsi surnommée fortement couplée ou entièrement couplée. Cet élément piézoélectrique précis est ensuite utilisé pour étudier en détails les subtilités de la conception d’un système SHM en mettant l’accent sur la propagation des ondes de Lamb. Afin d’éviter la contamination des résultats par les réflexions sur les bords une nouvelle forme particulière d’élément absorbant a été développée et mise en œuvre. Les résultats de simulation dans le domaine fréquentiel jettent un éclairage nouveau sur les limites des modèles théoriques actuels pour l’excitation des ondes de Lamb par piézoélectriques. L’excitation par un élément piézoélectrique couplé est ensuite entièrement simulée dans le domaine temporel, et les résultats de simulation sont validés par deux cas de mesures expérimentales ainsi que par la simulation classique avec des éléments finis en utilisant le logiciel commercial ANSYS. // Abstract : Structural health monitoring (SHM) is a novel technology, and like any new technological advancement it has yet not realized its full potential. It builds on advancements in several disciplines including nondestructive evaluation, smart materials, and embedded sensors and actuators. One of the enabling disciplines is the numerical simulation. SHM encompasses a variety of techniques, vibration based, impedance and guided ultrasonic waves. Guided waves offers a wide repertoire of advantages. The major challenge facing the full utilization of the numerical simulation in designing a viable guided waves based SHM System is the formidable computational requirements for accurate simulation. The main reason for these requirements is the dispersion induced by numerical discretization as explained in the literature review. The spectral element (SEM) is a variant of the p-version finite element (FEM) that offers certain remedies to the numerical dispersion errors problem, yet it lacks a systematic study of the numerical dispersion errors and its dependence on the meshing parameters. The present work attempts to fill that gap for engineering theories. It starts by developing the formulation of the spectral element for different relevant engineering theories for guided waves propagation in various structural elements, like rods and plates. Then, extending the utility of a novel technique for computing the numerical dispersion errors, we systematically apply it in order to evaluate the numerically induced dispersion in terms of errors in the propagation speeds. This technique is employed for the various forms of guided waves propagation in structural elements covered in the present thesis in order to quantitatively assess the accuracy requirements in terms of the meshing parameters. The Lamb guided waves constitute a special case of the plane strain elastic waves, that is due to the presence of the double traction free boundaries, couple in the section plane and this coupling leads to an infinitude of propagating modes that are dispersive in nature. Lamb waves simulation have not been a subject of numerical dispersion analysis in the open literature neither for SEM nor FEM for that matter. We report here for the first time the numerical dispersion analysis results for Lamb waves propagation. For all the numerical dispersion analysis presented here, the analysis was done for both the frequency domain and time domain analysis. Based on the established understanding of the numerical discretization effects on the guided waves propagation, we utilize this knowledge to study the application of SEM to SHM simulations. In order to do so the piezoelectric excitation is developed, and a new static condensation technique is developed for the semidiscrete matrix equations, that eliminate the need for iterative solution, thus dubbed strongly coupled or fully coupled implementation. This accurate piezoelectric element are then used to study in details the intricacies of the design of an SHM system with specific emphasis on the Lamb waves propagation. In order to avoid the contamination of the results by the reflections from the edges a new special form of absorbing boundary was developed and implemented. The Simulation results in the frequency domain illuminated the limitations of the current theoretical models for piezoelectric excitation of Lamb waves. The piezoelectric excitation of a fully coupled element is then simulated in the time domain, and the results of simulation was verified against two cases of experimental measurements as well as conventional finite element simulation using the commercial software ANSYS

Dual-Porosity and Dual-Permeability Poromechanics Solutions for Problems in Laboratory and Field Applications

In this work, a study of anisotropic dual-porosity and dual-permeability poromechanics is presented through generalized analytical solutions for selected problems in laboratory and field applications. For example, the solution to the inclined wellbore geometry with standard applications in the oil and gas industry for drilling stability or consolidation in naturally fractured rock formations are derived and illustrated. In addition, the dual-porosity and dual-permeability poromechanics solutions to common laboratory testing setups in geomechanics and biomechanics for purposes of rock and bio-tissue characterization are developed for rectangular strip, solid and hollow cylinder geometries.The behaviors of naturally fractured rock formations or the responses of the well known dual-porosity bone structure are modeled as dual-porosity and dual-permeability poroelastic media that fully couples the secondary porosity medium's deformation, fluid flow and interporosity exchange processes. For chemically active fractured media, e.g., clay, shale, or biomaterial, chemical interaction effects including osmotic and solute transport in both the primary porosity (matrix) and secondary porosity (fracture) are addressed based on non-equilibrium thermodynamics. Thermohydromechanical coupling under non-isothermal condition is incorporated by adopting a "single-temperature" approach in which a single representative thermodynamic continuum is argued to be sufficient to describe the thermally induced responses of a naturally fractured rock formation.The physical and mathematical models are used to find poromechanics analytical solutions for pore pressure, fluid flux, stress, and displacement, in addition to solute flux for chemically active material or temperature for non-isothermal condition to the above problem geometries. These solutions are general and can be tailored to simulate specific field problems or experimental testing. For instance, the inclined wellbore solutions include boundary conditions for simulating openhole drilling and fluid injection or withdrawal. On the other hand, the solutions for laboratory testing of rectangular and cylinder geometries account for two primary axial loading modes, namely, stroke control or stress relaxation and load control or creep test. The rectangular strip solution is also shown to simplify to the classical one-dimensional consolidation in soil mechanics.For non-reactive dual-porous material under isothermal condition, generic dual-poromechanics results are plotted and compared with single-poromechanics counterpart representing a homogenous isotropic medium when applicable. Parametric analyses are also carried out through the responses of a solid cylinder under unconfined compression to evaluate the effects of material anisotropy and dimensionless dual-poroelastic parameters such as permeability ratio, storage ratio, and interporosity coefficient. For chemically active fractured formation, the analyses is focused on the impacts of chemical salinity gradients via osmotic and solute transport on pore pressure and effective stress distributions near the wellbore or fluid/solute flux and displacement of solid cylinder under axial-flow-only oedometer testing setup. Finally, the effects of temperature gradients manifested through thermal expansion/contraction and conductive heat transport are assessed using the analytical solutions for inclined wellbore and rectangular strip geometries. Furthermore, the significance of heat convection is evaluated numerically and displayed.Application-wise, the inclined wellbore solution is used to perform time-dependent wellbore stability analysis for drilling through chemically active fractured rock formations under non-isothermal conditions. The hollow cylinder is applied to study elastic consolidation of a producing naturally fractured reservoir and associated implications on porosity and permeability reduction in the near-wellbore region. Finally, some realistic quasi-static loading conditions commonly encountered in experimental testing and field applications such as cyclic, linear ramping, and exponentially decayed are demonstrated via the solutions of unconfined solid cylinder