Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices

In the present work, we show that the linearized homogenized model for a pantographic lattice must necessarily be a second gradient continuum, as defined in Germain (1973). Indeed, we compute the effective mechanical properties of pantographic lattices following two routes both based in the heuristic homogenization procedure already used by Piola (see Mindlin, 1965; dell'Isola et al., 2015a): (i) an analytical method based on an evaluation at micro-level of the strain energy density and (ii) the extension of the asymptotic expansion method up to the second order. Both identification procedures lead to the construction of the same second gradient linear continuum. Indeed, its effective mechanical properties can be obtained by means of either (i) the identification of the homogenized macro strain energy density in terms of the corresponding micro-discrete energy or (ii) the homogenization of the equilibrium conditions expressed by means of the principle of virtual power: actually the two methods produce the same results. Some numerical simulations are finally shown, to illustrate some peculiarities of the obtained continuum models especially the occurrence of bounday layers and transition zones. One has to remark that available well-posedness results do not apply immediately to second gradient continua considered here

Isogeometric Finite Element Analysis of Nonlinear Structural Vibrations

In this thesis we present a new method for nonlinear frequency response analysis of mechanical vibrations. For an efficient spatial discretization of nonlinear partial differential equations of continuum mechanics we employ the concept of isogeometric analysis. Isogeometric finite element methods have already been shown to possess advantages over classical finite element discretizations in terms of exact geometry representation and higher accuracy of numerical approximations using spline functions. For computing nonlinear frequency response to periodic external excitations, we rely on the well-established harmonic balance method. It expands the solution of the nonlinear ordinary differential equation system resulting from spatial discretization as a truncated Fourier series in the frequency domain. A fundamental aspect for enabling large-scale and industrial application of the method is model order reduction of the spatial discretization of the equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information. We investigate the concept of modal derivatives theoretically and using computational examples we demonstrate the applicability and accuracy of the reduction method for nonlinear static computations and vibration analysis. Furthermore, we extend nonlinear vibration analysis to incompressible elasticity using isogeometric mixed finite element methods.In dieser Arbeit stellen wir eine neuartige Methode für die nichtlineare Frequenzanalyse von angeregten mechanischen Schwingungen vor. Zur effizienten Ortsdiskretisierung der partiellen Differentialgleichungen der nichtlinearen Kontinuumsmechanik wenden wir das Prinzip der isogeometrischen Analyse an. Die isogeometrische Finite Elemente Methode bietet im Vergleich zu klassischen Finiten Elemente-Diskretisierungen zahlreiche Vorteile, insbesondere eine exakte Geometriedarstellung und höhere Genauigkeit der numerischen Approximationen mittels Spline-Funktionen. Anschließend verwenden wir die Harmonic Balance Methode zur Berechnung der nichtlinearen Schwingungsantwort bei periodischen externen Anregungen. Dabei wird die Lösung des aus der Ortsdiskretisierung resultierenden, nichtlinearen gewöhnlichen Differentialgleichungssystems im Frequenzraum als abgeschnittene Fourierreihe entwickelt. Um eine effektive Anwendung der Methode auf große Systeme im Rahmen von industriellen Problemen zu ermöglichen, ist eine Modellreduktion der Ortsdiskretisierung der Bewegungsgleichung notwendig. Dazu schlagen wir eine modale Projektionsmethode vor, die mit modalen Ableitungen und damit Informationen zweiter Ordnung erweitert wird. Wir untersuchen das Prinzip der modalen Ableitungen theoretisch und demonstrieren anhand numerischer Beispiele die Anwendbarkeit und Genauigkeit der Reduktionsmethode bei der nichtlinearen Frequenzanalyse. Außerdem erweitern wir die nichtlineare Vibrationsanalyse mittels gemischter isogeometrischer Methoden auf inkompressible Elastizität

Superconvergence and error estimation of finite element solutions to fire-exposed frame problems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.When a fire reaches the point of flashover the hot gases inside the burning room ignite resulting in furnace-like conditions. Thereafter, the building frame experiences temperatures sufficient to compromise its structural integrity. Physical and mathematical models help to predict when this will happen. This thesis looks at both the thermal and structural aspects of modelling a frame exposed to a post-flashover fire. The temperatures in the frame are calculated by solving a 2D heat conduction equation over the cross-section of each beam. The solution procedure uses the finite element method with automatic mesh generation/adaption based on the Delaunay triangulation process and the recovered heat flux. With the Euler-Bernoulli assumption that the cross-section of a beam remains plane and perpendicular to the neutral line and that strains are small, an error estimator, based on the work of Bank and Weiser [9], has been derived for finite element solutions to small-deformation, thermoelastic and thermoplastic frame problems. The estimator has been shown to be consistent for all finite element solutions and asymptotically exact when the solution involves appropriate higher degree polynomials. The asymptotic exactness is shown to be closely related to superconvergence properties of the approximate solution in these cases. Specifically, with coupled bending and compression, it is necessary to use quadratic approximations, instead of linear, for the compression and twisting terms to get a global O(h2) rate of convergence in the energy norm, some superconvergence properties and asymptotic exactness with the error estimator

Geometric Mechanics of the Vertical Slice Model

The goals of the present work are to: (i) investigate the dynamics of oceanic frontogenesis by taking advantage of the geometric mechanics underlying the class of Vertical Slice Models (VSMs) of ocean dynamics; and (ii) illustrate the versatility and utility of deterministic and stochastic variational approaches by deriving several variants of wave-current interaction models which describe the effects of internal waves propagating within a vertical planar slice embedded in a 3D region of constant horizontal gradient of buoyancy in the direction transverse to the vertical pane.Comment: 32 pages, 3 figures, comments welcome by emai

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”