Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

This is the published version. Copyright © 2011 Scientific Research PublishingThe present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method (GM ), Galerkin method with weak form (/GMWF ), Petrov-Galerkin method (PGM), weighted residual method (WRY ), and least squares method or process (LSM or LSP ) to construct finite element approximations in time. A correspondence is estab- lished between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) com- putational processes for which types of operators and, 2) to establish which integral forms do not yield un- conditionally stable computations (variationally inconsistent integral forms, VIC). It is shown that varia- tionally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approxima- tions as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s method as well as Newmark method to demonstrate highly meritorious features of the pro- posed methodology

Thermodynamic Consistency of the currently used Beam Mathematical Models and Thermodynamically Consistent New Formulations for Bending of Thermoelastic, and Thermoviscoelastic Beams

In order to enhance currently used beam mathematical models in R^2 and R^3 to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism for the currently used beam mathematical models. This is the main motivation for the work presented in this dissertation. The currently used beam mathematical models for homogeneous, isotropic matter and reversible deformation physics are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler's equation(s) extracted from the first variation of this energy functional set to zero yield the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain these two approaches yield same mathematical models. The energy methods or the principle of virtual work cannot be used for irreversible process, thus precluding their use in the presence of dissipation and memory mechanisms. In this dissertation we examine whether the currently used beam mathematical models for reversible deformation physics and with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler-Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher order thermodynamically consistent beam mathematical models that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented here ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams. The new formulation presented for thermoelastic reversible mechanical deformation is extended for thermoviscoelastic beams with dissipation and for thermoviscoelastic beams with dissipation and memory. In each case model problem studies are presented using currently used mathematical models (when possible) and the results are compared with those obtained using the new thermodynamically consistent formulation presented here