Browsing by Author "Kloucek, Petr"

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    A nonlinear thermodynamic model for phase transitions in shape memory alloy wires
    (2003) Reynolds, Daniel Ryan; Kloucek, Petr
    Through a mathematical and computational model of the physical behavior of shape memory alloy wires, this study shows that localized heating and cooling of such materials provides an effective means of damping vibrational energy. The thermally induced pseudo-elastic behavior of a shape memory wire is modeled using a continuum thermodynamic description based on an improved Landau-Devonshire potential. Our construction of the potential function allows the model to account for particular alloys as well as the general solid-state phase transformation, improving over traditional potentials that idealize many of the material properties or focus only on individual processes. The material's thermodynamic response is modeled using a nonlinear conservation of momentum and a nonlinear heat equation. The heat equation introduces an inhomogeneous version of the Fourier heat flux, thereby describing the discontinuous heat flow associated with shape memory materials more thoroughly than standard, continuous heat dissipation mechanisms do. This continuum thermodynamic model is then solved computationally to determine the resulting state of the wire in time. Continuous time Galerkin methods and affine finite elements treat the temporal and spatial discretizations of the model, respectively. Traditional methods for solution of the resulting finite-dimensional, nonlinear, nonconvex system of equations must introduce a significant artificial dissipation to achieve existence of solutions. The solution of the discrete system here uses a novel combination of the damped Newton method and a homotopy method for minimizing the artificial dissipation. This combination, inspired by the well-known Method of Vanishing Viscosity for the solution of scalar hyperbolic conservation laws, reduces the artificial dissipation errors introduced by traditional approaches for such nonlinear, nonconvex thermomechanical models. Computational tests show that the proposed model successfully describes the relevant physical processes inherent in shape memory alloy behavior. Further computational experiments then confirm that up to 80% of an initial shock of vibrational energy can be eliminated at the onset of a thermally-induced phase transformation.
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    Approximation and computation of the solution to the magnetosphere-ionosphere coupling equation via a mixed formulation
    (2004) Wightman, Jennifer Lee; Kloucek, Petr; Toffoletto, Frank R.
    This study develops a numerical technique for the approximation of the magnetosphere-ionosphere (MI) coupling equation, which is a crucial step in the Rice Convection Model (RCM), a physical model that treats plasma in Earth's inner and middle magnetosphere via a multi-fluid approximation. The MI coupling equation is a second-order elliptic boundary value problem that describes conservation of current between the magnetosphere and the ionosphere. The current RCM solver is based on a finite difference scheme and produces unphysical results when the ionospheric conductance has large spatial gradients. We develop an alternative finite element approximation of the MI coupling equation, applying the method of fictitious domains to treat the high-latitude boundary condition along the immersed boundary Gamma, a boundary that varies in time and does not align with the computational grid. The result of using fictitious domains is a domain decomposition problem that we solve via a mixed finite element formulation. We compare both a nonconforming and conforming finite element approach within the framework of the mixed formulation. We are able to demonstrate that both the conforming and nonconforming methods generate solutions that are compatible with the current RCM solver when actual RCM data is used. Furthermore, we demonstrate on several analytic test examples that the finite element approximation is more accurate than the finite difference approximation. Therefore, we conclude that the finite element solver is more robust than the finite difference solver. In addition, we provide convergence results for the nonconforming approximation when the conductance coefficients are bounded and measurable, and we use spectral theory from the harmonic Steklov eigenproblem to derive a precise definition of the trace space on the interface Gamma. Our overall approximation technique is generalizable to a class of elliptic boundary value problems in which the boundary varies in time or does not align with a fixed grid. Finally, our numerical solver can be modified for use in the RCM-Jupiter that is currently being developed.
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    Computational modeling of internal surfaces in austenite-martensite system
    (2003) Melara, Luis Adolfo; Kloucek, Petr
    In this work, we present a new computational method based on a nonconforming domain decomposition technique for modeling of phase transitions. Phase transitions are the result of thermal or mechanical loading in ferromagnetic materials or shape memory materials. Modeling of phase transitions is important because it can help to predict or control the behavior of these materials. This thesis will focus on phase transitions characterized by two directions of magnetization in the case of ferromagnets and two variants of Martensite in the case of shape memory materials. In both types of materials, branching occurs near an internal surface which is characterized by complex microstructures. These microstructures occur at a minimum energy state. The new computational method simulates the branching behavior of these microstructures near an internal surface. We approximate the microstructures via energy minimization. We minimize the total stored energy stored in vicinity of internal surface with the minimizing function representing the microstructures. We compare the numerical results obtained by the new technique with those obtained by a more standard technique, one not incorporating nonconforming domain decomposition. Furthermore, we verify the various energy scaling laws used to predict the total stored energy near an internal surface. Among these laws, we verify the local-in-y scaling property which has been conjectured but not proven.
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    The Steepest Descent Minimization of Double-Well Stored Energies Does Not Yield Vectorial Microstructures
    (2001-03) Kloucek, Petr
    We prove that the Steepest Descent algorithm applied to the minimization of total stored energies with rank-one related rotationally symmetric energy wells does not produce relaxing vectorial microstructures with non-trivial Young measures.
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    Vibration damping and heat transfer using material phase changes
    (2009-03-24) Kloucek, Petr; Reynolds, Daniel R.; Rice University; United States Patent and Trademark Office
    A method and apparatus wherein phase changes in a material can dampen vibrational energy, dampen noise and facilitate heat transfer. One embodiment includes a method for damping vibrational energy in a body. The method comprises attaching a material to the body, wherein the material comprises a substrate, a shape memory alloy layer, and a plurality of temperature change elements. The method further comprises sensing vibrations in the body. In addition, the method comprises indicating to at least a portion of the temperature change elements to provide a temperature change in the shape memory alloy layer, wherein the temperature change is sufficient to provide a phase change in at least a portion of the shape memory alloy layer, and further wherein the phase change consumes a sufficient amount of kinetic energy to dampen at least a portion of the vibrational energy in the body. In other embodiments, the shape memory alloy layer is a thin film. Additional embodiments include a sensor connected to the material.