# Browsing by Subject "Lagrange equations"

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Item Discrete Lagrange equations for reacting thermofluid systems(2009-05) Hean, Charles Robert, 1960-; Fahrenthold, Eric P.Show more The application of Lagrange's equations to non-equilibrium reacting compressible thermofluid systems yields a modeling methodology for thermofluid dynamics compatible with the discrete energy methods used extensively in other energy domains; examples include mechanical systems simulations and molecular dynamics modeling. The introduction of internal energies as generalized coordinates leads to a thermomechanical model with a simple but general form. A finite element interpolation is used to formulate the ODE model in an ALE reference frame, without reference to any partial differential equations. The formulation is applied to highly nonlinear problems without the use of any time-splitting or shock-tracking methods. The method is verified via the solution of a set of example problems which incorporate a variety of reference frames, both open and closed control volumes, and moving boundaries. The example simulations include transient detonations with complex chemistry, piston-initiated detonations, canonical unstable overdriven detonations, high-resolution induction-zone species evolution within a pulsating hydrogen-air detonation, and the detonation of a solid explosive due to high-velocity impact.Show more Item Hamilton's equations with Euler parameters for hybrid particle-finite element simulation of hypervelocity impact(2002) Shivarama, Ravishankar Ajjanagadde; Fahrenthold, Eric P.Show more Hypervelocity impact studies (impact velocities > 1 km/sec) encompass a wide range of applications including development of anti-terrorist defense and orbital debris shield for the International Space Station (ISS). The focus of this work is on the development of a hybrid particle-finite element method for orbital debris shield simulations. The problem is characterized by finite strain kinematics, strong energy domain coupling, contact-impact, shock wave propagation and history dependent material damage effects. A novel hybrid particle finite element method based on Hamilton’s equations is presented. The model discretizes the continuum of interest simultaneously (but not redundantly) into particles and finite elements. The particles are ellipsoidal in shape and can translate and rotate in three dimensional space. Rotation is described using Euler parameters. Volumetric and contact impact effects are modeled using particles, while strength is modeled using conventional Lagrangian finite elements. The model is general enough to accommodate a wide range of material models and equations of state.Show more Item A model of the interaction of bubbles and solid particles under acoustic excitation(2008-05) Hay, Todd Allen, 1979-; Hamilton, Mark F.Show more The Lagrangian formalism utilized by Ilinskii, Hamilton and Zabolotskaya [J. Acoust. Soc. Am. 121, 786-795 (2007)] to derive equations for the radial and translational motion of interacting bubbles is extended here to obtain a model for the dynamics of interacting bubbles and elastic particles. The bubbles and particles are assumed to be spherical but are otherwise free to pulsate and translate. The model is accurate to fifth order in terms of a nondimensional expansion parameter R/d, where R is a characteristic radius and d is a characteristic distance between neighboring bubbles or particles. The bubbles and particles may be of nonuniform size, the particles elastic or rigid, and external acoustic sources are included to an order consistent with the accuracy of the model. Although the liquid is assumed initially to be incompressible, corrections accounting for finite liquid compressibility are developed to first order in the acoustic Mach number for a cluster of bubbles and particles, and to second order in the acoustic Mach number for a single bubble. For a bubble-particle pair consideration is also given to truncation of the model at fifth order in R/d via automated derivation of the model equations to arbitrary order. Numerical simulation results are presented to demonstrate the effects of key parameters such as particle density and size, liquid compressibility, particle elasticity and model order on the dynamics of single bubbles, pairs of bubbles, bubble-particle pairs and clusters of bubbles and particles under both free response conditions and sinusoidal or shock wave excitation.Show more Item Topics in Lagrangian and Hamiltonian fluid dynamics : relabeling symmetry and ion-acoustic wave stability(1998) Padhye, Nikhil Subhash, 1970-; Morrison, Philip J.Show more Relabeling symmetries of the Lagrangian action are found for the ideal, compressible fluid and magnetohydrodynamics (MHD). These give rise to conservation laws of potential vorticity (Ertel's theorem) and helicity in the ideal fluid, cross helicity in MHD, and a conservation law for an ideal fluid with three thermodynamic variables. The symmetry that gives rise to Ertel's theorem is generated by an infinite parameter group, and leads to a generalized Bianchi identity. The existence of a more general symmetry is also shown, with dependence on time and space derivatives of the fields, and corresponds to a family of conservation laws associated with the potential vorticity. In the Hamiltonian formalism, Casimir invariants of the noncanonical formulation are directly constructed from the symmetries of the reduction map from Lagrangian to Eulerian variables. Casimir invariants of MHD include a gauge-dependent family of invariants that incorporates magnetic helicity as a special case. Novel examples of finite dimensional, noncanonical Hamiltonian dynamics are also presented: the equations for a magnetic field line flow with a symmetry direction, and Frenet formulas that describe a curve in 3-space. In the study of Lyapunov stability of ion-acoustic waves, existence of negative energy perturbations is found at short wavelengths. The effect of adiabatic, ionic pressure on ion-acoustic waves is investigated, leading to explicit solitary and nonlinear periodic wave solutions for the adiabatic exponent r = 3. In particular, solitary waves are found to exist at any wave speed above Mach number one, without an upper cutoff speed. Negative energy perturbations are found to exist despite the addition of pressure, which prevents the establishment of Lyapunov stability; however the stability of ion-acoustic waves is established in the KdV limit, in a manner far simpler than the proof of KdV soliton stability. It is also shown that the KdV free energy (Benjamin, 1972) is recovered upon evaluating (the negative of) the ion-acoustic free energy at the critical point, in the KdV approximation. Numerical study of an ion-acoustic solitary wave with a negative energy perturbation shows transients with increased perturbation amplitude. The localized perturbation moves to the left in the wave-frame, leaving the solitary wave peak intact, thus indicating that the wave may be stable.Show more