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- Keyword
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- Cauchy problem for periodic initial data (1)
- Change of type (1)
- Critical buckling load, Hollomon’s law, Axial plastic columns, High strength metals, Work-hardening (1)
- Finite element method (1)
- Finite element solution, Ritz-Galerkin method, Nonlinear Euler-Bernoulli beam, Power-law, Work hardening material, Hollomon's equation, Convergence, Error estimate, Hermite elements (1)
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- Galerkin, Cauchy-Dirichlet (1)
- Generalized sine, Hamilton system, nonlinear spring, vibration, analytic solution (1)
- Global existence (1)
- Global solvability, fourth-order, nonlinear boundary value problems, monotone operator, Leray-Schauder fixed point theorem, coercivity (1)
- Heat transfer, Finite element solutions, Tubes, Power-law flows, Polymer flows, Viscous dissipation, Graetz-Nusselt (1)
- Hyperbolic-parabolic (1)
- Incompressible fluid (1)
- Infinite energy (1)
- L2 estimates (1)
- Method of lines (1)
- Nonstrict hyperbolicity, singularity formation, damping (1)
- Phase transition (1)
- Power-law (1)
- Proudman-Johnson equation, blow-up (1)
- Quasilinear parabolic problem (1)
- Stagnation point (1)
- Three dimensions (1)
Articles 1 - 14 of 14
Full-Text Articles in Physical Sciences and Mathematics
Analytic And Finite Element Solutions Of The Power-Law Euler-Bernoulli Beams, Dongming Wei, Yu Liu
Analytic And Finite Element Solutions Of The Power-Law Euler-Bernoulli Beams, Dongming Wei, Yu Liu
Mathematics Faculty Publications
In this paper, we use Hermite cubic finite elements to approximate the solutions
of a nonlinear Euler-Bernoulli beam equation. The equation is derived
from Hollomon’s generalized Hooke’s law for work hardening materials with
the assumptions of the Euler-Bernoulli beam theory. The Ritz-Galerkin finite
element procedure is used to form a finite dimensional nonlinear program
problem, and a nonlinear conjugate gradient scheme is implemented to find
the minimizer of the Lagrangian. Convergence of the finite element approximations
is analyzed and some error estimates are presented. A Matlab finite
element code is developed to provide numerical solutions to the beam equation.
Some …
Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton
Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton
Mathematics Faculty Publications
This paper examines the effect of damping on a nonstrictly hyperbolic 2 x 2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.
Critical Buckling Loads Of The Perfect Hollomon’S Power-Law Columns, Dongming Wei, Alejandro Sarria, Mohamed Elgindi
Critical Buckling Loads Of The Perfect Hollomon’S Power-Law Columns, Dongming Wei, Alejandro Sarria, Mohamed Elgindi
Mathematics Faculty Publications
In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant cross-sections. The associated critical buckling loads are calculated by Euler-type analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Holloman’s power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and …
On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei
On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei
Mathematics Faculty Publications
In this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains.
Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay
Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay
Mathematics Faculty Publications
In this work we present some analytic and semi-analytic traveling wave solutions of generalized Burger' equation for isothermal unidirectional flow of viscous non-Newtonian fluids obeying Gee-Lyon nonlinear rheological equation. The solution of Burgers' equation for Newtonian flow as a special case. We also derive estimates of shock thickness for non-Newtonian flows.
An H1 Model For Inextensible Strings, Stephen C. Preston, Ralph Saxton
An H1 Model For Inextensible Strings, Stephen C. Preston, Ralph Saxton
Mathematics Faculty Publications
We study geodesics of the H1 Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). This metric is a regularization of the usual L2 metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The H1 geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is C∞ in the Banach topology C1 ([0,1], R2), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one of the curves fixed, …
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton
Mathematics Faculty Publications
For arbitrary values of a parameter --- finite-time blowup of solutions to the generalized, inviscid Proudman Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.
Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi
Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi
Mathematics Faculty Publications
In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drabek, R. Manasevich and M. Otani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be defined by systems of first order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can …
Global Existence Of Some Infinite Energy Solutions For A Perfect Incompressible Fluid, Ralph A. Saxton, Feride Tiğlay
Global Existence Of Some Infinite Energy Solutions For A Perfect Incompressible Fluid, Ralph A. Saxton, Feride Tiğlay
Mathematics Faculty Publications
This paper provides results on local and global existence for a class of solutions to the Euler equations for an incompressible, inviscid fluid. By considering a class of solutions which exhibits a characteristic growth at infinity we obtain an initial value problem for a nonlocal equation. We establish local well-posedness in all dimensions and persistence in time of these solutions for three and higher dimensions. We also examine a weaker class of global solutions.
Phase Transitions And Change Of Type In Low-Temperature Heat, Ralph A. Saxton, Katarzyna Saxton
Phase Transitions And Change Of Type In Low-Temperature Heat, Ralph A. Saxton, Katarzyna Saxton
Mathematics Faculty Publications
Classical heat pulse experiments have shown heat to propagate in waves through crystalline materials at temperatures close to absolute zero. With increasing temperature, these waves slow down and finally disappear, to be replaced by diffusive heat propagation. Several features surrounding this phenomenon are examined in this work. The model used switches between an internal parameter (or extended thermodynamics) description and a classical (linear or nonlinear) Fourier law setting. This leads to a hyperbolic-parabolic change of type, which allows wavelike features to appear beneath the transition temperature and diffusion above. We examine the region around and immediately below the transition temperature, …
Finite Element Solutions Of Heat Transfer In Molten Polymer Flow In Tubes With Viscous Dissipation, Dongming Wei, Haibiao Luo
Finite Element Solutions Of Heat Transfer In Molten Polymer Flow In Tubes With Viscous Dissipation, Dongming Wei, Haibiao Luo
Mathematics Faculty Publications
This paper presents the results of finite element analysis of a heat transfer problem of flowing polymer melts in a tube with constant ambient temperature. The rheological behavior of the melt is described by a temperature dependent power-law model. Aviscous dissipation term is included in the energy equation. Temperature profiles are obtained for different tube lengths and different entrance temperatures. The results are compared with some similar results in the literature.
Decay Estimates Of Heat Transfer To Melton Polymer Flow In Pipes With Viscous Dissipation, Dongming Wei, Zhenbu Zhang
Decay Estimates Of Heat Transfer To Melton Polymer Flow In Pipes With Viscous Dissipation, Dongming Wei, Zhenbu Zhang
Mathematics Faculty Publications
In this work, we compare a parabolic equation with an elliptic equation both of which are used in modeling temperature profile of a power-law polymer flow in a semi-infinite straight pipe with circular cross section. We show that both models are well-posed and we derive exponential rates of convergence of the two solutions to the same steady state solution away from the entrance. We also show estimates for difference between the two solutions in terms of physical data.
A Priori Lρ Error Estimates For Galerkin Approximations To Porous Medium And Fast Diffusion Equations, Dongming Wei, Lew Lefton
A Priori Lρ Error Estimates For Galerkin Approximations To Porous Medium And Fast Diffusion Equations, Dongming Wei, Lew Lefton
Mathematics Faculty Publications
Galerkin approximations to solutions of a Cauchy-Dirichlet prob-
lem governed by a generalized porous medium equation.
Existence, Uniqueness, And Numerical Analysis Of Solutions Of A Quasilinear Parabolic Problem, Dongming Wei
Existence, Uniqueness, And Numerical Analysis Of Solutions Of A Quasilinear Parabolic Problem, Dongming Wei
Mathematics Faculty Publications
A quasilinear parabolic problem is studied. By using the method of lines, the existence and uniqueness of a solution to the initial boundary value problem with sufficiently smooth initial conditions are shown. Also given are L2 error estimates for the error between the extended fully discrete finite element solutions and the exact solution.