Dongming Wei

Numerical approximation of the first eigenpair of the p-laplacian using finite elements and the penalty method

For 1 consider the eigenvalue problem for the p-Laplacian where . The first eigenvalue λ 1 can be obtained by minimizing the functional over . A method for computing λ 1 numerically is presented. The technique uses a finite element approximation to the first eigenfunction and a penalty function to enforce the constraint. Convergence is proved and numerical results are presented. The numerical results are compared with exact values when known. A lower bound for p -Laplacian eigenvalues is also presented. In particular, this work provides a computational framework for obtaining precise approximations of the best constant for the Sobolev imbedding .

A priori L r error estimates for Galerkin approximations to porous medium and fast diffusion equations

Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation formulamath on bounded convex domains are considered. The range of the parameter ρ includes the fast diffusion case 1 < ρ < 2. Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in L∞(0,T; L ρ (Ω)) norm with an error controlled by O(Δt1/4) for 1 < ρ < 2 and O(Δt1/2ρ) for 2 < ρ < ∞. For the fully discrete problem, a global convergence rate of O(Δt1/4) in L 2 (0,T;L ρ (Ω)) norm is shown for the range 2N/N + 1 < ρ < 2. For 2 < ρ < ∞, a rate of O(Δt1/2ρ) is shown in L ρ (0, T; L ρ (Ω)) norm.

Penalty finite element approximations of the stationary power-law Stokes problem

Finite element approximations of the stationary power-law Stokes problem using penalty formulation are considered. A priori error estimates under appropriate smoothness assumptions on the solutions are established without assuming a discrete version of the BB condition. Numerical solutions are presented by implementing a nonlinear conjugate gradient method.

PENALTY APPROXIMATIONS TO THE STATIONARY POWER-LAW NAVIER-STOKES PROBLEM

In this work, penalty approximations to the steady state Navier-Stokes problem governed by the the power-law model for viscous incompressible non-Newtonian flows in bounded convex domains in Rd (2 ≤ d) are studied. Existence and uniqueness of solutions to the penalty approximations are proved, convergence is shown, and rates of convergence are derived.

Optimal numerical flux of power-law fluids in some partially full pipes

Consider the steady state pressure driven flow of a power-law fluid in a partially filled straight pipe. It is known that an increase in flux can be achieved for a fixed pressure by partially filling the pipe and having the remaining volume either void or filled with a less viscous, lubricating fluid. If the pipe has circular cross section, the fluid level which maximizes flux is the level which avoids contact with exactly 25% of the boundary. This result can be proved analytically for Newtonian fluids and has been verified numerically for certain non-Newtonian models. This paper provides a generalization of this work numerically to pipes with non-circular cross sections which are partially full with a power-law fluid. A simple and physically plausible geometric condition is presented which can be used to approximate the fluid level that maximizes flux in a wide range of pipe geometries. Additional increases in flux for a given pressure can be obtained by changing the shape of the pipe but leaving the perimeter fixed. This computational analysis of flux as a function of both fluid level and pipe geometry has not been considered to our knowledge. Fluxes are computed using a special discretization scheme, designed to uncover general properties which are only dependent on fluid level and/or pipe cross-sectional geometry. Computations use finite elements and take advantage of the variational structure inherent in the power-law model. A minimization technique for approximating the critical points of the associated non-linear energy functional is used. In particular, the numerical scheme for the non-linear partial differential equation has been proved to be convergent with known error estimates. The numerical results obtained in this work can be useful for designing pipes and canals for transportation of non-Newtonian fluids, such as those in chemical engineering and food processing engineering.

Solvability of the Brinkman-Forchheimer-Darcy Equation

The nonlinear Brinkman-Forchheimer-Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy’s range.

Nonlinear Waves in Rods and Beams of Power-Law Materials

Some novel traveling waves and special solutions to the 1D nonlinear dynamic equations of rod and beam of power-law materials are found in closed forms. The traveling solutions represent waves of high elevation that propagates without change of forms in time. These waves resemble the usual kink waves except that they do not possess bounded elevations. The special solutions satisfying certain boundary and initial conditions are presented to demonstrate the nonlinear behavior of the materials. This note demonstrates the apparent distinctions between linear elastic and nonlinear plastic waves.

Buckling and post-buckling of graphene tubes

Abstract Buckling and post-buckling of a long, thin, inextensible tube made of graphene material acted upon by a uniform normal pressure are considered. A system of differential equations governing the equilibrium states of such a tube is derived. Perturbation solutions are provided for the cases of pressure values close to the critical buckling pressures. A Matlab numerical solver based on Newton’s and shooting methods are developed. The buckling and post-buckling shapes of the deformed tube subject to various levels of pressure are presented.

Characterization of Corneal Indentation Hysteresis

Corneal indentation is adapted for the design and development of a characterization method for corneal hysteresis behavior - Corneal Indentation Hysteresis (CIH). Fourteen porcine eyes were tested using the corneal indentation method. The CIH measured in enucleated porcine eyes showed indentation rate and intraocular pressure (IOP) dependences. The CIH increased with indentation rate at lower IOP (<; 25 mmHg) and decreased with indentation rate at higher IOP (> 25 mmHg). The CIH was linear proportional to the IOP within an individual eye. The CIH was positively correlated with the IOP, corneal in-plane tensile stress and corneal tangent modulus (E). A new method based on corneal indentation for the measurement of Corneal Indentation Hysteresis in vivo is developed. To our knowledge, this is the first study to introduce the corneal indentation hysteresis and correlate the corneal indentation hysteresis and corneal tangent modulus.

An iterative finite difference scheme for buckling of graphene beam subject to axial compressive load

In this paper buckling loads and modes of an Euler beam made of graphene are considered. An eigenvalue problem is formulated with an additional parameter representing materials elastoplasticity. Analytical approximate solution using the perturbation method is presented. This approximate solution is used as a basis for an iterative finite difference scheme to develop a more accurate solution. Deflection at points along the beam are calculated using this iterative finite difference scheme. Convergence with good accuracy is achieved.