Nonlinear boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

In this paper, we consider the boundary control problem of the unforced generalized Korteweg–de Vries–Burgers (GKdVB) equation when the spatial domain is [0,1]. Three control laws are derived for this equation and the L ²-global exponential stability of the solution is proved analytically. Numerical results using the finite element method (FEM) are presented to illustrate the developed control schemes.     

MOSQUITO: An efficient finite difference scheme for numerical simulation of 2D advection

An explicit finite difference method for the treatment of the advective terms in the 2D equation of unsteady scalar transport is presented. The scheme is a conditionally stable extension to two dimensions of the popular QUICKEST scheme. It is deduced imposing the vanishing of selected components of the truncation error for the case of steady uniform flow. The method is then extended to solve the conservative form of the depth‐averaged transport equation. Details of the accuracy and stability analysis of the numerical scheme with test case results are given, together with a comparison with other existing schemes suitable for the long‐term computations needed in environmental modelling. Although with a truncation error of formal order 0(ΔxΔt, ΔyΔt, Δt²), the present scheme is shown actually to be of an accuracy comparable with schemes of third‐order in space, while requiring a smaller computational effort and/or having better stability properties. In principle, the method can be easily extended to the 3D case. Copyright © 1999 John Wiley & Sons, Ltd.     

An investigation of an Emden-Fowler equation from thin film flow

A third-order ordinary differential equation (ODE) for thin film flow with both Neumann and Dirichlet boundary conditions is transformed into a second-order nonlinear ODE with Dirichlet boundary conditions.Numerical solutions of the nonlinear second-order ODE are investigated using finite difference schemes.A finite difference formulation to an Emden-Fowler representation of the second-order nonlinear ODE is shown to converge faster than a finite difference formulation of the standard form of the second-order nonlinear ODE.Both finite difference schemes satisfy the von Neumann stability criteria.When mapping the numerical solution of the second-order ODE back to the variables of the original third-order ODE we recover the position of the contact line.A nonlinear relationship between the position of the contact line and physical parameters is obtained.     

Differential-equation-based representation of truncation errors for accurate numerical simulation

High-order compact finite difference schemes for two-dimensional convection-diffusion-type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross-derivatives, making the overall O(h⁴) schemes conform to a 3 × 3 stencil. We show that the two-dimensional constant coefficient scheme collapses to the optimal scheme for the one-dimensional case wherein the finite difference equation yields nodally exact results. The two-dimensional schemes are tested against standard model problems, including a Navier-Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h²) centred differencing and commonly used O(h) and O(h³) upwinding schemes.     

A FINITE DIFFERENCE METHOD FOR SIMULATING TRANSVERSE VIBRATIONS OF AN AXIALLY MOVING VISCOELASTIC STRING

A finite difference method is presented to simulate transverse vibrations of an axially moving string.By discretizing the governing equation and the equation of stress- strain relation at different frictional knots,two linear sparse finite difference equation systems are obtained.The two explicit difference schemes can be calculated alternatively, which make the computation much more efficient.The numerical method makes the nonlinear model easier to deal with and of truncation errors,O(Δt~2 Δx~2).It also shows quite good stability for small initial values.Numerical examples are presented to demonstrate the efficiency and the stability of the algorithm,and dynamic analysis of a viscoelastic string is given by using the numerical results.     

Numerical studies of the fingering phenomena for the thin film equation

We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h_t + (h²?h³)_x = ??·(h³?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h²?h³)_x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h³?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd.     

Invertibility and Non-Invertibility in Thin Elastic Structures

The nonlinear elastic energy of a thin film of thickness h is given by a functional E ^h . Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h ^−α E ^h for α ≧ 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.     

Wave regimes on a nonlinearly viscous fluid film flowing down a vertical plane

The flow of a thin film of a nonlinearly viscous fluid whose stress tensor is modeled by a power law, flowing down a vertical plane in the field of gravity, is considered. For the case of low flow rates, an equation that describes the evolution of surface disturbances is derived in the long-wave approximation. The domain of linear stability of the trivial solution is found, and weakly nonlinear, steady-state travelling solutions of this equation are obtained. The mechanism of branching of solution families at the singular point of the neutral curve is described. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 73–84, May–June, 2005.     

A COMPARATIVE STUDY OF TIME-STEPPING TECHNIQUES FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS: FROM FULLY IMPLICIT NON-LINEAR SCHEMES TO SEMI-IMPLICIT PROJECTION METHODS

We present a numerical comparison of some time-stepping schemes for the discretization and solution of the non-stationary incompressible Navier– Stokes equations. The spatial discretization is by non-conforming quadrilateral finite elements which satisfy the LBB condition. The major focus is on the differences in accuracy and efficiency between the backward Euler, Crank–Nicolson and fractional-step Θ schemes used in discretizing the momentum equations. Further, the differences between fully coupled solvers and operator-splitting techniques (projection methods) and the influence of the treatment of the nonlinear advection term are considered. The combination of both discrete projection schemesand non-conforming finite elementsallows the comparison of schemes which are representative for many methods used in practice. On Cartesian grids this approach encompasses some well-known staggered grid finite difference discretizations too. The results which are obtained for several typical flow problems are thought to be representative and should be helpful for a fair rating of solution schemes, particularly in long-time simulations.     

Adaptive mesh refinement and load balancing based on multi-level block-structured Cartesian mesh

We developed a framework for a distributed-memory parallel computer that enables dynamic data management for adaptive mesh refinement and load balancing. We employed simple data structure of the building cube method (BCM) where a computational domain is divided into multi-level cubic domains and each cube has the same number of grid points inside, realising a multi-level block-structured Cartesian mesh. Solution adaptive mesh refinement, which works efficiently with the help of the dynamic load balancing, was implemented by dividing cubes based on mesh refinement criteria. The framework was investigated with the Laplace equation in terms of adaptive mesh refinement, load balancing and the parallel efficiency. It was then applied to the incompressible Navier–Stokes equations to simulate a turbulent flow around a sphere. We considered wall-adaptive cube refinement where a non-dimensional wall distance y⁺ near the sphere is used for a criterion of mesh refinement. The result showed the load imbalance due to y⁺ adaptive mesh refinement was corrected by the present approach. To utilise the BCM framework more effectively, we also tested a cube-wise algorithm switching where an explicit and implicit time integration schemes are switched depending on the local Courant-Friedrichs-Lewy (CFL) condition in each cube.     

Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows

Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re _λ between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)³. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other. The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does. In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138, 2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which with increasing Reynolds number approach the empirical values.     

Criteria of finite element algorithm for a class of parabolic equation

In finite element analysis of transient temperature field, it is quite notorious that the numerical solution may quite likely oscillate and/or exceed the reasonable scope, which violates the natural law of heat conduction. For this reason, we put forward the concept of time monotony and spatial monotony, and then derive several sufficient conditions for monotonic solutions in time dimension for 3 - D passive heat conduction equations with a group of finite difference schemes. For some special boundary conditions and regular element meshes, the lower and upper bounds for t/x ² can be obtained from those conditions so that reasonable numerical solutions are guaranteed. Spatial monotony is also discussed. Finally, the lumped mass method is analyzed. We creatively give several new criteria for the finite element solutions of a class of parabolic equation represented by heat conduction equation.     

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ω_ε that is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H ¹(Ω_ε) as ε → 0 (N → +∞). Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006.     

Homogenization of a Boundary-Value Problem with Varying type of Boundary Conditions in a Thick Two-Level Junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ω _ɛ that is the union of a domain Ω₀ and a large number 2N of thin rods with thickness of order ɛ = (N ⁻¹). Depending on their lengths, the thin rods are divided into two levels. In addition, the rods from each level are ɛ-periodically alternated. Inhomogeneous Neumann boundary conditions are given on the vertical sides of the thin rods of the first level, and homogeneous Dirichlet boundary conditions are given on the vertical sides of the rods of the second level. We investigate the asymptotic behavior of a solution of this problem as ɛ → 0 and prove a convergence theorem and the convergence of the energy integral. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 241–257, April–June, 2005.     

Linear Stability of Steady States for Thin Film and Cahn-Hilliard Type Equations

We study the linear stability of smooth steady states of the evolution equation

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.     

Computational modeling of biomagnetic micropolar blood flow and heat transfer in a two-dimensional non-Darcian porous medium

We study theoretically and computationally the incompressible, non-conducting, micropolar, biomagnetic (blood) flow and heat transfer through a two-dimensional square porous medium in an (x,y) coordinate system, bound by impermeable walls. The magnetic field acting on the fluid is generated by an electrical current flowing normal to the x–y plane, at a distance l beneath the base side of the square. The flow regime is affected by the magnetization B ₀ and a linear relation is used to define the relationship between magnetization and magnetic field intensity. The steady governing equations for x-direction translational (linear) momentum, y-direction translational (linear) momentum, angular momentum (micro-rotation) and energy (heat) conservation are presented. The energy equation incorporates a special term designating the thermal power per unit volume due to the magnetocaloric effect. The governing equations are non-dimensionalized into a dimensionless (ξ,η) coordinate system using a set of similarity transformations. The resulting two point boundary value problem is shown to be represented by five dependent non-dimensional variables, f _ξ  (velocity), f _η (velocity), g (micro-rotation), E (magnetic field intensity) and θ (temperature) with appropriate boundary conditions at the walls. The thermophysical parameters controlling the flow are the micropolar parameter (R), biomagnetic parameter (N _H ), Darcy number (Da), Forchheimer (Fs), magnetic field strength parameter (Mn), Eckert number (Ec) and Prandtl number (Pr). Numerical solutions are obtained using the finite element method and also the finite difference method for Ec=2.476×10⁻⁶ and Prandtl number Pr=20, which represent realistic biomagnetic hemodynamic and heat transfer scenarios. Temperatures are shown to be considerably increased with Mn values but depressed by a rise in biomagnetic parameter (N _H ) and also a rise in micropolarity (R). Translational velocity components are found to decrease substantially with micropolarity (R), a trend consistent with Newtonian blood flows. Micro-rotation values are shown to increase considerably with a rise in R values but are reduced with a rise in biomagnetic parameter (N _H ). Both translational velocities are boosted with a rise in Darcy number as is micro-rotation. Forchheimer number is also shown to decrease translational velocities but increase micro-rotation. Excellent agreement is demonstrated between both numerical solutions. The mathematical model finds applications in blood flow control devices, hemodynamics in porous biomaterials and also biomagnetic flows in highly perfused skeletal tissue. Dedicated to Professor Y.C. Fung (1919-), Emeritus Professor of Biomechanics, Bioengineering Department, University of California at San Diego, USA for his seminal contributions to biomechanics and physiological fluid mechanics over four decades and his excellent encouragement to the authors, in particular OAB, with computational biofluid dynamics research.     

Hybrid stress analysis by digitized photoelastic data and numerical methods

A method is presented that determines photoelastic isochromatic values at the nodal points of a grid mesh which in turn is generated by a computer program that accepts digitized input. Values of σ₁ - σ₂ are computed from the digitized fringe orders. The Laplace equation is solved to separate the principal stresses at each nodal point. The method is extended to digitize isoclinics. Subsequently, σ _x - σ _y and τ _xy are calculated to be used as starting values for the solution of the pertaining partial differential equations to enhance convergence. For further accelerating the rate of convergence, superfluous boundary conditions are added from the digitized data; significant improvement is demonstrated. Estimated values of σ _x - σ _y from the digitized data are further used in conjunction with the solution of the Laplace equation to determine the state of stress without solving the boundary value problems. Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

A convenient way of interpreting steady shear rheo-optical data of semi-dilute polymer solutions

Rheo-mechanical and rheo-optical investigations were carried out with the aim of determining the influence of deformation and orientation or disentangling of polymer coils on the flow behavior in the non-Newtonian region of the flow curve, for a moderately concentrated network solution. To avoid the influence of polydispersity this was done on a series of narrowly distributed polystyrene standards (dissolved in toluene). By using steady state shear flow measurements it was possible to detect qualitatively a reduction in the entanglement density within the non-Newtonian flow region. Birefringence experiments were able to show that deformation of the polymer coils also occurs in the Newtonian flow region, which has no effect on the flow behavior in this range, whereas in the non-Newtonian flow region the increase in deformation is lower than in the Newtonian range. The flow birefringence and its orientation can be described over the whole range of the flow curve with a newly developed equation system (Eq. 8 and 14) derived from the stress states of a sheared solution using the stress-optical rule. Starting from these equations, it could be shown, that in the Newtonian flow region a mastercurve in form of a reduced birefringence Δn′/η₀=f(γ˙) and a reduced orientation φ= f(γ˙/γ˙ _crit) can be plotted, independent from concentration and molar mass. A comparison of the experimentally determined orientation angle and birefringence curve form with theoretical deformations and orientations of polymer coils in a solution state, without intermolecular interactions, was able to demonstrate that the flow behavior of a moderately concentrated network solution is determined decisively (approximately to 85%) by the disentanglement. Received: 8 May 2000 Accepted: 12 September 2000     

Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation

Summary A hypersingular integral equation or a differential-integral equation is used to solve the penny-shaped crack problem. It is found that, if a displacement jump (crack opening displacement COD) takes the form of (a ²−x ²−y ²)^1/2 x ^m y ⁿ , where a denotes the radius of the circular region, the relevant traction applied on the crack face can be evaluated in a closed form, and the stress intensity factor can be derived immediately. Finally, some particular solutions of the penny-shaped crack problem are presented in this paper. Received 1 July 1997; accepted for publication 13 October 1997