A new high-order accuracy explict difference scheme for solving three-dimensional parabolic equations

In this paper, a new three-level explicit difference scheme with high-orderaccuracy is proposed for solving three-dimensional parabolic equations. The stabilitycondition is r＝△t/△x² ＝△t/△y²＝△t/△z²≤1/4, and the trumcation error is O(△t²+△x⁴).     

A class of two-level explicit difference schemes for solving three dimensional heat conduction equation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｏｆｔｈｒｅｅ＿ｄｉｍｅｎｓｉｏｎａｌｈｅａｔｃｏｎｄｕｃｔｉｏｎｅｑｕａｔｉｏｎｉｎｔｈｅｒｅｇｉｏｎＤ :0≤ｘ,ｙ ,ｚ≤Ｌ ,0 ≤ｔ≤Ｔ ｕ ｔ= 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 ,ｕ｜ｘ=0 =ｆ1(ｙ,ｚ,ｔ) ,　ｕ｜ｘ=Ｌ =ｆ2 (ｙ ,ｚ,ｔ) ,ｕ｜ｙ=0 =ｇ1(ｚ,ｘ,ｔ) ,　ｕ｜ｙ=Ｌ =ｇ2 (ｚ,ｘ,ｔ) ,ｕ｜ｚ=0 =ｈ1(ｘ ,ｙ ,ｔ) ,　ｕ｜ｚ=Ｌ =ｈ2 (ｘ ,ｙ ,ｔ) ,ｕ｜ｔ=0 =φ(ｘ ,ｙ,ｚ) .(1 )(2 )…     

The high accuracy explicit difference scheme for solving parabolic equations 3-dimension

ＩｎｔｒｏｄｕｃｔｉｏｎＦｒｏｍｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍ,ｗｅｃｏｕｌｄｃｏｎｃｌｕｄｅｍａｎｙｐｒｏｂｌｅｍｓａｂｏｕｔｓｏｌｖｉｎｇｐａｒａｂｏｌｉｃｐａｒｔｉａｌｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ．Ｎｏｗｔｈｅｒｅａｒｅｍａｎｙｎｕ...     

The application of Moire interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip

This paper presents the application of Moire interferometry in measuring the displacement and strain field at notch-tip and crack-tip before and after crack propagation. The experiment is carried out using a three point bending beam with a notch. TheN _x andN _y fringe patterns representing displacement field, and the ΔN _x/Δx and ΔN _γ/Δγ fringe patterns representing the strain field are obtained. The sensitivity of the measured displacement is 0.417 μm per fringe order. The displacement and strain distribution along the sectionx=0 have been worked out according toN _x andN _γ fringe patterns. The project supported by Chinese Academy of Sciences and National Natural Science Foundation of China     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε²Δu − V(x) u + f(u) = 0, x ∈ R ^N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. An erratum to this article can be found at     

A class of three-level explicit schemes with higher stability properties for a dispersive equation ut=auxxx

In this paper, a class of three level explicit schemes for a dispersive equation u_t=au_xxx with stability condition |r|=|α|Δt/(Δx)³≤2.382484, are considered. The stability condition for this class of schemes is much better than |r|≤0.3849 in [1], [2] and |r|≤0.701659 in [3], and |r|≤1.1851 in [4].     

Nonsimilarity solutions for non-Darcy mixed convection from horizontal surfaces in a porous medium

Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ζ _f =Ra^* _x /Pe² _x is found to characterize the effect of buoyancy forces on the forced convection with K ^′ U _∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where the dimensionless parameter ζ _n =Pe _x /Ra^*1/2 _x is found to characterize the effect of the forced flow on the natural convection, with (K ^′ U _∞/ν)Ra^*1/2 _x /Pe _x characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the solution of the first regime is carried out for ζ _f =0, the pure forced convection limit, to ζ _f =1 and the solution of the second is carried out for ζ _n =0, the pure natural convection limit, to ζ _n =1. The two solutions meet and match at ζ _f =ζ _n =1, and R ^* _h =G ^* _h . Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξ _f =Ra _x /Pe _x ^3/2 is found to measure the buoyancy effects on mixed convection with Da _x Pe _x /ɛ as the wall effects. The second region covers the natural convection dominated region where ξ _n =Pe _x /Ra _x ^2/3 is found to measure the force effects on mixed convection with Da _x Ra _x ^2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature exponents are presented. Received on 8 July 1996     

A correction method for measuring turbulence kinetic energy dissipation rate by PIV

The accuracy of turbulent kinetic energy (TKE) dissipation rate measured by PIV is studied. The critical issue for PIV-based dissipation measurements is the strong dependency on the spatial resolution, Δx, as reported by Saarenrinne and Piirto (Exp Fluids Suppl:S300–S307, 2000). When the PIV spacing is larger than the Kolmogorov scale, η, the dissipation is underestimated because the small scale fluctuations are filtered. For the case of Δx smaller than the Kolmogorov scale, the error rapidly increases due to noise. We introduce a correction method to eliminate the dominant error for the small Δx case. The correction method is validated by using a novel PIV benchmark, random Oseen vortices synthetic image test (ROST), in which quasi-turbulence is generated by randomly superposing multiple Oseen vortices. The error of the measured dissipation can be more than 1,000% of the analytical dissipation for the small Δx case, while the dissipation rate is underestimated for the large Δx case. Though the correction method does not correct the underestimate due to the low resolution, the dissipation was accurately obtained within a few percent of the true value by using the correction method for the optimal resolution of η/10 < Δx < η/2.     

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Let be a domain. Suppose that f ∈ W^1,1_loc(Ω,R²) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J_f. We show that f^-1 ∈ W^1,1_loc(f(Ω),R²) and that Df⁻¹(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f⁻¹ ∈ W^1,q(f(Ω),R²) for some 1<q≤2 are also given.     

An Efficient Derivation of the Aronsson Equation

For 1<p<∞, the equation which characterizes minima of the functional u↦∫ _U |Du| ^p ,dx subject to fixed values of u on ∂U is −Δ _p u=0. Here −Δ _p is the well-known ``p-Laplacian'. When p=∞ the corresponding functional is u↦|| |Du|<sup>2</sup>|| <sub>L∞(U)</sub> . A new feature arises in that minima are no longer unique unless U is allowed to vary, leading to the idea of ``absolute minimizers'. Aronsson showed that then the appropriate equation is −Δ_∞ u=0, that is, u is ``infinity harmonic' as explained below. Jensen showed that infinity harmonic functions, understood in the viscosity sense, are precisely the absolute minimizers. Here we advance results of Barron, Jensen and Wang concerning more general functionals u↦||f(x,u,Du)|| _L∞(U) by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses. (Accepted September 6, 2002) Published online April 14, 2003 Communicated by S. Muller     

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

A Similarity Subgrid Model for Premixed Turbulent Combustion

The accuracy of large-eddy simulation (LES) of a turbulent premixed Bunsen flame is investigated in this paper. To distinguish between discretization and modeling errors, multiple LES, using different grid sizes h but the same filterwidth Δ, are compared with the direct numerical simulation (DNS). In addition, LES using various values of Δ but the same ratio Δ/h are compared. The chemistry in the LES and DNS is parametrized with the standard steady premixed flamelet for stochiometric methane-air combustion. The subgrid terms are closed with an eddy-viscosity or eddy-diffusivity approach, with an exception of the dominant subgrid term, which is the subgrid part of the chemical source term. The latter subgrid contribution is modeled by a similarity model based upon 2Δ, which is found to be superior to such a model based upon Δ. Using the 2Δ similarity model for the subgrid chemistry the LES produces good results, certainly in view of the fact that the LES is completely wrong if the subgrid chemistry model is omitted. The grid refinements of the LES show that the results for Δ = h do depend on the numerical scheme, much more than for h = Δ/2 and h = Δ/4. Nevertheless, modeling errors and discretization error may partially cancel each other; occasionally the Δ = h results were more accurate than the h ≤ Δ results. Finally, for this flame LES results obtained with the present similarity model are shown to be slightly better than those obtained with standard β-pdf closure for the subgrid chemistry.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

Periodic solutions of dynamical systems

Summary In this paper we look for T-periodic solutions of dynamical systems. Particularly we consider the system whereU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. We assume that the problem is asymptotically linear with a bounded nonlinearity. Under a resonance assumption, we find a multiplicity of T-periodic solutions for T large enough.

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Sommario In questo lavoro si cercano soluzioni periodiche di periodo T assegnato di sistemi dinamici. In particolare si considera un sistema di n equazioni differenziali del secondo ordine del tipo doveU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. Nel caso in cui il problema sia asintoticamente lineare, con termine nonlineare limitato e in condizioni di risonanza, troviamo che esiste tale che per il sistema ha una molteplicità di soluzioni.

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Presented at the VII A.I.M.E.T.A. and supported by M.P.I. (40% and 60%).     

Streamwise evolution of an inclined cylinder wake

The streamwise evolution of an inclined circular cylinder wake was investigated by measuring all three velocity and vorticity components using an eight-hotwire vorticity probe in a wind tunnel at a Reynolds number Re_d of 7,200 based on free stream velocity (U _∞) and cylinder diameter (d). The measurements were conducted at four different inclination angles (α), namely 0°, 15°, 30°, and 45° and at three downstream locations, i.e., x/d = 10, 20, and 40 from the cylinder. At x/d = 10, the effects of α on the three coherent vorticity components are negligibly small for α ≤ 15°. When α increases further to 45°, the maximum of coherent spanwise vorticity reduces by about 50%, while that of the streamwise vorticity increases by about 70%. Similar results are found at x/d = 20, indicating the impaired spanwise vortices and the enhancement of the three-dimensionality of the wake with increasing α. The streamwise decay rate of the coherent spanwise vorticity is smaller for a larger α. This is because the streamwise spacing between the spanwise vortices is bigger for a larger α, resulting in a weak interaction between the vortices and hence slower decaying rate in the streamwise direction. For all tested α, the coherent contribution to [`(v<sup>2</sup>)] \overline{{v^{2}}} is remarkable at x/d = 10 and 20 and significantly larger than that to [`(u²)] \overline{{u^{2}}} and [`(w<sup>2</sup>)]. \overline{{w^{2}}}. This contribution to all three Reynolds normal stresses becomes negligibly small at x/d = 40. The coherent contribution to [`(u²)] \overline{{u^{2}}} and [`(v²)] \overline{{v^{2}}} decays slower as moving downstream for a larger α, consistent with the slow decay of the coherent spanwise vorticity for a larger α.     

Solution of the equations for a regular beam emitted by a curvilinear surface in the nonstationary case

An analytical solution is given of the equations for a regular beam (§1) emitted by an arbitrary surface in the nonstationary case and the - and T-limited states for nonzero initial velocity (§ §2-4). It is assumed that the emitter is the coordinate surface x¹=0 in the orthogonal system x¹ (i=1, 2, 3), and the current density J, the electric field , and the magnetic field H are given functions J(t, x²,x³), (t, x², x³), and H (x¹, x², x³). The solution is given in the form of series in terms of (X¹) with coefficients that are functions of t, x², and x³. These coefficients are determined from recurrence relations ( =1/3, 1/2, 1, depending on the emission conditions). Plane, cylindrical, and spherical diodes are considered in § 5 in the case in which the high-frequency component of the current density J is not small in comparison with its constant components.     

The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

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Ellipsoidal Domains: Piecewise Nonuniform and Impotent Eigenstrain Fields

In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment. Consider the set of nested similar ellipsoidal domains {Ω₁, Ω₂,⋯,Ω _N+1}, which are embedded in an infinite isotropic medium. Suppose that

Existence of Singular Self-Similar Solutions of the Three-Dimensional Euler Equations in a Bounded Domain

A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t) ^{α, y = x/(t* ~ t)β,α,β>} 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H³(Ω,R³ X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a timet = t*, t* < +∞.