Viscous Inner and Outer Pressure Forming Method of Thin-walled Tube and Its Application

Aiming at overcoming the difficulties in integral forming of thin-walled tubes with complex shapes, a novel forming method by inner and outer pressure through viscous was proposed. In this method, by dividing large deformation of the part into inner and outer pressure forming deformations, the limit deformation of tube part can be increased by several times. Meanwhile, the principle of viscous inner and outer pressure forming was provided, and key problems during the forming process such as reduction of the wall-thickness and instability wrinkling were analyzed. Thereby, the complex curved surface super-alloy GH3044 thin-walled tube with varying diameter ratio of 1.35（the ratio between the maximum and minimum diameters of the part） can be integrally formed by this method. The experimental surface of the formed part is superior in quality and the wall-thickness distribution is uniform. The results show that the viscous inner and outer pressure forming can provide a new approach for integral forming of thin-walled tubes with complex shapes.

Notes on Canonical Forms of Integrable Vector Nonlinear Schr?dinger Systems

We present canonical forms of integrable vector nonlinear Sehr6dinger systems. Mathematically, it is enough to focus on these canonical forms.

Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.

CLASSIFICATION OF POSITIVE SOLUTIONS TO ASYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 < t_1, t_2 < min{α, k}, and 1 < p ≤ τ_1 :=(n+α-2t_1)/( n-α), 1 < q ≤ τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R^n) G_α(x, ξ)v^q(ξ)/|η|t^2 dξ v(x) =∫_(R^n) G_α(x, ξ)(u^p(ξ))/|η|^(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|^(n-α))is the Green's function of(-?)^(α/2) in R^n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z0 in R^(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1)> n-α,1 < p ≤ τ_1 and 1 < q ≤ τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).

NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SEMI-LINEAR EQUATION INVOLVING THE FRACTIONAL LAPLACIAN IN R^N

In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space R^n：（-△）^α/2u（x） ： f（xn）u^p（x）, x ∈R^n（0.1）in the subcritical case with 1〈 p〈n＋a/n-a.Instead of carrying out direct investigations on pseudo-differential equation （0.1）, we first seek its equivalent form in an integral equation as below：u（x）=∫R^nG∞（x, y） f（yn） u^p（y）dy,where G∞（x, y） is the Green＇s function associated with the fractional Laplacian in R^n. Employing the method of moving planes in integral forms, we are able to derive the nonexistence of positive solutions for （0.2） in the subcritical case. Thanks to the equivalence, same con- clusion is true for （0.1）.

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.

LOCAL PETROV-GALERKIN METHOD FOR A THIN PLATE

The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

Theta Function and Hilbert Modular Forms of Half Integral Weight

In this paper a kind of theta function is constructed by means of spherical function. And we also obtain some Hilbert modular forms of half integral weight.

Metal flow hysteresis behavior and shape control strategy during porthole die extrusion of micro heat pipe

The shape control strategy of micro grooves is still unclear and challenging during the porthole die extrusion of grooved micro heat pipe(MHP).Through the simulation and experiment of porthole die extrusion of a MHP profile,the metal flow hysteresis behavior within micro features and the effect of ram speed and extrusion temperature on it and the resulting forming integrity was elucidated.Innovatively,Taguchi design and variance analysis(ANOVA)were introduced to determine their influence magnitude on the metal flow uniformity calculated by simulation results.The main findings are given below.The metal flow hysteresis derives from part feature size effect.The negligible friction-affected area during conventional extrusion severely slows down the metal flow within micro features during the MHP profile extrusion,which is due to the surge in the area ratio of the friction-affected area to the region in which it is located.Neither ram speed nor extrusion temperature can change the distribution of the friction-affected area.However,increasing ram speed multiplies the metal flow hysteresis and severely reduces the forming integrity,whereas extrusion temperature has little effect.Following this strategy,batch extrusion of the profile with microgrooved width of 0.27±0.02 mm was achieved in industrialized conditions.

Some Inequalities for Geominimal Surface Areas

The concept of geominimal surface area is first intro- duced, and then, the affine surface area projection inequality is given by Petty. In recent years, associated with geominimal surface area, Lutwak obtained some significant results containing the extended affine projection inequality. In this paper, combining with curvature image and combinations of convex bodies, we get some inequalities for geominimal surface areas. Furthermore, the integral form of geominimal surface area is obtained.

Realization of Numbers As the Degrees of Maps between Manifolds

We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.