Plastic forming processes of transverse non-homogeneous composite metallic sheets

In this work an analysis of the radial stress and velocity fields is performed according to the J 2 flow theory for a rigid/perfectly plastic material. The flow field is used to simulate the forming processes of sheets. The significant achievement of this paper is the generalization of the work by Nadai & Hill for homogenous material in the sense of its yield stress, to a material with general transverse non-homogeneity. In Addition, a special un-coupled form of the system of equations is obtained where the task of solving it reduces to the solution of a single non-linear algebraic differential equation for the shear stress. A semi-analytical solution is attained solving numerically this equation and the rest of the stresses term together with the velocity field is calculated analytically. As a case study a tri-layered symmetrical sheet is chosen for two configurations: soft inner core and hard coating, hard inner core and soft coating. The main practical outcome of this work is the derivation of the validity limit for radial solution by mapping the “state space” that encompasses all possible configurations of the forming process. This configuration mapping defines the “safe” range of configurations parameters in which flawless processes can be achieved. Several aspects are researched: the ratio of material's properties of two adjacent layers, the location of layers interface and friction coefficient with the walls of the dies.

ON SOLUTIONS TO NONHOMOGENEOUS ALGEBRAIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

We consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then $$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$ With this in hand, we also prove that if $f$ is a transcendental entire function, then $f'p_k(f)+q_m(f)$ assumes every complex number $\alpha $, with one possible exception, infinitely many times, where $p_k(f), q_m(f)$ are polynomials in $f$ with degrees $k$ and $m$ with $k\ge m+1$. This result generalizes a theorem originating from Hayman [‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2)70(2) (1959), 9–42].

Algebroid Solutions of Second Order Complex Differential Equations

Using value distribution theory and maximum modulus principle, the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that our results are sharp.

Dynamic Analysis of a Dual-Mode CO2 Heat Pump With Both Hot and Cold Thermal Storage

We investigated the dynamics of a transcritical CO2 heat pump system including hot and cold thermal storages, which makes up the concept “thermal battery”. The analytical model is used for the study of the dynamics of the system involving simultaneous supply of heating and cooling for buildings. The model includes the dynamics of the gas cooler, evaporator and the thermal storages, while the compressor and the expansion valve are considered quasi-static. The heat transfer in the dynamically modeled components is described by partial differential equations (PDEs) consisting of heat conduction, convection, and source terms. Each component is divided into a number of volumes adjusted according to the required precision and reasonable computational time. We applied two discretization schemes in order to find a numerical solution to the PDEs. The spatial discretization for the heat exchangers is performed by using the upwind scheme, where the fluid properties are individually calculated within each volume. Due to the discrete events in form of tapping and loading (or charging and discharging) of the heat storages, the discretization approach takes into account the sharp spatial transitions within the thermal storages. Therefore, the method of lines in combination with the Superbee slope-limiter was applied for the spatial discretization for high resolution calculation. The modeling approach results in a set of algebraic and ordinary differential equations (ODEs), hence the model becomes an algebraic differential equation problem, which we solved by using MATLAB solver ODE15s. This extended model was used to simulate a dynamic response of the case with varying heating and cooling consumption over a period of 24 hours in a building. The heating and cooling energy consumption follow a sinusoidal and continuous pattern. The results include the effect on both the outlet temperatures and the system coefficient-of-performance (COP). The outlet energy from the hot storage and the cold storage is used for heating tap water and a chilled water space cooling application subject to temperature requirements. Dimensioning of both storages is crucial for obtaining the required temperatures. The model identifies the critical storage levels required to satisfy the periodic but out-of-phase combination of heating and cooling demands. The volume of the cold storage will have to be considerably larger than the hot storage due to the lower temperature difference.

Differential equations on complex projective hypersurfaces of low dimension

Let n=2,3,4,5 and let X be a smooth complex projective hypersurface of $\mathbb {P}^{n+1}$. In this paper we find an effective lower bound for the degree of X, such that every holomorphic entire curve in X must satisfy an algebraic differential equation of order k=n=dim X, and also similar bounds for order k>n. Moreover, for every integer n≥2, we show that there are no such algebraic differential equations of order k<n for a smooth hypersurface in $\mathbb {P}^{n+1}$.

UNCERTAINTY MODELING USING FUZZY ARITHMETIC BASED ON SPARSE GRIDS: APPLICATIONS TO DYNAMIC SYSTEMS

Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy-valued extension of any real-valued objective function. An efficient and accurate approach to compute expensive multivariate functions of fuzzy numbers is given by fuzzy arithmetic based on sparse grids. In this paper, we illustrate the general applicability of this new method by computing two dynamic systems subjected to uncertain parameters as well as uncertain initial conditions. The first model consists of a system of delay differential equations simulating the periodic outbreak of a disease. In the second model, we consider a multibody mechanism described by an algebraic differential equation system.

On a Problem of Rubel Concerning the Set of Functions Satisfying All the Algebraic Differential Equations Satisfied by a Given Function

For two functions ƒ and g, define g ≪ ƒ to mean that g satisfies every algebraic differential equation over the constants satisfied by ƒ . The order ≪ was introduced in one of a set of problems on algebraic differential equations given by the late Lee Rubel. Here we characterise the set of g such that g ≪ ƒ, when ƒ is a given Liouvillian function.