An Inverse Method for Controlling the Temperature Distribution in Infinite Wedge Domain

2012 ◽  
Author(s):  
Hossein Rastgoftar ◽  
Faissal A. Moslehy
Keyword(s):  
Temperature Distribution ◽  
Conformal Mapping ◽  
Laplace Equation ◽  
Lyapunov Functional ◽  
Time Derivative ◽  
Circular Domain ◽  
Control Technique ◽  
Infinite Domain ◽  
Unit Radius ◽  
Infinite Wedge

The paper presents an analytical solution for controlling the temperature distribution in infinite wedge domain. The objective is to assign the heat flux at the boundaries of the domain such that a desired temperature distribution inside the semi-infinite domain is achieved. Since the conduction equation (Laplace equation) retains its form when the infinite domain is transformed into a finite domain by conformal mapping, the infinite domain can be transformed into a disk of unit radius. Then the Laplace equation is investigated in the domain confined by a circle of unit radius. The control technique used in this paper is based on the Lyapunov approach. A Lyapunov functional is defined over the circular domain and the control heat fluxes at the boundary of the disk are assigned such that the time derivative of the Lyapunov functional becomes negative definite. Since the conformal mapping is invertible, attaining a desired temperature distribution in the circular domain leads to achieving the desired temperature distribution in the infinite domain.

A Study on Optimum Topology of Plate Structure Using Coordinate Transformation by Conformal Mapping

2002 ◽  
Author(s):  
Satoshi Kitayama ◽  
Hiroshi Yamakawa
Keyword(s):  
Conformal Mapping ◽  
Fluid Mechanics ◽  
Laplace Equation ◽  
Coordinate Transformation ◽  
New Method ◽  
Design Domain ◽  
Simple Design ◽  
Bending Moments ◽  
Complex Design ◽  
Optimum Topology

This paper presents a new method to determine an optimum topology of plate structure using coordinate transformation by conformal mapping. We have already proposed a method to determine an optimum topology of planar structure using coordinate transformation by conformal mapping. In that study first we defined simple design domain in which analysis and optimization were performed easily. We calculated optimum topology in this simple design domain. Then we applied coordinate transformation by conformal mapping to optimum topology calculated in simple design domain, and obtained some optimum topologies in complex design domain. We also showed that the invariants of stresses which were the sum and difference of principal stress satisfied Laplace equation and relationshi p between fluid mechanics and electromagnetic could be valid in the theory of elasticity. In this study we clarify two invariants of bending moments satisfy Laplace equation under a certain condition. We note the similarity between Airy stress function of 2-D elastic body and deflection of plate, and will show that the two invariants of bending moments which are the sum and difference of principal bending moments satisfy Laplace equation using this similarity. As a result we will show that corresponding relationship between fluid mechanics, electromagnetic and elasticity may be valid in the theory of plate. Then by using this relationship, we proposed a new method to determine optimum topology using coordinate transformation by conformal mapping. Our proposed method will be useful to determine optimum topology easily in complex design domain. Through numerical examples, we can examine the effectiveness of the proposed method.

scholarly journals A calculus for flows in periodic domains

2020 ◽  
Author(s):  
Peter J. Baddoo ◽  
Lorna J. Ayton
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Moving Boundaries ◽  
Asymptotic Approximations ◽  
Periodic Domain ◽  
Target Domain ◽  
Potential Flows ◽  
Flow Phenomena ◽  
Periodic Domains ◽  
Prime Function

AbstractPurpose: We present a constructive procedure for the calculation of 2-D potential flows in periodic domains with multiple boundaries per period window.Methods: The solution requires two steps: (i) a conformal mapping from a canonical circular domain to the physical target domain, and (ii) the construction of the complex potential inside the circular domain. All singly periodic domains may be classified into three distinct types: unbounded in two directions, unbounded in one direction, and bounded. In each case, we use conformal mappings to relate the target periodic domain to a canonical circular domain with an appropriate branch structure.Results: We then present solutions for a range of potential flow phenomena including flow singularities, moving boundaries, uniform flows, straining flows and circulatory flows.Conclusion: By using the transcendental Schottky-Klein prime function, the ensuing solutions are valid for an arbitrary number of obstacles per period window. Moreover, our solutions are exact and do not require any asymptotic approximations.

Formability Improvement Technique for Heated Sheet Metal Forming by Partial Cooling

2014 ◽  
Vol 622-623 ◽  
pp. 279-283 ◽  
Author(s):  
Eiichi Ota ◽  
Yasuhiro Yogo ◽  
Takamichi Iwata ◽  
Noritoshi Iwata ◽  
Kenjiro Ishida ◽  
...  
Keyword(s):  
Temperature Distribution ◽  
Sheet Metal ◽  
Deep Drawing ◽  
Initial Temperature ◽  
Hot Stamping ◽  
Local Deformation ◽  
Control Technique ◽  
Forming Limit ◽  
Forming Process ◽  
Initial Temperature Distribution

A forming process for heated sheet metal, such as hot-stamping, has limited use in deformable shapes. Higher temperature areas which have not yet come into contact with dies are more easily deformed; therefore, local deformation occurs at these areas which leads to breakage. To improve the formability of heated sheet metal, a deformation control technique utilizing the temperature dependence of flow stress is proposed. This technique can suppress local deformation by partial cooling around potential cracking areas to harden them before forming. In order to apply this technique to a variety of product shapes, a procedure to determine a suitable initial temperature distribution for deep drawing and biaxial stretching was developed with a coupled thermal structural simulation. In this procedure, finite elements exceeding forming limit strain are highlighted, and an initial temperature distribution is defined with areas of decreased temperature around the elements to increase their resistance to deformation. Subsequently, the partial cooling technique was applied to a deep drawing test with a heated steel sheet. The results of the experiment showed that the proposed technique improved 71% in the forming limit depth compared with results obtained using a uniform initial temperature distribution.

scholarly journals Generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130848 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Slit Domain ◽  
Combined Use ◽  
Polygonal Region ◽  
Upper Half Plane ◽  
Indefinite Integral ◽  
Detailed Derivation

A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

Gibbs–Thomson Effect on Spherical Solidification in a Subcooled Melt

2009 ◽  
Vol 131 (9) ◽  
Author(s):  
Yeong-Cheng Lai ◽  
Chun-Liang Lai ◽  
Hsieh-Chen Tsai
Keyword(s):  
Temperature Distribution ◽  
Liquid Phase ◽  
Energy Balance ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Solidification Front ◽  
Spherical Nucleus ◽  
Thomson Effect ◽  
Infinite Domain ◽  
Spherical Growth

This study aims to investigate theoretically the growth of a spherical nucleus due to solidification in an infinite domain of a subcooled melt. The effects on the spherical growth due, respectively, to the subcooling, the Gibbs–Thomson condition, and the density-difference induced convection are analyzed and discussed systematically. With the Gibbs–Thomson effect considered, no exact solutions can be found easily. Thus, a binomial temperature distribution in the liquid phase is reasonably assumed to approximate the actual one with the satisfaction of the energy balance at the solidification front and other boundary conditions.

scholarly journals A Lyapunov functional for a system with a time-varying delay

2012 ◽  
Vol 22 (2) ◽  
pp. 327-337 ◽  
Author(s):  
Józef Duda
Keyword(s):  
Lyapunov Functional ◽  
Linear Time ◽  
Time Derivative ◽  
Time Varying ◽  
Linear Time Invariant ◽  
Time Varying Delay ◽  
Analytical Formulas ◽  
Linear Time Invariant System ◽  
Time Invariant ◽  
Varying Delay

A Lyapunov functional for a system with a time-varying delay The paper presents a method to determine a Lyapunov functional for a linear time-invariant system with an interval time-varying delay. The functional is constructed for the system with a time-varying delay with a given time derivative, which is calculated on the system trajectory. The presented method gives analytical formulas for the coefficients of the Lyapunov functional.

scholarly journals Regularity of solutions of the parabolic normalized p-Laplace equation

2018 ◽  
Vol 9 (1) ◽  
pp. 7-15 ◽  
Author(s):  
Fredrik Arbo Høeg ◽  
Peter Lindqvist
Keyword(s):  
Viscosity Solution ◽  
Laplace Equation ◽  
Time Derivative ◽  
Regularity Of Solutions

Abstract The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to {L^{2}} .

A Study on Optimum Topology Using Conformal Mapping

2000 ◽  
Author(s):  
Satoshi Kitayama ◽  
Hiroshi Yamakawa
Keyword(s):  
Conformal Mapping ◽  
Fluid Mechanics ◽  
Laplace Equation ◽  
Coordinate Transformation ◽  
Theory Of Elasticity ◽  
Conformal Mappings ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Planar Structures ◽  
Optimum Topology

Abstract This paper presents a method to determine optimum topologies of two dimensional elastic planar structures by using conformal mappings. We use the conformal mappings which is known to be effective in two dimensional fluid mechanics, electromagnetics and elasticity by complex coordinate transformation. We show that two invariants of stress can satisfy the Laplace equation, and then we clarify that corresponding relationships between fluid mechanics and electromagnetics can also be valid in the theory of elasticity. Then, presented a method to obtain optimum topologies is easier than by the conventional methods. We treated several numerical examples by the presented method. Through numerical examples, we can examine the effectiveness of the proposed method.

scholarly journals DEVELOPMENT OF A COMPUTER MODEL OF TEMPERATURE DISTRIBUTION IN THE REACTOR

2020 ◽  
Vol 2020 (1) ◽  
pp. 105-106
Author(s):  
Evgeniya Bolotova ◽  
Svetlana Senotova
Keyword(s):  
Temperature Distribution ◽  
Laplace Equation ◽  
Stationary Distribution ◽  
Computer Model ◽  
Graphical Display

The paper considers the problem of the stationary distribution of heat in the reactor. The function giving the temperature distribution is a solution to the Laplace equation. For visual graphical display of the temperature distribution in the reactor, a program in C # was developed.

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