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.

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.

Improved Method Based on Fluid Mechanics Analysis for Buoy Gauge Design and Experimental Research

2012 ◽  
Vol 163 ◽  
pp. 133-137
Author(s):  
Ao Yu Chen ◽  
Xu Dong Pan ◽  
Guang Lin Wang
Keyword(s):  
Fluid Mechanics ◽  
Experimental Research ◽  
Traditional Method ◽  
New Method ◽  
Improved Method ◽  
Advanced Method ◽  
Inner Surface ◽  
Mechanics Analysis ◽  
Linear Property

Traditional method of buoy gauge design is rather complicated, so an advanced method by building and solving fluid mechanics equations is proposed in this paper. The curve of the taper pipe inner surface is calculated, according to different buoy gravity and diameter. In order to examine the effect of this improved method, an experiment is carried out. Results show that linear property of the buoy gauge improved by new method is excellent.

A Study on Shape Optimization Using Affine Transformation

2001 ◽  
Author(s):  
Satoshi Kitayama ◽  
Hiroshi Yamakawa
Keyword(s):  
Shape Optimization ◽  
Cantilever Beam ◽  
Coordinate Transformation ◽  
Affine Transformation ◽  
Sufficient Conditions ◽  
Optimal Shape ◽  
Design Domain ◽  
Shape Transformation ◽  
Numerical Examples ◽  
Inclined Beam

Abstract This paper presents a new method to determine an optimal shape using affine transformation which is used in the field of Computer Aided Design (CAD), linear programming, and etc. We use affine transformation as coordinate transformation. Affine transformation is a linear transformation, so that shapes transformed must be linearly. Shape optimization of a inclined beam for example, we can deal with in the following manner. We define a simple cantilever beam first in initial design domain, and calculate an optimal shape. Then we use affine transformation remaining with optimal shape calculated in simple design domain and get to an optimal shape of the inclined beam. To compare with an optimal shape obtained by our proposed method, we calculate an optimal shape directly by conventional method in the same design domain after coordinate transformation. We show that affine transformation plays a role as scaling to structural optimization by finite element method and that necessary and sufficient conditions between design variables and shape transformation matrix may exist to get an exact optimal shape. We treat some numerical examples by our proposed method. In numerical examples, we consider shape optimization of inclined cantilever beam for simplicity. We show that some stepwise linear optimal shapes could be expressed from an optimal shape of a simple cantilever beam by using affine transformation. Optimal shape calculated by our method can obtain easily and speedy. Through some numerical examples, we could examine effectiveness of our proposed method.

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.

scholarly journals The Latest Method for Surface Tension Determination: Experimental Validation

Energies ◽  
2020 ◽  
Vol 13 (14) ◽  
pp. 3629
Author(s):  
Tomasz Janusz Teleszewski ◽  
Andrzej Gajewski
Keyword(s):  
Surface Tension ◽  
Heat Exchange ◽  
Closed Form ◽  
Laplace Equation ◽  
Experimental Validation ◽  
Reference Data ◽  
New Method ◽  
Air Bubbles ◽  
Inflection Points

The highest effectiveness of heat exchange is under boiling; hence, surface tension is an important parameter and should be determined when new liquid substances are created. The most popular methods are based on numerically solving the Young–Laplace equation by applying the Bashforth and Adams algorithm, which fails at the poles and at the inflection points. The newest algorithm is based on the closed-form expressions that define a drop or bubble. It gives the accurate solutions for the fully created drops or bubbles. To validate it, the surface tension value is determined for the air bubbles in water and compared with the reference data. Because the relative discrepancies are extremely small, the new method may be thought of as positively validated.

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