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.

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.

Integral characteristics of elastic inclusions and cavities in the two-dimensional theory of elasticity

1992 ◽  
Vol 3 (1) ◽  
pp. 21-30 ◽  
Author(s):  
A. B. Movchan
Keyword(s):  
Conformal Mapping ◽  
Unit Disk ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
The Matrix ◽  
Integral Characteristics ◽  
Elastic Inclusions

Integral characteristics, such as elastic polarization matrices of elastic inclusions and cavities, are described. The matrix of elastic polarization of a finite cavity is constructed in the case of the two-dimensional Lamé operator under the assumption that the geometry of the domain occupied by the cavity is defined by a conformal mapping from the unit disk. Examples and applications of these integral characteristics in the theory of cracks are considered.

Analytical Solution for Solving Bending Problem of Orthotropic Beams

2014 ◽  
Vol 1065-1069 ◽  
pp. 2044-2048
Author(s):  
You Zhen Yang ◽  
Han Lin Ma ◽  
Hu Wang
Keyword(s):  
Hamiltonian System ◽  
Analytical Solution ◽  
Current Method ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Thickness Direction ◽  
Dimensional Theory ◽  
Zero Eigenvalue ◽  
Numerical Examples ◽  
End Conditions

Based on the two-dimensional theory of elasticity, Hamiltonian system is introduced to solve the bending of orthotropic beams and the original problems come down to solve the eigensolutions of zero eigenvalue. The symplectic concept makes no hypothesis of deformation along the thickness direction. Thus, the current method can precisely analyze beams with arbitrary depth-to-length ratio, and can deal with arbitrary end conditions. In additional, a new improved boundary conditions for fixed ends beam is presented. Numerical examples showing comparison with other methods are given to illustrate the accuracy of the present approach.

scholarly journals Harmonic shears and numerical conformal mappings

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2231-2241 ◽  
Author(s):  
Tri Quach
Keyword(s):  
Conformal Mapping ◽  
Numerical Algorithm ◽  
Minimal Surfaces ◽  
Conformal Mappings ◽  
Harmonic Mappings ◽  
Numerical Conformal Mapping ◽  
Numerical Examples ◽  
Numerical Example ◽  
Numerical Conformal Mappings

In this article we introduce a numerical algorithm for finding harmonic mappings by using the shear construction introduced by Clunie and Sheil-Small in 1984. The MATLimplementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.

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