Shape Optimization Approach to a Free Boundary Problem

We take a shape optimization approach to solve a free boundary problem of the Poisson equation numerically. A numerical method called traction method invented by one of the authors are applied. We begin by changing the free boundary problem to a shape optimization problem and define a least square functional as a cost function. Then shape derivative of the cost function is derived by using Lagrange multiplier method. Detail structures and profiles of exact solutions to a concrete free boundary problem due to A. Henrot are also illustrated with proofs. They are used to check the efficiency of the traction method.Selected Papers from the International Symposium on Computational Science - International Symposium on Computational Science Kanazawa University, Japa

A Proposal of a Shape-0ptimization Method Using a Constitutive Equation of Growth : In the Case of a Static Elastic Body

A simple method for analysis of uniform-strength shape is newly proposed. In this paper, the most fundamental case of a static elastic body is considered. The idea of the present method came from the growth behavior of living organisms by which they changed their own shapes to adapt themselves to the mechanical living environment. The scheme consists of the iteration of the two analytical steps : (1) conventional elastic analysis for evaluation of stress distribution, and (2) incremental growth analysis using a constitutive equation of growth. In the latter step, a shape is deformed with an incremental bulk strain which is generated according to an objective stress indicating strength of the material. Two examples of a cantilever beam under top shear loading and a column under top compressive loading and gravity are analyzed to show the effectiveness of the proposed method.・rights：日本機械学会・rights：本文データは学協会の許諾に基づきCiNiiから複製したものである・relation：isVersionOf：http://ci.nii.ac.jp/naid/110002347599

Shape optimization problems

This book provides theories on non-parametric shape optimization problems, systematically keeping in mind readers with an engineering background. Non-parametric shape optimization problems are defined as problems of finding the shapes of domains in which boundary value problems of partial differential equations are defined. In these problems, optimum shapes are obtained from an arbitrary form without any geometrical parameters previously assigned. In particular, problems in which the optimum shape is sought by making a hole in domain are called topology optimization problems. Moreover, a problem in which the optimum shape is obtained based on domain variation is referred to as a shape optimization problem of domain variation type, or a shape optimization problem in a limited sense. Software has been developed to solve these problems, and it is being used to seek practical optimum shapes. However, there are no books explaining such theories beginning with their foundations. The structure of the book is shown in the Preface. The theorems are built up using mathematical results. Therefore, a mathematical style is introduced, consisting of definitions and theorems to summarize the key points. This method of expression is advanced as provable facts are clearly shown. If something to be investigated is contained in the framework of mathematics, setting up a theory using theorems prepared by great mathematicians is thought to be an extremely effective approach. However, mathematics attempts to heighten the level of abstraction in order to understand many things in a unified fashion. This characteristic may baffle readers with an engineering background. Hence in this book, an attempt has been made to provide explanations in engineering terms, with examples from mechanics, after accurately denoting the provable facts using definitions and theorems

Proposal of a New Crack Model Considering the Discontinuity in the Cracked Plane and Its Application to the Evaluation of Crack Parameter

A new crack model which enables us to evaluate crack parameters such as crack energy density and its distribution, COD and COA under arbitrary load history is proposed and the availability of the proposed model is demonstrated through finite element analyses of elasto-plastic crack. The contents are as follows ; (1) A crack model considering the discontinuity in the cracked plane is introduced and the constitutive equation for a plane with discontinuity is formulated on as elasto-plastic case. (2) The finite element formulation of the model is carried out by introducing a plane element. (3) An elasto-plastic crack expressed by the proposed model under monotonic loading is analyzed by finite element method and the availability of the model is verified through the evaluation of crack energy density.・rights：日本機械学会・rights：本文データは学協会の許諾に基づきCiNiiから複製したものである・relation：isVersionOf：http://ci.nii.ac.jp/naid/110002358635

An Evaluation of the Fracture Resistance of a Stably Growing Crack by Crack Energy Density : 1st Report, Derivation of Fundamental Relations and Proposal of Evaluation Method

The objective of this study is to propose a practical method to evaluate the fracture resistance of a stably growing crack by crack energy density and to verify it through its applications to actual stable crack growth problems. The contents of this report are as follows: (1) More refined investigation of the relationship obtained before between initial crack length, present crack length, load-diplacement curves and the additional rate of crack energy density which holds until a crack starts to grow is made by using two crack models. (2) A relationship between initial crack length, present crack length, load-displacement curves and the additional rate of crack energy density caused by using the same crack models as sued in (1). (3) A method to evaluate the fracture resistance of a stably growing crack from load-displacement curves which can be easily obtained by experiments is proposed, based on the relations above.・rights：日本機械学会・rights：本文データは学協会の許諾に基づきCiNiiから複製したものである・relation：isVersionOf：http://ci.nii.ac.jp/naid/110002358883

Shape optimization method applied to room design based on an incompressible fluid model (Topology optimization theory and applications toward wide fields of natural sciences)

"Topology optimization theory and applications toward wide fields of natural sciences". May 7～9, 2014. edited by Takashi Nakazawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.This paper presents numerical results obtained, using a numerical method for a flow field shape optimization, for the design of a room from which people can evacuate smoothly. The domain of interest is a two-dimensional Poiseuille flow with sudden contraction in which a disk is located initially. In fact, the compressible fluid model has been used to describe the dynamics of people for the study of evacuation. However, we set a model involving the incompressible Navier-Stokes equation for the first trial. And the shape optimization to minimize the dissipation energy on the disk is demonstrated under the volume constraint. For reshaping numerically, the traction method is used. Numerical results reveal that the shape in the wake of the disk becomes an acute angle to decrease the dissipation energy monotonically, thereby satisfying the volume constraint. Such a shape has never been inferred from results of earlier studies of the evacuation problem in jamology

Proposal of New Stability-instability Criterion for Crack Extension Based on Crack Energy Density and Physical Systematization of Other Criteria

A new stability-instability criterion (named T_ε(T^*_ε) criterion) for crack growth which is applicable to small scale yielding cracks and also to large scale yielding cracks is proposed based on crack energy density concept. T_ε criterion is a criterion in which attention is paid to the rate of variation of crack energy density at a crack tip point of every moment, and T^*_ε criterion is another version of T_ε criterion and is a criterion in which attention is paid to the rate of variation of crack energy density at a fixed point which will be a new crack tip point after extension. The relations between T_ε(T^*_ε) criterion and other known criteria (criterion based on g-R-a curves; T_j, T_δ and T_W criteria) are also discussed and it is shown that the physical meanings of other criteria can be made clear and all criteria can be systematized based on the new criterion