2021 Fiscal Year Final Research Report
Deepening and applications of shape optimization theories
| Project/Area Number |
17K05140
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Computational science
|
|---|
| Research Institution | Nagoya University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2017-04-01 – 2022-03-31
|
| Keywords | 偏微分方程式 / 最適化 / 数値解析 |
|---|
| Outline of Final Research Achievements |
An optimization problem with respect to boundary shape of domain in which boundary value problem of partial differential equation is defined is called the shape optimization problem. In the previous study, defining the problem in an appropriate function space and solving it by the gradient and Newton methods had been presented. In this study, remaining theoretical and practical problems were resolved. As the theoretical problems, the existence theorems of the optimum solutions and the computation method of the second derivative of a cost function with respect to domain variation were revealed. As the practical problems, not only design problems in engineering, but also problems related to medical support and elucidation of biological functions were solved.
|
| Free Research Field |
計算科学
|
| Academic Significance and Societal Importance of the Research Achievements |
学術的意義：形状最適化問題は，密度や形状変動を表す関数を設計変数に選ぶことから，関数最適化問題として位置づけられる．本研究では，適切な関数空間上で許容集合を定義して，解がその中に入ることを保証するための条件を明確にすることができた．また，評価関数の2階微分を計算する方法として，これまで知られていなかった方法や領域変動型の問題においては定義を示すことができた．これらの成果は，同類の問題を考える上での基礎を与える．
社会的意義：形状最適化問題は，工学における様々な設計問題だけでなく，医療支援や生体機能の解明などに役立つさまざまな逆問題の解法としても使えることが明らかにされた．
|
