## Ohtsuka Lab.

Around 2007, at Kaken
, I made the plan to create a virtual laboratory with many staff for mathematical research, and created the site CoMFoS
. Having retired from the university
, I want to create my own virtual lab as a virtual world
. Here, the tool MaKR
for creating Web pages is recreated so that it can be used for mathematical expressions by MathJax
. MaKR = **Ma**thematical **K**nowledge **R**epository

We can see the building
in virtual world. The place is 30-minute walk from the ferry port to Miyajima
, where the World Heritage Itsukushima Shrine
is located. The building is virtual, but the room (office) and its place are real.

On the ground, the laboratory office, library, my office and cafe terrace. There is a computer room on the 2nd floor and a restaurant on the 3rd floor.

## Topics

*Mathematical research based on the concept of fracture mechanics*starting from graduate school (1980), that is a really difficult problem and has not been solved yet.- I proposed a gneralization (
*GJ-integral*) of J-integral to express*3-dimensional fracture phenomena*in 1981 - Systematic study of
*shape sensitivity analysis*such as Hadamard's variational formula using*GJ-integral*. - The method using GJ-integral has features such as being applicable even if the boundary value problem has
*singular points*and also*extracting local features*

## Fracture

Let $\Omega $ be a bounded domain with Lipschitz boundary in $\mathbb{R}^d, d=2,3$, and $\Sigma \subset \Omega $ be a surface with the edge. The elasticity is defined on the domain $\Omega _{\Sigma }:=\Omega \setminus \Sigma $ with stress free $\sigma _{ij}^{±}(u)\nu _j=0$ on $\Sigma $,where $u$ is the displacement permitting the jump across the crack , $\sigma _{ij}^{±}(u)$ the stress on the both($±$) side of $\Sigma $, and $\nu =(\nu _j)_{j=1,…,d}$ is the unit normal vector on $\Sigma $ directed from the plus side toward the minus. Notice that $u^+(x)=u^-(x)$ on $x\in \partial \Sigma $.Examining the singularity at the edge of the crack in an elastic body with a crack is called

*crack problem*.

*Fracture problem*is to simulate crack progresses, that is the initiation and propagation of failure, etc. Huge mathematical model describing fracture phenomena has been created, but there is no one to handle phenomena by systematically expressing, For now, part of fracture phenomenon can be analyzed mathematically based on energy balance theory of Griffith.

## Shape Optimization of boundary

Let $\Omega $ be a bounded domain with Lipschitz boundary in $\mathbb{R}^d, d=2,3$ and $u(\Omega )$ a minimizer of an energy functional $\mathcal{E}$ over a subset $V(\Omega )$ in the function space $W^{1,p}(\Omega ;\mathbb{R}^m),m\ge 1$, Consider a cost functional $F(u(\Omega ))$. The *shape optimization of boundary* is a problem of finding a method of deforming a given set $\Omega $ so as to lower the cost function, and *optimized shape* $\Omega ^O$ is given by $$ F(u(\Omega ^O)) \lt F(u(\Omega ))$$

### Energy Optimization in Dirichlet boundary conditions

A famous example is the Poisson's equation $-\Delta u=f$ with Dirichlet boundary conditions; $u=0$ on $\partial \Omega $. The minimizer $(u^{OD},\Omega ^{OD})$ of the energy $\mathcal{E}(u)=\int _{\Omega }\{|\nabla u|^2/2-fu\} dx$, that is, $$ F(u(\Omega ^{OD}))=\min _{\Omega }\{F(u(\Omega )):|\Omega |=constant\}$$ satisfy the condition $\partial u^{OD}/\partial n=\textrm{constant}$ on $\partial \Omega ^{OD}$. Here $F(u(\Omega ))=\mathcal{E}(u)$. The figure just below is the numerical example when $f=1$.Optimal design $\Omega ^{OD}$ obtained by iterative calculation from the polygon on the left

Just because the disk $\Omega ^{OD}$ is energy stable, not all shapes can be made into a disk. It is more practical to think about how to deform the shape to make it more energy stable. We say

*Shape Optimization Problem*to find the method how to deform the shape $\Omega $ to make more stable shape $\Omega ^{O}$. We call $\Omega ^O$

*optimized shape*, and $\Omega ^{OD}$ (if exists) is called

*optimal shape*(right hand side of the figure just above).

An optimized shape of a polygonal domain

Top

### Optimization in Mixed boundary condition

Select the unit disk $\Omega =\{(x_1,x_2):x_1^2+x_2^2 \lt 1\}$ as the initial, $u=0$ on $\Gamma _D=\partial \Omega \cap \{x_2 \gt 0\}$ and put $\partial u/\partial n=0$ on $\Gamma _N=\partial \Omega \cap \{x_2 \lt 0\}$ as shown in the figure just below. The right figure is an optimized shape $\Omega ^O$ of $\Omega $. The solution $u$ has high singularity at $\gamma _M^i,i=1,2$, that is, $u\notin H^{3/2}$ and $u\in H^{3/2-\epsilon }$ near $\gamma _M^i,i=1,2$ for all $\epsilon \gt 0$.Optimized shape in Poisson Eq. with mixed boundary condition

A series of optimizations can be obtained.

Notice that optimized shapes are determined by $f$ in $-\Delta u=f$. For example $f(x_1,x_2)=\sin (2\pi x_1)$, we get a series of optimized shapes as shown below..

A series of optimizations under $-\Delta u=\sin (\pi x_1)$.

## Mean Compliance Optimization

Cantilever by Allaire (see [Al07] 6.5 Algorithmes numériques) More*The initial domain $\Omega $ is just below. A series of domains $\Omega ^O_k$ satisfying the constraints $|\Omega ^O_k|=\textrm{constant}$ ($|D|$ represents the area of D.). $$ F(\Omega )=\int _{\Gamma _N}g·u\,ds$$

where $\Gamma _N$ is fixed and vertical deformation of $\Gamma _D$ B is allowed. In [Al07], $\Gamma _D$ is fixed.

The left side is the optimized shape in the figure just above, and the right side is the optimized shape with 7 holes.

*optimnal shape cannot be determined only by the constraint $|\Omega |$=constant*.

## Least Square Error

For a given functoin $u_d$, find $(u^O,\Omega ^O)$ such that $$ F(u(\Omega )):=\int _{\Omega }|u(\Omega )-u_d|^2dx$$ The example below is an optimized shape when $u_d = 0$ in Poisson's equation under Dirichlet boundary conditions starting from the square region. Since it corresponds to the problem of cooling, it is considered that the longer the boundary, the more optimized domains.The left side is the starting region, and an optimized domain in the right.

At the 400th iteration, the optimized shape looks like an elongated string.