## Ohtsuka Lab.

The virtual laboratory is the workplace of my independent research after retirement created by my tool MaKR which is recreated with mathematical expressions MathJax . The main topics are more details and additional pictures of*AMS-SoptS*

AMS-SoptS
is the paper *Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations*

published in the journal Sugaku Expositions
from American Mathematical Society
This article originally appeared in Japanese in Sugaku **70** **3** (2018), 255--274. However, the symbols have been improved and the results have been added in relation to fracture mechanics and recent results.

## What's SoptS

The theme is named **SoptS** (**S**hape **Opt**imisation of **S**ingular Points in BVPs) to study fracture phenomena and (boundary) shape optimisation in terms of *singularity of solutions* of boundary value problem for partial differential equations (BVPs). SoptS is closely related to *fracture mechanics* and *shape optimisation problems*. The main tool in SoptS, the GJ-integral, is a natural extension of the well-known *J-integral* in fracture mechanics.

(MaKR = **Ma**thematical **K**nowledge **R**epository, making since around 2000.)

## 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

### 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.

$\Gamma _D^O$ become small like a dot, and $\Gamma _N^O$ has a shape in which a thread is attached to a disk and connected to $\Gamma _D^O$.

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.). $$ F(\Omega )=\int _{\Gamma _N}g\cdot u\,ds$$

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*.

More..

## 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.

## Shape optimization of Interfaces

Consider a composite in which two different elastic plates are joined together. The static shape is divided into $\Omega _1$ and $\Omega _2$ as shown on the left in the figure below, and Young's modulus are $E_1 = 200, E_2=20$ and Poisson's ratio are $\nu _1=0.25, \nu _2=0.3$. The initial compliance is $14.2113$ and the compliance after optimization is $12.092$ as shown in the figure on the right. **Information about the page:** The current position is painted circle in the diagram below. Blue is the *main MaKR* and **orange** is a duplicate for *MaKR's public use*, where dashed line means the *connection* to the private area. The dashed lines are *only connections* to main MaKR.