Error in AMS-SoptS

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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 (Shape Optimisation of Singular 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 = Mathematical Knowledge Repository, making since around 2000.)



  • 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



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
As a feature, the angle at $(\gamma _M^i)^O,i=1,2$ becomes smaller than $\pi $, then $u\in H^{3/2}$ near $(\gamma _M^i)^O$. Second feature is that $|\Gamma _D^O| \lt |\Gamma _D|, |\Gamma _N^O| \gt |\Gamma _D|$ where $|\Gamma |$ denotes the length of the curve $\Gamma $. A series of optimizations as shown in the figure below can be obtained.

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.
Since the thinly stretched shape are more optimized, 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.
Next, consider the mixed boundary condition setting by $\Gamma _D=\Gamma _1,\Gamma _N=\Gamma _1\cup \Gamma _2\cup \Gamma _4$, and $u_d=0$. From the sequence below, it can be seen that $\Gamma _N^O$ approaches $\Gamma _D^O (u = 0)$ to reduce the error.

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.



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