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This web page is an evolving progress report on a joint research with Muruhan Rathinam and Alen Agheksanterian on issues related to random media and homogenization.
Please note: This page is very much under construction and as such, very incomplete. Visit again to watch as it takes shape.
In a domain $\D \subset \R^n$ consider the boundary value problem \begin{equation} \label{eq:BVP} \begin{aligned} -\div \big( A(x) \nabla u(x) \big) &= f(x) & & x \in \D, \\ u(x) &= g(x) & & x \in \partial\D, \end{aligned} \end{equation} where $f : \D \to \R$, $\:$ $g : \partial\D \to \R$ and $A : \D \to \Snn$ (the space of $n \times n$ symmetric matrices) are given and $u : \D \to \R$ is the unknown. We assume that $A$ is uniformly positive-definite, that is, there exist constants $0 < \nu_1 \le \nu_2$ such that \[ \nu_1 |\xi|^2 \le \xi \cdot A(x) \xi \le \nu_2 |\xi|^2 \quad \text{for all } x \in \D \text{ and } \xi \in \R^n. \]
The boundary value problem in $\ref{eq:BVP}$ may be interpreted variously as a description of conduction of heat, or distribution electric charges or potential, or diffusion of a chemical through a medium. The coefficient $A$ specifies the medium's physical properties. If $A$ is independent of $x$, then the medium is homogeneous, otherwise it is inhomogeneous. If $A(x)$ is a multiple of the identity matrix, then the medium is isotropic at $x$, otherwise it is anisotropic.
The (infinitesimal) displacement field $\vec{u} : \D \to \R^n$ of an elastic medium in $\D$ satisfies the boundary value problem: \begin{equation} \label{eq:BVP-elasticity} \begin{aligned} -\div \Big( \C \big[ E(\vec{u}) \big] \Big) &= \vec{f} & &\text{in } \D, \\ \vec{u} &= \vec{g} & &\text{on } \partial\D, \end{aligned} \end{equation} where $E(\vec{u}) = \frac{1}{2} \big(\nabla \vec{u} + (\nabla \vec{u})^T\big)$ is the (infinitesimal) strain corresponding the the displacement $\vec{u}$, and where $\vec{f} : \D \to \R^n$ is the body force, $\vec{g} : \partial\D \to \R^n$ is the boundary displacement, and $\C = \C(x)$, the elasticity tensor at point $x \in \D$, is a linear transformation on the space of second order symmetric tensors; see Gurtin [4,5]. We assume that $\C$ is uniformly positive-definite, that is, there exist constants $0 < \nu_1 \le \nu_2$ such that \[ \nu_1 |E|^2 \le E\cdot \C(x)[E] \le \nu_2 |E|^2 \quad \text{for all } x \in \D \text{ and } E \in \Snn. \]
If $\C$ is independent of position, then the material is homogeneous, else it is inhomogeneous. If the material at a point $x\in\D$ has no preferred directions, then it is isotropic at $x$, else it is anisotropic at $x$. It can be shown that if the material is isotropic, then: \begin{equation} \C [E] = 2\mu E + \lambda \tr E, \quad \text{for all } E \in \Snn. \end{equation} The scalar coefficients $\mu$ and $\lambda$ are the material's Lamé moduli.
Let $\Xi$ be the rectangle $\prod_{i=1}^n (0,l_i)$ and let $A : \Xi \to \Snn$ be given. Extend $A$ as a $\Xi$-periodic function to $\R^n$. Then define $A_\eps(x) = A(x/\eps)$ for arbitrary $\eps>0$. If $A$ is non-constant and $\eps$ is small, then the function $A_\eps(x)$ is highly oscillatory.
Consider the boundary value problem \begin{equation} \label{eq:BVP-eps} \begin{aligned} -\div \big( A_\eps\nabla u_\eps \big) &= f & &\text{in } \D, \\ u_\eps &= 0 & &\text{on } \partial\D. \end{aligned} \end{equation} It can be shown, see e.g. Bensoussan, Lions, Papanicolaou [1], Sánchez-Palencia [6], or Jikov, Kozlov, Olenik, [3], that as $\eps\to0$, the solution $u_\eps$ converges weakly in the Sobolev space $H^1_0(\D)$ to the solution $u_0$ of the boundary value problem \begin{equation} \label{eq:BVP-0} \begin{aligned} -\div \big( A_0\nabla u_0 \big) &= f & &\text{in } \D, \\ u_0 &= 0 & &\text{on } \partial\D, \end{aligned} \end{equation} where $A_0$, which is a constant positive-definite matrix, characterizes the effective conductivity of the material. The animation in Figure 1 illustrates the homogenization of a checkerboard.
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| Figure 1: An illustration of the homogenization of a checkerboard. |
The matrix $A_0$ is computed as follows. For arbitrary $\lambda \in \R^n$, solve the unit cell problem \begin{equation} \label{eq:BVP-unit-cell} -\div \big( A(\lambda + \nabla v_\lambda \big) = 0 \quad\text{in } \Xi, \end{equation} in the Sobolev space $H^1_{\mathrm{per}}(\Xi)$ which is the restriction to $\Xi$ of $\Xi$-periodic functions in $H^1_{\mathrm{loc}}(\R^n)$.
The solution $v_\lambda$ (which is determined modulo an additive constant,) is linear in $\lambda$. The coefficient matrix $A_0$ is obtained from \begin{equation} \label{eq:A0-per} A_0 \lambda = \frac{1}{|\Xi|} \int_\Xi A(\lambda + \nabla v_\lambda) \, dx, \end{equation} where $|\Xi|$ is the volume of $\Xi$.
Let $n=2$ and let the unit cell be the square $\Xi = (-1,1)\times(-1,1)$. Let $M = \begin{pmatrix} 2 & 0 \\ 0 & 1\end{pmatrix}$ and Let $Q(t) = \begin{pmatrix} \cos t & \sin t \\ -\sin t & \cos t\end{pmatrix}$. Define $A$ on $\Xi$ by: \begin{equation} \label{eq:A-2x2-per} A(x) = \begin{cases} Q\big(\frac{\pi}{6}\big)^T M Q\big(\frac{\pi}{6}\big) & \text{if $x$ is in the first or third quadrant}, \\ Q\big(\frac{2\pi}{3}\big)^T M Q\big(\frac{2\pi}{3}\big) & \text{if $x$ is in the second or fourth quadrant}. \end{cases} \end{equation} It can be shown that the corresponding homogenized matrix $A_0 = \sqrt{2} \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$.
Figure 2 shows the solution $v_\lambda$ of the unit cell problem for $\lambda = (1,0)$, computed using a finite element solver that we have developed for this purpose.
The problem needs to be solved twice, once with $\lambda = (1,0)$ and again with $\lambda = (0,1)$. Then the homogenized diffusivity $A_0$ is evaluated through the averaging formula $\ref{eq:A0-per}$. We obtain the predicted value of $A_0$ within the computer's floating point accuracy.
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| Figure 2: The solution of the boundary value problem $\ref{eq:BVP-unit-cell}$ on $\Xi=(-1,1)\times(-1,1)$ with $\lambda = (1,0)$ and $A$ as in $\ref{eq:A-2x2-per}$. |
Let $(\Omega,\mathcal{F},\mu)$ be a probability space. As usual, $\Omega$ is the set of all events, $\mu$ is a probability measure on $\Omega$, and $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$.
Let $T_x : \Omega \to \Omega$ be an ergodic and measure preserving dynamical system with parameter $x \in \R^n$. We define the diffusivity coefficient $A$ in $\ref{eq:BVP}$ in terms of a uniformly positive-definite function $\mathcal{A} : \Omega \to \Snn$, via \[ A(x,\omega) = \mathcal{A}\big(T_x(\omega)\big), \quad x \in \R^n, \: \omega \in \Omega. \] Then we let $A_\eps(x,\omega) = A(x/\eps,\omega)$. This defines a medium with random diffusivity of length scale $\eps$. We look at the boundary value problem: \begin{equation} \label{eq:BVP-eps-rand} \begin{aligned} -\div \big( A_\eps(x,\omega)\nabla u(x,\omega)\big) &= f(x) & &\text{in } \D, \\ u_\eps(x,\omega) &= 0 & &\text{on } \partial\D. \end{aligned} \end{equation} It can be shown that for almost all $\omega\in\Omega$, the solution $u_\eps(\cdot,\omega)$ of $\ref{eq:BVP-eps-rand}$ converges to the solution $u_0$ of the problem: \begin{equation} \label{eq:BVP-0-rand} \begin{aligned} -\div \big( A_0\nabla u_0 \big) &= f & &\text{in } \D, \\ u_0 &= 0 & &\text{on } \partial\D, \end{aligned} \end{equation} where $A_0$ is a constant matrix, independent of $x$ and $\omega$.
The homogenized matrix $A_0$ may be computed according to the following algorithm. Fix $\omega \in \Omega$. Extract a square patch of size $(0,\rho)\times(0,\rho)$ from the material. Tile $\R^n$ periodically with that patch. Apply the procedure described in Section 2 above to calculate the corresponding homogenized matrix $A_{\rho,\omega}$. Bourgeat and Piatnitski [2] have shown that as $\rho \to \infty$, the matrix $A_{\rho,\omega}$ converges to $A_0$ for almost all $\omega$.
Let $n=2$ and let the microstructure consists of an infinite checkerboard, each tile of which is a homogeneous material with diffusivity equal to 1 or 2, with equal probability. It can be shown that the corresponding homogenized matrix $A_0 = \sqrt{2} \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$.
The left subfigure of Figure 3 shows a $20\times20$ patch from a realization of such a random medium. The right subfigure of Figure 3 shows the solution of the corresponding unit cell problem. Computing $A_0$ from $\ref{eq:A0-per}$ we get: \[ A_0 \approx \begin{pmatrix} 1.4356768 & 0.0013898277 \\ 0.0013898277 & 1.4249341 \end{pmatrix}, \] which is somewhat close to the theoretical value of $\sqrt{2}I$. Computing on a larger sample will give a better approximation.
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| Figure 3: On the left, a $20\times20$ patch from a realization of the random checkerboard. On the right, the solution of the corresponding unit cell problem. | |
... under construction ...
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This web page was created on 2008–09–14.
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