3.2. Static micromagnetics

As summed up in [Abert, 2019], Static micromagnetics is the theory of stable magnetization configurations.

The magnetization is found as minimizer of the following optimization problem:

(3.1)\[\min_{\mbf \in \Mcal} E(\mbf) \qquad \text{subject to} \qquad | \mbf (\xbf) | = 1 \quad \forall \xbf \in \Omega \qquad \text{and} \qquad | \mbf (\xbf) | = 1 \quad \forall \xbf \in \partial\Omega. \]

As the minimizer \(\mbf\) is necessarily a critical point of \(E\), we derive the optimality conditions using the Lagrangian multiplier technique.

3.2.1. Energy derivative

To proceed, we need to evaluate the derivative of the energy \(E: \Mcal \to \Rbb\). As the energy is the sum of different components, we can write:

\[ dE(\mbf; \vbf) = dE^{\mathsf{zee}} (\mbf; \vbf) + dE^{\mathsf{ex}} (\mbf; \vbf) + dE^{\mathsf{dem}} (\mbf; \vbf) + dE^{\mathsf{ani}} (\mbf; \vbf). \]

Hence, we can derive the directional derivatives for all components following Definition 1.10.

3.2.1.1. Zeeman energy

\[\begin{split} dE^{\mathsf{zee}} (\mbf; \vbf) &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ - \mu_0 \int_{\Omega} \Ms \ \left( \mbf + \epsilon \vbf \right) \cdot \Hbf^{\mathsf{zee}} \dxbf + \mu_0 \int_{\Omega} \Ms \ \mbf \cdot \Hbf^{\mathsf{zee}} \dxbf \right] \\ &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ \cancel{- \mu_0 \int_{\Omega} \Ms \ \mbf \cdot \Hbf^{\mathsf{zee}} \dxbf} - \epsilon \mu_0 \int_{\Omega} \Ms \ \vbf \cdot \Hbf^{\mathsf{zee}} \dxbf + \cancel{\mu_0 \int_{\Omega} \Ms \ \mbf \cdot \Hbf^{\mathsf{zee}} \dxbf} \right] \\ &= - \mu_0 \int_{\Omega} \Ms \, \Hbf^{\mathsf{zee}} \cdot \vbf \, \dxbf \end{split}\]

3.2.1.2. Exchange energy

\[\begin{split} dE^{\mathsf{ex}} (\mbf; \vbf) &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ \int_{\Omega} A \left( \nabla \left( \mbf + \epsilon \vbf \right) \right)^2 \dxbf - \int_{\Omega} A \left( \nabla \mbf \right)^2 \dxbf \right] \\ &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ \int_{\Omega} A \left( \nabla \mbf + \epsilon \nabla \vbf \right)^2 \dxbf - \int_{\Omega} A \left( \nabla \mbf \right)^2 \dxbf \right] \\ &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ \int_{\Omega} A \left( \cancel{\left( \nabla \mbf \right)^2} + \epsilon^2 \left( \nabla \vbf \right)^2 + 2 \epsilon \sum_i \nabla m_i \cdot \nabla v_i \right) \dxbf - \cancel{\int_{\Omega} A \left( \nabla \mbf \right)^2 \dxbf} \right] \\ &= \lim_{\epsilon \to 0} \left[ \int_{\Omega} A \left( \epsilon \left( \nabla \vbf \right)^2 + 2 \sum_i \nabla m_i \cdot \nabla v_i \right) \dxbf \right] \\ &= 2 \sum_i \int_{\Omega} A \nabla m_i \cdot \nabla v_i \dxbf \end{split}\]

As we would like to unload all the derivatives on \(\mbf\) rather than \(\vbf\), we can go one step further. Integrating by parts and using the divergence theorem, we get

\[\begin{split} \int_{\Omega} A \nabla m_i \cdot \nabla v_i \dxbf &= \int_{\Omega} \nabla \cdot (v_i A \nabla m_i) \dxbf - \int_{\Omega} \nabla \cdot (A \nabla m_i) v_i \dxbf \\ &= \int_{\partial\Omega} v_i A \frac{\partial m_i}{\partial \bf{n}} \dsbf - \int_{\Omega} \nabla \cdot (A \nabla m_i) v_i \dxbf \end{split}\]

And, by summing for \(i=1,2,3\) we finally get

\[ dE^{\mathsf{ex}} (\mbf; \vbf) = - 2 \int_{\Omega} \left[ \nabla \cdot (A \nabla \mbf) \right] \cdot \vbf \, \dxbf + 2 \int_{\partial\Omega} A \frac{\partial \mbf}{\partial \bf{n}} \cdot \vbf \, \dsbf \]

3.2.1.3. Demagnetization energy

\[\begin{split} dE^{\mathsf{dem}} (\mbf; \vbf) &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ - \frac{\mu_0}{2} \int_{\Omega} \Ms \left( \mbf + \epsilon \vbf \right) \cdot \Hbf^{\mathsf{dem}} \left( \mbf + \epsilon \vbf \right) \dxbf + \frac{\mu_0}{2} \int_{\Omega} \Ms \mbf \cdot \Hbf^{\mathsf{dem}} (\mbf) \dxbf \right] \\ &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ - \frac{\mu_0}{2} \int_{\Omega} \Ms \, \mbf \cdot \Hbf^{\mathsf{dem}} \left( \mbf + \epsilon \vbf \right) \dxbf - \epsilon \frac{\mu_0}{2} \int_{\Omega} \Ms \, \vbf \cdot \Hbf^{\mathsf{dem}} \left( \mbf + \epsilon \vbf \right) \dxbf + \frac{\mu_0}{2} \int_{\Omega} \Ms \, \mbf \cdot \Hbf^{\mathsf{dem}} (\mbf) \dxbf \right]. \end{split}\]

As the demagnetization field \(\Hbf^{\mathsf{dem}}\) is linear in the magnetization \(\mbf\), we have

\[ \lim_{\epsilon \to 0} \Hbf^{\mathsf{dem}} (\mbf + \epsilon \vbf) = \Hbf^{\mathsf{dem}} (\mbf). \]

Therefore,

\[ dE^{\mathsf{dem}} (\mbf; \vbf) = - \frac{\mu_0}{2} \int_{\Omega} \Ms \, \Hbf^{\mathsf{dem}} (\mbf) \cdot \vbf \, \dxbf . \]

3.2.1.4. Anisotropy energy

For the uniaxial anisotropy, let us first calculate

\[\begin{split} \left( \left( \mbf + \epsilon \vbf \right) \cdot \ebf_{\mathsf{u}} \right)^2 &= \left(\mbf \cdot \ebf_{\mathsf{u}} + \epsilon \vbf \cdot \ebf_{\mathsf{u}} \right)^2 \\ &= \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^2 + 2 \epsilon \left(\mbf \cdot \ebf_{\mathsf{u}} \right) \left( \vbf \cdot \ebf_{\mathsf{u}} \right) + \epsilon^2 \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^2 \end{split}\]

and

\[\begin{split} \left( \left( \mbf + \epsilon \vbf \right) \cdot \ebf_{\mathsf{u}} \right)^4 &= \left[ \left( \left( \mbf + \epsilon \vbf \right) \cdot \ebf_{\mathsf{u}} \right)^2 \right]^2 \\ &= \left[ \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^2 + 2 \epsilon \left(\mbf \cdot \ebf_{\mathsf{u}} \right) \left( \vbf \cdot \ebf_{\mathsf{u}} \right) + \epsilon^2 \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^2 \right]^2 \\ &= \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^4 + 4 \epsilon \left(\mbf \cdot \ebf_{\mathsf{u}} \right)^3 \left( \vbf \cdot \ebf_{\mathsf{u}} \right) + 6 \epsilon^2 \left(\mbf \cdot \ebf_{\mathsf{u}} \right)^2 \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^2 + 4 \epsilon^3 \left(\mbf \cdot \ebf_{\mathsf{u}} \right) \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^3 + \epsilon^4 \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^4 \end{split}\]

Then,

\[\begin{split} dE^{\mathsf{aniu}} (\mbf; \vbf) &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ - \int_{\Omega} \left[ K_{\mathsf{u}1} \left( \left( \mbf + \epsilon \vbf \right) \cdot \ebf_{\mathsf{u}} \right)^2 + K_{\mathsf{u}2} \left( \left( \mbf + \epsilon \vbf \right) \cdot \ebf_{\mathsf{u}} \right)^4 \right] \dxbf + \int_{\Omega} \left[ K_{\mathsf{u}1} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^2 + K_{\mathsf{u}2} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^4 \right] \dxbf \right] \\ &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\Omega} \left[ \cancel{- K_{\mathsf{u}1} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^2} - \epsilon^2 K_{\mathsf{u}1} \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^2 - 2 \epsilon K_{\mathsf{u}1} \left(\mbf \cdot \ebf_{\mathsf{u}} \right) \left( \vbf \cdot \ebf_{\mathsf{u}} \right) - \cancel{K_{\mathsf{u}2} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^4} - 4 \epsilon K_{\mathsf{u}2} \left(\mbf \cdot \ebf_{\mathsf{u}} \right)^3 \left( \vbf \cdot \ebf_{\mathsf{u}} \right) - 6 \epsilon^2 K_{\mathsf{u}2} \left(\mbf \cdot \ebf_{\mathsf{u}} \right)^2 \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^2 - 4 \epsilon^3 K_{\mathsf{u}2} \left(\mbf \cdot \ebf_{\mathsf{u}} \right) \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^3 - \epsilon^4 K_{\mathsf{u}2} \left( \vbf \cdot \ebf_{\mathsf{u}} \right)^4 + \cancel{K_{\mathsf{u}1} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^2} + \cancel{K_{\mathsf{u}2} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^4} \right] \dxbf \\ &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\Omega} \left[ - 2 \epsilon K_{\mathsf{u}1} \left(\mbf \cdot \ebf_{\mathsf{u}} \right) \left( \vbf \cdot \ebf_{\mathsf{u}} \right) - 4 \epsilon K_{\mathsf{u}2} \left(\mbf \cdot \ebf_{\mathsf{u}} \right)^3 \left( \vbf \cdot \ebf_{\mathsf{u}} \right) \right] \dxbf \\ &= - \int_{\Omega} \left[ 2 K_{\mathsf{u}1} \left(\mbf \cdot \ebf_{\mathsf{u}} \right) \, \ebf_{\mathsf{u}} + 4 K_{\mathsf{u}2} \left(\mbf \cdot \ebf_{\mathsf{u}} \right)^3 \, \ebf_{\mathsf{u}} \right] \cdot \vbf \, \dxbf \end{split}\]

For the cubic anisotropy, (ignoring all monomials where \(\epsilon\) has a higher power than 1)

\[\begin{split} dE^{\mathsf{anic}} (\mbf; \vbf) &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[ \int_{\Omega} \left[ K_{\mathsf{c}1} \left( (m_1+\epsilon v_1)^2 (m_2+\epsilon v_2)^2 + (m_2+\epsilon v_2)^2 (m_3+\epsilon v_3)^2 + (m_3+\epsilon v_3)^2 (m_1+\epsilon v_1)^2 \right) + K_{\mathsf{c}2} (m_1+\epsilon v_1)^2 (m_2+\epsilon v_2)^2 (m_3+\epsilon v_3)^2 \right] \dxbf - \int_{\Omega} \left[ K_{\mathsf{c}1} \left( m_1^2 m_2^2 + m_2^2 m_3^2 + m_3^2 m_1^2 \right) + K_{\mathsf{c}2} m_1^2 m_2^2 m_3^2 \right] \dxbf \right] \\ % &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\Omega} \left[ K_{\mathsf{c}1} \left( (m_1^2 + 2 \epsilon m_1 v_1) (m_2^2 + 2 \epsilon m_2 v_2) + (m_2^2 + 2 \epsilon m_2 v_2) (m_3^2 + 2 \epsilon m_3 v_3) + (m_3^2 + 2 \epsilon m_3 v_3) (m_1^2 + 2 \epsilon m_1 v_1) \right) + K_{\mathsf{c}2} (m_1^2 + 2 \epsilon m_1 v_1) (m_2^2 + 2 \epsilon m_2 v_2) (m_3^2 + 2 \epsilon m_3 v_3) - K_{\mathsf{c}1} \left( m_1^2 m_2^2 + m_2^2 m_3^2 + m_3^2 m_1^2 \right) - K_{\mathsf{c}2} m_1^2 m_2^2 m_3^2 \right] \dxbf \\ % &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\Omega} \left[ K_{\mathsf{c}1} \left( m_1^2 m_2^2 + 2 \epsilon m_1^2 m_2 v_2 + 2 \epsilon m_1 m_2^2 v_1 + m_2^2 m_3^2 + 2 \epsilon m_2^2 m_3 v_3 + 2 \epsilon m_2 m_3^2 v_2 + m_1^2 m_3^2 + 2 \epsilon m_1 m_3^2 v_1 + 2 \epsilon m_1^2 m_3 v_3 \right) + K_{\mathsf{c}2} \left( \cancel{m_1^2 m_2^2 m_3^2} + 2 \epsilon m_1^2 m_2 m_3^2 v_2 + 2 \epsilon m_1 m_2^2 m_3^2 v_1 + 2 \epsilon m_1^2 m_2^2 m_3 v_3\right) - K_{\mathsf{c}1} \left( m_1^2 m_2^2 + m_2^2 m_3^2 + m_3^2 m_1^2 \right)^2 - \cancel{K_{\mathsf{c}2} m_1^2 m_2^2 m_3^2} \right] \dxbf \\ % &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\Omega} \left[ K_{\mathsf{c}1} \left( \cancel{m_1^2 m_2^2} + 2 \epsilon m_1^2 m_2 v_2 + 2 \epsilon m_1 m_2^2 v_1 + \cancel{m_2^2 m_3^2} + 2 \epsilon m_2^2 m_3 v_3 + 2 \epsilon m_2 m_3^2 v_2 + \cancel{m_1^2 m_3^2} + 2 \epsilon m_1 m_3^2 v_1 + 2 \epsilon m_1^2 m_3 v_3 \right) + K_{\mathsf{c}2} \left( \cancel{m_1^2 m_2^2 m_3^2} + 2 \epsilon m_1^2 m_2 m_3^2 v_2 + 2 \epsilon m_1 m_2^2 m_3^2 v_1 + 2 \epsilon m_1^2 m_2^2 m_3 v_3\right) - K_{\mathsf{c}1} \left( \cancel{m_1^2 m_2^2} + \cancel{m_2^2 m_3^2} + \cancel{m_3^2 m_1^2} \right)^2 - \cancel{K_{\mathsf{c}2} m_1^2 m_2^2 m_3^2} \right] \dxbf \\ % &= 2 \int_{\Omega} \left[ K_{\mathsf{c}1} \left( m_1^2 m_2 v_2 + m_1 m_2^2 v_1 + m_2^2 m_3 v_3 + m_2 m_3^2 v_2 + m_1 m_3^2 v_1 + m_1^2 m_3 v_3 \right) + K_{\mathsf{c}2} \left( m_1^2 m_2 m_3^2 v_2 + m_1 m_2^2 m_3^2 v_1 + m_1^2 m_2^2 m_3 v_3 \right) \right] \dxbf \\ % &= 2 \int_{\Omega} \left[ K_{\mathsf{c}1} \left( \begin{array}{c} m_1 (m_2^2 + m_3^2) \\ m_2 (m_1^2 + m_3^2) \\ m_3 (m_1^2 + m_2^2) \end{array} \right) \cdot \vbf + K_{\mathsf{c}2} \left( \begin{array}{c} m_1 (m_2 m_3)^2 \\ m_2 (m_1 m_3)^2 \\ m_3 (m_1 m_2)^2 \end{array} \right) \cdot \vbf \right] \dxbf \end{split}\]

3.2.1.5. Gateaux derivative of the energy

Following the definition from Definition 1.10, we define the Gateaux derivative as an operator \(E' : \Lcal (\Mcal, \Rbb)\):

\[ \langle E'(\mbf), \vbf \rangle_{\Mcal, \Rbb} := dE (\mbf; \vbf). \]

Taking as example an energy composed of demagnetization and exchange, we get:

\[\begin{split} \langle E'(\mbf), \vbf \rangle_{\Mcal, \Rbb} &= dE^{\mathsf{ex}} (\mbf; \vbf) + dE^{\mathsf{dem}} (\mbf; \vbf) \\ &= - 2 \int_{\Omega} \left[ \nabla \cdot (A \nabla \mbf) \right] \cdot \vbf \, \dxbf + 2 \int_{\partial\Omega} A \frac{\partial \mbf}{\partial \bf{n}} \cdot \vbf \, \dsbf - \frac{\mu_0}{2} \int_{\Omega} \Ms \, \Hbf^{\mathsf{dem}} (\mbf) \cdot \vbf \, \dxbf \\ &= - \int_{\Omega} \left[ 2 \nabla \cdot (A \nabla \mbf) + \frac{\mu_0}{2} \Ms \, \Hbf^{\mathsf{dem}} (\mbf) \right] \cdot \vbf \, \dxbf + 2 \int_{\partial\Omega} A \frac{\partial \mbf}{\partial \bf{n}} \cdot \vbf \, \dsbf \end{split}\]

In general, we can split this derivative as an operator \(E'_\Omega\) acting on the internal domain and an operator \(E'_{\partial\Omega}\) acting on the boundary. Depending on the type of energy considered, the operator \(E'_{\partial\Omega}\) might be null.

In this case, we get

\[ \langle E'_\Omega (\mbf), \vbf \rangle_{\Mcal, \Rbb} = - \int_{\Omega} \left[ 2 \nabla \cdot (A \nabla \mbf) + \frac{\mu_0}{2} \Ms \, \Hbf^{\mathsf{dem}} (\mbf) \right] \cdot \vbf \, \dxbf \]

and

\[ \langle E'_{\partial\Omega} (\mbf), \vbf \rangle_{\Mcal, \Rbb} = 2 \int_{\partial\Omega} A \frac{\partial \mbf}{\partial \bf{n}} \cdot \vbf \, \dsbf. \]

By the Riesz representation theorem, we can identify the operators in \(\Lcal(\Mcal, \Rbb)\) with magnetizations in \(\Mcal\):

\[ E'_\Omega (\mbf) = - \left[ 2 \nabla \cdot (A \nabla \mbf) + \frac{\mu_0}{2} \Ms \, \Hbf^{\mathsf{dem}} (\mbf) \right] \]

and

\[ E'_{\partial\Omega} (\mbf) = 2 A \frac{\partial \mbf}{\partial \bf{n}} \]

3.2.2. Discretize-then-optimize approach

In this approach, first we discretize the optimization problem (3.1) in \(n\) variables using the finite element approximation and the optimize the problem in \(\Rbb^n\).

3.2.2.1. Finite element discretization

Given a grid \(\Tcal\) of points \(\{ \xbf_i \}_{i=1}^n\) and a finite element basis \(\{ \varphi_i \}_{i=1}^n\) such that

\[ \varphi_i(\xbf_j) = \delta_{i,j} \]

3.2.2.2. Lagrangian approach

We proceed with the Lagrangian multiplier technique. Let

\[ \mathcal{L} (\mbf, \lambda, \mu) := E(\mbf) + \int_\Omega \lambda \left( |\mbf|^2 - 1 \right) \dxbf + \int_{\partial \Omega} \mu \left( |\mbf|^2 - 1 \right) \dsbf. \]

with \(\lambda: \Omega \to \Rbb\) and \(\mu: \partial\Omega \to \Rbb\) multipliers to enforce space contraints.

3.2.3. Optimize-then-discretize approach

Important

Work in progress. Not sure this will work.

If the constraints were convex in \(\mbf\), we could follow Section 1.7.1 in [Hinze et al., 2008], because \(\Mcal\) is a Banach space and \(dE: \Mcal \to \Rbb\) is Gateaux differentiable.