jorgensd
diff --git a/‎.github/workflows/publish_docker.yml‎
Lines changed: 73 additions & 23 deletions b/‎.github/workflows/publish_docker.yml‎
Lines changed: 73 additions & 23 deletions
diff --git a/‎_toc.yml‎
Lines changed: 5 additions & 1 deletion b/‎_toc.yml‎
Lines changed: 5 additions & 1 deletion
diff --git a/‎chapter2/elasticity_scaling.md‎
Lines changed: 2 additions & 2 deletions b/‎chapter2/elasticity_scaling.md‎
Lines changed: 2 additions & 2 deletions
diff --git a/‎chapter2/helmholtz.md‎
Lines changed: 125 additions & 0 deletions b/‎chapter2/helmholtz.md‎
Lines changed: 125 additions & 0 deletions
@@ -1,4 +1,5 @@
-name: Publish tutorial docker image
+# Recipe based on: https://docs.docker.com/build/ci/github-actions/multi-platform/#distribute-build-across-multiple-runners
+name: Build and publish platform dependent docker image
 on:
   push:
     branches:
@@ -15,8 +16,12 @@ env:
   IMAGE_NAME: ${{ github.repository }}
 
 jobs:
-  build-and-push-image:
-    runs-on: ubuntu-latest
+  build:
+    strategy:
+      matrix:
+        os: ["ubuntu-24.04", "ubuntu-24.04-arm"]
+    runs-on: ${{ matrix.os }}
+
     permissions:
       contents: read
       packages: write
@@ -25,43 +30,88 @@ jobs:
       - name: Checkout repository
         uses: actions/checkout@v4
 
-      - name: Set up QEMU
-        uses: docker/setup-qemu-action@v3
-
-      - name: Set up Docker Buildx
-        uses: docker/setup-buildx-action@v3
-
       - name: Log in to the Container registry
         uses: docker/login-action@v3
         with:
           registry: ${{ env.REGISTRY }}
           username: ${{ github.actor }}
           password: ${{ secrets.GITHUB_TOKEN }}
 
+      - name: Set up Docker Buildx
+        uses: docker/setup-buildx-action@v3
+
       - name: Extract metadata (tags, labels) for Docker
         id: meta
         uses: docker/metadata-action@v5
         with:
           images: ${{ env.REGISTRY }}/${{ env.IMAGE_NAME }}
 
-      - name: Build Docker image
+      - name: Set architecture tag (amd64)
+        if: ${{ matrix.os == 'ubuntu-24.04' }}
+        run: echo "ARCH_TAG=amd64" >> $GITHUB_ENV
+
+      - name: Set architecture tag (arm)
+        if: ${{ contains(matrix.os, 'arm') }}
+        run: echo "ARCH_TAG=arm64" >> $GITHUB_ENV
+
+      - name: Build and push by digest
+        id: build
         uses: docker/build-push-action@v6
         with:
-          context: .
-          load: true
-          push: false
           file: docker/Dockerfile
-          platforms: linux/amd64
-          tags: ${{ steps.meta.outputs.tags }}
+          platforms: ${{ env.ARCH_TAG }}
           labels: ${{ steps.meta.outputs.labels }}
+          outputs: type=image,"name=${{ env.REGISTRY }}/${{ env.IMAGE_NAME}}",push-by-digest=true,name-canonical=true,push=true
 
-      - name: Build (arm) and push (amd/arm) Docker image
-        uses: docker/build-push-action@v6
+      - name: Export digest
+        run: |
+          mkdir -p ${{ runner.temp }}/digests
+          digest="${{ steps.build.outputs.digest }}"
+          touch "${{ runner.temp }}/digests/${digest#sha256:}"          
+
+      - name: Upload digest
         if: github.event_name == 'push'
+        uses: actions/upload-artifact@v4
         with:
-          context: .
-          push: true
-          file: docker/Dockerfile
-          platforms: linux/amd64,linux/arm64
-          tags: ${{ steps.meta.outputs.tags }}
-          labels: ${{ steps.meta.outputs.labels }}
+          name: digests-${{ env.ARCH_TAG }}
+          path: ${{ runner.temp }}/digests/*
+          if-no-files-found: error
+          retention-days: 1
+  
+  merge-and-publish:
+    if: github.event_name == 'push'
+    runs-on: ubuntu-latest
+    needs:
+      - build
+    steps:
+      - name: Download digests
+        uses: actions/download-artifact@v4
+        with:
+          path: ${{ runner.temp }}/digests
+          pattern: digests-*
+          merge-multiple: true
+
+      - name: Log in to the Container registry
+        uses: docker/login-action@v3
+        with:
+          registry: ${{ env.REGISTRY }}
+          username: ${{ github.actor }}
+          password: ${{ secrets.GITHUB_TOKEN }}
+
+      - name: Extract metadata (tags, labels) for Docker
+        id: meta
+        uses: docker/metadata-action@v5
+        with:
+          images: ${{ env.REGISTRY }}/${{ env.IMAGE_NAME }}
+        
+
+      - name: Create manifest list and push
+        working-directory: ${{ runner.temp }}/digests
+        run: |
+          docker buildx imagetools create $(jq -cr '.tags | map("-t " + .) | join(" ")' <<< "$DOCKER_METADATA_OUTPUT_JSON") \
+            $(printf '${{ env.REGISTRY }}/${{ env.IMAGE_NAME}}@sha256:%s ' *)          
+
+      - name: Inspect image
+        run: |
+          docker buildx imagetools inspect ${{ env.REGISTRY }}/${{ env.IMAGE_NAME}}:${{ steps.meta.outputs.version }}  
+
@@ -35,6 +35,10 @@ parts:
           - file: chapter2/ns_code1
           - file: chapter2/ns_code2
       - file: chapter2/hyperelasticity
+      - file: chapter2/helmholtz
+        sections:
+          - file: chapter2/helmholtz_code
+
   - caption: Subdomains and boundary conditions
     chapters:
       - file: chapter3/neumann_dirichlet_code
@@ -48,4 +52,4 @@ parts:
       - file: chapter4/solvers
       - file: chapter4/compiler_parameters
       - file: chapter4/convergence
-      - file: chapter4/newton-solver
+      - file: chapter4/newton-solver
@@ -18,6 +18,6 @@ where $\beta = 1+\frac{\lambda}{\mu}$ is a dimensionless elasticity parameter an
 ```
 is a dimensionless variable reflecting the ratio of the load $\rho g$ and the shear stress term $\mu \nabla^2u \sim \mu \frac{U}{L^2}$ in the PDE.
 
-One option for the scaling is to chose $U$ such that $\gamma$ is of unit size ($U=\frac{\rho g L^2}{\mu}$). However, in elasticity, this leads to displacements of the size of the geometry. This can be achieved by choosing $U$ equal to the maximum deflection of a clamped beam, for which there actually exists a formula: $U=\frac{3}{2} \rho g L^2\frac{\delta^2}{E}$ where $\delta=\frac{L}{W}$ is a parameter reflecting how slender the beam is, and $E$ is the modulus of elasticity. Thus the dimensionless parameter $\delta$ is very important in the problem (as expected $\delta\gg 1$ is what gives beam theory!). Taking $E$ to be of the same order as $\mu$, which in this case and for many materials, we realize that $\gamma \sim \delta^{-2}$ is an appropriate choice. Experimenting with the code to find a displacement that "looks right" in the plots of the deformed geometry, points to $\gamma=0.4\delta^{-2}$ as our final choice of $\gamma$.
+One option for the scaling is to choose $U$ such that $\gamma$ is of unit size ($U=\frac{\rho g L^2}{\mu}$). However, in elasticity, this leads to displacements of the size of the geometry. This can be achieved by choosing $U$ equal to the maximum deflection of a clamped beam, for which there actually exists a formula: $U=\frac{3}{2} \rho g L^2\frac{\delta^2}{E}$ where $\delta=\frac{L}{W}$ is a parameter reflecting how slender the beam is, and $E$ is the modulus of elasticity. Thus the dimensionless parameter $\delta$ is very important in the problem (as expected $\delta\gg 1$ is what gives beam theory!). Taking $E$ to be of the same order as $\mu$, which in this case and for many materials, we realize that $\gamma \sim \delta^{-2}$ is an appropriate choice. Experimenting with the code to find a displacement that "looks right" in the plots of the deformed geometry, points to $\gamma=0.4\delta^{-2}$ as our final choice of $\gamma$.
 
-The simulation code implements the problem with dimensions and physical parameters $\lambda, \mu, \rho, g, L$ and $W$. However, we can easily reuse this code for a scaled problem: Just set $\mu=\rho=L=1$, $W$ as $W/L(\delta^{-1})$, $g=\gamma$ and $\lambda=\beta$.
+The simulation code implements the problem with dimensions and physical parameters $\lambda, \mu, \rho, g, L$ and $W$. However, we can easily reuse this code for a scaled problem: Just set $\mu=\rho=L=1$, $W$ as $W/L(\delta^{-1})$, $g=\gamma$ and $\lambda=\beta$.
@@ -0,0 +1,125 @@
+# The Helmholtz equation
+Author: Antonio Baiano Svizzero 
+  
+The study of computational acoustics is fundamental in fields such as noise, vibration, and harshness (NVH), noise control, and acoustic design. In this chapter, we focus on the theoretical foundations of the Helmholtz equation - valid for noise problems with harmonic time dependency - and its implementation in FEniCSx to compute the sound pressure for any acoustic system.
+
+## The PDE problem
+The acoustic Helmholtz equation in its general form reads
+
+$$
+\begin{align}
+\nabla^2 p + k^2 p = -j \omega \rho_0 q \qquad\text{in } \Omega,
+\end{align}
+$$
+
+where $k$ is the acoustic wavenumber, $\omega$ is the angular frequency, $j$ the imaginary unit and $q$ is the volume velocity ($m^3/s$) of a generic source field.
+In case of a monopole source, we can write  $q=Q \delta(x_s,y_s,z_s)$, where $\delta(x_s,y_s,z_s)$ is the 3D Dirac Delta centered at the monopole location. 
+
+This equation is coupled with the following boundary conditions: 
+
+- Dirichlet BC:  
+
+    $$
+    \begin{align}
+    p = \bar{p} \qquad \text{on  }  \partial\Omega_p,
+    \end{align}
+    $$
+
+- Neumann BC:  
+
+    $$
+    \begin{align}
+    \frac{\partial p}{\partial n} = - j \omega \rho_0 \bar{v}_n\qquad \text{on  }  \partial\Omega_v,
+    \end{align}
+    $$
+
+- Robin BC:  
+
+    $$
+    \begin{align}
+    \frac{\partial p}{\partial n} = - \frac{j \omega \rho_0 }{\bar{Z}} p \qquad \text{on  }  \partial\Omega_Z,
+    \end{align}
+    $$
+
+where we prescribe, respectively, an acoustic pressure $\bar{p}$ on the boundary $\partial\Omega_p$,
+a sound particle velocity $\bar{v}_n$ on the boundary $\partial\Omega_v$ and
+an acoustic impedance $\bar{Z}$ on the boundary $\partial\Omega_Z$ where $n$ is the outward normal.
+In general, any BC can also be frequency dependant, as it happens in real-world applications.
+
+## The variational formulation
+Now we have to turn the equation in its weak formulation.
+The first step is to multiplicate the equation by a *test function* $v\in \hat V$,
+where $\hat V$ is the *test function space*, after which we integrate over the whole domain, $\Omega$:
+
+$$
+\begin{align}
+\int_{\Omega}\left(\nabla^2 p + k^2 p \right) \bar v ~\mathrm{d}x = -\int_{\Omega} j \omega \rho_0 q \bar v ~\mathrm{d}x.
+\end{align}
+$$
+
+Here, the unknown function $p$ is referred to as *trial function* and the $\bar{\cdot}$ is the complex conjugate operator.
+
+In order to keep the order of derivatives as low as possible, we use integration by parts on the Laplacian term: 
+
+$$
+\begin{align}
+\int_{\Omega}(\nabla^2 p) \bar v ~\mathrm{d}x =
+-\int_{\Omega} \nabla p  \cdot \nabla \bar v ~\mathrm{d}x
++ \int_{\partial \Omega} \frac{\partial p}{\partial n} \bar v ~\mathrm{d}s.
+\end{align}
+$$
+
+Substituting in the original version and rearranging we get: 
+
+$$
+\begin{align}
+\int_{\Omega} \nabla p  \cdot \nabla \bar v ~\mathrm{d}x
+- k^2 \int_{\Omega} p \bar v ~\mathrm{d} x = \int_{\Omega} j \omega \rho_0 q \bar v ~\mathrm{d}x
++ \int_{\partial \Omega} \frac{\partial p}{\partial n} \bar v ~\mathrm{d}s.
+\end{align}
+$$
+
+Since we are dealing with complex values, the inner product in the first equation is *sesquilinear*,
+meaning it is linear in one argument and conjugate-linear in the other,
+as explained in [The Poisson problem with complex numbers](../chapter1/complex_mode).
+
+The last term can be written using the Neumann and Robin BCs, that is: 
+
+$$
+\begin{align}
+\int_{\partial \Omega} \frac{\partial p}{\partial n} \bar v ~\mathrm{d}s =
+-\int_{\partial \Omega_v}  j \omega \rho_0  \bar v ~\mathrm{d}s
+- \int_{\partial \Omega_Z}  \frac{j \omega \rho_0 \bar{v}_n}{\bar{Z}} p \bar v ~\mathrm{d}s.
+\end{align}
+$$
+
+Substituting, rearranging and taking out of integrals the terms with $j$ and $\omega$ we get the variational formulation of the Helmholtz.
+Find $u \in V$ such that: 
+
+$$
+\begin{align}
+\int_{\Omega} \nabla p  \cdot \nabla \bar v ~\mathrm{d}x
++ \frac{j \omega }{\bar{Z}} \int_{\partial \Omega_Z}   \rho_0 p \bar v ~\mathrm{d}s 
+- k^2 \int_{\Omega} p \bar v ~\mathrm{d}x
+= j \omega \int_{\Omega}  \rho_0 q \bar v ~\mathrm{d}x
+-j \omega\int_{\partial \Omega_v}   \rho_0 \bar{v}_n \bar v ~\mathrm{d}s \qquad  \forall v \in \hat{V}.
+\end{align}
+$$
+
+We define the sesquilinear form $a(p,v)$ is
+
+$$
+\begin{align}
+a(p,v) = \int_{\Omega} \nabla p  \cdot \nabla \bar v ~\mathrm{d}x
++ \frac{j \omega }{\bar{Z}} \int_{\partial \Omega_Z}  \rho_0  p \bar v ~\mathrm{d}s,
+- k^2 \int_{\Omega} p \bar v ~\mathrm{d}x 
+\end{align}
+$$
+
+and the linear form $L(v)$ reads
+
+$$
+\begin{align}
+L(v) =  j \omega \int_{\Omega}\rho_0 q \bar v ~\mathrm{d}x - j \omega \int_{\partial \Omega_v}  \rho_0 \bar{v}_n \bar v ~\mathrm{d}s.
+\end{align}
+$$