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Partitioned heat conduction with Schwarz-type domain decomposition (#381)
* Add heat equation Schwarz domain decomposition. * Apply suggestions from code review --------- Co-authored-by: Benjamin Uekermann <[email protected]> Co-authored-by: Gerasimos Chourdakis <[email protected]>
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‎partitioned-heat-conduction-overlap/README.md

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---
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title: Partitioned heat conduction with overlapping Schwarz-type domain decomposition
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permalink: tutorials-partitioned-heat-conduction-overlap.html
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keywords: FEniCS, heat conduction, overlap
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summary: We solve a simple heat equation. The domain is partitioned and the coupling is established in an overlapping-Schwarz-type domain decomposition.
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---
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{% note %}
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Get the [case files of this tutorial](https://github.com/precice/tutorials/tree/master/partitioned-heat-conduction-overlap). Read how in the [tutorials introduction](https://www.precice.org/tutorials.html).
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{% endnote %}
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## Setup
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We solve a partitioned heat equation, but apply an overlapping Schwarz-type domain decomposition method in this tutorial.
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![Case setup of partitioned-heat-conduction case with overlapping Schwarz-type domain decomposition](images/tutorials-partitioned-heat-conduction-overlap-setup.png)
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Note that this case applies an overlapping Schwarz-type coupling method and not (like most other tutorials in this repository) a Dirichlet-Neumann coupling. This results in a symmetric setup of the solvers.
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## Running the simulation
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This tutorial is for FEniCS.
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For choosing whether you want to run the left or right participant, please provide the following commandline input:
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* `python3 heat.py left` flag will run the left participant.
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* `python3 heat.py right` flag will run the right participant.
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Like for the case `partitioned-heat-conduction` (using Dirichlet-Neumann coupling), we can also expect for the overlapping domain decomposition applied here to recover the analytical solution. `errorcomputation.py` checks this explicitly, by comparing the numerical to the analytical solution and raising an error, if the approximation error is not within a given tolerance.

‎partitioned-heat-conduction-overlap/clean-tutorial.sh

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../tools/clean-tutorial-base.sh

‎partitioned-heat-conduction-overlap/images/tutorials-partitioned-heat-conduction-overlap-setup.pdf

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‎partitioned-heat-conduction-overlap/images/tutorials-partitioned-heat-conduction-overlap-setup.png

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‎partitioned-heat-conduction-overlap/images/tutorials-partitioned-heat-conduction-overlap-setup.tex

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\documentclass{standalone}
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\usepackage{tikz}
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\usetikzlibrary{fit,positioning,calc}
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\usepackage{pgfplots}
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\pgfplotsset{
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compat=1.16,
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}
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\usepgfplotslibrary{external}
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\tikzexternalize[
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only named,
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]
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\definecolor{pblue}{RGB}{0,102,189} % precice blue
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\definecolor{porange}{RGB}{243,98,33} % precice orange
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% Make the 'export as png' a seperate style, with default density 200
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\tikzset{
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export as png/.style={
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external/system call/.add={}{
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&& convert -density #1 -transparent white "\image.pdf" "\image.png"
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},
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},
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export as png/.default={200},
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}
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\begin{document}
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\tikzset{export as png}
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\tikzsetnextfilename{tutorials-partitioned-heat-conduction-schwarz-setup}
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\begin{tikzpicture}
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\draw
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(0,0) coordinate (a)
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rectangle
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++(3,3) coordinate (b);
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\node[fill=pblue!50, draw=black, fit=(a) (b),inner sep=0pt] (participantL) {$\Omega_\mathcal{L}$};
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\draw
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($(participantL.south east)+(2,0)$) coordinate (a)
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rectangle
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++(3,3) coordinate (b);
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\node[fill=porange!50, draw=black, fit=(a) (b),inner sep=0pt] (participantR) {$\Omega_\mathcal{R}$};
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\draw
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($(participantL.south east)-(1,0)$) coordinate (a)
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rectangle
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($(participantL.north east)$) coordinate (b);
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\node[fill=porange!50,opacity=0.5, fit=(a) (b),inner sep=0pt] (overlapL){};
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\draw
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($(participantR.south west)+(1,0)$) coordinate (a)
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rectangle
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($(participantR.north west)$) coordinate (b);
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\node[fill=pblue!50,opacity=0.5, fit=(a) (b),inner sep=0pt] (overlapR){};
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\node[below = 0.5cm of participantR,xshift=-0.2cm](overlapLabel){Overlapping region: $\Omega_\mathcal{R}\cap\Omega_\mathcal{L}$};
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\draw[dashed] ([yshift=-.5cm]overlapL.center) -- (overlapLabel);
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\draw[dashed] ([yshift=-.5cm]overlapR.center) -- (overlapLabel);
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\draw[very thick, pblue]($(participantL.south east)-(1,0)$) -- ($(participantL.north east)-(1,0)$);
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\draw[very thick, porange]($(participantR.south west)+(1,0)$) -- ($(participantR.north west)+(1,0)$);
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\draw[very thick, porange](participantL.south east) -- node[right]{$\Gamma_\mathcal{R}$}($(participantL.north east)$);
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\draw[very thick, pblue](participantR.south west) -- node[left]{$\Gamma_\mathcal{L}$} ($(participantR.north west)$);
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\node[left = 0cm of participantL,align=center] {Dirichlet solver $\mathcal{L}$\\ acts on left part $\Omega_\mathcal{L}$};
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\node[right = 0cm of participantR,align=center] {Dirichlet solver $\mathcal{R}$\\ acts on right part $\Omega_\mathcal{R}$};
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\draw[->]($(participantR.north west)+(1,0)$) to[out=150,in=30] node[above right,align=center]{Temperature $u$ on $\Gamma_\mathcal{R}$ for $\mathcal{L}$} (participantL.north east);
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\draw[->]($(participantL.north east)-(1,0)$) to[out=30,in=150] node[above left,align=center]{Temperature $u$ on $\Gamma_\mathcal{L}$ for $\mathcal{R}$} (participantR.north west);
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\draw[draw=none] (participantL.south east) -- node[below]{$\Gamma_D$} (participantL.south west);
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\draw[draw=none] (participantR.south east) -- node[below]{$\Gamma_D$} (participantR.south west);
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\end{tikzpicture}
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\end{document}

‎partitioned-heat-conduction-overlap/left-fenics/clean.sh

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#!/bin/sh
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set -e -u
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. ../../tools/cleaning-tools.sh
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clean_fenics .

‎partitioned-heat-conduction-overlap/left-fenics/precice-adapter-config.json

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{
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"participant_name": "Left",
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"config_file_name": "../precice-config.xml",
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"interface": {
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"coupling_mesh_name": "Left-Mesh",
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"write_data_name": "Temperature-Left",
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"read_data_name": "Temperature-Right"
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}
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}

‎partitioned-heat-conduction-overlap/left-fenics/run.sh

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#!/bin/sh
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set -e -u
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python3 ../solver-fenics/heat.py Left

‎partitioned-heat-conduction-overlap/precice-config.xml

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<?xml version="1.0" encoding="UTF-8" ?>
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<precice-configuration>
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<log>
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<sink
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filter="%Severity% > debug and %Rank% = 0"
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format="---[precice] %ColorizedSeverity% %Message%"
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enabled="true" />
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</log>
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<data:scalar name="Temperature-Left" />
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<data:scalar name="Temperature-Right" />
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<mesh name="Left-Mesh" dimensions="2">
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<use-data name="Temperature-Left" />
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<use-data name="Temperature-Right" />
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</mesh>
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<mesh name="Right-Mesh" dimensions="2">
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<use-data name="Temperature-Left" />
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<use-data name="Temperature-Right" />
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</mesh>
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<participant name="Left">
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<provide-mesh name="Left-Mesh" />
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<receive-mesh name="Right-Mesh" from="Right" />
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<write-data name="Temperature-Left" mesh="Left-Mesh" />
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<read-data name="Temperature-Right" mesh="Left-Mesh" />
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<mapping:nearest-neighbor
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direction="read"
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from="Right-Mesh"
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to="Left-Mesh"
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constraint="consistent" />
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</participant>
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<participant name="Right">
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<provide-mesh name="Right-Mesh" />
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<receive-mesh name="Left-Mesh" from="Left" />
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<write-data name="Temperature-Right" mesh="Right-Mesh" />
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<read-data name="Temperature-Left" mesh="Right-Mesh" />
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<mapping:nearest-neighbor
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direction="read"
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from="Left-Mesh"
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to="Right-Mesh"
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constraint="consistent" />
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</participant>
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<m2n:sockets acceptor="Left" connector="Right" exchange-directory=".." />
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<coupling-scheme:serial-implicit>
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<participants first="Left" second="Right" />
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<max-time value="1.0" />
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<time-window-size value="0.1" />
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<max-iterations value="100" />
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<exchange data="Temperature-Left" mesh="Left-Mesh" from="Left" to="Right" initialize="true" />
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<exchange data="Temperature-Right" mesh="Right-Mesh" from="Right" to="Left" initialize="true" />
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<relative-convergence-measure data="Temperature-Left" mesh="Left-Mesh" limit="1e-5" />
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<relative-convergence-measure data="Temperature-Right" mesh="Right-Mesh" limit="1e-5" />
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<acceleration:IQN-ILS>
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<data name="Temperature-Right" mesh="Right-Mesh" />
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<initial-relaxation value="0.1" />
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<max-used-iterations value="100" />
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<time-windows-reused value="5" />
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<filter type="QR2" limit="1e-3" />
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</acceleration:IQN-ILS>
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</coupling-scheme:serial-implicit>
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</precice-configuration>

‎partitioned-heat-conduction-overlap/right-fenics/clean.sh

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#!/bin/sh
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set -e -u
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. ../../tools/cleaning-tools.sh
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clean_fenics .

‎partitioned-heat-conduction-overlap/right-fenics/precice-adapter-config.json

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{
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"participant_name": "Right",
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"config_file_name": "../precice-config.xml",
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"interface": {
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"coupling_mesh_name": "Right-Mesh",
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"write_data_name": "Temperature-Right",
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"read_data_name": "Temperature-Left"
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}
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}

‎partitioned-heat-conduction-overlap/right-fenics/run.sh

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#!/bin/sh
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set -e -u
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python3 ../solver-fenics/heat.py Right

‎partitioned-heat-conduction-overlap/solver-fenics/errorcomputation.py

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from fenics import inner, assemble, dx, project, sqrt
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def compute_errors(u_approx, u_ref, v, total_error_tol=10 ** -4):
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# compute pointwise L2 error
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error_normalized = (u_ref - u_approx) / u_ref
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# project onto function space
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error_pointwise = project(abs(error_normalized), v)
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# determine L2 norm to estimate total error
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error_total = sqrt(assemble(inner(error_pointwise, error_pointwise) * dx))
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error_pointwise.rename("error", " ")
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assert (error_total < total_error_tol)
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return error_total, error_pointwise

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