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<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no">
<title>Getting started with OpenFOAM-preCICE for fluid-structure interaction simulations</title>
<link rel="stylesheet" href="dist/reset.css">
<link rel="stylesheet" href="dist/reveal.css">
<link rel="stylesheet" href="dist/theme/precice.css" id="theme">
<!-- Theme used for syntax highlighted code -->
<link rel="stylesheet" href="plugin/highlight/atom-one-light.css" id="highlight-theme">
<style>
.container{
display: flex;
}
.col{
flex: 1;
}
</style>
<!-- FontAwesome -->
<link href="//netdna.bootstrapcdn.com/font-awesome/4.1.0/css/font-awesome.min.css" rel="stylesheet">
<style> .reveal i.fa { font-family:FontAwesome; font-style: normal; } </style>
</head>
<body>
<div class="reveal">
<div class="slides">
<!-- Title slide -->
<section>
<small>18th OpenFOAM Workshop - Genoa, July 11-14, 2023</small>
<h1>Getting started with<br/>OpenFOAM-preCICE<br/> for FSI simulations</h1>
<div style="margin-top:50pt">
<div>
<small>Gerasimos Chourdakis, Technical University of Munich<br/><a href="mailto:[email protected]">[email protected]</a></small><br/>
<img data-src="images/logos/tum.png" style="border:none; box-shadow:none; height:50px;">
</div>
</div>
</section>
<section data-markdown="sections/intro.md" data-separator-vertical="vvv"></section>
<!-- Level 1: General -->
<section data-markdown data-separator-vertical="vvv">
<textarea data-template>
# Using the preCICE library
Coupling two toy Python solvers
vvv
## Dependencies
- [preCICE](https://precice.org/installation-overview.html) v2 (e.g. [packages for Ubuntu](https://github.com/precice/precice/releases))
- Python 3
- Python packages `numpy`, `matplotlib`
- [preCICE Python bindings](https://precice.org/installation-bindings-python.html):
```bash
pip3 install --upgrade pip
pip3 install --user pyprecice
```
---
## "Generator" and "Propagator"
<div class="container" style="margin-top:50pt">
<div class="col">
- `generator.py`: generates random data in a 1D domain
- `propagator.py`: propagates boundary data over a 2D domain
</div>
<div class="col">
<img data-src="images/level1/propagator_description.png" style="border:none; box-shadow:none; max-width:100%;">
</div>
</div>
---
## Unidirectional coupling
<img data-src="images/level1/serial-explicit.svg" style="border:none; box-shadow:none; max-width:80%;">
---
## generator.py
```python
import numpy
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
t = t + dt
if(t > 0.1):
break
```
vvv
## propagator.py
```python
# generate mesh
n = 20
x = numpy.linspace(0, 1, n+1)
y = numpy.linspace(0, 1, n+1)
# initial data, associated to cell centers
u = numpy.zeros([n, n])
dt = 0.01
t = 0
# boundary condition for u (arbitrary)
u[:,-1] = y[:-1]
while True:
print("Propagating data")
# some (arbitrary) convection-diffusion rule
# top row
u[0, :-1] = 0.3 * u[0, :-1] + 0.6 * u[0, 1:] + 0.1 * u[1, :-1]
# inner domain
u[1:-1, :-1] = 0.2 * u[1:-1, :-1] + 0.6 * u[1:-1, 1:] + 0.1 * u[2:, :-1] + 0.1 * u[:-2, :-1]
# bottom row
u[-1, :-1] = 0.3 * u[-1, :-1] + 0.6 * u[-1, 1:] + 0.1 * u[-1, :-1]
t = t + dt
if(t > 0.2):
break
```
</textarea>
</section>
<section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/generator-propagator-uncoupled.mp4"><video data-autoplay src="videos/generator-propagator-uncoupled.mp4"></video></a></p>
</section>
<section>
<h2>Uncoupled simulation</h2>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/propagator_constant.webm"><video data-autoplay src="videos/propagator_constant.webm"></video></a></p>
</section>
</section>
<!--- Level 1: Code step-by-step -->
<!-- Base -->
<section>
<section data-auto-animate>
<h2>Import preCICE</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers>
import numpy
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
t = t + dt
if(t > 0.1):
break
</code></pre>
</section>
<!-- Import preCICE -->
<section data-auto-animate>
<h2>Import preCICE</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="2">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
t = t + dt
if(t > 0.1):
break
</code></pre>
</section>
<!-- Configure preCICE -->
<section data-auto-animate>
<h2>Configure preCICE</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="2|8-19">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
t = t + dt
if(t > 0.1):
break
</code></pre>
</section>
<!-- Define the coupling mesh -->
<section data-auto-animate>
<h2>Define the coupling mesh</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="8-19|21-27">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
t = t + dt
if(t > 0.1):
break
</code></pre>
</section>
<!-- Initialize and finalize preCICE -->
<section data-auto-animate>
<h2>Initialize and finalize preCICE</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="21-27|29,42">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
interface.initialize()
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
t = t + dt
if(t > 0.1):
break
interface.finalize()
</code></pre>
</section>
<!-- Advance the coupling: focus -->
<section data-auto-animate>
<h2>Advance the coupling</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="29,44|34-42">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
interface.initialize()
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
t = t + dt
if(t > 0.1):
break
interface.finalize()
</code></pre>
</section>
<!-- Advance the coupling: call -->
<section data-auto-animate>
<h2>Advance the coupling</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="34-42">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
interface.initialize()
dt = 0.01
t = 0
while True:
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
interface.advance(dt)
t = t + dt
if(t > 0.1):
break
interface.finalize()
</code></pre>
</section>
<!-- Advance the coupling: end -->
<section data-auto-animate>
<h2>Advance the coupling</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="34-42">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
interface.initialize()
dt = 0.01
t = 0
while interface.is_coupling_ongoing():
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
interface.advance(dt)
t = t + dt
interface.finalize()
</code></pre>
</section>
<!-- Advance the coupling: time -->
<section data-auto-animate>
<h2>Advance the coupling</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="34-42">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
interface.initialize()
dt = 0.01
t = 0
while interface.is_coupling_ongoing():
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
precice_dt = interface.advance(dt)
t = t + dt
interface.finalize()
</code></pre>
</section>
<!-- Advance the coupling: time 2-->
<section data-auto-animate>
<h2>Advance the coupling</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="34-42|29,34-42">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
precice_dt = interface.initialize()
dt = 0.01
t = 0
while interface.is_coupling_ongoing():
dt = np.minimum(dt, precice_dt)
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
precice_dt = interface.advance(dt)
t = t + dt
interface.finalize()
</code></pre>
</section>
</section>
<!-- Run and results -->
<section>
<section>Not there yet, but let's run it<br/><small>(similar changes in <tt>propagator.py</tt> - try it at home)</small></section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/generator-propagator-steering.mp4"><video data-autoplay
src="videos/generator-propagator-steering.mp4"></video></a></p>
</section>
<section>
<h2>Nothing happening?</h2>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/propagator_constant.webm"><video data-autoplay src="videos/propagator_constant.webm"></video></a></p>
<small>Did we forget something?</small>
</section>
</section>
<section>
<!-- Write & read data: focus -->
<section data-auto-animate>
<h2>Write & read data</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="25-31">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
precice_dt = interface.initialize()
dt = 0.01
t = 0
while interface.is_coupling_ongoing():
dt = np.minimum(dt, precice_dt)
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
precice_dt = interface.advance(dt)
t = t + dt
interface.finalize()
</code></pre>
</section>
<!-- Write & read data: implement -->
<section data-auto-animate>
<h2>Write & read data</h2>
<pre data-id="code-animation"><code class="language-python" data-trim data-line-numbers="25-31|44-46">
import numpy
import precice
# generate mesh
n = 20
y = numpy.linspace(0, 1, n + 1)
# preCICE setup
participant_name = "Generator"
config_file_name = "../precice-config.xml"
solver_process_index = 0
solver_process_size = 1
interface =
precice.Interface(
participant_name,
config_file_name,
solver_process_index,
solver_process_size
)
# Get the preCICE mesh id
mesh_name = "Generator-Mesh"
mesh_id = interface.get_mesh_id(mesh_name)
# Define the coupling mesh
vertices = [[1, y0] for y0 in y[:-1]]
vertex_ids = interface.set_mesh_vertices(mesh_id, vertices)
# Get the exchanged data id
data_name = "Data"
data_id = interface.get_data_id(data_name, mesh_id)
precice_dt = interface.initialize()
dt = 0.01
t = 0
while interface.is_coupling_ongoing():
dt = np.minimum(dt, precice_dt)
print("Generating data")
u = 1 - 2 * numpy.random.rand(n)
interface.write_block_scalar_data(data_id, vertex_ids, u)
precice_dt = interface.advance(dt)
# interface.read_block_scalar_data(data_id, vertex_ids, u)
t = t + dt
interface.finalize()
</code></pre>
</section>
</section>
<!-- Run and results, part 2 -->
<section>
<section>It should work now!</small></section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/generator-propagator-uncoupled.mp4"><video
data-autoplay src="videos/generator-propagator-coupled.mp4"></video></a></p>
</section>
<section>
<h2>Data is being transfered!</h2>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/propagator_coupled.webm"><video data-autoplay src="videos/propagator_coupled.webm"></video></a></p>
<small>Most basic case: uni-directional, serial-explicit, nearest-neighbor mapping, ...</small>
</section>
<!-- <section>
<h2>Side note: Subcycling</h2>
<img src="images/level1/subcycling.svg" />
</section> -->
</section>
<!-- preCICE config -->
<section>
<section>
<h2>preCICE configuration</h2>
<img src="images/level1/config-original.png" alt="Visual representation of the precice-config.xml"/>
<small>Visual representation of precice-config.xml using the <a href="https://precice.org/tooling-config-visualization.html" title="preCICE config visualizer">config visualizer</a>.</small>
</section>
</section>
<!-- OpenFOAM + CalculiX -->
<section data-markdown="sections/level3.md" data-separator-vertical="vvv"></section>
<section>
<section>
<h2>Configure OpenFOAM</h2>
<pre><code class="language-cpp" data-trim data-line-numbers="">
// fluid-openfoam/0/U
flap
{
type movingWallVelocity;
value uniform (0 0 0);
}
</code></pre>
<pre><code class="language-cpp" data-trim data-line-numbers="">
// fluid-openfoam/0/pointDisplacement
flap
{
type fixedValue;
value $internalField;
}
</code></pre>
<pre><code class="language-cpp" data-trim data-line-numbers="">
// fluid-openfoam/constant/dynamicMeshDict
solver displacementLaplacian;
</code></pre>
</section>
<section>
<h2>Load the OpenFOAM adapter</h2>
<pre><code class="language-cpp" data-trim data-line-numbers="">
// fluid-openfoam/system/controlDict
functions
{
preCICE_Adapter
{
type preciceAdapterFunctionObject;
libs ("libpreciceAdapterFunctionObject.so");
}
}
// Don't forget to build the adapter with Allwmake
</code></pre>
</section>
<section>
<h2>Configure the OpenFOAM adapter</h2>
<pre><code class="language-cpp" data-trim data-line-numbers="">
// fluid-openfoam/system/preciceDict
preciceConfig "../precice-config.xml";
participant Fluid;
modules (FSI);
interfaces {
Interface1 {
mesh Fluid-Mesh;
patches (flap);
locations faceCenters;
readData (Displacement);
writeData (Force);
};
};
FSI {
rho rho [1 -3 0 0 0 0 0] 1;
}
</code></pre>
</section>
<section>
<h2>Configure the CalculiX adapter</h2>
<div class="col">
<pre><code class="language-yaml" data-trim data-line-numbers="">
// solid-calculix/config.yml
participants:
Solid:
interfaces:
- nodes-mesh: Solid-Mesh
patch: surface
read-data: [Force]
write-data: [Displacement]
precice-config-file: ../precice-config.xml
</code></pre>
</div>
</section>
<section>
<h2>Configure preCICE</h2>
<img src="images/level3/precice-config.png"/>
</section>
<section>
<h2>Configure preCICE</h2>
<div class="col">
<pre><code class="language-xml" data-trim data-line-numbers="2-3|5-13|15-31|15,19-20,31|15,21-25,31|40|42-48">
<solver-interface dimensions="2">
<data:vector name="Force" />
<data:vector name="Displacement" />
<mesh name="Fluid-Mesh">
<use-data name="Force" />
<use-data name="Displacement" />
</mesh>
<mesh name="Solid-Mesh">
<use-data name="Displacement" />
<use-data name="Force" />
</mesh>
<participant name="Fluid">
<export:vtk directory="results" />
<use-mesh name="Fluid-Mesh" provide="yes" />
<use-mesh name="Solid-Mesh" from="Solid" />
<write-data name="Force" mesh="Fluid-Mesh" />
<read-data name="Displacement" mesh="Fluid-Mesh" />
<mapping:rbf-thin-plate-splines
direction="write"
from="Fluid-Mesh"
to="Solid-Mesh"
constraint="conservative" />
<mapping:rbf-thin-plate-splines
direction="read"
from="Solid-Mesh"
to="Fluid-Mesh"
constraint="consistent" />
</participant>
<participant name="Solid">
<use-mesh name="Solid-Mesh" provide="yes" />
<write-data name="Displacement" mesh="Solid-Mesh" />
<read-data name="Force" mesh="Solid-Mesh" />
<watch-point mesh="Solid-Mesh" name="Flap-Tip" coordinate="0.0;1" />
</participant>
<m2n:sockets from="Fluid" to="Solid" exchange-directory=".." />
<coupling-scheme:serial-explicit>
<participants first="Fluid" second="Solid" />
<max-time value="1.5" />
<time-window-size value="0.01" />
<exchange data="Force" mesh="Solid-Mesh" from="Fluid" to="Solid" />
<exchange data="Displacement" mesh="Solid-Mesh" from="Solid" to="Fluid" />
</coupling-scheme:serial-explicit>
</solver-interface>
</code></pre>
</div>
<small>Notice the serial-explicit coupling scheme</small>
</section>
</section>
<section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/execution-serial-explicit.mp4"><video
data-autoplay src="videos/execution-serial-explicit.mp4" style="max-height:800px;"></video></a></p>
</section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/video-explicit.ogv"><video
data-autoplay src="videos/video-explicit.ogv" style="max-height:800px;"></video></a></p>
</section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/video-explicit.ogv"><img
src="images/level3/screenshot-explicit.png" style="max-height:800px;"></img></a></p>
</section>
<section>
<h2>What we did so far: serial-explicit scheme</h2>
<img src="images/level3/scheme-serial-explicit.png" style="max-height:800px;"><img>
</section>
<section>
<h2>Let's try an implicit coupling scheme</h2>
<img src="images/level3/scheme-serial-implicit.png" style="max-height:800px;"><img>
</section>
<section>
<h2>Configure the coupling scheme</h2>
<div class="col">
<pre><code class="language-xml" data-trim data-trim data-line-numbers="1,13|2|3-4|5-6|7-9|10-12">
<coupling-scheme:serial-implicit>
<participants first="Fluid" second="Solid" />
<max-time value="1.5" />
<time-window-size value="0.01" />
<exchange data="Force" mesh="Solid-Mesh" from="Fluid" to="Solid" />
<exchange data="Displacement" mesh="Solid-Mesh" from="Solid" to="Fluid" />
<relative-convergence-measure limit="5e-3" data="Displacement" mesh="Solid-Mesh" />
<relative-convergence-measure limit="5e-3" data="Force" mesh="Solid-Mesh" />
<max-iterations value="50" />
<acceleration:constant>
<relaxation value="0.5" />
</acceleration:constant>
</coupling-scheme:serial-implicit>
</code></pre>
</div>
<small>Simplest case: constant under-relaxation</small>
</section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/video-implicit.ogv"><video
data-autoplay src="videos/video-implicit.ogv" style="max-height:800px;"></video></a></p>
</section>
<section>
<p><a href="https://github.com/MakisH/ofw18-training/blob/master/videos/video-implicit.ogv"><img
src="images/level3/screenshot-implicit.png" style="max-height:800px;"></img></a></p>
</section>
<section>
<h2>Too slow! Can we do better?</h2>
</section>
<section>
<h2>Improvement 1: Aitken under-relaxation</h2>
<div class="col">
<pre><code class="language-xml" data-trim>
<acceleration:aitken>
<data name="Displacement" mesh="Solid-Mesh" />
<initial-relaxation value="0.5" />
</acceleration:aitken>
</code></pre>
</div>
<small>Average iterations drop from 3.85 (constant) to 3.49 (Aitken)<br/>
Note: This simulation is reaching a steady-state, thus differences diffuse</small>
</section>
<section>
<h2>Improvement 2: Anderson acceleration</h2>
<div class="col">
<pre><code class="language-xml" data-trim data-line-numbers="1-3,8|4-7">
<acceleration:IQN-ILS>
<data name="Displacement" mesh="Solid-Mesh" />
<initial-relaxation value="0.5" />
<preconditioner type="residual-sum" />
<filter type="QR2" limit="1e-2" />
<max-used-iterations value="100" />
<time-windows-reused value="15" />
</acceleration:IQN-ILS>
</code></pre>
</div>
<small>Average iterations drop from 3.49 (Aitken) to 2.91 (IQN)</small>
</section>
<section>
<h2>There are also parallel schemes</h2>
<img src="images/level3/scheme-parallel-implicit.png" style="max-height:800px;"><img>
<small>Can use fields from both / all participants</small>
</section>
<section>
<h2>Improvement 3: Parallel coupling schemes</h2>
<div class="col">
<pre><code class="language-xml" data-trim data-line-numbers="1,19|10-12,18">