Merge branch 'develop' into 'master'

Update to version 1.3

See merge request !7

.dockerignore

+**/.git

.gitlab-ci.yml

-image: docker:stable
 variables:
-  DOCKER_TLS_CERTDIR: ""
-  DOCKER_HOST: tcp://docker:2375
-  DOCKER_DRIVER: overlay2
   APP_DIRECTORY: .
   DOCS_DIRECTORY: ${APP_DIRECTORY}/docs
-  DOCS_LATEX_NAME: fenicsr13
-services:
-- docker:dind # docker in docker
 stages:
+- prepare
 - build
 - test
 - deploy
 
 # **************************************************************************** #
-# build
+# prepare
 # **************************************************************************** #
 
-build:environment:
-  stage: build
+prepare:docker:
+  stage: prepare
   before_script:
     - docker login -u $CI_REGISTRY_USER -p $CI_REGISTRY_PASSWORD $CI_REGISTRY
   script:
     - docker pull $CI_REGISTRY_IMAGE:latest || true
-    - docker build --cache-from $CI_REGISTRY_IMAGE:latest --tag $CI_REGISTRY_IMAGE:$CI_COMMIT_SHA --tag $CI_REGISTRY_IMAGE:latest .
-    # - docker push $CI_REGISTRY_IMAGE:$CI_COMMIT_SHA # no need
-    - docker push $CI_REGISTRY_IMAGE:latest
-  when: manual
+    - docker pull $CI_REGISTRY_IMAGE:$CI_COMMIT_SHA || true
+    - docker pull $CI_REGISTRY_IMAGE:$CI_COMMIT_REF_NAME || true
+    - docker build
+      --cache-from $CI_REGISTRY_IMAGE:$CI_COMMIT_REF_NAME
+      --tag $CI_REGISTRY_IMAGE:latest
+      --tag $CI_REGISTRY_IMAGE:$CI_COMMIT_SHA
+      --tag $CI_REGISTRY_IMAGE:$CI_COMMIT_REF_NAME .
+  tags:
+    - shell
+
+# **************************************************************************** #
+# build
+# **************************************************************************** #
 
-build:doc:
+build:docs:
   stage: build
+  dependencies:
+    - prepare:docker
   image:
-    name: $CI_REGISTRY_IMAGE:latest
+    name: $CI_REGISTRY_IMAGE:$CI_COMMIT_SHA
     entrypoint: [""]
   script:
     - cd ${DOCS_DIRECTORY}
@@ -47,6 +52,8 @@ build:doc:
       - ${DOCS_DIRECTORY}/_build/html/
       - ${DOCS_DIRECTORY}/_build/latex/
     expire_in: 6 month
+  tags:
+    - docker
 
 # **************************************************************************** #
 # test
@@ -57,8 +64,16 @@ build:doc:
   before_script:
     - pip install -e . # local install to have right coverage
   image:
-    name: $CI_REGISTRY_IMAGE:latest # use prebuilt image in Gitlab's registry
-    entrypoint: [""] # == Repo. Has to be done if Dockerfile has own entrypoint
+    name: $CI_REGISTRY_IMAGE:$CI_COMMIT_SHA
+    entrypoint: [""]
+  tags:
+    - docker
+
+test:flake8:
+  extends: .test
+  script:
+    - cd ${APP_DIRECTORY}
+    - flake8 fenicsR13 examples tests
 
 test:pytest:
   extends: .test
@@ -77,6 +92,26 @@ test:pytest:
 # deploy
 # **************************************************************************** #
 
+.deploy:docker:
+  stage: deploy
+  before_script:
+    - docker login -u $CI_REGISTRY_USER -p $CI_REGISTRY_PASSWORD $CI_REGISTRY
+  tags:
+    - shell
+
+deploy:docker:dev:
+  extends: .deploy:docker
+  script:
+    - docker push $CI_REGISTRY_IMAGE:$CI_COMMIT_REF_NAME
+
+deploy:docker:latest:
+  extends: .deploy:docker
+  script:
+    - docker push $CI_REGISTRY_IMAGE:latest
+  only:
+    - master
+    - tags
+
 deploy:latex:
   stage: deploy
   image:
@@ -84,7 +119,7 @@ deploy:latex:
     name: aergus/latex:latest
     entrypoint: [""]
   dependencies:
-    - build:doc
+    - build:docs
   script:
     - cd ${DOCS_DIRECTORY}/_build/latex/
     - make
@@ -95,11 +130,13 @@ deploy:latex:
   only:
     - master
     - tags
+  tags:
+    - shell
 
 pages:
   stage: deploy
   dependencies:
-    - build:doc
+    - build:docs
   script:
     - mv ${DOCS_DIRECTORY}/_build/html/ ${CI_PROJECT_DIR}/public/
   artifacts:
@@ -107,4 +144,6 @@ pages:
       - public
   only:
     - master
-    - tags
\ No newline at end of file
+    - tags
+  tags:
+    - shell

CHANGELOG.rst

 Change log
 ----------
 
+1.3 (2020-08-18)
+~~~~~~~~~~~~~~~~
+
+- Add option to use Garlerkign Least Squares (GLS) stabilization
+- Allow for different chi_tilde on boundaries
+- Add option for multiple mesh regions:
+    - Allows for different Knudsen numbers in regions
+    - Add new example: Lid-driven cavity with two mesh regions
+- Optimization of CI pipeline and scripts in Gitlab:
+    - Assert Flake8 compliance
+- Add option to output solution vectors (for model order reduction experiments)
+- Improve README with R13 equation set and list of features
+
 1.2 (2020-02-17)
 ~~~~~~~~~~~~~~~~
 

Dockerfile

@@ -34,13 +34,14 @@ RUN \
 
 # Install any needed packages specified in requirements.txt
 # RUN pip install --trusted-host pypi.python.org -r requirements.txt
-COPY requirements.txt /tmp/
-RUN pip install --requirement /tmp/requirements.txt
+WORKDIR /fenicsR13
+ADD ./requirements.txt /fenicsR13/requirements.txt
+RUN pip install -r /fenicsR13/requirements.txt
 
 # Install the fenicsR13 package (puts it into the PATH)
-COPY . /tmp/
-RUN pip install --editable /tmp/.
+ADD . /fenicsR13
+RUN pip install --editable /fenicsR13/.
 
 # Replace default FEniCS Docker WELCOME screen with custom WELCOME screen
-COPY WELCOME .
-RUN echo "Built: $(date)" >> WELCOME
\ No newline at end of file
+ADD WELCOME .
+RUN echo "Built: $(date)" >> WELCOME

README.rst

@@ -5,16 +5,35 @@
 .. inclusion-marker
 
 fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform
-================================================================================
+===================================================================================================================
 
 |pipeline| |coverage| |version| |zenodo| |website|
 
 ``#extendedGasDynamics`` ``#using`` ``#FEniCS``
 
-Installation
+.. math::
+    \scriptsize \nabla \cdot u \, & \scriptsize = \dot{m} \\ \scriptsize \nabla p + \nabla \cdot \sigma \, & \scriptsize = b \\ \scriptsize \nabla \cdot u + \nabla \cdot s \, & \scriptsize = r \\ \scriptsize \frac{4}{5} {(\nabla s)}_{\text{stf}} + 2 {(\nabla u)}_{\text{stf}} + \nabla \cdot m \, & \scriptsize = - \frac{1}{\mathbb{K}} \sigma \\ \scriptsize \frac{5}{2} \nabla \theta + \nabla \cdot \sigma + \frac{1}{2} \nabla \cdot R + \frac{1}{6} \nabla \Delta \, & \scriptsize = - \frac{1}{\mathbb{K}} \frac{2}{3} s \\ \scriptsize m \, & \scriptsize = - 2 \mathbb{K} {(\nabla\sigma)}_{\text{stf}} \\ \scriptsize R \, & \scriptsize = - \frac{24}{5} \mathbb{K} {(\nabla s)}_{\text{stf}} \\ \scriptsize \Delta \, & \scriptsize = - 12 \mathbb{K} \left( \nabla \cdot s \right)
+
+Main Features
 --------------------------------------------------------------------------------
 
-This repository contains all main and auxiliary scripts to solve the linear R13 equations for rarefied gas flows with the `FEniCS`_ software in 2D, including examples and some convergence tests.
+- Solving Steady 2D Linear R13 Equations in Normalized Form
+- Stabilization with: Continous Interior Penalty (CIP) or Galerkin Least Squares (GLS)
+- Option for convergence study with input for exact solution
+- Output in XDMF_/HDF5 and PDF
+- Arbitray finite element combinations (thanks to FEniCS_)
+- Easy setup of parameter studies
+- Structured/Documented YAML_ input file format
+- Rarefied gas flow effects predictable: Knudsen paradox, Knudsen pump, ...
+- Interface for Gmsh_ meshes
+
+.. _FEniCS: https://fenicsproject.org/
+.. _YAML: https://de.wikipedia.org/wiki/YAML
+.. _XDMF: http://www.xdmf.org/index.php/XDMF_Model_and_Format
+.. _Gmsh: http://gmsh.info/
+
+Installation
+--------------------------------------------------------------------------------
 
 Download the repository as a `zip-file`_ and un-zip, or use `git`_ with
 
@@ -49,7 +68,7 @@ Make sure you installed Docker and it is running on your system. You can start t
     # Run (and possibly pull) fenicsr13_release service
     docker-compose run --rm fenicsr13_release
     # Only for developers, install the lastest version using pip
-    # pip install -e .
+    # sudo pip install -e .
 
 When you run this for the first time, docker will pull (download and extract) the container image from our repository which is roughly 800MB and the download may require some patience. After the initial download the docker image will be stored (2-3 GB) on your system and any new run will start the container immediately.
 
@@ -376,7 +395,7 @@ Contact
     :target: https://git.rwth-aachen.de/lamboo/fenicsR13/pipelines
     :alt: Test coverage
 
-.. |version| image:: https://img.shields.io/badge/version-1.2-blue.svg
+.. |version| image:: https://img.shields.io/badge/version-1.3-blue.svg
     :target: https://git.rwth-aachen.de/lamBOO/fenicsR13/-/tags
     :alt: Documentation Website
 

docs/conf.py

@@ -25,7 +25,7 @@ copyright = '2020'
 author = 'Lambert Theisen'
 
 # The full version, including alpha/beta/rc tags
-release = '1.2'
+release = '1.3'
 
 # -- General configuration ---------------------------------------------------
 

examples/channel_flow_force/input.yml

@@ -15,12 +15,19 @@ meshes:
 #   - fields: List of FEM parameters (shape, degree)
 #     - shape: Element shape, e.g. Lagrange
 #     - degree: Element degree, e.g. 2
-# - stabilization: Must contain cip
+# - stabilization: Must contain cip and gls
 #   - cip: Collection of Continous Interior Penalty (CIP) parameters
 #     - enable: Enable CIP stabilization
 #     - delta_theta: Stabilization of grad(T)*grad(T_test) over edge
 #     - delta_u: Stabilization of grad(u)*grad(u_test) over edge
 #     - delta_p: Stabilization of grad(p)*grad(p_test) over edge
+#   - gls: Collection of Garlerkin Least Squares (GLS) parameters
+#     - enable: Enable GLS stabilization
+#     - tau_energy: Stabilization with energy eq. residual
+#     - tau_heatflux: Stabilization with heatflux eq. residual
+#     - tau_mass: Stabilization with mass eq. residual
+#     - tau_momentum: Stabilization with momentum eq. residual
+#     - tau_stress: Stabilization with stress eq. residual
 elements:
   theta:
     shape: Lagrange
@@ -43,28 +50,41 @@ stabilization:
     delta_theta: 1.0
     delta_u: 1.0
     delta_p: 0.1
+  gls:
+    enable: False
+    tau_energy: 0.001
+    tau_heatflux: 0.001
+    tau_mass: 0.01
+    tau_momentum: 0.01
+    tau_stress: 0.01
 
 # Formulation Parameters
 # ======================
 # - nsd: Number of spatial dimensions == 2
 # - mode: Formulation mode, one of heat, stress, r13
-# - kn: Knudsen number
-# - chi_tilde: Refaction coefficient in Maxwell accomodation model
 # - heat_source: Heat source function for mode==heat||r13
 # - mass_source: Mass source function for mode==stress||r13
 # - body_force: Body force for mode==stress||r13
 nsd: 2
 mode: r13
-kn: 1.0
-chi_tilde: 1.0
 heat_source: 0
 mass_source: 0
 body_force: [1,0]
 
+# Region Parameters
+# =================
+# - regs: Dictionary of all mesh regions
+#   - reg_id: Must contain the following parameters:
+#     - kn: Knudsen number
+regs:
+  4000:
+    kn: 0.1
+
 # Boundary Conditions
 # ===================
 # - bcs: Dictionary of all boundary IDs from mesh
-#   - bc_id: must contain theta_w, u_t_w, u_n_w, p_w, epsilon_w
+#   - bc_id: must contain the following parameters
+#     - chi_tilde: Refaction coefficient in Maxwell accomodation model
 #     - theta_w: Value for temperature at wall
 #     - u_t_w: Value for tangential velocity at wall
 #     - u_n_w: Value for normal velocity at wall
@@ -72,24 +92,28 @@ body_force: [1,0]
 #     - epsilon_w: Inflow-model parameter <=> Weight of pressure prescription
 bcs:
   3001: # bot
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0
     p_w: 0 # No pressure gradient, flow shall be bodyforce-induced!
     epsilon_w: pow(10,-3)
   3002: # right
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0 # No slip
     p_w: 0
     epsilon_w: pow(10,+3) # No slip
   3003: # top
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0
     p_w: 0 # No pressure gradient, flow shall be bodyforce-induced!
     epsilon_w: pow(10,-3)
   3004: # left
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0 # No slip
@@ -115,9 +139,11 @@ convergence_study:
 # Postprocessing
 # ==============
 # - write_pdfs: Write all solution fields as PDF plot
+# - write_vecs: Write all solution fields as vectors
 # - massflow: List of BC IDs to compute massflow J=int_bc dot(u,n) ds
 postprocessing:
   write_pdfs: True
+  write_vecs: False
   massflow: [3002]
 
 # Parameter Study
@@ -127,5 +153,5 @@ postprocessing:
 # - parameter_values: List of value for parameter, e.g. [0.01,0.1,1,10]
 parameter_study:
   enable: True
-  parameter_key: ["kn"]
-  parameter_values: [0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0]
\ No newline at end of file
+  parameter_key: ["regs", 4000, "kn"]
+  parameter_values: [0.03125, 0.0625, 0.125, 0.25, 0.5, 1.0, 2.0] # 2^-5 to 2^1

examples/channel_flow_force/plot_table.py

@@ -10,7 +10,7 @@ if not gui:
 
 fig = plt.figure()
 ax = fig.add_subplot(1, 1, 1)
-ax.plot(x,y, "-o", label="Channel Flow")
+ax.plot(x, y, "-o", label="Channel Flow")
 ax.set_xscale("log")
 
 plt.xlabel("Knudsen number")

examples/channel_flow_force/postprocessing.sh

@@ -2,10 +2,10 @@
 
 rm table.csv
 
-for KNUDSEN in 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
+for KNUDSEN in 0.03125 0.0625 0.125 0.25 0.5 1.0 2.0
 do
   MASSFLOW=$(cat results_channel_flow_force$KNUDSEN/massflow_3002)
   echo "$KNUDSEN, $MASSFLOW" >> table.csv
 done
 
-cat table.csv
\ No newline at end of file
+cat table.csv

examples/channel_flow_pressure/input.yml

@@ -15,12 +15,19 @@ meshes:
 #   - fields: List of FEM parameters (shape, degree)
 #     - shape: Element shape, e.g. Lagrange
 #     - degree: Element degree, e.g. 2
-# - stabilization: Must contain cip
+# - stabilization: Must contain cip and gls
 #   - cip: Collection of Continous Interior Penalty (CIP) parameters
 #     - enable: Enable CIP stabilization
 #     - delta_theta: Stabilization of grad(T)*grad(T_test) over edge
 #     - delta_u: Stabilization of grad(u)*grad(u_test) over edge
 #     - delta_p: Stabilization of grad(p)*grad(p_test) over edge
+#   - gls: Collection of Garlerkin Least Squares (GLS) parameters
+#     - enable: Enable GLS stabilization
+#     - tau_energy: Stabilization with energy eq. residual
+#     - tau_heatflux: Stabilization with heatflux eq. residual
+#     - tau_mass: Stabilization with mass eq. residual
+#     - tau_momentum: Stabilization with momentum eq. residual
+#     - tau_stress: Stabilization with stress eq. residual
 elements:
   theta:
     shape: Lagrange
@@ -43,28 +50,41 @@ stabilization:
     delta_theta: 1.0
     delta_u: 1.0
     delta_p: 0.1
+  gls:
+    enable: False
+    tau_energy: 0.001
+    tau_heatflux: 0.001
+    tau_mass: 0.01
+    tau_momentum: 0.01
+    tau_stress: 0.01
 
 # Formulation Parameters
 # ======================
 # - nsd: Number of spatial dimensions == 2
 # - mode: Formulation mode, one of heat, stress, r13
-# - kn: Knudsen number
-# - chi_tilde: Refaction coefficient in Maxwell accomodation model
 # - heat_source: Heat source function for mode==heat||r13
 # - mass_source: Mass source function for mode==stress||r13
 # - body_force: Body force for mode==stress||r13
 nsd: 2
 mode: r13
-kn: 1.0
-chi_tilde: 1.0
 heat_source: 0
 mass_source: 0
 body_force: [0,0]
 
+# Region Parameters
+# =================
+# - regs: Dictionary of all mesh regions
+#   - reg_id: Must contain the following parameters:
+#     - kn: Knudsen number
+regs:
+  4000:
+    kn: 0.1
+
 # Boundary Conditions
 # ===================
 # - bcs: Dictionary of all boundary IDs from mesh
-#   - bc_id: must contain theta_w, u_t_w, u_n_w, p_w, epsilon_w
+#   - bc_id: must contain the following parameters
+#     - chi_tilde: Refaction coefficient in Maxwell accomodation model
 #     - theta_w: Value for temperature at wall
 #     - u_t_w: Value for tangential velocity at wall
 #     - u_n_w: Value for normal velocity at wall
@@ -72,24 +92,28 @@ body_force: [0,0]
 #     - epsilon_w: Inflow-model parameter <=> Weight of pressure prescription
 bcs:
   3001: # bot
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: pow(10,-3)
   3002: # right
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: pow(10,+3)
   3003: # top
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: pow(10,-3)
   3004: # left
+    chi_tilde: 1.0
     theta_w: 1
     u_t_w: 0
     u_n_w: 0
@@ -115,9 +139,11 @@ convergence_study:
 # Postprocessing
 # ==============
 # - write_pdfs: Write all solution fields as PDF plot
+# - write_vecs: Write all solution fields as vectors
 # - massflow: List of BC IDs to compute massflow J=int_bc dot(u,n) ds
 postprocessing:
   write_pdfs: True
+  write_vecs: False
   massflow: [3002]
 
 # Parameter Study
@@ -127,5 +153,5 @@ postprocessing:
 # - parameter_values: List of value for parameter, e.g. [0.01,0.1,1,10]
 parameter_study:
   enable: True
-  parameter_key: ["kn"]
-  parameter_values: [0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0]
\ No newline at end of file
+  parameter_key: ["regs", 4000, "kn"]
+  parameter_values: [0.03125, 0.0625, 0.125, 0.25, 0.5, 1.0, 2.0] # 2^-5 to 2^1

examples/channel_flow_pressure/plot_table.py

@@ -10,7 +10,7 @@ if not gui:
 
 fig = plt.figure()
 ax = fig.add_subplot(1, 1, 1)
-ax.plot(x,y, "-o", label="Channel Flow")
+ax.plot(x, y, "-o", label="Channel Flow")
 ax.set_xscale("log")
 
 plt.xlabel("Knudsen number")

examples/channel_flow_pressure/postprocessing.sh

@@ -2,10 +2,10 @@
 
 rm table.csv
 
-for KNUDSEN in 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
+for KNUDSEN in 0.03125 0.0625 0.125 0.25 0.5 1.0 2.0
 do
   MASSFLOW=$(cat results_channel_flow_pressure$KNUDSEN/massflow_3002)
   echo "$KNUDSEN, $MASSFLOW" >> table.csv
 done
 
-cat table.csv
\ No newline at end of file
+cat table.csv

examples/knudsen_pump/input.yml

@@ -16,12 +16,19 @@ meshes:
 #   - fields: List of FEM parameters (shape, degree)
 #     - shape: Element shape, e.g. Lagrange
 #     - degree: Element degree, e.g. 2
-# - stabilization: Must contain cip
+# - stabilization: Must contain cip and gls
 #   - cip: Collection of Continous Interior Penalty (CIP) parameters
 #     - enable: Enable CIP stabilization
 #     - delta_theta: Stabilization of grad(T)*grad(T_test) over edge
 #     - delta_u: Stabilization of grad(u)*grad(u_test) over edge
 #     - delta_p: Stabilization of grad(p)*grad(p_test) over edge
+#   - gls: Collection of Garlerkin Least Squares (GLS) parameters
+#     - enable: Enable GLS stabilization
+#     - tau_energy: Stabilization with energy eq. residual
+#     - tau_heatflux: Stabilization with heatflux eq. residual
+#     - tau_mass: Stabilization with mass eq. residual
+#     - tau_momentum: Stabilization with momentum eq. residual
+#     - tau_stress: Stabilization with stress eq. residual
 elements:
   theta:
     shape: Lagrange
@@ -44,28 +51,41 @@ stabilization:
     delta_theta: 1.0
     delta_u: 1.0
     delta_p: 0.1
+  gls:
+    enable: False
+    tau_energy: 0.001
+    tau_heatflux: 0.001
+    tau_mass: 0.01
+    tau_momentum: 0.01
+    tau_stress: 0.01
 
 # Formulation Parameters
 # ======================
 # - nsd: Number of spatial dimensions == 2
 # - mode: Formulation mode, one of heat, stress, r13
-# - kn: Knudsen number
-# - chi_tilde: Refaction coefficient in Maxwell accomodation model
 # - heat_source: Heat source function for mode==heat||r13
 # - mass_source: Mass source function for mode==stress||r13
 # - body_force: Body force for mode==stress||r13
 nsd: 2
 mode: r13
-kn: 0.1
-chi_tilde: 1.0
 heat_source: 0
 mass_source: 0
 body_force: [0,0]
 
+# Region Parameters
+# =================
+# - regs: Dictionary of all mesh regions
+#   - reg_id: Must contain the following parameters:
+#     - kn: Knudsen number
+regs:
+  4000:
+    kn: 0.1
+
 # Boundary Conditions
 # ===================
 # - bcs: Dictionary of all boundary IDs from mesh
-#   - bc_id: must contain theta_w, u_t_w, u_n_w, p_w, epsilon_w
+#   - bc_id: must contain the following parameters
+#     - chi_tilde: Refaction coefficient in Maxwell accomodation model
 #     - theta_w: Value for temperature at wall
 #     - u_t_w: Value for tangential velocity at wall
 #     - u_n_w: Value for normal velocity at wall
@@ -73,48 +93,56 @@ body_force: [0,0]
 #     - epsilon_w: Inflow-model parameter <=> Weight of pressure prescription
 bcs:
   3010: # inner top
+    chi_tilde: 1.0
     theta_w: 0.5 * x[0] + 1.0
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: 0
   3011: # inner left
+    chi_tilde: 1.0
     theta_w: -0.5 * (atan2(x[1],-(x[0] + 1.0)))/(M_PI/2.0) + 1.0
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: 0
   3012: # inner bottom
+    chi_tilde: 1.0
     theta_w: -0.5 * x[0] + 1.0
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: 0
   3013: # inner right
+    chi_tilde: 1.0
     theta_w: 0.5 * (atan2(x[1],(x[0]-1.0)))/(M_PI/2.0) + 1.0
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: 0
   3020: # outer top
+    chi_tilde: 1.0
     theta_w: 0.5 * x[0] + 1.0
     u_t_w: 0
     u_n_w: 0
     p_w: 0
     epsilon_w: 0
   3021: # outer left