Merge branch 'master' of https://github.com/feelpp/book.feelpp.org

docs/dev/modules/reference/pages/Integrals/README.adoc

@@ -1,14 +1,12 @@
 // -*- mode: adoc -*-
-
-Integration
------------
+= Integration
 
 You should be able to create a mesh now. If it is not the case, get
 back to the section <<../Mesh/README.adoc,Mesh>>.
 
 Prerequisites::
-* <<../Mesh/iterators.adoc,Iterate over a region of a mesh elements, faces, edges and points>>
-* <<../Keywords/keywords.adoc,Define expressions>>
+* xref:Mesh/iterators.adoc[Iterate over a region of a mesh elements, faces, edges and points]
+* xref:Keywords/keywords.adoc[Define expressions]
 
 include::integrate.adoc[]
 

docs/math/modules/fem/pages/ch-problemes-mixtes.adoc

@@ -230,8 +230,8 @@ Thanks to the Lemma of Céa applied to Saddle-Point Problems, the unique solutio
 ++++
         \begin{array}[c]{rl}
           ||u-u_h||_X & \leq c_{1h} \inf_{v_h \in X_h}  ||u-v_h||_X + c_{2}
-          \inf_{q_h \in M_h}  ||q-q_h||_M\\
-          ||p-p_h||_X & \leq c_{3h} \inf_{v_h \in X_h}  ||u-v_h||_X + c_{4h} \inf_{q_h \in M_h}  ||q-q_h||_M
+          \inf_{q_h \in M_h}  ||p-q_h||_M\\
+          ||p-p_h||_M & \leq c_{3h} \inf_{v_h \in X_h}  ||u-v_h||_X + c_{4h} \inf_{q_h \in M_h}  ||p-q_h||_M
           \end{array}
 ++++
 where
@@ -396,7 +396,7 @@ Look for stem:[(u,p) \in [H^1_0(\Omega)\]^d \times L^2_0(\Omega)] such that
         \begin{array}[c]{rl}
           \int_\Omega \nabla u : \nabla v -\int_\Omega p \nabla \cdot v  & =
           \int_\Omega f \cdot v, \quad \forall v \in [H^1_0(\Omega)]^d\\
-          \int_\Omega q \nabla \cdot u & = - \int_\Omega g q, \quad \forall q \in L^2_0(\Omega)
+          - \int_\Omega q \nabla \cdot u & = - \int_\Omega g q, \quad \forall q \in L^2_0(\Omega)
         \end{array}
         \right.
 ++++
@@ -430,9 +430,9 @@ Look for stem:[(u_h,p_h) \in X_h \times M_h] such that
 ++++
       \left\{
         \begin{array}[c]{rl}
-          \int_\Omega \nabla u_h : \nabla v_h + \int_\Omega p_h \nabla \cdot v_h
+          \int_\Omega \nabla u_h : \nabla v_h - \int_\Omega p_h \nabla \cdot v_h
           & = \int_\Omega f \cdot v_h, \quad \forall v_h \in X_h\\
-          \int_\Omega q_h \nabla \cdot u_h & = -\int_\Omega g q_h, \quad \forall q_h \in M_h
+         - \int_\Omega q_h \nabla \cdot u_h & = -\int_\Omega g q_h, \quad \forall q_h \in M_h
         \end{array}
         \right.
 ++++

docs/math/modules/fem/pages/elasticity/numerical-experiments.adoc

@@ -82,8 +82,7 @@ feelpp_qs_elasticity_pure_traction_2d --config-file cantilever.cfg --functions.g
 <script type="text/javascript">
 feelppVtkJs.createSceneImporter( vtkVisuSection3, {
                                  fileURL: "https://girder.math.unistra.fr/api/v1/file/5affc155b0e9574027048059/download",
-                                 objects: { "reference":[ { scene:"cantilever-pure-traction", name:"Cantilever" } ],
-                                            "displacement":[ { scene:"displacement", name:"Rigid Rotation" } ],
+                                 objects: { "reference":[ { scene:"cantilever-pure-traction", name:"Cantilever" }, { scene:"displacement", name:"Displacement" } ],
                                             "deformed":[ { scene:"deformed", name:"Cantilever Rotated" } ] }
                                } );
 </script>

docs/math/modules/fem/pages/elasticity/pure-traction.adoc

@@ -58,7 +58,7 @@ a(\disp{d},\disp{d})=\int_\Omega \lambda(\nabla \cdot u)^2 + \int_\Omega 2 \mu \
 ce qui signifie que stem:[\deformt{\disp{d}} = 0], donc que les composantes de stem:[\disp{d}] sont des polynomes du premier degré et donc que
 [stem]
 ++++
-\disp(x) = \tau + R \times \disp{x}
+\disp{d}(x) = \tau + R \disp{x}
 ++++
 avec stem:[\tau \in \RR^3]  et stem:[R \in \RR^{3\times 3}].
 De plus remarquons que stem:[R] est anti-symmétrique, _i.e_ stem:[R+R^T=0], du fait que stem:[\deformt{\disp{d}} = 0].

docs/toolboxes/antora.yml

@@ -8,6 +8,7 @@ nav:
 - modules/csm/nav.adoc
 - modules/fsi/nav.adoc
 - modules/thermoelectric/nav.adoc
+- modules/heatfluid/nav.adoc
 - modules/multifluid/nav.adoc
 - modules/maxwell/nav.adoc
 - modules/hdg/nav.adoc

docs/toolboxes/modules/cfd/nav.adoc

@@ -8,6 +8,6 @@
 *** xref:toolbox.adoc#_materials[Materials]
 *** xref:toolbox.adoc#_boundary_conditions[Boundary Conditions]
 *** xref:toolbox.adoc#_body_forces[Body Forces]
-*** xref:toolbox.adoc#_post_pro[Post-Processing]
-*** xref:toolbox.adoc#_stab[Stabilisation methods]
-*** xref:toolbox.adoc#_run[Running a CFD simulation]
+*** xref:toolbox.adoc#_post_processing[Post-Processing]
+*** xref:toolbox.adoc#_stabilization_methods[Stabilisation methods]
+*** xref:toolbox.adoc#_run_simulation[Running a CFD simulation]

docs/toolboxes/modules/cfd/pages/toolbox.adoc

@@ -619,13 +619,15 @@ NOTE: Documentation pending
 
 == Run simulation
 
-programme avalaible :
+The computational fluid dynamics applications available are
 
-* `feelpp_toolbox_fluid_2d`
+* [x] 2D CFD toolbox : `feelpp_toolbox_fluid_2d`
 
-* `feelpp_toolbox_fluid_3d`
+* [x] 3D CFD toolbox : `feelpp_toolbox_fluid_3d`
 
+Here is an example of execution of the 2D CFD toolbox on 4 processors using the configuration script `<myfile.cfg>`
 ----
 mpirun -np 4 feelpp_toolbox_fluid_2d --config-file=<myfile.cfg>
 ----
 
+

docs/toolboxes/modules/fsi/pages/theory.adoc

@@ -219,34 +219,89 @@ Different coupling strategies (semi-implicit and semi-explicit) can be set betwe
 
 * using a Dirichlet-Neumann scheme
 * using a Robin-Robin coupling scheme between the fluid and the structure.
+* using a Generalized Robin-Neumann scheme
 
 NOTE: The list needs to be updated
 
 == Coupling strategies
+
+Discrete coupling conditions must mimic the continuous coupling conditions that impose velocity, stress and geometric continuity on the FSI interface.
+
+FSI coupling conditions can be practically enforced in different ways. A key element determining the numerical scheme to be used is the added-mass effect. This phenomenon depends on the ratio stem:[\frac{\rho_s}{\rho_f}] and refers to the different acceleration that a body immersed in a fluid experiences, as opposed to the acceleration it would experience in vacuum. This effect is explained by imagining that, as it moves, the solid carries with it a certain volume of fluid, and the higher the fluid density, the stronger the added mass. The added-mass effect is strong when fluid and solid densities are approximately equal.
+
+Initially, when FSI simulations were carried out for aerodynamic purposes, this effect was negligible and a Dirichlet-Neumann approach was employed; lately, FSI modeling started employing denser fluids, and numerical schemes that are independent from added mass effects became necessary: two examples are Robin-Robin and Generalized Robin-Neumann schemes. They are all explicit schemes.
+
+=== Dirichlet-Neumann
+
+Dirichlet-Neumann coupling strategy is the easiest and directest one. It discretizes the FSI coupling conditions as follows: at the stem:[n]-th time iteration, on the fluid sub-problem the following conditions are imposed
+
+[stem]
+++++
+
+\mathbf{u}_f^n = \frac{\boldsymbol{\eta}_s^{n-1} - \boldsymbol{\eta}_s^{n-2}}{\Delta t} \quad \textrm{in stem:[\Omega_f^t]}
+
+++++
+
+They are the discretized Dirichlet prescription on the velocity suggested by the coupling conditions.
+
+On the solid sub-problem, instead, the Neumann condition on the stresses' continuity is imposed in a discrete form
+[stem]
+++++
+\boldsymbol{\sigma}_s^n \mathbf{n}_s = -\boldsymbol{\sigma}_f^n \mathbf{n}_f \quad \textrm{in stem:[\Omega_s^*]}
+++++
+
 === Robin-Robin
 
-The conditions read
+Robin-Robin (RR) coupling strategy imposes coupling conditions in Robin form, in two steps: at the stem:[n]-th time iteration, on the solid sub-problem the following conditions are imposed
 [stem]
 ++++
-\begin{aligned}
-      \boldsymbol{\sigma}_{s}^{n} \boldsymbol{n}_s^n + \frac{\gamma
-      \mu_f }{h} \boldsymbol{u}_{s}^n  &=&  \frac{\gamma \mu_f }{h}
-                                           \boldsymbol{u}_{f}^{n-1} -
-                                           \boldsymbol{\sigma}_{f}^{n-1}
-                                           \boldsymbol{n}_f  \quad
-                                           \quad \text{in}\ \Omega^*_s \\
-      \boldsymbol{\sigma}_{f}^{n} \boldsymbol{n}_f + \frac{\gamma
-      \mu_f }{h} \boldsymbol{u}_{f}  &=& \frac{\gamma \mu_f }{h}
-                                         \boldsymbol{u}_{s}^n +
-                                         \boldsymbol{\sigma}_{f}^{n-1}
-                                         \boldsymbol{n}_f \quad \quad
-                                         \text{in}\ \Omega^t_f
-    \end{aligned}
+\boldsymbol{\sigma}_s^n\mathbf{n}_s + \frac{\gamma \mu_f}{h} \mathbf{u}_s^n = \frac{\gamma \mu_f}{h} \mathbf{u}_f^{n-1}- \boldsymbol{\sigma}_f^{n-1}\mathbf{n_f} \quad \textrm{in stem:[\Omega_s^*]}
 ++++
-with
+where
 [stem]
 ++++
-\boldsymbol{u}_{s}^n = \frac{\partial
-      \boldsymbol{\eta}_{s}^n }{\partial t}
+\mathbf{u}_s^n= \frac{\partial \boldsymbol{\eta}_s^n}{\partial t}
 ++++
-and latexmath:[\gamma] a parameter to choose.
+The right-hand side depends only on quantities coming from the stem:[(n-1)]-th time iteration.
+
+Once the solid sub-problem has been solved, one uses stem:[\mathbf{u}_s^n ] to impose on the fluid sub-problem:
+[stem]
+++++
+\boldsymbol{\sigma}_f^n\mathbf{n}_f + \frac{\gamma \mu_f}{h} \mathbf{u}_f^n = \frac{\gamma \mu_f}{h} \mathbf{u}_s^{n} +\boldsymbol{\sigma}_f^{n-1} \mathbf{n_f} \quad \textrm{in stem:[\Omega_f^t]}
+++++
+Here, stem:[\gamma] is a penalization parameter to be chosen.
+
+It should be noticed that in (RR) stability and accuracy are subject to a CFL-condition: hyperbolic-type CFL, for which stem:[\Delta t = \mathcal{O}(h)], is sufficient to have stability, but parabolic-type CFL, for which stem:[\Delta t = \mathcal{O}(h^2)], is necessary to have optimal accuracy.
+
+=== Generalized Robin-Neumann
+
+This coupling strategy, at the stem:[n]-th time iteration, imposes on the fluid sub-problem:
+[stem]
+++++
+\boldsymbol{\sigma}_f^n \mathbf{n} + \frac{\rho_s}{\Delta t} \mathbf{B}\mathbf{u}_f^n = \frac{\rho_s}{\Delta t} \mathbf{B}\mathbf{u}_s^{n-1} - \boldsymbol{\sigma}_s^{n-1}\mathbf{n}_s \quad \textrm{in stem:[\Omega_f^t]}
+++++
+where
+[stem]
+++++
+\mathbf{u}_s^{n-1}= \frac{\partial \boldsymbol{\eta}_s^{n-1}}{\partial t}
+++++
+and stem:[\mathbf{B}] is a self-adjoint operator coming from a mass lumping procedure. The fluid-sided condition is called generalized Robin because of stem:[\bf{B}] that multiplies fluid and solid velocities.
+
+On the solid sub-problem, the following discrete Neumann condition is imposed
+\[
+\boldsymbol{\sigma}_s^n \mathbf{n}_s = -\boldsymbol{\sigma}_f^n \mathbf{n}_f \quad \textrm{in stem:[\Omega_s^*]}
+\]
+Also in this case there are CFL-type constraints that link space and time discretization steps: under parabolic-type CFL condition, stability is guaranteed; however, if adequate extrapolation techniques are employed, stability can be obtained under weaker requirements. 
+
+These succint explanations were taken from the articles:
+
+* Erik Burman, Miguel Angel Fernández. Explicit strategies for incompressible fluid-structure inter-
+action problems: Nitsche type mortaring versus Robin-Robin coupling. International Journal for
+Numerical Methods in Engineering, Wiley, 2014, 97 (10), pp.739–758. <10.1002/nme.4607>. <hal-
+00819948v2> https://hal.inria.fr/hal-00819948v2
+
+* Miguel Angel Fernández, Jimmy Mullaert, Marina Vidrascu. Generalized Robin-Neumann explicit
+coupling schemes for incompressible fluid-structure interaction: stability analysis and numerics.
+International Journal for Numerical Methods in Engineering, Wiley, 2015, 101 (3), pp.199-229.
+<10.1002/nme.4785>. <hal-00875819> https://hal.inria.fr/hal-00875819
+

docs/toolboxes/modules/hdg/pages/mixedpoisson.adoc

@@ -34,7 +34,7 @@ The models are not considered for now.
 [source,json]
 .Model section
 ----
-"Model": "HDG"
+"Models": { "equations":"hdg" }
 ----
 
 ==== Materials
@@ -153,7 +153,10 @@ Two fields can be exported, the potential stem:[p] and the flux stem:[\boldsymbo
 ----
 "PostProcess":
 {
-    "Fields":["potential","flux"]
+    "Exports":
+    {
+        "fields":["potential","flux"]
+    }
 }
 ----