# What is a Toolbox

Feel++ comes with toolboxes. What are they ?

Definition: A Feel++ Toolbox

For a given set of partial differential equations for a given physics (e.g. computational fluid dynamics) or set of physics (e.g. fluid structure interaction), a Feel++ toolbox provides:

• the configuration the set of equations through one or several (`.json`) files and (`.cfg`) files

• one or more solution strategies to solve the set of equations

Using the JSON and CFG, it is possible to configure and run models by defining the relevant physical quantities—such as material properties, loads, constraints, sources, and fluxes.

## 1. JSON files

The Model JSON (`.json`) files allow to configure a set of partial differential equations, more precisely they define:

### 1.1. Names

A model JSON file starts by giving names (long and short).

``````"Name": "Turek-Hron cfd2", (1)
"ShortName":"cfd2", (2)``````
 1 Name of the example, usually printed on-screen and in log files during simulations 2 Short name of the example, it is used to create directories to store the results of the simulation of the model
 These names are not used currently, but it should be the case in the future.

### 1.2. Models

A section `Models` can appear if the toolbox has several kinds of model (typically for select the PDE used). In most toolboxes, this section is optional and a model by default is choosen. The latter allows to configure the set of equations associated with fluid toolbox.

``````"Models": (1)
{
"equations":"Navier-Stokes" (2)
}``````
 1 Section `Models` defined by the toolbox to define the main configuration of a toolbox and in particular the set of equations to be solved 2 toolbox specific option to define set of equations to be solved, read the toolbox manual to learn about the possible options.

### 1.3. Parameters

This section of the Model JSON file defines the parameters that may enter expressions used in the subsequent sections.

Example of a `Parameters` section
``````"Parameters": (1)
{
"ubar":"1.0" (2)
}``````
 1 name of the section 2 defines a new parameter `ubar` and its associated value

### 1.4. Materials

This section of the Model JSON file defines material properties linking the Physical Entities in the mesh data structures to these properties.

Example of Materials section
``````"Materials":
{
"Water": (1)
{
"physics":"fluid", (2)
"markers":"[marker1,marker2]", (3)
"rho":"1.0e3", (4)
"mu":"1.0" (5)
}
}``````
 1 gives the name of the physical entity (here `Physical Surface`) associated to the Material. 2 defined which kind of physics is applied in this material. This is an optional section, by default all physics are applied. The value can be also a vector of physic. 3 defined mesh marker(s) where the material properties are applied. This is an optional section, by default the marker is take as the name <1>. 4 density $\rho$ is called `rho` and is given in SI units 5 viscosity $\mu$ is called `mu` and is given in SI units

### 1.5. BoundaryConditions

This section of the Model JSON file defines the boundary conditions.

Example of a `BoundaryConditions` section
``````"BoundaryConditions":
{
"velocity":  (1)
{
"Dirichlet": (2)
{
"inlet": (3)
{
"expr":"{ 1.5*ubar*(4./0.1681)*y*(0.41-y),0}:ubar:y" (4)
},
"wall1": (5)
{
"expr":"{0,0}" (6)
},
"wall2": (7)
{
"expr":"{0,0}" (8)
}
}
},
"fluid": (9)
{
"outlet": (10)
{
"outlet": (11)
{
"expr":"0" (12)
}
}
}
}``````
 1 the field name of the toolbox to which the boundary condition is associated 2 the type of boundary condition to apply, here `Dirichlet` 3 the physical entity (associated to the mesh) to which the condition is applied 4 the mathematical expression associated to the condition, note that the parameter `ubar` is used 5 another physical entity to which `Dirichlet` conditions are applied 6 the associated expression to the entity 7 another physical entity to which `Dirichlet` conditions are applied 8 the associated expression to the entity 9 the variable toolbox to which the condition is applied, here `fluid` which corresponds to velocity and pressure $(\mathbf{u},p)$ 10 the type of boundary condition applied, here outlet or outflow boundary condition 11 the hysical entity to which outflow condition is applied 12 the expression associated to the outflow condition, note that it is scalar and corresponds in this case to the condition $\sigma(\mathbf{u},p) \normal = 0 \normal$

### 1.6. PostProcessing

This section allows to define the output fields and quantities to be computed and saved for e.g. visualization.

Template of a `PostProcess` section
``````"PostProcess":
{
"Exports":
{
"fields":["field1","field2",...]
},
"Measures":
{
"<measure type>":
{
....
}
}
}``````

### 1.7. Exports

The `Exports` section is implemented when you want to visualize some fields with ParaView software for example. The entry `fields` should be filled with names which are available in the toolbox used.

### 1.8. Measures

Several quantities can be computed after each time step for transient simulation or after the solve of a stationary simulation. The values computed are stored in a CSV file format and named <toolbox>.measures.csv. In the template of `PostProcess` section, `<measure type>` is the name given of a measure. In next subsection, we present some types of measure that are common for all toolbox. Other types of measure are available but depend on the toolbox used, and the description is given in the specific toolbox documentation.

The common measures are :

TODO

#### 1.8.2. Statistics

The next table presents the several statistics that you can evaluate :

Statistics Type Expression

min

$\underset{x\in\Omega}{\min} u(x)$

max

$\underset{x\in\Omega}{\max} u(x)$

mean

$\frac{1}{ | \Omega |} \int_{\Omega} u$

integrate

$\int_{\Omega} u$

with `u` a function and $\Omega$ the definition domain where the statistic is applied.

The next source code shows an example of `Statistics` section with several kinds of computation. The results are stored in a CSV file at columns named `Statistics_mystatA_mean`, `Statistics_mystatB_min`, `Statistics_mystatB_max`, `Statistics_mystatB_mean`, `Statistics_mystatB_integrate`.

Example of a `Statistics` section
``````"Statistics":
{
"mystatA": (1)
{
"type":"mean", (2)
"field":"temperature" (3)
},
"mystatB": (4)
{
"type":["min","max","mean","integrate"], (5)
"expr":"2*x+y:x:y", (6)
"markers":"omega" (7)
}
}``````
 1 the name associated with the first Statistics computation 2 the Statistics type 3 the field `u` evaluated in the Statistics (here the temperature field in the heat toolbox) 4 the name associated with the second Statistics computation 5 the Statistics type 6 the field `u` evaluated in the Statistics 7 the mesh marker where the Statistics is computed ($\Omega$ in the previous table). This entry can be a vector of marker

The function `u` can be a finite element field or a symbolic expression. We use the `field` entry for a finite element field and `expr` for symbolic expression. `field` and `expr` can not be used simultaneously.

All expressions can depend on specifics symbols related to the toolboxes used. For example, in the heat toolboxes :

``"expr":"2*heat_T+3*x:heat_T:x"``

where `heat_T` is the temperature solution computed at last solve. It can also depend on a parameter defined in the `Parameters` section of the JSON.

The quadrature order used in the statistical evaluation can be specified. By default, the quadrature order is 5. For example, use a quadrature order equal to 10 is done by adding :

``"quad":10``
 Quadrature order is also used with `min` and `max` statistics. We get the min/max values by evaluating the expression on each quadrature points.
 In the `mean` and `integrate` Statistics, the quadrature order is automatically chosen when `field` is used. In this case, the `quad` entry has no effect.

The expression can be a scalar, a vector or a matrix. However, there is a particularity in the case of `mean` or `integrate` statistics with non-scalar expression. The result is not a scalar value but a vector or matrix. We store in the CSV file each entry of this vector/matrix.

#### 1.8.3. Norm

The next table presents the several norms that you can evaluate :

Norm Type Expression

L2

$\| u \|_{L^2} = \left ( \int_{\Omega} \| u \|^2 \right)^{\frac{1}{2}}$

SemiH1

$| u |_{H^1} = \left ( \int_{\Omega} \| \nabla u \|^2 \right)^{\frac{1}{2}}$

H1

$\| u \|_{H^1} = \left ( \int_{\Omega} \| u \|^2 + \int_{\Omega} \| \nabla u \|^2 \right)^{\frac{1}{2}}$

L2-error

$\| u-v \|_{L^2} = \left ( \int_{\Omega} \| u-v \|^2 \right)^{\frac{1}{2}}$

SemiH1-error

$| u-v |_{H^1} = \left ( \int_{\Omega} \| \nabla u-\nabla v \|^2 \right)^{\frac{1}{2}}$

H1-error

$\| u-v \|_{H^1} = \left ( \int_{\Omega} \| u-v \|^2 + \int_{\Omega} \| \nabla u-\nabla v \|^2 \right)^{\frac{1}{2}}$

where $\| . \|$ represents the norm of the generalized inner product. The symbol `u` represents a field or an expression and `v` an expression.

The next source code shows an example of Norm section with two norm computations. The results are stored in a CSV file at columns named `Norm_mynorm_L2` and `Norm_myerror_L2-error`.

Example of a `Norm` section
``````"Norm":
{
"mynorm": (1)
{
"type":"L2", (2)
"field":"velocity" (3)
},
"myerror": (4)
{
"type":"L2-error", (5)
"field":"velocity", (6)
"solution":"{2*x,cos(y)}:x:y", (7)
"markers":"omega" (8)
}
}``````
 1 the name associated with the first norm computation 2 the norm type 3 the field `u` evaluated in the norm (here the velocity field in the fluid toolbox) 4 the name associated with the second norm computation 5 the norm type 6 the field `u` evaluated in the norm 7 the expression `v` with the error norm type 8 the mesh marker where the norm is computed ($\Omega$ in the previous table). This entry can be a vector of marker
 with the `H1-error` or `SemiH1-error` norm, the gradient of the solution must be given with `grad_solution` entry. Probably this input should be automatically deduced in the near future.

Several norms can be computed by listing it in the type section :

``````"type":["L2-error","H1-error","SemiH1-error"],
"solution":"{2*x,cos(y)}:x:y",

An expression (scalar/vector/matrix) can be also passed to evaluate the norm. But in this case, the `field` entry must be removed and this expression replaces the symbol `u`.

``"expr":"2*x*y:x:y"``
 As before, in the case of `H1` or `SemiH1` norm type, the `grad_expr` entry must be given.
``"grad_expr":"{2*y,2*x}:x:y"``

All expressions can depend on specifics symbols related to the toolboxes used. For example, in the heat toolboxes :

``"expr":"2*heat_T+3*x:heat_T:x"``

where `heat_T` is the temperature solution computed at last solve. It can also depend on a parameter defined in the `Parameters` section of the JSON.

The quadrature order used in the norm computed can be also given if an analytical expression is used. By default, the quadrature order is 5. For example, use a quadrature order equal to 10 is done by adding :

``"quad":10``

### 1.9. An example

``````"PostProcess": (1)
{
"Exports": (2)
{
"fields":["velocity","pressure","pid"] (3)
},
"Measures": (4)
{
"Forces":"wall2", (5)
"Points": (6)
{
"pointA": (7)
{
"coord":"{0.6,0.2,0}", (8)
"fields":"pressure" (9)
},
"pointB": (10)
{
"coord":"{0.15,0.2,0}", (11)
"fields":"pressure" (12)
}
}
}
}``````
 1 the name of the section 2 the `Exports` identifies the toolbox fields that have to be exported for visualisation 3 the list of fields to be exported 4 the `Measures` section identifies outputs of interest such as 5 `Forces` applied to a surface given by the physical entity `wall2` 6 `Points` values of fields 7 name of the point 8 coordinates of the point 9 fields to be computed at the point coordinate 10 name of the point 11 coordinates of the point 12 fields to be computed at the point coordinate

Here is a biele example from the Toolbox examples.

## 2. CFG files

The Model CFG (`.cfg`) files allow to pass command line options to Feel++ applications. In particular, it allows to

• setup the output directory

• setup the mesh

• setup the time stepping

• define the solution strategy and configure the linear/non-linear algebraic solvers.

• other options specific to the toolbox used

``````directory=toolboxes/fluid/TurekHron/cfd3/P2P1G1 (1)

case.dimension=2 (2)
case.discretization=P2P1G1 (3)

[fluid] (4)
filename=$cfgdir/cfd3.json (5) mesh.filename=$cfgdir/cfd.geo (6)
gmsh.hsize=0.03 (7)

solver=Newton (8)

pc-type=lu (9)

bdf.order=2 (10)

[ts]
time-step=0.01 (11)
time-final=10 (12)``````
 1 the directory where the results are stored 2 the dimension (2d/3d) of the application 3 the discretization choosen (specific to the toolbox) 4 the prefix of option 5 the path of the json file 6 the path of geo/mesh file 7 characterist size of the mesh 8 type of non linear solver (specific to fluid toolbox) 9 type of preconditioner 10 order of BDF tiem scheme (specific to fluid toolbox) 11 time step 12 time final

If the mesh file is stored on a remote storage as Girder, the `mesh.filename` option in the previous listing can be replaced by

``mesh.filename=girder:{file:5af862d6b0e9574027047fc8}``

where `5af862d6b0e9574027047fc8` is the id of the mesh file in the Girder platform. All options for Girder access are listed here :

Option Default value Description

`url`

girder.math.unistra.fr

url of a Girder platform

`file`

<no default value>

one or several file id(s)

`api_key`

<no default value>

an authentication key

## 3. Run an application

Each toolbox are compiled in an executable named `feelpp_toolbox_<tbname>` where `<tbname>` is name of a toolbox. A non-exhaustive list of toolbox executables is :

• feelpp_toolbox_fluid

• feelpp_toolbox_solid

• feelpp_toolbox_heat

• feelpp_toolbox_heatfluid

• feelpp_toolbox_thermoelectric

• feelpp_toolbox_fsi_2d

• feelpp_toolbox_fsi_3d

With this executable, a cfg file must be given in the command line thanks to the `config-file` option :

``feelpp_toolbox_fluid --config-file myfile.cfg``

Another way is to use the `case` option, where case represents a folder containing a cfg, json files and eventually a geometry or mesh file.

``feelpp_toolbox_fluid --case mydir``

If the folder contains only one cfg, the programme use this one. Else it’s possible to specify the cfg file to choose by adding `case.config-file` option

``feelpp_toolbox_fluid --case mydir --case.config-file myfile.cfg``

The `case` option can also define a folder which represents a remote data in a github repository.

``feelpp_toolbox_fluid --case "github:{path:toolboxes/fluid/TurekHron}" --case.config-file cfd2.cfg``

The remote data from github can be configured by several parameters :

Option Default value Description

`owner`

feelpp

the github organization

`repo`

feelpp

the github repository in organization

`branch`

<default in github>

the branch in the git repository

`path`

<root of repository>

the path in the git repository

`token`

<no default value>

an authentication token

``feelpp_toolbox_fluid --case "github:{owner:feelpp,repo:feelpp,branch:develop,path:toolboxes/fluid/TurekHron,token:xxxxx}" --case.config-file cfd2.cfg``