Modeling and Analysis using the JSON files
The Model JSON (.json
) files allow to configure a set of partial differential equations, more precisely they define:
1. Names
A model JSON file starts by giving names (long and short).
"Name": "TurekHron cfd2", (1)
"ShortName":"cfd2", (2)
1  Name of the example, usually printed onscreen 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. 
2. Models
A section Models
is present if the toolbox enables multiple physics, i.e. it allows to select the type of PDE/Model.
In most toolboxes, this section is optional and a default model is enabled.
Here is an example of this section for the fluid
toolbox.
"Models": (1)
{
"equations":"NavierStokes" (2)
}
1  Section Models defined by the toolbox to define the main configuration 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. 
3. Parameters
This section of the Model JSON file defines the parameters that may enter expressions used in the subsequent sections.
Parameters
section"Parameters": (1)
{
"ubar":"1.0", (2)
"chi":"t<2:t", (3)
"pIn": (4)
{
"type":"fit", (5)
"filename":"$cfgdir/pin.csv", (6)
"abscissa":"time", (7)
"ordinate":"pressure", (8)
"interpolation":"P1", (9)
"expr":"10*t+3:t" (10)
}
}
1  name of the section 
2  defines a new parameter ubar and its associated value 
3  defines a new parameter chi and its associated expression (currently symbols supported are x , y , z , nx , ny , nz , t 
4  defines a new parameter pIn and its definition is given in the subsection below 
5  the type of parameter is fit 
6  the filename of a csv file used for the fitting 
7  column name of csv file used in abscissa 
8  column name of csv file used in ordinate 
9  interpolation type of the fit. Possible values are : P0 , P1 , Spline , Akima 
10  expression used in order to read the fitted value 
4. Materials
This section of the Model JSON file defines material properties linking the Physical Entities in the mesh data structures to these properties.
"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 
5. InitialConditions
This section of the Model JSON file defines initial conditions. Depending on the type of model : * if we use a transient model, it corresponds to the initial conditions of the time scheme applied * if we use a steady model, it corresponds to the initial guess given to the solver
As presented below, there are two ways to define initial conditions either use a mathematical expressions or a file.
InitialConditions
defined from mathematical expressions"InitialConditions":
{
"temperature": (1)
{
"Expression": (2)
{
"myic1": (3)
{
"markers":"Omega1", (4)
"expr":"293" (5)
},
"myic2": (6)
{
"markers":["Omega2,Omega3]", (7)
"expr":"305*x*y:x:y" (8)
}
}
}
}
1  the field name of the toolbox to which the initial condition is associated 
2  the type of boundary condition to apply, here Expression 
3  a name that identifies an initial condition imposed on a field 
4  the name of marker (or a list of markers) where an expression is imposed as initial condition. The markers can represent any kind of entity (Elements/Faces/Edges/Points). If this entry is not given, the expression is applied on the mesh support of the field. 
5  an expression which is applied to the field 
6  another name that identifies an initial condition 
7  idem as <4> 
8  idem as <5> 
InitialConditions
section defined from a file"InitialConditions":
{
"temperature": (1)
{
"File": (2)
{
"myic": (3)
{
"filename":"$home/feel/toolboxes/heat/temperature.h5", (4)
"format":"hdf5" (5)
}
}
}
}
1  the field name of the toolbox to which the initail condition is associated 
2  the type of boundary condition to apply, here File 
3  a name that identifies an initial condition imposed on a field 
4  a file that represents a field saved (WARNING : must be compatible with the current mesh and partitioning) 
5  the format of the file read (possible values are "default","hdf5","binary","text"). It’s an optional entry, the default value is choosen by Feel++ (it’s "hdf5" if Feel++ was compiled with a hdf5 library). 
6. BoundaryConditions
This section of the Model JSON file defines the boundary conditions.
BoundaryConditions
section"BoundaryConditions":
{
"velocity": (1)
{
"Dirichlet": (2)
{
"inlet": (3)
{
"expr":"{ 1.5*ubar*(4./0.1681)*y*(0.41y),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 \(\sigma(\mathbf{u},p) \normal = 0 \normal\) 
7. PostProcessing
This section allows to define the output fields and quantities to be computed and saved for e.g. visualization.
PostProcess
section"PostProcess":
{
"Exports":
{
"fields":["field1","field2",...]
},
"Save":
{
"Fields":
{
"names":["field1","field2",...]
"format":"hdf5" }
},
"Measures":
{
"<measure type>":
{
....
}
}
}
7.1. 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.
7.2. Save
The Save
section is implemented when you want to store data using the Feel++ format.
For example, It can be useful to have access to these data and use them in another application.
Currently, there is only the possibility to save the fields (finite element approximation).
Save
section"Save":
{
"Fields":
{
"names": (1)
"format": (2)
}
}
1  the names of fields that we want to save (can be a name or a vector of name) 
2  the format used (possible values are "default","hdf5","binary","text"). It’s an optional entry, the default value is choosen by Feel++ (it’s "hdf5" if Feel++ was compiled with a hdf5 library). 
7.3. 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 :
7.3.2. 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
.
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 nonscalar 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.
7.3.3. 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 
\( \ u \_{H^1} = \left ( \int_{\Omega} \ u \^2 + \int_{\Omega} \ \nabla u \^2 \right)^{\frac{1}{2}} \) 
L2error 
\( \ uv \_{L^2} = \left ( \int_{\Omega} \ uv \^2 \right)^{\frac{1}{2}}\) 
SemiH1error 
\(  uv _{H^1} = \left ( \int_{\Omega} \ \nabla u\nabla v \^2 \right)^{\frac{1}{2}} \) 
H1error 
\( \ uv \_{H^1} = \left ( \int_{\Omega} \ uv \^2 + \int_{\Omega} \ \nabla u\nabla v \^2 \right)^{\frac{1}{2}} \) 
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_L2error
.
Norm
section"Norm":
{
"mynorm": (1)
{
"type":"L2", (2)
"field":"velocity" (3)
},
"myerror": (4)
{
"type":"L2error", (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 H1error or SemiH1error 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":["L2error","H1error","SemiH1error"],
"solution":"{2*x,cos(y)}:x:y",
"grad_solution":"{2,0,0,sin(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
8. 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.
9. The generator of cases by using the index definitions
Sometimes, it appears that a large part of a JSON section is duplicated many times and just a few words/letters of the syntax have changed.
In order to avoid this repetition, a generic block can be created and the expansion is controlled by entries called index(i)
(where (i)
is an integer > 0).
it’s currently available in PostProcess or in markers subtree.

9.1. A first example
We want to apply several postprocessings of type Statistics Measures
from an expression (always identical) on several mesh markers called top
, left
, bottom
and right
.
The classic way is to write theses measures for each marker. This implies a lot of duplication as illustrated in the next snippet JSON :
"Statistics":
{
"my_top_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"top"
},
"my_left_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"left"
},
"my_bottom_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"bottom"
},
"my_right_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"right"
}
}
The generic section that will generate exactly the same measures is :
"Statistics":
{
"my_%1%_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"%1%",
"index1":["top","left","bottom","right"]
}
}
The keyword %1%
can be placed in any location of the properties of Statistics Measures
and it will be replaced by the values given by index1
.
For this example of measures, an important thing is to be sure that the name of the measure is unique, else it will be overridden. 
9.2. A second example
The previous case is a little bit restrictive because only one value can be associated for each case generated. However, we can put several values by cases by using an array of array.
As an illustration, we have this JSON snippet that we want to factorize :
"Statistics":
{
"Check_HeatFlux_top":
{
"type":"integrate",
"expr":"heat_Concrete_k*heat_dnT  h_top*(heat_TT0_top):heat_Concrete_k:heat_dnT:heat_T:h_top:T0_top",
"markers":"top"
},
"Check_HeatFlux_bottom":
{
"type":"integrate",
"expr":"heat_Aluminium_k*heat_dnT  h_bottom*(heat_TT0_bottom):heat_Aluminium_k:heat_dnT:heat_T:h_bottom:T0_bottom",
"markers":"bottom"
},
"Check_HeatFlux_left":
{
"type":"integrate",
"expr":"heat_Wood_k*heat_dnT  h_left*(heat_TT0_left):heat_Wood_k:heat_dnT:heat_T:h_left:T0_left",
"markers":"left"
},
"Check_HeatFlux_right":
{
"type":"integrate",
"expr":"heat_Insulation_k*heat_dnT  h_right*(heat_TT0_right):heat_Insulation_k:heat_dnT:heat_T:h_right:T0_right",
"markers":"right"
}
}
The generic JSON section will be the following :
"Statistics":
{
"Check_HeatFlux_%1_1%":
{
"type":"integrate",
"expr":"heat_%1_2%_k*heat_dnT  h_%1_1%*(heat_TT0_%1_1%):heat_%1_2%_k:heat_dnT:heat_T:h_%1_1%:T0_%1_1%",
"markers":"%1_1%",
"index1":[ ["top", "Concrete"],["bottom", "Aluminium"], ["left","Wood"], ["right","Insulation"] ]
}
}
Compared to the previous case, the keywords used here are %1_1%
and %1_2%
. The number 1
placed in front corresponds to the fact that we use the index1
.
The second number (after the underscore) corresponds to the id in the subarray. Each subarray in the index1
array must have the same size.
In this example, the size of a subarray is 2. Consequently, we can only have here the value 1
or 2
for the id in the subarray.
In summary, this example generates 4 cases :
Case  %1_1% 
%1_2% 













9.3. Cases generated by cartesian product
We can also generate a set of case by a cartesian product of an arbitrary number of indexes.
For example, to generate several measures associated onebyone with the following markers :
matA3
, matA5
, matA7
, matB3
, matB5
, matB7
. As show just after in the snippet JSON,
the cartesian product is automaticallly apply when more than one index is given :
"Statistics":
{
"my_%1%_%2%_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"mat%1%%2%",
"index1":["A","B"],
"index2":["3","5","7"]
}
}
The keyword %1%
(resp %2%
) is replaced by the values given by index1
(resp index2
).
An arbitrary number of index can be put, but the ids should be contiguous and always start to 1 (index1
,index2
,index3
,…).
We can also use the array of array format for giving several values in a index :
"Statistics":
{
"my_%1%_%2_2%_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"mat%1%%2_1%",
"index1":["A","B"],
"index2":[ ["3","trois"],["5","cinq"],["7","sept"] ]
}
}
We retrieve here the symbol %2_1% and %2_2% because the index2 is build as an array of array.
Case  %1% 
%2_1% 
%2_2% 

























Therefore, this example generates the following 6 measures :

my_A_trois_eval
with markers assigned tomatA3

my_A_cinq_eval
with markers assigned tomatA5

my_A_sept_eval
with markers assigned tomatA7

my_B_trois_eval
with markers assigned tomatB3

my_B_cinq_eval
with markers assigned tomatB5

my_B_sept_eval
with markers assigned tomatB7
9.4. Range of integers
A special syntax is designed to generate an index representing a range of integers.
This sequence is defined by a start number, stop number (not include) and a progression step.
These parameters are separated by the symbol :
, as we can see here :

1:10
→ 1,2,3,4,5,6,7,8,9 
1:10:2
→ 1,3,5,7,9
This notation can be used in all index(i)
entries (and also in an array of array).
Therefore, we can rewrite the previous example with this syntax :
"Statistics":
{
"my_%1%_%2%_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"mat%1%%2%",
"index1":["A","B"],
"index2":["3:9:2"]
}
}
9.5. The markers
entry
In many contexts (Materials
, BoundaryConditions
, PostProcess
, …), it’s necessary to give the names of mesh markers.
Generally, an entry called markers
should be filled.
There are 3 ways to use it :
1  Only one string 
"markers":"matA3"
1  An array of string 
"markers":["matA3","matA5","matA7","matB3","matB5","matB7"]
1  A subtree with an entry called name that can be filled by one string or an array of string 
"markers": { "name":["matA3","matA5","matA7","matB3","matB5","matB7"] }
The subtree case has been introduced in fact in order to use a generator of names of mesh markers based on the index methodology explain previously. If we want to generate the previous example, we can also write this JSON snippet :
"markers":
{
"name":"mat%1%%2%",
"index1":["A","B"],
"index2":["3","5","7"]
}
9.6. Several levels of indexes
It’s also possible to combine the index at several levels of properties.
The important thing is to keep a contiguous progression of the indexes ids.
The following code JSON snippet generates some Statistics Measures
by using several indexes. And for each measure,
it uses also the generator of markers with other indexes.
"Statistics":
{
"my_%1%_%2%_eval":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":
{
"name":"mat%1%%2%_%3%",
"index3":["x","y","z"]
},
"index1":["A","B"],
"index2":["3:9:2"]
}
}
This example generates the following 6 measures :

my_A_3_eval
with markers assigned tomatA3_x
,matA3_y
,matA3_z

my_A_5_eval
with markers assigned tomatA5_x
,matA5_y
,matA5_z

my_A_7_eval
with markers assigned tomatA7_x
,matA7_y
,matA7_z

my_B_3_eval
with markers assigned tomatB3_x
,matB3_y
,matB3_z

my_B_5_eval
with markers assigned tomatB5_x
,matB5_y
,matB5_z

my_B_7_eval
with markers assigned tomatB7_x
,matB7_y
,matB7_z
We need to use index3
in the markers
subtree because index1
and index2
are already used in a parent property.
If several generators are completely independents, each section should start with the index1
. It’s the case with the following example :
"Statistics":
{
"my_%1%_eval1":
{
"type":"integrate",
"expr":"3.12*heat_dnT:heat_dnT",
"markers":"%1%",
"index1":["top","left","bottom","right"]
},
"my_%1%_eval2":
{
"type":"integrate",
"expr":"x*y:x:y",
"markers":"%1%",
"index1":["top","left","bottom","right"]
}
}