Field Definition: FEM

We support computations on fields defined by outside sources.

In Action

There are four steps to the implementation process:

• 1. Diderot code (observ.diderot)
• 2. C code that communicates to the generated Diderot code (observ_init.c)
• 3. Python code that initiates the C code and creates FEM data (observ.py)
• 4. Running the program (run.sh) For the most part steps 2-4 are the same for each example and code can be easily reused.

1.Diderot Code (observ.diderot)

Simple Definition

The user defines an input variable to represent a FEM field. The path included is a path to the relevant data file.

input fem#k(d)[α] F0;
string path = "fnspace_data/data.json";
ofield#4(2)[] F = convert(F0, path);

The final term is an ofield type that acts the same as the Diderot field type.

Include Function Space

The user can choose to define the field by describing the function space VF and by providing a path to a directory pathVF.

input fem#k(d)[α] F0;
fnspace VF = ....
string pathVF = ...
ofield#4(2)[] F = convert(F0,VF,pathVF);

The variable VF is a fnspace type. It is defined with a mesh, element, and int (to indicate order of coefficients). The current options for a mesh are UnitCubeMesh() and UnitSquareMesh(). An element type is an abstract representation of a reference element. It can be either Lagrange() or P(). A fnspace type represents a function space. It can be either FunctionSpace() for scalars or TensorFunctionSpace() for non-scalars. The following is an example of a 2-d scalar field:

input fem#k(2)[] F0;
mesh M = UnitSquareMesh(4,4);
element E = P();
int polyorder = 4;
fnspace VF = FunctionSpace(M,E,polyorder);
ofield#4(2)[] F = convert(F0,VF,pathVF);

To represent a 3-d scalar field the mesh and fem type need to be changed:

input fem#k(3)[] F0;
mesh M = UnitCubeMesh(4,4,4)

To define a vector field of length i use TensorFunctionSpace():

input fem#k(3)[i] F0;
fnspace VF = TensorFunctionSpace(M, E, polyorder,{i});

To accommodate a second-order (i,j) tensor field augment the shape parameter:

input fem#k(3)[i, j] F0;
fnspace VF = TensorFunctionSpace(M, E, polyorder,{i,j});

Sumary of syntax

• Define a Mesh, Element, and Function Space
  • Define a Unit Square Mesh- UnitSquareMesh(): int × int → mesh
  • Define a Unit Cube Mesh- UnitCubeMesh(): int × int × int → mesh
  • Define a Lagrange reference element- Lagrange(): →element
  • Define a P reference element- P(): → element
  • Define a function space for scalar fields- FunctionSpace(): mesh × element × int →fnspace
  • Define a function space for non-scalar fields- TensorFunctionSpace(): mesh × element × int × int sequence→fnspace
• Define an ofield with fem data
  • Define a fem field- convert(): fem#k(d)[α] × string →ofield#k(d)[α]
  • Define a fem field with the function space-convert(): fem#k(d)[α] × fnspace × string →ofield#k(d)[α]
• Other operations on ofield
  • Check if a position is inside a field-insideF(): tensor[d]×ofield#k(d)[α] →boolean
  • Probe the field at a position-inst(): tensor[d]×ofield#k(d)[α] → tensor[α]
  • Get the cell number the point is located in-GetCell(): tensor[d]×ofield#k(d)[α] → int

2. C code that communicates to the generated Diderot code (observ_init.c)

The C code is used to communicate with the generated Diderot code. The function callDiderot_observ() can be called by outside tools.

void callDiderot_observ(char *Outfile, void *valF)

The function takes the name of the output nrrd file and a pointer to the field data.

Otherwise, the code here is FEM independent and does not need augmentation.

3. Python code that initiates the C Code (observ.py)

The python code is used to create the fem field data. Otherwise, the code can be used verbatim.

It is necessary to have files *init.py *makejson.py included in the path. Currently saved to the dfn_fem/data.

sys.path.insert(0, '../data/') #path to init.py, makejson.py
from init import *
from makejson import *

The field is defined by an outside source, such as Firedrake.The field could be the solution to a pde equation or the result of interpolating an expression over a function space:

from firedrake import *
.....
f0 = Function(V).interpolate(Expression(expf0))

After defining the field f0 the following code is used to pass the field to Diderot.

data = organizeData(f)
_call = ctypes.CDLL('observ_init.so') 

Note 'observ_init.so' is the name of Diderot init program created.

_call.callDiderot_observ.argtypes =(ctypes.c_char_p,ctypes.c_void_p) #argument types
result =_call.callDiderot_observ(ctypes.c_char_p(name.encode('utf-8')),ctypes.cast(ctypes.pointer(data),ctypes.c_void_p))# call library

Note callDiderot_observ() is the name of the function in observ_init.c we want to call.

4. Running the whole thing (run.sh)

Make and run

make observ_init.so python observ.py

Details

• Branch: Diderot-Dev
• Syntax: “see summary above"
• Text: none

Directory Organization

• Gradient of a field (2-d, Scalar, Unit Square Mesh, element P) : X1
• ^ (3-d, Scalar, Unit Cube Mesh, Lagrange): X2
• Inner product between vector field and second order tensor field (3-d): X3