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1.2 CoefficientFunctions

In NGSolve, CoefficientFunctions are representations of functions defined on the computational domain \(\Omega\). Examples are expressions of coordinate variables \(x, y, z\) and functions that are constant on subdomains. Much of the magic behind the seamless integration of NGSolve with python lies in CoefficientFunctions. This tutorial introduces you to them.

import netgen.gui
%gui tk
from netgen.geom2d import unit_square
from ngsolve import *
mesh = Mesh (unit_square.GenerateMesh(maxh=0.2))

Define a function

myfunc = x*(1-x)
myfunc   # You have just created a CoefficientFunction

<ngsolve.fem.CoefficientFunction at 0x7f6e44cfad00>

x        # This is a built-in CoefficientFunction

<ngsolve.fem.CoefficientFunction at 0x7f6e44cfa518>

Draw the function

Use the mesh to visualize the function.

Draw(myfunc, mesh, "firstfun")

Evaluate the function

mip = mesh(0.2, 0.2)
myfunc(mip)

0.16000000000000003

Note that myfunc(0.2,0.3) does not work: You need to give points in the form of MappedIntegrationPoints like mip above. The mesh knows how to produce them.

Examining functions on sets of points

pts = [(0.1*i, 0.2) for i in range(10)]
vals = [myfunc(mesh(*p)) for p in pts]
for p,v in zip(pts, vals):
    print("point=(%3.2f,%3.2f), value=%6.5f"
         %(p[0], p[1], v))

point=(0.00,0.20), value=0.00000
point=(0.10,0.20), value=0.09000
point=(0.20,0.20), value=0.16000
point=(0.30,0.20), value=0.21000
point=(0.40,0.20), value=0.24000
point=(0.50,0.20), value=0.25000
point=(0.60,0.20), value=0.24000
point=(0.70,0.20), value=0.21000
point=(0.80,0.20), value=0.16000
point=(0.90,0.20), value=0.09000

We may plot the restriction of the CoefficientFunction on a line using matplotlib.

import matplotlib.pyplot as plt
px = [0.01*i for i in range(100)]
vals = [myfunc(mesh(p,0.5)) for p in px]
plt.plot(px,vals)
plt.xlabel('x')
plt.show()

<Figure size 640x480 with 1 Axes>

Interpolate a CoefficientFunction

We may Set a GridFunction using a CoefficientFunction:

fes = H1(mesh, order=1)
u = GridFunction(fes)
u.Set(myfunc)
Draw(u)         # Cf.: Draw(myfunc, mesh, "firstfun")

• The Set method interpolates myfunc to obtain the grid function u.
• Set does an Oswald-type interpolation as follows:
  • It first zeros the grid function;
  • It then projects myfunc in \(L^2\) on each mesh element;
  • It then averages dofs on element interfaces for conformity.

Integrate a CoefficientFunction

We can numerically integrate the function using the mesh:

Integrate(myfunc, mesh)

0.166666666666666

Differentiate

There is no facility to directly differentiate a CoefficientFunction. But you can interpolate it into a GridFunction and then differentiate the GridFunction.

u.Set(myfunc)
gradu = grad(u)
gradu[0]

<ngsolve.fem.CoefficientFunction at 0x7f6deaab8518>

Draw(gradu[0], mesh, 'dx_firstfun')

Obviously the accuracy of this process can be improved for smooth functions by using higher order finite element spaces.

Vector-valued CoefficientFunctions

Above, gradu provided an example of a vector-valued coefficient function. To visualize it, click on Visual menu in GUI and check Draw Surface Vectors.

Draw(gradu, mesh, "grad_firstfun")

You can also define vector coefficient expressions directly:

vecfun = CoefficientFunction((-y, sin(x)))

Draw(vecfun, mesh, "vecfun")

Expression tree

Internally, coefficient functions are implemented as an expression tree made from building blocks like x, y, sin, etc., and arithmetic operations.

E.g., the expression tree for myfunc = x*(1-x) looks like this:

print(myfunc)

coef binary operation '*', real
  coef coordinate x, real
  coef binary operation '-', real
    coef N5ngfem27ConstantCoefficientFunctionE, real
    coef coordinate x, real