Compute Integrals

We start with setting the Feel++ environment and loading the Feel++ library.

Set the Feel++ environment with local repository
import feelpp.core as fppc
import sys
app = fppc.Environment(["myapp"],config=fppc.localRepository(""))
Results 
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/PETSc.py:3
      2 from petsc4py.lib import ImportPETSc  # noqa: E402
----> 3 PETSc = ImportPETSc(ARCH)
      4 PETSc._initialize()

File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/lib/__init__.py:29, in ImportPETSc(arch)
     28 path, arch = getPathArchPETSc(arch)
---> 29 return Import('petsc4py', 'PETSc', path, arch)

File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/lib/__init__.py:95, in Import(pkg, name, path, arch)
     94 # import extension module from 'path/arch' directory
---> 95 module = import_module(pkg, name, path, arch)
     96 module.__arch__ = arch  # save arch value

File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/lib/__init__.py:73, in Import..import_module(pkg, name, path, arch)
     72 sys.modules[fullname] = module
---> 73 spec.loader.exec_module(module)
     74 return module

File petsc4py/PETSc.pyx:1, in init petsc4py.PETSc()

ValueError: numpy.dtype size changed, may indicate binary incompatibility. Expected 96 from C header, got 88 from PyObject

The above exception was the direct cause of the following exception:

SystemError                               Traceback (most recent call last)
File :1
----> 1 import feelpp.core as fppc
      2 import sys
      3 app = fppc.Environment(["myapp"],config=fppc.localRepository(""))

File /usr/lib/python3/dist-packages/feelpp/core/__init__.py:12
     10 from ._feelpp import  *
     11 from ._core import *
---> 12 from ._alg import *
     13 from ._mesh import *
     14 from ._plot import *

SystemError: initialization of _alg raised unreported exception

1. Integral of a function

\(\int_{\Omega} f(x) dx\) is the integral of the function \(f\) over the domain \(\Omega\). The domain \(\Omega\) is a subset of \(\mathbb{R}^n\) and \(f\) is a function defined on \(\Omega\). The domain \(\Omega\) is discretized as a mesh \(\mathcal{T}\) and the function \(f\) may be approximated by a finite element function \(\hat{f}\). The integral is approximated by the sum of the integrals over the elements of the mesh \(\mathcal{T}\). The integral over each element is computed using a quadrature formula.

We first load a mesh

Load a mesh \(\mathcal{T}\)
geo=fppc.download( "github:{repo:feelpp,path:feelpp/quickstart/laplacian/cases/feelpp2d/feelpp2d.geo}", worldComm=app.worldCommPtr() )[0]
print("geo file: {}".format(geo))
mesh = fppc.load(fppc.mesh(dim=2,realdim=2), geo, 0.1)
Results 
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 geo=fppc.download( "github:{repo:feelpp,path:feelpp/quickstart/laplacian/cases/feelpp2d/feelpp2d.geo}", worldComm=app.worldCommPtr() )[0]
      2 print("geo file: {}".format(geo))
      3 mesh = fppc.load(fppc.mesh(dim=2,realdim=2), geo, 0.1)

NameError: name 'fppc' is not defined

Then we can define functions \(f\) and compute the integral of \(f\) over the mesh \(\mathcal{T}\).

Compute integrals
from feelpp.core.integrate  import integrate

i1 = integrate(range=fppc.elements(mesh),expr="sin(x+y):x:y")
i2 = integrate(range=fppc.boundaryfaces(mesh),expr="x*nx+y*ny+z*nz:x:y:z:nx:ny:nz")
i3 = integrate(range=fppc.markedelements(mesh, "marker"), expr="1")

print(f"i1 = {i1}, i2 = {i2}, i3 = {i3}")
Results 
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/PETSc.py:3
      2 from petsc4py.lib import ImportPETSc  # noqa: E402
----> 3 PETSc = ImportPETSc(ARCH)
      4 PETSc._initialize()

File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/lib/__init__.py:29, in ImportPETSc(arch)
     28 path, arch = getPathArchPETSc(arch)
---> 29 return Import('petsc4py', 'PETSc', path, arch)

File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/lib/__init__.py:95, in Import(pkg, name, path, arch)
     94 # import extension module from 'path/arch' directory
---> 95 module = import_module(pkg, name, path, arch)
     96 module.__arch__ = arch  # save arch value

File /usr/lib/petsc/lib/python3/dist-packages/petsc4py/lib/__init__.py:73, in Import..import_module(pkg, name, path, arch)
     72 sys.modules[fullname] = module
---> 73 spec.loader.exec_module(module)
     74 return module

File petsc4py/PETSc.pyx:1, in init petsc4py.PETSc()

ValueError: numpy.dtype size changed, may indicate binary incompatibility. Expected 96 from C header, got 88 from PyObject

The above exception was the direct cause of the following exception:

SystemError                               Traceback (most recent call last)
File :1
----> 1 from feelpp.core.integrate  import integrate
      3 i1 = integrate(range=fppc.elements(mesh),expr="sin(x+y):x:y")
      4 i2 = integrate(range=fppc.boundaryfaces(mesh),expr="x*nx+y*ny+z*nz:x:y:z:nx:ny:nz")

File /usr/lib/python3/dist-packages/feelpp/core/__init__.py:12
     10 from ._feelpp import  *
     11 from ._core import *
---> 12 from ._alg import *
     13 from ._mesh import *
     14 from ._plot import *

SystemError: initialization of _alg raised unreported exception

2. Next steps