A FEniCS Tutorial - FEniCS Project

2. forgetting that the spatial coordinates are denoted by a vector x,
3. forgetting that the x, y, and z coordinates in space correspond to x[0],
x[1], and x[2], respectively.
Failure to initialize parameters in the expressions lead to a compilation error
where this problem is explicitly pointed out.
Problems in the Solve Step. Sometimes the problem lies in the solve step
where a variational form is turned into a system of algebraic equations. The
error message ”Unable to extract all indicies” points to a problem with the
variational form. Common errors include
1. missing either the TrialFunction or the TestFunction object,
2. no terms without TrialFunction objects.
3. mathematically invalid operations in the variational form.
The first problem implies that one cannot make a matrix system or system of
nonlinear algebraic equations out of the variational form. The second problem
means that there is no ”right-hand side” terms in the PDE with known quantities.
Sometimes this is seemingly the case mathematically because the ”righthandside”iszero.
VariationalformsmustrepresentthiscaseasConstant(0)*v*dx
where v is a TestFunction object. An example of the third problem is to take
the inner product of a scalar and a vector (causing in this particular case the
error message to be ”Shape mismatch”).
All Programs Fail to Compile. On Ubuntu Linux unfinished updates of
the system (run by Update Manager) may causes all compilations to fail. When
previously working programs no longer can be compiled, reboot Ubuntu, run
the Update Manager, and wait until it has finished. Try compiling a working
program again.
7.8 Books on the Finite Element Method
There are a large number of books on the finite element method. The books
typically fall in either of two categories: the abstract mathematical version
of the method and the engineering ”structural analysis” formulation. FEniCS
builds heavily on concepts in the abstract mathematical exposition. An easyto-read
book, which provides a good general background for using FEniCS,
is Gockenbach [8]. The book by Donea and Huerta [5] has a similar style,
but aims at readers with interest in fluid flow problems. Hughes [10] is also
highly recommended, especially for those interested in solid mechanics and heat
transfer applications.
Readers with background in the engineering ”structural analysis” version of
the finite element method may find Bickford [1] as an attractive bridge over to
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