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Finite elements concepts in Sundance

In the finite-element method, a PDE is written in weak form on an infinite-dimensional space and then discretized onto a finite-dimensional space as specified by the choice of Mesh and by the choice of basis function.

There are many possible weak formulations for any given PDE, for instance Galerkin or first-order least squares. Furthermore, there is then some freedom in choice of basis function (often constrained by stability considerations). Sundance is designed to allow great flexibility in choice of formulation and discretization method.

Function spaces and basis families

The unknown and variational (test) functions in a weak equation are represented by UnknownFunction and TestFunction objects, respectively. The first constructor argument to an UnknownFunction or TestFunction object is the BasisFamily with which it will be discretized. For example,

  Expr T = new UnknownFunction(new Lagrange(1), "T");

creates a function T that will be discretized with first-order Lagrange interpolation.

A BasisFamily such as Lagrange is a geometry-independent specification of a basis function; the basis family object automatically hook to the low-level functions appropriate to the cell being integrated (e.g. and for a line, and so on).

Weak equations

The continuous formulation of a finite-elements problem is as a weak equation, for example

plus possible restrictions on the variations on specified boundary segments (see Boundary conditions below).

To represent a weak equation, use the Sundance Integral object. Assuming that the boundary segment has been represented with a CellSet called gamma the equation above would be written

Integral myEqn = Integral(-(grad*v)*(grad*u)) 
   + Integral(-v*f)
   + Integral(gamma, v*G);

Integrals with no domain specified are taken to be over all maximal cells. Notice that Integral objects can be added using the overloaded \c+operator.

Since the integrals will be done with quadrature, you can give a specification of a quadrature family as an optional argument to Integral. For example,

Integral myEqn = Integral(someCellSet, v*sin(x), new GaussLegendre(8));

will cause 8th-order Gauss-Legendre quadrature to be used on whatever cells get integrated. You can use this feature on a term-by-term basis to assign high-order quadrature for complicated terms but low-order quadrature for all others, for example

Integral myEqn = Integral(v*sin(4*x), new GaussLegendre(16))
   + Integral(v*u, new GaussLegendre(2));

The default quadrature rule is 4th-order Gauss-Legendre.

Warning:: The Sundance Integral object is essentially just a bookkeeper that groups domains with expressions. It doesn't know how to do integration by parts, so for problems in which integration by parts is necessary you must do that step by hand before putting your problem into an Integral object.

Boundary conditions

There are many ways to handle boundary conditions in finite-elements problems.

In Galerkin methods, Neumann BCs are easy: the BC along a CellSet top is incorporated by adding

    Integral(top, v*G)

to the weak equation, where v is the variation of u and G is the expression which represents the flux. Natural BCs, which are zero-flux Neumann BCs, are especially easy, since the integral is indentically zero. To do natural BCs, you do nothing at all.

Dirichlet BCs can done by replacing the weak equation on the boundary nodes with another weak equation representing the BCs. In Sundance, this is done with the EssentialBC object. To apply the weak condition on the CellSet left, write

   EssentialBC bc = EssentialBC(left, v*(u-sin(y)));

EssentialBC objects can be combined using the "and" operator &&, for example

   EssentialBC bc = EssentialBC(left, v*(u-sin(y))) 
       && EssentialBC(right, v*(u-cos(y)));

In least-squares formulations, you can do dirichlet BCs in least-squares form (i.e., as just another Integral) or in essential form. Using the least-squares form has the advantage that your matrix remains SPD, but the BCs will not always be satisfied exactly.

You can also do Dirichlet BCs by introducing lagrange multipliers along the boundary. In this case, the BC becomes just another integral to be added to the weak equation. This method is sometimes useful, but produces indefinite matrices.

Next section: Linear solvers

Contact:

Kevin Long (krlong@ca.sandia.gov)

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