3.4 Boundary conditions

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3.4 Boundary conditions

The finite element method distinguishes between natural and essential boundary conditions. As an introduction to these concepts, consider the weak form of the left-hand side of Laplace's equation:

$\begin{displaymath} \int_G S_i\nabla^2 u dV = -\int_G \nabla S_i\cdot\nabla u dV +\int_{\partial G} S_i\mbox{\boldmath{$n$}}\cdot\nabla u dS\end{displaymath}$

The last term is an integral over the boundary of the normal derivative (or flux). This is called a natural boundary condition; the boundary integrands represent a physical quantity (for example, flux in a diffusion problem, or stress in a linear elasticity problem). The condition is implemented by substituting directly if the integrand is known, or by substituting an expression involving unknowns, as in a radiation condition for example. As shown in examples later in this Tutorial Guide, natural boundary conditions are specified at the time that the PDE problem is specified.

On the other hand, an absolute specification at some set of boundary nodes is called an essential boundary condition. This is enforced by including it in the set of equations, replacing the equation which had been formed by using as a test function.

To illustrate this point, suppose that node is a boundary node with value . The third row of the matrix is independent of other nodal values and is given by

$\begin{displaymath} \left[ \begin{array} {cccccc} 0 & 0 & 1 & 0 & \cdots & 0\... ...\ u_2 \\ u_3 \\ \vdots \\ u_N\end{array} \right] ~~=~~U\end{displaymath}$

Incorporation of this boundary condition into the matrix system gives the new system

$\begin{displaymath} \left[ \begin{array} {ccccc} A_{11} & A_{12} & A_{13} & A_... ...{c} R_1 \\ R_2 \\ U \\ R_4 \\ \vdots\end{array} \right]\end{displaymath}$

If the matrix is symmetric it is necessary to do a further elimination to regain symmetry:

$\begin{displaymath} \left[ \begin{array} {ccccc} A_{11} & A_{12} & 0 & A_{14} ... ...-UA_{23} \\ U \\ R_4-UA_{43} \\ \vdots\end{array} \right]\end{displaymath}$

To summarise the discussion so far, essential boundary conditions are implemented by modification of the global matrix and RHS vector, whilst natural boundary conditions are often accounted for in the RHS vector alone. These concepts are so important, however, that we now provide a more detailed commentary. Some of the language concepts will be defined more completely in later chapters.

In the Galerkin procedure, a term such as $D_m {\rm Exp}$ (where denotes partial differentiation with respect to , could be or , and ${\rm Exp}$ might or might not involve suffices, an unknown or another differentiation) is multiplied by a test function over the region with boundary $\partial G$ . For all second-order terms and some first-order terms, the integration is done by parts. This gives:

$\begin{displaymath} \int_G T ~ D_m {\rm Exp} ~ dV = ~-~ \int_G (D_m T) ~ {\rm Exp} ~ dV ~+~ \int_{\partial G} T n_m {\rm Exp} ~ dS \end{displaymath}$

The application of boundary conditions in the finite element method requires either that some information is used to replace the integrand resulting from second-order terms, when is non-zero there, or that for such test functions the whole equation is replaced by an essential condition. So when Fastflo implements such terms, it adds only the second integral into the equations - either to the sparse matrix or the right-hand side vector. The boundary integral is left for the user to supply, using a b statement. If no integral is supplied, the boundary condition enforced is that the integral is zero - a common condition in practice.

The boundary integrals often have physical significance, and it is best to try to formulate the equations to take advantage of this. In fact, many second-order equations correspond to one of the following patterns:

time rate of change of heat = div (flux)
time rate of change of momentum = div (stress)

For steady equations the rates would be zero. The divergences are integrated by parts, and the boundary integrands will be the normal components of either the flux or the stress. On interior boundaries, integrals are generated on each side, and the net integrand is the difference.

There are three possible specifications on any particular boundary.

1.: The user may do nothing. This is equivalent to asserting that the integrand (flux, stress, ...) is zero on an outside boundary or the integrand is continuous on an interior boundary. Note that for the integrand to be identifiable with flux or stress, all components of the derivatives of these must be present in a form which will be differentiated by parts.
2.: The user may set the boundary integral, by including the appropriate value in a b boundary statement, which will be added to the left-hand side.
3.: The user may specify a Dirichlet condition, in which case the equation, with its boundary integrals, will be overwritten.

There are some potential paradoxes here, typified by the Stokes equation. Here the divergence of stress may be written

$\begin{displaymath} D_j {\mu} D_i v_j + D_j {\mu} D_j v_i - D_i p\end{displaymath}$

where

is the velocity and

is the pressure. It generates a boundary integral

$\begin{displaymath} n_j {\mu} D_i v_j + n_j {\mu} D_j v_i - n_i p\end{displaymath}$

where

is the unit outward normal, and this is indeed the surface stress. In incompressible flow, with constant viscosity $\mu$ , the first term of the stress divergence will be zero, and is often omitted. However, in the boundary integral, the corresponding term is not zero. The boundary integrand would be

$\begin{displaymath} n_j {\mu} D_j v_i - n_i p\end{displaymath}$

and if the surface stress is set to zero as a boundary condition, the integrand is not equal to zero, but is

$\begin{displaymath} - n_j {\mu} D_i v_j \end{displaymath}$

Now it is perfectly possible, and even more efficient, to make that prescription, but it is more natural to include the full stress term in the equation. Section 15.4 contains an example where a downstream free stress condition is implemented in this way.

It is also important that the pressure term be in a form which would be integrated by parts, in order to appear in an implicit boundary stress. To divide by the density $\rho$ , for example, the term should be written as D_i {1/rho} U2 and not {1/rho} D_i U2 which by Fastflo's rules would not be integrated by parts.

In complicated systems of equations, care is needed to ensure that where Dirichlet conditions are specified on some components but not others, they do not have the effect of overwriting any unknown boundary integrals.