Page 328 - Process Modelling and Simulation With Finite Element Methods

Electrokinetic Flow 315

where 4 is the test function (conjugate to @, ds is the increment of arc length
along the curve B, and h is the Lagrange multiplier. The Galerkin method for
finite elements chooses the optimal value of h to balance the constraint r(@)=O
so that the error in satisfying it is minimized “in the energy” sense. Physically,
the Lagrange multiplier should have a meaning - inspection of (9.1) shows that it
has the same units as rh. In optimization theory, for instance, the Lagrange
multiplier conjugate to a constraint on supply of a commodity is the “shadow
price” for that supply - the price that would identically balance the supply and
demand. Since r(@) is a generalized Dirichlet condition, the Lagrange multiplier
is a generalized boundary flux of the field (dependent) variable.
In heat and mass transfer problems, the Lagrange multiplier conjugate to
fixed temperature or concentration on the boundary is the heat flux or mass flux
across the boundary. So the Lagrange multiplier his that value for each
individual element. In fluid dynamics, the flux of momentum across a boundary
is a force. But which force? It depends on the quantity rexpressed by the user
in general form: kfi. r for a Dirichlet condition with G=O. For the Navier-
Stokes equations, it is the viscous momentum dissipation in the PDE, so the
boundary flux is the viscous force on the boundary. In FEMLAB 3.0, there are
plans to include the pressure term in the r-vector as the default and to leave the
current arrangement (viscous stress) as an option in the application mode. In
general form, the PDE which we compute is the divergence of r($)), so the flux
computed by h is the normal component of l-. To illustrate this, we will revisit
the ECT forward problem shown in $7.3.2.

Example: Very accurate flux computations in the ECT forward problem
If you recall, this problem computes the boundary fluxes (charges) across the
electrodes held at given voltages on the boundary of a cylinder with badly
conducting rods placed axially when the duct is full of a much better conducting
substance. The heat transfer analogue is that the electrode surfaces are held at
fixed temperatures and we compute the heat flux across these surfaces. All other
external boundaries are insulated (no flux) and internal boundaries have
continuous temperature and flux. Rather than set up the whole problem again,
we start by reading in the MAT file ect.mat with the old solution (see Figure
7. I). We will investigate the accuracy of the original computations of flux with
two different meshes (coarse and fine) and then compare with the flux
calculation by the weak boundary constraint using the Lagrange multiplier.
Figure 9.1 shows the original coarse mesh used to compute the boundary fluxes
across the electrode surfaces.