Page 23 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 9

conditions are such that the electric and magnetic fields are separable (e.g., in station-
ary cases) than the magnetic flux law has to be included in the physical model of the
magnetic field.

Boundary conditions (external interactions) and initial conditions
(initial state)
The boundary conditions define the interactions of the system with the environment
during the entire process. The initial conditions define the initial state of the system
that executes a process. Initial and boundary conditions play an essential role in pro-
viding for the uniqueness of the solution: prior to actually solving the problem it is
necessary to verify, prove the existence and uniqueness of the solution of the mathe-
matical mode.
This stage may be difficult, not always possible, and for only few cases theorems of
existence and uniqueness are available, for instance, Laplace Poisson problems with
Dirichlet, Neumann, and Robin boundary conditions (Tikhonov and Samarskii,
1963). For scalar fields, Dirichlet condition defines the primitive, the unknown to be
solved for first, on the boundary—the same for the system and its environment. For
vector fields, this condition states that the boundary contains the field, or the field
vanishes there, for example, magnetic insulation for the magnetic vector potential.
Neumann condition defines the normal component of a flux, for example, either zero
to state that the boundary is insulated with respect to that flux, or nonzero, to define a
boundary source. A Robin condition provides for a linear combination between the
scalar and the normal component of its flux. It may define a convection boundary
condition (Newton law, in convection heat transfer) or a contact resistance for electro-
kinetic problems, and others.
As it will be seen next, a necessary and sufficient number of boundary conditions
have to be given for consistency with the PDE mathematical model. When no theo-
rem is available to ascertain boundary conditions that ensure the existence and unique-
ness of the solution, consistent boundary conditions may be observed out of
experiments, symmetries, and conservation laws. To this end, conservation laws of
momentum, mass, energy, fluxes, charges, are utilized or even statistical models and
methods are used to recast experimental data. This holds true, for instance, for
Navier Stokes momentum equation.
In such situations, the number of boundary conditions required to obtain a solution
is decided first. Then an approximate but efficient method consists in considering the
principal part of the PDE. For instance, for Navier Stokes equation the principal part
is the Laplacean, a second-order partial differential operator. This criterion is not
always sufficient. For instance, Neumann problem nUru 5 fx; yÞ (flux boundary con-
ð
2
ditions only) for Laplace equation, r u 5 0, has a solution if and only if, supplemen-
H
tary, a necessary closure condition for the fluxes is satisfied, that is, Σ fx; yÞdA 5 0,
ð