Page 29 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 15

introduced as a spectral method too (Peyret and Taylor, 1983). It may also be consid-
ered as belonging to the weighted residual class of methods. As with all numerical
methods, the PDE numerical solution is calculated in a finite number of points that
are obtained by dividing the computational domain into elements that form the discre-
tization mesh. The unknowns are related to the grid nodes (vertices), which are a subset
of the mesh nodes, called grid nodes. It should be noted that when automatic
(Delaunay, 1934) mesh generation algorithms are used only the geometry of the
computational domain (curvature, edges, voids, etc.) counts in the discretization of the
computational domain, and the underlying physics is of less concern. Either structured
or unstructured meshes may be used. Each of them has pros and cons.
The solution to the mathematical problem, the PDE is then represented locally by
using a set of trial analytic functions that, endowed with certain smoothness and regu-
larity properties, form a basis for the representation of the exact solution to the prob-
lem. For instance, they form a closed set defined on the Hilbert space of the functions
of integrable square, and it is desirable that they are orthonormal too (Peyret and
Taylor, 1983). Sturm Liouville singular problems are resources for the set of functions
that may form a basis, and the boundary conditions of the mathematical problem, in
general the principal part of the PDE may be used to single out one of the available
solutions. In general lower order polynomials are preferred—here, Lagrange polyno-
mials (Orszag, and Gotlieb, 1980).
FEM methods may use node elements for electromagnetic (vector) fields too.
In this situation the vector field quantities are described with their components at
the vertices, that is, their scalar projections are assigned to each vertex. However,
using the coordinates of the vectors, give rise to difficulties in implementing the
boundary conditions and satisfying the continuity of the numerical solution
(Webb, 1993). To overcome these difficulties, edge elements were proposed instead
by Takahashi et al. (1992). Their usage does not necessarily eliminate the unphysi-
cal, spurious modes though Schroeder and Wolf (1994). These maybeavoided
only by a proper finite element formulation (Mur, 1998).

1.7 Coupled (multiphysics) problems

In many circumstances, the systems are sieges of multiple concurring irreversible-flow trans-
port phenomena—mass, heat, electrical charges, etc. From a thermodynamic perspective,
unlike the Gibbsian formulation for systems in internal and external equilibrium, these inter-
actions are characterized through interactions rates. This finding neither introduces new
physical insight nor explains, per se, the couplings between the interaction rates and the local
thermodynamic properties of the system subject of such interactions (Bejan, 1988). The ana-
lytic forms of these couplings are just postulated, proposed by the irreversible thermodynamics
through relations between gradients of local thermodynamic properties and fluxes that