Page 16 - Computational Modeling in Biomedical Engineering and Medical Physics

2 Computational Modeling in Biomedical Engineering and Medical Physics

the scanner development, reduce the design costs, and enhance its safety margins.
Numerical experiments in simulacra to real-life circumstances provide a wealth of
knowledge, and numerical simulation qualifies as an extremely useful tool.

1.2 The system and its boundary

The starting point in any experimental or theoretical analysis has to be the precise def-
inition of the system, which is—for a simple, comprehensive definition—a collection
of matter in a region of interest (ROI) in space, to be observed, investigated, and mea-
sured. In particular in numerical modeling, the system substantiates and hosts the
thought experiment. With the system comes its environment,or surroundings, with
which the system interacts through work, heat, and mass transfer. The entity that sepa-
rates the system and its environment is thus the boundary, which belongs in the same
time to the system and to its surroundings. In fact the state of the system (i.e., the col-
lection of values that the state quantities have at a specific time moment) depends on
the interactions with its environment, which are perceived or stated at the boundary
level. The boundary is traditionally assimilated to a surface and not to another system.
Mathematically it is a two-dimensional geometric variety of zero thickness, that is, a
surface, if the system is a three-dimensional volume, a contour if the system is two-
dimensional, and two points if the system is one-dimensional. Whereas in solid-body
mechanics the system and its boundary may be evident, in electromagnetism, fluid
mechanics, heat transfer, the system emerges once its boundary is drawn and its inter-
actions with the surroundings (the boundary conditions) are set.
The history or the film of the states that an evolving system may “visit” while
undergoing internal and or external interactions is called the evolution path. The state
of a system, at any time, is defined through the ensemble of local quantities called
thermodynamic properties. For instance, physical quantities such as temperature, pressure,
electric potential, energy, and densities are thermodynamic properties. Their values
do not depend on the history of the system that evolves in time and depend strictly
on the instantaneous conditions in which they are measured or computed. Physical
quantities such as work, heat, and mass transfer interactions are not thermodynamic
properties. The thermodynamic state properties variation depends on the initial and
final states of the system that are associated with the path. The quantities that are not
thermodynamic properties depend on the initial and final states and on the path
between them—they are quantities of interaction.
Of concern in defining a system is the continuity, in the mathematical sense, of the
underlying properties across its boundary. Their discontinuity may trigger analytical
difficulties. For instance, the gradient of the temperature, (a scalar property, which is
proportional to the heat flux density, may not be well defined on the boundary should
this scalar be a discontinuous function there.