Biomedical Engineering Reference
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Fig. 3.25 Sources of errors reflecting the degree of approximation in the key simulation steps
modelling a physical system
The key steps from the physical system to an approximate solution of an
employed discrete model are depicted schematically in Fig. 3.25 , including error
sources in each simulation step.
While analytic solutions provide information about an unknown quantity at an
infinite number of locations in a particular region, numerical methods provide
values only at discrete points in that region.
With the finite element method, the considered continuum domain of the
problem is discretized (discretization: reduction of an infinite number of degrees of
freedom to a finite number) by subdividing the two or three dimensional domain
into a finite number of small non-overlapping subdomains of simple geometry.
These are called (finite) elements, which represent discrete portions of the
physical structure. The employed elements are geometrically simple, such as
segments for triangles or quadrilaterals for surfaces or two-dimensional problems
and segments for volumes such as tetrahedrons or hexahedrons. The boundaries of
adjacent elements are connected at a number of discrete points, designated as
nodes, which usually are the vertices of the element or midpoints of the element
boundary. Depending on the element type, additional node locations, other than
the ones mentioned, may be assigned.
As illustrated in Fig. 3.26 , aside from the approximation of the solution to a
problem on the domain of body B, the domain itself is approximated by a finite
element mesh: X X h ¼ S
n
X e (the subscript e refers to the element domain and
e ¼ 1
the subscript h denotes the fineness of the discretization and relates to the mesh
size) where X h is the approximate domain, n is the total number of elements, X e is
the element domain and S
n
denotes the assembly of n element domains, i.e.
e ¼ 1
summation of all element contributions as well as fulfillment of inter element
compatibility.
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