Biomedical Engineering Reference
In-Depth Information
(iii)
The equations that define the behavior of the
unknown variable, such as the equations of motion
or the relationships between stress and strain or
strain and displacement, are formulated for each
element in the form of matrices. These element
matrices are then assembled into a global system of
equations for the entire discretized domain. This
system is defined by a coefficient matrix, an
unknown vector of nodal values, and a known
''right-hand side'' (RHS) vector. Boundary
conditions in derivative form would already be
included in the RHS vector at this stage, but those
that set the unknown function to a known value at
the boundary have to be incorporated into the
system matrix and RHS vector by overwriting
relevant rows and columns. Since the RHS vector
contains information about the boundary conditions,
it is sometimes called the ''external load vector.''
(iv)
The final step in FEA involves solving the global
system of equations for the unknown vector. In
theory, this can be achieved by premultiplying the
RHS vector by the inverse of the coefficient matrix.
The result is the discrete (pointwise) solution to
the original problem. If the problem is linear and
isotropic, the elements of the matrix are constants
and the required matrix inversion can be done. If
the defining equations are nonlinear or the material
is anisotropic, the coefficient matrix itself will be
a function of the unknown variables and matrix
inversion is not straightforward. Some kind of
linearization is necessary before the matrix can be
inverted (e.g., successive approximation or
Newton's methods; see, for example,
Harris and
St¨cker, 1998
). In practice, the global system
matrix, whether linear or nonlinear, is seldom
inverted directly, usually because it is too large.
Some indirect method of solving the system of
equations is preferred [i.e., lower-upper (LU)
decomposition, Gaussian elimination; see, for
example,
Harris and St¨cker, 1998
].
The evaluation of element matrices, their assembly into
the global system, and the possible linearization and
eventual solution of the global system is a task that is always
passed on to a high-speed computer. This usually requires
complex computer programs written in a high-level lan-
guage, such as Fortran. Indeed, it is the advent of high-
speed computers and workstations and the continuous
improvements in processor speed, memory management,
and disk storage that have enabled large-scale FE problems
to be tackled with relative ease.
The modern-day FEA toolbox also includes facilities for
data pre- and postprocessing. Data preprocessing usually
involves input formatting and grid definition, the latter of
which may require some ingenuity, because mesh design
may affect the convergence and accuracy of the numerical
solution. Element size is governed by local geometry and the
rate of change of the solution in different parts of the
domain. Mesh refinement (a gradation of element size) in
the vicinity of sharp corners, boundary layers, high solution
gradients, stress concentrations or vortices is done routinely
to enhance the accuracy and convergence of the solution.
Adaptive procedures that allow the mesh to change with the
solution according to some error criteria are usually in-
corporated into the FE process (
George, 1991; Brebbia and
Aliabadi, 1993
;
Zienkiewicz andTaylor, 1994
). Typically, this
means that the mesh is refined in areas where the solution
gradient is high, and elements are removed from regions
where the solution is changing slowly. The result is usually
a dramatic improvement in convergence, accuracy, and
computational efficiency. Postprocessing of data involves the
evaluation of
ad hoc
variables such as strains, strain rates,
stresses; generating plots such as simple
xy
-plots, contour
plots, and particle paths; and solution visualization and ani-
mation. All of the additional information facilitates the un-
derstanding and interpretation of the results.
The importance of checking and validating FE solutions
cannot be overemphasized. The most basic validation in-
volves a ''patch test'' (
Zienkiewicz and Taylor, 1994
)in
which a few elements (i.e., a patch of the material) are an-
alyzed to verify the formulation of interpolating functions
and the consistency of the code itself. Second, a very simple
problem with known analytical solution is simulated with
a coarse grid to verify that the code reproduces the known
solution with acceptable accuracy. For example, parabolic
flow in a tube can be simulated with a very coarse grid and
the result quickly compared against the analytical solution.
We caution, however, that reproducing the solution in
a simpler problem does not guarantee that the code will
work in more realistic and complicated cases. It is also
recommended that numerical solutions be obtained from at
least three meshes with increasing degrees of mesh re-
finement. Such solutions should converge with mesh
refinement (
h
-convergence,
Strang and Fix, 1973
). Com-
parison of numerical results to experimental data should
always be made where possible. Last, especially in the ab-
sence of analytical solutions or experimental data, numerical
solutions should be compared across different numerical
methods, or across different numerical codes if the same
method is used. There is no gold standard for the number of
validation tests that are required for any particular problem.
The greater the variety of test problems and checks, the
greater the degree of confidenceonecanhaveintheresults
of the FEM.
The continuum equations
Whether we use FEA to compute the stress in a prosthetic
limb or to simulate blood flow in bifurcating arteries, the
