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The Finite Element Method:
A Four-Article Series
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FINITE ELEMENT ANALYSIS: Solution
by Steve Roensch, President, Roensch & Associates
...Third in a four-part series
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While the pre-processing and post-processing phases of the
finite element method are interactive and time-consuming for
the analyst, the solution is often a batch process, and is
demanding of computer resource. The governing equations are
assembled into matrix form and are solved numerically. The
assembly process depends not only on the type of analysis
(e.g. static or dynamic), but also on the model's element
types and properties, material properties and boundary
conditions.
In the case of a linear static structural analysis, the
assembled equation is of the form Kd = r, where K is the
system stiffness matrix, d is the nodal degree of freedom
(dof) displacement vector, and r is the applied nodal load
vector. To appreciate this equation, one must begin with
the underlying elasticity theory. The strain-displacement
relation may be introduced into the stress-strain relation
to express stress in terms of displacement. Under the
assumption of compatibility, the differential equations of
equilibrium in concert with the boundary conditions then
determine a unique displacement field solution, which in
turn determines the strain and stress fields. The chances
of directly solving these equations are slim to none for
anything but the most trivial geometries, hence the need for
approximate numerical techniques presents itself.
A finite element mesh is actually a displacement-nodal
displacement relation, which, through the element
interpolation scheme, determines the displacement anywhere
in an element given the values of its nodal dof.
Introducing this relation into the strain-displacement
relation, we may express strain in terms of the nodal
displacement, element interpolation scheme and differential
operator matrix. Recalling that the expression for the
potential energy of an elastic body includes an integral for
strain energy stored (dependent upon the strain field) and
integrals for work done by external forces (dependent upon
the displacement field), we can therefore express system
potential energy in terms of nodal displacement.
Applying the principle of minimum potential energy, we may
set the partial derivative of potential energy with respect
to the nodal dof vector to zero, resulting in: a summation
of element stiffness integrals, multiplied by the nodal
displacement vector, equals a summation of load integrals.
Each stiffness integral results in an element stiffness
matrix, which sum to produce the system stiffness matrix,
and the summation of load integrals yields the applied load
vector, resulting in Kd = r. In practice, integration rules
are applied to elements, loads appear in the r vector, and
nodal dof boundary conditions may appear in the d vector or
may be partitioned out of the equation.
Solution methods for finite element matrix equations are
plentiful. In the case of the linear static Kd = r,
inverting K is computationally expensive and numerically
unstable. A better technique is Cholesky factorization, a
form of Gauss elimination, and a minor variation on the
"LDU" factorization theme. The K matrix may be efficiently
factored into LDU, where L is lower triangular,
D is diagonal, and U is
upper triangular, resulting in LDUd = r.
Since L and D are easily inverted,
and U is upper
triangular, d may be determined by back-substitution.
Another popular approach is the wavefront method, which
assembles and reduces the equations at the same time. Some
of the best modern solution methods employ sparse matrix
techniques. Because node-to-node stiffnesses are non-zero
only for nearby node pairs, the stiffness matrix has a large
number of zero entries. This can be exploited to reduce
solution time and storage by a factor of 10 or more.
Improved solution methods are continually being developed.
The key point is that the analyst must understand the solution
technique being applied.
Dynamic analysis for too many analysts means normal modes.
Knowledge of the natural frequencies and mode shapes of a
design may be enough in the case of a single-frequency
vibration of an existing product or prototype, with FEA
being used to investigate the effects of mass, stiffness and
damping modifications. When investigating a future product,
or an existing design with multiple modes excited, forced
response modeling should be used to apply the expected
transient or frequency environment to estimate the
displacement and even dynamic stress at each time step.
This discussion has assumed h-code elements, for which the
order of the interpolation polynomials is fixed. Another
technique, p-code, increases the order iteratively until
convergence, with error estimates available after one
analysis. Finally, the boundary element method places
elements only along the geometrical boundary. These
techniques have limitations, but expect to see more of them
in the near future.
Next month's article will discuss the post-processing phase
of the finite element method.
© 2008-2013 Roensch & Associates. All rights reserved.
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This four-article series was published in a newsletter of
the American Society of Mechanical Engineers (ASME).
It serves as an introduction to the recent analysis discipline
known as the finite element method. The author
is an engineering consultant and expert witness specializing
in finite element analysis.
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