# MECH Finite Element Methods

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```					MECH3300 Finite element methods

Lecture 10 Issues in modelling
Solving the equations
Use of symmetry
• With increasing computer power, many analysts cannot be bothered
with symmetry. However, there are always problems that are too big,
that can be reduced in size at least 4 fold, by exploiting symmetry.

• Even if loading is unsymmetric, symmetry of geometry can be exploited
components and superimposing two solutions, modelling 1/2 the
structure, and changing restraints on the plane of symmetry..
F                      F/2           F/2       F/2
F/2

Symmetric case       Antisymmetric case
Plane of symmetry
Restraints for modelling symmetry
• With symmetric deformation, nodes in a plane of symmetry
remain there.

• To make this happen, fix displacement normal to the plane and
rotations about axes in-plane (if defined).

• Conversely, permit in-plane displacements and rotation about
the normal to the plane of symmetry.

• Anti-symmetry is the opposite - no motion in the plane of
symmetry.
Meshing solid models
• For machine components, it is possible to automatically mesh a
CAD solid model. Standard formats are needed to import
geometry into FE packages (eg IGES - International Graphic
Exchange Standard).

• Geometry needs to be checked, for duplicate edges, very short
edges that would produce too fine a mesh locally, and for details
that should be omitted, as it is not practical to mesh them.
Geometry must be checked for free edges, or faces, to ensure it
is connected properly.

• A surface mesh is often created first with plates, and this used to
guide generation of a solid mesh.

• A mesh created by automatic or other meshing must be checked
for duplicated nodes or elements, and to ensure all elements are
connected correctly.
Example of auto-meshing inappropriate
geometry (Strand7 tutorial problem)
•                         Nodes cluster to attempt
to match 2 very short
side-lengths present in
the geometry - these
“slivers” need removing.
Some simplification of the
real geometry is often
needed to get an
appropriate mesh.
A solid model subject to a point load will have very high stress at the
load. This makes it impossible to see the stress elsewhere on a
contour plot, and is unrealistic.
Hence loads must be distributed realistically. This can take some
Distribution?

Restraints also need realistic distribution. For the pin in a hole case,
polar coordinates at the centre of the hole are needed so as to fix
radial displacements around the hole, but not tangential ones. Over
what angle of arc?
Meshing arbitrary shapes
• This is something commercial packages still not not do well.

• A surface described by a stereolithography *.stl file can often be
imported, but this can be a very fine triangulation. It may be
possible to “unrefine” this to get a more suitable mesh (the
command exists in NASTRAN for Windows for instance), but
this may still take much to long to do to be practical.
Equation solvers
• The large set of equations in a linear elastic finite element model is
solved either by Gauss elimination or by an iterative algorithm (eg the

• The iterative solvers tend to become more efficient on very large
problems. (eg 100000 nodes). They use sparse matrix storage, that is
they store only non-zero terms along with their row and column

• There are different ways of organizing how Gauss elimination is done.
• It is always 2 stages - triangular decomposition [K] = [L][U], followed
by back-substitution F = [L]z, z = [U]u.

• (a) the equations can be written in an order that minimizes the
bandwidth of [K], so that only terms within the bandwidth are stored on
disk or processed by the solver.
• (b) the “wavefront” solver, implemented in ANSYS can be used.
The wavefront solver
• In this approach to solving the equations, the elements are
numbered so that elements connected together have similar
numbers. Only those elements connected to a node having its
equations processed must be assembled at any one time.

• The effect of solving equations one node at a time is that the
mesh is divided into 3 regions - nodes yet to be processed,
nodes being processed, and nodes having been processed.

• The boundaries between these regions sweep across the mesh,
hence the “wavefront” name.

• This approach may be more efficient than a bandwidth -
minimizing solver.
Explicit Solvers
One approach to highly nonlinear problems is to integrate in
time using Newton’s law ie
FEXT + FINT = Ma
FEXT = external forces (applied loads) at some time.
FINT = internal forces (due to elastic or plastic deformation etc.)
M = mass matrix
a = accelerations - solve for these. If M is diagonal, then
these are found “explicitly” from the forces.
The new accelerations give new velocities and displacements
(central difference method). These are used to find new
internal forces, and to again to find new accelerations, and so
on. Timestep size must be very small (eg microseconds).
Substructures or superelements
• Traditionally, very large models of ships, cars bodies etc. were solved by
a divide and conquer approach of separating the mesh into different
regions called either substructures or superelements.

• The equations for each region are “statically condensed” to give many
fewer equations just relating forces/displacements on the interfaces
between regions.

• These are solved, and the interface displacements provide boundary
conditions to solve the equations for each substructure separately.

• A typical substructure would be the wing of an aircraft - the rear fuselage
may be another, and the tail a third etc.

• These methods are still used where a model of part of a system is
prepared by a sub-contractor - eg a payload for the space shuttle.
Eigenvalue problems
• Finding critical loads for buckling, or finding natural frequencies for
vibration leads to matrix eigenvalue problems.

• For undamped natural frequencies [K]u = l[M]u must be solved, where
[M] is a mass matrix and the eigenvalue l is the angular natural
frequency squared w2 - more next semester in dynamics.

• For buckling [K]u = l[Kg]u where [Kg] is a geometric stiffness matrix
describing the change in stiffness due to finite deformation, worked out

• The eigenvalue l this time is a factor on the loading that will scale it up
to that causing buckling.

• This approach is as accurate as hand calculation of a critical load, but
not a true finite deformation analysis that keeps updating coordinates
of nodes - it is just a perturbation of a linear analysis.
Solving eigenvalue problems
• We usually only want to know the lowest natural frequencies of a
vibrating structure, as they are most easily excited.
• Similarly, only the lowest buckling eigenvalue is meaningful, as it
takes less work to make the structure collapse at this load, than at
the others predicted, which could only occur if buckling were
prevented at the lowest critical load.

• Hence the most appropriate solvers are algorithms that iteratively
refine estimates of u to find the lowest modes of deformation.
These can however, potentially miss a solution in an ill-conditioned
problem. An algorithm which counts how many eigenvalues exist in
a particular range of values can be used to check if solutions are
missing (a Sturm sequence check).

• The user can typically select how many modes of deformation to
estimate. The results are usually animated to see the motion more
clearly.
Solving non-linear problems
• Non-linear problems take much more computation to solve than
static problems, as the solvers work by successive linearization,
that is, the solution keeps going off on a tangent and is then
corrected.

• A static finite deformation analysis, for instance, is typically done
load increment, “equilibrium iterations” are performed to correct
the error between the external loads and the predictions of
internal loads, found from the latest linearized solution.

• The number of iterations, to predict a substantial change of
shape can be many hundreds, and the equivalent of a linear
static analysis is done every iteration.

• A solution for the post-buckled state of a structure can be quite
hard to obtain.
History of load-deflection in some element
• The progress of a non-linear solution can be pictured by the

Equilibrium
(eg tension in
graph
a particular
bar)

Deflection magnitude

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 views: 5 posted: 11/24/2011 language: English pages: 15