MECH3300 Finite Element Methods

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29 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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MECH3300 Finite Element Methods

Lecture 12


Other materials

Conduction heat transfer

Optimization

Related numerical methods

Orthotropic materials


Most packages accept data for orthotropic materials


those for
which 3 perpendicular axes


the principal material axes


exist.




These can have 3 Young’s modulii and 3 Poisson ratios. Often
both directions transverse to a fibre are the same, so that we can
specify just a longitudinal modulus
E
1
, a transverse modulus
E
2
, a
major Poisson ratio
n
12

(transverse strain for unit longitudinal
strain), and a minor Poisson ratio
n
21
.



If one of these is unknown, the relation
E
1
n
21

= E
2
n
12

can be used to
estimate it. This follows from work
-
energy arguments and leads to
a symmetric
[D]

matrix for the material.



Modulii may be effective values found by multiplying
E
i

for material
i

by the volume fraction of this material, and summing contributions
from different materials.

Fibre composite materials


These are typically a lay
-
up, with orthotropic properties specified for
each layer (or lamina).



Output may consist of indices giving the ratio of stresses in each
layer to those causing failure.



To find such summary output, a failure criterion is needed for an
individual ply. One commonly used theory is Tsai
-
Hill theory which
is a generalization of Von Mises’ theory:





FI

= failure index (OK if < 1). Note there are limits on direct stresses
in both principal material directions (
s
1


<
s
1b

)
and (
s
2

<


s
2b

) which
can be different in tension and compression, as well as a limit on the
shear stress in the principal material axes (
s
12

< t
12
).


Viscoelastic materials


These materials include most plastics. There is a time
-
dependent
response.



Creep occurs under a steady load (an increase in strain at steady
stress).



Stress relaxation occurs after a deflection is imposed (a decrease in
stress at constant strain).



Experimental data from creep tests is needed. A typical material
model needs a time constant to describe the short time response; the
primary creep, plus elastic and damping modulii to describe the later
response or secondary creep.

strain

time

At constant
load:

Secondary creep

Primary
creep

Conduction heat transfer


Structural packages often provide the option of a conduction
heat transfer analysis. This can be used to provide a
temperature field, which is then used by the structural package
to predict stresses caused by prevention of thermal expansion.



Conduction is a mathematically similar problem to linear
elasticity. It is simpler, as the unknown temperature at a node is
a scalar, not a vector like displacement.



There are many analogies:


An imposed heat flow is like a distributed load.


Temperature gradients are like strains.


An insulating boundary with zero temperature gradient is like a free
surface with no normal stress (ie the default boundary condition).


A known temperature is like an imposed displacement.


Conductivity is analogous to Young’s modulus, but there is no
analogy to Poisson’s ratio.

Distinctive features of conduction analysis


If there is radiation from a surface, this is a highly nonlinear
boundary condition (


T
4
), requiring a specialised iterative
solver. Radiation from one surface of a model to another may
not be available in a structural package.



Transient response (response versus time) is characterised by
exponential change in temperature, not oscillation as in a
dynamic solid mechanics problem. The specific heat is an
analogy to damping.



Adaptive time
-
stepping is often used in transient conduction
problems, as a small timestep is needed initially, when
temperature is changing rapidly, but a larger timestep is
sufficient once the temperature is only changing slowly.

Structural optimization problems


Many codes offer some features to optimize structures (eg
ANSYS, COSMOS, NASTRAN).



3 types of structural optimization are commercially available, the
usual aim being to minimize weight.



Property optimization

-

properties of elements such as plate
thickness, or beam cross
-
section are varied.



Shape optimization

-

coordinates of groups of nodes are changed
to alter the shape of a model in a prescribed way. This is most
easily implemented in a FE package embedded in a CAD system,
as dimensions of the CAD model become the variables.



Topology optimization

-

elements are added or removed from a
model, depending on how stressed they are
-

appropriate for solid
models of machine components.

Design sensitivity analysis


A key component of an algorithm for optimization, is finding how
sensitive responses of interest (such as stresses) are to changes in
design variables (such as a plate thickness).



This task resembles solving a statics problem. Given we wish to
vary plate thickness
t
, we can differentiate the equations
[K]
u
=

F

wrt
t
, that is
(
d/dt

[K])
u

+ [K]
d
u
/
dt

= 0
or



[K]
d
u
/
dt =
-

(
d/dt

[K])
u




That is, we can take known current nodal displacements
u

and use
them to find how they vary with
t
by solving for
d
u
/
dt.
This then
implies how stresses vary with

t.



These estimates of how much stresses change guide the solver on
which of several thicknesses to change, and by how much.

Data for an optimization problem


Design variables

-

what properties of elements or what
dimensions of a geometric (CAD) model do you want to vary?
Computational effort increases rapidly if too many are used.



Responses
-

what outputs are of interest (stresses, natural
frequencies etc.). These are used to define the design objective
(eg minimum weight) and to define constraints.



Constraints

-

what limits do you want to set on responses such
as stress or deflection or natural frequency?

Related numerical methods in solid mechanics


The finite element method is not well adapted to boundaries at
infinity. The related
boundary element method

is useful for such
problems (eg stresses in rocks around tunnels). It discretizes the
surface with elements, but not the volume. It uses an analytical
solution to reduce the dimensionality of the problem.



The
finite strip method

seeks to save computation by using a FE
model in one direction and a Fourier series model in another.



Mesh
-
free methods

have been developed that use nodes but avoid
defining any permanent elements. Nodes within range of a particular
node are influenced by it, the influence decaying with distance
according to a weighting function associated with the node.



Particle methods

for granular solids model rigid bodies linked by
springs and dampers representing contact properties. These
connections are continually updated.

Summary


Finite element packages are useful tools, but you always need
to remain critical.



Numerical models need checking with hand calculations, or in
critical cases they need benchmarking against test results.



Packages are designed with solving particular problems in mind.
A new type of problem may violate assumptions made in the
software development.



Watch the limits of what can be done. For example, a package
that can model a cable may not be able to model a very long
cable. Beam models will never capture stresses at joints
accurately. Etc.