Finite Elements in SVG with Subtitles
F R A Hopgood
January 2021
With nothing to do while locked in during the Corona Virus pandemic,
a reasonably accurate rendition of the original Finite Elements film was created
plus subtitles of the soundtrack in about two months.
s01 start
The practical importance of a physical equation lies in its
ability to predict the value of some unknown quantity.
Unfortunately not all physical equations are so simple.
Frequently we find situations where, although the equation
is known, we have no direct way of solving it.
The Finite Element method is one technique
for overcoming this problem.
s02 start
An engineer designing a bridge will need to know how the
proposed structure will behave under load.
The equations describing the distribution of structural stresses
are known. but they can't be directly solved for a complicated
shape such as a bridge.
However the equations can be solved for a very simple shape
like triangles or rectangles.
s03 start
The Finite Element Method takes advantage of this fact. We replace the
single complicated shape with an approximately equivalent network
of simple elements.
The overall pattern of elements is referred to as the
Finite Element Mesh
and this pattern will be unique to each new problem
The initial step is to design this mesh and for this we must
first decide what kind of elements to use.
One dimensional rods, or two-dimensional triangles
or quadrilaterals.
s04 start
or three-dimensional blocks or a combination of different
kinds of elements.
The accuracy of the calculation is going to depend on the
number of elements we choose to have in the mesh.
The more elements we have the smaller each one will be and
the more accurate the results.
Unfortunately more elements also means more calculations to be
done. So for reasons of economy therefore we look
for the happy medium.
Just enough elements to give adequate accuracy within a
reasonable computing time.
s05 start
In our example of the bridge we have chosen to use elements
defined by 8 points. These points are referred to as nodes.
In general we regard each node as being capable of moving
both horizontally and vertically.
The exception will be those points around the outside edge that
we can regard as immovably fixed.
These boundary conditions must be included to complete the
description of the physical problem so that the solution will be
uniquely defined.
Finally we must specify the elastic properties of the material and
what loading we wish to apply.
In this example we have chosen a large point load at the centre
of the bridge.
The mathematical treatment aims at deriving an equation
describing the whole system.
s06 start
We start from the basic relationship expressing the
displacement of any node as a function of the node's
coordinates X and Y.
In this example, the elements have 8 nodes and for each of these
we write an equation describing the displacement of the node as a
function of its coordinates.
This gives us 8 equations for the whole element which together
form a matrix
s07 start
This matrix is now the starting point for a series of steps
based on the fundamental laws of mechanics.
Step 1 relates to the displacement of stresses. From stresses
we obtain strain energy and from this we derive potential energy
s08 start
and finally from minimum potential energy we obtain a pair of
system equations for the complete element.
By analogy of the equation of a simple spring this new matrix
is called the stiffness matrix for the element
Instead of a single displacement X the matrix operates on the
vector X whose components give the displacement
for the whole element
s09 start
We carry out this process for every element in the mesh
so that we now have a stiffness matrix for each one.
The important step now is to combine all these individual
matrices into a single large matrix representing the stiffness
of the whole system.
Now, any two neighbouring elements will have nodes in common,
so values for common nodes will appear in both matrices.
The matrices can therefore be combined by a simple
merging technique.
In practice, the process of solving the overall system equation
is done concurrently with the combining of the matrices.
The whole process is known as reduction.
We now use a standard procedure to eliminate part of the matrix.
The rows of the matrix represent a set of simultaneous equations.
We solve the first equation and substitute a solution in the
remaining ones. Repeating until finally we are ready to add
on the matrix for the next element.
Eventually, when the last matrix has been added, we are
left with the solution for a single node.
s10 start
We can now use this result rather like a key working backwards
through all the equations of the system until at last the
displacement of every node is obtained.
From these results we can quickly calculate the corresponding
stresses. In this example we are showing the stresses as a
contour diagram
with the areas of greatest compression shown in red and
greatest tension in dark blue.
The process we have been watching requires vast numbers of
individual calculations to be done and for this reason
can only be done on the computer.
Furthermore, the calculations usually involve such large matrices
that a powerful computer with a large core store is needed.
A great advantage of the computer however is that now we have
completed the calculation for the load at one point it is a very simple
matter to repeat the whole thing with the load moved to another point.
If we keep repeating the process with a sequence of different points
we can eventually build up a continuous animated sequence showing
the behaviour of the bridge as the load moves across it.
This particular sequence was calculated using the PAFEC Finite
Element computer program developed in the Mechanical
Engineering Department at Nottingham University
and then was drawn onto film using the ANTICS graphics
animation program at the Atlas Computer Laboratory.
Many other users at the Atlas Laboratory have also been
working on applications of the Finite Element method.
s11 start
The Civil Engineering Department at Southampton University
is developing a computer program to study the behaviour of
fluids in motion.
When fully developed this program could find important
practical applications in the design of off-shore structures
such as North Sea Oil Rigs.
In this example we are simulating a channel containing a
rectangular block obstruction. The finite element mesh is based on
triangular elements.
We introduce a steady flow of liquid into the channel. The speed of
flow, the viscosity of the liquid and the dimensions of the channel
are all specified.
The upper diagram shows the path of the fluid, the distribution
of free stream lines. The lower diagram shows what are called
stationary stream lines.
It contains the same information as the upper diagram except that the
uniform velocity of unobstructed steady flow has been subtracted in such
a way as to exaggerate and clarify the small variations we wish to study.
At this point we have made a change in the conditions equivalent
to speeding up the flow. The streamlines change until again they
form a steady state.
Increasing the speed of flow still further however and a steady state
is no longer possible. Vortices begin to form and eventually a
dynamic equilibrium state is reached.
s12 start
Finally, one more example, also from Southampton. This program is designed
to study the dispersal of pollutants deposited in tidal waters.
The finite element mesh described here corresponds to the Eastern Solent
extending to Portsmouth and the thickness of the elements
correspond to the depth of the water.
We also include the rise and fall of the tide and insert a
specified outfall of pollutants.
The computer program then shows us clearly whether the pollutants
will be carried straight out to sea or whether tidal effects will
cause dangerous concentrations to build up near the beaches.
Once again from this example we can now continue.
The same program may be used for any other area of water
and any specified discharge of pollutants. We can investigate
varying the rate of discharge and timing it to fit in with
the movement of the tide.
We can take into account the biological decay of the pollutants and
we can also include the background level of pollution from the
rivers themselves.
s13 start
A major advantage of the Finite Element method is the ability
to work with arbitrary shapes.
This overcomes many of the limitations of older numerical
techniques and gives the method a flexibility comparable to the
use of physical models but without their limitations.
For example, different materials can be investigated easily by
specifying different material properties as program data.
Properties that are difficult to vary in physical models. We can
simulate life size structures on the computer. Whereas with the
use of small scale models we are left with the scaling up problems.
Engineering research today is becoming increasingly sophisticated.
It is only by using the largest and fastest computers that we can hope
to solve the complicated systems of equations representing a complex
physical situation.
With modern powerful computers it also possible to conduct a
comprehensive parameter search using the Finite Element program
for large numbers of different cases in order to optimise the system.
The Finite Element method can be applied to the widest possible
variety of engineering problems. Undoubtedly it can be said to
lead the field in modern engineering computing methods.
s14 start
>
The End
produced by the
Atlas Computer
Laboratory
in association with the
Royal College
of Art
animation and music
by the
ANTICS
computer
on an
ICL 1906A computer
and
SD4020
microfilm recorder
designed by
Alan Kitching
executive producer
Jean Crow
acknowledgements
Department of
Mechanical Engineering
University of Nottingham
Dick Henshell & Dave Parkes
Department of
Civil Engineering
University of Southampton
Carlos Brebbia Stuart Smith
Bob Adley
Peter Hadingham
Swift Film Productions
© Atlas
Computer
Laboratory
1975
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