on
9 people importance of a physical equation
lies in its ability to predict the value of some unknown pocketing
unfortunately not all physical questions also simple
frequently we find situations where all the equation is known
we have no direct way of solving it a finite element method
is one technique for overcoming this problem
an engineer designing a bridge
we need to know how the proposed structure will behave under load
the equations describing the distribution of structural stresses
armaan but they can't be directly solved for a complicated shapes such as a
bridge
however the equations
can be solved for the simple shapes like triangles all rectangles
the finite element method takes advantage of this fact
we replace the single complicated shape within approximately
coolant network a simple elements in all capital element is referred to as
a finite element mesh and those papillion would be unique to each new
problem
the initial step is to design the smash
and for this we must first decide what kind of elements will use
one-dimensional
robs all two-dimensional
triangles all quadrilaterals
all three dimensional blocks are a combination of different kinds of
elements
the accuracy at the calculation is going to depend
company number up elements we choose to head in the mash the more elements we
have
the smaller each one will be and the more accurate the results
unfortunately mall elements
also means model calculations to be done a reason to the company that'll
we look for the happy medium just enough elements to get
adequate accuracy within a reasonable computing time in our example the bridge
we've chosen to use elements defined by eight points
these points are referred to as nerds in general
we regard each node has been capable of moving both horizontally
emphatically the exception look at those points around the outside edge that we
can regard as
immovable it fixed these boundary conditions must be included
to complete the description of the physical problem so that the solution
will be uniquely defined
finally with the specified the elastic properties of the material
and what limiting we wish to apply in this example
we've chosen a large point load at the center the bridge
the mathematical treatment
aims at arriving in a question describing the whole system
we start from the basic relationship expressing the displacement any node
as a function of the notes coordinates X and why
in this example
the elements have eight nodes and for each of these we write an equation
describing the displacement of the node
as a functional its coordinates this gives us
aid the questions for the whole government which together form
a matrix this matrix
is now the starting point for a series of steps based on the fundamental laws
of mechanics
step 1:
relates displacements to stresses from stress is we obtain
strain energy and from this we derive
potential energy and finally from minimum potential energy
we obtain appear out system equations for the complete
element by analogy with 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 of the displacements for the whole element we carry out this
presents for every element in the mesh
so that we now have and stiffness matrix for each one
the important step now is to combine
all these individual matrices into a single large matrix
representing the stiffness up the whole system now
any to neighboring elements will head nodes in common
so bad news for common notes will appear in both matrices
matrices can therefore be combined by a simple matching technique
in practice the process of solving the overall system equation
is done concurrently with the combining with 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 sold for first question and substitute a solution in the remaining
ones
repeating until finally we are ready to head on the matrix for the next element
eventually
when the last matrix has been added well left with a solution for
a single node we can now use this result rather like it
key working backwards through all the equations with the system
until at last the displacement every node is obtained
from these results we can quickly calculate the corresponding stresses
in this example we're showing the stresses as a compliment diagram
with the areas of greatest compression shown in red
and greatest tension in dark blue the process we've 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
the powerful computer with a large cost of is needed
a great advantage at the computer however
is that now we've competed the calculation for a look at one point
it's a very simple matter to repeat the whole thing with the move 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 behavior of the bridge as the low moves across it
this particular sequence was calculated using the pic finite elements computer
program
developed in the mechanical engineering department at Nottingham University
and was then going on to film using the antics graphic animation program
at the Atlas computer laboratory
many other uses it for at least a boundary have also been working on
applications that the finite element method
the civil engineering department at Southampton University is developing a
computer program to study the behavior of fluids in motion
when fully developed this program could find important practical applications
in the design of offshore structures such as North Sea oil rigs
in this example West 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 light the viscosity of the liquid and dimensions that the channel
are all specified the upper diagram shows the part of the fluid
the distribution of free streamlines the ladder diagram shows what a cold
stationary streamlined 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've made a change in the conditions equivalent to speeding up the
flow
the streamlined change until again the phone the steady-state
increasing the speed of light is still further however and a steady state is no
longer possible
go to seize begin to form eventually a dynamic equilibrium state is reached
finally one more example also from Southampton this program is designed to
study the dispersion of pollutants deposited in tidal waters
the finite element mesh here
corresponds to the east in silent extending to Portsmouth and the
thickness of the elements cosponsor the depth of the water
we also include the rise and fall of the tide
and insert a specified out for pollutants the computer program then
shows as
clearly with the pollutants will be carried straight out to sea whether
tidal effects will cause dangerous concentrations to build up near the
beaches
once again from this example we could now continue
the same program may be used for any other area of water
and any specified discharge pollutants we can investigate leading the way to
discharge
and trying to fit in with the moment at the tide we can take into account the
biological to kill the pollutant and we can also include the background level of
pollution from the rivers themselves
a major advantage at the finite element method
is the ability to work with arbitration apes visitor comes many of the
limitations uphold a new medical techniques
against the method of flexibility comparable to the use a physical models
but without the limitations for example
different materials can be investigated easily by specifying different material
properties as program data
properties that a difficult ability and physical models
we can simulate life-size structures on the computer
well as with the use of small scale models we're left with the scaling up
problems
engineering research today is becoming increasingly sophisticated
it often ended by using the largest and fastest computers
but we can help to solve the complicated systems of equations
representing a complex physical situation with movie
powerful computers it's also possible to conduct a comprehensive parameters such
using a finite element program for large numbers of different cases
in order to optimize the system but finite element method
can be applied to the widest possible variety of engineering problems
undoubtedly they can be said to lead the field in modern engineering computing
methods and
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