Unifying two fields of Computational Mechanics:
Solid and Fluid
Invited Lecture at International Symposium on
Science and Society
March 12-14, 2005; St Petersburg, Russia
Since the late 1960's, computer-based methods have been employed for simulating,
by way of discretization, the behaviour of continua; but techniques employed
for solids and for fluids have been very different.
The lecture suggests that the difference derives from historical accident,
and that, from the point of view of society, a single approach is to be
preferred, especially when solids and fluids interact significantly.
Examples of using the unified approach will be presented.
- Modern society relies on computer simulation
- Who are the people involved?
- Why is the schism bad?
- Historical review
- Classical mathematics
- Numerical methods
- The advent of digital computers
- Solid-stress analysis
- Fluid-flow analysis
- Scientific appraisal
- Essential similarities and differences
- Comparison of FE (solids) and FV (fluids)
- Obstacles to unification
- A successful unification
- Main features
- Concluding remarks
Computer simulation is today essential for the design of:
- Will they withstand earthquakes?
- Are the heating and air-conditioning adequate?
- Will the water-sprinkler system control spread of a fire?
- energy-producing and -consuming equipment
- Will gas-turbine blades survive their high temperatures and
stresses for the stipulated time between overhauls?
- environmental-protection measures
- Will a power-station furnace satisfy low-emissions requirements.
- Will nuclear-waste containers resist corrosion and thermal stress
for hundreds of years?
The use of less-than-the-best simulation techniques endangers society.
- The process-simulation calculations are made by engineers (civil,
mechanical, aeronautical, chemical, fire-safety).
- The software which they use, is often supplied by commercial
- embodies the relevant scientific laws, mathematically expressed:
- takes in the engineer's specification of the scenario;
- predicts what is likely to result
from their conjunction
- The software uses material properties, models of turbulence and
of chemical reactions supplied by physicists and chemists.
The community of software creators is split, polarised, schismatic.
Its two clans (FE and FV) use:
| for stresses in solids|| for flow of fluids|
| finite-element methods || finite-volume methods |
Many phenomena of great interest involve both solids and fluids;
for example an
launch or a
Computer-simulation techniques are therefore needed which will simulate
Britain is an island; so we are forced to pay attention to such
Waves pound on the shore; cliffs crumble; breakwaters provide refuge for
ships; and the extraction of energy from tidal motion is the aim of many
a British inventor.
St Petersburg owes its existence to the sea's proximity, and to
the vision of Peter the Great.
The Bronze Horseman
На берегу пустынных волн
Стоял он дум великих полн
И вдаль глядул....
... и думал он...
Здесь будет город заложен
The empty wave's deserted strand
Around him, in his mind a grand
Idea swelled: ....
... here should arise
A town to open Europe's eyes
But the continuation of Pushkin's famous poem tells us that St Petersburg's
fluid-solid interactions were not always benign.
Many other such interactions are hazardous:
Fires destroy buildings; earthquakes cause tsunamis;
volcanoes erupt when groundwater reaches hot magma;
loosened snow creates avalanches.
The foundations of solid and fluid mechanics were laid down by two
Robert Hooke and
Isaac Newton, who much disliked each other;
Newton never acknowledged that Hooke first stated the inverse-square
law of gravity.
Their ideas were later harmonised and internationalised. The harmony is
happily symbolised by the
family names of the two textbook authors of my youth:
Love, who wrote about Elasticity; and
Lamb, whose subject was Hydrodynamics.
The internationalisation is exemplified by the Petersburgers,
Euler and Bernoulli,
who studied also in France, Germany and Switzerland.
Another famous Petersburger who must be mentioned is
Stepan Prokofievich Timoshenko. His textbook,
The Theory of Elasticity, became a world-renowned classic.
All mentioned so far regarded Nature's laws as embodied in
differential equations to which they sought analytical
However, Timoshenko added to his second edition (1951), an appendix
on "finite-difference" equations and their numerical
solution by the "relaxation" technique, recently developed by my own
This replaced analysis by arithmetic.
Finite-difference equations were approximate forms of
differential equations; and their use had been pioneered for solid-stress
problems by Runge, in 1908, and in 1910 by
Richardson, who later applied it to
fluid-flow, ambitiously hoping thereby to
predict the weather.
More successful, for the simpler boundary-layer flow problem, was the
so-called "continuation method" of H Goertler in 1939, who had picked up a
suggestion made by
L Prandtl in 1904.
Numerical methods, executed by humans, were too expensive for widespread
use; but in the 1950s the digital computer changed everything.
Now, such methods could be used; and experience quickly revealed
- Many different sets of equations could be derived
from the same differential equations, some less approximate than others.
- Many different procedures could be devised for solving them;
and these differed in efficiency, and success.
- Which were best seemed often to
depend on the problem in question.
- Mathematicians provided no guidance as to where superiority
was to be found.
It was left to the engineers, who needed the solutions, to find the
best technique by trial-and-error. I was one of those engineers.
The appendix to the 1970 edition of Timoshenko's book contains a
2-page section on "solutions by digital computer" and concludes with:
"Similar methods .... are included in what is now known as the
The stress-analysis community had now a flag round which to rally.
The originator of the name appears to have been the American,
RW Clough who used it in 1956.
However, I believe that the method itself was invented, in 1953, by my former
colleague at Imperial College, London,
John Argyris, although his name for it was different:
the "matrix displacement method".
Interestingly, Argyris worked in 1944 with HL Cox, who appears to have
used a "finite-stringer method" for stresses in aircraft-wing panels in
1936! We all climb on the shoulders of our predecessors.
Argyris was in the Aeronautical Department, I in Mechanical Engineering;
so we knew nothing of each others' work. My own development of
computational techniques had a different origin.
My first (1951) research was on the burning of liquid fuels. Like
Kruzhilin for heat transfer, and
Eckert and Lieblein for mass transfer, I exploited themethod of
von Karman and Pohlhausen.
This involved choosing a polynomial form of velocity profile, and
expressing its coefficients in terms of weighted integrals of the
But even high-order polynomials could not well express the complex
shapes of profile which appeared in flames; so why not, I thought, use
the infinitely flexible step-wise approximation instead?
I had thus stumbled upon the finite-volume technique; and my student
Suhas Patankar expressed the idea in Fortran, later
drawing crucially on the independent 1965 work of
Francis Harlow, of Los Alamos.
The solid-stress and fluid-flow problems are similar in that:
The solid-stress and fluid-flow problems are different in that:
- The differential equations for displacements in solids and
velocities in fluids are almost identical.
- The equations for each of the three direction have terms in common
("dilatation" for solids; "pressure" for fluids); so they must be solved
- The equations for displacement are linked by Poisson's ratio, to
which the velocity equations have no counterpart.
- On the other, the equations for velocity contain first-order differential
coefficients, which those for displacement are spared.
- Turbulence in fluids (in effect) causes material properties to
vary much more greatly than in solids.
Both the finite-element and computational-fluid-dynamic communities:
They differ greatly in terminology; but essentially only in that:
A trivial difference? No! of such are
lasting schisms made!
- "discretize" continua; i.e. imagine them to be made of a finite
number of adjoining pieces;
- derive equations for the displacements or velocities of each of
those pieces by integrating weighted differential equations over each
- solve the equations by iterative error-reduction procedures.
One of the first FE specialists to believe that finite-element
techniques could and should be used for fluids was
Olek Zienkiewicz of Swansea University.
It can be indeed; but it is most successful for
low-Reynolds-number flows, because the
first-order differential coefficient cannot be handled by the
None of the many attempts to market FE-based codes for general fluid-flow
purposes appear to have succeeded.
Attempts to apply FV methods to solid-stress problems have been fewer;
and my own of 1991, failed to handle bending satisfactorily.
It can thus be said that the obstacles to unification have been:
- incomplete understanding of the other-side's physics; and
- readiness among engineers to believe that unification is neither
possible nor desirable.
This is my second attempt; and its basis is unchanged, namely:
- Use a single grid to cover both solid and fluid regions.
- In fluid regions solve for velocity and pressure as usual.
- In solid regions solve for displacement and dilatation.
- Treat fluid pressure and shear at interfaces as loads on the solid
- Treat Poisson's ratio linkages between differently-directed
displacements as loads on solid boundaries.
- Treat thermal-expansion effects similarly.
The new feature is:
- solve additional equations for the components of
- use rotation gradients as sources of displacement.
(a) What the 1991 formulation could do
Results from an early study are shown below for a two-solid-material
block, heated by radiation from above, and cooled by a stream of air:
(1) velocity vectors,
(2) displacement vectors,
computed at the same time, and
(3) horizontal-direction stresses,
obtained by post-processing.
(b) What the 1991 formulation could not do
That bending can now be satisfactorily handled is shown by the
(c) Curved surfaces with a cartesian grid
One of the early arguments for preferring FE methods was that they
handled curved boundaries more easily.
But FV techniqes serve just as well, even when the grid is
cartesian, as shown here for flow in a
In this case there are heat sources in the solids, giving rise to these
Because of the unified-computational-mechanics technique, we are able
to compute simultaneously the velocity and displacement
(e) Forces on an under-water structure
Finally, deflection is shown of a sea-bed structure, resulting from the
What I have attempted to show is that:
- a single computational technique can simulate the behaviour of
solids and fluids simultaneously;
- there will be advantages to society if the current schism in
the computational-mechanics community can be healed.
Because many careers and reputations depend on retaining the schism, the healing process
will not be swift; but I hope that it will not be
so slow as the equally desirably rapprochements of:
- Shias and Sunnis,
- Protestants and Catholics, and
- Orthodox and Old Believers.
Благодарю за внимание
The End !!!
His disappointment, once he had recognised that
the mathematical representation of turbulence was not yet
possible, he expressed in his best-known (and perhaps only) poetic
Bigger whirls have little whirls
That feed on their velocity;
And little whirls have smaller whirls,
And so on to viscosity.
This is of course a parody of an earlier rhyme
(by Jonathan Swift) which runs (more or less):
Bigger fleas have little fleas
Upon their backs to bite 'em;
And little fleas have smaller fleas,
And so ad infinitum.
I have found no picture, and therefore substitute this test for
students of English:
How should Clough be pronounced?
| Kluff|| as in enough||?|
| Kloo|| as in through||?|
| Klow|| as in bough||?|
| Kloff|| as in cough||?|
| Klo|| as in although||?|
| Klokh|| as in Lough Ness||?|
I do not know which is the correct answer.