Abstract

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.

next

Contents

1. Introduction
   1. Modern society relies on computer simulation
   2. Who are the people involved?
   3. Why is the schism bad?
2. Historical review
   1. Classical mathematics
   2. Numerical methods
   3. The advent of digital computers
   4. Solid-stress analysis
   5. Fluid-flow analysis
3. Scientific appraisal
   1. Essential similarities and differences
   2. Comparison of FE (solids) and FV (fluids) techniques
   3. Obstacles to unification
4. A successful unification
   1. Main features
   2. Examples
5. Concluding remarks

1. Introduction

1.1 Modern society relies on computer simulation

• buildings
  • 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.
next

1.2 Who are the people involved?

• The process-simulation calculations are made by engineers (civil, mechanical, aeronautical, chemical, fire-safety).
• The software which they use, is often supplied by commercial companies; it
  • 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 representations of chemical reactions supplied by physicists and chemists.

The community of software creators is split, polarised, schismatic. Its two clans (FE and FV) use:

1.3 Why is the schism bad?

Computer-simulation techniques are therefore needed which will simulate fluid-solid interactions.

Britain is an island; so we are forced to pay attention to such interactions:
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.

next

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.
next

2. Historical review

2.1 Classical mathematics

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.

next

2.2 Numerical methods

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 Oxford professor, Richard Southwell. 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.
next

2.3 The advent of digital computers

Now, such methods could be used; and experience quickly revealed that:

1. Many different sets of equations could be derived from the same differential equations, some less approximate than others.
2. Many different procedures could be devised for solving them; and these differed in efficiency, and success.
3. Which were best seemed often to depend on the problem in question.
4. 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.
next

2.4 Solid-stress techniques

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.
next

2.5 Fluid-flow techniques

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 differential equation.

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. next

3. Scientific appraisal

3.1 Essential similarities and differences

• 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 simultaneously.

• 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.

3.2 Comparison of FE (solids) and FV (fluids) techniques

1. "discretize" continua; i.e. imagine them to be made of a finite number of adjoining pieces;
2. derive equations for the displacements or velocities of each of those pieces by integrating weighted differential equations over each piece.
3. solve the equations by iterative error-reduction procedures.

• FE specialists use a variety of "weighting functions", but most often that proposed long before computers existed by Galerkin; whereas

• FV specialists use as "weighting function".... 1.0 !!

3.3 Obstacles to unification

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 Galerkin weighting.

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:

1. incomplete understanding of the other-side's physics; and
2. readiness among engineers to believe that unification is neither possible nor desirable.

4. A successful unification

4.1 Main features

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 boundaries.
• 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 rotation.
• use rotation gradients as sources of displacement.

4.2 Examples

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 horizontal vectors here.
next

4.2 Examples (continued)

But FV techniqes serve just as well, even when the grid is cartesian, as shown here for flow in a turn-around duct.

In this case there are heat sources in the solids, giving rise to these thermal-expansion contours.

Because of the unified-computational-mechanics technique, we are able to compute simultaneously the velocity and displacement vectors.

(e) Forces on an under-water structure Finally, deflection is shown of a sea-bed structure, resulting from the action of surface waves.
next

5. Conclusions

1. a single computational technique can simulate the behaviour of solids and fluids simultaneously;
2. 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.

This is of course a parody of an earlier rhyme (by Jonathan Swift) which runs (more or less):

How should Clough be pronounced?

I do not know which is the correct answer.