Unifying two fields of Computational Mechanics:
Solid and Fluid


Brian Spalding

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.



  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

Computer simulation is today essential for the design of:

The use of less-than-the-best simulation techniques endangers society.

1.2 Who are the people involved?

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 

1.3 Why is the schism bad?

Many phenomena of great interest involve both solids and fluids; for example an under-water launch or a ski-jump.

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.


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.

2. Historical review

2.1 Classical mathematics

The foundations of solid and fluid mechanics were laid down by two contemporaneous Englishmen, Robert Hooke and Isaac Newton, who much disliked each other; for 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.


2.2 Numerical methods

All mentioned so far regarded Nature's laws as embodied in differential equations to which they sought analytical solutions.

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.

2.3 The advent of digital computers

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 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.

2.4 Solid-stress techniques

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 finite-element method."

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.

2.5 Fluid-flow techniques

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 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 solid-stress and fluid-flow problems are similar in that: The solid-stress and fluid-flow problems are different in that: next

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

Both the finite-element and computational-fluid-dynamic communities:
  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.
They differ greatly in terminology; but essentially only in that: A trivial difference? No! of such are lasting schisms made!

3.3 Obstacles to unification

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 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:

The new feature is:


4.2 Examples

(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 horizontal vectors here.

4.2 Examples (continued)

(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 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.

5. Conclusions

What I have attempted to show is that:
  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:

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 utterance:

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.