ModelMaker is designed as a model building package for solving differential equations numerically. At the heart of the package is a numerical integrator (called a �compartment�) around which the user builds a model of a system of interest. The compartments (there can be a number of them) interact with other components of the model system, such as �parameters�, �variables�, �defined values� and �events�, by means of �flows� and �influences�, which determine how each component interacts with the others. Flows and influences have a direction associated with them so that the result of a calculation in one component can be available to a second component, but the result of the second component may not be available to the first. Parameters and variables can be defined as �global� which means they are available to all components, but this fact is not obvious from a casual inspection of the screen alone. It is also possible to include �delays� and �lookups� into the model.
Figure 1. Model diagram produced by ModelMaker. Square boxes represent compartments (integrators), rounded boxes represent variables (no integration) and angular boxes represent defined values. The model in this case was a set of six particles moving in one dimension and coupled together by harmonic forces, (force constant k). Each particle has a position, h, a velocity, v, and an acceleration, a. In addition each particle is subject to a damping force proportional to velocity, with constant d. The problem is subject to periodic boundary conditions (particle 1 sees particle 6, and vice versa) and so is a 1-dimensional representation of a benzene ring. Particle 1 is subject to a sinusoidal oscillation, h1, whose frequency, F, increases linearly with time. The problem is to ascertain the motion of the remaining 5 particles.
In the main window is a diagram of the problem currently being solved which shows all the components (with different shaped boxes to signify the different component types) and arrows to signify the flows and influences that are defined. The direction of the arrow indicates which way the influence goes (see Figure 1). If there is a logical error in the way the model has been set up then the offending boxes remain shaded until the error is corrected. This is very helpful in ensuring that the model is internally consistent, but of course it does not ensure that the model represents what you want it to. The model will not �run� until all the errors are corrected, but once this is done and all initial values are defined the model can be run by simply pressing �run�. When the run is complete the results can be plotted using a limited graphics routine, from which hardcopy can be made if needed (see Figure 2). The length of the run varies considerably depending on the complexity of the problem being solved.
Installation on a PC system was quick and straightforward. The programme comes with a detailed manual which contains full descriptions of all the facilities available in the package. However the introductory part of the manual uses the example of a lake system to develop familiarity with the methods and terminology used. To go through this example in full would require a considerable amount of reading and testing and is not really suitable for someone in a hurry to set up a particular problem. The same example can be loaded as a tutorial item from disk. The description of the integrator as a �compartment� is fairly obscure to the novice - why not simply call it an �integrator�?. Other descriptions are perhaps not so obscure but there is certainly a case to try and standardize the use of terminology in the package to make the function of the different facilities more immediately obvious to the novice. In practice this did not appear to be a serious obstacle since the terminology became familiar enough once the function of each component was understood.
Figure 2 Graphic output from ModelMaker, using the model described in Figure 1. The bottom line shows the driving oscillation, h1, with a small amplitude of motion and a frequency which increases steadily with time. The other lines show the response of the remaining 5 particles. At a certain driving frequency (the first resonance) all the particles move in phase with an amplitude much larger than the driving amplitude - the system is absorbing a large amount of energy at this frequency. At another, higher frequency, again resonance is observed, but this time only half the particles have large amplitude motions and they are out of phase with one another. Thus the phenomena of resonance and normal motion are demonstrated by this simple technique.
The overall time taken to set up the problem shown in figures 1,2 plus a separate problem of integrating a complicated function, and to become fully familiar with running both problems, was about 10 hours for someone who had no prior experience of ModelMaker nor of other integration packages. For someone familiar with the package the time would be much less, (e.g. � hour), but for any particular problem the time consuming element will likely be the initial breakdown of the problem into its ModelMaker component parts. This may not be totally trivial when it is realised that each integrator can only solve an equation of the form:
dv/dt= (algebraic expression)
Therefore if the package were to be used to solve a complex problem the user
will need to have a good grasp of differential equations. It will probably
also help if he or she knows something about solving differential equations
numerically. This fact is not brought out in the manual, which is written as
if the user would need to have only a superficial understanding of
differentiation and integration. There is possibly a case for a �front end�
package to break down a complex problem into its ModelMaker components, but
this might be awkward to implement in practice.
The problem shown in Figure 1 contains a number of compartments which have identical algebraic expressions - only the variable names change. If the problem were to be extended to say double the number of particles it would be necessary to laboriously define each component one by one. There is no facility within the package to reproduce a given component several or many times with different variable names, in a manner that is possible in say a FORTRAN program where the same operation can be carried out on a large number of elements. The graphics routine is versatile in allowing the user to display any quantity in the model against another over any range of the variables, but has little scope for labelling the graph, altering the character size and changing line thicknesses. To do this the user would have to unload the results into a file and plot them off-line.
The availability of a reliable and accurate integrator on a desktop computer can be a powerful tool for developing models or parts of models, even if at the end of the day the full model is solved in the conventional way using computer programming. I would have the thought the applications of ModelMaker to the kind of problems encountered by crystallographers would be fairly limited, given the size of a typical crystallography dataset, but as a general learning tool about numerical integration and as means of developing solutions to particular problems it could be very powerful.
Alan K. Soper
27 September 1995
All calculated values from tables and graphs can be transferred to other Windows applications for further analysis, graphing etc. using the Windows clipboard. This is very much more straightforward than writing values to a file as suggested in the review
Global components are indicated in the diagram by green cross hatching which makes them more obvious than suggested in the review.
Components can be reproduced several times by using copy and paste
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