INTRODUCTION
Although a well-kept secret, finite element analysis is yet another productive Mac^[1] engineering application. Finite element methods, like computer systems, moved onto design engineering desktops from the realm of large mainframes and corporate MIS staff. Evolution from last desperate resort to engineering tool arose from productivity gains made possible by low cost, non-intrusive desktop computer systems. Apparently, real productivity gains are not associated with super-computer processing speeds, but with creative autonomous use of personal computers with relatively modest CPU speeds. Lightning fast number-crunching, although useful, affects productivity less than ability to organize and phrase problems and communicate significance within the pace of the product design cycle.

In the engineering world, finite element analysis is only useful in a design engineering context. Computational elegance usually takes a back seat to productivity, cost, schedule and other design imperatives. The Mac contribution is an integrated work environment and a convenient non-intrusive operating system. The following discussion illustrates the design engineering process with several individual analysis cases.

ABOUT FINITE ELEMENT ANALYSIS
Finite element analysis and its cousins, finite difference and boundary element methods consider physical systems, however complicated, as a composite of the individual responses of simple elements. Ultimately the composite behavior is characterized by a large system of simultaneous equations, solved by matrix methods. Manual methods for such analysis predate the U.S. Civil War, but general engineering use had to wait for FORTRAN and the digital computer in the late 50's. Commercial software evolved during the next 10--15 years into engineering tools for formulating and solving general problems in solid and fluid mechanics.

Analysis begins by modeling--defining an element mesh with geometry defined by node points. The engineer chooses boundary conditions and specifies element properties and loading, based on assumed physical behavior. Solution involves assembly of the individual elements into the structural stiffness matrix. The stiffness matrix is inverted and multiplied into the applied load vector to provide the displacement vector. Displacements go back into the individual element stiffnesses to provide stress and loading. At one time the model was formulated one node and one element at a time, but all modern software uses either stand-alone or integral modellers to create individual elements from geometric entities. Likewise graphic and tabular post-processors extract and summarize relevant results from a massive amount of numerical results. Engineering assessment usually requires additional arithmetic beyond that provided by the finite element software itself.

WHY MAC
Conventional wisdom sees finite element analysis strictly as number crunching, but finite element analysis is a mixed bag. Graphic modelling is interactive and paced by the user's engineering skill. Interpretation combines graphic displays and arithmetic processing of selected results, typically for comparison to design code requirements or experimental data. Documentation involves graphic and tabular presentation of the small fraction of the total output which is actually pertinent. Only matrix operations and the more esoteric graphic presentations involve heavy number crunching. Processor speed certainly matters, but a solution crunched in an hour may require days or sometimes weeks to model and interpret. Consequently labor savings in these areas impact productivity quite heavily.

The real power of the Mac is in ease of information flow between software packages. Finite element model diagrams saved as PICT files from analysis software or terminal emulation may have modelling assumptions and terminology applied (and clutter removed) in a drawing program. Pasted into the final report, the diagram documents assumptions concisely and clearly. Graphic results can be sifted and clarified in the same way to amplify important points in a form clearly understood by the laity--a significant obstacle to the use of finite element techniques.

Likewise, tabular bulk data moves easily through terminal emulators, spreadsheets, and charting software into a word processor to provide summary results justifying conclusions. The path might also detour into mathematical analysis software if comparative analysis such as a design-of-experiments approach were used. Specialized input such as a load history can also can be assembled with a spreadsheet or from a custom program, graphed, formatted and used with finite element software.

DESIGN EXAMPLES
Figures 1--4 show typical finite element models for determining stress, deflection and loading. The tables show particular engineering tasks required before and after numeric solution to make the results meaningful. The truss and dished cover in Figures 1 and 2 can be solved manually by an engineer with a day or so to waste, but finite element solution only takes an hour or two from boot-up to answers, depending on the user's skill and the design issues. The pressure bulkhead and forming tool are straightforward, but well past any manual analysis other than hip shot level.

Crane Truss
Conventional wisdom has it that engineering results spring full blown from graphic output--the stress plot is all. In fact, interpretation is usually numeric, comparative and always properly the job of the design engineer. The crane truss in Figure 1 requires both stability and strength assessment using allowable stresses and an interaction formula with steel construction code design rules. Stability analysis in particular is not done by finite element software because the detail involved is not found in the finite element model, and the mandated methodology is not as simple as matrix inversion. Connection design uses calculated member forces with design formulas appropriate to the type of connection and the materials involved.

Dished Cover
The dished cover in Figure 2 is a typical pressure vessel component made from a spherical shell welded to an edge ring. The figure shows a sectional view with the computed stress distribution. The model is easily solved, but the engineering assessment is more difficult. The high stresses at the joint between the shell and the ring require classification for comparison to allowable values relating to collapse, acceptable distortion and fatigue. This classification involves determination of the linearly varying stress components along the stress classification line. One approach uses a straight line and a higher order polynomial fit to the stress variation. The linear fit determines the margin against plastic collapse; the high order fit determines the margin against fatigue. This approach is typical of nuclear equipment and similar critical applications and is set forth in the ASME pressure vessel design codes.
Another important aspect of the design is the bolting and gasket loading. Bolting provides enough preload to prevent leakage as the pressure is applied, but not so much to crush the seal. The designer uses the loading and deformation output with fastener and seal design criteria to specify the gasket size and material and the bolt size and torque to obtain the preload. The arithmetic is again too specialized to be incorporated into general purpose finite element software, but quite well suited for a spreadsheet.

Pressure Bulkhead
Figure 3 shows stress contours in a portion of a flat plate used to separate chambers in a pressure vessel. The bulkhead must meet ASME Code standards for strength and must also be evaluated for seal integrity. The attachment weld between the face bar and the plate is assessed for strength using computed forces between the adjoining elements. The weld is not modeled explicitly, because its shape is indeterminate. Moreover existing engineering criteria for weld stress include margins for built-in stress, fatigue at permissible discontinuities and other effects not amenable to any realistic analysis.
The distortion of the face bar determines seal effectiveness. Rigid body movement has no effect, but the runout (elastic deformation out of flat) can cause gaps leading to leakage. The calculated overall deformations (including both rigid body and elastic components) are easily plotted and analyzed using spreadsheet software.

Composite Forming Tool
The forming tool shown in Figure 4 is used for spacecraft composite lay-up. The tool is subject to finite element analysis to determine tool distortion under its own weight, to insure that the required contour is produced. This particular tool was designed, manufactured and analyzed without drawings. The tool was designed on an MS-DOS workstation using an imported data base to define the desired curvature. The design geometry was transferred to a Mac for analysis using an IGES file. Displacements calculated by finite element analysis were converted to surface distortions using spreadsheet software.
The tasks noted in all the examples generally apply to the engineering assessment, not the usual communications from designer to analyst. Communications include sketching, documentation and perhaps faxing and formal submittal. Such tasks are also easier in the Mac environment, again because the flow of information between applications is invariably quick and uncomplicated.

FINITE ELEMENT SOFTWARE
Good finite element software is available for the Mac itself or software resident on a host system can be used. Resident software is expensive, but it provides seamless access. Remote access to a host frees up a desktop machine for other work and requires lower initial outlay. Difficult or costly on-line access makes for tough learning curves and lower comfort levels, both productivity dampers.

Mac-based finite element codes range from shareware to large scale programs to run a full range of static, thermal and dynamic and non-linear analysis. General purpose software includes a solid modeller, post-processing and graphic display modules, and an element library including beams, plates, solids and special purpose elements to simulate gaps, coulomb friction and local damping. Other useful features include translators for importing and exporting CAD models or to produce input to other finite element software. Mass and section property calculation, and kinematic analysis, fatigue and fracture calculations may also be available.

Limited capability programs and working demonstration versions of general purpose codes give new users experience without mortgaging the farm. Typical demo versions handle only small problems, but permit full use of the modeller and post-processing features and graphic capabilities. Small benchmark problems will check program accuracy and usability, and a potential user may also attempt deliberate crashes (e.g. with a distorted mesh) to test program stability. in fact, buying expensive general finite element software for any platform without a good solid wring-out is somewhat reckless.

Besides meeting the user's engineering requirements, Mac finite element software should have all the regular Macintosh interface features and disk file output. The realities of finite element analysis may dictate some departures from usual Mac practice, but the more Mac-like the interface, the lesser the likelihood that the program reflects the dreaded DOS-port syndrome. PICT file graphic output is much preferable to the FKEY screen shot which the uninitiated may consider equivalent. Text file command stream input and the ability to direct post-processor output to disk are indispensable.

DISTRIBUTED PROCESSING
As an alternative to resident software, all or part of an analysis may be handed off to a host machine, using the Mac as a terminal emulator. Main frame finite element software provides Tektronix or DEC graphic output for viewing and capture by Mac resident communication/emulation software. In one scenario, the Mac becomes a client to modeller and solver on a large server. Graphic displays and tabular results generated by the server are retrieved with a terminal emulation software and processed for engineering use by the client Mac. Another approach uses geometry in a neutral format (IGES or DXF for example) imported from a CAD work station to a Mac solid modeller for model assembly. The finite element model could either be run on the Mac or a host machine.

Product guides such as `Apple Engineering/Scientific Solutions Guide,' are good starting points for finding finite element software. Descriptions are brief and often optimistic, but shopping is fairly easy. The best resource is user comment, via on-line services or the Internet. Users tend to complain about bugs rather than applaud features which work as promised, so user comment is a useful counterbalance to sales reps who often know little about practical engineering.

CONCLUSION

• There is no question that the Mac offers major engineering productivity advantages because of a tight operating system and well-integrated applications. The cross-disciplinary nature of finite element analysis requires that information of all sorts flow smoothly between many applications.

• From a user standpoint, the future of engineering finite element software lies in improved modelling and post processing. Everyone wants faster solvers, but modeling real-world problems and the ability to communicate significance remain the chief productivity barriers. The ideal package might incorporate a single high powered solver engine run from modular pre-and post processors addressing a particular engineering discipline. For example piping networks and building structures are commonly analyzed as frameworks, but the engineering design and safety Code issues are quite different. Apple Computer's OpenDoc technology should make it possible for one developer to offer a basic modeller and solver and others to provide specialized 'driver' software addressing a host of engineering design specialties that would be unsupportable by a single developer.

• The dark side seems to be developer attitude. Engineering is a highly vertical market, and narrow perspectives abound among developers. Macs are sometimes seen as an anomaly. User support and service for engineering finite element software are difficult at best, regardless of platform, and a relatively low user base doesn't help matters. The blackest of all is the practice of cobbling Mac software from existing code, ignoring inherent Mac advantages while incorporating all the disadvantages of the original machinery. Such software is likely to be poorly supported, difficult to use and a good prospect for orphan-hood

SIMPLE CRANE TRUSS

Section property calculation                   HyperCard; spreadsheet            
Weight and mass property verification          Spreadsheet                       

AISC/CMAA stress and stability assessment      Spreadsheet                       
Connection design and stress computation       HyperCard; spreadsheet            
Reaction load statics check                    Spreadsheet                       

DISHED COVER FOR `QUICK ACTING' CLOSURE

CAD file input from designer; export to        Apple file exchange; file         
designer                                       transfer                          

ASME Code primary, secondary and peak stress   Spreadsheet; charting; custom     
classification and assessment                  program                           
Bolting design--preload and fatigue            Spreadsheet; custom program       
Seal design--gasket load separation at seal    Spreadsheet                       

FLATE PLATE PRESSURE BULKHEAD WITH OPENING

MODELING

CAD file input                                 Apple file exchange; file         
                                               transfer                          
Face bar section properties                    HyperCard; spreadsheet            
Weight for model verification                  Spreadsheet                       

ASME Code primary, secondary and peak stress   Spreadsheet; charting; custom     
classification and assessment                  program                           
Face bar weld joint loading and design         Spreadsheet; charting             
Seal design--face bar distortion               Spreadsheet; charting             

COMPOSITE FORMING TOOL¹

CAD file input                                 Apple file exchange; file         
                                               transfer                          
Stiffener section properties                   Spreadsheet; HyperCard            

Gross surface displacement (includes rigid     Spreadsheet; charting             
body movement)                                                                   
Local departure from shape (effect of          Charting                          
elastic distortion on molded shape)                                              
Weight breakdown                               Spreadsheet; charting