Improving productivity in finite element analysis through interactive processing

Finite Elements in Analysis and Design i (1985) 35-48 North-Holland

35

IMPROVING PRODUCTIVITY IN FINITE ELEMENT ANALYSIS THROUGH INTERACTIVE PROCESSING W.S. WOODWARD Advanced Energy Systems Division, Westinghouse Electric Corporation. Madison, PA 15663, U.S.A.

J.W. M O R R I S Research and Development Center, Westinghouse Electric Corporation, Pittsburgh, PA 15235, U.S.,,L Received May 1984 Revised September 1984 Abstract. "User friendly" interactive pre- and post-processing programs allow major increases in productivity in finite element analysis. These programs are designed to reduce the amount of effort required to develop input and summarize output for large finite element programs. The gains in productivity, however, are nc,t without cost. The use of this technology requires additional computer resources and user training. Although the increases in productivity are obvious to those who use this t~hnology, management decisions are based on economic considerations. Therefore, engineering management must understand the benefits and the value of the increases in productivity to effectively evaluate the use of these tools. This paper discusses the concepts which form the basis for improving productivity through the use of interactive pre- and post-processing. Examples, taken from the authors" experience, are presented which illustrate the improvements in productivity which can be obtained using this technology.

Introduction

Since the development of 'general purpose' finite element computer programs in the 1960's, the finite element method of analysis has become a major tool of the engineering practitioner. The parallel development of large mainframe computers has provided the engineer with the capability of solving extremely complex problems. Regardless of the program or the problem being solved, all finite element analyses are performed in the following three basic steps: (1) The data which are required to define geometry, loads, material properties, and boundary conditions are assembled and put into a form which can be interpreted by the computer~ (2) The finite element computer program is executed to generate a solution. (3) The results produced by the computer program are compiled in a form which is meaningful to ~he engineer. Steps (1) and (3) are referred to as "pre-' and "post-processing' respectively. For the first finite element programs these steps were performed manually and it was soon recognized that they are very labor intensive. A moderate sized problem required the definition of hundreds of elements and the inp~,t of thousands of node coordinates to define the geometry. The engineer had to first write out all data in code format and then have it manually key-punched onto cards which could then be read by the computer. Early general purpose programs employed 'pattern generation' schemes that enabled the user to ~-eplicate nodes and elements, which greatly reduced the effort req~,ired to construct a model. Pre-programmed tables of material properties were often included and special purpose mesh generator programs were written to serve an individual's or industry's need for model construction. In order to facilitate the checking of 0168-874X/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

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W.S. Woodward, J. W. Morris / Improving productivity in FE analysis

input data, graphical display of the model geometry was included as a feature of general purpose programs. The ability to plot results was also built into most general purpose programs, which greatly reduced the effort of post-processing. Even with these aids, the construction of a finite element model was a very time-consuming and error-prone task. The analysis of large models was mostly confined to the aerospace, defense, and nuclear industries, where accurate solutions were required and cost was a minor constraint. In the mid 1970's, the concept of 'general purpose' mesh generators emerged. These programs were distinctly different from previous programs in that they executed 'interactively'. The earliest versions employed 'drag-mesh' schemes where a portion of a model could be replicated along a user specified series of lines or curves. Additional developments have given the user a wide variety of construction techniques. Today, these programs have been extended to generate all the required input data for the finite element program, including the material properties, boundary conditions, and loads. They also have been extended to interactively reduce results. Recently pre- and post-processing programs such as PATRAN-G, MENTAT, GIFTS, FIGURES-II, and ANSYS PREP-7 have become available in the software marketplace. The availability of these programs, combined with the advent of the minicomputer and timesharing on large mainframe computers, has given users in all industries the capability of constructing and solving large finite element problems. A practice of using a hierarchical structure of computers to perform finite element analyses has emerged. The pre- and postprocessing functions are performed interactively on minicomputers and the finite element solution is obtained using a larger mainframe computer. The improvement in productivity which can be achieved through interactive pre- and post-processing is the subject of this paper. The use of this technology is not without cost. It requires additional computer time and hardware. User training is also required. Although most engineers who have used these new tools may consider the benefits obvious, engineering managers need comparative data relating the cost of the technology to the savings produced by it in order to assess its value. The purpose of this paper is to discuss how productivity can be increased using interactive pre- and post-processing, and to present examples which demonstrate the costs and benefits of this technology.

Finite element computer aided engineering (CAE) technology Interactive pre- and post-processing programs are specifically designed to reduce the effort and time required to prepare input and summarize output for general purpose finite element programs. They also minimize the possibility of error and allow the engineer to build more complex models with more optimal element distributions. Using interactive pre- and postprocessing programs, the engineer can quickly generate and verify node locations and element specifications which define a finite element model, prepare load and boundary conditions, and visually verify all specifications of material types for the model. Many programs can now begin with the geometry of models previously generated on a computer aided drafting system. Model outlines can then be used to generate the finite element nodes and elements. In other cases the engineer generates the geometry of the model using the pre-processing program. After obtaining the finite element solution, the engineer can visually check the resulting displacements, temperatures, or stress and strain components superimposed on the model.

Methods of model generation The construction of a single complex planar or 3-D finite element model usually requires a variety of mesh generation techniques. A single pre-processor must offer many generation options to the user in order to be sufficiently versatile in mesh generation. Several mesh

W.S. Woodward,J. W. Morris / Improvingproductivity in ,rE analysis

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generation techniques which are available in commercial programs are as follows: (1) Single node and element generation: This technique is a tedious procedure, however, it can be very effective on models with a few higher order elements. (2) Digitizing input: Many users have scaled drawings of a model. Outlines of the model or nodes and elements can be transferred directly into the program using an electronic digitizit~g tablet. (3) Pattern generation: Often an entire model can be generated by simply repeating a portion of the model. (4) Duplication: Portions of an existing model or the entire model can be translated a n d / o r rotated a n d / o r mirrored a n d / o r scaled to generate another portion of the model. (5) Region generation: This technique generates nodes and elements within a region bounded by previously defined lines.

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W.S. Woodward, J. I~: Morris / Improving productivity in FE analysis

(6) Dragging generation: This mesh generation technique is used to extrude a fixed cross section of elements. Dragging a line creates 2-D or shell elements. Dragging 2-D elements creates 3-D elements of constant cross section. (7) l~fting generation: This mesh gene~-ation technique is used for generating meshes of non-fixed ero~s section. Lofting a line through a series of cross sectional lines generates shell elem~ats. Lof~i~,g a set of 2-D elements through lines defining the outline of the mesh at various cro.~s sect.ions generates a smooth mesh of 3-D elements. S~rae of ghese methods of mesh generation are illustrated in Fig. 1. Recent developments in automatic mesh generation algorithms include blending functions [1-4], coordinate transformations [5-10], automatic triangularization [11-13] and quadtree and octree encoding [14]. Each of these techniques is designed for more automatic generation of finite element meshes using just the geometric description of the model. When there is a need to analyze many geometrically similar finite element models, interactive mesh generation commands can be combined in a user written program based on a generic model which contains all the necessary model features. Dimensions in the generic model are interactively specified by the user and the program automatically generates an e.tire finite element model. Using this method, a user can generate finite element models in minutes rather than hours. Fig. 2(a) illustrates a generic model of a electronic chip solder joint. By specifying the various dimensions shown in the figure the user con generate many different solder joint configurations. One such configuration is illustrated in Fig. 2(b). Other methods of finite element model generation involve 'adaptive' mesh generation techniques. Adaptive meshing pertains to methods which automatically improve the location of the finite element node,s and/or improve the shape of the elements once the finite elemem model is generated. These techniques ut.ilize criteria based on the minimum potential energy or strain energy density which are d..~rived from an existing fin:~te dement solution. Self-adaptive processors iteratively perform finite element analyses until the desired level of accuracy is obtair,ed. A re.cent study describes several adaptive meshing techniques [15i.

Methods of loads and bow/dary co,~dition specification Manual generation of loads and boundary conditions for finite element analy~:s can be a very tedious and time-consuming task. The engineer normally specifies the nodal point index

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W.S. Woodward.J. W. Morris / Improvingproductivity in FE analysis

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and the value of each force component for each nodal load. Element loads are specified by element name, face name, and load type. The following two techniques are used by interactive pre-processors to reduce the amount of effort required to specify load and boundary conditions: (1) Menuing: Node couples, node loads, element loads, and node constraints can be generated by specifying the load or constraint type, the value, and the appropriate node or element from the plotted model. (2) Location and association: Nodal points and elements have coordinates associated with them. This location can be used as a qualifier to determine if the node or element lies (i) on an arbitrary plane, (ii) within a specified range of the global cartesian or a local coordinate system, or (iii) on a specified surface. The node or element can then be loaded accordingly. The nodal point and element can also be associated with the specific region which generated the node or elements. Of these two techniques, the latter is by far the most powerful. With location and association, the user can apply loads and constraints to many nodes or elements with a single interactive command. Methods of verification

One of the most important features of an interactive pre-processing program is the interactive plotting capability. The user can readily find errors or omissions in the nodes, elements, or boundary conditions of the model. Other modeling parameters such as material types and real constant types (thickness specificatk i) can also be quickly verified with the selective plotting capability. Fig. 3 illustrates a sample graphical display of a simple lug which is constrained at the base and loaded with a radial point load applied to the interior of the hole. The elements are reduced in size for visual inspection of the model. Constraint and load symbols allow easy verification of the boundary conditions. Verification of the model is quick and more reliable than manual checking of input data. Methods of post-processing

Nearly all finite element programs generate large volumes of output. Each nodal point has one te six resulting displacement values. Each element can have from one to over 320 resulting values of stress and strain. For a large model with several loading steps the amount of output can be overwhelming. An interactive post-processing program can quickly summarize the output in one or more plots of the results superimposed on the model. Fig. 4 illustrates a post-processing plot for the lug problem shown in Fig. 3. The engineer can quickly see areas of high stresses and accept the design or make a decision to redesign the component. If the

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W.S. Woodward, J. IV. Morris / Improving productivity in FE analysis

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Interactive vs. batch computing

The key to reducing the calendar time required to perform a finite element analysis is working 'interactively'. Interactive pre- and post-processing reduces calendar time because the interactive mode of computing greatly streamlines the process of obtaining valid engineering solutions from finite element programs. Batch mode computing contains steps which directly affect the productivity of the engineer as well as increase the calendar time required to obtain solutions, Interactive computing provides a more direct path for information exchange, which eliminates many of the time-consuming steps associated with batch processing and thus reduces total labor requirements. Bonine [16] illustrates the batch process in terms of 'engineering turnaround time' and the 'apparent response time' of a computer system. Engineering turnaround time is the total time required to obtain a valid solution and the apparent response time is the time required to process a single run through the computer. These concepts are illustrated in Figs. 5 and 6 (taken from Bonine [16]). The inefficiencies attributed to batch processing are primarily due to the required steps which determine the apparent response time. Formatting data, travel to and from the computer facility, and the batch and local queue steps can all be eliminated using interactive processing. Formatting data is very tedious and time-consuming, and it requires no engineering judgment. It does require extreme concentration for extended periods of time if errors are to be minimized. With interactive processing the data entry step remains and it is performed by the engineer or technician who is responding to the interactive program. The format and keypunch steps are combined into one data entry step. The tedium is reduced and the process is greatly accelerated through interactive prompting and free format input. Batch job queue time and local queue time are eliminated because the input data is submitted to the computer electronically. If the solution is obtained on a remote computer, then the batch processing time is real. Finally, interactive post-processing aids the engineer in the interpretation of results by providing graphic display of results in real time. The calendar time associated with output queues and travel to and from the computer terminal is eliminated. Interactive processing also decreases the engineering turnaround time by eliminating the multiple apparent response times usually realized in the batch mode (Fig. 6). Interactive pre-processing all but eliminates the chance of data entry and modeling errors if used properly.

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W.S. Woodward, J. tV. Morris / Improving Froductivity in FE analysis

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Fig. 6. Valid engineering solutions require multiple "apparent response times'. Data entry errors are usually caught by the engineer by prompting the p,vogram to display the data after it is entered. Once it is checked, there are no intermediate steps which could introduce error. In batch mode, modeling errors are not usually caught untii one run has been returned from the computer. Model refinements and corrections are usually dealt with after a plot of the geometry is obtained from a single batch run. With interactive pre-proce~sing the analyst is given a graphic display of the model as it is generated. Modeling errors are caught immediately in the generation process, before a solution is attempted. This reduces the number of apparent response times required to obtain a valid solution. Therefore, interactive prccessing decreases the time required for analysis by reducing both the number of time-consuming steps inherent in the batch process and the number of batch mode analyses required to obtain a valid solution.

Improvements in productivity Examples that illustrate the increases in productivity which can be achieved using interactive processing are examined here in terms of the following three quantities: (1) the time required to perform an analysis, (2) the total cost of an analysis, and (3) the quality of the analysis. Reduced analysis time (both engineering time and calendar time), reduced analysis cost, and

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W.S. Woodward.J. W. Morris / Improving productivity in FE analysis

better quality analysis all constitute improvement in productivity. While many may agree that all three of these quantities are important to defining productivity, the ranking of these quantities will vary depending on an individual's objectives for a given problem. Interactive preand post-processing has been found to reduce the required a~alysis time, reduce the cost of analysis, and provide better quality results.

Reduced analysis time Interactive processing reduces both the calendar time and the engineering time required for analysis. The reduction in calendar time can be directly attributed to the interactive process which reduces both the 'apparent response time' for turnaround and reduces the number of multiple response times required for a valid solution. Reduced engineering time is primarily due to the labor saving power of the interactive processors themselves. An example which demonstrates both reduced calendar time and reduced engineering time for pre-processing is illustrated in Fig. 7. Two comparable 3-D models of steam generator steamheads are shown. The first is a steamhead for an LMFBR steam generator which was generated using interactive pre-processing. This model was generated with the FIGURES-If program [17] and contains 1565 20-node isoparametric elements with 24783 degrees of freedom. Six man-weeks of effort were required for pre-processing and valid elastic solutions were obtained for six load conditions within two man-months of effort. A total of ninety hours of interactive computer time was required. These pre-processing times include time used to experiment with different mesh densities and construction techniques to obtain a mesh which was appropriate for the problem. Pre-processing was performed on a PRXMEcomputer and the solutions were generated on a CRAV computer. Each load case required twenty-three minutes of CRAY CPU time. Also shown in Fig. 7 is a comparable model of a PWR steam generator steamhead which contains 5378 linear elements with 13 290 degrees of freedom. This model was constructed without the use of a modern interactive pre-processor and required foarteen tan-months to build and verify the model [18]. The comparison of these models demonstrates over a ten-to-one savings in both engineering and calendar time for pre-processing. A second example is illustrated in Fig. 8. Five axisymmetric models, which contain ap-

ATOR STEAMHEAD 6 MAN WEEKS

PWR STEAM GENERATOR STEAMHEAD 14 MAN MONTHS

Fig. 7. Comparable 3-D models can be generated and verified interactively in 1/10 the time required using conventional methods.

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proximately 500-600 8-node isoparametric elements~ are shown. Each model required four engineering hours to construct using the FIGURES-II pre-processor. Earlier models of the same component were constructed manually using only the pattern generation technique and required thirty-two hours of effort. Thus, the modeling time was reduced by a factor of eight. Each model in Fig. 8 was analyzed elastically for the internal pressure load and three thermal transients. All five designs were analyzed in 2½ calendar months which is approximately the time which would have been allocated to completely analyze one design without interactive r, rocessing. In this example the reduction in calendar time for a single an~.lysis was used to optimize the design by studying four additional design concepts. Thus, the savings in time was translated into a better product. Such an undertaking would no! be possible in most production environments without interactive processing. Incidently, the configuration initially thought to be the best turned out to be considerably less than optimal. These examples are representative of the savings in time which can be realized with interactive processing. Savings in both calendar time and engineering time range from 50 to 90~, depending on the complexity of the model. It is important to note that interactive processing provides the least savings in time for simple models and provides the most savings for complex models. R e d u c e d costs

While "time is money" is a phrase which certainly applies to engineering, the cost of computer technology is real too. The savings illustrated in the previous section required hours of use of a minicomputer as well as engineers "trained' in the use of the interactive tools. User training, the use of the software and additional computer time and hardware all add to the cost of the finite element analysis. User training is usually obtained through a one or two day course in the use of the software,

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W.S. Woodward, J. W. Morris / Improving productivity in FE analysis

followed by several weeks of hands-on effort. The hands-on effort is usually productive in that the user can be applying the program to an actual problem while developing skill in the use of the program. The first author's initial experience with interactive processing was directed towards analyzing the five designs illustrated in Fig. 8. Several days were required to develop the first model. This contrasts the four-hour construction time cited above which represents an average time for an experienced user. However, two days still represents a 50~ savings in modeling time relative to use of the manual pattern generation technique. Moreover, the two days of user instruction plus the two days of construction time illustrates that the increased productivity associated with the first use of interactive modeling can pay for the engineering time lost to user training. It must also be remembered that user training is a one time cost. Computer and software costs vary depending on how these resources are obtained. Software is sold or leased separately or obtained on a timesharing basis. A survey of programs [19] shows that one-time license fees range from $9500 to 77 300 depending on the program and size of the computer on which the software is placed. Yearly rentals are also available for some packages. One vendor provides timesharing on an hourly basis at $50.00/hour. This cost includes both the cost of the software and computeJ" time, and it will be used here as a basis for factoring computer and software costs into the calculation of the overall cost of pre-processing. Terminal costs must also be included. The use of interactive processing requires a terminal with graphics capability. Terminal costs are low. Today, a low priced color terminal and color copier may be purchased for approximately $5000. It will be demonstrated from the examples which follow that this cost can be recovered by the savings associated with using interactive pre-processing for several moderate sized problems. A comparison of the costs associated with the steam generator steamhead models in Fig. 7 provides the most dramatic illustration of the reduction of pre-processing cost which can be realized with interactive processing. The pre-processing of the LMFBR steamhead model required ninety hours of interactive computer time at a cost of $4500. At $60 per engineering hour, the six man-weeks of engineering time that were required to build and verify the model cost $14400. Therefore, the total pre-processing cost is $18900. In contrast, the PWR steamhead model required a total of fourteen man-months of engineering time at a cost of $145 600. This excludes the unknown batch computer costs which were realized in the debugging process. Thus, the pre-processing cost for the PWR steamhead model exceeds 7.7 times that of the LMFBR steamhead model. The cost of mesh generation for the axisymmetric models in Fig. 8 also demonstrates a reduction in cost. Each model required four hours to construct interactively for a cost of $440. A minimum thirty-two hours of engineering time was required to construct similar models manually using pattern generation for a cost of $1920. Thus, interactive mesh generation reduced the cost of modeling by a factor of 4.4. The savings obtained from using interactive pre-processing for this analysis is $1480. The savings from four such analyses would be sufficient to recover the cost of a color terminal and printer. It should be noted that the costs of pre- and post-processing can be further reduced by utilizing lower cost engineering technicians rather than engineers. In most cases, only minimal specifications from the engineer are required for a trained technician to produce satisfactory results.

Improved quality of analysis Better models with fewer errors can be constructed with interactive processing. The variety of mesh generation techniques available with interactive processors gives more control over element distribution. Models constructed with conventional pattern generation often exhibit inefficient element distributions because the effort required to produce the desired mesh is excessive. The models in Fig. 8 illustrate this point. While all the models were generated

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Fig. 9. Color display reduces verification time and reduces chance of error. (a) Missing element pressure is difficult to detect in black-and-whitedisplay. (b) Color i display enhances verification of missingelement pressure load.

interactively, the models vary in quality. Model 1 was generated first, and it exhibits large disparities in adjacent element sizes and inefficient placement of elements. This occurred because the user was still thinking only in terms of the conventional pattern generation technique. For each subsequent model the user made better use of the mesh generation techniques available in the interactive processor and the models improved in quality. The placement of elements in Model 5 are much improved compared to Model 1. Element sizes vary uniformly and the density is high only in the regions of high stress gradients. Fewer modeling errors are made with interactive pre-processing because of the extensive capabilities which allow visual checking of graphical displays. This point is illustrated with an actual error which was detected on the L M F B R superheater steamhead mode! in Fig. 7. Fig. 9 shows a portion of the interior surface of the steamhead which is loaded with an internal pressure. One element surface has a missing pressure load. This error was rapidly identified using the graphical display provided by the pre-processing program and the error was corrected in minutes. Such an error would most likely have gone undetected without the use of interactive pre-processing. Fig. 9 illustrates both black-an_~-white (a) and color ! (b) computer displays. The comparison of these displays illustrates the great advantage which color offers in model verification. Both verification time and the change of error are greatly reduced through the use of color.

Future trends Pre- and post-processing programs have now been installed on a variety of computers. The engineer can execute these programs on work stations, supermini, mainframe, or supermainframe computers. Today, a hierarchical system of computers is most widely used where pre- and post-processing are performed on minicomputers and the finite element solution is obtained on a large mainframe computer. The use of interactive pre- and post-processing on i The original color display in Fig. 9(b) is reproduced here in shades of gray.

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W.S. Woodward, J. W. Morris / Improving productivity in FE analysis

hierarchical systems is prevalent because most corporations currently use hierarchical systems with distributed terminal networks for scientific computing. Current trends involve running interactive finite element codes on both larger and smaller computers. There is a strong trend toward implementing total finite element packages, which include interactive pre-processors, equation solvers and post-processors, on stand-alone work stations and personal computers. This application is ideally suited for small and moderate sized problems where data base management and equation solving can be performed with reasonable response times. It also offers additional savings in time for small to moderate problems because it eliminates the queue time associated with batch computing and opens the door for interactive design and analysis. At the other end of the computing spectrum, interactive finite element computing is being performed on supermainframe computers. PATRAN-G is already operational on the CRAY computer [21,22]. The major advantage of using interactive supercomputing is reduced response time for processing data and obtaining solutions for large models. On the supercomputers, models with as many as 5000 degrees of freedom are small and may be solved in seconds. Developments are also directed towards merging the CAD and CAE technologies. Computer interfaces which allow the geometry from CAD data bases to be transferred directly to CAE data bases are being established. The development of an Initial Graphics Exchange Specification (IGES) was initiated in 1979 by the National Bureau of Standards for the purpose of achieving a practical means of exchanging product definition data. IGES has been adopted as an ANSI standard and is a candidate for an international standard. The latest version of IGES supports CAD vendor translation and fi'~ite element modeling capabilities. With this technology, drawings generated on CAD systems can be electronically input to pre-processing programs and model development can be performed without manual entry of geometric data. The development of artificial intelligence capabilities offers additional promise for improving analysis productivity. However, few intelligent systems have been designed to date. One such program, SACON, has been written to evaluate a structural designer's need for analysis and recommends appropriate analysis methods which can be performed on the MARC finite element computer program. However, SACON does not generate models or perform a~,e!yses. There is great potential for using artificial intelligence to further automate finite element analysis methods. Finite element modeling is amenable to rule-based expert systems as is the evaluation of analysis results. An expert system could be designed to begin with geometric data from a CAD system and the specification of materials, loads, environment and evaluation criteria determine the type of analyses which need to be performed (like SACON), design models, generate valid solutions using adaptive mesh generation and evaluate results relative to the criteria. Such a program would bring the general body of finite element expertise to the engineering practitioner. However, the implementation of a comprehensive intelligent system will require substantial lesources and years of development.

Conclusions Interactive pre- and post-processing increases productivity in finite element analysis by greatly reducing calendar time, engineering time and the cost required to perform an analysis. The savings in calendar time can be attributed directly to the interactive process which eliminates many time-consuming steps inherent in the batch process. Savings of engineering time are realized by the labor saving work performed by the interactive pre- and post-processing programs. The calendar time and engineering time were found to be reduced by factors of 8 and 10 for the analyses cited in this paper. Furthermore, the largest savings in time are realized for complex models..Pre-processing costs associated with analyses were found to be less than the

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cost of using conventional pattern generation by factors ranging from 4.4 to 7.7. Again, the largest cost savings are realiz~ for complex models. The cost of user training was found to be recovered in a single analysis of a moderate size problem. The use of this technology requires computer terminals with graphics capability. Interactive graphics allow quick and accurate checking of the model, boundary, conditions, loads, materials, and element types which reduces the change of error. The cost of inexpensive color terminals can also be recovered from the savings in engineering cos associated with the use of interactive processing for three or four moderate sized problems.

Acknowledgment This paper was presented at the ASME Pressure Vessel and Piping Conference held in San Antonio, Texas in June 1984. The authors are grateful to Dr. Nicholas Perrone for his support and encouragement to publish this work. The authors gratefully acknowledge Dr. D.S. Griffin, Mr. P. Falk and Mr. S. Gabrielse who supported the efforts which formed the basis of this paper. The authors also wish to acknowledge Dr. A.K. Dhalla and Mr. Timothy Butler who performed analyses cited in this paper. In addition, the authors are grateful for the many helpful comments provided by Dr. Wayne VanBuren.

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