Application of the BEM in conceptual mechanical design...some degree of banding for multiple zone problems. Thus, only a comparatively small part of the system matrix will change

Application of the BEM in conceptual

mechanical design

J. Trevelyan

University of Durham, School of Engineering, South Road,

Durham, DH1 3LE, UK

Email: jon. trevelyan@durham. ac. uk

Abstract

The engineer working on conceptual design of engineering components is restricted by theprecise requirements of most analysis methods. The development of a design can be greatlyenhanced by the use of an analysis technique which allows a sketched geometric definition,automated development of the corresponding analysis model and rapid re-analysis given adesign perturbation. This combination of features allows designers to experiment with radicallydifferent designs and to control an intuitive design optimisation based on their own experiencebut guided by the analysis results. The Boundary Element Method (BEM) is particularly wellsuited in this area because of the increased speed and reliability of automatically modifiedmeshing. This paper describes the implementation of an Object-Oriented BEM system whichfulfils this need.

Introduction

Conventional software packages for stress/displacement analysis are best usedfor the detailed design of components and structures. They can provide anaccurate prediction of the structural stability and durability of a design beforeprototype testing need take place. However, they are of limited use to theengineer involved in conceptual design. The limitations arise from themethodology generally applied in commercial packages based on the FiniteElement Method (FEM) and those of similar technology available today. Therequirements of the concept designer are largely not met by such systems. Theprecision required in the definition of a mathematical model for this type ofsoftware is inappropriate for conceptual design, and inevitably impedes thedetermination of response results in a timely manner. Furthermore, and perhapsmore importantly, the difficulty of changing a pre-existing finite element model

Transactions on Modelling and Simulation vol 18, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X

810 Boundary Elements

to reflect a design change can often be so time-consuming that the creation of atotally new model for the revised geometry may be the faster option.

The proposed research puts forward a solution based on the BoundaryElement Method (BEM). Its use of a boundary representation not only lends itselfwell to the methodology used in this work, but also integrates well with otherComputer Aided Design and Manufacturing systems which use a similardescription of the component geometry, as well as with systems which performdiverse other tasks such as cost estimation, tolerance analysis and design formanufacturability. Since a boundary representation is used throughout theprocess, the likelihood of errors in translation of the geometry from one system toanother will be much reduced.

While the use of the BEM is widespread in both industrial and academicorganisations, in the area of mechanical design the method has not fulfilled itsgreat potential. There are a number of contributing factors to this situation, butone which may be identified is that the structure of commercial codes remainsbased on the methodology of early (i.e. pre-graphic) finite element modelling.That is, a mathematical model of a component or structure may be generated inthe following steps:

(i) define the basic geometry by creating a set of points with known geometriccoordinates, linking them together with straight lines, circular arcs and/or splinesand surfaces.

(ii) define a set of 'elements' on each geometric entity according to some rulesand guidelines which can be difficult to satisfy.

(iii) define a set of material properties and boundary conditions which areattached either to given geometric entities or to specific elements.

(iv) if the model needs to be changed, return to step (i) and repeat the entireprocess.

(v) write a data file.

(vi) exit the preprocessor and run the analysis program.

This type of model creation process is contrary to the ideal process of conceptualdesign since it forces the designer to postulate a detailed design before he/she isready to do so. As a result, for components or structures which require anumerical analysis to determine their stress/displacement response, only a fewdesign options may be tried in the time available.

The model creation and model modification processes may be greatlysimplified. The following are required for a realistic conceptual design/analysissystem:


Boundary Elements 811

• both analysis and graphical display functions to be contained in thesame program and to share a common database

• simple sketching and rubber-banding of geometric primitives• a high degree of automation in preparation of an analysis model

Thus the aim of this work is to bridge the gap between conventional analysissystems and graphical capabilities which have become established in thepreparation of presentation material, etc. Such a link has been proposed by Bettigand Han\ who developed an interactive Windows environment for some finitedifference and elementary beam finite element calculations.

By its nature, conceptual design is based around what-if questions. Thestructured approach used by conventional analysis systems does not lend itselfwell to this environment. It is preferable to use an Object-Oriented technique.This has been described at length in the literature , and has a number of essentialfeatures which are important to the work described in this paper. Notably, there isa clear link between the data relating to a mathematical model and the routineswhich operate on that data. In addition, the structure of Object-OrientedProgramming (OOP) makes it highly suitable for an event-driven process ratherthan a sequential analysis process. Further, as has been shown in various OOPimplementations of the finite element method , it is possible to take advantageof class hierarchy to manage efficiently the large amounts of different (butrelated) data which abound in numerical elasticity analysis.

The speed of providing analysis solutions is of prime importance. If theanalysis results are not available to the designer in a 'reasonable' time theirusefulness is limited. At this stage, then, it is necessary to consider two distinctstages in the design analysis process discussed in this paper.

1. the initial analysis2. the analysis of a perturbed design

It is recognised that the speed of an initial BEM solution to a sketched andautomatically meshed representation of a design is confined within certainbounds. Integration has to be performed and a set of simultaneous equationssolved. In spite of certain performance enhancements which can be brought aboutin these activities, it should be recognised that a stress analysis will take sometime. However, the second stage in the design analysis process considers theeffects of a design change, whether it be geometric or relating to boundaryconditions. In this activity, it is possible to make significant performanceimprovements over conventional analysis systems, and at the same time offer theease of use which is required to allow the designer to concentrate on his/herdesign. In particular, the approach discussed in this paper relies heavily on a rapidre-analysis and the automatic updating of results following a design change.

The idea of static re-analysis has been considered by various researchers,largely with respect to the finite element method and with the aim of updating



analysis results as part of a shape optimisation procedure. The early work onthese methods has been reviewed by Arora and later Abu Kassim and Topping .The re-analysis concept was developed further and applied to BEM matrixstructures by Kane et al , who described two iterative schemes for the re- solutionof a perturbed system by partitioning the matrix into the original and perturbedportions. The schemes presented are advantageous in that they reuse a triangulardecomposition of the initial system matrix in the iterative procedure. However, itis not so clear how the methods could go on to consider a second perturbation ofa different part of the model with equal efficiency. Clearly, the availability of theoriginal solution as a starting point for the perturbed solution makes an iterativescheme overwhelmingly attractive to re-analysis. Iterative solution schemes suchas CG, Bi-CG, GMRES, etc. are well established in the literature "^ and notdiscussed in detail here. At present, the reanalysis performed in theimplementation described here solves the full (revised) matrix using a directGauss elimination procedure with partial columnal pivoting. It is recognised thatsubstantial performance gains are to be realised by using an iterative scheme suchas the diagonal preconditioned GMRES, which has been shown to give superiorperformance for many BEM solutions even without the head-start provided bythe good first

2. BEM and partial re-analysis

The BEM is a matrix method of analysis. For the problem of stress analysis itis well established that the stress and displacement response of a structure may bedetermined by finding the solution to an equation

Hu = Gt (1)

in which H and G are respectively square and (for continuous elements)rectangular matrices depending on the geometry and material properties underconsideration, and u and t are vectors of displacements and tractions, from whichstresses may be derived^. It follows that a change in design geometry will causethe H and G matrices to change, thereby changing the response of the structure.The concept of partial reanalysis relies on the assumption that any single, andsmall, design change will modify the location of only a few of the elementscomprising the mathematical model. The change must be small in the sense thatthe number of elements on each geometric entity remains constant, allowing thesame matrix structure and node numbering to be re-used. If the model is moreradically modified, requiring the addition or subtraction of degrees of freedom,similar concepts may be used with some extra complication and performancepenalties, though for most practical situations it may well be more efficient torestart a new analysis.

By consideration of the boundary conditions equation (1) is reduced to thesolvable square form

A x = B y = z (2)



A is the system matrix, which for the purposes of this study will be assumed to benonsymmetric and fully populated, though it is well known that this can be givensome degree of banding for multiple zone problems. Thus, only a comparativelysmall part of the system matrix will change. It can readily be shown that, for aBEM system having N nodes in which M nodes are moved, the proportion a ofthe system matrix which needs to be updated is given by

- M)

For simple modifications of small models the proportion a has been found tolie between 0.15 and 0.6. For larger models, it is anticipated that typical designchanges would give values of a towards the lower end of this range, and possiblylower than 0.15. It is possible to update the terms in these parts of the matrix,leaving the other terms as before, and solve the system to compute the newdisplacement and traction vectors relating to the changed geometry. By solvingequation (2), the new solution to the perturbed geometry may be determinedquickly and accurately, enabling the rapid updating of the results contours whichare being displayed. It is important that the reanalysis of the perturbed modelshould give as accurate a solution as a normally executed BEM model of thesame geometry.

In the implementation described below, the performance of the first analysis isenhanced by direct assembly of the A matrix and z vector during the integrationstage, precluding the need to store H and G. It should be noted that the B matrixneeds to be stored also at this stage as it is required to build a revised right handside vector z during the reanalysis. In addition, a rigid body translation methodfor computing the diagonal coefficients of H can still be used in the reanalysiswithout the need to store the original H matrix, since the only rows in which anew diagonal is required are those for the moved nodes, and the entire row willbe recalculated.

3. OOP Implementation

The above ideas have been programmed into an application called CODA(Concept Oriented Design Analysis) using the Microsoft Visual C++ compilerwith the contained Microsoft Foundation Classes. The application performs two-dimensional stress analysis and interactive re-analysis using the BEM. Theinteractive sketching capabilities are performed using a class CCodaView,derived from the foundation class CView. The model information is stored in adocument class CCodaDoc, which itself contains references to classes CElement,CNodePoints, CMeshPoints, CBC, etc. Cross referencing is readily achieved by,for example, including in an object of class CElement pointers to the node andmesh point objects which correspond. The use of class inheritance is made by, for



example, including a class for points (CPts) and then deriving classes fordifferent types of points, notably CNodePoints, CMeshPoints andCInternalPoints. The base class CPts contains (;t,;y) coordinate information, so thederived classes can use these variables, but each derived class can add its owndata and routines as appropriate. For example, the class CNodePoints can includevariables defining the boundary condition types, whereas CInternalPoints caninclude variables which contain the components of the stress and displacementsolution ready for postprocessing.

Given this framework, the process of implementing the analysis part of thesystem is very similar to that in conventional sequential coding using, forexample, C or even Fortran. However, the Graphical User Interface (GUI) is non-sequential by nature and requires the event driven methodology which is wellsupported by the OOP framework. GUI toolbars, icons and other graphical aids to'user-friendly' programming are also straightforward within a compiler such asVisual C++, and (for all the initial difficulties involved in OOP for an engineerschooled in Fortran) it is eventually recognised that this is the approach of choicefor interactive applications.

In the CODA application, the geometry is represented by 'shapes', which areinitially rubber-banded rectangles or circles (note that the 'model' coordinates ofthe cursor location are continually displayed on the screen so that although thegeometry may be roughly sketched it may also be accurately defined).Rectangular shapes may then be modified by adding or deleting vertices, and/orby interactively moving vertices using a click-and-drag rubber-banding operation.Entire shapes may be moved, for example to change the position of a circular bolthole in a rectangular plate. Boundary conditions (x or y or x+y) may be applied todefine traction or displacement conditions over shape segments. Materialproperties are selected by material name in preference to the definition ofYoung's modulus and Poisson ratio as in a conventional analysis system. In total,the entire model may be sketched and prepared for analysis very quickly. Theanalysis is initiated by another toolbar menu item and a progress bar is used todisplay the percentage run completion. The results may be displayed in contourform or as animated deformed shapes. Prior to analysis, the programautomatically and invisibly meshes the shapes and generates a triangulated grid ofinternal points for contour displays.

The reanalysis is identified by the user displaying a contour plot of results,selecting a command 'Change Geometry' from the menu, and then by rubber-banding to move a shape vertex. The updated contours are automaticallydisplayed.



4. Examples

4.1 Rectangular plate with circular hole under uniaxial stress

Figure 1 shows the standard test case of a rectangular plate with a centralcircular hole. A tensile stress of lOOMPa is applied in the horizontal direction.This model may be constructed for analysis using nine cursor-click operations:

(i) click the rectangle icon(ii) rubber-band a rectangle(iii) click the circle icon(iv) rubber-band a circle(v) click the traction icon(vi) click the line segment on the right side of the rectangle and type the

traction value 1000(vii) click the displacement constraint icon(viii) click the line segment on the left side of the rectangle and type the x-

displacement value 0(ix) click the line segment on the bottom of the rectangle and type the y-

displacement value 0

Notice that the default material properties (those of mild steel) have beenadopted. Clicking the analysis icon initiates the analysis. A total of 48 quadraticelements was generated automatically (though more or fewer could be used if theuser sets the 'fine mesh' or 'coarse mesh' options in preference to the default'standard mesh'), and the results gave maximum and minimum tangentialstresses of +302.7 MPa (figure 2) and -101.3 MPa, indicating excellentcorrelation with the analytical solution for this simple problem.

4.2 CPU time comparisons

A number of small-to-medium size models have been run, including areanalysis of each. Table 1 shows the important cpu times obtained for the runs.The application was running under Windows95 on a 166MHz Pentiumprocessor. It is clear that for this size model the integration stage of the reanalysiscan be performed quickly. For model 1 this was achieved in 16% of the run timefor the first analysis, but for the other models this reduced to 6-8% of the firstanalysis total cpu time. The remaining time for the reanalysis was taken inrepeating a full direct solution. It is anticipated that this will be reduced verysubstantially by the implementation of a iterative solution scheme for this phaseof the work. However, the total reanalysis time was less than half the total cputime for the first analysis in every case.



Model

12345

No. ofnodes

100148152176210

No. ofmovednodes

3715353747

First analysiscpu (s)

4.239.5612.3615.7125.98

Re-analysis cpu (s)

Integration0.660.551.041.271.87

Total1.643.794.616.8711.31

Table 1: Cpu time breakdown for analysis and re-analysis

5. Conclusions

Use of a graphical C++ compiler for Object-Oriented Windows applicationshas produced a design analysis tool which overcomes the main limitations ofconventional analysis systems when applied to conceptual design tasks. Acomponent geometry may be defined as a set of sketched 'shape' objects anddeformed by adding, moving and deleting vertices as required. Boundaryelements and internal points are defined automatically according to inbuilt rules.By consideration of the relatively small proportion of the system matrix whichneeds to be updated following a design change, a re-analysis may be performed inwhich the integration phase is reduced substantially in time. At present the full,direct solution increases the cpu time of a re-analysis beyond that which mightreasonably be expected with the use of iterative re-solution techniques. Work willbe continuing on this project in this direction. In spite of this, the re-analysis timeis still less than half of the cpu time for the first analysis for all models run. Thusthe CODA application, with its simple presentation-graphics style GUI andautomated reanalysis, already offers a potential for major benefits to engineersinvolved in conceptual mechanical design.

6. References

1. Bettig, B.P. & Han, R.P.S., An Object-Oriented Framework for InteractiveNumerical Analysis in a Graphical User Interface Environment, Int. J. forNum. Meth. in Engg., Vol. 39, 2945-2971, 1996.

2. Barton, J.J. & Nackman, L.R., Scientific and Engineering C++: AnIntroduction with Advanced Techniques and Examples, Addison Wesley,1994.

3. Lafore, R., Object-Oriented Programming in Turbo C++, Waite GroupPress, 1991.



4. Forde, B.W.R., Foschi, R.O. & Stiemer, S.F., Object-Oriented FiniteElement Analysis, Comput. Struct., Vol. 34, 355-374, 1990.

5. Miller, G.R., An Object-Oriented Approach to Structural Analysis andDesign, Comput. Struct., Vol. 40, 75-82, 1991.

6. Mackie, R.I., Object-Oriented Programming of the Finite Element Method,Int. J. for Num. Meth. in Engg., Vol. 35, 425-436, 1992.

7. Arora, J.S., Survey of Structural Reanalysis Techniques, /. Struct. Divn.,ASCE, Vol. 102, No. ST4, 783-802, 1976.

8. Abu Kassim, A.M. & Topping, B.H.V., Static Reanalysis: A Review, J.Struct. Engg., ASCE, Vol. 113, No. 5, 1029-1045, 1987.

9. Kane, J.H., Keshava Kumar, B.L. & Gallagher, R.H., Boundary-ElementIterative Reanalysis for Continuum Structures, J. Eng. Mechanics, ASCE,Vol. 116, No. 10,2293-2309, 1990.

lO.Bettess, J.A., Solution Techniques for Boundary Integral Matrices, fromNumerical Methods for Transient and Coupled Problems, (eds. R.W.Lewis,E.Hinton. P.Bettess, B.A.Schrefler), Wiley, 1987.

ll.Saad, Y. & Schultz, M.H., GMRES: A Generalized Minimum ResidualAlgorithm for Solving Nonsymmetric Linear Systems, SIAM J. Sci. Statist.

f., Vol. 7, No. 3, 856-869, 1986.

12.Guru Prasad, K., Kane, J.H., Keyes, D.E. & Balakrishna, C., PreconditionedKrylov Solvers for BEA, Int. J. for Num. Meth. in Engg., Vol. 37, 1651-1672, 1994.

13.Davey, K. & Rosindale, L, An Iterative Solution Scheme for Systems ofWeakly Connected Boundary Element Equations, Int. J. for Num. Meth. in

., Vol. 39, 3933-3951, 1996.

14. Kane, J.H., Keyes, D.E. & Guru Prasad, K., Iterative Solution Techniques inBoundary Element Analysis, Int. J. for Num. Meth. in Engg., Vol. 31, 1511-1536, 1991.

15.Brebbia,C.A. & Dominguez, J., Boundary Elements: An IntroductoryCourse, 2nd edition, Computational Mechanics Publications and McGrawHill, 1992.



lOOMPa

Figure 1: Simple test case

Figure 2: Screen layout showing maximum principal stress contours for platewith hole


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Application of the BEM in conceptual mechanical design...some degree of banding for multiple zone problems. Thus, only a comparatively small part of the system matrix will change