- Email: [email protected]
Zen and the art of finite element design
Finite Elements in Analysis and Design 3 (1987) 85-91 North-Holland
85
ZEN AND T H E ART OF FINITE E L E M E N T DESIGN * Richard H. MACNEAL (Chairman) The MacNeal-Schwendler Corporation, 815 Colorado Boulevard, Los Angeles, CA 90041, U.S.A. Received February 1987 Abstract. The paper reviews Robert M. Pirsig's ideas concerning the concept of quality as expressed in his book Zen and the Art of Motorcycle Maintenance and applies them to the problem of designing finite elements. Using quadrilateral membrane elements as a typical example, it is conclude that finite elements differ remarkably in the quality of their performance and that improvements in quality occur in response to the dedication and insight which finite element designers brings to their task.
Introduction
Those who have read Robert M. Pirsig's Zen and the Art of Motorcycle Maintenance [12] (or Z A M M for short) know that it has almost nothing to do with Zen and only uses motorcycle maintenance as a vehicle for the expression of larger ideas as is hinted by its subtitle, "An Inquiry into Values". What the book is actually about, to put it as briefly as possible, is the concept of 'quality' and its manifestations in many fields, including science and technology. For this reason, if for no other, a brief review of Z A M M might interest those engineers who are concerned with the quality of the technology they produce. The book is superficially a novel about a motorcycle trip, but the plot and character development have been deliberately weakened in order to focus attention on the 'chautauquas' which the principal character composes over the long uneventful miles. The subject matter of the chautauquas ranges from technology to art to science to philosophy, but always the purpose is to explore the manifestations of quality so as to capture its essence. One of the principal points made is that quality, which the author equates with excellence, can easily be recognized but cannot be defined. About the closest the author comes to defining quality is to say that it is the inverse side of caring and that it lies in the relationship between people and the things they produce. Examples are taken largely from motorcycle maintenance but also from chemical engineering, rhetoric, mathematics, and other fields. Pirsig finds much evidence of a general lack of quality in the m o d e m world. For this he ultimately blames Western philosophy as inherited from classical Greece for putting too much evidence on truth and reason and not enough on values. The result is an outpouring of technology that is all too often ugly, shoddy, unsafe, or inhuman. While the latter point may have been more in evidence when Pirsig was writing ZAMM in the early seventies, it is arguable that today's technology has achieved a modestly higher level of quality. Environmental concerns are beginning to be heeded and terms such as human engineering and user friendliness have entered the jargon. The word has even begun to be heard on mahogany row where In Search of Excellence [11] is likely to be spotted on an executive's bookshelf. * A preliminary version of this paper was presented at the Fourth Chautauqua on Productivity in Engineering and Design, Coronado, CA, October 27-29, 1986. 0168-874X/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
R.H. MacNeal / Zen and the art of finite element design
86
To come closer to home, there is no question that the quality of finite element systems have improved in the last fifteen years nor can it be doubted that there is still a long way to go. The emphasis today is much more than formerly on the satisfaction of the finite element user's needs and on the minimization of his frustrations. Some of the manifestations of improved quality include computers which have become more available, more powerful, and easier to use; interactive graphics which has greatly facilitated the user's tasks of preparing input and examining output; and the finite element codes themselves which now address more applications in greater detail while becoming more rugged and more useable with fewer errors. Although finite element systems may have grown in size and scope and ease of use, their central feature for interpreting engineering mechanics continues to be the finite elements. One might expect of finite elements that their mathematical foundation is rigorous and unchanging but, unfortunately, the design of finite elements is much less of an exact science than, for example, the formulation of algorithms for the solution of the resulting equations. The element is, in fact, the principal repository of assumptions and approximations in finite element analysis. A succeeding section will examine the ways in which changes in the mathematical formulation of finite elements have improved the quality of finite element systems, but first it will be inquired whether finite element design qualifies as an art form in the sense that Pirsig applies the term to motorcycle maintenance.
Finite element design as an art form
Fig. 1 shows a typical finite element as it might appear in a textbook or in the graphical representation of a finite element model. For all but a very few specialists, Fig. 1 is a perfectly adequate representation of a finite element because they neither know nor care what goes on inside the element. Indeed, it is no more necessary to know what goes on inside a finite element in order to use it than it is to know what goes on inside a motorcycle in order to ride it. What goes on inside does, however, matter critically as far as performance and reliability are concerned. The task of designing a finite element is not a cut-and-dried operation as is indicated by the great number of finite elements that have been proposed in the last thirty years. Even though a lot of mathematical tools and physical principals are available to assist the specialist, finite element design also requires rather arbitrary design choices in all but a few exceptionally simple cases. An example of the latter is the three-noded membrane triangle shown in Fig. 2 which has two displacement degrees of freedom (dof's) at each external node. The resulting six external degrees of freedom must be balanced by six internal degrees of freedom which might, for example, be coefficients in a series representation of the internal displacement field. Two principals which were learned the hard way early in the history of
v3 l
vlV ,, v2 _
I
2
Fig. 1. A typical finite element.
II
Ul
Fig. 2. Constant strain triangle.
u2
R.H. Macneal / Zen and the art of finite element design
I
87
V4u4 ~ v3 u
Vl
v2 uI
u2
Fig. 3. Four-noded membrane element.
finite element design can account for all of the internal degrees of freedom. They are that rigid body motions (three internal dof,s) and constant strain states (three internal dof,s) must be exactly represented in order to achieve minimum acceptable performance. If these principals are invoked, then there is no need for other design choices. As a result, the design of the three-noded, six-degree-of-freedom membrane triangle has remained essentially unaltered since it was first introduced [16] in 1956. It is a staple of all finite element programs. The case of the four-noded membrane element shown in Fig. 3 involves additional choices. Since it has eight external degrees of freedom and since the rigid body motions and constant strain states account for six of them, it follows that two additional internal degrees of freedom must be selected. One might have imagined that the best possible choices for these two degrees of freedom would have been found and cataloged long ago but that is not the case. Prior to 1960, everyone simply filled the quadrilateral with triangles [16], then along came Taig [15] with his famous isoparametric element, then dozens of others right up to the present moment [2], and no doubt into the future. If optimum solutions for the minimal complexity offered by the membrane quadrilateral have proven so elusive, it can only be imagined how much more challenging are the more complex plate, shell, and solid elements. The point of this discussion is not merely that the quality of element performance depends on making appropriate choices but also that the act of choosing is a conscious effort by the finite element designer using not only the available tools but also his experience, imagination, and intelligence. That is where the art comes in, just as it does in motorcycle maintenance or in any other practical art. The job can be done well or badly, with more or less quality, which may be less apparent to the eye than it is in the utility of the result. The ways in which the quality of finite elements can vary will be examined next.
Quality in finite element performance and design The only way the average user can see evidence of quality in finite elements is by comparing the quality of output for runs using different finite elements. He might then find differences in accuracy or run time which could be attributed to the elements or he might find situations where one element had features not present in another or where one element occasionally produced completely misleading results. From such comparisons he would probably conclude that one element was generally better, or even with enough data that one element was better in some instances while another element was better in others. The performance measures which are of interest to finite element users can be summarized as follows:
R.H. MacNeal / Zen and the art of finite element design
88
• Accuracy. • Economy: • in element formation, • in problem solution. • Features: • for convenience, • for breadth of application. • Reliability or ruggedness. Accuracy and economy clearly provide a design trade-off because both accuracy and cost should be expected to increase with element complexity. The challenge is to provide both at the same time. The provision of features such as different options for materials properties increases development cost and requires sensitivity to user needs but it is not the same order of challenge. Reliability, as it applies to finite elements is probably the most important but also the most elusive of finite element qualities. An element may, for example, have excellent accuracy in some applications and very poor accuracy in others. As an extreme example, situations exist where a small change in element shape or loading can result in a drastic reduction in accuracy. As a response to such concerns, more attention has been directed in recent years to the quality of finite element analysis in general and to the reliability of finite elements in particular. This has led to the formation of committees and organizations devoted to the improvement of the finite element method (such as N A F E M S in the U.K.), conferences and forums on standards, quality, and reliability (such as forums at the 1984 and 1985 SDM meetings, a conference on reliability in July 1986 at Swansea, Harry Schaeffer's Fourth Chautauqua on Productivity in Engineering and Design in October 1986, and John Robinson's Fifth World Conference in 1987), and papers or reports comparing the accuracy of finite elements for a number of so-called 'standard' tests [3,10,13,14]. The latter are usually selected so that at least some of the elements exhibit gross inaccuracies. Such tests are beneficial if the purpose is to alert users to element configurations which should be avoided or to alert developers to defects which should be corrected. Dissatisfaction with element reliability is probably the most important motivating force for element improvement. At least that has been the case at MacNeal-Schwendler. The evolution of the quadrilateral membrane element mentioned earlier is a good example of the fruits of that motivation. As will be recalled, quadrilateral membrane elements vary chiefly in the way in which they account for the internal degrees of freedom which remain after rigid body motions and constant strain states have been correctly represented. The most important degrees of freedom beyond the constant strain states are those associated with inplane bending. At the very least, reasonably accurate results should be expected when a square element is subjected to inplane bending in one of its principal directions as shown in Fig. 4. Several quadrilateral membrane elements, including homegrown varieties, have been introduced into N A S T R A N and M S C / N A S T R A N and it is a simple matter to subject them to the test indicated in Fig. 4. The results of the test are tabulated in Table 1 along with the year in which the element was released for general use in M S C / N A S T R A N or its predecessor.
el2
el2
_\
/
)
Fig. 4. Inplane bending of a square element.
R.H. Macneal / Zen and the art of finite element design
89
Table 1 Results for inplane bending of a square element Element name
Date released
O/M
QDMEM QDMEM1 QDMEM2 QUAD4 a QUAD8 Q4S Exact
1970 1973 1973 1974 1980 1987 -
0.224 0.675 0.585 0.910 1.000 1.000 1.000
Called the modified QDMEM1 prior to 1976.
The evolutionary improvement is quite apparent. The earlier elements, particularly the QDMEM, are intolerably inaccurate while the later elements score perfect results. The excessive stiffness exhibited by the earlier elements has been diagnosed as a locking problem which is caused by the presence of parasitic shear strain. The symptoms become worse as the element's shape is modified from a square to an increasingly slender rectangle. It should also be mentioned, in fairness, that the QUAD8 has twice as many degrees of freedom as the first four elements and the Q4S has fifty percent more than the first four, as illustrated in Fig. 5. Thus the higher accuracy of the latter two elements has been purchased at the cost of increased run time. The QUAD8 has the further drawback that it does not exactly represent constant strain states for arbitrarily-shaped quadrilaterals. In spite of its less than perfect accuracy, the QUAD4 has been the workhorse of the MSC/NASTRAN element family. While its accuracy for inplane bending holds up with increasing slenderness, it has been discovered that the accuracy deteriorates badly if the element is distorted into parallelogram or trapezoidal shapes. This behavior can be investigated by the sequence of test problems shown in Fig. 6. The results for four MSC/NASTRAN quadrilateral membrane elements are presented in Table 2. They clearly show the hopelessness of QDMEM and the deterioration of QUAD4 accuracy with shape change. The other two elements perform well with the edge going to Q4S. Two major points emerge from the example of the quadrilateral membrane element. The first is that all elements are n o t alike and they can, in fact, exhibit important differences in quality of performance. The second point is that the quality of performance tends to increase with the passage of time due to the accumulation of design improvements. Each design improvement is the product of a creative act of caring by some designer, somewhere whether it be a breakthrough or a modest incremental improvement. The NASTRAN quadrilateral membrane elements again provide examples of both.
Av
U
w
U
v
a
v
b
w
c
Fig. 5. External degrees of freedom of NASTRAN and MSC/NASTRAN quadrilateral degrees of freedom. (a) QDMEM, QDMEM1, QDMEM2, QUAD4. (b) QUAD8. (c) Q4S.
90
R.H. MacNeal / Zen and the art of finite element design
Rectangular ShapeElements
Trapezoidal ShapeElements
Parallelogram ShapeElements
P
l~
P
Fig. 6. Inplane bending of cantileverbeam with three differentelement shapes. As has been mentioned, the earliest quadrilateral membrane elements, of which the Q D M E M and QDMEM2 are examples, consisted of groupings of triangular elements. The design of Q D M E M 2 is perhaps the earliest of such elements [16] and dates from 1956. The Q D M E M element was conceived by the present author about 1967 during the initial development of NASTRAN [7]. Since its performanc e is inferior to that of the earlier design, it represents the kind of backward step which can occur when ignorance and expediency prevail. The first breakthrough in the design of quadrilateral membrane elements came with the four-noded isoparametric element published by Taig [15] in 1961. This design was later copied into the QDMEM1 element in 1973. It embodies an efficient method for calculating the coefficients of an assumed internal displacement field and guarantees satisfaction of the rigid body and constraint strain conditions for any arbitrary quadrilateral shape. The Taig element was the model for all the assumed displacement elements designed thereafter. The membrane part of the QUAD4 element [8], for example, includes a minor modification to the Taig element which averages inplane shear strain in order to ameliorate the locking problem noted in Table 1. An important secondary breakthrough occurred in 1966 when Irons [6] generalized the Taig element to arbitrarily curved two- and three-dimensional shapes with any number of extra nodes per edge. In so doing, he generated the entire family of standard isoparametric elements. Irons was followed by other designers who introduced accuracy enhancing modifications such as those embodied in M S C / N A S T R A N ' s QUAD8. The four-node Q4S element [5], which is not yet in production, is the by-product of a lengthy effort to find a modification to QUAD4 which would successfully remove the locking problem Table 2 Results for inplane bending of a cantileverbeam (tip displacement due to tip load) Element n a m e QDMEM QUAD4 QUAD8 Q4S Exact
Rectangular elements 0.032 0.904 0.987 0.993 1.000
Trapezoidal elements 0.016 0.071 0.946 0.988 1.000
Parallelogram elements 0.014 0.080 0.995 0.984 1.000
R.H. Macneal / Zen and the art of finite element design
e x h i b i t e d for n o n r e c t a n g u l a r shapes while c o n t i n u i n g to satisfy the A f t e r n u m e r o u s failures we c a m e to the conclusion a n d were able to was i m p o s s i b l e w i t h o u t the a d d i t i o n of a d d i t i o n a l external degrees insight we were very quickly able to achieve the desired result r o t a t i o n a l degree of f r e e d o m at each corner, as suggested b y some designs [1,4].
91 c o n s t a n t strain c o n d i t i o n . d e m o n s t r a t e that the task of f r e e d o m [9]. W i t h this b y a d d i n g an a d d i t i o n a l recent t r i a n g u l a r e l e m e n t
Concluding remarks W h i l e the r e a d e r m a y n o t have been c o n v i n c e d that finite e l e m e n t design is an art form, it is h o p e d that he will at least agree that w h a t goes on inside finite elements matters, that all finite elements are not alike in quality, a n d that the quality of finite e l e m e n t p e r f o r m a n c e d e p e n d s on the d e d i c a t i o n , insight, a n d a b i l i t y which the finite e l e m e n t designer brings to his task. I n these respects, finite elements d o not differ p e r c e p t i b l y f r o m a n y other p r o d u c t of h u m a n skill a n d intelligence.
References [1] ALLMAN, D.J., "A compatible triangular element including vertex rotations for plane elasticity analysis", Computers & Structures 19 (12) pp. 1-8, 1984. [2] BELVTSCIIKO,T. and W.E. BACHRACH,"Efficient implementation of quadrilaterals with high coarse-mesh accuracy", Comput. Meths. in Appl. Mech. Engrg. 54, pp. 279-301, 1986. [3] BELYTSCHKO,T., and W. LIU, "Test problems for shell finite elements", Finite Element Standards Forum, 26th Structures, Structural Dynamics, and Materials Conf., Orlando, FL, April 15, Book 1, pp. 25-44, 1985. [4] BERGAN,P.G. and C.A. FELIPPA,"A triangular membrane element with rotational degrees of freedom", Comput. Meths. Appl. Mech. Engrg. 50, pp. 25-69, 1985. [5] HARDER,R.L. and R.H. MACNEAL,"A new membrane option for the QUAD4", M S C / N A S T R A N Users' Conf., Pasadena, CA, March 1985. [6] IRONS, B.M., "Engineering application of numerical integration in stiffness method", AIAA Journal 14, pp. 2035-2037, 1966. [7] MACNEAL, R.H., editor, The N A S T R A N Theoretical Manual, Level 15.5, National Aeronautics and Space Administration, NASA SP-221(01), 1922. [8] MACNEAL,R.H., "A simple quadrilateral shell element", Computers & Structures, 8, pp. 175-183, 1978. [9] MACNEAL,R.H., Locking and the Patch Test for Four-Noded Trapezoidal Elements, MacNeal-Schwendler Memo. RHM-87, January 1985. [10] MACNEAL,R.H. and R.L. HARDER,"A proposed standard set of problems to test finite element accuracy", J. Finite Elements in Analysis & Design 1 (1), pp. 3-20, 1985. [11] DETERS,J. and R.H. WATERMAN,JR., In Search of Excellence, Warner Books, 1982. [12] PIRSlG,R.M., Zen and the Art of Motorcycle Maintenance, Morrow Quill Paperbacks, New York, 1974. [13] ROBINSON,J. and S. BLACKHAM,An Evaluation of Lower Order Membranes as Contained in the M S C / N A S T R A N , SAS, and PAFEC FEM Systems, Robinson & Associates, Dorset, U.K., September 1979. [14] ROBINSON,J. and S. BLACKHAM,An Evaluation of Plate Bending Elements: M S C / N A S T R A N , ASAS, PAFEC, ANSYS, & SAP4, Robinson & Associates, Dorset, U.K., August 1981. [15] TAIG, I.C., Structural Analysis by the Matrix Displacement Method, Engl. Electric Aviation Rept. No. S017, 1961. [16] TURNER, M.J., R.W. CLOUGH, H.C. MARTIN and L.J. ToPP, "Stiffness and deflection analysis of complex structures", J. Aero. Sci. 23, pp. 805-823, 1956.
85
ZEN AND T H E ART OF FINITE E L E M E N T DESIGN * Richard H. MACNEAL (Chairman) The MacNeal-Schwendler Corporation, 815 Colorado Boulevard, Los Angeles, CA 90041, U.S.A. Received February 1987 Abstract. The paper reviews Robert M. Pirsig's ideas concerning the concept of quality as expressed in his book Zen and the Art of Motorcycle Maintenance and applies them to the problem of designing finite elements. Using quadrilateral membrane elements as a typical example, it is conclude that finite elements differ remarkably in the quality of their performance and that improvements in quality occur in response to the dedication and insight which finite element designers brings to their task.
Introduction
Those who have read Robert M. Pirsig's Zen and the Art of Motorcycle Maintenance [12] (or Z A M M for short) know that it has almost nothing to do with Zen and only uses motorcycle maintenance as a vehicle for the expression of larger ideas as is hinted by its subtitle, "An Inquiry into Values". What the book is actually about, to put it as briefly as possible, is the concept of 'quality' and its manifestations in many fields, including science and technology. For this reason, if for no other, a brief review of Z A M M might interest those engineers who are concerned with the quality of the technology they produce. The book is superficially a novel about a motorcycle trip, but the plot and character development have been deliberately weakened in order to focus attention on the 'chautauquas' which the principal character composes over the long uneventful miles. The subject matter of the chautauquas ranges from technology to art to science to philosophy, but always the purpose is to explore the manifestations of quality so as to capture its essence. One of the principal points made is that quality, which the author equates with excellence, can easily be recognized but cannot be defined. About the closest the author comes to defining quality is to say that it is the inverse side of caring and that it lies in the relationship between people and the things they produce. Examples are taken largely from motorcycle maintenance but also from chemical engineering, rhetoric, mathematics, and other fields. Pirsig finds much evidence of a general lack of quality in the m o d e m world. For this he ultimately blames Western philosophy as inherited from classical Greece for putting too much evidence on truth and reason and not enough on values. The result is an outpouring of technology that is all too often ugly, shoddy, unsafe, or inhuman. While the latter point may have been more in evidence when Pirsig was writing ZAMM in the early seventies, it is arguable that today's technology has achieved a modestly higher level of quality. Environmental concerns are beginning to be heeded and terms such as human engineering and user friendliness have entered the jargon. The word has even begun to be heard on mahogany row where In Search of Excellence [11] is likely to be spotted on an executive's bookshelf. * A preliminary version of this paper was presented at the Fourth Chautauqua on Productivity in Engineering and Design, Coronado, CA, October 27-29, 1986. 0168-874X/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
R.H. MacNeal / Zen and the art of finite element design
86
To come closer to home, there is no question that the quality of finite element systems have improved in the last fifteen years nor can it be doubted that there is still a long way to go. The emphasis today is much more than formerly on the satisfaction of the finite element user's needs and on the minimization of his frustrations. Some of the manifestations of improved quality include computers which have become more available, more powerful, and easier to use; interactive graphics which has greatly facilitated the user's tasks of preparing input and examining output; and the finite element codes themselves which now address more applications in greater detail while becoming more rugged and more useable with fewer errors. Although finite element systems may have grown in size and scope and ease of use, their central feature for interpreting engineering mechanics continues to be the finite elements. One might expect of finite elements that their mathematical foundation is rigorous and unchanging but, unfortunately, the design of finite elements is much less of an exact science than, for example, the formulation of algorithms for the solution of the resulting equations. The element is, in fact, the principal repository of assumptions and approximations in finite element analysis. A succeeding section will examine the ways in which changes in the mathematical formulation of finite elements have improved the quality of finite element systems, but first it will be inquired whether finite element design qualifies as an art form in the sense that Pirsig applies the term to motorcycle maintenance.
Finite element design as an art form
Fig. 1 shows a typical finite element as it might appear in a textbook or in the graphical representation of a finite element model. For all but a very few specialists, Fig. 1 is a perfectly adequate representation of a finite element because they neither know nor care what goes on inside the element. Indeed, it is no more necessary to know what goes on inside a finite element in order to use it than it is to know what goes on inside a motorcycle in order to ride it. What goes on inside does, however, matter critically as far as performance and reliability are concerned. The task of designing a finite element is not a cut-and-dried operation as is indicated by the great number of finite elements that have been proposed in the last thirty years. Even though a lot of mathematical tools and physical principals are available to assist the specialist, finite element design also requires rather arbitrary design choices in all but a few exceptionally simple cases. An example of the latter is the three-noded membrane triangle shown in Fig. 2 which has two displacement degrees of freedom (dof's) at each external node. The resulting six external degrees of freedom must be balanced by six internal degrees of freedom which might, for example, be coefficients in a series representation of the internal displacement field. Two principals which were learned the hard way early in the history of
v3 l
vlV ,, v2 _
I
2
Fig. 1. A typical finite element.
II
Ul
Fig. 2. Constant strain triangle.
u2
R.H. Macneal / Zen and the art of finite element design
I
87
V4u4 ~ v3 u
Vl
v2 uI
u2
Fig. 3. Four-noded membrane element.
finite element design can account for all of the internal degrees of freedom. They are that rigid body motions (three internal dof,s) and constant strain states (three internal dof,s) must be exactly represented in order to achieve minimum acceptable performance. If these principals are invoked, then there is no need for other design choices. As a result, the design of the three-noded, six-degree-of-freedom membrane triangle has remained essentially unaltered since it was first introduced [16] in 1956. It is a staple of all finite element programs. The case of the four-noded membrane element shown in Fig. 3 involves additional choices. Since it has eight external degrees of freedom and since the rigid body motions and constant strain states account for six of them, it follows that two additional internal degrees of freedom must be selected. One might have imagined that the best possible choices for these two degrees of freedom would have been found and cataloged long ago but that is not the case. Prior to 1960, everyone simply filled the quadrilateral with triangles [16], then along came Taig [15] with his famous isoparametric element, then dozens of others right up to the present moment [2], and no doubt into the future. If optimum solutions for the minimal complexity offered by the membrane quadrilateral have proven so elusive, it can only be imagined how much more challenging are the more complex plate, shell, and solid elements. The point of this discussion is not merely that the quality of element performance depends on making appropriate choices but also that the act of choosing is a conscious effort by the finite element designer using not only the available tools but also his experience, imagination, and intelligence. That is where the art comes in, just as it does in motorcycle maintenance or in any other practical art. The job can be done well or badly, with more or less quality, which may be less apparent to the eye than it is in the utility of the result. The ways in which the quality of finite elements can vary will be examined next.
Quality in finite element performance and design The only way the average user can see evidence of quality in finite elements is by comparing the quality of output for runs using different finite elements. He might then find differences in accuracy or run time which could be attributed to the elements or he might find situations where one element had features not present in another or where one element occasionally produced completely misleading results. From such comparisons he would probably conclude that one element was generally better, or even with enough data that one element was better in some instances while another element was better in others. The performance measures which are of interest to finite element users can be summarized as follows:
R.H. MacNeal / Zen and the art of finite element design
88
• Accuracy. • Economy: • in element formation, • in problem solution. • Features: • for convenience, • for breadth of application. • Reliability or ruggedness. Accuracy and economy clearly provide a design trade-off because both accuracy and cost should be expected to increase with element complexity. The challenge is to provide both at the same time. The provision of features such as different options for materials properties increases development cost and requires sensitivity to user needs but it is not the same order of challenge. Reliability, as it applies to finite elements is probably the most important but also the most elusive of finite element qualities. An element may, for example, have excellent accuracy in some applications and very poor accuracy in others. As an extreme example, situations exist where a small change in element shape or loading can result in a drastic reduction in accuracy. As a response to such concerns, more attention has been directed in recent years to the quality of finite element analysis in general and to the reliability of finite elements in particular. This has led to the formation of committees and organizations devoted to the improvement of the finite element method (such as N A F E M S in the U.K.), conferences and forums on standards, quality, and reliability (such as forums at the 1984 and 1985 SDM meetings, a conference on reliability in July 1986 at Swansea, Harry Schaeffer's Fourth Chautauqua on Productivity in Engineering and Design in October 1986, and John Robinson's Fifth World Conference in 1987), and papers or reports comparing the accuracy of finite elements for a number of so-called 'standard' tests [3,10,13,14]. The latter are usually selected so that at least some of the elements exhibit gross inaccuracies. Such tests are beneficial if the purpose is to alert users to element configurations which should be avoided or to alert developers to defects which should be corrected. Dissatisfaction with element reliability is probably the most important motivating force for element improvement. At least that has been the case at MacNeal-Schwendler. The evolution of the quadrilateral membrane element mentioned earlier is a good example of the fruits of that motivation. As will be recalled, quadrilateral membrane elements vary chiefly in the way in which they account for the internal degrees of freedom which remain after rigid body motions and constant strain states have been correctly represented. The most important degrees of freedom beyond the constant strain states are those associated with inplane bending. At the very least, reasonably accurate results should be expected when a square element is subjected to inplane bending in one of its principal directions as shown in Fig. 4. Several quadrilateral membrane elements, including homegrown varieties, have been introduced into N A S T R A N and M S C / N A S T R A N and it is a simple matter to subject them to the test indicated in Fig. 4. The results of the test are tabulated in Table 1 along with the year in which the element was released for general use in M S C / N A S T R A N or its predecessor.
el2
el2
_\
/
)
Fig. 4. Inplane bending of a square element.
R.H. Macneal / Zen and the art of finite element design
89
Table 1 Results for inplane bending of a square element Element name
Date released
O/M
QDMEM QDMEM1 QDMEM2 QUAD4 a QUAD8 Q4S Exact
1970 1973 1973 1974 1980 1987 -
0.224 0.675 0.585 0.910 1.000 1.000 1.000
Called the modified QDMEM1 prior to 1976.
The evolutionary improvement is quite apparent. The earlier elements, particularly the QDMEM, are intolerably inaccurate while the later elements score perfect results. The excessive stiffness exhibited by the earlier elements has been diagnosed as a locking problem which is caused by the presence of parasitic shear strain. The symptoms become worse as the element's shape is modified from a square to an increasingly slender rectangle. It should also be mentioned, in fairness, that the QUAD8 has twice as many degrees of freedom as the first four elements and the Q4S has fifty percent more than the first four, as illustrated in Fig. 5. Thus the higher accuracy of the latter two elements has been purchased at the cost of increased run time. The QUAD8 has the further drawback that it does not exactly represent constant strain states for arbitrarily-shaped quadrilaterals. In spite of its less than perfect accuracy, the QUAD4 has been the workhorse of the MSC/NASTRAN element family. While its accuracy for inplane bending holds up with increasing slenderness, it has been discovered that the accuracy deteriorates badly if the element is distorted into parallelogram or trapezoidal shapes. This behavior can be investigated by the sequence of test problems shown in Fig. 6. The results for four MSC/NASTRAN quadrilateral membrane elements are presented in Table 2. They clearly show the hopelessness of QDMEM and the deterioration of QUAD4 accuracy with shape change. The other two elements perform well with the edge going to Q4S. Two major points emerge from the example of the quadrilateral membrane element. The first is that all elements are n o t alike and they can, in fact, exhibit important differences in quality of performance. The second point is that the quality of performance tends to increase with the passage of time due to the accumulation of design improvements. Each design improvement is the product of a creative act of caring by some designer, somewhere whether it be a breakthrough or a modest incremental improvement. The NASTRAN quadrilateral membrane elements again provide examples of both.
Av
U
w
U
v
a
v
b
w
c
Fig. 5. External degrees of freedom of NASTRAN and MSC/NASTRAN quadrilateral degrees of freedom. (a) QDMEM, QDMEM1, QDMEM2, QUAD4. (b) QUAD8. (c) Q4S.
90
R.H. MacNeal / Zen and the art of finite element design
Rectangular ShapeElements
Trapezoidal ShapeElements
Parallelogram ShapeElements
P
l~
P
Fig. 6. Inplane bending of cantileverbeam with three differentelement shapes. As has been mentioned, the earliest quadrilateral membrane elements, of which the Q D M E M and QDMEM2 are examples, consisted of groupings of triangular elements. The design of Q D M E M 2 is perhaps the earliest of such elements [16] and dates from 1956. The Q D M E M element was conceived by the present author about 1967 during the initial development of NASTRAN [7]. Since its performanc e is inferior to that of the earlier design, it represents the kind of backward step which can occur when ignorance and expediency prevail. The first breakthrough in the design of quadrilateral membrane elements came with the four-noded isoparametric element published by Taig [15] in 1961. This design was later copied into the QDMEM1 element in 1973. It embodies an efficient method for calculating the coefficients of an assumed internal displacement field and guarantees satisfaction of the rigid body and constraint strain conditions for any arbitrary quadrilateral shape. The Taig element was the model for all the assumed displacement elements designed thereafter. The membrane part of the QUAD4 element [8], for example, includes a minor modification to the Taig element which averages inplane shear strain in order to ameliorate the locking problem noted in Table 1. An important secondary breakthrough occurred in 1966 when Irons [6] generalized the Taig element to arbitrarily curved two- and three-dimensional shapes with any number of extra nodes per edge. In so doing, he generated the entire family of standard isoparametric elements. Irons was followed by other designers who introduced accuracy enhancing modifications such as those embodied in M S C / N A S T R A N ' s QUAD8. The four-node Q4S element [5], which is not yet in production, is the by-product of a lengthy effort to find a modification to QUAD4 which would successfully remove the locking problem Table 2 Results for inplane bending of a cantileverbeam (tip displacement due to tip load) Element n a m e QDMEM QUAD4 QUAD8 Q4S Exact
Rectangular elements 0.032 0.904 0.987 0.993 1.000
Trapezoidal elements 0.016 0.071 0.946 0.988 1.000
Parallelogram elements 0.014 0.080 0.995 0.984 1.000
R.H. Macneal / Zen and the art of finite element design
e x h i b i t e d for n o n r e c t a n g u l a r shapes while c o n t i n u i n g to satisfy the A f t e r n u m e r o u s failures we c a m e to the conclusion a n d were able to was i m p o s s i b l e w i t h o u t the a d d i t i o n of a d d i t i o n a l external degrees insight we were very quickly able to achieve the desired result r o t a t i o n a l degree of f r e e d o m at each corner, as suggested b y some designs [1,4].
91 c o n s t a n t strain c o n d i t i o n . d e m o n s t r a t e that the task of f r e e d o m [9]. W i t h this b y a d d i n g an a d d i t i o n a l recent t r i a n g u l a r e l e m e n t
Concluding remarks W h i l e the r e a d e r m a y n o t have been c o n v i n c e d that finite e l e m e n t design is an art form, it is h o p e d that he will at least agree that w h a t goes on inside finite elements matters, that all finite elements are not alike in quality, a n d that the quality of finite e l e m e n t p e r f o r m a n c e d e p e n d s on the d e d i c a t i o n , insight, a n d a b i l i t y which the finite e l e m e n t designer brings to his task. I n these respects, finite elements d o not differ p e r c e p t i b l y f r o m a n y other p r o d u c t of h u m a n skill a n d intelligence.
References [1] ALLMAN, D.J., "A compatible triangular element including vertex rotations for plane elasticity analysis", Computers & Structures 19 (12) pp. 1-8, 1984. [2] BELVTSCIIKO,T. and W.E. BACHRACH,"Efficient implementation of quadrilaterals with high coarse-mesh accuracy", Comput. Meths. in Appl. Mech. Engrg. 54, pp. 279-301, 1986. [3] BELYTSCHKO,T., and W. LIU, "Test problems for shell finite elements", Finite Element Standards Forum, 26th Structures, Structural Dynamics, and Materials Conf., Orlando, FL, April 15, Book 1, pp. 25-44, 1985. [4] BERGAN,P.G. and C.A. FELIPPA,"A triangular membrane element with rotational degrees of freedom", Comput. Meths. Appl. Mech. Engrg. 50, pp. 25-69, 1985. [5] HARDER,R.L. and R.H. MACNEAL,"A new membrane option for the QUAD4", M S C / N A S T R A N Users' Conf., Pasadena, CA, March 1985. [6] IRONS, B.M., "Engineering application of numerical integration in stiffness method", AIAA Journal 14, pp. 2035-2037, 1966. [7] MACNEAL, R.H., editor, The N A S T R A N Theoretical Manual, Level 15.5, National Aeronautics and Space Administration, NASA SP-221(01), 1922. [8] MACNEAL,R.H., "A simple quadrilateral shell element", Computers & Structures, 8, pp. 175-183, 1978. [9] MACNEAL,R.H., Locking and the Patch Test for Four-Noded Trapezoidal Elements, MacNeal-Schwendler Memo. RHM-87, January 1985. [10] MACNEAL,R.H. and R.L. HARDER,"A proposed standard set of problems to test finite element accuracy", J. Finite Elements in Analysis & Design 1 (1), pp. 3-20, 1985. [11] DETERS,J. and R.H. WATERMAN,JR., In Search of Excellence, Warner Books, 1982. [12] PIRSlG,R.M., Zen and the Art of Motorcycle Maintenance, Morrow Quill Paperbacks, New York, 1974. [13] ROBINSON,J. and S. BLACKHAM,An Evaluation of Lower Order Membranes as Contained in the M S C / N A S T R A N , SAS, and PAFEC FEM Systems, Robinson & Associates, Dorset, U.K., September 1979. [14] ROBINSON,J. and S. BLACKHAM,An Evaluation of Plate Bending Elements: M S C / N A S T R A N , ASAS, PAFEC, ANSYS, & SAP4, Robinson & Associates, Dorset, U.K., August 1981. [15] TAIG, I.C., Structural Analysis by the Matrix Displacement Method, Engl. Electric Aviation Rept. No. S017, 1961. [16] TURNER, M.J., R.W. CLOUGH, H.C. MARTIN and L.J. ToPP, "Stiffness and deflection analysis of complex structures", J. Aero. Sci. 23, pp. 805-823, 1956.