Practical Convergence-Divergence Checks for Stresses from FEA

Transcription

1 Pratial Convergene-Divergene Cheks for Stresses fro FEA G.B. Sinlair Departent of Mehanial Engineering, Louisiana State University, Baton Rouge, LA J.R. Beishei Developent Departent, ANSYS In., Canonsburg, PA S. Sezer Departent of Mehanial Engineering, Louisiana State University, Baton Rouge, LA Abstrat In pratie in finite-eleent stress analysis, the engineer first needs to know if key stresses are onverging, and seond if they have onverged to a reasonable level of auray. Then these stresses an be reliably used in design. The engineer further needs to know if, instead, key stresses are diverging beause of singularities present. Then these stresses an be of no diret use in design. This paper desribes soe straightforward heks for assessing onvergene or divergene of stresses fro FEA. The perforane of the onvergene-divergene heks suggested here is evaluated analytially with a siple error odel, and with series analogues. These heks are also evaluated on an extensive set of diverging trial probles and onverging test probles. Soe alternative heks put forward elsewhere are likewise evaluated. The evaluation of the suggested onvergene-divergene heks shows that they an fairly onsistently disern orretly whether stresses fro FEA are onverging or diverging. In addition, if onverging, the evaluation shows that these heks an reasonably aurately and typially onservatively gauge the degree to whih stresses have atually onverged. In ontrast, the evaluation shows that the alternative heks an onlude stresses are onverging when, in fat, they are diverging. Thus these alternative heks an be seriously isleading. Introdution Bakground and otivation Inherent in deterining stresses with finite eleent analysis (FEA) is disretization error. Only when disretization error is ontrolled is it possible to obtain suffiiently aurate stresses for oparison with strengths in designing for strutural integrity. Needed therefore are onvergene heks to ontrol disretization error. Coparably iportant for stresses fro FEA is to know if, in atuality, they are diverging beause of the presene of stress singularities. Then stresses annot be eaningfully opared with strengths in designing for strutural integrity. Needed therefore are divergene heks to asertain when a stress singularity is present. One exaple of the iportane of onvergene-divergene heks in the FEA of stresses oes fro the jet engine industry. This exaple onerns the attahent of blades to disks in jet engines. Shown in Figure 1 is a setion of a blade base and a segent of a disk to whih it is attahed. As a result of disk rotation, the blade wants to ove vertially upwards (indiated by the arrow in Figure 1), and this tendeny is restrained by ontat with the disk (on CC' in Figure 1 after the gap loses). The partiular detail in Figure 1 reflets the teahing of U.S. Patent 5,141,401 (Referene 1), whih lais that FEA shows the ontat stresses between the blade and the disk (on CC') are redued as a result of underutting the disk by an angle φ (just above C). In fat, with the underut, loal stresses are infinite as in

2 Eφ = ln r as r 0 (1) ( ν )( π φ) where r is a diensionless radial distane fro the underut vertex (Figure 1), and E, ν are Young s odulus, Poisson s ratio (taken as oon for the blade and the disk). Stress singularities of this type are identified by Sneddon (Referene 2, Setion 48.4): Fields for the speifi onfiguration of Figure 1 an readily be assebled using fields in Referene 3, on page 282. Clearly, then, the stress-based reason put forward for the advantage of underutting in Referene 1 is inorret. Given effetive onvergenedivergene heks, suh errors ould and should be avoided. Figure 1. Dovetail blade attahent with underutting Unfortunately the foregoing exaple of a singular onfiguration in elasti stress analysis is far fro a rare ourrene: Referene 4 furnishes a reent review of the asyptoti identifiation of stress singularities in 2D and 3D elastiity and undersores their abundant ourrene. Furtherore, while reourse to the referenes in Referene 4 an aid in the detetion of stress singularities, there reains the possibility that the onfiguration for FEA has a singularity that is yet to be identified asyptotially. In general, then, effetive onvergene-divergene heks are essential for eaningful stresses fro FEA. Literature review The last dozen years or so has seen an extensive aount of researh on onvergene of FEA for answers in general, and for stresses in partiular. For the ost part, this ativity is well refleted in reent finite eleent texts: Akin (Referene 5, 2005), Cook et al. (Referene 6, 2002), Reddy (Referene 7, 2006), and Zienkiewiz et al. (Referene 8, 2005). All of these referenes disuss onvergene of disretization error with esh refineent. Most of this disussion enters on rates of onvergene: Referene 6 provides an espeially good disussion of this aspet of onvergene. There are soe ipliit onvergene heks for test probles in these texts. However, there are no expliit onvergene heks for key stresses in appliations in any of these texts. Too, while Referenes 6 and 8 do disuss onvergene in the presene of a stress singularity, there are no expliit divergene heks in any of these texts. These two oissions in

3 these texts reflet the dearth in the literature at large of expliit onvergene-divergene heks for finite eleent stress analysis of appliations. Convergene heks in pratie onsist of two parts: an assessent of whether the FEA is onverging, then, if it is, an assessent of whether it has onverged. That is, is the FEA oving in the diretion of the true key stresses sought with esh refineent, then, if it is, has it got lose enough? The latter assessent requires an estiate of the ultiate error in the key stresses sought. That is, a loal a posteriori error estiate for the finest esh used. All of Referenes 5-8 give onsiderable attention to disretization error and furnish expliit a posteriori error easures. These easures are global in nature (e.g., Zienkiewiz and Zhu, Referenes 9,10). Suh error easures an be effetive in guiding eleent size gradation within a esh so that aurate alulations of peak loal stresses result (e.g., as in Referene 11). However, they annot be used diretly to estiate error in loal stresses aurately. On the one hand, this is beause an FEA of a speifi nonsingular onfiguration an have different loal stresses onverge at different effetive rates, so no one global error easure an apture these distint errors. On the other hand, this is beause an FEA of a speifi singular onfiguration an have a loal stress of interest diverge while a global error easure onverges. Needed for loal stresses are loal error easures. Of reent ties, there has been ore attention foused on loal error easures rather than just global: A fairly reent review is given in Ainsworth and Oden (Referene 12, Chapter 8). In tie, the results of this researh an be expeted to find their way into standard odes and assist stress analysts: For the present, their ipleentation typially requires developent of additional oputer algorithis used in onert with standard algoriths. Here, instead, we seek to provide siple loal error easures that an be used with onvergene tests using standard odes without any adjunt ode. Outline of reainder of paper We begin with soe siple suggestions for deiding if loal stresses of interest are onverging with esh refineent; that is, are h onvergent where h is a easure of eleent size. If judged onverging, we offer a opanion diret estiate of the extent to whih loal stresses have onverged; that is, a loal disretization-error estiate. If judged not onverging, we offer soe singularity signatures to gauge if the lak of onvergene is beause of a stress singularity, or beause the FEA siply has yet to onverge suffiiently. We also desribe soe alternative onvergene heks that ould take less oputational effort to ipleent. As a preliinary evaluation of the suggested heks, we next exaine their perforane with a siplified disretization-error odel. This evaluation suggests soe iproveents with a view to being onservative. As a first 1D evaluation of the iproved suggested heks, we exaine their analogous perforane when used in series suation. As a seond 2D evaluation, we exaine perforane on a series of singular trial probles and nonsingular test probles (soe detailed results for these probles are appended). These probles have analytially known singularities and known exat solutions, respetively. Here, though, we treat the as if they were appliations with unknown solutions in applying the iproved onvergenedivergene heks. Then we an draw on the known solutions to evaluate how well they atually work. We also subit the alternative heks to a like evaluation. We lose with rearks on both the iproved suggested heks and the alternatives in light of the results found. Candidate onvergene-divergene heks Siple onvergene heks In a test proble with a known exat solution, it is straightforward to assess onvergene of an FEA. With a first oarse esh and a seond refined esh, diret oparison with the exat result for the stress of interest reveals whether the error is reduing with esh refineent; that is, whether the FEA is onverging. Then oparison of the result for this stress fro the refined esh reveals whether the error is suffiiently low; that is, whether the FEA has onverged.

4 In an appliation, however, the true answer for the stress of interest is sought but, of ourse, not known a priori. Under these irustanes, heking for onverging requires at least two suessively-refined eshes for a total of three eshes: a oarse (C), a ediu (M), and a fine (F). Furtherore, the ediu and fine eshes should not be the outoe of inor refineents if a reasonably stern test of onverging is to result. To avoid this shortoing, at the outset we systeatially refine eshes throughout by saling eleent lengths by a sale fator λ with λ 2 (2) For λ =2, a saple sequene of eshes for a retangular plate is illustrated in Figure 2. For a general λ, if h is a linear easure of representative eleent size in the originating oarse esh, we have the following sequene of esh sizes: C - h, M - h, F - h 2 λ λ (3) This leads to nubers of eleents for different diensional probles as in Table 1.

5 Figure 2. Saple esh refineent ( λ =2) for a retangular elasti plate (a) oordinates and oarse esh, (b) ediu esh, () fine esh

6 Table 1: Eleent nubers in esh sequenes for heks Mesh 1D 2D 3D C N N N M λ N λ 2 N λ 3 N F λ 2 N λ 4 N λ 6 N On suh esh sequenes, we initially adopt the following onvergene heks. We judge the stress FEA to be onverging if 2 provided ( ) ( ) 2 - > - (4) f , where is the stress of interest and subsripts distinguish the esh f used to alulate it (if = = f, we judge the result to not only be onverging, but also have onverged) 1. Given opliane with (4), we judge the stress FEA to have onverged if - < e (5) f f s provided f 0, where es is the relative error level sought. In pratie, usually es less than 0.01 (1%) serves as an exellent level, less than 0.05 (5%) as a good level, and less than 0.1 (10%) as a satisfatory level, though ertainly ore stringent levels an be set. We have used onvergene heks of this ilk for soe tie (e.g., Referene 13, 1982), though only relatively reently expliitly stated the (Referene 14, 1999). We ake no lai of originality, either for Referene 14 or the present paper. It sees ertain that other finite eleent users have eployed siilar, if not opletely equivalent, heks in appliations (though, to date, we have not found an expliit stateent of suh heks). Before turning to divergene, it is iportant to set expetations re the likely suess of the onvergene heks in (4), (5). No suh heks an rigorously be guaranteed to predit onvergene when it truly ours. More iportantly, onversely no suh heks an rigorously be guaranteed to predit divergene when it truly ours. Given, therefore, soe eleent of judgent is involved, aution is appropriate to try and realize onservativeness in pratie. That is, to avoid prediting onvergene when in fat key stresses fro the FEA are diverging. With this aveat in ind, we adjoin the following additional requireent for onverging to (4): ( - )( - ) 0 > (6) f unless = f (if = f, we judge the result to be onverging and onverged). When (4) holds but (6) does not, further eshes are to be run to deide if FEA stresses are onverging. This added ondition reoves osillatory onvergene fro onsideration, a response for whih it is diffiult to estiate disretization errors in FEA stresses diretly. It also redues the probability of prediting onvergene when in fat stresses are osillatory and diverging, as on rare oasions they an be in elastiity. 1 One ight think that f ould siply play the role of the exat solution and take f - > f - as a onverging hek instead of (4): Suh a hek fails to detet the presene of stress singularities.

7 Singularity signatures When (6) is oplied with but (4) is not, we have two possibilities: the FEA is not yet apparently onverging/onverged on the esh sequene at hand, or the FEA is diverging. The forer ours when the fine esh result for the stress of interest is still a long way fro the true result beause of the oarseness of even this esh. The latter ours when stress singularities are present. Being able to distinguish whih irustane is appliable is useful beause it deterines whether ore refined FEA is useful, as in the forer ase, or useless, as in the latter ase. The ost reliable eans of distintion is via asyptoti identifiation of stress singularities (see Referene 4 for a reent review). Absent asyptotis, we use the following singularity signatures to deterine their presene. Stress singularities our in two predoinant fors in elastiity: power singularities and logarithi singularities. For power singularities, the loal stress behaves like - ( γ 0 ) = O r as r 0 (7) wherein 0 is an applied stress, r is a diensionless radial distane fro the singular point, and γ is the singularity exponent. For logarithi singularities, behaves like 2 ( ) = O ln r as r 0 (8) 0 With loal FEA values of being typially extrapolated fro nearby points in the eleents adjoining the singular point, and these eleents being refined as in (3), the following singularity signatures result: for power singularities f γ ~ ~ λ as h 0 (9) and for log singularities - ~ - ~ ln λ as 0 (10) f 0 h The result in (10) is the underlying reason for the stritly greater than sign in ondition (4) for onverging. To ipleent the asyptoti results in (9) and (10), we proeed as follows. For power singularities, we obtain suessive estiates of the singularity exponent via ( ) ( f ) ˆ γ = ln ln λ, % γ = ln lnλ (11) provided, and f are all of the sae sign and 0. Then we judge exponent estiates to be onstant and a power singularity present if 2 ˆ γ - % γ < 0.1 ˆ γ + % γ (12) That is, the hange in exponent is less than 10% of its average value. This perentage is provisional: Ultiately we need to validate it on nuerial experients. Given validation, then we judge there to be no power singularity if (12) does not hold. For logarithi singularities, we obtain suessive estiates of the inreent in the stress when (6) holds via = -, = - (13) f Then we judge inreents to be onstant and a log singularity present if 2 In (1), 0 effetively is Eφ ( ν 2 )( π φ )

8 2 - + < 0.1 (14) That is, the sae provisional perentage hange as for (12). In effet, (14) sets the range for whih nuerial equality is judged to hold between the ters on either side of (4), and an FEA is deeed to be diverging beause of a log singularity. If neither (12) nor (14) hold when (4) does not hold, we judge the FEA to be not yet apparently onverging/onverged, heneforth tered nononvergent. Typially with stress singularities, other ontributions to the stress at the singular point are o(1) as r 0. On oasion, however, stress singularities our in onert with a hydrostati pressure that an ask their presene. When this is the ase, (13), (14) an still be used to identify log singularities, but (11), (12) require adaptation to be effetive in identifying power singularities. If the agnitude of the hydrostati pressure is known, it an siply be subtrated out and (11), (12) then used. If the agnitude is not known, (11) needs to be replaed by f - f - f ˆ γ = ln ln λ, % γ = ln ln λ (15) - f - then (15), (12) used. In (15), f is alulated on a yet ore refined esh than the fine. For the oasional singular onfiguration, it is also possible for there to be a onstant ontribution to soe but not all noral stress oponents whih an ask the singular behavior in these oponents: Then siply onsidering other noral stresses helps reveal the singularities presene. Even with suh adjustents, regular stresses an aouflage singular stresses to soe extent. Aordingly we judge a singularity to our if any stress oponent satisfies (12) or (14) at the loation of the key stress of interest, and thus key stress values there to be unaeptable. Alternative onvergene-divergene heks When λ = 2, stern onvergene heks result, but the nubers of eleents in the fine esh are 4N, 16N and 64N for 1D, 2D and 3D probles, respetively (see Table 1). With soe planning when onstruting the initial oarse esh, and given today s oputer apabilities, this should be feasible for 1D and 2D probles. However, for 3D it ay prove to be too oputationally taxing. One eans of reduing oputational effort is to onfine esh refineent to the viinity of where the stress of interest ats: We investigate this possibility further with our test probles. Another eans is to use a saller sale fator, but still one that results in a fairly stern onvergene hek. A hoie of λ = 32would see to be a reasonable oproise in this regard. Then eleent nubers for the esh sequene of (3) are N, 3.4N and 11.4N in 3D. If this still proves too taxing, a esh sequene fored by first oarsening the oarse esh then refining it ay ake oputation feasible: that is, a sequene with 0.3N, N and 3.4N in 3D. Presuably also with a view to reduing oputational effort, soe alternative onvergene-divergene heks are soeties pratied in FEA. The first of these alternatives eploys a esh sequene with linearly inreasing nubers of eleents. Thus onvergene-divergene heks on the esh sequene C- N, M-2 N, F-3N (16) and so on. This sequene is used irrespetive of the diension of the proble, so 3D probles are ore oputationally tratable. Here we would use it in onjuntion with (4), (5), (6) to try to gauge onverging and onverged, and (12), (14) to try to distinguish diverging fro nononverging. Clearly as the sequene in (16) ontinues, we have diinishing hanges in eshes to the point of being inor (e.g., 100N, 101N, ). This akes it all too easy for (4), (5), (6) to be oplied with when the FEA ay be not onverged, or worse, not onverging. Thus suh linearly inreasing sequenes have the potential of being nononservative. However, possibly in pratie at the outset, the sequene of (16) suffies and thereby redues oputation. We investigate this possibility subsequently. The seond of these alternatives eploys just a two-esh hek. Here, then, onvergene-divergene heks on the first two eshes of (3) using just the ounterpart of (5) on C and M eshes to deide if

9 onverged or not. This abbreviated sequene is used irrespetive of the diension of the proble, so potentially taxing oputations for fine eshes in 3D are avoided. This two-esh approah would appear to be quite vulnerable to being nononservative. With just two eshes, nononservative estiates of final disretation error an easily result with osillatory onvergene, soething not guarded against by (6) beause it annot be applied. Further, there is no assessent of onverging. Consequently there is the inreased possibility of judging an FEA to have onverged, when in fat it is diverging beause of a weak singularity with sall, but not dereasing, stress inreents. We investigate these onerns for two-esh heks subsequently. Evaluation via an error odel Error odel We define the loal disretization error e for the stress of interest by e = ( - ) sgn ( ) (17) a h a where a is the atual or true value of, h the value as deterined via FEA on a esh of size h, and sgn is the signu funtion. With this definition, e is positive whenever < < f < a. That is, whenever a in agnitude is approahed fro below, the nor in FEA. Then we adopt the following siplified odel for e: ( ) e= e h h (18) 0 0 wherein e0 is the value of e on an initial esh of size h 0, and is the effetive onvergene rate (>0 for onvergene). Asyptotially as h 0, values of are known (see, e.g., Cook et al., Referene 6, Chapters 4, 9). For exaple, for a four-node quadrilateral eleent in plane elastiity, typially ~1. This is so provided the proble for FEA is suffiiently ontinuous. Then an even be inreased, using superonvergent reovery tehniques, to ~2(see Zienkiewiz et al., Referene 8, Chapter 13). However, even asyptotially, is redued absent suffiient ontinuity. For exaple, for onforing elasti ontat probles, ~½near the edge of ontat one ontat extent is established, and for a stress-free proud orner subtending an angle of 150º in an elasti solid, ~1/5, 0.534, in the viinity of the orner for states of antiplane shear, plane strain, respetively (fro, e.g., Referene 4, Setions 4.1, 2.1). Moreover, these three lower rates apply to higherorder eleents suh as the eight-node quadrilateral when used on the given exaples, despite the fat that typially ~2for eight-node eleents. In general, then, quite a range of asyptoti onvergene rates an result in the FEA of stresses. Suh variations are opounded for the effetive onvergene rate,, of the siplified error odel of (18). This odel approxiates atual power series in h for e with but one ter. With h suffiiently sall, the effetive of (18) does approah asyptoti values. However, for h as used in pratie, of (18) an differ appreiably fro asyptoti values. For exaple, if has two ontributions of O(h) and O(h 2 ) of the sae sign, an be lose to 2 on a given three-esh sequene, rather than its asyptoti value of 1. Further, if these two ontributions are of opposite sign, an be arkedly less than 1 and even negative on a given three-esh sequene. Hene, in pratie, the deterination of an effetive onvergene rate an be quite sensitive to the esh size used. Siple heks: Modified onverged heks Despite the soewhat apriious nature of in the siplified error odel of (18), we apply this odel to our esh sequene of (3) with h therein replaed by h 0. Assuing a >0 to aid the exposition, we thus have 2 a - = e0, a - = e0 λ, a - f = e0 λ (19)

10 Eliinating a by subtrating the seond of (19) fro the first and the third fro the seond, thereafter eliinating e 0 by foring a quotient of the results, then taking logs, gives - = ln ln λ f - (20) The sae result holds for a <0. Under (6), of (20) is guaranteed real. Further, onvergene with >0 ours in (20) if and only if (4) holds. Hene a justifiation of our onverging riterion when the siplified error odel of (18) applies. In effet in (5) we have an estiate of the absolute value of the relative disretization error e where Denoting this estiate by e ˆ, we have e = e a (21) eˆ = f - f (22) Now using the siplified error odel of (18) and substituting in (21), (22) fro (19) gives eˆ λ -1 = e 1- e (23) provided e 1, where e here is the relative disretization error for the fine esh. If λ >2and e> 0, the ratio in (23) is greater than 1, and ê is a onservative estiate of e. To guard against λ < 2, and thereby ê leading to a nononservative estiate, we adopt the odified estiate e% where e% = eˆ λ -1, λ < 2 (24) with as estiated by (20). That is, if λ =2, we use (24) instead of (22) when <1, and if λ =3 2, when < Now, therefore, our onverged hek opares ê of (22) with e s when λ 2, e% of ( 24 ) with es when λ < 2. We use this odified onverged hek on low-order eleents (fournode quadrilaterals, three-node triangles). For seond-order eleents (eight-node quadrilaterals, six-node triangles), we try to take advantage of their potentially iproved onvergene rates in probles other than ontat probles by taking, instead of (24), e% = eˆ λ -1, 2 e% eˆ λ > 2 = -1, 2 (25) with ontinuing to be estiated by (20). The first of (25) inludes (24). The seond of (25) liits the degree to whih we use estiated to extrapolate error estiates. 3 Even with these odifiations, our onverged hek an still be nononservative ( ê or e% < e ). This an happen when e < 0 and peak stresses are approahed fro above (see (23)). While this is not the nor in FEA, it an happen. However, then we should at least be alerted to the possibility of a nononservative error estiate by having > > f. It an also happen when of (20) overestiates the atual effetive onvergene rate. We an obtain an appreiation of how often this atually happens fro our later nuerial experients. 3 For yet higher-order eleents, one ould extend the range of appliability of the first of (25) to yet higher values of. This possibility is not investigated here.

11 Alternative heks: Divergene detetion? We also use the siplified error odel of (18) to evaluate the alternative onvergene-divergene heks, paying speial attention to when these alternatives an be nononservative. For the linearly inreasing esh sequene of (16), the ounterpart of (3) is 1 i 1 i h h h (26) C-, M- 2,F- 3 where i = 1, 2, 3 for 1D, 2D, 3D, respetively, is the nuber of diensions involved. Then applying (18) to the esh sequene of (26) with h therein replaed by h 0, and proeeding as previously, yields i = 3 i - -2 f i i (27) Fro (27) when (6) holds, we find our onverging hek (4) to be oplied with when >-i (28) Clearly sine < 0 orresponds to divergene, this alternative an be nononservative: That is, predit onvergene when in fat FEA stresses are diverging. Moreover, the higher the diension of the proble being analyzed, the ore likely suh nononservative preditions are to result. For the two-esh hek, applying (18) to the ounterpart of (22) gives ( λ ) e eˆ = a ( 1-e ) where ê and e are the estiated and atual, relative, disretization errors for the ediu esh. Thus the two-esh ounterpart of (5) an be et if λ 1, < 0, and FEA stresses are atually diverging, albeit slowly. Further, noralizing (29) by e for the ediu esh leads to (23) exept that now ê, e are for the ediu esh. As for (23), this an be nononservative whenever λ < 2. Now, though, absent a third esh we do not have available an estiate of as in (20) to help reedy the situation as in (24) or (25). Consequently this alternative an give nononservative estiates of disretization errors. As pointed out earlier, it an also be nononservative when (6) does not hold and we have osillatory onvergene. (29) Evaluation via a series analogy Series analogy and test series As a 1D analogy, we reall lassial series suation. We denote partial sus N by S N N n=1 S of the sequene { a } = an (30) Thus in our series analogy, N beoes the nuber of ters in the su instead of the nuber of eleents. To aid the exposition, we take a n >0. Hene S N is absolutely onvergent if the liit N is bounded. A neessary ondition for S N to be so onvergent is that n li a n =0 n (31) Further, fro d Alebert s ratio test,

12 li n li n a a n+1 n a a n+1 n < 1 onvergent series > 1 divergent series This test fails if the ratio equals 1. We now view the value of stress of interest,, as the outoe of a sequene of FEA deterinations sued in aordane with ( ) ( ) (32) = (33) f To interpret these suessive ters as a n in (30) with a n >0, we assue 0 < < < f < a. Then our onverging hek (4) under (6) is analogous to the ratio test of (32), exept that we do ake a deterination of divergene when the ratio is 1, and our onverged test is analogous to the neessary ondition of (31). To explore this analogy further, we seek to apply our heks of (4), (5), and (6), as well as (12), (14), (22), (24) and (25), to a set of series whih are independently known either to be onvergent or divergent, and see how well our heks predit this behavior. A first andidate pair of series to this end are the lassial aritheti progression (AP) and geoetri progression (GP): an = n, SN = N( N +1) 2 (34) n N a = α, S = α 1-α 1-α n N ( ) ( ) whereα is a onstant. The GP of (34) is onvergent if α <1, divergent otherwise, while the AP is siply divergent. Soe other series that an be used in this way are given in Table 2, together with asyptoti values as N fro integral estiates, S % N. In S % N, C1 = 1.459, C2 = 2.613, and Γ is Euler s onstant: These asyptoti values apply for N 4 and are atually aurate to within 0.1% for all suh N. The first of the series in Table 2 diverges as N, and so is like a power singularity: The seond diverges as lnn, and so is like a log singularity. The reaining series are onvergent with slow onvergene akin to a ontat proble, linear onvergene as an our in a regular proble with low-order eleents, and rapid onvergene as an be the ase with high-order eleents.

13 Table 2: Test series a n S % N R 1 pn 1 n 2 ( N +½) -C 2 1 ln (N+½) + Γ 1 1 C 32 2 n p( N +½) p2 1 2 n 1 4 n 2 π N +½ 4 π ( N +½) 2 8 Siple heks with odifiations Beause all a n >0, (6) is autoatially oplied with. To hek (4), we for the ratio S R = S 4N 2N 2N - S - S N (35) Then (4) for λ = 2, in effet, has the series onverging if R <1, diverging or nononvergent if R 1. Fro (34) for the AP, R ~ 4 > 1 as N, so (4) orretly predits not onverging for large N. In fat, R > 1 for all N, so that this orret predition holds for any N. Further, (12) holds for N 2 and γ ~2. This series, therefore, ould be interpreted as equivalent to the power singularity assoiated with a onentrated oent. Irrespetive of how appropriate suh an interpretation is, (12) ertainly predits that the series is divergent rather than just nononvergent. Fro (34) for the GP, Hene (4) predits the series is onvergent if N ( ) R = α N 1+ α (36) N 2 α < 1+ 5 While (37) is true, it is onservative: In reality, the GP is onvergent for any α <1. For the other series of Table 2, the asyptoti values of R of (35) given in this table show that (4) predits the first two to be not onverging and the last three to be onverging, as is indeed the ase. Moreover (4) does this for all N 1, not just N. ½ For a =1 n, (12) is oplied with for N 1, so divergene analogous to a power singularity (with n γ ~½) is predited rather than just a nononvergent series. For an =1 n, (14) is oplied for N 2, so divergene analogous to a log singularity is predited rather than just a nononvergent series. Given the asyptoti expressions S % N in Table 2, both of these preditions would see to be appropriate. 1 (37)

14 32 For a =1 n, the analogue of (22) with λ =2 has n In fat e ˆ ~ p2-1 as N C pn 2 e ~ 1 as C pn 2 N (38) (39) so ê is nononservative. Now, though, (20) gives ~ ½, so our odified estiate of (24) applies. This gives e~ % 1 as N (40) C pn 2 Hene an asyptotially orret estiate and support for our adoption of the odified error estiate of (24) when onvergene is slow. 2 For a =1 n, we have, fro the analogues of (22), (21), n 3 eˆ ~ e ~ as N (41) 2 2π N Hene an asyptotially orret error estiate. Here (20) gives ~1, so with λ =2there is no need to apply (24). 4 For a =1 n, we have n e ˆ ~, ~ as π N e 32π N N (42) Hene a onservative estiate by a fator of 7. Now, though, (20) gives ~3, so applying (25) we have 35 e % ~ as N (43) π N Hene still a onservative estiate but a ore aurate one. Thus applying our onvergene-divergene heks orretly predits divergene whenever series atually diverge. Further, these heks typially predit onvergene whenever series atually onverge, the exeption being for soe onvergent GP for whih they predit not onverging. Copanion error estiates, odified where appropriate, are asyptotially aurate, or if not, at least onservative. All told, a satisfatory and onservative perforane by these heks. Alternative heks: Divergene detetion? For the first alternative with linearly inreasing nubers of eleents, the analogue of (4) has series onverging when an >0if a = S - S > S - S = a (44) N+1 N+1 N N+2 N+1 N+2 That is, requiring an be dereasing in agnitude, as in (31). While (31) is a neessary ondition for onvergene, it is not suffiient. Aordingly, although (44) an predit onvergene when series are indeed onvergent, it an also predit onvergene when series are divergent. An exaple of the forer is the GP; for this series, (44) orretly predits onvergene when α <1. An exaple of the latter is the haroni progression of Table 2, an =1 n; for this series, (44) inorretly predits onvergene when, in ½ fat, the series is divergent. A further exaple is a =1 n where again (44) predits onvergene when n

15 the series is divergent. As with the previous error odel evaluation, therefore, this alternative hek exhibits an inability to detet divergene. Suh an inability represents a serious nononservative failure for this alternative hek. For the seond alternative with a two-esh hek, the analogue for series would have the series being satisfatorily sued ( e <0.1) when a n >0 if s ( ) S - S S < 0.1 (45) 2N N 2N For the AP, (45) is never oplied with so that this series would never be regarded as sued. However, for the haroni progression ( ) ( ) ( ) S - S S ~ ln 2 ln N as N (46) 2N N 2N Hene for N>1024, (45) is et, and the series is predited to be satisfatorily sued when, in fat, it is divergent. For still larger N, (46) with a two-esh hek ould even indiate suation to within an exellent error level. Analogously to the previous error odel evaluation, therefore, this alternative hek exhibits an inability to rejet diverging results. Suh an inability represents a serious nononservative failure for this alternative hek. Nuerial experients: Diverging stresses with power singularities Siple heks with odifiations Here we apply our siple onvergene-divergene heks, odified when appropriate, to see if they an detet divergene on a set of trial probles with power singularities. These are trial probles rather than true test probles beause they do not have known exat solutions throughout the elasti solid for analysis: However, these trial probles do have known, asyptotially-identified, stress singularities at a point in the elasti solid, and this level of analytial knowledge suffies for the present assessent. These singular trial probles are for: biaterial reentrant orners under tension, butt joints under tension, reentrant orners under tension and both in-plane and out-of-plane shear, and elasti half-spaes indented by rigid flat-ended punhes. All told, 14 different probles with known power singularities. A variety of 2D ANSYS eleents are used in the analysis of these singular trial probles: four-node quadrilaterals (4Q, PLANE42 of Referene 15), eight-node quadrilaterals (8Q, PLANE82 of Referene 15), six-node triangles (6T, PLANE2 of Referene 15), and three-node triangles (3T, triangle option in PLANE42 of Referene 15). This variety is eployed to assess the robustness of our onvergenedivergene heks with respet to eleent hoie. Typially these eleents are run in the plane strain ode, the axisyetri ode only being used for one ontat proble. Given the general asyptoti equivalene of plane strain and axisyetri singular stresses in elastiity (Zak, Referene 16), we do not expet uh differene between the FEA of these two states; This axisyetri, singular, ontat proble is erely run to onfir this expetation by oparison with its plane strain ounterpart. For the biaterial orners, free eshes with nearly uniforly-sized eleents are used (AMESH, Referene 15). For the other probles, unifor strutured eshes are used. All probles are analyzed on at least three eshes with systeati refineent as in (3) with λ =2(see Referenes for details). There are, of ourse, superior approahes to the FEA of singular stress probles given one ats on an awareness of a stress singularity s presene at the outset. Then for inverse-square-root singularities, there are quarter-point eleents (Henshell and Shaw, Referene 21; Barsou, Referene 22), and for other singularities, other id-side node plaeents (Wait, Referene 23). Alternatively, eleents adjoining the singular point an be enrihed with analytial expressions refleting asyptoti singular harater, or, at the very least, eleent sizes an be signifiantly redued in the viinity of the singularity to apture it better. All suh approahes would presuably then seek to deterine the oeffiient of the stress singularity, rather than the singular, and thus loally infinite, stress itself (see Referene 24 for a fairly reent review of various eans of extrating singularity oeffiients). With suh oeffiients aurately deterined, it ight then be possible to use the in a generalized frature ehanis treatent.

16 This is not what we are doing here. Instead we are preeding as if we had no awareness of the stress singularity s presene and asking our onvergene-divergene heks to reveal its presene. Hene the use of unbiased unifor or nearly unifor eshes to provide a fair assessent of the effetiveness of our onvergene-divergene heks in this regard. A total of 60 suh eshes are run on the trial probles with power singularities to assess the ability of our onvergene-divergene heks to detet FEA divergene. Viewing one three-esh sequene as a single nuerial experient, these eshes give rise to 26 experients. Here we present soe seleted results, then erely suarize the perforane of our onvergene-divergene heks re divergene detetion (opanion detailed results are set out in Appendix A). As a first seleted exaple, we onsider an elasti biaterial plate with a right-angled reentrant orner under tension (Figure 3, φ = 90º ). In addition to boundary onditions applying a noral transverse tration 0, the plate has stress-free onditions on the orner flanks and syetry onditions on the boundary at x = l (Figure 3). The upper and lower halves are perfetly bonded on y = 0 (Figure 3). The halves have a oon Poisson s ratio of ¼, but distint Young s oduli of E+, E - for y >0, y <0, respetively. Here we onsider E+ E - =16. Fro Bogy (Referene 25), this hoie ay be shown to lead to two singularities: l l 0 0 s = O + O as r 0 r r where s is any singular stress oponent at the singular point, and r is now as in Figure 3. With esh refineent, then, we an expet the stronger singularity to doinate and ˆ, γ γ% of (11) to approah with (12) holding: It reains to be seen how true this is for atual FEA results, espeially given the presene of a seond singularity. (47) Figure 3. Biaterial elasti plate with a reentrant orner under tension The FEA of this biaterial onfiguration uses free eshes of 6T eleents (see Referene 17 for further speifis). We thus have results as in Table 3 for the iu noralized stress

17 ( ) = y at x = y = 0 (48) 0 where oordinates are as in Figure 3. Like results for other orners are given in Appendix A, Table 15. Table 3: Divergene detetion for a biaterial orner under tension ( E+ E - =16) ˆ, γ γ% In Table 3, the esh nuber,, reflets the nuber of eleents used. Speifially N = N (49) where N 1 =128is the nuber of eleents in esh 1, the initial oarse esh, and N the nuber in esh. Hene for the of Table 3, refineent as in (3) with λ =2. Further, is the differene between on suessive eshes, hene the ounterpart of the differenes in (4). Clearly in Table 3, (6) is oplied with but (4) is not, so our onvergene-divergene heks do not predit onvergene. Then (11) has suessive exponent estiates as in Table 3, yielding 2 ˆ γ - % γ =0.01 (50) ˆ γ + % γ in opliane with (12), so our onvergene-divergene heks orretly predit the presene of a power singularity. Moreover, despite the presene of a seond singularity of weaker yet oparable strength, the singularity exponent ultiately estiated is lose to the atual exponent (3% higher, see (47)). This first exaple is for a strong singularity. A sterner test of our onvergene-divergene heks results if we have a weaker power singularity. Aordingly, as a seond seleted exaple, we onsider an elasti biaterial plate with a butt joint under tension (Figure 4). Aside fro the appliation of 0, the plate is stress free on its edges. The adherend (Young s odulus E + ) and adhesive ( E - ) are perfetly bonded on their interfae at y = l 20(Figure 4). The adherend and adhesive have a oon Poisson s ratio of 1/3, but distint Young s oduli with E+ E - =2. Absent a reentrant orner and with a saller aterial disontinuity, this onfiguration should be less singular than our first: Indeed, this is the ase. Fro Bogy (Referene 26), we now have l s = O 0 as r 0 r where r is as in Figure 4. The FEA of this biaterial onfiguration uses strutured unifor eshes of 4Q eleents. The initial oarse esh has N1 =110on R via syetry (see Figure 4). Subsequent eshes have (51) 2( -1) N =2 N (52) 1

18 eleents, = 2-5 being the esh nuber for this analysis. Thus eshes in opliane with (3) for λ =2. Results for this partiular butt joint are as in Table 4 wherein now ( ) = y at x = l, y = l 20 0 (53) with oordinates as in Figure 4. Like results for a butt joint with a stronger singularity are given in Appendix A, Table 15. Figure 4. Butt joint between elasti plates under tension

19 Table 4: Divergene detetion for a butt joint under tension ( E+ E- =2 ) ˆ, γ γ% Clearly in Table 4, again (6) is oplied with but (4) is not, so our onvergene-divergene heks do not predit onvergene. Then (11) has suessive exponents as in Table 4. Using these pairwise for eshes 1, 2 and 2, 3, then 2, 3 and 3, 4, and finally for 3, 4 and 4, 5, there results the following values for the quotient of (12): 2 ˆ γ - % γ = 0.21, 0.15, 0.08 ˆ γ + % γ (54) Hene on eshes 1-4, (12) is not oplied with and our onvergene-divergene heks have that the FEA does not, yet anyway, reveal a power singularity. While this represents a lak of resolution on the part of the heks on these results, nonopliane with (4) at least eans that a stress analyst would not aept the as either onverging or onverged. On eshes 4, 5, (12) is oplied with and our onvergenedivergene heks orretly predit the presene of a power singularity. Again the singularity exponent ultiately estiated is quite lose to the atual exponent (7% lower, see (51)). 4 As a third and final seleted exaple, we onsider another weak power singularity. This onerns an elasti plate with a right-angled reentrant orner as in Figure 3 when φ = 90º, but now oprised of a single aterial and under a shear tration of agnitude τ 0 on the upper and lower edges. The orner flanks ontinue to be stress free and now a ounterbalaning oent is supplied via onstant noral trations on the plate edge at x = l. Absent a aterial disontinuity and with shear loading, a weaker singularity than in (47) should result: Indeed this is the ase. Fro Willias (Referene 27), we now have where r reverts to as in Figure l s = O τ 0 as r 0 r (55) 4 This partiular onfiguration was also subjeted to a subsequent subodel around the singular point. The subodel used 4Q eleents and followed the proedure in Referene 11. It resulted in suessive estiates of γ of 0.041, 0.042; hene a deonstration that our onvergene-divergene heks an reveal power singularities with just loal refineent.

20 The FEA of this reentrant orner onfiguration uses strutured unifor eshes of 3T eleents. The initial esh has N1 =192and subsequent eshes have eleent nubers as in (52) (see Referene 18 for further speifis). Results for this partiular orner are given in Table 5 wherein ( ) τ = τxy at x = y = 0 τ (56) 0 with oordinates as in Figure 3. Results for other reentrant orner onfigurations with stronger singularities are given in Appendix A, Table 17. Table 5: Divergene detetion for a 90º reentrant orner under in-plane shear τ τ ˆ, γ γ% In Table 5, both (6) and (4) are oplied on eshes 1-4, so our hek has the iu shear stress onverging despite it being singular. However, alulating fro (20) results in the suessive low values (<1) given in Table 5, whene suessive error estiates fro (24) of e% = 0.25, 1.12 (57) Thus unsatisfatory error levels, and our odified onvergene-divergene heks would at least not predit τ had onverged. In fat, on eshes 2, 3, 4, (14) is oplied with and a log singularity predited. While this is atually an errant predition, it would ertainly not see τ being viewed as aeptable. On the 3, 4, 5 esh sequene, (6) is oplied with but (4) is not, so ˆ, γ γ% values are alulated and inluded in Table 5. These values satisfy (12), the quotient therein being 0.03, so on these ore refined eshes the weak power singularity present is deteted. Again the singularity exponent ultiately estiated is quite lose to the atual exponent (1% higher, see (55)). It transpires that the results in Table 5 are the only ones that oply with (4) in the trial probles with power singularities. Thus (4) detets divergene or nononvergene in 24/26 experients (see Appendix A, Tables 15, 17, 18 for the additional results). Further, one (4) is not oplied with, (12) indiates a power singularity in 23/24 experients. 5 For the present set of nuerial experients, this perforane is not signifiantly effeted by the use of free eshes instead of strutured (Apendix A, Table 15 f. Table 17a), or different 2D stress states (plane strain, antiplane shear, axisyetri; Appendix A, Tables 17, 18). 5 In 11 other experients on 6 probles with power singularities (Referenes 3, 20, 28), (4) indiated not onverging in 11/11, then (12) revealed power singularities in 9/11.

21 Alternative heks: Divergene detetion? We now use soe of our trial probles with power singularities to evaluate the alternative onvergenedivergene heks. In this evaluation, we are interested to see if these alternatives rejet peak stresses beause they are diverging, or at least beause error levels are unsatisfatory. We first apply the linearly inreasing esh sequene of (16) to the sae biaterial orner proble as in Figure 3 and Table 3. This proble has two quite strong singularities (see (47)): Hene diverging stresses should be readily deteted by an effetive onvergene-divergene hek. The FEA eploys free eshes of 6T eleents with N 1 = 128 (see Referene 17 for further speifis). Results for peak stresses are given in Table 6 wherein is as in (49). Table 6: Divergene detetion for a biaterial orner under tension ( E+ E - =16) ê Clearly in Table 6, the stress inreents ˆ are onotonially dereasing with esh refineent. Thus (4) and (6) are oplied with on either the esh sequene N 1,2 N1, 3 N1 or 2 N1, 3 N1, 4 N 1, and these FEA stresses are predited to be onverging. Then applying (22) gives ê as in Table 6, and a satisfatory ultiate error level on the first sequene, a satisfatory and approahing good level on the seond. Hene here this alternative onvergene-divergene hek would have the stress analyst aept finite stress values as satisfatory when in fat the stress is infinite. Furtherore, the sae erroneous onlusions are reahed with this hek on the sae esh sequene for two other orner probles with strong singularities (see Appendix A, Table 15). As earlier, therefore, an inability of this alternative hek to detet divergene and a nononservative failure as a hek. Given this ontinued failure, we heneforth dispense with any further assessent of this linearly-inreasing-esh-sequene alternative. We next apply the two-esh hek to the stresses in Table 4. Then the ounterpart of (22) realizes the following suessive estiates as the esh nuber inreases: e ˆ = 0.02, 0.02, 0.02, 0.03 (58) Hene all stresses would be judged onverged to within a good level of auray. Siilarly, applying the two-esh hek to the stresses in Table 5 realizes: e ˆ = 0.10, 0.07, 0.06, 0.06 (59) Hene, while the first estiate, being unsatisfatory, would inur further refineent, subsequent stresses would be judged onverged to within a satisfatory level of auray. As earlier, therefore, an inability of this alternative hek to rejet divergent stresses and a nononservative failure as a hek. Given this ontinued failure, we heneforth dispense with any further assessent of this two-esh alternative.

22 Nuerial experients: Diverging stresses with logarithi singularities Contat probles We start an assessent of the perforane of the odified onvergene-divergene heks on trial probles with log singularities by exaining FEA results for peak ontat stresses when an elasti halfspae is indented by a sooth rigid wedge (Figure 5). In the FEA of this plane strain proble, the finite region R for analysis siply has roller supports on its underside, and syetry onditions on its vertial boundary down the enter of the onfiguration with stress-free onditions on its other vertial boundary. With oordinates as in Figure 5, the iu, noralized ontat stress sought is ( ) =- z at x = z =0 p (60) wherein p is the average pressure on the wedge. This stress has a logarithi singularity akin to that of (1). Fro Sneddon (Referene 2, Setion 48.4), ( ) s =ord Eφ ln r as r 0 (61) where E is Young s odulus of the half-spae, and now φ is the gap angle shown in Figure 5 and r is a diensionless radial oordinate as in the sae figure. Analysis uses 4Q host eleents in onjuntion with surfae-to-surfae ontat eleents (CONTA171 and TARGE169, Referene 15), and a ontat algorith with a Lagrange ultiplier on the ontat noral and penalty on tangent (Referene 15). 6 The rigid wedge is siulated with 4Q eleents and a Young s odulus of 10 6 E. The initial esh for the half-spae is unifor with N 1 =400, while the esh for the wedge has but 10 eleents with lower nodes aligned with half-spae surfae nodes. Subsequent eshes for both have eleent nubers as in (52). Thus eshes in opliane with (3) for λ =2. Results for -3 when p E =10 and φ =1º and 2º are given in Table 7 (orresponding gap angles are indiated in parentheses). 6 The default ontat algorith in ANSYS (the augented Lagrangian algorith) an produe siilar results to those in Table 7, but requires finer eshes to rid the noral ontat stress of spurious tensile results.

23 Figure 5. Elasti half-spae indented by a sooth rigid wedge Table 7: Divergene detetion for wedge indentation ( ) 1º ( ) 2º For the 1º stresses in Table 7 on esh sequenes 1,2,3 and 2,3,4, (6) is oplied with but (4) is not, so diverging stresses are predited. Cheking (12) and (14) shows that neither are oplied with, so nononvergent stresses are predited rather than a log singularity. Nonetheless the stresses fro either of these two sequenes would still be not aeptable. On the esh sequene 3,4,5, both (4) and (6) are stritly oplied with and onverging stresses predited. However, fro (20), is low (0.04), and onsequently fro (24), e% =9.58, a ost unsatisfatory level. Further, heking (14) finds it to hold (as has to be the ase whenever < 1 7 for λ = 2 ), and a log singularity is thus orretly predited.