Bridging Methods for Atomistic-to-Continuum Coupling and Their Implementation

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1 Commun. Comput. Phys. doi:.428/ip Vol. 7, No. 4, pp April 2 Bridging Methods for Atomisti-to-Continuum Coupling nd Their Implementtion Pblo Seleson nd Mx Gunzburger Deprtment of Sientifi Computing, Florid Stte University, Tllhssee, FL , USA. Reeived 7 Mrh 29; Aepted (in revised version 4 September 29 Avilble online 2 Deember 29 Abstrt. Severl issues onneted with bridging methods for tomisti-to-ontinuum (AtC oupling re exmined. Different oupling pprohes using vrious energy blending models re studied s well s the influene tht model prmeters, blending funtions, nd grids hve on simultion results. We use the Lgrnge multiplier method for enforing onstrints on the tomisti nd ontinuum displements in the bridge region. We lso show tht ontinuum models re not pproprite for deling with problems with singulr lods, wheres AtC bridging methods yield orret results, thus justifying the need for multisle method. We investigte models tht involve multiple-neighbor intertions in the tomisti region, prtiulrly fousing on omprison of severl pprohes for deling with Dirihlet boundry onditions. AMS subjet lssifitions: 74B5, 74G65, 74S3, 74S5, 7-8, 7C2 PACS: x, 2.7.Ns, 2.7.-, y, 2.7.Dh Key words: Multisle modeling, tomisti-to-ontinuum oupling, boundry tretment. Introdution Atomisti models suh s moleulr dynmis re n epted pproh for urtely desribing mteril proesses tht our t the mirosopi level. Unfortuntely, mny systems of interest involve too mny prtiles to be fesibly treted using suh methods. As result, pproximtions to tomisti models tht re more effiient yet hve suffiient ury re of interest. Severl pprohes hve been proposed in tht sense; prtiulr mbitious pproh, lled MAAD ( mrotomisti b initio dynmis, tht ttempts to ouple ontinuum to sttistil to quntum mehnis is desribed in []. In generl, the methods desribed in the literture ttempt to ouple between two sles (e.g., miron- nd nno-sles. Some of the methods pply domin deomposition using Corresponding uthor. Emil ddresses: ps6@fsu.edu (P. Seleson,gunzburg@fsu.edu (M. Gunzburger Globl-Siene Press

2 832 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp the sme physil desription, i.e., the sme type of equtions, on the whole domin; this is the se of the qusi-ontinuum method [, 3, 4]. Other methods implement domin deomposition using different models in different domins, pplying some sort of oupling mehnism between them; some exmples of this type of pproh re found in [ 4, 8, 5 7]. For review of multisle mteril methods, the reder is referred to [8 2]. In tomisti-to-ontinuum (AtC oupling tehniques, n tomisti model is used in regions where mirosle resolution is neessry but elsewhere, (disretized ontinuum model is pplied. Severl methods were proposed in the mnner; for omprison of different multisle methods for the oupling of tomisti nd ontinuum models see [9]. The entrl question in AtC oupling methods is how to ouple the models, tking into ount their different ntures. In [, 2, 4], fore-bsed blending model is pplied to ouple tomisti nd ontinuum models. Blending is effeted in bridge region (lso lled interfe or blending or overlp region over whih the tomisti displement is onstrined by the interpoltion of the ontinuum displement. Seemingly, suh n pproh over-onstrins the system nd uses the omputtionl solution to devite from wht is expeted. Insted, we follow similr pproh to tht in [3] nd use Lgrnge multiplier method to enfore onstrints, thus reduing the number of onstrints. The fous in [3] is on omprison between overlpping nd non-overlpping domin deomposition methods, wheres we exmine severl omponents of overlpping domin deomposition methods (lso lled hndshke models [2] feturing two different blending shemes; we lso study issues relted to the implementtion of those methods. In ontrst to [], where oupling is implemented t the fore level, we blend the models t the energy level nd use the minimiztion of the blended potentil energy to determine the equilibrium onfigurtion of the system; n pproh, lled the Arlequin method, for whih the energy of the system is ssumed to be shred between o-existent models ws studied in [5 7]. This pper fouses on implementtion detils nd diffiulties of AtC oupling methods. In prtiulr, we study severl issues relted to the pplition of n ugmented Lgrnge multiplier method, inluding the effets of nonuniformity of the Lgrnge multiplier grid nd the vlue hosen for the penlty prmeter. Another issue of interest is the pplition of boundry onditions. In physil systems, long rnge intertions re the generl se; therefore, multiple-neighbor intertions hve to be implemented. Thus, n pproprite tretment is needed to orretly desribe system intertions ner the boundry where only few toms re vilble for intertion. In this pper, severl different pprohes for the se of Dirihlet, i.e., displement, boundry onditions, in the ontext of multiple-neighbor intertions, re disussed nd ompred in Setion 7. The outline of this pper is s follows. In Setion 2, we present the generl frmework of the AtC oupling method, s well s its implementtion in one dimension. We desribe the different omponents of the model s well s physil interprettion for the energy blending tehnique. In Setion 3, we introdue the quntittive tools implemented in the

3 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp model nlysis, s well s the different hoies for some of the model omponents whose effets we investigte; in ddition, we present n lterntive energy blending model. In Setion 4, we present omprison between the energy blending models, nd explin the orret pplition of the models to void the lk of prtil intertions round the bridge boundries. In Setion 5, we study the onvergene behvior of AtC blended models with respet to the different hoies of the model omponents, inluding the blending funtions, Lgrnge multiplier bsis, Lgrnge multiplier grid, finite element grid, nd penlty prmeter vlue. In Setion 6, we present singulr lod simultions showing tht the finite element method is not pproprite for the tretment of regions with singulr phenomen, wheres the AtC blended model produes orret results. In Setion 7, we provide mens for deriving model prmeters for tomisti nd AtC blended models for whih one hs multiple-neighbor tomisti intertions, nd lso ompre the effetiveness of different wys to impose Dirihlet boundry onditions in suh settings. We lso show tht for problems with singulr lods, some nomlies n rise in omputtionl solutions. Finlly, in Setion 8, we summrize the min onlusions of this reserh. 2 The model 2. The generl frmework The generl pproh to find the equilibrium onfigurtion of system in the presene of externl nd internl fores involves the minimiztion of the totl potentil energy. Beuse we desribe different prts of our domin Ω using different models, we need to find wy to ombine both the ontinuum nd the tomisti desriptions into single potentil energy expression. We define (in the referene onfigurtion three subregions: Ω C : the ontinuum region, : the tomisti region, Ω M Ω bri = ΩM ΩC : the bridge region. Using blending funtions, we determine the ontribution to the globl potentil energy of eh representtion (ontinuum nd tomisti in eh subregion. We n express the totl potentil energy of the system s where we hve the internl potentil energy of the system W =W int W ext, (2. W int = ξ(w dω C Ω C + 2 θ α,β w α,β, (2.2 α M β N α The tomisti model is ssumed to be vlid on the whole domin Ω, but to mke simultions trtble, is only used in the tomisti region Ω M ; the ontinuum model is ssumed to be vlid only on ΩC. We ssume there re no time dependent effets nd our system is t zero temperture.

4 834 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp nd the externl potentil energy of the system W ext = ξ(b udω C Ω C + ξ(t udγ C Γ C + θ( α f ext α d α. (2.3 α M M = {α α Ω M } is the set of indexes of the tom positions in the tomisti region, nd, for some given l (whih represents the length of the intertomi intertion, N α = {β β Ω : β α < l} is the set of indexes of the position of the toms interting with tom α; w α,β is the potentil energy of the tomisti bond α β; w = w (F is the potentil energy per unit volume of the ontinuum (s funtion of the deformtion grdient F; d α = x α α is the displement of the prtile α Ω M, with x α its position in the urrent onfigurtion, nd α its position in the referene onfigurtion; u u( the ontinuum displement of point t Ω C; fext α is the externl fore pplied on the prtile α; Γ C the boundry of ΩC ; B the externl body fore (fore per unit volume; T the boundry trtion (the dependeny of B, T, nd u on is omitted for simpliity; θ α,β n intertomi intertion blending funtion depending on θ(, α, nd β. The energy blending funtions ξ( nd θ( hve the form ξ(= Ω M\Ωbri α( Ω bri Ω C\Ωbri nd θ( = Ω M\Ωbri α( Ω bri Ω C\Ωbri so tht ξ(+θ( =, with α( hosen funtion. The energy blending funtions re used to divide the energy in the bridge region Ω bri between the o-existent models, so tht eh model ontributes prtilly to the totl energy in the bridge region. To pply displement onstrints between the ontinuum nd tomisti desriptions in the bridge region, we use the ugmented Lgrngin method; in ddition to Lgrnge multipliers for onstrint enforement, this method dds penlty term to the potentil energy. The totl potentil energy is repled by the expression, W AL =W int W ext +λ T g+ 2 pgt g, (2.4 with λ={λ γ } vetor of Lgrnge multipliers for the onstrints nd g={g γ } the onstrint eqution vetor. Note tht the penlty term is positive qudrti form; thus, minimiztion renders this term smll; p is referred to s the penlty prmeter. 2.2 A liner one-dimensionl se To better fous on the effets produed by the new pprohes, we onsider simple onedimensionl model with liner onstrints. The expressions for the internl nd externl

5 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp energies, i.e., (2.2 nd (2.3 respetively, redue to W int f = W ext f = i N d α= ξ(w d+ θ α,β w α,β, i 2 β N α N d ξ(bu d+ξ(tu f + i α= θ( α f ext α d α, where N d is the number of prtiles in Ω M ; i, f re the boundries of the domin. The liner onstrints over the bridge region re given by g γ = u( γ d γ = γ F {α α Ω bri }, (2.5 i.e., the onstrints re pplied to tomisti prtiles in the bridge region. To obtin the disrete equtions, we implement finite element (FE method in Ω C nd moleulr mehnis desription in Ω M ; thus, we n express the pproximte displement field u h ( in Ω C in terms of the FE bsis funtions {ωh i (}, i=,,n u, s follows: u( u h (= N u i= ω h i (uh i, (2.6 where the u h i s denote the FE displements t the FE nodes, nd N u is the number of FE nodes in Ω C. The Lgrnge multiplier (LM field is lso expressed in term of bsis funtions {Λ K (}, K =,,N λ, (ultimtely resulting in redution in the number of onstrint equtions λ(= N λ K= Λ K ( λ K, with λ K the LM oeffiients nd N λ the number of LM grid nodes in Ω bri. A generl piture of our domin (ssuming Ω M =[ i,], Ω C =[, f], nd Ω bri =[,] is presented in Fig., with the thin vertil brs representing the FE nodes, the thik vertil brs the LM grid nodes, nd the irles the tomisti prtiles. Stble onfigurtions of the AtC blended system re found by minimizing the energy (2., subjet to the onstrint (2.5, i.e., by solving the following set of equtions: W AL u h j =; j=,,n u, W AL d ζ =; ζ=,,n d, W AL λ L =; L=,,N λ. (2.7 The integrls in the ontinuum region Ω C re extended to the entire domin, i.e., Ω = [ i, f ], sine the energy blending funtion ξ( vnishes outside Ω C.

6 836 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp i Atomisti Bridge Continuum Figure : Atomisti-to-ontinuum oupling multisle grid showing the toms (irles, finite element nodes (thin vertil brs nd Lgrnge multiplier grid nodes (thik vertil brs. Three regions re defined in the entire domin Ω=[ i, f ]: the tomisti region Ω M=[ i,], the ontinuum region Ω C=[, f], nd the bridge region Ω bri =[,] where both the tomisti nd the ontinuum models o-exist. The resulting system of equtions is f 2 i N d α= ξ( w u h j f = i d+ γ F ( Nλ Λ K ( γ λ K ω h j ( γ+ p K= γ F ξ(b(ω h j (d+ξ(t(ωh j ( f w α,β θ α,β β N α d ζ N λ K= = θ( ζ fζ ext, ζ=,...,n d, Λ L ( γ γ F ( Nu ωi h ( γu h i d γ i= where the inditor funtion Λ K ( ζ λ K I F (ζ p ( Nu f ωi h( γu h i d γ ω h j ( γ i=, j=,...,n u, i ωi h( ζu h i d ζ I F (ζ (2.8 i= ( Nu =, L=,...,N λ, { if ζ F, I F (ζ= if ζ / F, nd F is defined in (2.5. Above nd in the reminder of the pper we dopt the onvention of using Greek subsripts, i.e., α, β,γ,ζ, to refer to tom numbers, Ltin lowerse subsripts, i.e., i,j, to refer to FE node numbers, nd Ltin upperse subsripts, i.e., L,K, to refer to LM node numbers. There re severl hoies we hve to mke in order to implement our model; in the following, we present the min hoies nd reltions for our model omponents. FE bsis funtions. We pproximte the displements of our system in the ontinuum region through FE method. The stndrd ht funtions re hosen s bsis for For the unknowns orresponding to the displements of the first tom nd the lst FE node, we reple the equtions by Dirihlet boundry onditions, i.e., d =u( nd u h N u =u( h N u respetively, with u( the ext solution of our problem. In ddition, in the se of nerest-neighbor intertions, the lst tom of the tomisti region of the AtC blended system, i.e., α= N d, is supposed to intert with prtile t the position Nd + Nd +s, where s is the tomisti sping; the position Nd + is in the ontinuum region beyond the tomisti region of the AtC blended system, i.e., Ω C\Ωbri, thus, n pproprite tretment in this se would be to ssume the displement d Nd + of prtile t Nd + is obtined by the interpoltion of the ontinuum pproximtion t tht point (see Setion (2.9

7 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp ontinuous pieewise liner funtions with respet to grid { h j }, j=,,n u: ω h j (= h j h j h j h j+ h j+ h j for h j h j for h j h j+ ω h j (= otherwise. Potentil energy. We hoose liner elstiity model for the potentil energy density w, nd, ordingly, liner spring model for the tomisti potentil w α,β : w = 2 K ( du d 2, w α,β= 2 K ( 2, dβ d α (2. s with K the elsti modulus, K /s the spring onstnt, nd s the tomisti sping; we ssume nerest-neighbor intertion, i.e., w α,β = for β α =. Furthermore, symmetry of the intertomi intertion blending funtion is ssumed, i.e., θ α,β = θ β,α. The tomisti nd (disretized ontinuum models result in the sme elsti energy, if we hoose K = K for nerest-neighbor tomisti intertion, uniform FE grid with resolution identil to tht of the tomisti system, i.e., = s, nd bsis of ontinuous pieewise liner funtions for the FE method. Disretized system. A more speifi disretized system of equtions is obtined by pplying our ontinuum nd tomisti intertion models (2. into the system of equtions (2.8; then, we obtin, for the se of nerest-neighbor intertion nd pieewise liner FE bsis funtions, the disretized system of equtions (2.-(2.3. { ( } h j K ξ(d h j (j h + p ω h h j 2 j ( γω h j ( γ γ F { [( ( h j h + K ξ(d h ( h + j+ ξ(d j j h j 2 h j } ( 2 +p ω h j ( γ u h j γ F u h j ( h j+ h j 2 { ( } h j+ + K ξ(d h j (j+ h + p ω h h j 2 j+ ( γω h j ( γ γ F ( p γ F ω h j ( γd γ + N λ K= Λ K ( γ ω h j ( γ γ F f = ξ(b(ω h j (d+ξ(t(ωh j ( f ; j=,2,,n u, (2. i i λ K u h j+ ]

8 838 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp ( [ K θ α,α d α + (θ α,α +θ α,α+ s p N u i= ω h i ( αu h i I F(α ( N u Λ L ( γ ωi h( γ i= γ F with θ α θ( α. N λ K= ( K 2.3 A physil interprettion for energy blending s ] ( K + pi F (α d α θ α,α+ d α+ s Λ K ( α λ K I F (α=θ α f ext α ; α=,2,,n d, (2.2 u h i Λ L ( γ d γ = ; L=,2,,N λ, (2.3 γ F So fr we hve presented model bsed on energy blending. One n energy blending form is imposed, the minimiztion of the potentil energy provides the equilibrium onfigurtion in the presene of externl fores, given n internl potentil energy funtion. In order to give possible physil interprettion to our blending sheme, we strt from modified version of the ontinuum equilibrium eqution, introduing the energy blending funtion ξ( in prtiulr wy s follows: d (ξ(p(+ξ(b(=. (2.4 d In the se of hyperelsti mterils, the Piol stress is given by P = Ψ F, with Ψ = Ψ(F the strin-energy funtion nd F = I+Grdu the deformtion grdient. In our onedimensionl se, Ψ w = ( du 2 2 K = d 2 K (F 2 ; P= Ψ F = K du (F =K d. (2.5 Let us develop numeril sheme using FE method, strting from the equilibrium eqution ( d du ξ(k +ξ(b( =. d d Multiplying by the test funtion w h j (, integrting nd then using integrtion by prts, we obtin f i ξ(k du d (wh j (d= f i ξ(b(w h j (d+ξ(t(wh j ( f i, (2.6 du with T=K d the boundry trtion (in the -D se, T=±P. Assuming the displement is pproximted by FE interpoltion, i.e., u( u h ( s in (2.6, we n rewrite (2.6 s f ξ( w f i u h d= ξ(b(w h j (d+ξ(t(wh j ( f ; j i i

9 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp this is the ontribution of the ontinuum model to the system (2.8. In (2.4, blending ws introdued in the physil quntities of the system; thus, it n be interpreted s hnge in the elsti modulus, i.e., K ξ(k, nd in the body fore, i.e., B( ξ(b(; therefore, those omponents hnge from full ontribution in the ontinuum region Ω C\Ωbri to null ontribution in the tomisti region Ω M\Ωbri. 3 Model nlysis omponents Below, we nlyze our model performne in terms of the errors of the AtC blended model when ompred to pure-tomisti one. We investigte the sensitivity of the results with respet to severl omponents of our model, i.e.,. energy blending funtions form in the bridge region: liner vs. ubi, 2. form of the intertomi intertion blending funtion, 3. Lgrnge multiplier grid properties: uniformity nd resolution, 4. Lgrnge multiplier bsis funtions hoie: pieewise liner vs. onstnt, 5. finite element grid resolution, 6. penlty prmeter vlue. For these purposes, we introdue, in Setion 3., some quntittive tools. In Setions , we provide different implementtion hoies for some of the model omponents, i.e., points -4 of the list bove. Then, in Setion 3.6, we disuss n lterntive AtC blended model. 3. Quntittive mesurements Totl error. We re interested in mesuring the globl error of our numeril simultions. Assuming the pure-tomisti model gives the orret solution for our system, we would like to lulte the error of the simultions produed by our AtC oupling methods in omprison to the pure-tomisti one. In the region [ i,], we ompre the displements between the toms in the pure-tomisti nd the AtC blended models, wheres in (, f ], the omprison is between the displements of the toms in the pure-tomisti model nd the interpoltion of the FE solution of the AtC blended models t the sme positions. The lultion of the error ǫ is done using the L 2 -like norm ( N d ǫ= N α u α α=(d 2 + N N α=n d + (u h ( α u α 2 /2, (3. with N the totl number of toms in [ i, f ] in the pure-tomisti model, N d the number of toms in [ i,], i.e., the number of toms in the tomisti region of the AtC blended model, d α the displement in the AtC blended model of the tom whih ws originlly t the position α, u α the displement in the pure-tomisti model of the tom originlly t the position α, nd u h ( α the FE pproximtion of the ontinuum displement t the position α.

10 84 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Cuhy strin. Devitions of the displements for the se of smooth externl fores n be reltively smll in the sense tht it is diffiult for the eye to pereive them. An lterntive wy to visulize those devitions is through the slope of the displements, i.e., the Cuhy strin. We mesure the strin t i s e( i =(u( i+ u( i /( i+ i. This is pplied to the tomisti displements of both the pure-tomisti nd AtC blended models s well s to the FE pproximtion of the ontinuum displement in the ontinuum domin of the AtC blended model. 3.2 Energy blending funtions One hoie for the energy blending funtions ξ( nd θ( is pieewise liner funtions: <, ξ(=, (3.2 >, with θ(= ξ(. This hoie is not C -ontinuous beuse the derivtives re disontinuous t the boundries of the bridge region, i.e., t = nd =. To see if smoother trnsition improves the ury of AtC simultions, we lso use pieewise ubi funtions with the requirement of C -ontinuity on [ i, f ], i.e., ξ( =, ξ( =, ξ ( =, nd ξ (=. Using these onditions, we obtin the pieewise ubi blending funtion with ξ(= <, αx 3 +βx 2 +γx+δ, >, α= 2 ( 3, β= 2 α(+, γ=3α, δ= 2 α2 (. The reson we onsider ubi s well s liner blending funtions θ nd ξ is tht the C ontinuity of the ubi funtions fores θ (respetively, ξ to be smll, reltive to the liner se, in the bridge region ner the ontinuum (respetively, tomisti boundry = (respetively, =. This results in wekened effet of the tomisti (respetively, ontinuum model in the bridge region ner the ontinuum (respetively, tomisti boundry whih is perhps desirble beuse presumbly we wnt the ontinuum model to dominte in the bridge region ner the ontinuum boundry nd we ertinly wnt the tomisti model to dominte in the bridge region ner the tomisti boundry. 3.3 Intertomi intertion blending funtion We use two different options for the intertomi intertion blending funtion θ α,β in the tomisti region; eh one is onsistent with different integrtion rule for the integrl, (3.3

11 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp whih ontins the energy blending funtion ξ(, in the ontinuum portion of W int. To show these reltions, we strt from the disretized ontinuum eqution (2., set p =, nd void the terms orresponding to the onstrints, in order to fous only on the ontributions from the internl nd externl energy expressions; the term ontining the trtion is negleted beuse we look t the internl nodes. The expression obtined is s follows: K ( h j h j ξ(d u h j u h ( h j (j h K j+ u h j+ u h j h j 2 ξ(d h j (j+ h h j 2 f = ξ(b(w h j (d. (3.4 i Trpezoidl rule. Using the trpezoidl qudrture rule for the integrtions, we get ( ξj +ξ j u h j u h ( j ξj +ξ j+ u h j+ u h j K K 2 h 2 h = ξ j B( h j h, with ξ j ξ(j h, where we hve used uniform FE grid with resolution h. Assuming h=s nd j=α, we n write the equivlent expression for the tomisti intertion: ( ( ( ( θα +θ α dα d α θα +θ α+ dα+ d α K K = θ α fα ext. 2 s 2 s This gives some insight for the hoie of θ α,β = 2 (θ α+θ β ; we refer to this intertomi intertion blending funtion pproh s the verge rule. Furthermore, we obtin the reltions K = K nd fα ext =B( α s tht re implemented through our model. Midpoint rule. We now onsider (3.4 with midpoint qudrture rule to pproximte the integrls to obtin ( h j + h j u h j u h j K ξ 2 h K ξ ( h j +j+ h u h j+ u h j 2 h = ξ( j B( h j h. Similrly to the verge rule derivtion, we obtin the midpoint rule for the intertomi intertion blending funtion, i.e., θ α,β = θ ( α + β Lgrnge multiplier grid: uniformity nd resolution We investigte the sensitivity of the AtC blended model on the number of LM grid nodes nd the differene between the results obtined when using uniform nd nonuniform We use the midpoint rule for the integrls on the left-hnd side of (3.4, but still use the trpezoidl rule for the integrl on the right-hnd side.

12 842 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp ( Uniform LM grid (b Nonuniform LM grid Figure 2: Comprison between uniform ( nd nonuniform (b Lgrnge multiplier (LM grid. The LM grid nodes re represented by the thik vertil brs; the thin vertil brs orrespond to finite element grid nodes, nd the irles to toms. The region [,] represents the bridge Ω bri. LM grids. The ide behind the implementtion of nonuniform LM grid is to strongly onstrin the tomisti displements by the ontinuum pproximtion ner the ontinuum region Ω C\Ωbri, wheres leving the toms less onstrined lose to the tomisti region Ω M\Ωbri. To hieve tht, we hoose the number N λ of LM grid nodes; let =( /(N λ, with [,] the bridge region, nd, for uniform grid, simply hoose the grid points λ (i=+ (i, i=,,n λ. (3.5 For the nonuniform grid, we pply mpping to (3.5 so tht ( π λ (i=+( sin 2 λ (i, i=,,n λ determines the LM grid points. In Fig. 2, we present, for illustrtion, omprison between uniform nd nonuniform LM grids, implemented on n AtC oupling multisle grid with N d =, N u =, nd N λ =7; the bridge domin is [,]=[.4,.6]. In this exmple, equivlent pure-tomisti nd pure-ontinuum FE grids would hve 67 toms nd 6 nodes respetively, in the entire domin, i.e., [,], resulting in toms per finite element. 3.5 Lgrnge multiplier bsis funtions hoie: pieewise liner vs. onstnt We use two different pprohes for the LM bsis funtions: pieewise onstnt nd pieewise liner, nd see how the results ompre. Note tht the number of LM bsis funtions is one less for the pieewise onstnt hoie thn for the pieewise liner hoie, i.e., we hve one less eqution in our system.

13 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Blending models Model I. The pproh we strted from is blending of the energy in the form of (2.2- (2.3 with the totl potentil energy given by (2.. We hve shown, in the one-dimensionl se, tht, in the ontinuum, this is onsistent with introduing the energy blending funtion ξ( in the equilibrium eqution s hnge in the elsti modulus nd in the body fore s in (2.4. In [], it ws shown tht blending similr to the one presented in Model I leds to the stisftion of Newton s third lw. To mke the system of equtions we re deling with more ler, we write down the ontributions of the internl nd externl potentil energies to the disretized equtions (2. nd (2.2, i.e., we neglet the LM nd penlty expressions. Using trpezoidl rule for the integrls on the left-hnd side (ssuming pieewise liner energy blending funtion ξ( nd orrespondingly the verge rule for the intertomi intertion blending funtion we obtin ( [ ( ( ] ξj +ξ j K 2 (j h h j uh j + ξj +ξ j K 2 (j h h j + ξj +ξ j+ K 2 (j+ h h j u h j ( ξj +ξ j+ K 2 (j+ h h j uh j+ = f ξ(b(ω h j (d+t( f ω h j ( f, i j=,2,,n u, (3.6 ( ( [( ( ]( θα +θ α K θα +θ d α + α θα +θ + α+ K d α 2 s 2 2 s ( ( θα +θ α+ K d α+ = θ α fα ext ; α=,2,,n d. (3.7 2 s In ddition, the disretized ontinuum eqution (3.6, in the limit of the tomisti resolution, ssuming K =K nd B( α s= fα ext (implementing the trpezoidl rule for integrting the right-hnd side nd negleting the trtion term, dds to the tomisti eqution (3.7 to give the pure-tomisti fore blne eqution, i.e., K s d α +2 K s d α K s d α+= f ext α. (3.8 Model II. A different pproh for the blending of the ontinuum nd tomisti models is obtined by letting the energy blending funtions multiply the whole equilibrium equtions. The ontinuum expression is { } d ξ( d (P(+B(=. (3.9 We multiply (3.9 by the test funtion w h j ( nd integrte; using integrtion by prts nd pplying the expliit form of the nominl stress ppering in (2.5, i.e., P(=K du d, we

14 844 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp get = f i f i du f du ξ(k d (wh j (d+ K i d ξ (w h j (d ξ(b(w h j (d+ξ(t(wh j ( f i, (3. tht is similr to (2.6 but with n extr term, i.e., the seond term on the left-hnd side of (3.. A disretiztion of (3. is [ ] K ξ j (j h h j uh j +ξ K j (j h h j + K (j+ h h j u h j ξ K j (j+ h h j uh j+ f = ξ(b(ω h j (d+t( fω h j ( f ; j=,2,,n u, (3. i ssuming pieewise liner energy blending funtion ξ(, nd tking its derivtive s ξ (=(ξ j ξ j /(j h h j for (h j,h j. In the tomisti region, the orresponding expression is obtined by multiplying the pure-tomisti fore blne eqution (3.8 by θ α. The resulting eqution is K θ α s d K α +2θ α s d K α θ α s d α+= θ α fα ext ; α=,2,,n d. (3.2 Notie tht we hve implemented the ontinuum nd tomisti intertion models ppering in (2.. In the se of nerest-neighbor intertion, nd ssuming FE grid resolution identil to the tomisti one, it is possible to show tht Newton s third lw is stisfied when tking into ount both the tomisti nd ontinuum fore ontributions to eh tom/node. In prtiulr, under those ssumptions, dding the tomisti nd disretized ontinuum expressions, i.e., (3. nd (3.2, results in the pure-tomisti fore blne eqution (3.8. The tomisti nd ontinuum ontributions to the equilibrium/fore blne equtions, both in Model I nd Model II, dd, under pproprite ssumptions, to the puretomisti fore blne eqution (3.8. This motivtes the following sttement. The AtC blended model n be redued to pure-ontinuum FE pproximtion with nonuniform mesh. This n be obtined if we hoose, in our AtC blended model, the sme FE grid resolution in the bridge region, s in the tomisti model, i.e., j h h j = s for j h,h j Ωbri, nd prtiulr qudrture rule for the right-hnd side, so tht the externl fores in the ontinuum nd tomisti models re the sme (see, e.g., the trpezoidl rule in Setion 6.2; in ddition, we hoose K = K. In this se, the resulting model n lso be obtined by using FE method with non-uniform mesh, with resolution identil to the tomisti one in the tomisti region, i.e., in Ω M, while different resolution is used in Ω C\Ωbri (equl to the one hosen for the FE pproximtion of the AtC blended model in tht region. Whether similr onnetion n be mde in the se of multiple-neighbor tomisti system nd higher-order FE pproximtion, is n open question.

15 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Figure 3: Multisle grid omposed of toms (irles nd finite element (FE nodes (vertil brs. The FE grid resolution is identil to the tomisti one; there is no tom on the right boundry of the bridge region. The domin of the problem is Ω=[,], nd the bridge region is hosen s Ω bri =[,]=[.5,.7]. 4 Compring Model I nd Model II in the bridge region In this setion, we fous on the bridge boundries, in order to point out some differenes between Model I nd Model II. For tht purpose, we run prtiulr se for whih the FE nodes overlp the toms in the bridge region, but there is missing tom t the lst FE node of the bridge region, i.e., no tom is present on the right boundry of tht region. The grid looks s in Fig. 3, with 2 FE nodes in Ω C=[.5,], nd 28 toms in ΩM =[,.7]. The toms inside the bridge region orrespond to toms hving the referene positions { α }, α = 2,,28, nd the FE nodes tht overlp them orrespond to the FE nodes t {j h }, j =,,8. We tke lose look t the ssembled mtrix for our disretized system of equtions; in the ssembled mtrix A, the rows re ordered suh tht the first N u rows orrespond to the FE equtions, the following N d rows orrespond to the tomisti equtions, nd the next N λ rows orrespond to the onstrints. We exmine the resulting mtrix fter dding A(42:49,42:49+A(:8,:8, i.e., the sum of the ontributions to the toms/nodes in the bridge region; this ontributions rise from the rows to 8, orresponding to the FE nodes in the bridge region (i.e., the first 8 FE nodes in Ω C nd from the rows 42 to 49, orresponding to the toms in the bridge region (i.e., the lst 8 toms in Ω M. In this numeril exmple, we used the following model hoies: N u = 2, N d = 28, N λ =2, =.5, =.7, p=, pieewise liner ξ(, nd pieewise liner Λ L (. Models I nd II both produe the mtrix entries In both models, the sum of the ontributions of the disretized ontinuum equilibrium eqution nd the tomisti fore blne eqution is onsistent with the pure-tomisti.

16 846 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp intertion expression K s d α +2 K s d α K s d α+, with K =. nd s=/4; the totl number of toms in pure-tomisti model hving the sme resolution is 4. Right boundry. We look t the ssembled mtrix A of the system, but this time only t the ontinuum ontributions. We fous on the elements orresponding to the first FE nodes of the ontinuum region, inluding eight overlpping FE nodes, FE node t the right bridge boundry, nd FE node inside the ontinuum region outside the bridge region, i.e., Ω C\Ωbri ; this orresponds to the mtrix blok A(:,:. For Model I, we hve the mtrix blok The first 8 rows orrespond to FE nodes tht overlp toms in the bridge region, thus we know tht they hve omplementry ontribution from the tomisti expression (nd we hve verified tht the ontinuum nd tomisti ontributions dd together to the sme pure-tomisti system intertion vlues. Row 9 is FE node in the bridge domin (on the right boundry, but without n overlpping tom. Beuse of the energy blending funtion ξ(, the elements A(9,8 = 7.5 nd A(9,9 = 77.5 hve vlues with smller bsolute vlue thn those orresponding to the pure-tomisti model (whih should be the sme s the elements A(,9 = nd A(, = 8, respetively, wheres there is no omplementry tomisti ontribution. For Model II, we hve the mtrix blok

17 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp In this se, the FE node 9 hs the orret intertion vlues. In ontrst to Model II, Model I suffers from n imblne of the intertion vlues on the right boundry of the bridge region, beuse the blending is performed using n verge of energy blending funtions evluted t different points; the lk of presene of n tom t the right boundry of the bridge region produes, s result, only prtil ontribution from the ontinuum model. Foring the presene of n tom t the right boundry of the bridge region resolves this imblne problem. Left boundry. The problem ppering on the right boundry of the bridge region, orresponding to the system illustrted in Fig. 3, motivtes similr study in reltion to the left boundry, but this time, for generl AtC onfigurtion. An nlogous problem to the one ourring t the right boundry of the bridge region, where the intertion rises only from prtil ontribution of the ontinuum model, n our on the left boundry of the bridge region; in prtiulr, beyond it, but lose to the bridge region, the tomisti ontribution is only prtil (beuse of the intertomi intertion blending funtion θ α,β nd there is no dditionl ontribution from the ontinuum model. Atoms inside the bridge region hve prtil ontributions from eh of the tomisti nd ontinuum models; tht is not the se for toms outside the bridge region, whih hve only n tomisti ontribution. In Model I, the intertomi intertion blending funtion θ α,β hs n verged form; thus, toms outside the bridge region, but still interting with toms inside the bridge region, hve n intertion weighted by θ α,β <, giving smller ontribution in omprison to the pure-tomisti model. In order to illustrte this, we exmine the tomisti eqution (3.7, for given prtile α, where the verge rule is used for θ α,β. Assuming α [ i,] nd α+ (,], then θ α = θ α =, wheres θ α+ <. We then obtin ( K s d α +2 ( K s d α ( K s d α+ = fα ext 2 ( θ α+ ( K s (d α+ d α ; thus, n extr rtifiil fore term ppers on the right-hnd side. A wy to void this problem is to set the intertomi intertion blending funtion θ α,β in the tomisti fore blne eqution in (2.8 orresponding to toms in the tomisti region outside the bridge region, i.e., toms with n index ζ / F. 5 Convergene studies In this setion, we nlyze, through omputtionl experiments, the performne of models nd their dependene on prmeters, using two bsi settings: I. Zero-lod se: B(= ; d u( i =; u h N u u( f =, II. Constnt-lod se: B(=; d u( i =; u h N u u( f =, where u( represents, in this se, the ext solution of our problem. In order to void boundry issues, we ple n tom on the right boundry of the bridge region nd use n intertomi intertion blending funtion θ α,β for toms in the tomisti region outside

18 848 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp u(.6 u( ( Zero lod (b Constnt lod Figure 4: Comprison of the displement profiles between the different models: tomisti-to-ontinuum (AtC blended (with n tomisti solution in [ i,]=[,.64], nd ontinuum finite element (FE pproximtion in d [, f ]=[.4,.], pure-tomisti, pure-ontinuum FE, nd the PDE K 2 u = B(, for zero-lod se ( d 2 nd onstnt-lod se (b. The prmeters used for the simultions re given in Tble long with N λ = 7, N u =2, nd p=, using uniform Lgrnge multiplier (LM grid for Model II; pieewise ubi energy blending funtion ξ( hoie with the verge rule (f. Setion 3.3 is implemented together with pieewise liner LM bsis funtions Λ L (. In ddition to the displement profiles, the multisle grid is shown in the plots, with the irles representing toms, the thin vertil brs FE nodes, nd the thik vertil brs LM grid nodes. The plots show qulittive greement between the models. the bridge region, i.e., Ω M\Ωbri. The prmeters used in ommon for the simultions in Setions re given in Tble. Tble : Model prmeters for onvergene studies. N d i f K K Qulittive results ompring the displement profiles of the pure-tomisti model, the AtC blended model (with n tomisti solution in [ i,], nd ontinuum FE pproximtion in [, f ], the pure-ontinuum FE model, nd the ext solution of the PDE d K 2 u =B( re presented in Fig. 4; qulittive greement is found between the different models. In the following setions, we investigte the error onvergene s fun- d 2 tion of different prmeters, treting ll possible ombintions of the hoies for the LM bsis funtions (Λ L ( pieewise liner or onstnt nd the energy blending funtion (ξ( pieewise liner or ubi. The vlue of =.64 is hosen so tht we obtin FE node t the right boundry of the interfe.

19 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Log 6 8 Log log (N λ ( Λ L (:p.liner - ξ(: p.liner log (N λ (b Λ L (:p.liner - ξ(: p.ubi Log log (N λ ( Λ L (:p.onst - ξ(: p.liner Log log (N λ (d Λ L (:p.onst - ξ(: p.ubi Figure 5: Totl error of the tomisti-to-ontinuum blended model s funtion of the number of Lgrnge multiplier (LM grid nodes N λ, for different ses, for the zero-lod se. The plots (, b,, d present different ombintions for the hoie of LM bsis funtions Λ L (: pieewise onstnt (p.onst or pieewise liner (p.liner, nd the hoie of the energy blending funtion ξ(: pieewise liner (p.liner or pieewise ubi (p.ubi. On eh plot, the results of 6 simultions re shown, s it is desribed in the lbel of the figure; uniform/nonuniform refer to the LM grid (f. Setion 3.4, midpoint/verge refer to the hoie of the intertomi intertion blending funtion θ α,β (f. Setion 3.3, nd Model I/Model II refer to the energy blending model (f. Setion 3.6. Figures (b,,d use subplots beuse of the lrge differenes in mgnitude of the errors (y-xes between the results of Model I nd Model II (the results of Model I pper in the top plot of eh subplot, wheres the results of Model II pper in the orresponding bottom plot. Notie tht in (, there is n overlpping between the results of the verge nd midpoint rules, both in the se of uniform nd nonuniform LM grid, for Model I; s onsequene (, seem to present only 4 urves, insted of 6. In (b,d similr behvior, though not identil, results from the midpoint nd verge rules in Model I. 5. Lgrnge multiplier grid resolution We investigte the error behvior of the AtC blended model s funtion of the number of LM grid nodes; we use in this se the vlues N u =2 nd p=. For the zero-lod se, the results re shown in Fig. 5; in Fig. 6, we present the results for the onstnt-lod The vlues of N d =29 (see Tble nd N u =2, orrespond to resolutions equivlent to 2 toms nd 34 FE nodes in [,]; this hoie gives proportion of 6 toms per finite element.

20 85 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Log 2.85 Log log (N λ log (N λ ( Λ L (:p.liner - ξ(: p.liner (b Λ L (:p.liner - ξ(: p.ubi Log 2.85 Log log (N λ log (N λ ( Λ L (:p.onst - ξ(: p.liner (d Λ L (:p.onst - ξ(: p.ubi Figure 6: Totl error of the tomisti-to-ontinuum blended model s funtion of the number of Lgrnge multiplier (LM grid nodes N λ, for different ses, for the onstnt-lod se. The model hoies nd lbel interprettion re the sme s in Fig. 5. As in the zero-lod se (Fig. 5, in (, there is n overlpping between the results of the verge nd midpoint rules, both in the se of uniform nd nonuniform LM grids, for Model I; in (b,d similr behvior, though not identil, results from the midpoint nd verge rules in Model I. Furthermore, in (b,d similr results re obtined, for Model II, between uniform nd nonuniform LM grid hoies; in prtiulr, in (b the urves overlp eh other. As onsequene, the plots seem to present less thn 6 urves. se. The uniform nd nonuniform LM grids follow the onstrution presented in Setion 3.4; the mximum number of LM grid nodes tken in the simultions, both for the uniform (N λ =48 nd nonuniform (N λ =8 LM grids, orresponds to the mximum number of nodes we n hoose so tht the LM grid resolution does not exeed the tomisti one. The min onlusions re s follows. For the pieewise liner ξ( hoie (Figs. 5(, nd 6(,, Model I produes the sme results for the verge nd midpoint rules. The reson for tht is tht both rules re equivlent for liner funtions. In the zero-lod se (Fig. 5, in ontrst to Model I, Model II seems to produe the ext solution for ll ombintions (see the y-xes sles, similr to wht hppens in pure-ontinuum FE se when the ext solution belongs to the FE spe; however, in

21 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp the onstnt-lod se (Fig. 6, the errors for both models re of the sme order. In order to understnd the differenes in behvior, we note tht the displement profile in the onstnt-lod se hs qudrti form (see Fig. 4(b, so tht the hoie of pieewiseliner interpoltion for the FE method introdues n error of similr mgnitude in both models; in ontrst, for the liner profile of the zero-lod se, pieewise-liner interpoltion produes no errors. In the zero-lod se (Fig. 5, for the pieewise liner ξ( pieewise liner Λ L ( hoie (, Model I onverges to the ext solution for lrge number of LM grid nodes. In the zero-lod se (Fig. 5, for Model I, the nonuniform LM grid gives better onvergene results (,,d, though in the se of pieewise ubi ξ( pieewise liner Λ L ( hoie (b, the differene between uniform nd nonuniform LM grids is less notieble. In the onstnt-lod se (Fig. 6, for Model I, the use of nonuniform LM grid does not improve the results in ll the ses; on the other hnd, in Model II, the uniform LM grid gives better results for the pieewise liner ξ( hoie (,, wheres for the pieewise ubi ξ( hoie (b,d, both LM grid hoies produe similr results. In the zero-lod se (Fig. 5, for the pieewise onstnt Λ L ( hoie (,d, we see onvergene, for the uniform LM grid se in Model I, inluding n pproximte stepwise behvior of the solution. In the onstnt-lod se (Fig. 6, in most of the ses, there seems to exist n optiml hoie for N λ for whih the error is minimized. Tht does not seem to be the se for Model II, for the pieewise ubi ξ( hoie, in prtiulr, when using pieewise onstnt LM bsis funtions Λ L ( (d, where n pproximte monotoni behvior for the error inrese ppers. 5.2 Finite element grid resolution We next investigte the error behvior of the AtC blended model s funtion of the number of FE nodes; for tht purpose, we hoose N λ = 7, i.e, log (N λ.85, whih we observed in Setion 5. gve result lose to optiml, in most of the ses, for the onstnt-lod se (see Fig. 6, nd p =. The mximum number of FE nodes tken for the simultions is N u = 2 whih gives FE grid with the sme resolution s the tomisti one. For the zero-lod se, the results re shown in Fig. 7; in Fig. 8, we present the results for the onstnt-lod se. The min onlusions re s follows. For the pieewise liner ξ( hoie (Figs. 7(, nd 8(,, Model I produes the sme results for the verge nd midpoint rules, s in Setion 5.. For the zero-lod se (Fig. 7, Model II seems to produe the ext solution for ll ombintions (see the y-xes sles, in ontrst to Model I, wheres in the onstnt-lod se (Fig. 8, the errors for both models re of the sme order, s in Setion 5.. On the ontrry, in Fig. 7 (zero-lod se the errors in Model I seem to be independent of the FE grid resolution, nd in Fig. 8 (onstnt-lod se the errors reh plteu, wheres the errors in Model II ontinue deresing when inresing the number of FE nodes. As in Se-

22 852 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Log Log N u N u ( Λ L (:p.liner - ξ(: p.liner (b Λ L (:p.liner - ξ(: p.ubi Log Log N u N u ( Λ L (:p.onst - ξ(: p.liner (d Λ L (:p.onst - ξ(: p.ubi Figure 7: Totl error of the tomisti-to-ontinuum blended model s funtion of the number of finite element grid nodes N u, for different ses, for the zero-lod se. The model hoies nd lbel interprettion re the sme s in Fig. 5. Figures (,b,,d use subplots beuse of the lrge differenes in mgnitude of the errors (yxes between the results of Model I nd Model II (the results of Model I pper in the top plot of eh subplot, wheres the results of Model II pper in the orresponding bottom plot. In (,, there is n overlpping between the results of the verge nd midpoint rules, both in the se of uniform nd nonuniform Lgrnge multiplier grids, for Model I; s onsequene (, seem to present only 4 urves, insted of 6. In (b,d similr behvior, though not identil, results from the midpoint nd verge rules in Model I. tion 5., we n rgue tht the pieewise liner FE pproximtion introdues dditionl errors in the onstnt-lod se, so tht both models hve errors of the sme mgnitude; on the other hnd, it is ler tht n dditionl soure of error is present in Model I nd beomes dominnt for lrge number of FE nodes. A possible interprettion is tht there is some error rising from noise in the bridge region tht prevents onvergene. To show the noise reted in the bridge region, we present, in Fig. 9, plot ompring the strin profiles for Model I ( nd Model II (b. In Model I, the nonuniform LM grid gives lower errors thn the uniform one (see Figs. 7 nd 8. In the onstnt-lod se (Fig. 8, for pieewise ubi ξ( hoie (b,d, Model II produes smller errors thn Model I; tht is not extly the se for the pieewise liner ξ( hoie (,.

23 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Log Log N u N u ( Λ L (:p.liner - ξ(: p.liner (b Λ L (:p.liner - ξ(: p.ubi Log Log N u N u ( Λ L (:p.onst - ξ(: p.liner (d Λ L (:p.onst - ξ(: p.ubi Figure 8: Totl error of the tomisti-to-ontinuum blended model s funtion of the number of finite element grid nodes N u, for different ses, for the onstnt-lod se. The model hoies nd lbel interprettion re the sme s in Fig. 5. As in the zero-lod se (Fig. 7, there is n overlpping between some of the urves presented in the results; in (,, there is n overlpping between the results of the verge nd midpoint rules, both in the se of uniform nd nonuniform Lgrnge multiplier (LM grids, for Model I; in (b,d similr behvior, though not identil, results from the midpoint nd verge rules in Model I. Furthermore, in (b,d, for Model II, there is n overlpping between the results of uniform nd nonuniform LM grid hoies. As onsequene, the plots seem to present less thn 6 urves. In Model II, in the onstnt-lod se (Fig. 8, we get the sme results for uniform nd nonuniform LM grids in the se of pieewise ubi ξ( hoie (b,d, wheres uniform LM grid performs better thn the nonuniform one in the se of pieewise liner ξ( hoie (,. 5.3 Penlty prmeter We investigte the error, of the AtC blended model, s funtion of the penlty prmeter. We hoose N λ = 7, s in Setion 5.2, nd N u = 2. For the zero-lod se, results re shown in Fig., wheres in Fig. the results in semi-log sle show the symptoti behvior for lrge vlues of p; in Fig. 2, we present the results for the onstnt-lod se, wheres in Fig. 3 the results in semi-log sle show the symptoti behvior for lrge

24 854 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp e( e( ( Model I (b Model II Figure 9: Comprison of the strin profiles between the tomisti-to-ontinuum (AtC blended model (with n tomisti solution in [,.64], nd ontinuum finite element pproximtion in [.4,.], nd the pure-tomisti model, for the zero-lod se, in Model I ( nd Model II (b (f. Setion 3.6. The simultion prmeters re presented in Tble ; N λ = 7, N u = 6, p=; we use pieewise ubi energy blending funtion ξ( hoie, the verge rule (f. Setion 3.3 for Model I, pieewise liner Lgrnge multiplier (LM bsis funtions Λ L ( hoie, nd uniform LM grid. vlues of p. The min onlusions re s follows. In the pieewise liner ξ( hoie (Figs. (,, (,, 2(,, nd 3(,, Model I produes the sme results for the verge nd midpoint rules, s in Setions 5. nd 5.2. In the zero-lod se (Figs. nd, Model II seems to produe the ext solution for ll ombintions (see the y-xes sles, in ontrst to Model I, wheres in the onstntlod se (Figs. 2 nd 3, the errors for both models re of the sme order, s in Setions 5. nd 5.2. In the zero-lod se (Figs. nd, for Model I, the error seems to derese monotonilly with inresing p; furthermore, the nonuniform LM grid, in most of the ses, results in smller error thn the uniform one. In the onstnt-lod se (Figs. 2 nd 3, there exists n optiml vlue of p 4 for the error onvergene in the se of pieewise liner ξ( hoie (, in Model II, wheres in the pieewise ubi ξ( hoie (b,d the error bsilly inreses with inresing p; in Model I, the optiml vlue for p in the pieewise liner ξ( se is round. In the onstnt-lod se (Figs. 2 nd 3, for the pieewise ubi ξ( hoie (b,d, fter some vlue of p, the behvior is similr for both models. On the ontrry, for the pieewise liner ξ( hoie (,, fter some vlue of p, Model II outperforms Model I. In the onstnt-lod se (Figs. 2 nd 3, the minimum vlue for the error obtined in the pieewise liner ξ( hoie (, is smller thn tht obtined in the pieewise ubi ξ( hoie (b,d.

25 P. Seleson nd M. Gunzburger / Commun. Comput. Phys., 7 (2, pp Log Log log [p] ( Λ L (:p.liner - ξ(: p.liner log [p] (b Λ L (:p.liner - ξ(: p.ubi Log Log log [p] log [p] ( Λ L (:p.onst - ξ(: p.liner (d Λ L (:p.onst - ξ(: p.ubi Figure : Totl error of the tomisti-to-ontinuum blended model s funtion of the penlty prmeter p, for different ses, for the zero-lod se. The model hoies nd lbel interprettion re the sme s in Fig. 5. Figures (,b,,d use subplots beuse of the lrge differenes in mgnitude of the errors (y-xes between the different ses. In (,, there is n overlpping between the results of the verge nd midpoint rules, both in the se of uniform nd nonuniform Lgrnge multiplier grids, for Model I; s onsequene (, seem to present only 4 urves, insted of 6. In (b,d similr behvior, though not identil, results from the midpoint nd verge rules in Model I. 5.4 Min onlusions regrding the error onvergene Following the bove results we n rrive t the following generl onlusions. Model II outperforms Model I (or is t lest s good for most ses. In prtiulr, for the zero-lod se, Model II seems to reprodue the ext solution. Liner energy blending (pieewise liner ξ( hoie, pplied to Model I, does not distinguish between the verge nd midpoint rules. For Model I, nonuniform LM grids result, in most of the ses, in smller errors ompred to uniform LM grids; on the ontrry, for Model II, better results re obtined, in most of the ses, when pplying uniform LM grid. In severl ses (in prtiulr for the onstnt-lod simultions, optiml vlues for N λ nd p re found.