CD-EXCALIBUR-FMS0069-M6c.2_ExcaliburFusionModellingSystem
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.. meta::
   :description: technical note

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ExCALIBUR,Finite,Element,Models,Complementary,Actions:,Code,Inte-,gration,,Acceptance,and,Operation,1.,M6c.2,Version,1.00,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Milestone,6c.2.,It,provides,ma-,terial,on,ﬁnite,element,exterior,calculus,,techniques,which,are,seen,as,effectively,indispensible,in,modern,numerical,models,involving,coupling,of,particles,to,electromagnetic,ﬁelds,viz.,particle-,in-cell,simulations.,A,small,collection,of,illustrative,examples,,taken,from,a,key,textbook,([1]),in,the,ﬁeld,,is,exhibited,,with,reference,to,implementations,in,the,Firedrake,PDE,solver,which,are,available,via,a,public,GitHub,repository.,Also,included,is,a,brief,overview,of,recent,developments,within,the,Nektar++,solver,framework,intended,to,produce,a,suitable,Nektar++-based,library,for,the,handling,of,continuum,ﬁeld,aspects,of,NEPTUNE.,This,material,covers,the,removal,of,the,need,for,an,XML,session,ﬁle,for,each,simulation,,in-progress,modiﬁcations,needed,to,harmonize,the,library,with,a,PIC,implementation,,and,also,recent,work,by,the,holder,of,the,grant,T/NA078/20.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0069,Issue:,Date:,1.00,27,September,2022,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Reviewed,By:,Name,and,Department,Ed,Threlfall,Owen,Parry,Will,Saunders,Signature,N/A,N/A,N/A,BD,James,Cook,BD,Date,27,September,2022,27,September,2022,27,September,2022,27,September,2022,2,1,Introduction,The,ﬁnite,element,method,for,solving,PDEs,is,mature,and,encompasses,many,different,variants,,for,example,there,is,a,wide,choice,of,basis,functions,and,assignments,of,degrees,of,freedom,,so,apparently,a,selection,of,methods,capable,of,solving,the,same,problem,(see,,for,example,,the,table,of,over,ﬁfty,ﬁnite,element,types,at,[2]).,In,recent,years,,works,such,as,[3],have,provided,a,unifying,theory,of,the,choice,of,ﬁnite,element,method,that,is,appropriate,,or,most,appropriate,,to,a,particular,type,of,problem.,This,framework,,going,under,the,designation,of,ﬁnite,element,exterior,calculus,(hereafter,FEEC),,has,clariﬁed,the,role,of,hitherto-discovered,discretizations,and,also,has,led,to,the,discovery,of,novel,element,types;,a,pleasing,distillation,of,these,ideas,is,the,‘periodic,table’,of,the,ﬁnite,elements,[4].,Another,favourable,aspect,is,that,most,of,the,constituent,element,types,have,known,order-p,implementations,and,thus,the,path,to,exponential,(spectral),convergence,,at,least,for,smooth,solutions,,is,straightforward.,These,developments,,which,have,been,led,by,mathematicians,and,numerical,analysts,,stem,from,a,deep,understanding,of,the,underlying,structure,of,both,the,continuum,PDEs,and,their,discrete,form.,In,particular,,the,geometrical,nature,of,the,dependent,variable,comes,to,the,fore,,with,the,equations,themselves,being,expressed,in,the,language,of,exterior,calculus,,viz.,differential,forms,and,the,exterior,derivative.,These,structures,encode,important,aspects,of,the,various,problems,,such,as,topological,invariants,and,conserved,quantities,,and,,with,the,correct,choice,of,discretiza-,tion,,these,properties,are,also,manifest,in,the,discretized,system,-,for,this,reason,,FEEC,is,referred,to,as,a,‘structure-preserving’,,‘mimetic’,,or,‘compatible’,,method.,Note,that,the,exterior,calculus,encompasses,the,more,familiar,‘ﬁelds’,of,physics,,e.g.,a,0-form,is,simply,a,scalar,ﬁeld,,a,1-form,is,a,(covariant),vector,ﬁeld,,and,a,2-form,is,an,antisymmetric,rank-2,covariant,tensor,of,which,a,good,example,is,the,Maxwell,tensor,of,classical,electrodynamics;,higher-k,forms,correspond,to,fully-antisymmetric,tensors,of,higher,rank.,One,small,cautionary,note,is,that,other,,non-ﬁnite,element,,structure-preserving,discretizations,exist,,going,under,the,wider,banner,of,discrete,exterior,calculus,(hereafter,DEC).,These,may,not,have,elements,satisfying,all,of,the,requirements,of,the,Ciarlet,deﬁnition,([5]),and,they,usually,involve,a,pair,of,spatial,grids,-,the,primal,and,the,Voronoi,dual.,The,prototypical,example,is,the,Yee,algorithm,widely,used,in,computational,electromagnetics,,in,which,the,electric,degrees,of,freedom,are,associated,to,one,spatial,grid,and,the,magnetic,degrees,of,freedom,the,other.,To,attempt,an,overview,(which,incorporates,advice,given,in,[6]),,there,are,two,main,reasons,to,use,FEEC,(or,DEC):,1),they,give,naturally,stable,discretizations,which,do,not,need,artiﬁcial,numerical,dissipation,(e.g.,upwinding,or,least-squares),to,remain,stable;,2),they,enable,the,construction,of,discretizations,with,conserved,quantities.,In,particular,,FEEC,/,DEC,has,been,used,extensively,to,provide,stable,discretizations,for,Maxwell’s,equations,,and,structure-preserving,methods,for,electromagnetism,coupled,to,particle,methods,have,been,shown,to,produce,effective,solvers,for,the,Vlasov-Poisson,system.,For,NEPTUNE,it,is,intended,that,structure-preserving,methods,are,incorporated,into,the,Nektar++,PDE,solver,framework,and,this,is,currently,under,consideration,by,the,main,developers,of,that,code,in,conjunction,with,UKAEA,NEPTUNE,team,members.,The,integration,of,the,families,of,ﬁnite,elements,used,by,FEEC,into,the,Nektar++,framework,in,terms,of,code,structure,and,optimizations,for,efﬁcient,calculation,is,an,area,for,further,research,,though,this,would,build,on,an,existing,body,of,knowledge,(see,e.g.,[7],for,efﬁcient,implementations,of,H(div)-,and,H(curl)-conforming,elements,,and,[8],for,an,application,of,the,sum-factorization,optimization,3,technique,to,these,element,types).,Some,instructive,examples,of,the,utility,of,FEEC,methods,are,presented,in,Section,2,as,a,prelude,to,the,implementation,of,similar,techniques,in,NEPTUNE,software.,The,spectral,element,framework,Nektar++,is,currently,being,modiﬁed,such,that,it,will,form,a,library,supplying,continuum,ﬁeld,capabilities,to,higher-level,NEPTUNE,applications.,The,main,work,in,this,direction,is,summarized,in,Section,3,,and,includes,the,removal,of,the,need,for,an,XML,ses-,sion,ﬁle,,modiﬁcations,to,interface,Nektar++,with,the,UKAEA,in-house,PIC,code,NESO-Particles,,preliminary,implementation,of,Nvidia,GPU,kernels,,modiﬁcations,to,the,NekMesh,mesh,generator,aimed,at,producing,meshes,suited,to,fusion-relevant,scenarios,,and,a,comment,on,the,amount,of,memory,used,by,Nektar++.,2,Finite,element,exterior,calculus,Taken,from,[1],and,associated,papers,,the,examples,presented,here,,which,illustrate,problems,where,conventional,FEM,may,perform,poorly,,focus,on,the,Hodge,Laplacian,-,a,generalization,2,operator,to,include,action,on,differential,forms,of,order,k,=,0,,1,,...,,D,(in,D,of,the,familiar,space,dimensions).,The,k,=,1,Hodge,Laplacian,is,the,familiar,vector,Laplacian,and,study,of,this,case,is,motivated,by,the,appearance,of,this,operator,in,the,Navier-Stokes,equations,(equivalently,,the,vector,vorticity,equation),,the,Maxwell,equations,in,the,frequency,domain,(which,can,give,eigenproblems,of,the,type,examined,in,Subsection,2.4),,and,the,vector,induction,equation.,r,The,exterior,derivative,operator,d,acts,to,convert,differential,forms,between,the,space,V,k,of,k-,forms,and,that,V,k+1,of,(k,+,1)-forms;,the,formal,adjoint,of,d,,denoted,d⇤,,is,given,by, ,,,⇤,where,is,the,Hodge,star,,and,called,the,co-differential.,The,sign,is,ﬁxed,by,the,adjoint,property,(see,[1]);,for,this,operator,in,two,dimensions,and,acting,on,0-,and,1-forms,,the,negative,sign,V,k,given,by,is,appropriate.,The,Hodge,Laplacian,is,then,the,unbounded,operator,from,V,k,Lk,=,d ,+, d.,⌘±⇤,!,d,⇤,The,Hodge,Laplace,problem,Lku,=,f,Phf,, ,Hk,u,?,(1),W,k.,The,possibility,of,a,null,has,a,unique,solution,(in,the,appropriate,space;,see,[1]),for,any,f,space,(denoted,Hk),for,Lk,(i.e.,harmonics;,see,Subsection,2.4),is,the,reason,for,the,condition,u,2,Hk,and,the,subtraction,of,the,projection,Phf,of,f,onto,this,space.,?,Two,weak,formulations,(used,because,they,are,numerically,convenient,and,allow,existence,proofs,of,well-behaved,solutions),of,this,problem,are,the,primal,weak,formulation,du,,dv,+,h,i,h,d⇤u,,d⇤v,=,f,h,i, ,PHf,,v,i,(2),where,u,is,a,trial,function,and,v,is,a,test,function,(in,a,Galerkin,approximation,,taken,from,the,same,ﬁnite-dimensional,function,spaces),,and,the,mixed,weak,formulation,,in,which, ,d⇤u,(note,the,adjoint),and,p,=,PHf,⌘,4,d ,,v,+,h,i,h, ,,⌧,h,du,,dv,i, ,h,+,i,u,,d⌧,p,,v,h,u,,q,h,=,0,,(3),f,,v,=,h,=,0.,,,i,i,i,i,According,to,[1],,there,are,four,possible,‘good’,choices,of,(pairs,of),discrete,spaces,for,this,problem,(though,some,may,in,practice,be,equivalent),,which,are,(V,k,h,denotes,the,function,space,used,Pr⇤k,to,discretize,the,space,of,k-forms;,denotes,the,complete,space,of,piecewise,polynomial,k-forms,of,polynomial,order,at,most,r,over,P, r,⇤k,denotes,a,similar,but,slightly,smaller,space,of,polynomial,k-forms,,called,the,domain,and,the,trimmed,space):,Th,represents,a,simplicial,triangulation,of,the,domain;,and,V,k,1,h,=,8, ,><,>:,Pr⇤k,or,P, r,⇤k, ,1(,Th),,1(, ,Th),V,k,h,=,8,><,P, r,⇤k(,or,Th),,1⇤k(,Pr, ,Th),(if,r,>,1).,>:,This,is,the,notation,of,the,‘periodic,table’,of,the,ﬁnite,elements,[4].,Elements,of,these,types,are,implemented,in,the,Firedrake,PDE,solver,framework,([9],,and,see,Appendix,A),and,so,it,is,straightforward,to,explore,these,methods,using,that,code.,It,is,worth,a,brief,note,to,say,that,Nektar++,has,a,narrower,range,of,element,types,and,is,currently,restricted,to,‘scalar’,elements,(the,framework,currently,treats,vector,quantities,on,a,scalar-per-component,basis).,Changes,to,this,situation,are,currently,being,considered,by,the,Nektar++,development,team.,2.1,Simple,one-dimensional,example,of,an,unstable,discretization,Section,1.1,of,[3],contains,a,particularly,straightforward,demonstration,of,the,need,for,a,numerically-,stable,method,in,the,case,of,a,very,simple,ODE.,A,yet,more,straightforward,example,is,found,in,the,reference,in,the,same,paper,to,work,by,Brezzi,and,Bathe,([10],,of,which,p.49,in,particular),,in,which,the,ODE,is,d2u,dx2,=,f, ,(4),on,the,interval,[, ,f,=,1;,the,solution,is,trivially,u,=,1,2,(1,1,:,1],with,homogeneous,Dirichlet,conditions,at,the,endpoints,,with,forcing,term,x2)., ,5,Figure,1:,Plots,of,the,derivative,for,the,example,from,[3],showing,sensible,output,for,a,linear,discretization,of,the,derivative,and,clear,nonsense,in,the,quadratic,case.,A,mixed,formulation,is,used,with,the,auxiliary,variable, ,the,discrete,approximation, h,and,uh:,⌘, ,u0,,giving,the,weak,form,obeyed,by,1,Z,1, ,1,1,Z, , h⌧dx,=, 0hv,dx,=,1,Z,1, ,1,1,Z, ,uh⌧,0,dx,,v,dx.,(5),(6),Consider,a,discretization,into,N,equal,ﬁnite,elements,with,u,represented,by,(discontinuous),piece-,wise,constants,and, ,by,continuous,piecewise,quadratics.,It,is,not,difﬁcult,to,show,that,the,L2-,norm,of,the,error, h, , ,is, h, ,||, ,||,2,0,=,1,N,2,4, ,5,9,✓,.,◆,(7),The,point,here,is,that,this,error,tends,to,a,constant,as,N,converges,to,the,analytic,one.,!,1,i.e.,the,numerical,solution,never,This,example,corresponds,to,the,script,unstable,1d,interval,laplacian.py,in,the,repository,[11].,That,reference,demonstrates,that,a,linear,space,for, ,is,stable,but,that,enriching,this,space,to,include,quadratic,functions,gives,the,problem,indicated,above;,the,script,includes,also,the,example,from,[3],which,has,a,slightly,less,trivial,cosine,forcing,term,and,which,produces,the,output,in,Fig.1.,6,2.2,k,=,2,Hodge,Laplacian,on,square,Section,2.3,of,[3],contains,an,example,somewhat,similar,to,the,one,in,the,preceding,section,,but,in,two,dimensions.,This,is,a,source,problem,on,the,square,with,homogeneous,Dirichlet,boundary,conditions;,the,equation,is,2u,=,2y(1,r,y),+,2x(1,x),, , ,(8),with,solution,x(1,x)y(1,y)., , ,Here,u,appears,to,be,a,scalar,but,it,actually,corresponds,to,a,2-form,(which,clearly,has,a,single,component,in,two,dimensions).,A,mixed,formulation,is,used,with, ,grad,u,,giving,⌘, , ,⌧d,2x,=,u,div,⌧d,2x,,·,Z,div, v,d,2x,=,Z,Z,Z,f,v,d2x.,(9),(10),In,this,case,it,is,demonstrated,that,the,combination,of,piecewise,linear,elements,for,(the,com-,ponents,of), ,and,piecewise,constants,for,u,is,not,numerically,stable.,A,stable,discretization,is,,however,,provided,by,choosing,the,lowest,order,Raviart-Thomas,elements,to,represent, ,and,a,DG,Lagrange,element,to,represent,u.,This,choice,of,elements,has,the,property,that,the,divergence,operator,maps,the,Raviart-Thomas,basis,functions,onto,the,space,of,piecewise,polynomials,of,one,order,less,([12]);,the,Raviart-Thomas,element,therefore,is,a,natural,candidate,for,the,discretization,of,H(div),i.e.,representing,vectorial,quantities,which,need,to,have,their,divergence,taken.,The,outputs,in,unstable,and,stable,cases,are,shown,in,Fig.2.,This,example,corresponds,to,the,script,k,equals,2,hodge,laplacian,square.py,in,[11].,2.3,k,=,1,Hodge,Laplacian,,re-entrant,corner,This,example,is,taken,from,[1],(there,Example,5.1).,Consider,the,L-shaped,domain,consisting,of,the,region,[,1,:,1],with,the,lower,left,quadrant,removed.,Further,consider,the,1-form,forced,Hodge,Laplacian,problem,on,this,domain, ,with,f,=,(1,,0),and,u,·,n,=,curl,u,=,0,on,the,boundary.,2u,=,(1,,0),,,r,(11),It,is,known,that,a,re-entrant,corner,produces,a,singularity,in,the,output,,of,the,type,↵,=,1,a,zero-degree,interior,angle,,is,↵,=,1,for,corner,internal,angle,'0,,so,a,r ,↵,with,1,3,singularity,in,this,case,(the,worst,case,,for,2,).,The,singularity,causes,problems,for,conventional,FEM,,1,'0,⇡,r ,|/, ,u, ,2,|,7,Figure,2:,Numerical,solutions,to,Eq.8,obtained,using,the,unstable,(left),and,stable,(right),methods,explained,in,the,text.,in,particular,the,conventional,FEM,basis,functions,are,unable,to,represent,it,as,it,lies,without,the,space,of,representable,functions.,The,left-hand-side,plots,in,Fig.3,show,output,generated,using,the,primal,weak,formulation,using,a,vector,CG,discretization,(note,that,this,requires,the,imposition,of,a,Dirichlet,condition,on,the,normal,component,of,u,,which,is,straightforward,in,Firedrake,only,if,the,boundaries,are,all,aligned,to,Cartesian,axes).,The,right-hand-side,plots,were,generated,using,the,mixed,weak,formulation,with,a,combination,of,elements,that,is,known,to,give,convergence,to,the,correct,solution.,A,lucid,explanation,of,why,this,primal,weak,formulation,fails,in,this,case,is,given,in,Section,5.1,of,[1],and,a,sketch,of,the,argument,may,serve,to,enlighten,here.,The,CG(N,),elements,are,used,to,discretize,the,space,H,1(⌦),of,functions,which,are,themselves,square-integrable,(L2),over,⌦,and,whose,ﬁrst,derivatives,(i.e.,gradients),are,also,L2,-,that,is,,the,discrete,subspace,spanned,by,H,1,n,contains,only,functions,which,are,contained,within,H,1.,In,the,case,of,the,re-entrant,corner,,the,behaviour,↵,where,↵,was,speciﬁed,above,and,the,of,the,vector,function,near,the,corner,is,as,a,singularity,r ,derivative,of,this,function,is,not,L2,-,as,a,result,,the,output,in,this,representation,can,never,con-,verge,to,the,true,answer.,(Note,also,that,turning,to,DG,elements,is,unlikely,to,yield,nice,results,due,to,the,non-uniqueness,of,ﬁelds,at,element,boundaries.),Consider,instead,discretizing,the,vector,ﬁeld,into,a,representation,that,is,a,subspace,of,H(curl),and,recall,that,the,mathematical,deﬁnition,of,a,discrete,curl,is,as,a,line,integral,over,the,element,boundary,divided,by,the,element,area.,Now,the,integrals,of,the,singular,functions,along,any,given,element,boundary,are,actually,ﬁnite,(as,an,2,despite,the,singularity,of,the,integrand),and,so,elements,contain-,illustration,,consider,ing,functions,in,Hn(curl),are,able,to,represent,the,corner,singularity.,Therefore,,one,is,invited,to,consider,for,the,representation,of,the,1-form,H(curl)-conforming,elements,,for,which,see,the,list,at,[13],,and,note,that,the,implementations,on,simplices,in,Firedrake,are,the,Raviart-Thomas,‘edge’-,type,(called,RTE),,also,known,as,N,´ed,´elec,curl,elements,of,the,ﬁrst,kind,(N1curl),and,also,Whitney,elements;,or,the,Brezzi-Douglas-Marini,‘edge’-type,(called,BDME),,alternately,known,as,N,´ed,´elec,dx,px,⌘,1,0,R,8,Figure,3:,Outputs,for,the,re-entrant,corner,problem,treated,using,a,standard,Galerkin,discretization,(left-hand-side,plots),and,a,combination,of,Raviart-Thomas,‘edge’,elements,for,the,1-form,and,CG,elements,for,the,divergence,(right-hand-side).,The,colour,maps,(top),show,the,magnitude,of,the,1-form,ﬁeld,and,the,lower,plots,are,the,streamlines.,The,scale,in,upper,ﬁgures,is,identical,(the,scale,is,meaningful,because,the,problem,is,inhomogeneous).,Note,that,the,colour,map,for,the,spurious,solution,on,the,left,looks,plausible,-,though,the,dominance,of,one,Cartesian,component,of,the,1-form,might,raise,suspicions.,9,curl,elements,of,the,second,kind,(N2curl).,The,mixed,variational,form,(Eq.4),is,used,(note,there,are,no,nontrivial,harmonics,in,this,case,as,the,domain,is,simply-connected).,It,is,found,that,clean,and,consistent-looking,plots,are,obtained,for,both,the,above,sets,of,spaces,us-,ing,the,default,solver,parameters,of,Firedrake;,see,Fig.3.,An,implementation,of,the,primal,formu-,lation,(which,generates,the,spurious,solution),can,be,found,in,the,script,re,entrant,corner,primal.py,,and,a,working,FEEC,implementation,in,re,entrant,corner.py,,both,in,[11].,2.4,The,k,=,1,Hodge,Laplacian,in,non-trivial,topologies,The,de,Rham,cohomology,space,is,that,of,solutions,to,the,equations,and,subject,to,a,boundary,condition,e.g.,u,du,=,0,, u,=,0,n,=,0.,·,(12),In,the,k,=,1,case,,the,Eqs.13,mean,respectively,curl,u,=,div,u,=,0,with,the,additional,conditions,that,the,trace,(here,the,normal,component,of,u),vanishes,on,the,boundary.,It,is,clear,that,these,conditions,together,mean,that,any,solutions,lie,in,the,kernel,of,the,Hodge,Laplacian,operator,and,are,solutions,to,Eqs.4,with,f,=,0,(and,no,p,or,q,needed).,It,is,shown,in,[1],that,the,dimension,of,the,cohomology,space,is,equal,to,the,ﬁrst,Betti,number,of,the,manifold,,which,in,two,dimensions,means,simply,the,number,of,holes,in,it;,this,gives,the,number,of,linearly-independent,solutions.,The,textbook,[1],also,explains,that,the,harmonics,give,a,representation,of,the,k,=,1,cohomology,of,the,domain,and,proves,that,the,correct,choice,of,discretization,allows,the,discrete,form,to,retain,the,same,(Theorem,5.1,,isomorphism,of,cohomology).,In,a,domain,with,non-zero,ﬁrst,Betti,number,,there,are,one,or,more,zero-eigenvalue,solutions,to,2u,=, u,,r,(13),because,the,harmonics,are,in,the,kernel,of,the,Hodge,Laplacian.,Some,additional,material,on,2D,harmonics,is,included,in,Appendix,B.,2.4.1,Harmonics,of,doubly-connected,regions,On,a,circular,annular,domain,,there,is,one,harmonic,solution,(up,to,multiplication,by,a,constant),which,is,the,usual,vortex,(here,written,in,familiar,vector,notation),u,=,✓,y,x2,+,y2,,, ,x,x2,+,y2,.,◆,10,(14),It,is,not,currently,straightforward,to,implement,in,Firedrake,a,primal,formulation,of,this,circular,annular,vortex,,due,to,the,need,to,apply,a,Dirichlet,boundary,condition,on,the,normal,component,of,the,vector,ﬁeld,at,the,boundaries,(see,[14]).,It,is,thus,easier,to,explore,numerically,the,same,problem,on,a,square,annulus,where,the,essential,boundary,condition,is,straightforwardly,imposed,(note,the,solution,has,re-entrant,corners,but,it,will,be,seen,at,least,that,the,FEEC,method,chosen,handles,these,correctly).,Consider,attempting,to,use,conventional,FEM,to,ﬁnd,the,vortex,solution,as,the,solution,to,the,eigenvalue,problem,deﬁned,by,Eq.13.,A,standard,primal,discretization,using,a,vector,CG,space,leads,to,the,solution,shown,in,the,left-hand-side,plots,of,Fig.4.,Additionally,,the,primal,weak,for-,mulation,gives,the,results,shown,in,Table,1,for,the,lowest,eigenvalue,,the,latter,(as,with,other,eigenvalue,computations,in,this,section),obtained,using,the,Scalable,Library,for,Eigenvalue,Prob-,lem,Computations,(SLEPc),developed,at,the,Universitat,Polit,`ecnica,de,Val,`encia,,Spain,[15],,which,is,called,from,within,Firedrake.,element,order,p,1,2,3,4,5,6,7,8, min,3.8877955,3.7499638,3.7083853,3.6880550,3.6761782,3.6684848,3.6631459,3.6592514,Table,1:,Lowest,eigenvalue,for,the,k,=,1,Hodge,Laplacian,problem,on,the,square,annulus,,using,the,primal,weak,formulation,with,CG,Lagrange,elements,of,order,p.,The,result,,though,appearing,to,converge,,is,completely,spurious,as,the,true,lowest,eigenvalue,is,exactly,zero.,It,is,known,that,the,true,lowest,eigenmode,must,have,an,eigenvalue,of,exactly,zero,and,hence,the,eigenmode,found,here,is,a,numerical,artifact.,A,contrasting,implementation,that,ﬁnds,the,harmonic,correctly,,using,a,mixed,formulation,with,Raviart-Thomas,‘edge’,elements,to,represent,the,1-form,and,CG,elements,to,represent,the,divergence,,can,be,found,in,eigenvalue,problem,square,vortex.py,at,[11];,the,primal,formulation,is,eigenvalue,problem,square,vortex,primal.py.,Note,that,it,is,possible,to,obtain,these,solutions,by,conventional,methods,(no,FEEC),by,combining,a,known,vortex,solution,(e.g.,Eq.18),and,a,scalar,Laplacian,problem.,The,trick,is,to,realize,that,,unless,the,solution,topology,changes,,two,(harmonic!),solutions,of,the,Hodge,Laplacian,problem,on,two,domains,are,related,by,a,gradient;,the,resulting,ﬂow,will,automatically,satisfy,the,zero-curl,equation,(because,the,curl,of,a,gradient,vanishes),and,also,the,zero-div,condition,if,the,scalar,obeys,the,scalar,Laplace,equation.,The,boundary,conditions,are,patched,up,by,applying,a,Neumann,condition,to,remove,unwanted,normal,components,at,the,boundaries.,The,scalar,Laplacian,problem,avoids,the,problems,associated,to,harmonics,(as,long,as,the,null,space,of,constants,is,dealt,with),and,also,the,non-L2,issue,because,the,scalar,problem,does,not,have,corner,divergences,(they,appear,in,the,gradient,of,the,scalar).,This,technique,is,demonstrated,in,the,script,square,vortex,cohomology,alternate.py,in,[11].,11,Figure,4:,Lowest,eigenmode,for,the,square,annulus,obtained,using,a,primal,CG,discretization,(left),and,using,a,mixed,discretization,with,Raviart-Thomas,‘edge’,elements,to,represent,the,1-,form,(right).,Note,that,the,overall,scale,is,meaningless,for,this,homogeneous,problem.,12,2.4.2,Harmonics,of,triply-,and,higher-connected,regions,The,triply-connected,region,considered,here,is,deﬁned,as,a,bounded,or,unbounded,region,of,the,plane,with,two,circular,discs,removed.,Because,the,dimension,of,the,cohomology,space,is,equal,to,the,ﬁrst,Betti,number,of,the,manifold,,two,harmonics,are,expected.,For,an,unbounded,plane,domain,,one,of,these,is,the,counter-rotating,vortex,pair,solution,2xy, ,(x2,+,y2)2,+,a4,+,2a2(y2,u,=,✓,,,x2), ,x2,y2,a2, ,(x2,+,y2)2,+,a4,+,2a2(y2, ,.,x2),◆, ,(15),This,is,in,fact,just,the,gradient,of,the,multiple-valued,coordinate, ,in,the,bipolar,coordinate,system,([16]),(cf.,the,single,circular,vortex,,which,is,the,gradient,of,the,polar,angle).,Cohomology,theory,means,there,must,be,a,second,solution,and,a,co-rotating,vortex,solution,(possessing,non-circular,inner,boundaries),is,easily,found,by,considering,the,streamfunction,for,two,same-sense,vortices,,and,is,given,by,(see,,for,example,,[17]),y(x2,+,y2,+,a2),(x2,+,y2)2,+,a4,+,2a2(y2, ,u,=,✓,,,x2), ,x(x2,+,y2,(x2,+,y2)2,+,a4,+,2a2(y2,a2), ,.,x2),◆, ,(16),It,is,straightforward,to,map,this,co-rotating,vortex,pair,back,onto,the,domain,with,two,circles,re-,moved,,and,also,to,incorporate,an,outer,boundary,,using,the,‘adding,a,gradient’,technique,men-,tioned,earlier.,These,solutions,are,exhibited,in,Fig.5,for,a,domain,with,a,square,outer,boundary.,The,necessary,gradient,can,be,obtained,using,Firedrake,,or,alternatively,using,the,advection-,diffusion-reaction,solver,of,Nektar++,and,post-processing,in,ParaView,(i.e.,calculating,gradient,and,adding,back,the,analytic,solution),-,see,the,script,double,vortex,cohomology,alternate.py,at,[11],for,the,former.,Note,also,that,the,higher-order,geometry,element,of,Nektar++,can,be,used,to,improve,the,accuracy,of,the,gradient,solution;,for,this,,the,second-order,element,geometry,capability,of,Gmsh,can,be,used,for,the,curved,boundaries,when,generating,the,mesh,(meshing,command:,gmsh,-2,filename.geo,-format,msh2,-order,2).,The,technique,using,FEEC,is,exhibited,in,the,script,eigenvalue,problem,double,vortex.py,at,[11].,Note,that,the,eigenstates,found,by,SLEPc,do,not,correspond,exactly,to,the,counter-,and,co-rotating,vortex,solutions,discussed,above,but,rather,to,two,mutually-orthogonal,linear,com-,binations,of,them,(also,,the,symmetry,of,the,two-vortex,example,above,invited,classiﬁcation,of,the,states,by,symmetry,but,a,more,general,logic,is,to,take,a,Lagrange-like,basis,where,for,each,eigenstate,the,ﬂow,circulation,vanishes,for,all,but,one,domain,hole).,A,more,complex,example,,consisting,of,a,square,domain,with,eight,holes,and,cohomology,space,of,corresponding,dimen-,sion,,is,exhibited,in,Fig.6.,I,n,this,Firedrake,example,,the,SLEPc,solver,ﬁnds,eight,zero-eigenvalue,eigenstates,in,a,few,seconds,and,again,they,seem,to,be,rather,arbitrary,in,terms,of,basis,choice.,2.4.3,Source,problems,in,the,presence,of,harmonics,Source,problems,on,domains,with,non-trivial,cohomologies,also,encounter,problems,associated,with,the,presence,of,harmonics.,An,example,,taken,from,[1],,is,13,Figure,5:,Streamlines,of,1-form,harmonics,on,the,square,with,two,circles,removed.,These,plots,were,created,using,the,semi-analytic,technique,outlined,in,the,text,in,order,to,obtain,pure,counter-,(left),and,co-rotating,(right),vortex,pair,solutions.,2u,=,(0,,x),r,(17),on,the,annulus,1,p2,,x2,+,y2,,p2,and,subject,to,the,boundary,conditions,u,n,=,(curl,u),n,=,0.,·,·,p,The,obvious,problem,with,this,is,the,ill-posedness:,any,multiple,of,the,vortex,solution,can,be,added,to,a,solution,and,the,problem,is,still,satisﬁed,(indeed,,a,naive,implementation,in,Firedrake,produces,obviously-nonsensical,output).,This,difﬁculty,can,be,removed,using,the,nullspace,fea-,ture,of,Firedrake,-,the,software,is,given,the,harmonic(s),as,the,basis,for,the,null,space.,However,,a,standard,primal,Galerkin,discretization,of,the,problem,still,fails,,for,the,reason,that,the,solution,obtained,is,polluted,by,some,multiple,of,the,harmonic,,even,though,the,solver,has,been,informed,of,the,harmonic,mode.,Results,from,the,standard,method,(using,per-component,order-6,CG,ele-,ments),can,be,seen,in,the,left-hand,plots,of,Fig.7.,On,the,right-hand,side,of,the,same,ﬁgure,are,the,outputs,from,a,FEEC,discretization,that,preserves,the,cohomology,of,the,continuum,problem,,in,this,case,order-2,Brezzi-Douglas-Marini,‘edge’,elements,for,the,1-form,and,order-3,CG,elements,for,the,divergence.,These,outputs,can,be,compared,with,the,very,similar,results,in,Figs.,5.1,of,[1],and,2.4,of,[3].,The,numerical,outputs,,as,well,as,an,analytic,solution,derived,starting,from,the,scalar,vorticity,2⌦=,1,,,are,produced,by,the,script,source,problem,on,annulus.py,in,[11].,equation,r,3,Nektar++,developments,It,is,an,ongoing,goal,of,NEPTUNE,to,shape,a,version,of,Nektar++,into,a,library,to,provide,ﬁnite-,element,capabilities,tailored,to,the,needs,of,the,project.,There,have,been,a,number,of,develop-,14,Figure,6:,The,1-cohomology,of,a,domain,containing,eight,holes,,as,calculated,by,Firedrake;,plots,show,ﬂow,streamlines.,The,calculation,uses,order-3,Raviart-Thomas,‘edge’,elements,to,represent,u,and,order-3,CG,Lagrange,elements,for,the,divergence, ,and,ﬁnds,all,eight,representatives,in,a,few,seconds,on,a,laptop.,15,Figure,7:,Outputs,for,the,source,problem,on,annulus,described,in,the,text,using,a,primal,CG,discretization,(left),and,using,a,mixed,discretization,with,order-2,Brezzi-Douglas-Marini,‘edge’,el-,ements,to,represent,the,1-form,and,order-3,CG,elements,for,the,divergence,(right).,The,primal,formulation,gives,a,solution,dominated,by,the,harmonic,vortex,solution,(the,amplitude,is,many,orders,of,magnitude,larger,than,that,of,the,true,solution).,The,analytic,solution,to,this,inhomoge-,neous,problem,is,visually,identical,to,and,matches,in,scale,the,FEEC,numerical,solution,except,that,it,avoids,the,artifacts,at,the,facet,boundaries,visible,in,the,numerical,solution,(curved-sided,elements,in,Firedrake,are,not,currently,as,straightforward,as,they,are,in,Nektar++).,Note,the,exis-,tence,of,four,stagnation,points,on,each,boundary,of,the,true,solution.,16,ments,during,the,period,covered,by,this,report,,by,UKAEA,personnel,and,by,the,developers,of,Nektar++,and,brief,descriptions,of,these,follow.,3.1,Removal,of,XML,session,ﬁle,One,factor,that,complicates,the,integration,of,Nektar++,functionality,in,NEPTUNE,is,the,require-,ment,that,conﬁguration,options,be,speciﬁed,in,XML,‘session’,ﬁles.,Running,multiple,solver,in-,stances,over,the,course,of,a,NEPTUNE,calculation,,therefore,,would,mean,keeping,track,of,several,external,ﬁles,and/or,manipulating,their,contents,on-the-ﬂy.,To,address,this,,a,number,of,changes,to,the,Nektar++,core,libraries,have,been,made,,separating,out,code,that,deﬁnes,a,Session,object,from,code,that,parses,XML.,This,allows,runtime,options,to,be,supplied,programmatically,instead,,signiﬁcantly,simplifying,the,process,of,using,Nektar++,as,a,library,in,NEPTUNE.,The,‘Session’,class,now,contains,only,core,member,variables,,getter,and,setter,functions,,with,all,XML-based,functionality,moved,into,a,subclass,,called,‘SessionXml-,Reader’.,To,set,up,a,Nektar++,solver,from,calling,code,,it,is,now,possible,to,instantiate,a,‘Session’,object,using,a,constructor,that,accepts,various,C++,standard,template,library,containers,as,argu-,ments.,For,instance,,ﬂuid,ﬁeld,names,can,be,supplied,as,a,vector,of,strings,and,solver,options,as,a,string-to-string,map.,The,mesh,can,either,be,read,from,XML,,as,before,,or,instantiated,programmatically,by,supplying,vertex,positions,and,vectors,of,vertex,indices,that,determine,the,shape,of,each,element.,Finally,,some,example,code,that,demonstrates,programmatic,solver,set,up,for,1D,and,2D,problems,has,been,added.,The,two,examples,double,as,regression,tests,for,the,framework,,ensuring,that,it,continues,to,work,as,intended,as,the,Nektar++,code,base,evolves.,A,merge,request,has,been,created,and,is,currently,under,review,by,the,Nektar++,core,develop-,ment,team;,the,changes,are,expected,to,be,incorporated,into,a,release,of,the,software,in,the,near,future.,3.2,Interfacing,with,particles,NESO,(NEPTUNE,Exploratory,SOftware),is,a,UKAEA-developed,proxyapp,coupling,particles,and,ﬁnite,element,methods.,It,is,expected,to,form,the,core,of,a,more,developed,NEPTUNE,code.,The,particles,aspect,of,NESO,is,called,NESO-Particles,(repository,at,[18]);,it,is,written,in,an,abstract,manner,that,is,independent,of,the,choice,of,FEM,library.,The,interface,layer,that,sits,between,Nektar++,and,NESO-Particles,has,been,written.,Fundamentally,this,layer,allows,particles,to,logically,exist,on,and,traverse,2D,linear,Nektar++,meshes.,The,interface,layer,uses,functionality,to,copy,Nektar++,geometry,objects,between,MPI,ranks.,By,copying,relevant,geometry,objects,between,the,subdomains,that,result,from,the,mesh,decomposition,process,,this,interface,layer,builds,the,halo,regions,that,are,required,in,order,to,map,positions,in,space,to,Nektar++,geometry,objects.,The,following,subsections,contain,additional,relevant,details.,17,3.2.1,Projection,The,functionality,to,convert,a,particle,representation,to,a,continuum,representation,proceeds,via,an,L2,Galerkin,projection.,This,projection,interface,is,expected,to,be,on,the,critical,path,of,the,simulation,and,should,be,implemented,with,computational,efﬁciency,in,mind.,Consider,a,source,point,cloud,of,particles,where,each,particle,p,has,a,position,xp,,quantity,(scalar,or,vector),qp,(e.g.,electric,charge,,current),and,a,Dirac-delta,shape, (x,xp).,This,point,cloud,can,be,represented,in,the,FEM,function,space,via,an,L2,Galerkin,projection,to,produce,a,function,q(x),=,ci i(x),,(scalar,or,vector,valued,,where, i,is,the,ith,basis,function,,here,with,coefﬁcient,ci),,which,may,be,used,as,a,source,or,coefﬁcient,term,in,some,other,equations.,The,projection,takes,the,form,(using,the,Einstein,convention;,i,is,the,free,index), , i j,dV,cj,⌘,ZV,Mijcj,=,qp, (x,p,X,ZV,xp) i,dV, ,⌘,p,X,qp i(xp).,This,is,an,Ax,=,b,problem,where,A,corresponds,to,the,mass,matrix,with,components,Mij,,x,to,p,qp i(xp),,i.e.,the,vector,where,the,ith,component,is,the,sum,of,the,vector,of,DOFs,cj,,and,b,to,the,values,of,the,ith,basis,function, i,evaluated,at,each,and,every,particle’s,position,xp.,P,The,interface,and,implementation,may,be,written,to,assume,or,require,that,the,particles,that,form,the,point,cloud,are,already,binned,into,mesh,cells,,i.e.,particles,may,be,stored,on,a,per-cell,basis,,or,a,map,between,cell,identiﬁers,to,particle,indices,may,exist.,Furthermore,,as,part,of,the,cell,binning,process,,the,reference,position,of,the,points,in,the,mesh,cell,may,be,known,by,the,calling,implementation,and,reused,by,the,Nektar++,implementation.,For,future,consideration,,note,that,simulations,could,involve,fast-moving,particles,that,are,treated,in,a,time-averaged,manner,where,the,particle,shape,function,and,weight,is,a,trajectory,over,a,time,step,,e.g.,the,contribution,from,a,particle,exists,over,a,line,through,space,rather,than,a,single,point,and,quadrature,along,these,trajectories,will,be,required.,3.2.2,Evaluation,Considering,a,source,ﬁeld,in,Nektar++,,the,evaluation,interface,should,evaluate,the,function,at,a,set,of,arbitrary,points.,As,in,the,projection,case,,these,arbitrary,points,may,already,be,binned,into,mesh,cells,and,the,reference,coordinates,known.,Also,as,in,the,projection,case,,this,function,will,be,called,at,every,time,step,for,every,particle,and,hence,is,expected,to,be,performance-critical.,3.2.3,Mesh,augmentation,(halos),The,NESO/NESO-Particles,implementation,duplicates,mesh,objects,to,enable,the,transfer,of,par-,ticle,data,over,the,globally,decomposed,mesh,-,this,mesh,augmentation,is,critical,to,how,NESO,determines,where,to,send,particles.,The,details,of,which,mesh,objects,are,duplicated,and,where,is,currently,determined,by,NESO,by,using,serialization/de-serialization,also,implemented,in,NESO.,Exactly,which,mesh,elements,are,duplicated,and,why,is,a,PIC-speciﬁc,choice,and,so,this,logic,18,may,not,sit,naturally,within,a,native,FEM,code,base.,Regardless,of,who,retains,ownership,of,this,portion,of,code,,certain,functions,to,handle,Nektar++,mesh,objects,are,needed:,1.,(De)Serialization,of,mesh,objects.,NESO,(or,any,other,implementation),can,currently,inspect,the,geometric,properties,of,Nektar++,mesh,objects,and,use,this,geometric,information,to,deter-,mine,which,objects,are,to,be,communicated,between,MPI,ranks.,Helper,methods,to,serialize,an,existing,object,(into,bytes),and,reconstruct,the,object,from,bytes,would,abstract,user,codes,(i.e.,NESO),from,the,exact,details,of,each,type,of,geometry,object.,2.,Construction,of,Nektar++,tree,data,structures,for,geometry,objects.,The,primary,use,of,these,duplicated,geometry,objects,is,to,map,a,point,in,space,to,a,geometry,object,(and,as,a,by-product,compute,the,reference,coordinates,for,use,in,projection/evaluation),which,determines,the,owning,If,Nektar++,has,data,structures,that,already,facilitate,mapping,points,to,geometry,MPI,rank.,objects,(e.g.,R-trees),then,it,would,be,advantageous,to,be,able,to,construct,these,data,structures,using,the,duplicated,objects.,3.,Future,time,stepping,methods,in,NEPTUNE,/,NESO,may,be,(semi-)implicit,and,iteratively,compute,new,particle,positions,via,a,sub-stepping,process,that,proposes,new,particle,positions,which,may,not,be,ﬁnal.,In,this,scenario,it,is,advantageous,to,be,able,to,evaluate,Nektar++,ﬁelds,from,cells,in,the,halo,region,on,each,rank.,This,halo,evaluation,allows,particles,to,be,stepped,into,the,ﬁnal,position,for,each,(full),time,step,before,particles,are,transferred,between,MPI,ranks.,3.3,GPU,kernels,Nektar++,has,been,augmented,with,the,capability,to,run,selected,kernels,(e.g.,transforms,between,mode,coefﬁcient,and,spatial,value,,Helmholtz,operator,,derivative,evaluations),on,an,NVIDIA,GPU,,via,kernel,implementations,in,CUDA.,The,implementation,includes,a,diagnostic,of,the,performance,achieved,(in,GFLOPS).,One,restriction,of,these,kernels,is,that,the,polynomial,element,order,is,currently,capped,at,a,maximum,of,11.,This,capability,is,to,be,seen,as,a,stepping-stone,to,a,longer-term,goal,of,making,the,entire,Nek-,tar++,framework,concordant,with,modern,approaches,to,performance,portability,,especially,efﬁ-,cient,operation,on,current,and,forthcoming,GPU-dominated,systems,from,AMD,,Intel,,and,Nvidia.,3.4,Python,interface,and,h-,and,r-adaptation,for,curves,Deliverable,1.4,of,Grant,T/NA078/20,speciﬁes,a,Python,interface,,documentation,and,CI,for,the,new,additions,the,the,NekMesh,mesh,generator.,The,code,can,be,found,at,[19].,Brieﬂy,,this,new,code,is,capable,of,local,h-reﬁnement,in,the,vicinity,of,a,speciﬁed,CAD,curve,(shown,in,Fig.8),and,also,anisotropic,r-reﬁnement,meaning,moving,mesh,nodes,closer,to,a,speciﬁed,CAD,curve,(Fig.9),,the,curves,here,being,proxies,for,magnetic,ﬁeld,lines.,These,features,are,demonstrated,in,the,ﬁgures,and,can,be,generated,by,running,the,script,run-adaption.py.,Note,that,the,example,includes,a,simple,illustration,of,a,higher-order,(curved-sided),geometry,for,the,representation,of,the,embedded,circle.,19,Figure,8:,h-reﬁnements,of,a,simple,2D,mesh,consisting,of,a,circle,embedded,in,a,square,domain.,The,left-most,plot,shows,the,mesh,prior,to,local,h-reﬁnement,in,the,vicinity,of,the,circle.,To,the,right,are,plots,h-reﬁned,locally,to,the,circle,using, ,parameters,of,0.0035,and,0.0015,respectively.,Figure,9:,r-reﬁnements,of,a,simple,2D,mesh,consisting,of,a,circle,embedded,in,a,square,domain.,The,left-most,plot,shows,the,mesh,prior,to,local,r-reﬁnement,in,the,vicinity,of,the,circle.,To,the,right,are,plots,r-reﬁned,with,radaptscale,parameters,of,0.7,and,0.4,respectively,such,that,the,mesh,nodes,are,displaced,toward,the,circle,without,alteration,to,the,mesh,connectivity.,20,It,is,also,possible,to,use,the,Python,interface,to,generate,simple,structured,meshes,of,quadrilat-,erals,or,triangles;,see,the,script,gen,mesh.py,in,the,repository,[20].,3.5,Memory,use,in,Nektar++,It,has,been,determined,([21]),that,the,incompressible,Navier-Stokes,solver,of,Nektar++,retains,in,memory,two,copies,of,the,system,matrix,-,pre-,and,post-static,condensation,versions,of,the,same,object.,Only,one,is,strictly,needed,for,the,computation.,This,results,in,larger-than-expected,memory,use,by,this,solver;,modiﬁcations,are,being,considered,by,the,developers.,4,Summary,The,ﬁnite,element,exterior,calculus,approach,offers,advantages,for,numerical,methods,in,providing,guarantees,of,stability,and,convergence.,This,was,demonstrated,in,examples,taken,from,the,literature,,which,were,implemented,in,Firedrake,due,to,the,wide,range,of,ﬁnite,element,types,implemented,in,that,package.,The,scripts,were,committed,to,a,public,GitHub,repository,which,is,intended,as,a,pedagogical,resource.,The,examples,in,this,report,focussed,on,the,Hodge,Laplacian,problem,and,demonstrated,the,need,for,FEEC,in,cases,resistant,to,traditional,approaches,,for,several,different,reasons:,problems,where,poorly-chosen,combinations,of,ﬁnite,elements,gave,unstable,discretizations,,problems,containing,singular,ﬁelds,,and,problems,on,domains,with,non-,trivial,topology.,For,NEPTUNE,,the,desirable,properties,of,FEEC,as,applied,to,coupled,particle-,continuum,systems,are,likely,to,be,the,main,area,of,interest,and,will,be,the,focus,now,that,some,of,the,foundations,of,the,theory,are,understood.,Developments,to,the,Nektar++,code,base,intended,to,align,it,to,use,as,a,FEM,library,for,NEP-,TUNE,were,exhibited;,this,is,an,area,of,ongoing,work.,In,particular,,the,coupling,to,a,PIC,im-,plementation,is,progressing,,via,the,UKAEA,in-house,NESO,code,,reinforcing,the,case,for,the,implementation,of,structure-preserving,methods,such,as,FEEC.,ET,acknowledges,assistance,from,Colin,Cotter,,Patrick,Farrell,,and,other,members,of,the,Fire-,drake,community.,Acknowledgement,The,support,of,the,UK,Meteorological,Ofﬁce,and,Strategic,Priorities,Fund,is,acknowledged.,A,Firedrake,Firedrake,is,a,framework,for,the,solution,of,PDEs.,The,mathematical,problem,is,speciﬁed,using,the,Uniﬁed,Form,Language,from,the,FEniCS,Project,([22]),,which,represents,a,powerful,domain-,speciﬁc,language,for,expressing,PDEs.,The,solvers,themselves,leverage,PETSc,([23]).,21,Figure,10:,Temperature,ﬁeld,of,a,convection,problem,in,a,unit,square,cavity,with,the,right-hand,104,side,maintained,at,unit,temperature,and,the,left,at,zero,temperature,,Rayleigh,number,5,and,Prandtl,number,7.0.,The,left-hand,plot,is,obtained,from,a,steady-state,solution,in,Firedrake,and,the,right-hand,one,from,time-evolving,to,the,steady,state,in,the,incompressible,Navier-Stokes,solver,of,Nektar++.,The,outputs,are,visually,identical.,⇥,Firedrake,has,major,strengths,in,that,the,problem,speciﬁcation,mirrors,very,closely,the,mathe-,matical,notation,for,the,weak,form,,and,the,approach,of,using,such,a,DSL,in,conjunction,with,behind-the-scenes,code,generation,facilitates,the,implementation,of,a,wide,variety,of,problems,,which,can,be,speciﬁed,very,economically,by,Python,script.,This,is,somewhat,in,contrast,to,Nek-,tar++,,in,which,new,solvers,must,be,implemented,in,C++.,Firedrake,also,implements,geometrical,constructs,such,as,vectorial,function,spaces,,and,supports,a,wide,range,of,vector,and,scalar,ﬁnite,element,types.,Conversely,,known,relative,strengths,of,Nektar++,are,tools,for,building,curved,ge-,ometry,meshes,and,the,ability,to,mix,element,geometries,in,the,same,mesh,(e.g.,hexahedra,and,tetrahedra),where,the,hexahedra,are,often,used,to,mesh,boundary,layers,efﬁciently.,Another,particular,strength,of,Firedrake,is,the,ability,to,handle,equations,with,null,spaces,,a,tech-,nique,used,in,the,script,source,problem,on,annulus.py,in,[11].,For,scalar,Neumann,problems,,Nektar++,deals,with,this,issue,by,ﬁxing,one,particular,value,on,the,boundary,but,this,approach,has,in,the,past,been,seen,to,generate,problems,(see,Appendix,B,of,[24]).,Equivalent,output,generated,by,Firedrake,and,by,Nektar++,is,shown,in,Fig.,10.,The,Nektar++,data,is,generated,by,using,the,incompressible,Navier-Stokes,solver,to,time-evolve,an,initial,con-,dition,to,a,laminar,convective,steady,state.,The,Firedrake,example,is,essentially,the,same,as,the,example,at,[25],and,performs,a,linear,solve,to,obtain,the,steady,state,,using,Taylor-Hood,elements,which,are,known,to,provide,a,convergent,solution.,Close,correspondence,is,seen,in,the,resulting,temperature,ﬁeld,outputs.,22,B,More,on,harmonics,and,vortices,B.1,Single,vortex,The,single-vortex,solution,on,a,circular,annulus,is,u,=,✓,y,x2,+,y2,,, ,x,x2,+,y2,.,◆,(18),It,is,a,property,of,this,solution,,when,viewed,as,a,ﬂuid,ﬂow,,that,the,differential,rotation,(with,radial,distance),causes,a,paddle-wheel,probe,to,circulate,the,origin,in,such,a,way,that,its,orientation,remains,constant,(this,is,the,zero-curl,property).,It,is,worth,noting,that,this,is,a,stationary,solution,to,the,Navier-Stokes,equations,of,ﬂuid,mechanics,that,has,zero,viscous,dissipation,(even,though,there,is,non-uniform,velocity),and,where,the,pressure,gradient,obeys,the,simple,Newtonian,law,i.e.,it,provides,the,centripetal,force.,Note,also,that,the,zero-curl,equation,means,that,|r,the,above,vortex,has,zero,vorticity,except,at,the,origin,,where,there,is,a,singularity,in,the,ﬂuid,velocity,,a,difﬁculty,that,can,be,avoided,by,considering,only,annular,domains,which,do,not,contain,the,origin.,=,⇢v2,r,p,|,The,solution,can,be,seen,to,be,the,gradient,of,the,polar,angle,',,but,it,is,not,an,exact,differential,form,because,',is,not,single-valued,on,the,domain.,It,is,also,the,curl,of,the,streamfunction,given,by,the,familiar,=,K,2⇡,ln,r,,where,K,is,the,vortex,strength,(and,recall,the,Riemann,surface,for,the,function,w,=,ln,z).,It,is,also,possible,to,deﬁne,the,same,problem,with,the,alternate,boundary,condition,that,the,angular,component,of,the,velocity,vanish,on,the,boundaries.,The,solution,to,this,is,,up,to,overall,sign,,the,Poincar,´e,dual,of,Eq.18,(apply,the,Hodge,star!),,under,which,the,streamlines,and,equipotentials,are,swapped,,i.e.,x,x2,+,y2,,, ,y,x2,+,y2, ,u,=,✓,.,◆,(19),This,is,the,radial,ﬂow,‘draining,bathtub’,solution,(which,ﬁnds,interesting,applications,in,the,area,of,ﬂuid-mechanical,black,hole,analogues);,it,is,clearly,curl-free,and,is,divergence-free,(radial,velocity,1,r,),except,at,the,origin.,As,an,aside,,it,has,not,yet,proven,possible,to,obtain,the,‘draining,/,bathtub’,solutions,directly,in,Firedrake,(this,problem,is,technically,the,‘de,Rham,complex,with,boundary,conditions’;,see,[26],for,a,discussion),B.2,Double,vortex,The,triply-connected,region,considered,is,deﬁned,as,a,bounded,or,unbounded,region,of,the,plane,with,two,circular,discs,removed;,two,harmonics,are,expected.,For,an,unbounded,plane,domain,,one,of,these,is,easily,found,using,the,bipolar,coordinate,system,[16],(in,this,and,later,,a,is,the,distance,of,each,focus,from,the,origin),23,ds2,=,(cosh,⌧,a2, ,cos, )2,d⌧,2,+,d 2,., , ,(20),Writing,down,the,curl,and,div,equations,in,these,coordinates,,an,obvious,solution,is,(translated,to,Cartesians),Eq.15,,which,is,in,fact,just,the,gradient,of,the,multiple-valued,coordinate, ,(cf.,the,single,circular,vortex).,It,turns,out,that,the,streamlines,and,equipotentials,of,this,solution,correspond,to,the,Apollonian,circles,known,since,antiquity,(these,are,the,curves,of,constant, ,or,⌧,as,deﬁned,in,Eq.20).,Phys-,ically,,this,solution,corresponds,to,the,ﬂow,associated,to,a,pair,of,counter-rotating,vortices,in,uniform,translation,and,possessing,zero,aggregate,angular,momentum;,note,this,is,also,the,so-,lution,for,the,magnetic,ﬁeld,lines,surrounding,two,inﬁnite,wires,carrying,counter-ﬂowing,currents.,(A,somewhat,similar,ﬂuid,ﬂow,with,two,counter-rotating,vortices,can,be,generated,by,dragging,a,teaspoon,in,a,cup,of,ﬂuid,but,there,the,non-slip,boundary,conditions,lead,to,additional,dynamics.),The,circles,corresponding,to,the,streamlines,are,found,by,integrating,the,expression,for,the,tangent,velocity,,dy,dx,=, ,x2,+,y2,+,a2,2xy,,,(21),a,Bernoulli-type,ODE,with,solution,x2,+,y2,+,a2,=,cx,for,constant,c.,Cohomology,theory,means,there,must,be,a,second,solution,,though,this,is,not,naturally,found,using,the,bipolar,coordinates,(the,serendipitous,Apollonian-circles,property,is,not,shared,by,this,other,solution).,A,co-rotating,vortex,solution,(possessing,non-circular,inner,boundaries),is,,how-,ever,,easily,found,by,considering,the,streamfunction,given,by,(see,,for,example,,[17]),=,ln,(x,+,a)2,+,y2,+,ln,(x,a)2,+,y2, ,(22),p,p,(which,is,obviously,the,superposition,of,two,same-sense,vortices,in,a,Green’s,function,approach;,note,that,a,similar,formula,,tanh,=,a2+x2+y2,,,can,be,used,to,deﬁne,the,Apollonian,circles,of,the,counter-rotating,case),,giving,the,velocity,ﬁeld,u,=,curl,(equivalent,to,and,often,called,‘rot,’,in,the,literature),corresponding,to,Eq.16.,ez,⇥,r, r?,⌘, ,2ax,That,the,streamlines,are,not,circles,can,be,seen,by,solving,the,equation,dy,dx,x(x2,+,y2,a2),y(x2,+,y2,+,a2), ,=, ,(23),to,x2,+,y2,+,a2,=,pc,+,a4,+,4a2x2,(in,general,,not,a,circle,,and,note,c,=,a4,corresponds,to,the,circles,of,zero,radius,at,the,vortex,cores).,This,solution,represents,two,co-rotating,vortices,and,as,such,has,zero,linear,momentum,and,a,ﬁnite,angular,momentum,(it,is,also,the,solution,for,the,magnetic,ﬁeld,lines,surrounding,two,inﬁnite,wires,carrying,co-ﬂowing,currents).,The,streamlines,close,to,the,vortex,cores,at,x,=,a,,y,=,0,are,bound,to,their,respective,vortex,,which,further,out,,the,streamlines,merge,and,encompass,the,pair;,the,boundary,between,the,two,classes,of, ,±,24,Figure,11:,Harmonics,as,in,Fig.5,except,for,different,boundary,conditions,corresponding,to,an,equal,linear,combination,of,the,‘vortex’-,and,‘draining,bathtub’-type,solutions,,showing,streamlines,and,equipotentials,(for,the,left-hand,plot,derived,from,the,counter-rotating,vortex,solution,,one,hole,is,a,source,and,the,other,is,a,sink,,hence,the,existence,of,streamlines,connecting,the,holes).,The,choice,of,boundary,condition,is,basically,a,choice,of,angle,at,which,the,streamlines,meet,the,boundaries,and,this,angle,is,the,same,for,all,the,curves.,streamline,is,demarcated,by,a,separatrix.,As,mentioned,in,the,main,text,,it,is,straightforward,to,map,the,latter,solution,back,to,the,domain,with,two,removed,circles,by,adding,a,gradient.,B.3,Generating,2D,vector,harmonics,from,functions,of,a,complex,variable,In,two,dimensions,,there,exists,a,simple,method,for,generating,divergence-,and,curl-free,vector,ﬁelds.,The,Cauchy-Riemann,equations,of,complex,analysis,are,,for,a,holomorphic,function,w(z,=,x,+,iy),=,u,+,iv,,@u,@x,@u,@y,=,=,@v,@y,,,@v,@x,., ,(24),(25),(26),Note,that,if,the,sign,of,the,y-derivative,terms,are,ﬂipped,,the,equations,become,respectively,the,zero-divergence,and,zero-curl,conditions,for,a,2D,vector,ﬁeld,with,Cartesian,components,(u,,v).,It,follows,that,the,real,and,imaginary,parts,of,any,antiholomorphic,function,form,the,x-,and,y-,components,of,a,divergence-,and,curl-free,two-dimensional,vector,ﬁeld,in,the,complex,z-plane.,25, ,In,particular,,the,single,vortex,solution,is,given,by,w(¯z),=,i,¯z,and,the,draining,bathtub,solution,1,¯z,.,The,action,of,Poincar,´e,duality,is,basically,represented,by,multiplication,by,i,-,by,w(¯z),=,repeated,multiplications,switch,the,solution,between,anticlockwise,vortex,,sink,,clockwise,vortex,,drain;,more,general,pure,phases,ei',give,solutions,with,mixed,vortex,and,drain,character,as,shown,in,Fig.11,for,the,two-vortex,case.,The,counter-,and,(non-circular,streamline),co-rotating,vortex,pair,solutions,(15,,16),exhibited,in,the,main,text,are,given,simply,by,w(¯z),=,i,2a,1,¯z,✓, ,a,⌥,1,¯z,+,a,,,◆,(27),in,which,the,factor,of,1,limit,represents,a,double,pole,in,the,complex,z-plane).,a,means,that,the,solution,reduces,to,a,point,dipole,in,the,limit,a,!,0,(this,It,is,apparent,that,the,vorticity,corresponds,to,the,residue,of,w(¯z),at,its,poles;,in,fact,functions,with,non-zero,residues,correspond,to,cohomology,representatives,of,domains,with,‘holes’,localized,at,poles,with,nonzero,residue.,Functions,lacking,residues,in,the,domain,of,interest,correspond,to,gradients,(equivalently,,exact,1-forms).,It,is,possible,to,illustrate,additional,vortex,solutions,simply,by,plotting,the,real,and,imaginary,parts,of,antiholomorphic,functions,,for,example,the,function,w(¯z),=,i (¯z),represents,an,inﬁnite,se-,2,,...,and,w(¯z),=,i,cosec(¯z),an,inﬁnite,quence,of,vortices,of,alternating,sign,located,at,¯z,=,0,, ,sequence,of,vortices,of,alternating,sign,at,Re(z),=,n⇡,i.e.,along,the,real,axis,for,integer,n;,a,related,solution,corresponding,to,co-rotation,is,given,by,w(¯z),=,i,cot(¯z).,A,doubly-periodic,lattice,of,vortices,can,be,generated,from,an,elliptic,function,,e.g.,w(¯z),=,i,sn(¯z,,2),which,is,a,solution,to,the,pendulum,equation,w00(z),=,sin,w(z).,These,are,shown,in,Fig.12.,1,, ,±, ,References,[1],D.N.,Arnold.,Finite,Element,Exterior,Calculus.,CBMS-NSF,regional,conference,series,in,applied,mathematics,93,,SIAM,,2018.,[2],DefElement.,https://defelement.com/elements/index.html,,2022.,Accessed:,Septem-,ber,2022.,[3],Arnold,D.N.,,Falk,R.S.,,and,Winther,R.,Finite,element,exterior,calculus:,from,Hodge,theory,to,numerical,stability.,Bull.,Amer.,Math.,Soc.,(N.S.),47,(2010),no.2,,281-354.,MR2594630.,[4],Periodic,table,of,the,ﬁnite,elements.,https://www-users.cse.umn.edu/~arnold/femtable/,,2022.,Accessed:,September,2022.,[5],P.G.,Ciarlet.,The,ﬁnite,element,method,for,elliptic,problems.,[6],Cotter,C.J.,Private,communication,,2022.,[7],Rognes,M.E.,,Kirby,R.C.,,and,Logg,A.,Efﬁcient,assembly,of,H(div),and,H(curl),conforming,ﬁnite,elements.,SIAM,Journal,on,Scientiﬁc,Computing,31,,no.,6,(2010):,4130-4151,,2010.,26,Figure,12:,Harmonics,generated,using,the,antiholomorphic,functions,w(¯z),=,i (¯z),(top,left),,w(¯z),=,i,cosec(¯z),(top,right),,w(¯z),=,i,cot(¯z),(bottom,left),,and,the,Jacobi,elliptic,function,w(¯z),=,i,sn(¯z,,2),(bottom,right),and,plotted,using,Wolfram,Alpha,[27].,27,[8],Homolya,M.,,Kirby,R.C.,,and,Ham,D.A.,Exposing,and,exploiting,structure:,optimal,code,generation,for,high-order,ﬁnite,element,methods.,submitted,to,ACM,Transactions,on,Mathe-,matical,Software,,2017.,[9],Florian,Rathgeber,,David,A.,Ham,,Lawrence,Mitchell,,Michael,Lange,,Fabio,Luporini,,An-,drew,T.,T.,Mcrae,,Gheorghe-Teodor,Bercea,,Graham,R.,Markall,,and,Paul,H.,J.,Kelly.,Fire-,drake:,automating,the,ﬁnite,element,method,by,composing,abstractions.,ACM,Trans.,Math.,Softw.,,43(3):24:1–24:27,,2016.,URL:,http://arxiv.org/abs/1501.01809,,arXiv:1501.01809,,doi:10.1145/2998441.,,2015.,[10],Brezzi,F.,and,Bathe,K.-J.,A,discourse,on,the,stability,conditions,for,mixed,ﬁnite,element,formulations.,Comput.,Methods.,Appl.,Mech.,Eng.,82,(1990),,27-57.,MR,1077650,,1990.,[11],Finite,element,exterior,calculus.,https://github.com/ethrefall/,Finite-element-exterior-calculus,,2022.,Accessed:,September,2022.,[12],Raviart-Thomas,ﬁnite,elements.,https://en.wikipedia.org/wiki/Raviart-Thomas_,basis_functions.,Accessed:,September,2022.,[13],DefElement,H-Curl,conforming,elements.,https://defelement.com/lists/categories/,Hcurl.html,,2022.,Accessed:,September,2022.,[14],GitHub,discussions,thread,re,homogeneous,Dirichlet,boundary,condition,in,Firedrake.,https://github.com/firedrakeproject/firedrake/discussions/2552,,2022.,Accessed:,September,2022.,[15],Scalable,Library,for,Eigenvalue,Problem,Computations.,https://slepc.upv.es/.,Accessed:,September,2022.,[16],Bipolar,coordinate,system.,https://en.wikipedia.org/wiki/Bipolar_coordinates.,Ac-,cessed:,September,2022.,[17],H.,Lamb.,Hydrodynamics.,Cambridge,University,Press,,1932.,[18],NESO-Particles,code,repository.,https://github.com/ExCALIBUR-NEPTUNE/,NESO-Particles.,Accessed:,September,2022.,[19],Python,interface,demo,for,NekMesh.,https://github.com/EXCALIBUR-NEPTUNE/,nekmesh-python-demo.,Accessed:,September,2022.,[20],Sandbox,for,exploring,NekMesh.,https://github.com/oparry-ukaea/nekmesh_sandbox.,Accessed:,September,2022.,[21],Moxey,D.,Private,communication,,2022.,[22],FEniCS,Project.,https://fenicsproject.org/.,Accessed:,September,2022.,[23],Portable,,Extensible,Toolkit,for,Scientiﬁc,Computation.,https://petsc.org/release/.,Ac-,cessed:,September,2022.,28,[24],E.,Threlfall,and,W.,Arter.,Finite,Element,Models:,Technical,Report,CD/EXCALIBUR-FMS/0051-M6.1,,UKAEA,,2021.,I.,//github.com/ExCALIBUR-NEPTUNE/Documents/blob/main/reports/ukaea_reports/,CD-EXCALIBUR-FMS0051-M6.1.pdf.,Complementary,Activities,https:,[25],Implementation,of,Rayleigh-B,´enard,convection,in,Firedrake.,https://www.,firedrakeproject.org/demos/rayleigh-benard.py.html.,Accessed:,September,2022.,[26],GitHub,Discussions,thread,re,de,Rham,complex,with,boundary,conditions,in,Firedrake.,https://github.com/firedrakeproject/firedrake/discussions/2501,,2022.,Accessed:,September,2022.,[27],WolframAlpha.,www.wolframalpha.com.,Accessed:,September,2022.,29

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