TN-02_RAdaptedMeshesTokamakEdgeRegion
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   :description: technical note

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Report,2048465-TN-02,D1.1:,r-adapted,meshes,for,the,tokamak,edge,region,Mashy,Green,&,David,Moxey,,King’s,College,London,Chris,Cantwell,,Bin,Liu,&,Spencer,Sherwin,,Imperial,College,London,28th,March,2022,Contents,1,Executive,summary,2,Introduction,3,New,features,in,NekMesh,3.1,CAD,curve,deﬁned,reﬁnement,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,3.2,CAD,curve,based,r-adaptation,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,4,User,documentation,4.1,CAD,requirement,for,NekMesh,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,4.2,Conﬁguration,ﬁle,(.mcf),for,reﬁnement,deﬁnitions,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,4.3,Command,line,options,for,r-adaptation,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,5,Test,cases,5.1,Circle,in,square,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,5.1.1,CAD,preparation,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,5.1.2,Curve,reﬁnement,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,r-adaptation,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,5.1.3,5.2,Plasma,core,separatrix,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,6,Conclusions,and,future,work,1,Executive,summary,1,2,2,2,5,8,9,9,10,11,11,12,12,14,17,20,This,report,focuses,on,the,initial,phase,of,work,undertaken,by,King’s,College,London,and,Imperial,College,London,to,investigate,challenges,for,generation,of,high-order,meshes,for,the,NEPTUNE,project.,In,this,second,report,that,comprises,Deliverable,1.1,of,our,workplan,,we,extend,the,two-dimensional,high-order,mesh,generation,process,for,producing,meshes,for,the,tokamak,edge,region,,with,a,particular,focus,on,anisotropic,element,clustering,leveraging,the,r-adaptation,method.,1,2,Introduction,The,main,objective,of,Task,1.1,is,to,modify,NekMesh,with,capabilities,to,generate,high-order,meshes,with,adequate,resolution,for,modelling,the,X-point,,plasma,separatrix,and,,if,required,,the,ﬂux,surfaces,in,the,tokamak,edge,region.,It,is,paramount,that,the,reﬁned,mesh,elements,conform,exactly,to,the,geometry,of,these,internal,features.,We,propose,the,use,of,internal,CAD,curves,which,will,guide,the,mesh,generation,process,in,NekMesh,on,how,reﬁne,and,adapt,the,mesh,appropriately,,using,modiﬁed,mesh,reﬁnement,techniques,and,leveraging,the,r-adaptation,scheme,of,[1],within,the,variational,framework,outlined,in,[2].,This,report,discusses,the,extensions,added,to,NekMesh,to,achieve,the,goals,stated,above,beyond,those,already,reported,on,in,our,previous,deliverable,[3].,We,assume,the,reader,is,familiar,with,the,processes,described,there,and,,in,order,to,reduce,the,length,of,this,report,,we,do,not,repeat,the,same,information,here,and,will,refer,to,speciﬁc,sections,of,the,previous,report,instead.,The,report,is,structured,as,follows:,ﬁrst,,we,discuss,the,technical,modiﬁcations,implemented,in,NekMesh,for,both,the,mesh,reﬁnement,and,r-adaptation,in,Section,3,,followed,by,detailed,user,documentation,of,how,to,use,the,new,features,in,Section,4.,Two,examples,are,then,provided,in,Section,5:,ﬁrst,of,a,simple,circle,in,a,square,,where,the,CAD,preparation,and,pipeline,process,are,discussed,in,detail,,followed,by,a,prototype,high-order,mesh,of,the,tokamak,edge,region,with,the,plasma,separatrix,and,X-points,used,for,reﬁnement,and,clustering,of,mesh,elements.,Finally,,conclusions,and,future,work,are,discussed,in,Section,6.,3,New,features,in,NekMesh,To,achieve,the,requirements,speciﬁed,above,,two,additional,features,have,been,added,to,NekMesh.,The,ﬁrst,extends,the,ability,to,specify,mesh,spacing,along,CAD,curves,which,in-,creases,the,mesh,density,in,a,desired,regions,,as,explained,in,Section,3.1.,The,second,additional,feature,uses,the,r-adaptation,scheme,,initially,developed,for,shock,capturing,in,compressible,ﬂuid,simulations.,This,process,shifts,mesh,nodes,towards,regions,of,interest,,whilst,preserving,the,connectivity,and,the,number,of,degrees,of,freedom,in,the,mesh.,This,scheme,is,modiﬁed,to,shift,the,mesh,towards,user-selected,CAD,curves,,resulting,in,anisotropic,elements,that,conform,exactly,the,curve.,These,changes,are,documented,in,Section,3.2.,3.1,CAD,curve,deﬁned,reﬁnement,The,current,release,version,of,NekMesh,uses,an,automatic,reﬁnement,technique,based,on,cur-,vature,where,the,mesh,spacing,is,obtained,with,the,help,of,an,underlying,octree,,as,explained,in,Section,3.2,of,[3],and,in,more,detail,in,reference,[4].,In,addition,to,the,automatic,mesh,spacing,,an,option,to,deﬁne,reﬁnement,lines,in,the,mesh,is,provided,where,a,user,can,specify,two,points,,the,mesh,spacing,,D,,and,a,radius,of,inﬂuence,around,the,line,,R,,to,provide,ad-,ditional,control,of,the,mesh,reﬁnement,to,the,user.,To,achieve,the,objectives,of,task,1.1,,it,is,paramount,to,provide,additional,reﬁnement,control,where,D,and,R,can,be,speciﬁed,over,more,complex,geometries,than,just,straight,lines.,As,such,,we,extended,this,functionality,to,work,on,any,CAD,edge,or,curve.,The,approach,we,have,taken,is,similar,to,the,line,reﬁnement,technique,and,informs,the,surface,meshing,routines,of,the,minimum,mesh,spacing,between,that,obtained,by,the,octree,and,the,manually,speciﬁed,one,in,the,deﬁned,region.,When,the,surface,mesh,is,generated,using,the,adapted,version,of,the,code,Triangle,,it,queries,the,octree,where,the,triangle,is,to,be,formed,for,its,spacing,,δ,,which,is,obtained,using,equation,(1),in,report,D1.3,[3].,We,note,that,δ,is,deﬁned,2,by,the,curvature,of,the,CAD,curve,to,set,the,mesh,spacing,not,only,for,triangles,formed,on,the,CAD,curve,but,for,all,triangles,within,the,same,octant.,The,user,deﬁned,reﬁnement,adds,an,additional,constraint,on,the,mesh,spacing,,where,if,any,of,the,triangle’s,vertices,are,a,distance,of,R,or,less,from,a,line,or,a,CAD,curve,,the,triangle’s,spacing,is,the,minimum,between,the,speciﬁed,value,and,the,one,assigned,from,the,octree,,i.e.,δ,=,min(D,,δ).,Querying,the,octree,for,the,mesh,spacing,in,the,MeshGen,module,NekDouble,Octree::Query(Array<OneD,,NekDouble>,loc),{,//,starting,at,master,octant,0,move,through,succsesive,m_octants,which,//,contain,the,point,loc,until,a,leaf,is,found.,//,first,search,through,sourcepoints,NekDouble,tmp,=,numeric_limits<double>::max();,//,search,line,sourcepoints,for,(int,i,=,0;,i,<,m_lsources.size();,i++),{,if,(m_lsources[i].withinRange(loc)),{,tmp,=,min(m_lsources[i].delta,,tmp);,},},//,search,curve,sourcepoints,for,(int,i,=,0;,i,<,m_csources.size();,i++),{,if,(m_csources[i].withinRange(loc)),{,tmp,=,min(m_csources[i].delta,,tmp);,},},OctantSharedPtr,n,=,m_masteroct;,bool,found,=,false;,while,(!found),{,//,...,A,lot,more,code,to,find,the,leaf,octant,...,n,=,n->GetChild(quad);,if,(n->IsLeaf()),{,found,=,true;,},},//,obtain,the,minimum,delta,between,the,leaf,octant,and,the,refinement,//,lines,/,curves,return,min(n->GetDelta(),,tmp);,},3,The,query,uses,a,new,struct,for,the,CAD,curves,used,as,reﬁnement,sources.,curvesource,struct,used,by,the,Octree,in,the,MeshGen,module,struct,curvesource,{,CADCurveSharedPtr,curve;,NekDouble,R,,delta;,curvesource(CADCurveSharedPtr,c,,NekDouble,r,,NekDouble,d),:,curve(c),,R(r),,delta(d),{,},//,Tests,if,a,point,is,within,a,specified,range,,R,,from,the,curve,bool,withinRange(Array<OneD,,NekDouble>,p),{,if,(curve->GetMinDistance(p),<=,R),{,return,true;,},else,{,return,false;,},},};,We,note,that,this,modiﬁcation,required,the,ability,to,obtain,distances,from,an,arbitrary,point,in,the,domain,to,a,CAD,curve,,implemented,in,the,CAD,engine,interface,of,NekMesh,as,a,function,CADCurveOCE::GetMinDistance.,Although,NekMesh,is,designed,to,support,multiple,backends,(speciﬁcally,OpenCASCADE,and,the,ITI,interface,CFI),,at,present,we,have,only,implemented,this,capability,in,the,OpenCASCADE,backend,and,hence,is,not,currently,supported,in,other,CAD,systems.,Obtaining,the,minimum,distance,of,a,point,from,the,CAD,curve.,//,Nektar++,distances,are,in,m,,OpenCASCADE,are,in,mm,NekDouble,CADCurveOCE::GetMinDistance(Array<OneD,,NekDouble>,xyz),{,gp_Pnt,loc(xyz[0],*,1000.0,,xyz[1],*,1000.0,,xyz[2],*,1000.0);,GeomAPI_ProjectPointOnCurve,proj(loc,,m_c,,m_b[0],,m_b[1]);,if(proj.NbPoints()),{,return,proj.LowerDistance(),/,1000.0;,},else,{,return,std::min(loc.Distance(m_c->Value(m_b[0])),,loc.Distance(m_c->Value(m_b[1]))),/,1000.0;,4,(a),(b),Figure,1:,Portion,of,a,mesh,using,curve,reﬁnement,(a),without,octree,manipulation;,(b),with,octree,manipulation.,},},A,limitation,of,the,above,approach,is,that,it,works,on-top,of,the,underlying,octree,structure,(by,specifying,the,minimum,spacing,in,a,speciﬁc,region),which,often,results,in,an,undesirably,abrupt,decrease,in,mesh,spacing,,and,consequently,some,level,of,anisotrapy,,as,can,be,seen,in,Figure,1(a).,To,rectify,this,issue,and,obtain,better,meshes,,we,have,implemented,additional,logic,to,manipulate,the,underlying,octree,structure.,This,is,done,by,encouraging,the,octants,where,a,reﬁnement,line,or,curve,is,found,to,continue,subdividing,until,their,spacing,is,equal,to,or,smaller,than,the,user,deﬁned,spacing,D,of,the,line,or,CAD,curve.,This,results,in,a,more,gradual,decrease,of,mesh,spacing,at,the,expense,of,additional,degrees,of,freedom,as,seen,in,Figure,1(b).,We,note,that,the,octree,manipulation,exerts,the,inﬂuence,of,the,reﬁnement,beyond,the,strictly,deﬁned,region,in,a,manner,that,is,hard,to,predict,due,to,the,nature,of,the,octree’s,subdivision,process.,The,source,code,for,this,change,is,heavily,fragmented,and,left,out,of,this,report.,It,required,signiﬁcant,changes,to,the,source,point,structs,in,Octree/SourcePoints.hpp,and,most,notably,the,algorithms,in,Octree::CompileSourcePointList(),and,Octant::Octant.,3.2,CAD,curve,based,r-adaptation,r-adaptation,is,a,novel,scheme,available,in,NekMesh,,whereby,the,position,of,mesh,vertices,are,adjusted,to,obtain,additional,resolution,where,it,is,needed,in,the,domain,without,changing,the,connectivity,of,the,mesh,,consequently,ﬁxing,the,number,of,degrees,of,freedom.,The,application,of,this,process,can,be,seen,in,[1],,where,this,was,combined,with,p-adaptation,to,better,resolve,discontinuities,in,the,ﬂow,region,by,squeezing,elements,toward,the,shock,region,,resulting,in,anisotropic,elements,surrounding,the,shock,structure.,The,method,utilises,the,variational,mesh,optimisation,technique,[2],brieﬂy,described,in,Section,3.3.3,of,report,D1.3,[3],,while,periodically,scaling,the,Jacobian,of,the,element,mappings,based,on,the,discontinuity,found,in,the,solution,ﬁeld,during,the,optimisation,iterations.,For,our,purposes,of,generating,highly,anisotropic,mesh,elements,conforming,to,the,shape,of,the,plasma,core,separatrix,and,ﬂux,surfaces,,we,have,adapted,the,r-adaptation,method,to,be,able,5,to,scale,and,shift,elements,towards,user,speciﬁed,CAD,curves.,In,this,way,the,user,can,control,a,priori,where,additional,resolution,will,be,required.,The,user,needs,to,specify,the,scaling,factor,and,,optionally,,a,radius,of,inﬂuence,around,the,curve,where,elements,are,scaled.,This,process,is,aided,by,the,ability,of,the,optimisation,procedure,to,slide,mesh,nodes,(excluding,the,end,vertices,which,remain,ﬁxed),along,CAD,curves,which,ensures,the,mesh,remains,conformal,to,the,curve,during,the,adaptation,process,,thus,exactly,capturing,the,desired,feature,within,the,domain.,To,allow,the,mesh,nodes,more,freedom,of,movement,,it,is,best,to,apply,the,r-adaptation,on,a,linear,mesh,and,elevate,it,to,a,high,order,mesh,after,the,adaptation,procedure.,The,r-,adaptation,procedure,when,applied,to,the,CAD,curves,using,a,linear,mesh,however,is,very,sensitive,to,the,scaling,factor,and,radius,of,inﬂuence,,which,are,problem,speciﬁc,parameters,that,require,manually,tuning.,If,the,scaling,factor,is,too,small,,this,can,result,in,such,highly,stretched,elements,that,the,high-order,optimisation,scheme,cannot,be,guaranteed,to,resolve,under,lower,resolutions.,Thus,,depending,on,the,desired,level,of,anisotropy,,a,ﬁner,resolution,might,be,required,using,the,CAD,curve,reﬁnement,process,described,above,in,Section,3.1.,Some,discussion,on,safe,values,found,through,trial,and,error,can,be,found,in,Section,5.,How-,ever,,a,detailed,study,of,the,parameters,is,left,for,future,work,,for,the,main,reason,that,in,upcoming,Task,1.2,,we,will,consider,quad-based,mesh,generation,approaches,to,this,problem.,One,signiﬁcant,part,of,this,is,to,utilise,isoparametric,boundary,layer,mesh,generation,in,these,regions,,based,on,the,work,in,[5].,This,may,give,better,control,over,the,anisotropy,required,in,this,region.,From,the,implementation,perspective,,several,changes,to,the,elements,class,of,the,optimisation,module,,ElUtil,,were,required.,The,current,function,where,the,mapping,is,updated,was,overhauled,to,become,CAD,aware,and,to,allow,for,either,the,shock-based,or,the,CAD-based,adaptation,to,work,seamlessly.,The,CAD-based,r-adaptation,was,further,split,for,performance,reasons,into,cases,where,elements,have,a,vertex,on,a,CAD,curve,and,those,where,a,radius,of,inﬂuence,extending,from,the,curve,is,deﬁned.,In,the,latter,,a,brute,force,approach,is,utilised,to,check,the,distance,of,each,vertex,of,each,element,from,the,CAD,curve.,UpdateMapping,function,using,CAD,curves,inside,the,ProcessVarOpti,module’s,elements,utility,ElUtil,if,(m_adapt_scale),{,//,Using,elements,with,a,node,with,a,radius,from,the,curve,if(m_adapt_radius),{,[&]{,for,(auto,&vert,:m_el->GetVertexList()),{,Array<OneD,,NekDouble>,x(3);,=,vert->m_x;,x[0],=,vert->m_y;,x[1],x[2],=,vert->m_z;,for,(auto,&curve,:,m_adaptcurves),{,if,(curve->GetMinDistance(x),<,m_adapt_radius),{,scaling,=,m_adapt_scale;,6,return;,//,return,lambda,function,to,break,loop,},},},}();,},//,Using,only,elements,with,a,node,on,the,curve,else,{,[&]{,for,(auto,&vert,:m_el->GetVertexList()),{,for,(auto,&curve,:,vert->GetCADCurves()),{,std::vector<CADCurveSharedPtr>::iterator,it,=,std::find(m_adaptcurves.begin(),,m_adaptcurves.end(),,if,(it,!=,m_adaptcurves.end()),{,scaling,=,m_adapt_scale;,return;,//,return,lambda,function,to,break,loop,},},(cid:44)→,curve);,},}();,},},if,(scaling),{,for,(int,i,=,0;,i,<,m_maps.size();,++i),{,for,(int,j,=,0;,j,<,9;,++j),{,maps[i][j],mapsStd[i][j],=,m_mapsStd[i][j],/,scaling;,=,m_maps[i][j],/,scaling;,},if,(m_dim,==,2),{,maps[i][9],mapsStd[i][9],=,m_mapsStd[i][9],*,scaling,*,scaling;,=,m_maps[i][9],*,scaling,*,scaling;,},else,if,(m_dim,==,3),{,maps[i][9],mapsStd[i][9],=,m_mapsStd[i][9],*,scaling,*,scaling,*,scaling;,=,m_maps[i][9],*,scaling,*,scaling,*,scaling;,},else,7,ASSERTL0(false,,"not,coded");,{,},},},else,{,maps,=,m_maps;,mapsStd,=,m_mapsStd;,},This,approach,is,suitable,for,the,prototype,phase,,but,further,improvements,such,as,utilising,the,octree,structure,to,reduce,the,search,space,can,be,made.,However,,within,the,2D,framework,considered,in,this,report,,and,taking,into,account,the,adaptation,process,that,takes,place,on,a,linear,mesh,,we,found,the,performance,to,be,acceptable.,Additionally,,a,bug,related,to,the,identiﬁcation,of,nodes,that,are,attached,to,the,CAD,curves,was,found,,whereby,the,end,vertices,of,CAD,curves,were,not,identiﬁed,and,were,allowed,to,be,move,as,regular,optimisation,nodes,(i.e.,not,attached,to,a,CAD,curve).,Since,the,mesh,is,generated,in,a,bottom-up,fashion,,this,can,lead,to,some,elements,no,longer,conforming,to,the,curve,and/or,general,degradation,of,the,mesh,quality,,as,demonstrated,by,Figure,2(a).,To,address,this,issue,,the,end,vertices,of,CAD,curve,are,now,removed,from,the,optimisation,and,are,considered,to,be,permanently,ﬁxed,in,space.,This,avoids,any,degradation,of,the,geometry,during,the,optimisation,process,,as,seen,in,2(b).,(a),(b),Figure,2:,Section,of,mesh,showing,the,intersection,between,two,edges,after,applying,r-,reﬁnement,to,a,linear,mesh,then,upgrading,it,to,higher,order:,(a),bug,-,CAD,curve,vertices,unidentiﬁed,and,allowed,to,move;,(b),bug,ﬁx,-,CAD,curve,vertices,ﬁxed,in,space.,4,User,documentation,In,the,following,sections,,a,description,of,how,to,use,the,additional,NekMesh,features,is,pro-,vided.,First,,the,CAD,requirements,for,generating,meshes,with,internal,curves,is,given,followed,by,a,description,of,the,additional,ﬁelds,used,in,the,NekMesh,conﬁguration,.mcf,ﬁle,used,for,the,curve,reﬁnement,feature.,Finally,the,additional,command,line,options,used,for,the,curve-based,r-adaptation,feature,are,outlined.,8,4.1,CAD,requirement,for,NekMesh,NekMesh,mainly,supports,the,OpenCASCADE,CAD,system,[6],and,the,use,of,STEP,ﬁles.,A,discussion,on,the,CAD,representation,can,be,found,in,Section,3.1,of,report,D1.3,[3].,Here,,we,discuss,the,requirements,for,generating,a,valid,STEP,ﬁle,for,NekMesh,with,internal,edges,(or,curves).,CAD,curves,read,into,NekMesh,must:,1.,not,be,periodic,,i.e.,has,distinct,two,distinct,end,vertices,,or,in,other,words,that,for,the,parametrisation,of,the,curve,,r(t),,there,does,not,exist,a,constant,T,such,that,r(t,+,T,),=,r(t);,2.,form,an,edge,loop,,i.e.,both,its,end,vertices,must,also,be,an,end,vertex,of,another,curve,,or,in,other,words,it,is,part,of,a,set,of,edges,that,together,form,a,closed,loop.,The,latter,restriction,also,implies,that,the,internal,curves,have,to,split,the,domain,into,multiple,unique,faces,that,share,some,of,the,same,edges,with,the,neighbouring,faces.,This,requires,the,decomposition,of,the,computational,domain,in,a,particular,manner,that,may,not,seem,intuitive,from,the,CAD,software,perspective.,Two,examples,,one,of,a,simple,circle,in,a,square,and,one,of,a,geometry,more,representative,of,the,cross,section,of,tokamak,reactor,with,the,plasma,separatrix,are,provided,in,Sections,5.1,and,5.2,respectively.,4.2,Conﬁguration,ﬁle,(.mcf),for,reﬁnement,deﬁnitions,The,.mcf,ﬁle,is,the,conﬁguration,ﬁle,for,the,import,module,in,NekMesh.,It,uses,a,xml,ﬁle,formal.,In,principle,,for,most,aeronautics,applications,,meshing,is,done,mostly,automatically,and,only,a,few,basic,options,have,to,be,explicitly,provided.,As,an,example,,a,typical,basic,2D,.mcf,ﬁle,would,contain,the,following:,<NEKTAR>,<MESHING>,<INFORMATION>,<I,PROPERTY="CADFile",VALUE="case.stp"/>,<I,PROPERTY="MeshType",VALUE="2D"/>,</INFORMATION>,<PARAMETERS>,<P,PARAM="MinDelta",VALUE="0.01"/>,<P,PARAM="MaxDelta",VALUE="0.01"/>,VALUE="0.1"/>,<P,PARAM="EPS",VALUE="4"/>,<P,PARAM="Order",</PARAMETERS>,</MESHING>,</NEKTAR>,where,the,parameters,for,MinDelta,,MaxDelta,and,EPS,are,explained,in,Section,3.2,of,report,D1.3,[3],and,Order,is,the,order,of,the,mesh,,in,this,example,4th,order.,For,more,advanced,meshes,containing,boundary,layers,,additional,options,can,be,deﬁned,,which,are,described,in,the,NekMesh,user,guide,[7].,Reﬁnement,lines,use,a,separate,XML,tag,with,the,following,format,,where,the,values,follow,the,same,notation,use,in,Section,3.1.,The,following,example,shows,how,a,reﬁnement,line,between,9,points,(x1,,y1,,z1),and,(x2,,y2,,z2),is,deﬁned,with,radius,R,=,0.005,,wherein,the,mesh,spacing,is,forced,to,a,value,δ,=,0.001.,<REFINEMENT>,<LINE>,<X1>,-0.01,</X1>,<Y1>,-0.01,</Y1>,<Z1>,0.0,</Z1>,<X2>,0.01,</X2>,<Y2>,0.01,</Y2>,<Z2>,0.0,</Z2>,<R>,0.005,</R>,0.001,</D>,<D>,</LINE>,</REFINEMENT>,For,consistency,reasons,,we,have,followed,the,same,format,for,CAD,curve,reﬁnement,,except,we,have,replaced,the,start,and,end,points,of,the,line,with,the,edge,ID,as,deﬁned,in,the,STEP,ﬁle.,<REFINEMENT>,<CURVE>,<ID>,1,</ID>,<R>,0.005,</R>,<D>,0.001,</D>,</CURVE>,</REFINEMENT>,Any,number,of,lines,and,curves,can,be,speciﬁed,within,the,same,reﬁnement,tag,in,any,order,,one,at,a,time.,An,example,using,several,curves,and,lines,can,be,found,in,the,provided,plasma,core,separatrix,test,case,attached,to,this,report.,4.3,Command,line,options,for,r-adaptation,In,contrast,to,the,curve,reﬁnement,which,is,read,by,the,input,module,,r-adaptation,is,part,of,the,variational,optimisation,module,,varopti.,Its,parameters,are,set,using,the,command,line,options,for,varopti,,which,is,important,as,it,allows,for,multiple,instances,of,the,module,to,be,piped,to,each,other,using,diﬀerent,parameters.,The,option,for,r-adaptation,using,curves,is,activated,when,a,curve,is,speciﬁed,using,the,option,radaptcurves,that,can,take,a,combination,of,curve,ids,in,list,and,range,format,as,follows:,NekMesh,-m,varopti:<optimisation_method>:radaptcurves=<int,id_item1>,<int,id_item2>,<int,id_range1_start>-<int,id_range1_end>:nq=<order_of_mesh>,(cid:44)→,where,optimisation,method,is,the,functional,used,for,the,optimisation,process,as,described,in,[2],and,nq,is,the,number,of,integration,points,per,element,edge,,which,is,equivalent,to,the,order,of,the,mesh,plus,1,,i.e.,nq=2,for,a,ﬁrst,order,linear,mesh,and,nq=5,for,a,4th,order,mesh.,Additionally,,a,scaling,factor,must,be,set,for,the,adaptation,to,take,eﬀect,,using,the,option,10,-m,varopti:<previous_options>:radaptscale=<double,scaling_factor>:...,Setting,the,radius,is,optional,and,can,be,set,with,-m,varopti:<previous_options>:radaptrad=<double,scaling_radius_value>:...,Finally,,an,additional,parameter,for,selecting,the,number,of,optimisation,iterations,between,scaling,updated,can,be,set,with,-m,varopti:<previous_options>:subiter=<int,number_of_iterations>,This,option,is,required,as,applying,the,scaling,too,often,can,prevent,the,optimisation,process,to,reach,the,desired,tolerance.,Crucially,,NekMesh,can,run,a,pipeline,modules,in,a,single,command,,so,multiple,passes,through,varopti,can,be,easily,run.,For,instance,NekMesh,-m,varopti:linearelsatic:radaptcurves=1-4:radaptscale=0.4:subiter=5:nq=2,-m,varopti:linearelsatic:radaptcurves=1-4:radaptscale=0.45:radaptrad=0.0,(cid:99),(cid:44)→,05:subiter=15:nq=2,-m,varopti:hyperelastic:nq=5,<case_name>.mcf,<mesh_name>.xml,would,be,a,typical,way,to,use,the,r-adaptation,process,,where:,the,ﬁrst,use,of,varopti,uses,the,linear,elastic,method,for,optimisation,,curves,1-4,are,selected,for,the,adaptation,,the,scaling,factor,is,0.4,applied,every,5th,iteration,and,the,optimisation,is,done,on,a,linear,mesh;,the,second,use,of,varopti,is,identical,to,the,ﬁrst,only,with,a,larger,scaling,factor,applied,every,15th,iteration,to,all,elements,with,a,vertex,within,5,mm,from,the,selected,curves;,the,last,call,to,varopti,uses,the,hyper-elastic,model,on,a,4th,order,mesh,without,applying,any,further,adaptation,and,optimising,the,high,order,mesh,nodes.,5,Test,cases,The,following,section,contains,two,test,cases.,The,ﬁrst,is,a,simple,circle,embedded,in,a,square,used,to,illustrate,in,detail,the,pipeline,process,to,generate,a,high-order,anisotropic,mesh,reﬁned,and,adapted,towards,the,CAD,curves,of,the,circle.,The,second,case,resembles,a,tokamak,cross,section,with,the,plasma,separatrix.,Due,to,the,unavailability,of,the,real,geometry,,this,is,our,best,attempt,at,recreating,one,by,tracing,an,image,in,FreeCAD.,5.1,Circle,in,square,This,test,case,consists,of,a,square,of,width,100,mm,with,an,embedded,circle,of,radius,70,mm,in,its,centre.,Although,simple,,it,serves,as,a,demonstration,of,the,entire,pipeline,,exploring,the,various,modiﬁcations,and,their,individual,inﬂuences,on,the,mesh,generation,process,when,considering,internal,curves,for,clustering,elements,in,regions,of,interest.,11,Two,ﬁles,,a,STEP,ﬁle,containing,the,CAD,information,,Cases/2d,circle,square.stp,,and,a,conﬁguration,ﬁle,,Cases/2d,circle,square.mcf,,are,attached,to,this,report,in,order,to,recreate,any,steps,of,this,example,and,made,available,as,part,of,the,NekMesh,test,suite.,5.1.1,CAD,preparation,Following,Section,4.1,,the,domain,is,split,into,two,non-overlapping,faces,with,eight,distinct,edges,(see,Figure,3).,Edges,1–4,represent,the,curves,of,the,circle,and,are,present,in,both,the,square,and,the,circle,faces,,as,illustrated,in,Figures,4.,Figure,3:,Illustration,of,the,faces,and,edges,of,the,circle,in,square,test,case.,5.1.2,Curve,reﬁnement,Meshing,the,above,case,with,the,unmodiﬁed,NekMesh,using,a,large,initial,mesh,spacing,of,10,mm,,as,seen,in,the,2d,circle,square.mcf,ﬁle,below,,we,execute,the,following,commands:,NekMesh,-v,-m,varopti:hyperelastic:nq=5,2d_circle_square.mcf,(cid:44)→,2d_circle_square.xml,<NEKTAR>,<MESHING>,<INFORMATION>,<I,PROPERTY="CADFile",VALUE="2d_circle-square.stp"/>,<I,PROPERTY="MeshType",VALUE="2D"/>,</INFORMATION>,<PARAMETERS>,<P,PARAM="MinDelta",VALUE="0.0015"/>,<P,PARAM="MaxDelta",VALUE="0.01"/>,VALUE="0.1"/>,<P,PARAM="EPS",12,(a),(b),Figure,4:,Illustration,of,the,separate,faces,and,associated,edges,of,the,circle,in,square,test,case,showing,(a),the,square,region,without,the,embedded,circle;,(b),the,circle,region,without,the,surrounding,square.,<P,PARAM="Order",VALUE="4"/>,</PARAMETERS>,</MESHING>,</NEKTAR>,This,results,in,a,sparse,4th,order,mesh,consisting,of,294,elements,and,4,470,degrees,of,freedom,,as,shown,in,Figure,5(a).,We,can,see,that,no,automatic,reﬁnement,took,place,due,to,the,curvature,being,adequately,captured,with,the,speciﬁed,MaxDelta,of,10,mm,in,mesh,spacing,,resulting,in,10,discrete,segments,per,edge,of,the,square.,To,better,resolve,the,circle,,we,select,curves,1–4,as,reﬁnement,curves,and,set,the,mesh,spacing,to,D,=,0.15mm,with,a,radius,of,inﬂuence,R,=,0.5,mm,,by,adding,the,following,<REFINEMENT>,tag,to,the,2d,circle,square.mcf.,<REFINEMENT>,<CURVE>,<ID>,1,</ID>,<R>,0.005,</R>,<D>,0.0015,</D>,</CURVE>,<CURVE>,<ID>,2,</ID>,<R>,0.005,</R>,<D>,0.0015,</D>,</CURVE>,13,<CURVE>,<ID>,3,</ID>,<R>,0.005,</R>,<D>,0.0015,</D>,</CURVE>,<CURVE>,<ID>,4,</ID>,<R>,0.005,</R>,<D>,0.0015,</D>,</CURVE>,</REFINEMENT>,Note,that,in,the,provided,conﬁguration,ﬁle,the,<REFINEMENT>,tag,is,already,present.,Using,the,modiﬁed,NekMesh,with,the,CAD,curve,reﬁnement,and,executing,the,same,command,as,the,previous,case,with,the,unmodiﬁed,code,,we,can,see,in,Figure,5(b),that,the,resolution,around,the,circle,drastically,increases.,Additionally,,the,octree,smoothing,is,exerting,its,inﬂu-,ence,throughout,most,of,the,domain,,which,further,increases,the,resolution,,resulting,with,a,mesh,containing,4,184,elements,and,66,022,degrees,of,freedom.,(a),(b),Figure,5:,Mesh,of,the,circle,in,square,test,case,(a),the,unmodiﬁed,version,of,NekMesh,with,no,reﬁnement;,(b),the,modiﬁed,version,of,NekMesh,using,reﬁnement,around,the,CAD,curves,of,the,circle.,5.1.3,r-adaptation,Despite,the,increase,in,resolution,,the,mesh,produced,by,employing,reﬁnement,is,still,isotropic,and,there,is,no,clustering,within,the,reﬁned,zone,around,the,circle’s,curves.,To,achieve,that,,we,employed,the,r-adaptation,method,,which,shall,be,explored,next.,Keeping,in,mind,Section,4.3,,we,will,adopt,a,pipeline,process,using,multiple,executions,of,the,variational,optimisation,mod-,ule.,The,diﬀerent,parameters,will,be,demonstrated,in,the,following,steps.,First,,we,attempt,to,generate,a,mesh,using,r-adaptation,only,on,elements,adjacent,to,the,circle’s,curve,,i.e.,without,specifying,a,radius.,This,is,done,with,following,command:,14,NekMesh,-v,-m,varopti:linearelastic:radaptcurves=1-4:radaptscale=0.275:subiter=5:re,(cid:99),(cid:44)→,stol=1e-5:nq=2:numthreads=10,-m,varopti:hyperelastic:nq=5:numthreads=10,2d_circle_square.mcf,2d_circle_square.xml,We,note,the,use,of,additional,options,,namely,restol,and,numthreads,which,were,not,pre-,viously,explained.,The,former,is,the,minimum,residual,required,to,halt,the,iterations,in,the,variational,optimisation,process,,and,the,latter,is,number,of,execution,threads,on,a,multi-core,CPU.,As,explained,in,Section,3.2,,the,r-adaptation,is,executed,on,a,linear,mesh,by,using,nq=2,,followed,by,a,second,pass,of,the,variational,optimisation,module,without,r-adaptation,and,using,nq=5,to,promote,the,mesh,to,4th,order,whilst,ensuring,the,process,of,increasing,the,polynomial,order,does,not,degrade,the,mesh,quality.,The,resulting,mesh,contains,the,same,number,of,elements,and,degrees,of,freedom,as,the,reﬁned,mesh,,but,as,seen,in,Figure,6,,anisotropic,mesh,elements,are,now,formed,on,the,circle,,while,the,elements,connected,to,them,are,still,isotropic,,resulting,in,a,sharp,transition.,The,scaling,factor,of,0.275,was,obtained,by,trial,error,by,decreasing,the,value,until,the,resulted,elements,were,too,skewed,for,the,selected,resolution.,(a),(b),Figure,6:,Circle,in,square,mesh,using,curve,reﬁnement,and,r-adaptation,applied,only,to,ele-,ments,adjacent,to,the,circle’s,curve:,(a),full,view;,(b),zoomed,in,view,around,a,curve.,To,address,the,sharp,transition,and,to,cluster,more,elements,around,the,circle,,a,radius,of,inﬂuence,of,0.2mm,wide,is,used,with,the,following,commands:,NekMesh,-v,-m,varopti:linearelastic:radaptcurves=1-4:radaptscale=0.275:radaptrad=0.,(cid:99),(cid:44)→,002:subiter=20:restol=1e-5:nq=2:numthreads=10,-m,varopti:hyperelastic:nq=5:numthreads=10,2d_circle_square.mcf,2d_circle_square.xml,Once,again,,it,was,found,that,a,scaling,factor,value,smaller,than,0.275,resulted,in,invalid,15,elements,when,upgrading,the,mesh,to,higher-order.,As,observed,in,Figure,7,,the,resulting,mesh,displays,a,wider,band,of,anisotropic,elements,around,the,circle’s,curves,,however,the,elements,adjacent,to,the,curves,are,less,stretched,than,before.,It,is,also,noted,that,the,value,for,subiter,is,relaxed,from,applying,the,scaling,every,5,iterations,to,every,20,iterations.,This,is,due,to,the,scaling,of,more,elements,,which,decreases,the,value,of,the,functional,being,optimised,within,the,variational,setting,(see,reference,[2]),,and,the,desired,residual,set,for,the,optimisation,not,being,reached,between,the,Jacobian,mapping,scaling,updates.,(a),(b),Figure,7:,Circle,in,square,mesh,using,curve,reﬁnement,and,r-adaptation,applied,to,elements,within,a,ﬁxed,radius,from,the,circle’s,curve:,(a),full,view;,(b),zoomed,in,view,around,a,curve.,Ideally,,we,wish,to,combine,the,highly,stretched,anisotropic,elements,around,the,curve,from,the,case,without,using,a,radius,with,the,wider,band,of,anisotropic,elements,obtained,from,applying,the,radius.,To,do,so,,we,can,chain,both,execution,of,the,variational,optimiser,using,r-adaptation,,ﬁrst,without,a,radius,and,then,with,using,the,following,commands:,NekMesh,-v,(cid:44)→,(cid:44)→,-m,varopti:linearelastic:radaptcurves=1-4:radaptscale=0.4:subiter=5:rest,(cid:99),ol=1e-5:nq=2:numthreads=10,-m,varopti:linearelastic:radaptcurves=1-4:radaptscale=0.4:radaptrad=0.00,(cid:99),2:subiter=15:restol=1e-5:nq=2:numthreads=10,-m,varopti:hyperelastic:nq=5:numthreads=10,2d_circle_square.mcf,2d_circle_square.xml,Finally,,we,obtain,the,desired,mesh!,As,seen,in,Figure,8,,the,resulting,mesh,has,both,highly,stretched,elements,adjacent,to,the,curve,as,well,as,a,band,of,anisotropic,elements,surrounding,the,curves,with,a,smoother,transition,towards,the,isotropic,elements,in,the,rest,of,the,domain.,In,this,ﬁnal,trial,,the,scaling,factor,was,increased,to,0.4,to,account,for,the,fact,that,the,scaling,is,applied,twice,to,the,elements,adjacent,to,the,curves.,To,summarise,,the,pipeline,process,used,to,generate,the,desired,meshes,,starting,from,deﬁning,reﬁnement,curves,in,the,.mcf,ﬁle,to,the,multiple,executions,of,the,variational,optimisation,module,have,been,demonstrated,,with,each,step,in,the,pipeline,explained.,The,values,used,in,the,diﬀerent,points,in,the,pipeline,are,problem,dependent,,although,from,our,experience,16,(a),(b),Figure,8:,Circle,in,square,mesh,using,curve,reﬁnement,and,r-adaptation,applied,ﬁrst,only,to,elements,adjacent,to,the,circle’s,curves,,then,to,elements,within,a,ﬁxed,radius,from,the,circle’s,curve:,(a),full,view;,(b),zoomed,in,view,around,a,curve.,the,selected,scaling,factor,values,are,similar,for,most,cases,and,range,between,0.3,to,0.65,when,employing,r-adaptation,once,without,and,once,with,a,radius.,Further,reduction,of,the,reﬁnement,mesh,spacing,can,allow,for,the,smaller,limit,of,scaling,factor,range,,although,further,studies,into,the,relation,between,the,various,parameters,is,still,required,and,left,for,future,work,as,explained,in,Section,3.2.,5.2,Plasma,core,separatrix,The,second,test,case,is,a,schematic,cross,section,of,the,tokamak,ﬁrst,wall,,where,the,CAD,is,a,very,rough,and,simpliﬁed,approximation,of,the,geometry,obtained,by,tracing,the,illustration,in,Figure,9,in,FreeCAD.,We,used,mm,as,the,units,when,tracing,which,gave,an,enclosing,domain,with,a,height,of,222,mm,and,width,of,106,mm.,We,are,aware,that,this,does,not,represent,reality,,however,scaling,the,units,is,trivial,for,the,mesh,generation,process.,Only,the,plasma,separatrix,of,the,edge-region,was,modelled,internally.,The,ﬂux,surfaces,were,left,out,at,this,stage,as:,a),it,is,unclear,if,these,would,form,part,of,the,requirements;,b),due,to,the,CAD,creating,process,,creating,separate,faces,for,each,gap,between,the,surfaces,could,be,extremely,time,consuming,and,was,judged,to,not,be,worth,the,eﬀort,without,the,availability,of,an,accurate,geometry.,The,resulting,STEP,ﬁle,consists,of,three,distinct,and,non-overlapping,faces,and,thirty,edges,,shown,in,Figure,10.,Edges,1,,2,,4,and,7–22,represent,the,walls,,edge,3,the,pump,and,edges,5,,6,and,23–30,the,internal,plasma,separatrix.,The,latter,group,is,mainly,of,interest,for,reﬁnement,and,r-adaptation,,however,some,outer,edges,also,require,reﬁnement,to,enable,a,smooth,mesh,transition,between,the,internal,regions,and,the,walls.,The,two,ﬁles,for,this,test,case,are,attached,to,this,report,in,Cases/cross,section.stp,for,the,CAD,ﬁle,and,Cases/cross,section.mcf,for,the,conﬁguration,ﬁle.,It,is,highly,recommended,that,the,reader,refers,to,the,conﬁguration,ﬁle,when,following,this,section,as,the,full,content,of,the,reﬁnement,deﬁnitions,is,too,lengthy,to,include,in,the,report.,The,basic,mesh,conﬁguration,(excluding,reﬁnement),is,similar,to,the,previous,test,case.,The,minimum,and,maximum,mesh,spacing,are,both,set,to,10mm,,noting,that,all,reﬁnement,will,17,Figure,9:,Illustrative,example,of,a,tokamak,cross,section.,Taken,from,reference,[8].,be,user,deﬁned,,and,the,(cid:15),=,1.,The,order,is,set,to,1,as,we,know,in,advance,that,reﬁnement,is,to,take,place,before,the,last,pass,of,the,variational,optimisation,upgrades,the,mesh,to,higher-,order.,Reﬁnement,is,then,applied,to,all,internal,edges,,i.e.,edges,5,,6,and,23–30,with,a,mesh,spacing,of,0.75,mm,and,a,radius,of,inﬂuence,of,3.5,mm.,These,parameters,were,carefully,chosen,so,that,the,reﬁnement,zones,around,the,internal,curves,do,not,contain,parts,of,the,outer,edges,while,obtaining,a,smooth,transition,in,the,mesh.,However,,in,the,bottom,region,surrounding,the,X-point,where,the,wall,edges,are,in,close,proximity,to,the,internal,ones,,further,tuning,was,required.,The,wall,edges,were,split,into,two,groups,with,slightly,diﬀerent,reﬁnement,parameters,as,follows:,edges,1–4,,7–10,,12,and,13,form,the,ﬁrst,group,of,edges,in,close,proximity,the,X-,point,,with,a,mesh,spacing,set,to,1.5mm,and,diﬀerent,radius,of,inﬂuence,values,to,obtain,a,smooth,mesh;,while,edges,11,and,14,which,are,further,away,,as,subscribed,a,mesh,spacing,of,0.25,mm,,again,with,diﬀerent,radius,values.,Lastly,,to,further,increase,the,resolution,just,around,the,X-point,,two,additional,reﬁnement,lines,were,deﬁned,using,a,resolution,of,0.2,mm,as,follows:,<LINE>,<X1>,-0.092,</X1>,18,Figure,10:,Illustration,of,the,faces,and,edges,of,the,tokamak,cross,section,test,case.,19,<Y1>,-0.062,</Y1>,<Z1>,0.0,</Z1>,<X2>,-0.090,</X2>,<Y2>,-0.066,</Y2>,<Z2>,0.0,</Z2>,<R>,0.00020,</R>,<D>,0.00020,</D>,</LINE>,<LINE>,<X1>,-0.0925,</X1>,<Y1>,-0.0645,</Y1>,<Z1>,0.0,</Z1>,<X2>,-0.088,</X2>,<Y2>,-0.063,</Y2>,<Z2>,0.0,</Z2>,<R>,0.00020,</R>,<D>,0.00020,</D>,</LINE>,For,r-adaptation,,only,the,internal,curves,are,considered.,In,the,ﬁrst,pass,of,the,variational,optimisation,module,,the,scaling,factor,is,set,to,0.4,,only,targeting,elements,which,are,adjacent,to,the,curves.,In,the,second,pass,the,scaling,factor,is,slightly,increased,to,0.45,and,a,radius,of,inﬂuence,of,1mm,is,set.,Lastly,,in,the,third,and,ﬁnal,pass,no,r-adaptation,is,employed,and,the,mesh,gets,upgraded,to,5th,polynomial,order.,The,full,execution,command,is,given,below:,NekMesh,-v,-m,varopti:linearelastic:radaptcurves=5,6,23-30:radaptscale=0.4:subiter=,(cid:99),5:restol=1e-5:nq=2:numthreads=10,-m,varopti:linearelastic:radaptcurves=5,6,23-30:radaptscale=0.45:radaptr,(cid:99),ad=0.0010:subiter=15:restol=1e-5:nq=2:numthreads=10,(cid:44)→,(cid:44)→,-m,varopti:hyperelastic:nq=6:numthreads=10,cross_section.mcf,cross_section.xml,The,resulting,mesh,is,shown,in,Figures,11–13.,All,the,selected,values,for,both,the,reﬁnement,curves,and,lines,,as,well,as,the,r-adaptation,,were,manually,tuned,through,a,lot,of,trial,and,error,and,were,the,best,combination,we,have,tried.,6,Conclusions,and,future,work,The,goal,of,Task,1.1,is,the,generation,of,meshes,capable,of,node,clustering,around,and,conform-,ing,to,internal,features,,with,the,aim,of,generating,a,mesh,suitable,for,the,tokamak,cross-section,edge,case.,In,this,report,,we,have,described,in,detail,the,additional,features,added,to,NekMesh,to,achieve,said,goal.,First,,we,extended,and,improved,the,user-deﬁned,reﬁnement,in,NekMesh,by,introducing,a,feature,that,allows,CAD,curves,to,be,used,as,sources,for,reﬁnement,and,improving,the,oc-,tree,subdivision,to,generate,smoother,meshes,when,user,deﬁned,reﬁnement,is,used.,Next,,r-adaptation,,a,method,originally,designed,to,cluster,nodes,and,form,anisotropic,elements,20,Figure,11:,5th,polynomial,order,mesh,for,the,tokamak,cross,section,test,case,with,elements,clustering,around,the,plasma,separatrix.,around,ﬂuid,discontinuities,in,order,to,better,resolve,shock,regions,,was,modiﬁed,to,work,on,user-deﬁned,CAD,curves.,21,(a),(b),Figure,12:,5th,polynomial,order,mesh,for,the,tokamak,cross,section,test,case,with,elements,clustering,around,the,plasma,separatrix.,Zoomed,in,views,of,the,lower,region,where,the,X-point,and,surrounding,wall,edges,can,be,observed.,(a),(b),Figure,13:,5th,polynomial,order,mesh,for,the,tokamak,cross,section,test,case,with,elements,clustering,around,the,edge-case,of,the,plasma,separatrix.,Zoomed,in,views,of,the,top,region.,22,User,documentation,for,using,the,new,features,is,then,provided,for,each,of,the,individual,features,as,well,the,CAD,preparation,requirements.,A,pipeline,process,is,then,described,from,CAD,preparation,to,mesh,generation,explaining,each,step,in,the,process,in,detail,through,the,use,a,simple,test,case,of,a,circle,embedded,in,a,square.,A,brief,discussion,of,the,parameters,and,their,choices,is,also,presented,,highlighting,some,of,the,limitations,of,the,new,features.,Finally,,a,simpliﬁed,geometry,mimicking,a,tokamak,cross-section,where,the,plasma,separatrix,is,modelled,using,an,internal,curve,is,presented.,The,resulting,mesh,satisﬁes,the,goals,set,out,for,Task,1.1,and,paves,the,way,for,Task,1.2.,However,,the,manual,nature,of,selecting,reﬁnement,and,scaling,parameters,by,trial,and,error,is,highlighted,as,a,signiﬁcant,drawback.,Further,understanding,of,the,reﬁnement,and,scaling,parameters,and,their,combination,is,clearly,needed.,This,is,left,for,future,work,with,the,knowledge,of,the,inevitable,changes,following,the,completion,of,Task,1.2.,The,next,obvious,next,step,is,to,mesh,an,accurate,geometry,of,the,tokamak,cross,section,once,it,becomes,available,to,us.,Further,,a,decision,of,whether,or,not,the,ﬂux,surfaces,are,part,of,the,requirements,should,be,made,to,enable,proper,validation,of,the,meshing,capabilities.,The,variational,optimisation,code,also,requires,a,refactoring,for,better,readability,,maintainability,and,,crucially,,performance.,For,instance,,the,current,implementation,uses,boost,pthreads,for,parallelisation,,which,for,2D,cases,is,suﬃcient,although,could,become,a,bottleneck,when,moving,to,3D.,Additional,work,falls,under,Task,1.2,where,the,generation,of,thin,boundary,layer,meshes,will,be,explored,as,a,means,of,reﬁning,curves,with,anisotropic,quadrilateral,elements,surrounding,the,curves.,This,will,require,signiﬁcant,modiﬁcations,to,the,current,boundary,layer,genera-,tion,which,breaks,down,on,meshes,containing,internal,curves,and,sharp,corners.,Lastly,,full,quadrilateral,mesh,generation,will,be,attempted,to,improve,the,overall,mesh,quality.,References,[1],Julian,Marcon,,Giacomo,Castiglioni,,David,Moxey,,Spencer,J,Sherwin,,and,Joaquim,Peir´o.,rp-adaptation,for,compressible,ﬂows.,International,Journal,for,Numerical,Methods,in,En-,gineering,,121(23):5405–5425,,2020.,[2],M.,Turner,,J.,Peir´o,,and,D.,Moxey.,Curvilinear,mesh,generation,using,a,variational,frame-,work.,Computer-Aided,Design,,103:73–91,,2018.,[3],D.,Moxey,,S.,Sherwin,,and,C.,Cantwell.,NEPTUNE,Report,for,D1.3:,Surface,Mesh,Gen-,eration.,Technical,Report,D1.3,,University,of,Exeter,&,Imperial,College,London,,May,2021.,[4],M.,Turner,,D.,Moxey,,and,J.,Peir´o.,Automatic,mesh,sizing,speciﬁcation,of,complex,three,dimensional,domains,using,an,octree,structure.,In,24th,International,Meshing,Roundtable,,2015.,[5],D.,Moxey,,M.,D.,Green,,S.,J.,Sherwin,,and,J.,Peir´o.,An,isoparametric,approach,to,high-,order,curvilinear,boundary-layer,meshing.,Comp.,Meth.,Appl.,Mech.,Eng.,,283:636–650,,2015.,[6],Open,Cascade,-,software,development,company.,https://www.opencascade.com/.,[7],Nektar++:,Spectral/hp,Element,Framework,–,User,Guide.,https://doc.nektar.info/userguide/latest/.,23,[8],G,Federici,,C.H,Skinner,,J.N,Brooks,,J.P,Coad,,C,Grisolia,,A.A,Haasz,,A,Hassanein,,V,Philipps,,C.S,Pitcher,,J,Roth,,W.R,Wampler,,and,D.G,Whyte.,Plasma-material,inter-,actions,in,current,tokamaks,and,their,implications,for,next,step,fusion,reactors.,Nuclear,Fusion,,41(12):1967–2137,,dec,2001.,24

:pdfembed:`src:_static/TN-02_RAdaptedMeshesTokamakEdgeRegion.pdf, height:1600, width:1100, align:middle`