in a peer - to - peer overlay network , each member maintains the identifiers of its overlay neighbors in its memory while leaving message routing to the underlay .a peer - to - peer overlay network is inherently decentralized and scales up easily .peer - to - peer overlays are well suited for distributed content storage and delivery. recent applications of such overlays to internet telephony  and digital cryptocurrencies  further enhance interest in scientific studies of the principles of peer - to - peer overlay networks .  due to the lack of central authority and the volunteer nature of overlay network participation , _churn _ , or joining and leaving of peers , is a particularly vexing problem affecting peer - to - peer overlay networks .churn may be cooperative , if departing processes execute a prescribed departure algorithm ; or adversarial , if they just exit.   + * infinite and unlimited churn . *every peer - to - peer overlay network has to handle churn .usually , while the topological changes in the overlay required by the churn requests occur , the primary services of the overlay ( _ e.g. _ content retrieval ) are either considered suspended or they are disregarded altogether . in other words ,the churn is considered finite , and the overlay network users just wait until join / leave requests stop coming , then the overlay network recovers and restores services . if churn happens again , the service suspension repeats .this may be tolerable if churn is infrequent since the overlay network is available most of the time .however , at the scales that peer - to - peer overlay networks achieve , peers nearly always wish to join or leave the overlay network . in this case, the service degradation caused by intermittent suspensions may become unacceptable .that is , it is necessary to consider _ infinite churn _ under which the overlay network has maintain services while handling it .  one way to handle churnis to engineer sufficient redundancy in the overlay network topology , so that if the peers leave , there are enough alternative paths to enable service operation . in this approach ,the amount of redundancy necessarily places a limit on the number of processes that churn concurrently : the churning processes must not sever all redundant paths . however , under heavy churn load and large scale , a peer - to - peer overlay network may breach this limit and collapse ( that is , partition itself ) .also , the necessity to add redundancy leads to wasted resources .hence , there is an interest in studying the possibility of _ unlimited churn _ , where this is no bound on the number of concurrently joining or leaving processes .this paper is an attempt to define and study infinite unlimited churn in peer - to - peer overlay networks .+ * unfair and local churn . * a request to join and , in cooperative churn , leave the overlay network is submitted to the overlay by the churning process .a churn handling algorithm is _ fair _ if it eventually satisfies every request .by contrast , a churn algorithm that allows the possibility , under infinite churn , to bypass indefinitely some requests ( still guaranteeing progress , that is , satisfying some churn requests indefinitely ) , is _unfair_. unfair algorithms are possibly more efficient .potentially , a churn handling solution may be straightforward and _ global _ : have some distinguished process manage all churn requests and process them .however , such a centralized global solution is not practical for large scale overlay networks as it creates a performance bottleneck and a single point of failure .in contrast , in a _solution , only processes in the immediate vicinity of the churning process are involved in processing the request .  in this paper , we study fairness and locality of churn solutions .   + * our contribution .* specifically , we consider churn in the asynchronous message passing system model .we prove that there does not exist an algorithm that can handle unlimited adversarial churn .we then focus on cooperative unlimited churn .we define the infinite unlimited churn problem by specifying the properties of a churn handing algorithm . we distinguish fair and unfair types of the problem .we prove that there is no local solution to the fair infinite unlimited churn problem .we then present an algorithm that solves the unfair version of the problem while maintaining a linear topology , i.e. topological sort .we extend our algorithm to handle skip lists and skip graphs . to the best of our knowledge, this paper is the first systematic study of unlimited infinite churn .+ * related work . * independently of peer - to - peer overlay networks , several papers  address determination of the rate of churn , which is a difficult task itself .fundamental problems in distributed computing , such as agreement , were also studied in the context of churn  , however their inherently global setting makes them unsuitable for building peer - to - peer overlays .  usually , peer - to - peer overlay networks are designed to have redundant links so that they can withstand limited peer departure ( that is , limited adversarial churn in our terminology )  .many papers address repairing the topology after determining a process unexpectedly left the overlay network  .limited churn also enables the possibility to maintain overlay services while adjusting  .another approach deals with self - stabilizing peer - to - peer overlay maintenance algorithms  that enable the peer - to - peer overlay network to recover from an arbitrary topology disruption , once it stops .that is , self - stabilizing algorithms may handle unlimited but finite adversarial churn . in these algorithms , churn is handled implicitly : topology changes are assumed to eventually stop , after this the system is designed to recover . to handle the possibility of initial incorrect state, the use of oracles ( that is , abstract entities that provide information about the overlay network ) may be necessary  .the recent trend is to provide a general framework that can be instantiated for various overlay networks  .overall , previous studies in the context of peer - to - peer overlay networks consider _ limited _ and/or _ finite _ churn , while this paper focuses on unlimited and infinite churn .* peer - to - peer overlay networks , topology . *a peer - to - peer overlay network consists of a set of processes with unique identifiers .we refer to processes and their identifiers interchangeably .processes communicate by message passing .a process stores identifiers of other processes in its memory .process is a _ neighbor _ of process if stores the identifier of .note that is not necessarily a neighbor of .a process may send a message to any of its neighbors .message routing from the sender to the receiver is carried out by the underlying network .a process may send a message only to the receiver with a specific i d , i.e. we do not consider broadcasts or multicasts .communication channels are fifo with unlimited message capacity .a _ structured _ peer - to - peer overlay network maintains a particular topology .one of the basic topologies is _ linear _ , or a topological sort , where each process has two neighbors and such that is the highest i d in the overlay network that less than and is the lowest i d greater than . consider a particular topology .cut - set _ is a ( proper ) subset of processes of the network such that the removal of these processes and their incident edges disconnects the network .it is known that if a network topology is not a complete graph , it has a cut - set . sincea peer - to - peer overlay network maintains its connectivity by storing identifiers in the memory of other processes , once disconnected it may not re - connect .hence , a peer - to - peer overlay network must not become disconnected either through the actions of the algorithm or through churn actions .+ * searching , joining and leaving the overlay network . *the primary use of a peer - to - peer overlay network is to determine whether a certain identifier is present in the overlay network .a search request message bearing an identifier , may appear in the incoming channel of any process that has already joined the overlay network .the request is routed until either the identifier is found or its absence is determined .a process may request to join the overlay network .we abstract bootstrapping by assuming that a join request , bearing the joining process identifier , appears in an incoming channel of any process that has already joined the overlay network .a process that joined the overlay network may leave it in two ways . in _adversarial churn _ a leaving process just exits the overlay network without participating in further algorithm actions . in _cooperative churn _ a leaving process sends a request to leave the overlay network .the leaving process exits only after it is allowed to do so by the algorithm .a process may join the overlay network and then leave .however , a process that left the overlay network may not join it again with the same identifier . a join or leave request is a _ churn request _ and the corresponding join or leave message is a _ churn message_. when a leaving process exits the overlay network , the messages in its incoming channels are lost .however , the messages sent from this process before exiting remain in the incoming channel of the receiving process .   + * infinite and unlimited churn definitions .* churn is _ infinite _ if the number of churn requests in a computation may not be bounded by a constant either known or unknown to the algorithm .churn is _ unlimited _ if the number of concurrent churn requests in the overlay network is not bounded by a constant either known or unknown to the algorithm .note that unlimited churn means that potentially all processes that are presently in the overlay network may request to leave .note also that the two properties are orthogonal .for example , churn may be finite but unlimited : all processes may request to leave but no more join or leave requests are forthcoming .alternatively , in infinite limited churn , there may be a infinite total number of join or leave requests but only , for example , five of them in any given state .  in this paperwe only consider infinite unlimited churn .+ * churn algorithm . *a churn algorithm handles churn requests . for each process, an algorithm specifies a set of variables and actions . an _action _ is of the form where _ label _ differentiates actions , _ guard _ is a predicate over local variables , and _ command _ is a sequence of statements that are executed _ atomically_. the execution of an action transitions the overlay network from one state to another .an algorithm _ computation _ is an infinite fair sequence of such states .we assume two kinds of fairness of computation : weak fairness of action execution and fair message receipt ._ weak fairness _ of action execution means that if an action is enabled in all but finitely many states of the computation then this action is executed infinitely often . _fair message receipt _ means that if the computation contains a state where there is a message in a channel , this computation also contains a later state where this message is not present in the channel , i.e. the message is received .we place no bounds on message propagation delay or relative process execution speeds , i.e. we consider fully asynchronous computations .+ * algorithm locality .* a churn request may potentially be far , i.e. a large number of hops , from the place where the topology maintenance operation needs to occur ._ place of join _ for a join request of process , is the pair of processes and that already joined the overlay network , such that has the greatest identifier less than and has the smallest identifier greater than . in every particular state of the overlay network , for any join request , there is a unique place of join .note that as the algorithm progresses and other processes join or leave the overlay network , the place of join may change ._ place of leave _ for a leave request of process is defined similarly ._ place of churn _ is a place of join or leave .a network topology is _ expansive _ if there exists a constant independent of the network size such that for every pair of processes and where the distance between and is greater than , a finite number of processes may be added hops away from to increase the distance between and by at least one .this constant is the _ expansion vicinity _ of the topology . in other words , in an expansive topology , every pair of processes far enough away may be further separated by adding more processes to the network without modifying the expansion vicinity of one member of the pair .note that a completely connected topology is not expansive since the distance between any pair of processes is always one .however , a lot of practical peer - to - peer overlay network topologies are expansive .for example , a linear topology is expansive with expansion vicinity of since the distance between any pair of processes at least two hops away may be increased by one if a process is added outside the neighborhood of one member of the pair .a churn algorithm is _ local _ if there exists a constant independent of the overlay network size , such that only processes within hops from the place of churn need to take steps to satisfy this churn request .the minimum such constant is the _ locality _ of the algorithm .note that a local algorithm may maintain only an expansive topology , and that the expansive vicinity of this topology must be greater than the locality of the algorithm .   + * the infinite churn problem .* a _ link _ is the state of channels between a pair of neighbor processes . as a churn algorithm services requests, it may temporarily violate the overlay network topology that is being maintained .a _ transitional link _ violates the overlay network topology while a _ stable link _ conforms to it .an algorithm that solves the infinite churn problem conforms to a combination of the following properties :  if there is a churn request in the overlay network , some churn request is eventually satisfied ;  if there is a churn request in the overlay network , this churn request is eventually satisfied ;  every transitional link eventually becomes stable ;  a message in a stable link is either delivered or forwarded closer to the destination ;  a message in a transitional link is not lost .note that the fair request property implies the request progress property .the converse is not necessarily true .[ defunfair ] _ the unfair infinite unlimited churn problem _ is the combination of request progress , terminating transition , message progress and message safety properties .[ deffair ] _ the fair infinite unlimited churn problem _ is the combination of fair request , terminating transition , message progress and message safety properties .  in other words ,fair infinite unlimited churn guarantees that every churn request is eventually satisfied while unfair infinite unlimited churn does not .* adversarial churn .*  [ trmnoadversarial ] there does not exist a solution for unlimited adversarial churn if the maintained topology is not fully connected .assume there is an algorithm that maintains a topology that is not fully connected such that is resilient against unlimited adversarial churn .let be the number of processes in the overlay network in some global state .if the topology is not fully connected , there exists a cut - set of processes whose cardinality is less than .since handles unlimited churn , it should handle the departure of every process in .however , since is a cut - set , its departure disconnects the overlay network , which is not possible to handle .hence , is not able to handle unlimited adversarial churn .  since adversarial unlimited churn can not be handled , in the rest of the paper we are considering cooperative ( unlimited ) churn .   +* fair local churn . *intuitively , the proof of the below theorem describes the continuous servicing of join requests before a distant churn request can reach its place of churn .this join chain keeps building in front of the distant churn request , precluding it from ever reaching the appropriate place .[ impossible ] there does not exist a local solution to the fair infinite unlimited churn problem for an expansive overlay network topology .assume there exists a local algorithm with locality that satisfies the fair infinite unlimited churn problem while maintaining an expansive topology with expansion vicinity of .let us add a churn request with i d to the incoming channel of another process whose distance to the place of churn for is greater than and , therefore , .since the topology is expansive , there is a set of processes that are not part of the overlay network such that every process in this set can be added hops away from such that the distance between and the place of churn for increases by at least one .observe that is assumed local with locality that is less than expansion vicinity .this means that every process in may be added to the overlay network without having to take a step .let us add the processes of to the overlay network .let us then have receive the churn request for . since is a solution to the fair infinite unlimited churn problem , must forward this request closer to its place of churn .suppose forwards this request to some process .however , since the distance from to s place of churn is at least , the distance from to this place of churn is at least .we continue this procedure ad infinitum .the resultant sequence is a computation of algorithm . yet, there is a churn request of process that is never satisfied .this means that violates the fair request property of the fair infinite unlimited churn problem , which contradicts our initial assumption .  since fair local ( infinite unlimited ) churn is impossible , in the next section we address unfair local ( infinite unlimited ) churn* linear topology under churn . * in a linear topology , each process maintains two identifiers : _ left _ , where it stores the largest identifier greater than and _ right _ , where it stores the smallest identifier less than .processes are thus joined in a chain .for ease of exposition , we consider the chain laid out horizontally with higher - id processes to the right and lower - id processes to the left .the largest process stores positive infinity in its _ right _variable ; the smallest process stores negative infinity in _left_. a _ left end _ of a link is the smaller - id neighbor process .right end _ is the greater - id process .as a process joins or leaves the overlay network it may change the values of its own or its neighbors variables thus transitioning the link from one state to another . in a linear topology , a link is _ transitional _ if its left end is not a neighbor of its right end or vice versa .the link is _ stable _ otherwise . the largest and smallest processes may not leave .the links to the right of the largest process and to the left of the smallest processes are always stable .a process may leave the overlay network only after it has joined .we assume that in the initial state of the overlay network , all links are stable .+ * algorithm description .* we present a local algorithm _ unfair infinite unlimited churn _( ) that satisfies the four properties of the unfair infinite unlimited churn problem while maintaining a linear topology .the basic idea of the algorithm is to have the _ handler _ process with the smaller identifier coordinate churn requests to its immediate right .this handler considers one such request at a time .this serializes request processing and guarantees the accepted request s eventual completion .the algorithm is shown in figure [ figalgo ] . to maintain the topology ,each process has two variables : _ left _ and _ right _ with respective domains less than and greater than . read - only variable _ leaving _ is set to * true * by the environment once the joined process wishes to leave the overlay network .variable _ busy _is used by the handler process to indicate whether it currently coordinates a churn request , or is initialized to * true * for a joining process .the incoming channel for process is variable .the request is sent in the form of a single _ join _ or _ leave _ message .we assume that a _ join _ and , for symmetry , a _ leave _ message is inserted into an incoming channel of an arbitrary joined process in the overlay network .message _ join _ carries the identifier of the process wishing to join the overlay network .leave _ carries the i d of the leaving process as well as the i d of the process immediately to its right . actions _joinrequest _ and _ leaverequest _ describe the processing of the two types of requests .if the receiver realizes that it is to the immediate right of the place of join or leave , and the receiver is not currently handling another request , i.e. , and it does not want to leave , it starts handling the arrived request .otherwise , the recipient process forwards the request to its left or right .request handling is illustrated in figure [ figjoinleave ] .it is similar for join and leave and is divided into five stages .the first two stages are _ setup _ stages : they set up the channels for the links of the the joining process or for the processes that are the neighbors of the leaving process .the third and forth stages are _ teardown stages _ : they remove the channels of the links being replaced .the last stage informs either the leaving process that it may exit , or the joining process that it may start coordinating its own churn requests . in the case of join ,two links need to be set up , hence the setup stages are divided into two substages 1.1 , 1.2 , 2.1 and 2.2 .similarly , in the case of leave , the teardown stages are divided into substages because two links need to be torn down .the messages transmitted during corresponding stages are 1 . set up a * sua * , 2 . set up b * sub * , 3 .tear down a * tda * , 4 .tear down b * tdb * and 5 . finish teardown * ftd*.   + * correctness proof .* we denote message * tda * or * tdb * as * td**. similarly , * su * * is * sua * or * sub*. lemmas [ lemmessages ] and [ lemstages ] follow immediately from the operation of the algorithm .[ lemmessages ] message * td * * is the last message in the channel in every teardown stage .message * su * * is the first message in a channel in every set up stage .[ lemstages ] the two links of a churning process transition through stages 1 through 5 .no link participates in concurrent transitions .the below corollary follows from lemma [ lemstages ] .[ lemterminatingtransition ] every transitional link is eventually stable .note that after the link is stable , it may transition again .[ lemmessagesafety ] no message in a transitional link is lost .observe that according to lemma [ lemmessages ] , in a teardown stage , the last message in the channel is a * td**. by lemma [ lemstages ] , each stage , including the teardown stages , eventually completes . by the design of the algorithm , a teardown stage completes whena * td * * message is received .since each channel is fifo , this teardown message is received only after all other messages in the channel are received .that is , after the teardown stage completes , there are no messages in the channel that is torn down .according to lemma [ lemstages ] , every channel in a transitional link is eventually torn down .hence the lemma.  [ lemmessageprogress ] unless the link starts transitioning , a message in a stable link is either eventually delivered or forwarded to a process closer to destination .consider a message in a channel of a stable link .we assume fair channel message receipt .this means , unless the channel is altered due to transitioning , the message is eventually processed by the recipient process .if this process is the destination , the message is delivered ; if not , the recipient forwards it closer to the destination .[ lemrequestprogress ] if there is a churn request in the overlay network , some churn request is eventually satisfied .  consider state of the computation where there is a churn request message in a channel of the overlay network . in this state, some other churn request processing may be under way . according to lemma [ lemterminatingtransition ] , the processing of all these requests eventually ends .let be the state of the computation where all requests under way in are done .if a new request processing has started in , then by applying lemma [ lemterminatingtransition ] to it , we obtain the claim of this lemma .let us consider the case where no request processing has occurred between and .according to lemma [ lemmessagesafety ] , message is sill in some channel of .since no requests are processed in , there are no transitional links . that is, is in a stable link . according to lemma [ lemmessageprogress ] , is forwarded towards its destination . continuing this way, we observe that either encounters a transitional link or arrives at a process that handles this request .if encounters a transitional link , this transition started after and the claim of the lemma follows .if arrives at , may either be busy or not . if is busy ,it is handling a request that started after .if is not busy , it starts processing .hence the lemma .  only processes to the immediate right and left of the churning process are involved in the processing of the churn request .hence the following lemma .[ lemlocality ] the locality of is 1 .the following theorem follows from corollary [ lemterminatingtransition ] and lemmas [ lemterminatingtransition],[lemmessagesafety],[lemmessageprogress],[lemrequestprogress ] and [ lemlocality ] .[ theoremuiuc ] algorithm is local and it solves the unfair infinite unlimited churn problem .churn algorithm extends to more complicated topologies such as skip lists  and skip graphs  . in these topologies ,the processes have links on multiple levels .the processes are linearized in the lowest level . in the higher levels ,the processes have links to progressively more distant peers .these higher level links accelerate overlay network searches and other operations .see figure [ figskiplist ] for an example skip list .to extend to such a structure a separate version of should be run at each level .the churn request should bear the level number to differentiate which level they belong to .the churning process should proceed up and down the levels as follows .a joining process first joins the first , linear , level , then the next and so on until it joins all the levels appropriate to the particular structure .the leaving process should proceed in reverse : the leaving process requests to leave the levels in decreasing order .for example , in figure [ figskiplist ] , a process needs to first leave from level , then and so on .while a joining process needs to first join the overlay network at linearized level , then proceed to join level and so on until it reaches the level appropriate for the particular structure .  as further researchit is interesting to consider extensions of to ring structures such as chord  or hyperring  .another important area of inquiry is addition of limited adversarial churn .this problem is difficult to address in the asynchronous message passing model where the exited process may not be differentiated from a slow one .oracles determining a process exit  may have to be used .james aspnes , jonathan kirsch , and arvind krishnamurthy .load balancing and locality in range - queriable data structures . in _ proceedings of the twenty - third annual acm symposium on principles of distributed computing _ , pages 115124 .acm , 2004 .john augustine , anisur rahaman molla , ehab morsy , gopal pandurangan , peter robinson , and eli upfal .storage and search in dynamic peer - to - peer networks . in _ proceedings of the twenty - fifth annual acm symposium on parallelism in algorithms and architectures _ , pages 5362 .acm , 2013 .john augustine , gopal pandurangan , and peter robinson .fast byzantine agreement in dynamic networks . in _ proceedings of the 2013 acm symposium on principles of distributed computing _ , pages 7483 .acm , 2013 .john augustine , gopal pandurangan , peter robinson , scott roche , and eli upfal . enabling robust and efficient distributed computation in dynamic peer - to - peer networks . in _foundations of computer science ( focs ) , 2015 ieee 56th annual symposium on _ , pages 350369 .ieee , 2015 .baruch awerbuch and christian scheideler .the hyperring : a low - congestion deterministic data structure for distributed environments . in _ soda _ , pages 318327 , philadelphia , pa , usa , 2004 . society for industrial and applied mathematics .andrew berns , sukumar ghosh , and sriram v pemmaraju .building self - stabilizing overlay networks with the transitive closure framework . in _symposium on self - stabilizing systems _ , pages 6276 .springer , 2011 .maximilian drees , robert gmyr , and christian scheideler .churn - and dos - resistant overlay networks based on network reconfiguration . in _ proceedings of the 28th acm symposium on parallelism in algorithms and architectures , spaa 2016 , california usa , july 11 - 13 , 2016. proceedings _ , pages 417427 .acm , 2016 .dianne foreback , andreas koutsopoulos , mikhail nesterenko , christian scheideler , and thim strothmann . on stabilizing departures in overlay networks .symposium on self - stabilizing systems _ , pages 4862 .springer , 2014 .dominik gall , riko jacob , andra w. richa , christian scheideler , stefan schmid , and hanjo tubig .time complexity of distributed topological self - stabilization : the case of graph linearization . in_ latin _ , pages 294305 , 2010 .sbastien gambs , rachid guerraoui , hamza harkous , florian huc , and anne - marie kermarrec .scalable and secure polling in dynamic distributed networks . in_ reliable distributed systems ( srds ) , 2012 ieee 31st symposium on _ , pages 181190 .ieee , 2012 .michael t goodrich , michael j nelson , and jonathan z sun . the rainbow skip graph : a fault - tolerant constant - degree distributed data structure . in_ proceedings of the seventeenth annual acm - siam symposium on discrete algorithm _ ,pages 384393 . society for industrial and applied mathematics , 2006 .rachid guerraoui , florian huc , and anne - marie kermarrec .highly dynamic distributed computing with byzantine failures . in _ proceedings of the 2013 acm symposium on principles of distributed computing _ , pages 176183 .acm , 2013 .steven y ko , imranul hoque , and indranil gupta . using tractable and realistic churn models to analyze quiescence behavior of distributed protocols . in _ reliable distributed systems , 2008 .ieee symposium on _ , pages 259268 , 2008 .andreas koutsopoulos , christian scheideler , and thim strothmann . towards a universal approach for the finite departure problem in overlay networks . in andrzej pelc and alexander a. schwarzmann , editors , _ stabilization , safety , and security of distributed systems - 17th international symposium , sss 2015 , edmonton , ab , canada , august 18 - 21 , 2015 , proceedings _, volume 9212 of _ lecture notes in computer science _ , pages 201216 .springer , 2015 .giuliano mega , alberto montresor , and gian pietro picco . on churn and communication delays in social overlays .in _ peer - to - peer computing ( p2p ) , 2012 ieee 12th international conference on _ , pages 214224 .ieee , 2012 .rizal mohd nor , mikhail nesterenko , and christian scheideler .corona : a stabilizing deterministic message - passing skip list . in _13th international symposium on stabilization , safety , and security of distributed systems ( sss ) _ , pages 356370 , grenoble , france , october 2011 .rizal mohd nor , mikhail nesterenko , and sbastien tixeuil .linearizing peer - to - peer systems with oracles . in _ stabilization , safety , and security ofdistributed systems - 15th international symposium , sss 2013 , osaka , japan , november 13 - 16 , 2013. proceedings _ , pages 221236 , 2013 .antony i. t. rowstron and peter druschel .pastry : scalable , decentralized object location , and routing for large - scale peer - to - peer systems . in _middleware _ , pages 329350 , london , uk , 2001 .springer - verlag .jared saia and amitabh trehan .picking up the pieces : self - healing in reconfigurable networks . in _parallel and distributed processing , 2008 .ipdps 2008 .ieee international symposium on _ , pages 112 .ieee , 2008 .ion stoica , robert morris , david liben - nowell , david r. karger , m. frans kaashoek , frank dabek , and hari balakrishnan . : a scalable peer - to - peer lookup protocol for internet applications ., 11(1):1732 , 2003 .

Abstract: we study unlimited infinite churn in peer - to - peer overlay networks . under this churn , arbitrary many peers may concurrently request to join or leave the overlay network ; moreover these requests may never stop coming . we prove that unlimited adversarial churn , where processes may just exit the overlay network , is unsolvable . we focus on cooperative churn where exiting processes participate in the churn handling algorithm . we define the problem of unlimited infinite churn in this setting . we distinguish the fair version of the problem , where each request is eventually satisfied , from the unfair version that just guarantees progress . we focus on local solutions to the problem , and prove that a local solution to the fair infinite unlimited churn is impossible . we then present and prove correct an algorithm that solves the unfair infinite unlimited churn problem for a linearized peer - to - peer overlay network . we extend this solution to skip lists and skip graphs .


conventional continuum theories discretized by finite element ( fe ) approaches become problematic at small length - scales and complex material behaviours . in these cases , the underlying microstructure or even the atomistic crystal structure comes into play .this introduces nonlocality , and requires discrete simulations such as structural lattice computations or molecular statics ( ms ) in order to capture the physics properly .discrete conservative systems are in their full description conveniently formulated within a variational framework , in which their behaviour follows a minimization of a potential energy  , i.e. where  denotes an arbitrary admissible vector collecting the positions of all lattice atoms ( or particles ) , denotes a configuration space , and  a suitable minimizer , see e.g.  , section 6 . for application - scale problems, the construction of  and the solution of  entails excessive computational efforts because of two facts :  1 .a large number of atoms and bonds contained in fully - resolved systems leads to considerable expenses associated with the solution of the euler lagrange equations involving large - scale energy gradients and hessians .2 .  for the assembly of energies , gradients , and hessians , all atoms or bondshave to be individually taken into account .the quasicontinuum ( qc ) methodology , originally formulated by  , and extended in various aspects later on , e.g.  , overcomes  and  in two steps .first , _ interpolation _ , based on a number of selected representative atoms , or _repatoms _ for short , constrains the displacements of the remaining atoms in the lattice , where  stores the positions of all the repatoms , and  denotes a subspace of the original configuration space  . because the dimension of  is usually much smaller than that of  , deficiency  is mitigated .the second involves _ summation _ , in which the energy and governing equations of the reduced model are determined by collecting the contributions only from so - called _ sampling atoms _ , in analogy to numerical integration of fe method . as a result , an approximation  to  in  is minimized , which resolves .section [ sect:3 ] of this paper presents a more detailed discussion of the two qc approximation steps .other techniques and further details can be found e.g. in  .  also at length scales larger than the nanoscale ( atomistic length scale ) ,many materials possess discrete underlying structures regular , irregular , or random at the micro- or meso - scale ; typical representatives are fibrous materials such as paper  or textile  . in such materialsthe bonds between the fibres ( or yarns ) take the role of atoms in atomistics . however , since the involved length - scales are larger , the interactions of these `` atoms '' ( i.e. particles ) often comprise dissipative processes . hence , the original qc formulation developed for purely conservative interactions can not be employed .initial theoretical developments to lift this limitation have been provided by  for fibre plasticity and bond - sliding failure . for the derivationthe authors have used a non - variational thermodynamically - consistent framework that employs the following virtual - power statement in eq ., the dot denotes the derivative with respect to time ; the vectors  and  store components of resulting internal and external forces .this means that the left- and right - hand sides can be identified as the internal and external powers ; for further details see  , section 2.1 .let us note that in the ideal , smooth and consistent case , the formulation of eq .would be connected to the one of  via the relation  . throughout this paper ,the qc approach based on eq .  will be referred to as the _ virtual - power - based qc_. the virtual - power - based qc framework has been employed in various contexts and proven to be efficient while accurate , see e.g.  . however , from this formulation , it is not entirely clear whether the governing equations derived from eq .  are also energetically consistent ; it may happen that some terms are missing , cf .e.g.  , tab . 1 , for an example in continuum gradient plasticity .variational approaches may furthermore be considered to provide finer information about system evolution such as microstructure pattern formation or phase transition , see e.g.  , , and  . in the case of adaptivity ,better error estimates and mesh refinement capabilities for localized phenomena ( such as damage ) can be explored in highly nonlinear problems , cf .e.g.  . from a broader perspective ,the variational formulation offers a consistent framework convenient for , e.g. , the rigorous treatment of evolutions that exhibit discontinuities in time , investigations of structural stability using energy landscapes arising from time - incremental minimization , or direct employment of non - linear optimization algorithms .finally , the variational formulation allows us to extend the conservative qc methodology to an entire class of rate - independent internal mechanisms in a natural way .the goal of this paper is therefore to reformulate the virtual - power - based qc framework for internal dissipative processes in terms of variational principles and show that the obtained solutions _ coincide _ for both formulations in the case of plasticity with isotropic hardening . to that end, a suitable potential  will be constructed such that describing the state of the system in analogy to . here, however , denotes a general state variable that also includes internal dissipative variables .furthermore , is an abstract state space , and the inclusion sign  indicates that the potential  is generally nonsmooth or may have multiple minima . in analogy to standard qc , a reduced variable  and an approximate energy  will be introduced in order to alleviate  and . in what follows ,the approach based on eq .  will be referred to as the _ variational qc_. its construction falsifies the statement presented in  , section 1 , claiming that the solution to eq ._ can not _ be obtained by direct minimization of an energy potential .  in order to construct the full energy potential , we employ the variational formulation of rate - independent processes as introduced in an _ abstract setting _ by  that is closely related to applications in continuum mechanics .earlier studies were provided e.g. by  , , , , , , , and  .section [ sect:2 ] of this paper first briefly introduces definitions and basic principles of the theory .second , the approach is reformulated in the particular context of discrete lattice systems .  the governing equations associated with  will be addressed in section [ sect:4 ] , where we recall the alternating minimization ( am ) method , see also  . since the energy potential  for plasticity is nonsmooth , we will also briefly discuss the return - mapping algorithm suitable for its minimization .  before closing this contribution by a summary and conclusions in section [ sect:6 ] ,we perform numerical tests on three benchmark examples presented in section [ sect:5 ] , two of which have been adopted from  , section 4 , and  , section 4.2 .we demonstrate that both approaches , represented by eqs .  and , lead to energetically - consistent solutions for the exact and central summation rules presented in  and  .the third example then presents both global as well as local quantities for an indentation test .finally , we show that despite the significant dimension reduction and time savings achieved by the qc method , the obtained errors in stored and dissipated energies due to interpolation and summation are rather low : the relative errors in energies do not exceed  , while the simulation time is decreased by a factor of 4  30 depending on the triangulation , loading , and geometry .the variational formulation for rate - independent processes comprises several steps and relies on two principles  and , which are described below ( for details see  ) .the state of the system within a fixed time horizon  ] , where  denotes a pseudo - time parametrizing the quasi - static evolution process .the state of the system is fully characterized by the state variable  .furthermore , we consider the total free ( helmholtz type ) energy of the system \times\mathscr{q}\rightarrow\mathbb{r} ] is called an energetic solution to the initial - value problem described by  if it satisfies the following two principles  and , together with an initial condition :  1 .* global stability : * for all  ] which expresses the energy balance in terms of the internal energy , the dissipated energy  , and the time - integrated power of external forces  .* initial condition : *   in the second principle , the dissipation along a curve  is expressed as where the supremum is taken over all  and all partitions of the time interval  ], the two principles  and  along with initial condition  naturally give rise to an  * * incremental problem : * for   amenable to a numerical solution in which each step is realized as a minimization problem of an incremental energy the main conceptual difficulty with  is that it represents a global minimization problem , which is computationally cumbersome for non - convex energies .it is reasonable , however , to assume that stable solutions of  are associated with local minima .then , multiple minimizers can exist , but only the ones satisfying  are adopted as proper solutions .it has been shown that as long as the solution  remains continuous in time , the energy balance holds ( see  ) . in the examples in section [ sect:5 ]we verify that this is indeed the case for hardening plasticity , thereby justifying the local minimization approach adopted in this work . as a result, the energy balance  need not be enforced explicitly .   to specify the above formulation for lattice systems , we start by introducing a geometrical setting and necessary notation ( depicted in fig .[ disslatt : fig:1 ] ) , followed by explicit definitions of energies . before doing so , let us mention that the term `` atoms '' is meant to represent individual particles or nodes of the underlying ( non - atomistic ) lattice , consistently with the original qc methodology developed for atomistic systems .furthermore , for clarity we confine our exposition to two spatial dimensions ; the extension to three dimensions is straightforward .the domain  in a reference configuration contains a set  of  atoms , where  returns the cardinality of a set  .the reference spatial position of each atom  is specified by a vector  , and can be expressed as a linear combination of primitive vectors in analogy to the bravais lattices since we confine our attention to regular structures only .all positions  are collected in a column matrix ^\mathsf{t} ] , , that represents also the abstract non - dissipative variable for appropriate  and  , where  . because such an affine transformation does not affect the results , we adopt  instead of  as our primal variables from now on for convenience . ] .furthermore , we introduce the distance measure between two atoms  , and the set of all distances  , cf .due to the time - dependent dirichlet boundary conditions and possible kinematic constraints , is a function of time and it forms a manifold in  .each bond is further endowed with two internal variables : the plastic slip or elongation of the bond  , and the cumulative plastic slip ( a hardening variable ) . for clarity ,we introduce column matrices  and  , both in  , collecting  and  of all bonds .then , the abstract dissipative variable can be specified as  , and  as  ( recall that  ) .having described the physical variables , we proceed to the formal definitions of the energies .because no external forces will be used , the elastic part of the incremental energy is not an explicit function of time  and reads as note that hatted variables  represent an arbitrary admissible configuration of the system , whereas the absence of hats indicates that these state variables are minimizers of , see also eq . .the interatomic potential energy  in  specifies a recoverable part of the energy stored in all atom interactions .it is sufficient to adhere to pair potentials , for which where the first equality holds as a consequence of the principle of potential invariance .note that permutation symmetry requires  , and that  corresponds to the elastic portion of the energy stored in a single bond stretched to a length  with a plastic elongation  .further , reflects an unrecoverable part of the stored energy , locked in all bonds due to hardening effects , where  denotes the hardening pair potential of a single bond . the dissipation distance for a single bond , , between two different states  and  defined as where  is an initial yield force .the total dissipation distance then collects the contributions of all bonds , namely note that setting  and  reduces  to standard ms defined in eq .; for further details we refer the interested reader to , e.g. , , chapter 6 .let us close this section with the observation that the total incremental energy  can be expressed in two equivalent forms : as a sum over all atom sites such as described above , or as a sum over all bonds .this allows us , therefore , to introduce the incremental bond , , and site , , energies of the form  [ subsect : disslatt : eq:7 ]   and write the reason for introducing two equivalent expressions for  is that the site energies  are convenient for the minimization of  with respect to the kinematic variable  , whereas the bond energies  are more suitable for the minimization with respect to the internal variable  .let us proceed to the two qc steps introduced to mitigate excessive computational demands implied by  and  that are inherently associated with the minimization problems  and  for conservative and dissipative systems .in the two sections below , on interpolation and summation , we explain how these steps apply on the incremental energy  .   upon specifying a subset of atoms  , that determine the deformation state of the system, one can reconstruct the positions of all the remaining atoms through interpolation : cf .eq . . here, represents a column matrix of all representative atoms position vectors for any admissible configuration , and the matrix  stores the basis vectors spanning  by columns .analogously , represents a vector of repatoms positions in the reference configuration ..  basically introduces a geometric equality constraint for  which , upon substitution into eq ., entails that the incremental energy becomes a function of  and  , i.e. reducing the number of degrees of freedom associated with the kinematic state of the system from  to  , which is substantial if  .minimization in  with respect to  then changes to a minimization over the subspace  which , due to the linearity in , effectively yields a projection of the full solution to that subspace . in order to specify  ,one usually introduces a triangulation of  equipped with piecewise - affine shape functions with compact support ( in analogy to the fe methodology , where the nodes are the repatoms ) .this triangulation is finely resolved ( to fully recover the underlying lattice ) in the region of interest and coarsely elsewhere .evaluations of these shape functions at the positions of all atoms then provide the base vectors , where  denotes the -th element of the set  , and  represents a shape function associated with a repatom  that is evaluated for an atom  in the undeformed configuration . in addition , we assume that the shape functions satisfy the kronecker - delta property , i.e.  for  , as well as partition - of - unity property , i.e.  for each  . note that higher - order or meshless interpolations are possible as well , see e.g.  , , , or  .similarly , we could introduce internal variables associated with repbonds  , and through interpolation represent all the remaining internal variables of the system , where  stands either for `` p '' or `` c '' .eq .  then would constrain the admissible vector of either plastic elongations  or cumulative plastic elongations  .this approach would , however , lead to a different technique than introduced by  , with possible benefits in error estimation , cf .e.g.  , since the procedure treats all unknowns equally and resembles a structure - preserving reduced - order - modelling method , cf .e.g.  or  . some similarities with the models for bond - sliding by  can be observed .however , let us emphasize that in their case , the internal variables associated with sliding are attributed to nodes rather than to bonds , so that they can be approximated in the same way as the displacements . within the virtual - power - based qc framework for lattices with plasticity ,only a subset of unknowns ( denoted  ) is sought as well , but this is an assumption of the _ summation _ rule , yielding an approximation to incremental energy  rather than a geometrical constraint , see section [ subsect:3.2 ] below .equation  is therefore not included in further considerations and is left as a possible future challenge .the interpolation step with piece - wise affine shape functions ensures that all atomic bonds inside triangles follow a homogeneous deformation , and therefore yields constant site energies for atoms that have all their nearest neighbours inside the same triangle , cf .[ subsect:3.2:fig:1 ] .this observation was significantly used during the construction of summation rules briefly recalled below .-axis with an inhomogeneity in the central region ; energy peaks near the inhomogeneity are trimmed for visualisation purposes . in the - plane below the surface ,contour lines are indicated for better clarity . ]let us express the approximation to the incremental energy defined in eqs .  and  as where  in general has the form ^\mathsf{t} ] stores all individual incremental site energies .choosing weight factors  , , clearly recovers the full sum presented in . introducing a set of sampling atoms  such that  for  and  for  , , one can rewrite  also as analogously to sampling atoms , we introduce a set of sampling bonds , , , defined as those bonds that are connected to all the sampling atoms  with removed duplicity , recall eq . .the approximation of the incremental energy can then be expressed again as a sum over all sampling atoms ( eq . ) or as a sum over all sampling bonds ( in analogy to eq . ) .consequently , the dimensionality of the internal variable reduces from  to  , because all variables associated with  become irrelevant .the reduced dissipative variable is denoted as  , where  is identified with  .let us recall eq .  and the discussion related to it . from that, it may be clear that a mapping from  back to the full solution  ( i.e.  in analogy to eq . )is not necessarily unique .the reduction of the global state variable from  to  is thus formed by the combination of the two qc steps that can not be separated .  in order to compute the total energy exactly ( i.e. to integrate the site energy function in fig .[ subsect:3.2:fig:1 ] exactly ) , it is necessary to incorporate all variations along the triangles edges . for a large triangle , the energy of only one atom site multiplied by the number of atoms within that triangle suffices to represent the plateau , whereas the atoms near the edges and in the fully resolved region must all be taken into account explicitly .this yields the so - called _ exact summation rule _ proposed by  , in which the explicit procedure how to compute weight factors ( ) and how to determine the set of sampling atoms  can be found .another sampling scheme focuses on the centres of the triangles and is therefore referred to as _ central summation rule _, as only the plateaus and vertex atoms are sampled .see  or  for further details and fig .[ subsect:3.2:fig:2 ] for a schematic representation of the two summation rules .compared to the exact summation rule , the central summation rule is cheaper , but introduces approximation errors .directly approaching  with respect to both variables  and  can be cumbersome . instead , it is relatively straightforward to perform the minimization in a staggered way , i.e. with respect to kinematic and internal variables sequentially , resulting in the am method .this scheme appears in the literature also under names such as block - nonlinear gauss seidel method or block - coordinate descent method . for the original formulationthe reader is referred to  , for recent works to e.g.  , and for applications in variational fracture to e.g.  .  in the context of the variational qc method ,the am procedure needs to be applied to the incremental problem  with the incremental energy defined by eq . .for a fixed time step  , it results in the following scheme  a.  * initialization : * , , and  b.  * general iteration : * + [ am ]   in the first time increment , i.e. for  , the initial condition  is used .let us note that a similar scheme applies also to more general definitions of the incremental energy  applicable to multi - body potentials or long - range interactions .   for hardening plasticity , i.e. for  and , the minimization in  can be performed in closed form . for arbitrary time step  and iteration , the solution reads    this relation follows from the bond - wise formulation of the energy in eqs .  and , and from the definition of the dissipation distance in eq . .the minimization then decomposes into independent problems solved by .substituting  into definition  provides the _ reduced _ incremental energy , cf .e.g.  , section 3.1.2 , or  , and the am algorithm simplifies to steps  for  .continuing with step , is sufficiently smooth with respect to  , so that the standard newton s algorithm can be employed . in what follows ,two nested iteration cycles will be used where  relates to am and  to newton s algorithm . assuming that  and , and  are fixed in , the second - order taylor expansion in  , in the vicinity of  , is applied to obtain the following stationarity conditions where  [ sect:4:eq:4 ]   condition  supplies a system of linear equations for increments  . iterating  and  until convergence of  then yields  . as usual ,dirichlet boundary conditions are imposed through known increments for constrained atoms . for tying conditions ,the constrained primal - dual minimization procedure is applied , cf .section [ sect:5 ] .we refer also to , section 6.4.4 , for similar approaches used for ms systems .the gradients  and hessians  are provided in detail in eqs .  in [ sect : a ] for the reader s convenience .  before dealing with the non - smooth step ,the energy is again rewritten into the bond - wise form , cf .eqs .  and .consequently , can be treated in analogy to , i.e. where  denotes the subdifferential with respect to  , cf .e.g.  or ( * ? ? ? * section 8.1 ) , and  represents a reduced bond energy . in mechanics terms , eq.  represents the karush kuhn tucker complementarity conditions of a stretched uniform bar with isotropic hardening , and can thus be solved with the standard return - mapping algorithm described in detail e.g. in  , section 1.4.2 .   instead of minimizing the exact incremental energy  , its approximation  in terms of the reduced variable  minimized in the variational qc formulation .using the chain rule in the taylor series expansion applied in step , recall eqs .  and, provides the stationarity conditions with  [ subsect:4.2:eq:2 ]   where the partial derivatives are expressed as  [ subsect:4.2:eq:3 ]   for definitions and explicit expressions of  and  see eqs .  and .the converged solution of  is denoted by  .  in order to minimize in , again the bond - wise version of the approximate incremental energyis employed , yielding where  denotes the weight factor associated with a bond . because the problems in  are independent , the weights  are irrelevant ( though they can be easily established from  ) and the problem can be solved sequentially using the return - mapping algorithm in analogy to eq . , but only over a subset of bonds  .interestingly , the resulting governing equations _ exactly _ coincide with those provided by  , indicating that the virtual - power - based formulation is also variationally - consistent .let us note that the introduction of constraints for the dissipative variables , as outlined in eq ., would lead to coupled systems of equations for  and a different ( non - local ) minimization strategy would be needed instead .this section demonstrates the previously discussed theory for three benchmark examples , two of which were originally introduced in  and  .the results will show that the energy balance  holds along the entire loading paths for all computed solutions . in all cases , we employ the following pair potential ( in eqs . , the superscripts  are dropped for brevity ) i.e. the bond stiffness reads  , in accordance with the standard truss theory .note that this definition corresponds to the rotated engineering deformation measure .the hardening potential reads as which by virtue of  and yields the power - law hardening rule in the form . \label{sect:5:eq:3}\ ] ] here , denotes the current yield force of a bond connecting atoms  and  , whereas the initial yield force reads  .all physical constants used throughout this section are specified in tab .[ sect:5:tab:1 ] . made above may seem academic , note that the power hardening law  has been used by  to model the mechanical behaviour of woven fabrics . ].dimensionless material and geometric parameters for all examples .[ cols="<,>,<",options="header " , ]   [ sect:5:tab:1 ]   as a first example a uniform loading test is presented .the domain  occupies  lattice spacings of 1 unit length ; the fully resolved system consists of  atoms and  bonds .a stiff region occupying  lattice spacings in the centre represents an inhomogeneity within an otherwise homogeneous medium .the young s modulus of the springs in the stiff domain is  times larger than elsewhere and the initial yield force  is infinite to prevent plastic yielding in the inclusion .all bonds on the boundary  ( see fig .[ sect:5:fig:1 ] ) , have cross - sectional areas reduced to  .the boundary conditions are set according to  , section 4.1 :  [ sect:5:eq3 ] ,\nonumber\\ r_x(\gamma_4 ) & = r_{0,x}(\gamma_4 ) = -50,\label{sect:5:eq3a}\\ { { \boldsymbol{r}}}(\gamma_1\cap\gamma_4)&={{\boldsymbol{r}}}_0(\gamma_1\cap\gamma_4)= \left[\begin{array}{c } -50 \\-50 \end{array}\right],\nonumber\\ r_y(\gamma_3 ) & = r_y(\gamma_3\cap\gamma_4),\label{sect:5:eq3b}\end{aligned}\ ] ]  where  denotes the deformed configurations of all the atoms on line segment  ( ^\mathsf{t} ] is divided uniformly into  increments ( i.e.  and  , cf . ) .the numerical example is performed for the exact and central summation rules , utilizing nine meshes ; eight of them are depicted in fig .[ sect:5:fig:2 ] while the ninth one represents the fully - resolved lattice .in order to demonstrate the importance of the mesh quality , we use two groups of triangulations : structured `` s '' ( figs . [ sect:5:fig:2a ]  [ sect:5:fig:2d ] ) and unstructured `` u '' ( figs .[ sect:5:fig:2e ]  [ sect:5:fig:2h ] ) , constructed such that the number of repatoms ( ) is pairwise nearly the same , see tab .[ sect:5:tab:3 ] .for the unstructured meshes , the coarsening immediately starts outside the fully - refined region with a mild coarsening gradient .consequently , the sizes of the fully - resolved regions differ .+  l|rrrrrrrr : r & & & & & & & & & + & & & & & & & & & + & 349 & 337 & 597 & 537 & 893 & 969 & 1,277 & 1,113 & 10,201 + & ex & 5,113 & 5,621 & 5,425 & 8,209 & 5,889 & 8,169 & 6,217 & 8,161 & + & c & 597 & 593 & 929 & 1,193 & 1,245 & 1,541 & 1,697 & 1,875 & + & ex & 25,389 & 27,076 & 26,814 & 36,360 & 28,373 & 36,288 & 29,474 & 36,504 & + & c & 3,492 & 3,536 & 5,076 & 7,636 & 6,482 & 8,544 & 8,396 & 10,388 & +  [ sect:5:tab:3 ]  concerning the implementation of step , the dirichlet boundary conditions  are applied in the standard way. the tying condition  is enforced by first collecting all atoms or repatoms on  ( except for atom  ) in a set  .then , eq .  is rewritten as which are globally assembled to and imposed by lagrange multipliers , i.e. by the primal - dual formulation .this yields an iterative solution of saddle - point problems ( cf . , section 14 ) , that reads \left[\begin{array}{c } \widehat{{{\boldsymbol{r}}}}^{i+1}-\widehat{{{\boldsymbol{r}}}}^i\\ \widehat{{{\boldsymbol{\lambda}}}}^{i+1 } \end{array}\right ] =  -\left[\begin{array}{c } { { \boldsymbol{f}}}^i \\ { { \boldsymbol{b}}}^i \end{array}\right ] , \quad { { { { \boldsymbol{b}}}^i = { { \boldsymbol{c}}}\widehat{{{\boldsymbol{r}}}}^i } } , \label{sect:5:eq6}\ ] ] for time step  and am iteration  , providing us with  and  upon convergence . for qc systems, is replaced with  and  with  .  the energy profiles corresponding to the loading program  for all the meshes and both summation rulesare depicted in fig .[ sect:5:fig:3a ] . here , we notice that the curves corresponding to the individual solutions are identical and that the errors are extremely small .moreover , we can check that all solutions satisfy the energy balance  along the entire loading path , since the thin dotted line corresponding to  lies on top of the thick dashed line representing  . upon zooming in ( fig .[ sect:5:fig:3b ] , where only the results for the mesh  are presented for clarity ) , we observe that the exact summation rule increases the system s energy with respect to the full lattice simulation , whereas the central summation rule causes the energy to be slightly lower .this behaviour is not surprising as the overall energy increases when the geometric constraints of the qc system are introduced ( while using the exact summation rule ) .further , since the site energies of atoms lying near the triangles edges are higher compared to internal atoms ( recall fig . [ subsect:3.2:fig:1 ] ) , the approximate energy for the central summation rule is slightly lower .( energy ) at ( 0,0 ) ; ( zoom ) at ( 8,0.325 ) ;  ( 2.4,1.8 ) rectangle ( 3.8,2.5 ) ;  ( 3.1,1.8 ) ( 5.5,-1.13 ) ; ( 3.1,2.5 ) ( 5.5,2.63 ) ;  because the exact and approximate energy profiles are indistinguishable by the naked eye , we introduce a relative error measure )$ ] for  , and  .namely , where  either stands for  , or  , and  stands for  , or  .the subscript `` qc '' denotes results obtained from the qc simulations , whereas the subscript `` full '' represents the results computed for the fully - resolved system .the error measure  is presented for the various meshes in fig .[ sect:5:fig:4 ] .we further distinguish between the error due to interpolation ( denoted as `` int . '' ) and the total error due to interpolation plus summation ( denoted as `` tot . '' ) .thus , `` int . '' relates to the error obtained for the exact summation rule , whereas `` tot . ''relates to the error obtained for the central summation rule .the largest value amounts to  and corresponds to the dissipation  , cf .[ sect:5:fig:4b ] .whilst the error due to interpolation behaves in a reasonable way , i.e. decreases with an increasing dimension of the projection basis , the interpolation plus summation error surprisingly behaves in the opposite way ( see fig .[ sect:5:fig:4 ] ) , i.e. increases with an increasing number of sampling atoms . since the total error is dominated by summation , this behaviour can be related to the mesh topology .namely , to the number of triangles for which the central sampling atom and all its neighbours are not contained in the same triangle .such a conjecture is supported by the perfect match between the total error profiles in fig .[ sect:5:fig:4c ] and the mesh characteristics in fig .[ sect:5:fig:5 ] .numerically quantified , the pearson s correlation coefficient for these data amounts to  ( structured mesh ) and  ( unstructured mesh ) .this result supports the idea of a direct transition between the fully - resolved and interpolated regions , or motivates the use of rapid mesh coarsening rather than a mild one . on the other hand ,the overall error is so small that this behaviour can also be deemed singular . finally , let us note that while the energy errors ( global error measures ) are negligible , the relative errors of the internal variables ( local error measures ) are still noticeable , cf . , section 4.3 .but this is consistent with one s expectations as the qc aims to closely approximate the global incremental energy  , recall section [ subsect:3.2 ] .  concerning the mesh types , we conclude from fig . [ sect:5:fig:4 ] that the performance in terms of the interpolation error is nearly the same , though the unstructured meshes are slightly more accurate .in terms of the total error , however , the structured meshes perform evidently better .the structured ones are also more efficient , as for them the corresponding numbers of sampling atoms ( ) and sampling bonds ( ) are systematically lower , see tab .[ sect:5:tab:3 ] . recall that the numbers of repatoms ( ) remain comparable .such behaviour can again be related to the mesh topology and the mesh coarsening gradient .combined results from fig .[ sect:5:fig:4 ] and tab .[ sect:5:tab:3 ] reveal that by accepting energy errors up to  , the number of degrees of freedom reduces up to the factor of  . in the case of sampling atoms ,the reduction is up to the factor of  .( eq . ) for the meshes presented in fig . [ sect:5:fig:2 ] .`` int . '' relates to interpolation ( exact summation rule ) and `` tot . '' to interpolation plus summation ( central summation rule).,title="fig : " ] +   used . ]the highly accurate energy reconstruction of the qc method for the uniform loading test is attributed to two aspects .first , the ( plastic ) deformation field is more or less piecewise constant over the deformed domain  .hence , it is well captured by constant approximations within triangles .second , perturbations near the inhomogeneity are resolved accurately by all employed meshes . consequently , the summation error outweighs the interpolation error .these two reasons and the example presented in  , sections 4.2 and 5.2 motivate the following test , in which the deformation field due to bending is linear rather than constant and more pronounced interpolation errors are expected .   in this example, the domain  is exposed to pure bending around the -axis , cf .[ sect:5:fig:6 ] .an inhomogeneity of  lattice spacings is situated at the bottom edge , in which the trusses have a -times higher young s modulus and an infinite initial yield force  prevents plastic yielding .the entire domain occupies  lattice units and comprises  atoms connected by  bonds .the boundary conditions for  read as follows  [ subsect:6.2:eq1 ] ,\label{subsect:6.2:eq1a}\\   \frac{r_x^{\alpha_{j+1}}-r_x^{\alpha_j}}{r_y^{\alpha_{j+1}}-r_y^{\alpha_j } } & = \tan\theta,\ j=1,\dots , n_{\gamma_2}-1,\label{subsect:6.2:eq1b }   \end{aligned}\ ] ]  where we have collected  atoms or repatoms on  in the set  ; note that  denotes the -th element of  .all atoms in  are sorted from bottom to top , i.e.  , and hence , and  .condition  therefore requires that  follows a straight line that is allowed to freely move in space and has a slope  , cf .[ sect:5:fig:6b ] .note that atoms can slide frictionlessly along this line but can not move in the perpendicular direction .equation  can then be rewritten as and globally assembled as in analogy to eqs .  and .since the constraints now depend on  through eq .below , matrix  changes for each time step  . to accomodate this , is replaced by  in eq . .the overall deformation process is finally parametrized as , \label{subsect:6.2:eq4}\ ] ] and the time interval is divided into 100 uniform increments ( i.e.  , ) .  the numerical example is again studied for the two summation rules , but now only for five unstructured meshes .four of them are shown in fig .[ subsect:5.2:fig:2 ] , the fifth one represents the full lattice .the corresponding numbers of atoms , repatoms , and sampling bonds can be found in tab .[ sect:5:tab:4 ] .+  l|rrrr : r & & & & & + & & & & & + & 507 & 812 & 894 & 1,194 & 20,301 + & ex & 10,709 & 13,989 & 10,693 & 11,301 & + & c & 941 & 1,558 & 1,342 & 1,872 & + & ex & 51,094 & 66,990 & 51,136 & 52,432 & + & c & 5,789 & 9,574 & 7,215 & 10,120 & +  [ sect:5:tab:4 ]  the energy evolution paths for all meshes are shown in fig .[ subsect:5.2:fig:3 ] for the exact summation rule , and in fig .[ subsect:5.2:fig:4 ] for the central summation rule . here , the differences between the solutions are more pronounced compared to the previous example , though they are still quite small .the results show that the energy equality  holds again . upon closer inspection, we notice that the interpolation error ( fig . [ subsect:5.2:fig:3b ] ) is smallest for mesh  , which has the smallest overall element size .so the error is dominated by the limitation of constant strain triangles to capture bending , and is almost insensitive to the size of the fully - resolved region . instead of refining the triangulation in the coarse part of the domain , an alternative approach for decreasingthe interpolation error would be to use higher - order approximations , as shown e.g. in  .furthermore , the total error due to interpolation and summation ( fig . [ subsect:5.2:fig:4b ] ) is smaller than the interpolation error alone , meaning that the two errors partially compensate .( energy ) at ( 0,0 ) ; ( zoom ) at ( 8,0.325 ) ;  ( 2.2,1.5 ) rectangle ( 3.6,2.2 ) ;  ( 2.9,1.5 ) ( 5.35,-1.08 ) ; ( 2.9,2.2 ) ( 5.35,2.35 ) ;  ( energy ) at ( 0,0 ) ; ( zoom ) at ( 8,0.2 ) ;  ( 2.1,1.6 ) rectangle ( 3.5,2.3 ) ;  ( 2.8,1.6 ) ( 5.35,-1.2 ) ; ( 2.8,2.3 ) ( 5.35,2.44 ) ;  the relative error  , defined in eq . , can be found in fig .[ subsect:5.2:fig:5 ] for the four meshes and the two summation rules . the best agreement achieved for mesh is due to the finer mesh resolution over the entire domain  ; the largest error amounts to  attained for mesh  , for dissipation  presented in fig .[ subsect:5.2:fig:5b ] . comparing the increase in the number of repatoms with respect to error, we conclude from fig .[ subsect:5.2:fig:5 ] and tab .[ sect:5:tab:4 ] that by accepting an energy error below  , we gain a reduction in the number of degrees of freedom up to the factor of  , and in the number of sampling atoms up to the factor of  .the dependency of the error in fig .[ subsect:5.2:fig:5 ] on the mesh topology is more complicated than in the previous example .specifically , the pearson s correlation coefficient between the number of triangles with central sampling atoms that do not have all the neighbours in the same triangle and the `` tot . ''error in fig .[ subsect:5.2:fig:5c ] drops to  , indicating no dependence .finally , the following quantity is examined measuring the relative error between full - lattice `` full '' and approximate `` qc '' solutions for extreme plastic deformations attained at  alongside the inhomogeneity , cf .[ sect:5:fig:6a ] .recall that for some  , the norm in eq .reads as  .quantity  is presented for the different meshes and summation rules in fig .[ subsect:5.2:fig:6 ] .it can be seen that the size of the fully resolved region has again less influence on the error than the mesh refinement in the coarse domain . because the interpolation increases the energy of the system and the summation slightly underestimates it , the two effects again cancel each other to some extent .notice that this does not hold for mesh  , for which the summation error dominates .   defined in eq .  for various meshes depicted in fig . [ subsect:5.2:fig:2 ] .`` int . '' relates to interpolation ( exact summation rule ) and `` tot . '' to interpolation plus summation ( central summation rule).,title="fig:"][subsect:5.2:fig:6a ]   in the final example , we consider a homogeneous x - braced lattice occupying a rectangular domain  of  lattice spacings , containing  atoms and  bonds , cf .[ subsect:6.3:fig:1 ] . in accordance with the previous examples , the lattice spacings in both directions are of unit length .the boundary conditions are prescribed as and the remaining part of the boundary , , is left free for potential frictionless contact with a rigid circular indenter .the indenter is characterized by its radius , , and centre , \quad t \in [ 0 , 1 ] .\label{subsect:6.3:eq2}\ ] ] the time interval is again uniformly divided into  increments , i.e.  and  .the inequality constraints associated with contact between the lattice and the indenter read where we have collected the  atoms or repatoms lying on  part of the boundary in the set  ; recall that  is the -th element of  .these constraints are incorporated through the primal - dual formulation as before , . to that purpose , we iteratively form a set of active constraints for inactive constraints , i.e. for  , the associated  . for all active constraints , the inequality conditions  are enforced as equality constraints and implemented using eq ., where we substitute because the tangent  now depends on  , it must be updated every newton iteration , in contrast to the previous two examples .possible violations of the inequality constraints are checked after convergence of newton s algorithm .if a change in  occurs , newton s algorithm is called again and the entire procedure is repeated until convergence of  .the same strategy applies also to qc systems .the numerical example is studied for the exact and central summation rule and again for five meshes .four of them are depicted in fig .[ subsect:6.3:fig:2 ] , whereas the fifth represents the full lattice .two meshes are regular and two irregular , pairwise with nearly the same number of repatoms and identical sizes of the fully - resolved regions , cf .[ sect:5:tab:5 ] .+  l|rrrr : r & & & & & + & & & & & + & 603 & 1,487 & 617 & 1,510 & 33,153 + & ex & 6,249 & 7,383 & 7,531 & 8,771 & + & c & 876 & 1,857 & 818 & 1,807 & + & ex & 32,066 & 36,746 & 36,388 & 41,224 & + & c & 4,300 & 8,352 & 3,922 & 8,153 & +  [ sect:5:tab:5 ]  a detailed view of the deformed configuration at  for the full lattice solution can be found in fig .[ subsect:6.3:fig:3a ] .note that only one half is shown thanks to symmetry .the dissipation localizes below the indenter through a plastic shear band , indicated in red .the reaction force of the indenter is presented in fig .[ subsect:6.3:fig:3b ] , where we notice a good agreement between the qc simulations with the full lattice solution , especially for structured meshes  and  . because the differences between the solid and dashed lines are negligible, we can conclude that the choice of the summation rule does not influence the results and the errors are dominated by interpolation , similarly to the pure bending test in section [ subsect:6.2 ] .( bcrack ) at ( 1,0 ) ; ( breact ) at ( 9,0.2 ) ; ( breactzoom ) at ( 5.4,1.55 )  ;  ( 9.75,2.8 ) rectangle ( 10.45,2 ) ;  ( 3.8,0.3 ) rectangle ( 6.95,2.9 ) ;  ( 10.1,2.8 ) ( 6.95,2.9 ) ; ( 10.1,2 ) ( 6.95,0.3 ) ;  the energy evolution profiles are presented in fig .[ subsect:6.3:fig:4 ] for the exact , and in fig .[ subsect:6.3:fig:5 ] for the central summation rule .the results show that the energy balance  holds in all cases , and that the differences between the two summation rules are practically negligible as already suggested by fig .[ subsect:6.3:fig:3b ] . upon closer investigation ( in figs [ subsect:6.3:fig:4b ] and [ subsect:6.3:fig:5b ] ) ,a better performance of the structured meshes compared to the unstructured ones can be observed again .( energy ) at ( 0,0 ) ; ( zoom ) at ( 7.9,0.015 ) ;  ( 2.2,1.5 ) rectangle ( 3.6,2.2 ) ;  ( 2.9,1.5 ) ( 5.4,-1.2 ) ; ( 2.9,2.2 ) ( 5.4,2.4 ) ;  ( energy ) at ( 0,0 ) ; ( zoom ) at ( 7.9,0.015 ) ;  ( 2.2,1.5 ) rectangle ( 3.6,2.2 ) ;  ( 2.9,1.5 ) ( 5.4,-1.2 ) ; ( 2.9,2.2 ) ( 5.4,2.4 ) ;  fig .[ subsect:6.3:fig:6 ] shows the energy errors  of eq .  for all systems .here we see that the best agreement with the solution of the full lattice is obtained using regular fine mesh  , whereas the largest error is obtained for unstructured coarse mesh  . because the total error is dominated by the interpolation , no significant correlation between the number of triangles with central sampling atoms that do not have all the neighbours in the same triangle ( fig .[ subsect:6.3:fig:7b ] ) and the `` tot . ''error ( fig . [ subsect:6.3:fig:6c ] ) is observed .pearson s correlation coefficient amounts to  in this case . for the summation error itself ( i.e. `` tot . '' minus `` int . ''line in fig .[ subsect:6.3:fig:6c ] ) , the correlation increases to  , indicating only a mild dependence . by committing errors in energies of less than , we achieve a reduction up to a factor of  in terms of the number of degrees of freedom .the computational gain in terms of the number of sampling atoms is around a factor of  .finally , fig .[ subsect:6.3:fig:7a ] shows the local error  of eq . .these results confirm that the size of the fully resolved region significantly influences the accuracy , whereas the choice of the summation rule has little effect on the local errors , except for mesh  .let us emphasize , however , that although the errors are acceptable ( below  ) , all qc approaches fail to identify in which bond the largest history variable occurs .this can be verified in fig .[ subsect:6.3:fig:3a ] , where the bonds with the extreme plastic slips are presented by thick lines for the different systems .note that the presented results correspond to the central summation rule only .an analysis of the quasicontinuum ( qc ) approach for regular lattice systems with internal dissipative processes is presented based on the variational , energy - consistent formulation of  .the main results can be summarized as follows :  1 .the virtual - power - based qc method introduced by  can be equivalently derived from an appropriate energy potential . as both qc schemes satisfy the energy balance , the variational structure of the virtual - power - based qc methodis confirmed by this study .2 .  as a consequence of the central summation rule , the number of internal variables is highly reduced .an alternative approach may be to introduce geometrical constraints for the internal variables , possibly allowing a unique reconstruction of all unknowns ( instead of only the kinematic variables ) .we have outlined briefly this perspective and emphasized the differences from the summation rule assumption .3 .  in the section describing the solution strategies , an alternating minimization method for nonsmooth potentials was specified for the qc framework .the example section has demonstrated the energy consistency of different qc schemes for three benchmark examples involving uniform loading , pure bending , and indentation test . in spite of large computational savings , all approximate solutions for all examples proved to be in a good agreement with the full - lattice solution in terms of stored and dissipated energies ( by accepting errors in energies only up to , the number of degrees of freedom may be reduced up to the factor of  ) .the presented variational qc framework generalizes the original energy - based conservative quasicontinuum methodology for an entire class of rate - independent interactions and can easily be adjusted to incorporate e.g. damage phenomena merely by appropriate modifications of the energies .in addition , our results provide a convenient framework for adaptive variational qc methods in regular lattice networks with dissipative interactions , with potential applications in , e.g. , fracture simulations of woven fabrics or polymers .all these topics are , nevertheless , beyond the scope of this contribution and will be reported separately .for the sake of completeness , we provide in this appendix the derivatives of  , recall eq . , with respect to  explicitly .the internal force associated with atom  , , reads as where  denotes the reduced incremental site energy of atom  with condensed variable  , see also eqs .  and .the global force is then expressed as note that if external force vector  is present , it is simply subtracted from the right - hand - side of eq . .the stiffness matrix associated with an atom site  , , reads as (\delta^{\beta\gamma}-\delta^{\alpha\gamma})(\delta^{\beta\delta}-\delta^{\alpha\delta}),\\ & \ m , n=1,2,\ \gamma,\delta=1,\dots , n_\mathrm{ato } , \end{aligned } \label{sect : a : eq:3}\ ] ] whereas global stiffness is expressed as here , we have used the relation and for brevity we have denoted above , relate to spatial dimensions , relate to atoms , denotes the kronecker - delta product with respect to spatial coordinates , denotes the kronecker - delta product with respect to atoms , and  denotes the tensor product of vectors  and  .financial support for this work from the czech science foundation ( gar ) under project no .14 - 00420s is gratefully acknowledged .49 natexlab#1#1[1]`#1 ` [ 2]#2 [ 1]#1 [ 1]http://dx.doi.org/#1 [ ] [ 1]pmid:#1 [ ] [ 2]#2 , , , . . ,http://www.sciencedirect.com/science/article/pii/s0022509615000630 , ., , , , a. . , .http://www.sciencedirect.com/science/article/pii/s0045782514002047 , ., , , . . ,http://dx.doi.org/10.1002/nme.3134 , ., , , b. . , .http://www.sciencedirect.com/science/article/pii/s0022509614001100 , . , , ,http://www.sciencedirect.com/science/article/pii/s0022509613002445 , ., , , d. . , .http://www.sciencedirect.com/science/article/pii/s004578251300279x , ., , , , a. . , .http://www.sciencedirect.com/science/article/pii/s0167663614001495 , ., , , , b. . , .http://dx.doi.org/10.1002/gamm.201510018 , ., , , , . . ,http://www.sciencedirect.com/science/article/pii/s0022509699000289 , . , . .http://dx.doi.org/10.1007/s10957-012-0134-2 , ., , , . . ,, , , , . , in : . http://www.gillesfrancfort.com/site/publications_files/published.version.cfmt00%20copy.pdf . , , , .http://epubs.siam.org/doi/abs/10.1137/140990644 , .http://www.sciencedirect.com/science/article/pii/s0022509607002220 , .http://stacks.iop.org/0965-0393/11/i=3/a=201 , .http://www.sciencedirect.com/science/article/pii/s0022509698000349 , ., , , . . ,http://www.sciencedirect.com/science/article/pii/s0045782514001510 , .http://rspa.royalsocietypublishing.org/content/464/2089/117 , , http://arxiv.org/abs/http://rspa.royalsocietypublishing.org/content/464/2089/117.full.pdf [ arxiv : http://rspa.royalsocietypublishing.org / content/464/2089/117.full.pdf ] . , , . .http://www.sciencedirect.com/science/article/pii/s0022509610002425 , .http://dx.doi.org/10.1007/s00161-010-0174-5 , .http://www.sciencedirect.com/science/article/pii/s0167663612000683 , ., , , , , , . . ,http://www.sciencedirect.com/science/article/pii/s002199910900165x , ., , , . . ,http://www.sciencedirect.com/science/article/pii/s016784421000039x , .http://journals.cambridge.org/article_s0962492913000068 , . , .. chapter .http://dx.doi.org/10.1007/0-387-21791-6_2 , . , .http://www.sciencedirect.com/science/article/pii/s0045782504002683 , . .http://www.springer.com/us/book/9781493927050 , .http://dx.doi.org/10.1007/s00030-003-1052-7 , . , , .http://stacks.iop.org/0965-0393/17/i=5/a=053001 , .http://dx.doi.org/10.1023/a%3a1026098010127 , .http://www.sciencedirect.com/science/article/pii/s0022509697000963 , .http://www.sciencedirect.com/science/article/pii/s0045782598002199 , .http://dx.doi.org/10.1007/s00161-011-0228-3 , ., , , . . ,http://www.sciencedirect.com/science/article/pii/s002250961100055x , .http://www.sciencedirect.com/science/article/pii/s1359835x06001813 , ., ( eds . ) , . . .http://www.sciencedirect.com/science/article/pii/s0045782598002308 , ., , , . . ,http://www.sciencedirect.com/science/article/pii/s099775381500128x , . , .international series of numerical mathematics , .http://www.springer.com/us/book/9783034805124?token=prtst0416p , .http://www.springer.com/us/book/9783709116241 , .interdisciplinary applied mathematics , .http://www.springer.com/gp/book/9780387975207?token=prtst0416p , ., , , . . ,http://dx.doi.org/10.1080/01418619608243000 , .http://dx.doi.org/10.1002/nme.1841 , .http://www.sciencedirect.com/science/article/pii/s0045782514003545 , .

Abstract: lattice systems and discrete networks with dissipative interactions are successfully employed as meso - scale models of heterogeneous solids . as the application scale generally is much larger than that of the discrete links , physically relevant simulations are computationally expensive . the quasicontinuum ( qc ) method is a multiscale approach that reduces the computational cost of direct numerical simulations by fully resolving complex phenomena only in regions of interest while coarsening elsewhere . in previous work ( , _ j. mech . phys . solids _ 64 , 154169 , 2014 ) , the originally conservative qc methodology was generalized to a virtual - power - based qc approach that includes local dissipative mechanisms . in this contribution , the virtual - power - based qc method is reformulated from a variational point of view , by employing the energy - based variational framework for rate - independent processes ( , _ rate - independent systems : theory and application _ , springer - verlag , 2015 ) . by construction it is shown that the qc method with dissipative interactions can be expressed as a minimization problem of a properly built energy potential , providing solutions equivalent to those of the virtual - power - based qc formulation . the theoretical considerations are demonstrated on three simple examples . for them we verify energy consistency , quantify relative errors in energies , and discuss errors in internal variables obtained for different meshes and two summation rules .   lattice model , quasicontinuum method , variational formulation , plasticity , multiscale modelling