cryptography is used to transmit a message from a sender ( referred to as alice ) to a receiver ( bob ) without leaking useful information to others .it has been proved that a message can be transmitted securely if it is coded and decoded by a sequence of random bits ( key ) whose length is equal to that of the message .the problem of secure transmission is then reduced to that of generating a secret key shared by alice and bob .classical cryptosystems rely on computational complexity and may be broken by an effective algorithm or a powerful computer .quantum key distribution ( qkd ) protocols , in contrast , provide an unconditionally secure key , the security of which is inherent in the laws of quantum mechanics .this remarkable advantage of qkd protocols has been attracting increasing research interest since the proposal by bennett and brassard . although qkd has been demonstrated in over-20-km - fiber communication channels , its application in practical communication systems is not straightforward .the qkd protocols require single - photon transmission to guarantee the security , and thus are vulnerable to loss and noise inherent in actual transmission channels .optical amplifiers will not solve this problem , because the noise of the optical amplifiers inevitably destroys the quantum correlation .single - photon transmission requires the use of complicated and inefficient photon counting techniques instead of conventional analog detection , besides a truly practical single - photon source is not yet available .the qkd protocols are therefore not fully compatible with the current optical fiber communication systems .a secure key distribution protocol compatible with the current systems is desirable .this would be a protocol that uses more than one photon and allows optical amplifiers to be used .such a protocol would be based on coherent state photons , or classical light .maurer has shown perfect cryptographic security can be obtained in a classical noisy channel with the help of a noiseless feedback channel .yuen and kim examined the principles underlying the qkd protocol with two non - orthogonal quantum states ( b92 protocol . )the security of the b92 protocol relies on two facts : ( i ) an eavesdropper ( eve ) can not accurately determine the value of each transmitted bit ( i.e. , no efficient opaque eavesdropping . )( ii ) eve can not closely correlate bob s measurement results with her own ( i.e. , no efficient translucent eavesdropping . )yuen and kim pointed out that these two conditions can be satisfied in a classical transmission system , where the detectors of bob and eve are under independent additive noise and show a small signal - to - noise - ratio ( snr . )they proposed a classical noise - based protocol for key distribution ( referred here to as the yk protocol . )the yk protocol working with classical light , would have advantages in practical implementations .proving the security in yk protocol is , however , subtler than in qkd protocols . sincemany photons are transmitted to carry one - bit information , the conditions specified above will not be satisfied if the snr of eve s detection is sufficiently high .eve s snr can be increased by using low - noise detection equipment , or simply by moving closer to alice than bob ( because of the fiber loss . )the original analysis of the security of yk protocol assumed the same snr for bob and eve . for practical implementations ,it is important to determine the design rules of eve s snr and bob s .  in this articlewe quantitatively examine the security of the yk protocol , and show that the yk protocol is secure , even if eve s snr is 9 db better than bob s .we also show experimental results that demonstrate secure key distribution against translucent attack .section 2 provides the condition for secure key distribution in terms of the secure key distribution rate .section 3 describes the experiment on the yk protocol using conventional fiber optics .section 4 discusses the implementation issues .we first define the secure key distribution rate .suppose alice transmits an equally probable binary string to bob .shannon information between alice and bob is expressed by where denotes the error rate of bob s decision . because of the decision errors , bob has the information in only of the sifted bits .alice and bob exchange redundant information over the public channel in order to obtain the reconciled key .this procedure is called _ error correction _ , the best known practical protocol for which was given by brassard and salvail .for successful error correction with brassard - salvail s protocol , the error rate should be less than 0.15 . to establish a secret key , alice and bob use _privacy amplification_ , random hashing of the reconciled key into a shorter key .if they shorten the reconciled key of length by the fraction and sacrifice bits as a safety parameter , eves s shannon information on the final key of length is bounded by the fraction is given by where the collision probability of is defined as follows : let be a random variable with an alphabet and distribution .the collision probability is the probability that takes the same value twice in two independent experiments , that is , .the logarithm of the collision probability thus refers to eve s information on the key .the collision probability can be expressed by the probability that is the -th signal of bob s string and the joint probability that is the -th signal of bob s string and is the -th signal of eve s string .we have the following formula for the fraction : .\label{fraction}\ ] ] according to bru and ltkenhaus , bob can generate secure bits from his sifted bits at the rate of we refer this rate as the secure key distribution rate , and for secure key distribution its value be positive .the actual key generation rate is further reduced by multiplying the generation rate of the sifted key . in the following part of this section, we derive the conditions under which is positive .  in the yk protocol ,the bit values ( `` 0 '' and `` 1 '' ) are encoded so as to make the probability distribution of the received signal symmetric .alice sends encoded bits on a weak classical light .signal is transmitted for `` 0 '' , and is transmitted for `` 1 '' , where .we here measure the signal value as the voltage on the load resistance of a photodiode .mean signal voltage is defined by , and represents the signal energy over the duration ( signal energy per bit . )the output of the detector contains the noise , so . if the noise is white gaussian noise with spectral density , the probability distribution of the detected signal is expressed by   \qquad \left ( for\ `` 0"\right ) } { \left ( 1/\sqrt{2\pi }\right ) \exp \left [ -\left ( v+s\right ) ^{2}/\left ( 2\sigma ^{2}\right ) \right ] \qquad \left ( for\ `` 1"\right ) } } % % beginexpansion { \left ( 1/\sqrt{2\pi } \right ) \exp \left [ -\left ( v - s\right ) ^{2}/\left ( 2\sigma ^{2}\right ) \right ] \qquad \left ( for\ `` 0"\right )  \atopwithdelims\{. \left ( 1/\sqrt{2\pi }\right ) \exp \left [ -\left ( v+s\right ) ^{2}/\left ( 2\sigma ^{2}\right ) \right ] \qquad \left ( for\ `` 1"\right ) } % % endexpansion ,  \label{pbd}\ ] ]  where the signal is averaged over the duration as .the snr in this system is define by . in a conventional decision scheme the bit valuesare determined to be `` 0 '' if and `` 1 '' if .decision errors will occur at the rate of , where is the scaled complementary error function defined by we set a threshold to make a decision:``0 '' if  and `` 1 '' if , but leave _ inconclusive _ if . the probability of making a decisionis given by the following decision rate : and the error rate is the sifted key is generated from the raw bit string by the bob s decision .the decision rate thus refers to the generation rate of the sifted key . as seen in eqs .( [ decision ] ) and ( [ error ] ) , the decision rate and the error rate  are determined by the values of the snr and the threshold .as described below , this error rate determines the joint probabilities and therefore the secure key distribution rate .the system is thus fully characterized by the snr and the threshold .the error rate can be reduced by increasing the threshold , but , a high threshold will also reduce the decision rate . since , as we can see by comparing fig .[ figone ] and fig .[ figtwo ] , the decision rate decrease faster than the error rate , the threshold value should not be set too high .  as in the b92 protocol ,the inconclusive results play a essential role in guaranteeing the security of the key distribution .a finite threshold value of bob enables him to make accurate decisions on his sifted key at a cost of the generation rate .eve , on the other hand , should make a decision with zero threshold in order to obtain conclusive results for all the transmitted bits .if eve uses a finite threshold in her decision , she will obtain the inconclusive results on the sifted bits .the assumption of independent noise prevents eve from predicting which bit bob will obtain a conclusive result .eve can acquire no information from these inconclusive bits .since eve s error rate is less than 1/2 , she will obtain more information by making a decision with zero threshold .therefore , bob can make more accurate decisions on the sifted key bits than eve can . that is , bob has more information than eve , and can distill secure key bits with alice .now we will examine the conditions for security against eavesdropping .we here consider only two simple kind of eavesdropping , translucent attack and opaque attack .a translucent attack can be made by simply putting a beam splitter in the transmission channel .the translucent attack to the yk protocol , in contrast to those to the qkd protocols , will not change the state of the transmitted light .the probability distribution of bob s bits is the same as that of alice s , , because after error correction alice and bob share completely correlated results .the joint probabilities are and .the fraction is calculated from eq .( [ fraction ] ) as the secure key distribution rate can be estimated by using eqs .( [ iab ] ) , ( [ rate ] ) , and ( [ trans - t ] ) .figure [ figthree ] shows eve s required error rate as a function of bob s . as bob s error rate increases , eve s error rate should be increased in order to obtain a positive secure key distribution rate .for example , if bob s error rate is 0.15 , eve should make errors at a rate greater than 0.27 .this implies that snr of eve s system should be less than 0.38 for white gaussian noise .on the other hand , bob s snr should be better than 0.057 to keep his error rate smaller than 0.15 and his decision rate at 10 .the secure key distribution is therefore possible even if eve s snr is six times ( 8 db ) as large as bob s .the tolerance of the snr increases as bob s error rate decreases , and it reaches 10 db for .  in an opaque attack , eve receives all the photons in out of the bits sent by alice .then eve sends the bits to bob according to her decision .eve never touches the rest of the bits ( bits ) and forwards them to bob . to protect information from opaque attack , bob should determine his threshold according to the average signal intensity of each bit .if he observes only the average intensity over many bits , eve can set a finite decision threshold to reduce her error rate and will then obtain conclusive results for bits ( ) if she sends only the conclusive results with signals times as intense as received , bob will obtain the same long - time average signal intensity he would if eve did not intercept the photons . if bob observes the signal intensity of each bit , eve must send every bit with the same intensity as she receives it .eve then should make a decision with zero threshold , otherwise she will lose the information on the inconclusive results .there is a trade - off for eve on the fraction : a large will increase eve s information gain , but will also make her easily detectable from the increase of bob s error rate .bob s error rate on the unintercepted bits is , but the error rate on the intercepted bits is .the eavesdropping thus increases bob s error rate to .\ ] ] to calculate the secure key distribution rate by using eqs .( [ iab ] ) , ( [ rate ] ) , and ( [ trans - t ] ) , we estimate the joint probabilities after the error correction , bob has bits . the probability distributioni symmetric : eve obtains  correct results and incorrect results on bob s bits .the joint probabilities are obtained as nf_{+}}{2\left ( 1-e_{b}^{\prime } \right ) nf_{+ } } \nonumber \\ & = & \frac{\left [ \left ( 1-e_{e}\right ) \eta + \left ( 1-\eta \right ) /2\right ] \left ( 1-e_{b}\right ) } { 2\left ( 1-e_{b}^{\prime } \right ) }  \nonumber \\ p\left ( 1,0\right )  & = & \frac{e_{e}e_{b}\eta nf_{+}+(1-e_{b})(1-\eta ) nf_{+}/2% } { 2\left ( 1-e_{b}^{\prime } \right ) nf_{+ } }  \label{jointp } \\ & = & \frac{e_{b}e_{e}\eta + ( 1/2)\left ( 1-e_{b}\right ) \left ( 1-\eta \right ) } { % 2\left ( 1-e_{b}^{\prime } \right ) }  \nonumber \\ p\left ( 0,0\right )  & = & p\left ( 1,1\right )  \nonumber \\ p\left ( 0,1\right )  & = & p\left ( 1,0\right ) .\nonumber\end{aligned}\ ] ] figure [ figfour ] shows the minimum required values of eve s error rate for secure key distribution as a function of bob s error rate .though bob can observe only values , he can estimate from the snr of his detection system .eve will be detected if .the detection is easy if bob s error rate is much lower than eve s .a high error rate for bob may hide eve , but secure key distribution is possible even in this case . suppose and .as shown in fig .[ figfour ] , the secure key distribution rate is positive if eve s error rate is larger than 0.12 .if the system is under white gaussian noise , this condition on the error rate is satisfied when eve s snr is smaller than 1.35 ( 1.3 db . ) since bob s snr should be better than 0.089 ( -10.5 db ) to keep the decision rate at 10 and , the tolerance in snr is 11.8 db .this small snr for eve implies that the signal should be sent on a weak light .increasing the light intensity reduces eve s error rate , and makes the secure key distribution impossible .in implementing the yk protocol , we should code the bit values in such a way that the probability distribution of the received signals is symmetric . in this experiment we used the unipolar manchester code .this code represents 1 as a change from on to off and 0 as a change from off to on .it can be decoded as follows : divide the incident light into two paths , one of which is set one half of the pulse width longer than the other . then take a difference of the two light intensities by a balanced detector .the latter half of the pulse slot yields a negative signal for 1 and a positive signal for 0 .binary phase shift keying ( bpsk ) also yields a symmetric distribution by homodyne detection , and would be more sensitive , but unipolar manchester code is easier to implement .figure [ figfive ] shows the experimental setup .two distributed feedback ( dfb ) laser diodes ( lds ) served as 1.3 m light sources . a pattern generator provided a signal pulse string to modulate one dfb ld ( signal ld ) directly .the second pattern generator was synchronized to the first , and provided an 8-ns pulse at the beginning of each pulse string .this pulse modulated the other dfb ld ( trigger ld ) directly to generate a trigger light pulse .the output of the signal ld was set weaker than that of the trigger ld .the clock frequency in the present experiment was 25 mhz .only a fixed pattern of 101010 was transmitted .the coded signal light then became a square wave with a duty of 50 % and a pulse duration of 20 ns .we sent strings of 30.8 kbits .the outputs of the two lds were combined and attenuated by an attenuator ( att1 . ) to simulate the translucent attack by an eavesdropper , we inserted a 50:50 divider .an attenuator ( att2 ) was placed in one arm of the divider to examine the snr tolerance for the secure key distribution .the signals of both outputs were detected by the receivers .each receiver consisted of a 50:50 divider , a fiber delay of a half pulse width , and a balanced detector .the balanced detectors made of two commercial ingaas _ pin _photodiodes loaded by 50 resisters were operated in analog mode .the catalog data ( typical values ) for the quantum efficiency and the dark current of the photodiodes at 25 c were 90 % and 5 na . the photodiodes ware not cooled .the output signals of the receivers were led to amplifiers ( db ) and then to analog - digital converters .figure [ figsix ] shows a typical probability distribution of the output signal from the amplifier .it is well represented by the sum of two gaussians .the intensity of the optical signal was 0.380 ( -34.2 dbm ) at the input port of the receiver , and the snr of this signal was 1.0 ( 0 db . ) we averaged the output pulse over the duration ( 10 ns ) , and evaluated the decision rate and error rate as a function of the snr and the threshold .the results are shown in fig .[ figone ] and fig .[ figtwo ] .the experimental results agree well with the theory assuming white gaussian noise .these indicated that white gaussian noise dominated the present receiver sensitivity , and that the security analysis described in sec .2 can be applied to the experiment .the number of the sifted bits became small when the threshold value is high .we had less than 30 bits , if the decision rate is less than 10 .this insufficient sample number caused the error rate fluctuation observed for large s in fig .[ figtwo ] . for snr valuesup to 0 db , the noise level was almost same as the dark noise level , but for larger snr , it increased with the signal intensity .dark current of the photodiodes was negligible compared to the thermal noise .these indicates that , for snr values up to 0 db , the sensitivity of the system was dominated by thermal noise , which is constant to the input photon number .as the intensity increased , the thermal noise was exceeded by shot noise , which is proportional to the input photon number .snr was proportional to the square of the input power for weak signals , and tended to be proportional to the input power as the signal intensity increased.  the error rate shown in fig .[ figtwo ] provides a criterion for key distribution secure against opaque attack .a low error rate of 0.038 was obtained for weak signals by setting the threshold at , where the snr was -9.25 db . the decisions were made at the rate of 0.0008 , slightly lower than 10 .this error rate was lower than the theoretical value of 0.072 because of the fluctuation described above .using line in fig .[ figfour ] , we conclude that the key distribution is secure if eve s error rate is larger than 0.1 , where we use the theoretical value of the error rate ( 0.072 ) for bob .this condition is satisfied if eve s snr is 0 db , because we obtained the error rate of 0.15 in the experiment .bob s advantage in snr was thus greater than 9.25 db .this advantage was almost constant for large snr signals .security against the translucent attack was examined as follows .we assigned one receiver that followed att2 as bob , and the other receiver as eve .att1 affected the snrs of both bob and eve , whereas att2 determined the ratio of the snrs . the decisions in bobwere recorded with several values of the threshold , while the eve s decisions were recorded with the threshold fixed at zero .we measured the error rate and decision rate of bob and the error rate of eve .we estimated the joint probabilities  , and from the bit data about which bob made correct decisions .finally , we calculated the secure key distribution rate by using eqs .( [ iab ] ) , ( [ fraction ] ) , and ( [ rate ] ) .figure [ figthree ] shows the secure key distribution rate as a function of the error rates of eve and bob .the symbols in fig .[ figthree ] show the secure key distribution rates estimated from the experiment .experimental results agree well with theoretical results ( lines . )the secure key distribution was achieved if error rates of eve and bob are in the region above the line in fig .[ figthree ] .the decrease in bob s snr reduced the range of the signal intensity for secure key distribution .secure key distribution was impossible when bob s snr was -9 db smaller than that of eve .this result also agrees well with the prediction .we obtained the largest actual secure key distribution rate when the snrs of both bob and eve were unity ( 0 db ) and bob s threshold was set to .the observed error rates were 0.01 for bob and 0.15 for eve .the secure key distribution rate was .higher secure key distribution rates were obtained by setting larger threshold values , but the reduction in the decision rate decreased the product .alice transmitted signals at 50 mb / s , so that the key transmission rate in the present experiment was 2 mb / s .this is a hundred times as fast as the key transmission rate reported in the qkd experiments .the transmission rate was limited only by the electric circuits .the secure key would be transmitted at 400 mb / s if a 10-gb / s transmission channel were used .this theoretical analysis has shown that the secure key distribution is possible as long as the ratio of bob s snr to eve s is better than -9 db , and the experimental results presented here confirmed it .a practical cryptosystem should thus be designed to satisfy this condition .eve may stay much closer to alice than bob , and her signal may be larger than bob s because of the fiber loss .we estimate a limit of the transmission distance in the following. it would be very difficult to use complicated networks , where the path of a traffic is not fixed .we have to construct a cryptosystem on a simple network or a point - point channel .suppose , for simplicity , we construct it on a point - to - point channel .snr is proportional to the square of the light intensity when the system sensitivity is limited by thermal noise .then bob s advantage of 9 db refers to the fiber length of 22.5 km using a lowest loss fiber ( 0.2 db / km ) and neglecting connection loss .snr is proportional to the light intensity in systems limited by shot noise , and the fiber can be as long as 45 km in those systems .this values would be increased assuming the translucent attack , because eve would tap the channel and receive small part of the signal .amplifiers can be used in yk protocol as long as the snr permits .they will improve the snr by reducing the effect of the thermal noise , and therefore will be useful when the system is limited by the thermal noise . in the shot noise limit , even an ideal amplifier reduces the snr by 3 db .the use of amplifiers is restricted by this degradation in the snr . at most three amplifiersare thus possible .the above estimation assumed that bob and eve use the same detectors .bob should reduce system noise as possible to guarantee the security , by cooling the receiver , for example .if he can suppress all the thermal noise , the snr of his system will be limited by shot noise , the standard quantum limit . the mean photon number transmitted in this systemshould be reduced to unity , because the snr of 0 db refers to mean photon number of unity in the shot noise limited systems .the error rate can be reduced below the standard quantum limit by optimum decision .the improvement will be apparent for small mean photon numbers .security analysis based on quantum detection theory as well as practical implementation of the optimum decision are open for further study. it would be noteworthy that the security analysis described in the present article will provide a security criteria for the b92 protocols employing dim coherent lights .the yk protocol provides more efficient key distribution at higher bit rates than do other qkd protocols , but it requires that the signal intensity be controlled to keep eve s snr advantage smaller than 9 db .this may be a disadvantage compared to the qkd protocols like bb84 , where the unconditional security is proved if the photons are generated by a perfect single photon source . however , it has been shown that the eve s advantage in snr will limit the efficiency of the protocol in a lossy channel .the snr control would be also required in actual bb84 systems .quantitative analysis of the security of the yuen - kim protocol shows that the secure key distribution is possible even if the eavesdropper receives signals with a signal - to - noise - ratio better than that with which the legitimate receiver receives them .it has been shown that the signal - to - noise - ratio of the legitimate receiver may be -9 db smaller than that of the eavesdropper .the results of an experiment using conventional fiber optics agrees well with the analysis results .these results have demonstrated a practical implementation of a secure key distribution protected by the laws of physics .we think the yk protocol would be a solution for practical cryptography systems .

Abstract: we examine the security of a protocol on cryptographic key distribution proposed by yuen and kim ( 1998 _ phys . lett . _ _ a _ * 241 * 135 ) . theoretical and experimental analysis shows that , even if the eavesdropper could receive more photons than the legitimate receiver , secure key distribution is possible as long as the signal - to - noise - ratio of the eavesdropper does not exceed eight times ( 9 db ) that of the receiver . secure key distribution was demonstrated using conventional fiber optics . the secure key transmission rate in the experiment was estimated to be 2 mb / s at its maximum ( 0.04 bit per sender s bit . ) the present protocol has advantages over other quantum key distribution protocols in that it is more efficient and more easily implemented , but careful design and management are necessary to ensure the security of the cryptosystem .   [ theorem]acknowledgement [ theorem]algorithm [ theorem]axiom [ theorem]claim [ theorem]conclusion [ theorem]condition [ theorem]conjecture [ theorem]corollary [ theorem]criterion [ theorem]definition [ theorem]example [ theorem]exercise [ theorem]lemma [ theorem]notation [ theorem]problem [ theorem]proposition [ theorem]remark [ theorem]solution [ theorem]summary


nowadays , large - scale reservoir simulations are becoming more and more popular in the oil and gas industry in order to simulate complex geological models .however , when a model is large enough , a simulator may take days or even weeks to finish one run using regular workstations and personal computers . this problem can also be observed in black oil , compositional and thermal simulations .efficient computational methods and fast reservoir simulators should be investigated .reservoir simulations have been studied for decades and various models and methods have been developed .coats studied black oil , compositional and thermal models , and he also investigated numerical methods , linear solver , preconditioner , grid effects and stability issues in his publications .kaarstad et al . implemented a parallel two - dimensional two - phase oil - water simulator , which could solve problems with millions of grid cells .rutledge et al . implemented a compositional simulator for parallel computers using the impes ( implicit pressure - explicit saturation ) method .shiralkar et al . developed a portable parallel production qualified simulator , which could run on a variety of parallel systems .killough et al . studied locally refined grids in their parallel simulator .dogru et al . developed a parallel black oil simulator , which was highly efficient and was capable of simulating models with up to one billion cells .zhang et al . developed a scalable general - purpose platform to support adaptive finite element and adaptive finite volume methods , which was also applied to reservoir simulations using discontineous galerkin method .for many reservoir simulations , most of the running time is spent on the solution of linear systems .we know that the most important is to develop efficient preconditioners .many preconditioners have been proposed , such as constrained pressure residual ( cpr ) methods , multi - stage methods , multiple level preconditioners and fast auxiliary space preconditioners ( fasp ) .chen et al . studied parallel reservoir simulations and developed a family of cpr - like preconditioners for black oil simulations and compositional simulations , including cpr - fp , cpr - fpf and cpr - ffpf methods .a black oil simulator has been developed based on our in - house parallel platform .the black oil model has three mass conservation equations for three components ( water , gas and oil ) .the system is fully coupled nonlinear system , which is solved by inexact newton - raphson methods , and structured grids and finite difference methods are applied .the performance of the black oil simulator is studied on ibm blue gene / q system using large - scale reservoir models for standard black oil model and two - phase oil - water model .numerical experiments show that our simulator is scalable and it is capable of simulating models with hundreds of millions of grid cells .the black oil model and its simplified model , two - phase oil - water model , are briefly introduced here .the black oil model has three phases ( water , oil and gas ) , and three components .the model assumes that there is no mass transfer between water phase and the other two phases , and gas component can exist in gas and oil phases .oil component is also assumed that it can exist in oil phase only .the reservoir is isothermal and no energy change is considered .the darcy s law is applied for black oil model , which establishes a relationship between volumetric flow rates of three components and their pressure changes in a reservoir , which is described as : where is the absolute permeability of rock , is a cross - section area , is the pressure difference , is viscosity of fluid , and is the length of a porous medium . in three - dimensional space ,the differential form of darcy s law is :   by combining darcy s law , black oil model has the following mass conservation equations for each component : where , for phase  , is its potential , and are porosity and permeability of a resevoir , and , , , , and are its saturation , phase viscosity , phase pressure , density , relative permeability and production ( injection ) rate , respectively . and are density of the oil component in the oil phase and the density of the solution gas in the oil phase , respectively .they have the following relations : where is reservoir depth , is capillary pressure between water phase and oil phase , is pressure of phase , and is capillary between gas phase and oil phase .the properties of fluids and rock are functions of pressure and saturation .the pressures of water and gas phases are functions of oil phase pressure and saturation ; see equation .the density of water is a function of its pressure : and the density of the oil phase is a function of its phase pressure and the bubble point pressure : where is bubble point pressure .the bubble point pressure is the pressure at which infinitesimal gas appears . the water viscosity is assumed to be a constant .the oil phase viscosity is a function of its pressure and the bubble point pressure : the relative permeabilities , and are functions of water and gas saturations and : where is calculated using the stone ii formula .  for real simulations ,the relative permeabilities are given by tables or analytic formulas .other properties , such as density , viscosity and capillary pressure , have analytic formulas and they can be calculated by table input too . with proper boundary conditions and initial conditions , a close system is given .here no flow boundary condition is assumed .this model is a simplified model of the standard black oil model , which assumes that the reservoir has two phases , oil and water , and they are immiscible .the model is similar to black oil model , which is written as :    different well constraints can be set for each active well .one commonly - used method is a sink - source model . for each perforation block , its well rate ( production or injection ) calculated by : where is bottom hole pressure of a well , is its well index , is reference depth of bottom hole pressure , is depth of the perforated grid block , and is phase pressure of the perforated grid block , such as oil , gas and water . can be calculated by several different models . in our simulator ,a peaceman model is chosen .many operation constraints and their combinations may be applied to each well at different time stages , such as a fixed bottom hole pressure constraint , a fixed oil rate constraint , a fixed water rate constraint , a fixed liquid rate constraint and a fixed gas rate constraint . when the fixed bottom hole pressure condition is applied to some well , its bottom hole pressure , , is known and its well rate is known if we have phase pressure of the perforated block .the constraint equation for the well is where is a constant set by the user input .no known exists for this constraint .when a fixed rate constraint is applied to a well , its bottom hole pressure is an unknown .for the fixed water rate constraint , the equation is where is constant . for the fixed oil rate constraint ,its equation is where is constant and known .a well may be applied different constraints at different time period .a schedule can be set by input , in which users can set operation changes for each well .   in this paper ,conservative finite difference schemes are employed to discretize these models .the inexact newton method is employed to solve the nonlinear equations .the time term is discretized by the backward euler difference scheme .if we let represent the value of a function at any time step , then its derivative at time step is approximated by   the space terms are discretized by cell - centered finite difference method . hereif we assume is a space direction and is the area of the corresponding face of a grid cell , the transmissibility term can be written as :    the nonlinear system can be represented by where is unknown vector , including oil phase pressure , water saturation and well bottom hole pressure .for black oil model , a gas saturation ( or bubble point pressure ) is also included .after linearization , a linear system , , is obtained in each newton iteration , where is jacobian matrix , is unknown to be determined , and is right - hand side .the standard newton method solves the linear system accurately .however , it is computationally expensive and it is not necessary sometimes . in our implementation , the inexact newton method is applied , whose algorithm is described in algorithm [ inewton - alg ] .give an initial guess and stopping criterion , let , and assemble right - hand side .assemble the jacobian matrix .find and such that let and . is the solution of the nonlinear system .the only difference between standard newton method and inexact newton method is how to choose .usually the parameter , , for standard newton method is fixed and small , such as .the parameter , , for inexact newton method is automatically adjusted .three different choices are listed as follows : where and are residual and right - hand side of -th iteration , respectively .the residual is defined as ,    if a proper matrix ordering ( numbering of unknowns ) is applied , the matrix derived from each newton iteration can be written as    where is the matrix corresponding to oil phase pressure unknowns , is the matrix corresponding to other unknowns in each grid cell , such as water saturation , gas saturation and bubble point pressure , and is the matrix coefficients corresponding to well bottom hole pressure unknowns , and other matrices are coupled items .the matrix is hard to solve in large - scale reservoir simulations .many multi - stage preconditioners have been developed to overcome this problem , such as cpr , fasp , cpr - fp and cpr - fpf methods .the key idea is to solve a sub - problem ( ) using algebraic multi - grid methods ( amg ) . in this paper ,the cpr - fpf method developed by chen et al . is applied .matrix decoupling techniques are also employed , such as abf decoupling and quasi - impes decoupling .an blue gene / q from ibm is employed to run reservoir simulations .the system , wat2q , is located in the ibm thomas j. watson research center .each node has 32 computer cards ( 64-bit powerpc a2 processor ) , which has 17 cores .one of them is for the operation system and the other 16 cores for computation .the system has 32,768 cpu cores for computation .the performance of each core is really low compared with intel processors .however , the system has strong network relative to cpu performance , and the system is scalable .the spe10 model is described on a regular cartesian grid , whose dimensions are ( ft ) .the model has cells ( cells ) .it has one injection well and four production wells .the original model is designed for two - phase oil - water model and it has around 2.244 millions of unknowns .the model is highly heterogeneous .its permeability is ranged from 6.65e-7 darcy to 20 darcy , and the x - direction permeability , , is shown in figure [ fig - spe10-perm ] .its porosity shown in figure [ fig - spe10-poro ] , ranges from 0 to 0.5 .data sets for porosity and permeability can be downloaded from spe10 s official website .the relative permeability of water phase is calculated by and the relative permeability of oil phase is calculated by where .capillary pressure is ignored .[ ex2 ] the original spe10 project is simulated .the termination tolerance for inexact newton method is and its maximal newton iterations are 20 . the linear solver bicgstab is applied and its maximal inner iterations are 50 .the quasi - impes decoupling strategy is used .simulation period is 2,000 days and maximal time step is 100 days .summaries of numerical results are shown in table [ tab - ex2 - 4 ] , and its scalability is shown by figure [ fig - ex2-cpr ] ..summaries of example [ ex2 ] [ cols="<,<,<,<,<,<,<",options="header " , ]   [ tab - ex9 - 1 ]  ]  the original spe1 project is a small one with 300 grid cells ( ) and the project is refined to 100 millions of grid cells ( ) .it has a homogeneous geological model .table [ tab - ex9 - 1 ] presents numerical summaries for nonlinear method , linear solver and performance .all four simulations use around 27 time steps .the simulation with 512 mpi tasks uses 140 newton iterations , which is more than other cases due to one time step cut .the linear solver and preconditioner are robust , which can solve a linear system in a few iterations . again , running time and figure [ fig - ex9-cpr ] show our simulator has excellent scalability .parallel reservoir simulations are studied in the paper , which are based on our in - house parallel platform .the platform provides grids , data , linear solvers and preconditioners for reservoir simulators .a black oil model is implemented .numerical experiments show that our simulator has excellent scalability and simulations can be sped up thousands of times faster .the paper also demonstrates that parallel computing techniques are powerful tools for large - scale reservoir simulations and ibm blue gene / q system is scalable .the support of department of chemical and petroleum engineering , university of calgary and reservoir simulation group is gratefully acknowledged . the research is partly supported by nserc / aiee / foundation cmg and aitf chairs .hu , x. , liu , w. , qin , g. , xu , j. , zhang , z. , development of a fast auxiliary subspace pre - conditioner for numerical reservoir simulators , spe reservoir characterisation and simulation conference and exhibition . 2011 .feng , c. , shu , s. , xu , j. , and zhang , c. , a multi - stage preconditioner for the black oil model and its openmp implementation , 21st international conference on domain decomposition methods , 2012 , france .baohua wang , shuhong wu , qiaoyun li , xiaobo li , hua li , chensong zhang , jinchao xu , a multilevel preconditioner and its shared memory implementation for new generation reservoir simulator , spe-172988-ms , spe large scale computing and big data challenges in reservoir simulation conference and exhibition , 15 - 17 september , istanbul , turkey , 2014 .wang , k. , zhang , l. , and chen , z. , development of discontinuous galerkin methods and a parallel simulator for reservoir simulation , spe-176168-ms , spe / iatmi asia pacific oil & gas conference and exhibition , 20 - 22 october , nusa dua , bali , indonesia , 2015 .

Abstract: this paper presents our work on simulation of large - scale reservoir models on ibm blue gene / q and studying the scalability of our parallel reservoir simulators . an in - house black oil simulator has been implemented . it uses mpi for communication and is capable of simulating reservoir models with hundreds of millions of grid cells . benchmarks show that our parallel simulator are thousands of times faster than sequential simulators that designed for workstations and personal computers , and the simulator has excellent scalability .   -scale reservoir , simulation , blue gene / q , parallel computing


the _ isogeometric analysis _ ( iga ) approach , which was proposed by hughes et al . , offers the possibility of seamless integration of computational analysis and geometric design .two major challenges in the current development of iga are volumetric parameterization and computational efficiency . in the recent book of cottrell andhughes  , it has been pointed out that the most significant challenges towards isogeometric analysis is how to construct analysis - suitable parameterizations from given cad models . on the other hand ,high - order basis functions are often employed to achieve smooth solution fields with high continuity , which however also increases the computational cost when filling stiffness matrices . in this paper , we propose a method for _ computation reuse _ in iga on three - dimensional models with similar semantic features , by which the computational efficiency can be significantly improved .applications in computational design that can be benefit from this research include : 1 ) the physical analysis on a family of products having the same topology but different shapes ; 2 ) using as the inner loop of physics - based shape optimization , where the computation can be greatly speeded up after applying a complete iga in the first iteration .+ ( a ) model a   + ( b)iga result of ( a )   + ( c ) model b   + ( d ) iga solution of ( c )   +   + ( e ) direct mapping result from ( b )   + ( f ) the difference between ( e)and ( d )   + ( g ) solution w.r.t the proposed method   + ( h ) the difference between ( g ) and ( d )   +  it should be mentioned that the result of iga - solution can not be directly reused by the mapping of a consistent parameterization ( e.g. ,  ) between models having similar semantic features as illustrated in fig.[fig : rabbit](a , b ) and ( c , d ) , a heat conduction problem with homogeneous boundary condition and the following source function is solved separately over two different planar domains that have the same tensor - product b - spline space .when directly mapping the result from fig.[fig : rabbit](b ) into the domain of ( c ) , the result of heat distribution is as shown in ( e ) which is quite different from iga result ( i.e. , fig.[fig : rabbit](d ) ) .fig.[fig : rabbit](f ) shows the difference in color between ( e ) and ( d ) . on the other aspects , when using the computation reuse approach developed in this paper , the corresponding solution is presented in fig.[fig : rabbit](g ) , where approximation errors brought in are trivial see the difference in color shown in fig.[fig : rabbit](h ) .the main contribution of our work can be summarized as follows :  * an efficient quadrature - free method is proposed to compute the entries of stiffness matrix with the help of bzier extraction and polynomial approximation techniques applying to trivariate rational bzier functions .* we present a framework of computational reuse in iga and the method for reuse when imposing boundary conditions in this framework . up to times speedupcan be observed by using our method on problems with large number of degree of freedom ( dof ) .the rest of our paper is organized as follows . in section [ sec : related ] , the related work on volumetric parameterization and computational efficiency of iga will be reviewed .the method to construct consistent b - spline based volumetric parameterization is presented in section [ sec : volcon ] .section [ sec : analysis ] presents the quadrature - free iga method using the bzier extraction and polynomial approximation techniques . by combining techniques presented in sections [ sec : volcon ] and [ sec : analysis ] , the computation reuse framework for a set of models with similar semantic features is presented in section [ sec : example ] .lastly , we conclude this paper and discuss possible future works in section [ sec : conclude ] .* volumetric parameterization * from the viewpoint of graphics applications , volume parameterization of 3d models has been studied in . on the other aspect , there are also some recent work on volume parameterization in the literature of iga .martin et al . proposed a method to fit a genus- triangular mesh by b - spline volume parameterization , based on discrete volumetric harmonic functions .a variational approach for constructing nurbs parameterization of swept volumes is proposed by aigner et al .escobar et al . proposed a method to construct a trivariate t - spline volume of complex genus - zero solids by using an adaptive tetrahedral meshing and mesh untangling technique .zhang et al . proposed a robust and efficient algorithm to construct injective solid t - splines for genus - zero geometry from a boundary triangulation .based on the morse theory , a volumetric parameterization method of mesh model with arbitrary topology is proposed in . in ,a constraint optimization framework is proposed to obtain analysis - suitable planar and volumetric parameterization of computational domain .pettersen and skytt proposed the spline volume faring method to obtain high - quality volume parameterization for isogeometric applications .zhang et al . studied the construction of conformal solid t - spline from boundary t - spline representation by octree structure and boundary offset . for volume parameterization of three - manifold solid models having homeomorphictopology , kwok and wang proposed an algorithm to constructing volumetric domains with consistent topology .the generated volumetric parameterizations share the same set of base domains and are constrained by the corresponding semantic features in the form of anchor points . in this paper, a compactly supported radial basis function method is proposed to construct consistent volumetric b - spline parameterization for models with similar semantic features .  *efficiency issues of isogeometric analysis * high - order basis functions are used to represent the geometry and the physical field in iga to achieve high - accuracy simulation results .hence , computational efficiency is a key issue in the field of isogeometric analysis . in order to improve the efficiency ,several methods have been proposed .there is a trend to use _ graphic possessing units _ ( gpu ) to improve the computational efficiency of assembling the stiffness matrix ( e.g. , ) . on the other hand ,efficiency improvement on integral computing has also been studied .hughes et al . proposed an efficient quadrature rules for nurbs - based isogeometric analysis .antolin et al . developed a sum - factorization approach to save the quadrature computational cost significantly based on the tensor - product structure of splines and nurbs shape functions .mantzaflaris and jttler presented a quadrature - free integration method by interpolation and look - up table for galerkin - based isogeometric analysis . in this paper , we propose the concept of _ computation reuse _ to improve the efficiency of iga on a set of cad models with consistent topology .  , and a target surface model , , the surface cross - parameterization can be applied to partition consistently to the surface of . after obtaining a surface model by mapping to ,the vertices on and are used as handles to map the sampled points on the interior patches to by volumetric parameterization .b - spline solid for can be obtained by fitting it to the boundary surface and the sampled points of . ]to prepare for the computation reuse in iga , we need to partition a model into a set of base domains consistent to the pre - defined one , where each base domain will be represented by a trivariate spline .we assume the reuse of iga is applied to the models with similar semantic features , e.g. , a whole sequence of products having different shapes but the same topology .the analysis will respect the semantic features , which are specified as anchor points , such that the boundary conditions can also be reused .  given a connectivity of volumetric base domains ( ) , e.g. , the template base domain in fig.[fig : volparaoverview ] , we can partition a target surface model ( ) into a set of volumetric sub - domains consistent to .the boundary surface is first partitioned consistently according to anchor points , and the boundary surface is used as the constraints to construct volumetric parameterization using csrbf - based elastic functions .the volumetric parameterization is used to wrap the sample points on the interior surface of to .the trivariate spline solids are constructed in each domain by fitting the boundary surface of and the sample points on above determined interior surfaces .based on the anchor points specified on , the surface of is partitioned consistently according to the nodes in using the method in .the basic idea is to trace shortest paths between anchor points on for each edge on the boundary surface of . to ensure topological consistency , we need to ensure the paths are free of intersections , blocking and wrong cyclical order when tracing the paths .surface cross - parameterization can then be computed through this partition , and a surface model is obtained by mapping the boundary surface of to . once the boundary surface is partitioned , the next step is to construct the interior patches .   due to the reason that is a surface model without volumetric information , we need to construct the interior patches for .a sampling is first taken on the interior patches in . by the volumetric parameterization presented in , using the vertices on and as handles , sample points on the interior patches of can be mapped to the interior of .specifically , the volumetric parameterization can be expressed by the elastic function : where are the locations of the constraint vertices on , are the weights , and is a linear polynomial that accounts for the linear and constant portions of .  to solve for the set of that will satisfy the constraints , , on the elastic function, we can substitute the right side of eq.([eq : csrbf ] ) for and yield to since this equation is linear with respect to the unknowns and the coefficients of , the unknowns can be formulated and solved as a linear system .we take the wendland s compactly supported radial basis function as , it is given by which has a compact support with the radius , and has -continuity . as a compactly supported kernel function is used , the linear equation system will become sparse and can be efficiently solved by cholesky decomposition or lu decomposition .  nowthe volumetric parameterization has been established .the sample points on the interior patches in are mapped to by .the trivariate spline solid ( i.e. , b - spline in our implementation ) for each base domain inside can be constructed by fitting the boundary surface of to its corresponding boundary surface on and the interior sample points obtained by the nonlinear elastic function . by this way , we can convert the target model into a set of connected trivariate spline solids with consistent topology as . when applying this to a sequence of models , all models will have spline solids in the same connectivity but different control points ( i.e. , different shapes ) . with the help of this setup, we will show how to reuse the computation of iga taken on one model in the iga of other models .in this section , an efficient quadrature - free method is proposed to compute the entries of stiffness matrix with the help of bzier extraction and polynomial approximation techniques of trivariate rational bzier functions .here we use heat conduction problem as an example to demonstrate the functionality of our approach .the quadrature - free method can be applied to many other problems of computational physics such as the linear elasticity problem in solid mechanics .some preliminary on bernstein polynomials will be used in our method .they are reviewed below .product of bernstein polynomials   integration of bernstein polynomials   degree elevation of bernstein polynomials   [ prop : prop ] let and be parametric function defined by and where and are scale values .then the product of and can be defined as where   this proposition can be proved directly by eq.([eq : product1 ] ) .   in order to achieve an efficient computation ,the isogeometric analysis problem is solved with bzier extraction , in which piece - wise b - spline representation is first converted into a bzier form .  without loss of generality , b - spline basis defined on a knot vectorcan be written as a linear combination of the bernstein polynomials , that is where denotes the bzier extraction operator and are the bernstein polynomials which are defined on  ] is either very complex or has no analytically solution .gaussian - quadrature method is usually employed in general iga to compute the integration of rational function in eq.([eq : system_iso ] ) approximately . as shown in eq.([eq : integration ] ) , the integration of polynomial bzier functions ( non - rational ) has an explicit and exact form . as a classical problem in cagd , approximating rational bzier curves and surfaces with polynomial bzier curves and surfaces has been studied in . in this paper , we further extend the weighted least - squares approach to trivariate splines and approximate the trivariate rational function with a trivariate polynomial bzier function .suppose that can be rewritten as , our bzier approximation problem can be stated as that to find the control variables , which can make the trivariate bzier representation become the best approximation of .that is , to minimize the following objective function when is set to be the weight function hence , the coefficient can be obtained by letting which is   eq.([eq : gijk ] ) can be rewritten as   from the product and integral computation properties of bzier polynomials , we have in which   then , from eq.([eq : linear ] ) for all the , we can obtain a linear system in the following form , in which {(\alpha+1 ) \times ( \beta+1 ) \times ( \gamma+1)} ] is a matrix with  ] is a matrix with and  ] with as entries defined in eq.([eqn : f(u ) ] ) .after the local stiffness matrix for each element are filled , the global stiffness matrix can be obtained by assembling . in the assembly process, the boundary condition described in the governing equation must be imposed and the corresponding entries related to the boundary condition will also be evaluated . in the following subsection , we will discuss the reuse for the imposition of boundary conditions .   as the bernstein basis functions do not have interpolating property at control points, we can not impose the essential boundary conditions directly onto the control variables on the boundary .special treatments need to be implemented to achieve the specified boundary conditions , such as the least square approach , the penalty function method or the nitsche method . in this paper ,a collocation method is employed to impose boundary conditions . for a set of computational models with topology - consistent volume parameterization, the control variables on the boundary can be reused for the same boundary conditions , and the collocation matrix can be reused for different boundary conditions .suppose that are collocation points on the boundary surface and are corresponding parametric coordinates in the parametric domain , the dirichlet boundary condition can be defined as in which are boundary values to be interpolated , and are control variables to be solved. this equation can be rewritten into a matrix form as in which the entry of matrix is , =[h(\{{\bf x}_i\})]$ ] , .then the boundary control variables can be solved by   in this paper , since the models with topology - consistent volume parameterization have the same basis functions for the corresponding blocks , the boundary control variables which is determined by eq.([imposition ] ) can be used for all other models for a pde problem with the same boundary conditions .furthermore , for the pde problem with different boundary conditions , the inverse of collocation matrix can also be reused for the models with topology - consistent volume parameterization . in practice, we apply lu - decomposition on the sparse matrix , and re - use the decomposition in eq.([imposition ] ) to determine the value of .   in this subsection, experimental results will be presented to show the advantage of the proposed computation reuse method .three sets of cad models are tested in this paper .the first set has two hand models ( fig .[ fig : hand ] ) , four airplane models are tested in the second set ( fig .[ fig : airplane ] ) , and the third set consists of four human models with consistent topology ( fig .[ fig : human ] ) . for each model in figs .[ fig : hand]-[fig : human ] , from left to right we show the input surface model with patch - partition information , the results of boundary b - spline surface fitting , the results of volume parameterization , and the iga results for a heat conduction problem respectively .  in order to illustrate the effectiveness of the proposed computation reuse approach , the corresponding average assembling time of the iga and iga - reuse approach are shown in table [ table : data ] for the models in figs . [fig : hand]-[fig : human ] respectively .the corresponding extra storages of the proposed iga - reuse method are also given .all the computations are implemented in c++ and timed on a macbook pro with a quad core 2.4 ghz intel core i7 processor and 8 gb ram . from the performance statistic shown in the table, we can find that the computational costs in assembling stiffness matrix can be reduced significantly ( i.e , up to 12 times ) by our method while keeping the nice trade - off between efficiency and storage .furthermore , as shown in fig .[ fig : datacurve ] , the acceleration ratio keeps increasing while increasing the degree of freedom in iga .overall , since the evaluation of high - order basis function for one model can be reused for other models with consistent topology , a similar performance of classical linear finite element method can be achieved for isogeometric analysis on a set of models .this addresses a main shortcoming of isogeometric analysis and makes it cost - efficient in solving large - scale problems in computational mechanics ..quantitative data and average assembling time for computation reuse in iga [ cols= " < , < , < , < , < , < , < , < " , ]   + # dof : the degree of freedom ; : the degree of basis function ; : -refinement step ; + nnz : number of none - zero elements in matrix .[ table : data ]   + ( a ) hand model i   +   + ( b ) hand model ii   +   + ( a ) airplane model i   +   + ( b ) airplane model ii   +   + ( c ) airplane model iii   +   + ( d ) airplane model iv   +   + ( a ) human body model i   +   + ( b ) human body model ii   +   + ( c)human body model iii   +   + ( d ) human body model iv   +   + ( a ) hand models   + ( b ) airplane models   + ( c ) human models   +in this paper , in order to improve the computational efficiency of isogeometric analysis , the concept of _ computation reuse _ is proposed for three - dimensional models with similar semantic features . for a set of models with consistent topology, a csrbf - based method is firstly applied to construct topology - consistent volumetric b - spline parameterization from given template domains . after obtaining the consistent volumetric parameterization, we propose an efficient quadrature - free method to compute the entries of stiffness matrix with the techniques of bzier extraction and polynomial approximation . with the help of our method , evaluation on the stiffness matrix and imposition of the boundary conditionscan be pre - computed and partially reused for models having consistent volumetric parametrization .effectiveness of the proposed methods has been verified on several examples with complex geometry , and the computation efficiency similar to classical finite element method can be achieved for iga on these models .we plan to test the proposed computation reuse approach on other physical simulation problems in the future .problems such as linear elasticity and flow simulation for a set of models with consistent topology have been widely practiced in the design of a family of products , which will potentially be benefited from our approach . on the other aspect, this technique can be used as the inner loop of physics - based shape optimization , in which the computation can be significantly speeded up after the first isogeometric analysis on the original shape .gang xu is partially supported by the national nature science foundation of china ( no .61472111 ) and zhejiang provincial natural science foundation of china ( no .lr16f020003 ) .m. aigner , c. heinrich , b. jttler , e. pilgerstorfer , b. simeon and a .- v .swept volume parametrization for isogeometric analysis . in e.hancock and r. martin ( eds . ) , the mathematics of surfaces ( mos xiii 2009 ) , lncs vol .5654(2009 ) , springer , 19 - 44 .p. antolin , a. buffa , f. calabr , m. martinelli , g. sangalli .efficient matrix computation for tensor - product isogeometric analysis : the use of sum factorization .computer methods in applied mechanics and engineering 2015 ; 285 : 817 - 828 .m. j. borden , m. a. scott , j.a .evans , t. j. r. hughes .isogeometric finite element data structures based on bzier extraction of nurbs .international journal for numerical methods in engineering 2011 ; 87(1 - 5 ) : 15 - 47 .e. cohen , t. martin , r.m .kirby , t. lyche and r.f .riesenfeld , analysis - aware modeling : understanding quality considerations in modeling for isogeometric analysis .computer methods in applied mechanics and engineering 2010 ; 199 : 334 - 356 .  j.a .cottrell , t.j.r .hughes , y. bazilevs .isogeometric analysis : toward integration of cad and fea .wiley , 2009 j.m .escobara , j.m .cascnb , e. rodrgueza , r. montenegro . a new approach to solid modeling with trivariate t - spline based on mesh optimization .computer methods in applied mechanics and engineering 2011 ; 200 : 3210 - 3222 .hughes , j.a .cottrell , y. bazilevs .isogeometric analysis : cad , finite elements , nurbs , exact geometry , and mesh refinement .computer methods in applied mechanics and engineering 2005 ; 194 : 4135 - 4195 .m. a. scott , m. j. borden , c. v. verhoosel , t. w. sederberg , and t. j. r. hughes .isogeometric finite element data structures based on bzier extraction of t - splines .international journal for numerical methods in engineering 2011 ; 88(2):126 - 156 .g. xu , b. mourrain , r. duvigneau , a. galligo .parameterization of computational domain in isogeometric analysis : methods and comparison . computer methods in applied mechanics and engineering 2011 ; 200 : 2021 - 2031 .

Abstract: volumetric spline parameterization and computational efficiency are two main challenges in _ isogeometric analysis _ ( iga ) . to tackle this problem , we propose a framework of _ computation reuse _ in iga on a set of three - dimensional models with similar semantic features . given a template domain , b - spline based consistent volumetric parameterization is first constructed for a set of models with similar semantic features . an efficient quadrature - free method is investigated in our framework to compute the entries of stiffness matrix by bzier extraction and polynomial approximation . in our approach , evaluation on the stiffness matrix and imposition of the boundary conditions can be pre - computed and reused during iga on a set of cad models . examples with complex geometry are presented to show the effectiveness of our methods , and efficiency similar to the computation in linear finite element analysis can be achieved for iga taken on a set of models .   computation reuse ; isogeometric analysis ; consistent volume parameterization


cooperation within groups of selfish individuals is ubiquitous in human and animal societies . to explain and understand the origin of this phenomenon , evolutionary games , providing a suitable theoretical framework ,have been studied extensively by many researches from various disciplines over the past decades .the evolutionary prisoner s dilemma game in particular , illustrating the social conflict between cooperative and selfish behavior , has attracted considerable attention both in theoretical as well as experimental studies . in a typical prisoner s dilemma , two players simultaneously decide whether they wish to cooperate or defect .they will receive the reward r if both cooperate , and the punishment p if both defect .however , if one player defects while the other decides to cooperate , the former will get the temptation t while the latter will get the sucker spayoff s. the ranking of these four payoffs is t , from where it is clear that players need to defect if they wish to maximize their own payoff , irrespective of the opponent s decision .resulting is a social dilemma , which typically leads to widespread defection . to overcome this unfortunate outcome , several mechanisms that support the evolution of cooperationhave been identified ( see for a review ) .  of particular renownare the investigations of spatial prisoner s dilemma games , which have turned out to be very inspirational over decades . in the first spatial prisoner s dilemma game introduced by nowak andmay , players were located on a square lattice , and their payoffs were gathered from the games with their neighbors .subsequently , players were allowed to adopt the strategy of their neighbors , providing their fitness was higher .it was shown that the introduction of spatial structure enables cooperators to form clusters , thereby promoting the evolution of cooperation . along this pioneering line of research , many different mechanisms aimed at sustaining cooperation were subsequently proposed and investigated .examples include the reward mechanism , simultaneous adoption of different strategies depending on the opponents , preferential selection of a neighbor , the mobility of players , heterogeneous teaching activity , differences in evolutionary time scales , neutral evolution , and coevolutionary selection of dynamical rules , to name but a few .looking at some examples more specifically , in a recent research paper , where players were allowed to either adjust their strategy or switch their defective partners , an optimal state that maximizes cooperation was reported . in was shown that the mobility of players can lead to an outbreak of cooperation , even if the conditions are noisy and do nt necessarily favor the spreading of cooperators .inspired by these successful research efforts , an interesting question posses itself , which we aim to address in what follows .namely , if we consider a simple addition to the prisoner s dilemma game that allows players to aspire to the fittest , _i.e. _ introducing the propensity of designating the most successful neighbor as being the role model , is this beneficial for the evolution of cooperation or not ?the answer is not straightforward since , as we have mentioned , defectors spread by means of their higher fitness .thus , the modification we consider might give them higher chances of replication .in the early pioneering works , nowak et al . have shown that increasing the probability to copy high payoff neighbors asymptotically leads to increased cooperation , yet this dependence was not monotonic over the whole parameter range . herewe aim to investigate this further in the presence of different levels of uncertainty by strategy adoptions and provide an interpretation of reported results .aside from the progress in promoting cooperation described above , another very important development came from replacing the square lattice with more complex interaction topologies ( see for a review ) , possibly reflecting the actual state in social networks more closely .recently , many studies have attested to the fact that complex networks play a critical role in the maintenance of cooperation for a wide range of parameters .quite remarkably , in the early investigations , it has been discovered that the scale - free network can greatly elevate the survivability of cooperators if compared to the classical square lattice . following this discovery ,many studies have built on it in order to extend the scope of cooperation on complex networks .for example , a high value of the clustering coefficient was found beneficial , while payoff normalization was found to impair the evolution of cooperation .motivated by these studies , we examine also how aspiring to the fittest in the prisoner s dilemma game fares on complex networks ; in particular , whether it promotes or hinders the evolution on cooperation .  herewe thus study the prisoner s dilemma game with the introduction of a mechanism that allows players to aspire to the fittest . comparing with previous works , where a neighbor was chosen uniformly at random from all the neighbors , the propensity of designating the most successful neighbor as the role model is the most significant difference .our aim is to study how this mechanism affects the evolution of cooperation on the square lattice , as well as on the scale - free network and the random regular graph , for different levels of uncertainty by strategy adoptions . by means of systematic computer simulationswe demonstrate , similarly as was reported already by nowak et al . , that this simple mechanism can actually promote the evolution of cooperation significantly .we give an interpretation of the observed phenomena and examine the impact of different levels of uncertainty by strategy adoptions and the impact of different interaction networks on the outcome of the modified prisoner s dilemma . in the remainder of this paperwe will first describe the considered evolutionary game , subsequently we will present the main results , and finally we will summarize our conclusions .we consider an evolutionary prisoner s dilemma game with the temptation to defect ( the highest payoff received by a defector if playing against a cooperator ) , reward for mutual cooperation , the punishment for mutual defection , and the sucker spayoff ( the lowest payoff received by a cooperator if playing against a defector ) . for positive we have , thus strictly satisfying the prisoner s dilemma payoff ranking . for simplicity , but without loss of generality , the payoffs can be rescaled such that , , and , where is the cost - to - benefit ratio .depending on the interaction network , the strategy adoption rule and other simulation details ( see _ e.g. _ ) , there always exists a critical cost - to - benefit ratio at which cooperators die out. we will be interested in determining to what extend does aspiring to the fittest , as we are going to introduce in what follows , affects this critical value under different circumstances .  throughout this work each player initially designated either as a cooperator ( ) or defector ( d ) with equal probability . as the interaction network, we use either a regular square lattice , the random regular graph constructed as described in , or the scale - free network with nodes and an average degree of four generated via the barabsi - albert algorithm .the game is iterated forward in accordance with the sequential simulation procedure comprising the following elementary steps .first , player acquires its payoff by playing the game with all its neighbors .next , we evaluate in the same way the payoffs of all the neighbors of player and subsequently select one neighbor via the probability where the sum runs over all the neighbors of player and is the newly introduced selection parameter .evidently , for the most frequently adopted situation is recovered where player is chosen uniformly at random from all the neighbors of player . for , however , eq .( 1 ) introduces a preference towards those neighbors of player that have a higher payoff .conversely , for players with a lower payoff are more likely to be selected as potential strategy donors .lastly then , player adopts the strategy from the selected player with the probability },\ ] ] where denotes the amplitude of noise or its inverse ( ) the so - called intensity of selection .irrespective of the value of one full iteration step involves all players having a chance to adopt a strategy from one of their neighbors once . here the evolutionary prisoner s dilemma game is thus supplemented by a selection parameters , enabling us to tune the preference towards which neighbor will be considered more likely as a potential strategy donor . for positive values of players are more likely to aspire to their most fittest neighbors , while for negative values of the less successful neighbors will more likely act as strategy donors .this amendment seems reasonable and is easily justifiable with realistic examples .for example , it is a fact that people are , in general , much more likely to follow a successful individual than someone who is struggling to get by .this is taken into account by positive values of . however ,under certain ( admittedly rare ) circumstances , it is also possible that individuals will be inspired to copy their less successful partners .indeed , the most frequently adopted random selection of a neighbor , retrieved in our case by , seems in many ways like the least probable alternative .it is also informative to note that aspiring to the fittest becomes identical to the frequently adopted `` best takes all '' rule if in eq .( 1 ) and in eq . ( 2 ) .this rule was adopted in the seminal work by nowak and may , as well as subsequently by huberman and glance who showed that under certain circumstances asynchronous updating is substantially less successful in ensuring the survivability of cooperators than synchronous updating .although in our simulations we never quite reach the `` best takes all '' limit , and thus a direct comparison is somewhat circumstantial , it is interesting to note that an additional uncertainty in the strategy adoption process via finite values of may alleviate the disadvantage that is due to asynchronous updating .results of computer simulations presented below were obtained on populations comprising to individuals , whereby the fraction of cooperators was determined within full iteration steps after sufficiently long transients were discarded . moreover , since the preferential selection of neighbors may introduce additional disturbances , final results were averaged over up to independent runs for each set of parameter values in order to assure suitable accuracy .we start by visually inspecting characteristic spatial distributions of cooperators and defectors for different values of the selection parameter .figure [ fig1 ] features the results obtained for and , whereat for ( upper middle panel ) a small fraction of cooperators can prevail on the square lattice by means of forming clusters , thereby protecting themselves against the exploitation by defectors .as evidenced in the upper leftmost panel , for negative values of even this small fraction of cooperators goes extinct , thus yielding as a results exclusive dominance of defectors . for positive values of ( upper right panel ), however , the cooperators start mushrooming , whereby clustering remains their mechanism of spreading and survivability .interestingly , large enough values of can facilitate the evolution of cooperation to the point of near - complete cooperator dominance ( bottom right panel ) , or at least equality with the defectors , as implied by in all lower panels of fig .these results suggest that when players aspire to adopt the strategy from their fittest neighbor the evolution of cooperation thrives . in what follows we will systematically examine the validity of this claim .to quantify the ability of particular values of the selection parameter to facilitate and maintain cooperation more precisely , we first calculate in dependence on the cost - to - benefit ratio for different values of .results presented in the top panel of fig. [ fig2 ] clearly attest to the fact that positive values of promote the evolution of cooperation , while on the other hand , negative values of impede it .note that the critical cost - to - benefit ratio , marking the extinction of cooperators , increases by a full order of magnitude at ( orange stars ) if compared to the ( black squares ) case .interestingly , the promotive effect on the survivability of cooperators becomes more potent monotonously with increasing , thus suggesting that a universally applicable mechanism is underlying the observed behavior .indeed , the monotonous increase of for increasing is obvious from the bottom panel of fig .[ fig2 ] , showing concisely the extend to which aspiring to the fittest promotes the evolution of cooperation on the square lattice .importantly , qualitatively identical results can be obtained on interaction networks other than the square lattice .results presented in fig . [ fig3 ] depict how cooperators fare on the random regular graph and the scale - free network for different values of .similarly as in fig . [ fig2 ] , it can be observed that positive values of promote the evolution of cooperation .conversely , negative values of promote the evolution of defection .this is in agreement with the observations made on the square lattice , thus designating as being universally effective in promoting the evolution of cooperation , in particular , working on regular lattices and graphs as well as highly heterogeneous networks .since the latter have been identified as potent promoters of cooperation on their own right , this conclusion is all the more inspiring .  in order to explain the promotive impact of positive values of on the evolution of cooperation , we examine time courses of for different values of the selection parameter . figure [ fig4 ] features results obtained for and , whereat cooperators die out if ( black line ; see also fig .[ fig2 ] ) . for positive values of ,on the other hand , the stationary state is a mixed c+d phase with cooperators occupying the larger portion of the square lattice .interestingly , however , in the most early stages of the evolutionary process ( note that values of were recorded also in - between full iteration steps ) it appears as if defectors would actually fare better for .in fact , the larger the value of , the deeper the initial downfall of cooperators .this is actually what one would expect , given that defectors are , as individuals , more successful than cooperators and will thus be chosen more likely as potential strategy donors if is positive .this in turn amplifies their chances of spreading and results in the decimation of cooperators ( only slightly more than 20 % survive ) .quite surprisingly though , the tide changes fast , and as one can observe from the presented time courses , the more so the deeper the initial downfall of cooperators . for we can observe instead of cooperator extinction their near - dominance with hoovering comfortably over ( orange line ) .we argue that for positive values of a negative feedback effect occurs , which halts and eventually reverts what appears to be a march of defectors towards their undisputed dominance .namely , in the very early stages of the game defectors are able to plunder very efficiently , which quickly results in a state where there are hardly any cooperators left to exploit .consequently , the few remaining clusters of cooperators start recovering lost ground against weakened defectors .crucial thereby is the fact that the clusters formed by cooperators are impervious to defector attacks even at high values of because of the positive selection towards the fittest neighbors acting as strategy sources ( occurring for ) . in a sea of cooperatorsthis is practically always another cooperator rather than a defector trying to penetrate into the cluster .this newly identified mechanism ultimately results in widespread cooperation that goes beyond what can be warranted by the spatial reciprocity alone ( see _ e.g. _ ) , and this irrespective of the underlying interaction network . as such, aspiration to the fittest , _i.e. _ the propensity of designating the most successful neighbor as being the role model , may be seen as a universally applicable promoter of cooperation .lastly , it is instructive to examine the evolution of cooperation for in dependence on the uncertainty by strategy adoptions .the latter can be tuned via , which acts as a temperature parameter in the employed fermi strategy adoption function .accordingly , when all information is lost and the strategies are adopted by means of a coin toss .the phase diagram presented in the top panel of fig .[ fig5 ] is well - known , implying the existence of an optimal level of uncertainty for the evolution of cooperation , as was previously reported in .in particular , note that the d c+d transition line is bell shaped , indicating that is the optimal temperature at which cooperators are able to survive at the highest value of . this phenomenon can be interpreted as an evolutionary resonance , albeit it can only be observed on interaction topologies lacking overlapping triangles .interestingly , positive eradicate ( as do interaction networks incorporating overlapping triangles ) the existence of an optimal , as can be observed from the phase diagram presented in the bottom panel of fig .the latter was obtained for and exhibits an inverted bell - shaped d c+d transition line , indicating the existence of the worst rather than an optimal temperature for the evolution of cooperation . this in turn implies that introducing a preference towards the fittest neighbors effectively alters the interaction network .while the square lattice obviously lacks overlapping triangles and thus enables the observation of an optimal , trimming the likelihood of who will act as a strategy source seems to effectively enhance linkage among essentially disconnected triplets and thus precludes the same observation .a similar phenomenon was observed recently in public goods games , where the joint membership in large groups was also found to alter the effective interaction network and thus the impact of uncertainly on the evolution of cooperation .in sum , we have shown that aspiring to the fittest promotes the evolution of cooperation in the prisoner s dilemma game irrespective of the underlying interaction network and the uncertainty by strategy adoptions .the essence of the identified mechanism for the cooperation promotion has been attributed to a negative feedback effect , occurring because of the formation of extremely robust clusters ( or groups on complex networks ) of cooperators that are impervious to defector attacks even at high temptations to defect .although initially the defectors appear to be heading to an undisputed victory , the fast exploitation and the consequent shortage of cooperators weakens the defectors and makes them susceptible to an overtake by the few remaining cooperators .further interesting is the fact that the introduction of a selection parameter , making the fittest neighbors more likely to act as sources of adopted strategies , effectively alters the interaction network .while in its absence there exists an intermediate uncertainty governing the process of strategy adoptions by which the largest cost - to - benefit ratio still warrants the survival of at least some cooperators , in its presence this feature vanishes and becomes qualitatively identical to what was observed previously on lattices that do incorporate overlapping triangles , such as the kagome lattice . since in fact the actual interaction topology remains unaffected by the different values of the selection parameter , we have argued that the differences in the evolution of cooperation are due to an effective transition of the interaction topology , which is brought about by the fact that some players are more likely to act as strategy sources than others .therefore , the bonds between certain player pairs appears stronger than average , although the interaction networks consist of links that are not weighted .  since aspiring to the fittest , _i.e. _ the propensity of designating the most successful neighbors as role models , appears to be both widely applicable as well realistically justifiable , we hope it will inspire future studies , especially in terms of understanding the emergence of successful leaders in societies via a coevolutionary process . an interesting interpretation of the selection parameter can also be obtained if the latter is considered as a measure of cognitive complexity of each individual . in particular, it is possible to argue that the more obtuse an individual is , the closer to random his choice of a role model will be . if individuals are to be able to aspire to the fittest , they should have some degree of information processing capabilities . on the other hand ,negative values of can be interpreted as a choice that is based on moral values , for example , when highly successful individuals are so by unethical actions and thus should not be imitated .matja perc acknowledges support from the slovenian research agency ( grant no .z1 - 2032 ) .zhen wang acknowledges support from the center for asia studies of nankai university ( grant no .2010 - 5 ) and from the national natural science foundation of china ( grant no .10672081 ) . helpful discussion with professor lianzhong zhangare gratefully acknowledged as well .this works has benefited substantially from the insightful comments of the physical review referees , and we are very grateful for their help .

Abstract: strategy changes are an essential part of evolutionary games . here we introduce a simple rule that , depending on the value of a single parameter , influences the selection of players that are considered as potential sources of the new strategy . for positive players with high payoffs will be considered more likely , while for negative the opposite holds . setting equal to zero returns the frequently adopted random selection of the opponent . we find that increasing the probability of adopting the strategy from the fittest player within reach , _ i.e. _ setting positive , promotes the evolution of cooperation . the robustness of this observation is tested against different levels of uncertainty in the strategy adoption process and for different interaction network . since the evolution to widespread defection is tightly associated with cooperators having a lower fitness than defectors , the fact that positive values of facilitate cooperation is quite surprising . we show that the results can be explained by means of a negative feedback effect that increases the vulnerability of defectors although initially increasing their survivability . moreover , we demonstrate that the introduction of effectively alters the interaction network and thus also the impact of uncertainty by strategy adoptions on the evolution of cooperation .