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New Nuclear Reaction Flow during r-Process Nucleosynthesis in Supernovae Critical Role of L

a r X i v :a s t r o -p h /0107368v 1 19 J u l 2001New Nuclear Reaction Flow during r-Process Nucleosynthesis

in Supernovae:

Critical Role of Light Neutron-Rich Nuclei

M.Terasawa a ?c ,K.Sumiyoshi c,d ,T.Kajino a,b ,

G.J.Mathews e ,and I.Tanihata c

Department of Astronomy,School of Science,University of Tokyo,Hongo,Bunkyo-ku,Tokyo 113-0033,Japan b National Astronomical Observatory,and Graduate University for Advanced Studies,Osawa,Mitaka,Tokyo 181-8588,Japan c Institute of Physical and Chemical Research (RIKEN),Hirosawa,Wako,Saitama 351-0198,Japan d Numazu College of Technology,Ooka,Numazu,Shizuoka 410-8501,Japan e Department of Physics and Center for Astrophysics,University of Notre Dame,Notre Dame,IN 46556,U.S.A.ABSTRACT We study the role of light neutron-rich nuclei during r-process nucleosynthesis in supernovae.Most previous studies of the r-process have concentrated on the reaction ?ow of heavy unstable nuclei.Although the nuclear reaction network includes a few thousand heavy nuclei,only limited reaction ?ow through light-mass nuclei near the stability line has been used in those studies.

However,in a viable scenario of the r-process in neutrino-driven winds,the

initial condition is a high-entropy hot plasma consisting of neutrons,protons,and electron-positron pairs experiencing an intense ?ux of neutrinos.In such environments light-mass nuclei as well as heavy nuclei are expected to play

important roles in the production of seed nuclei and r-process elements.Thus,we have extended our fully implicit nuclear reaction network so that it includes all nuclei up to the neutron drip line for Z ≤10,in addition to a larger network for Z ≥10.In the present nucleosynthesis study,we utilize a wind model of massive SNeII explosions to study the e?ects of this extended network.We

?nd that a new nuclear-reaction ?ow path opens in the very light neutron-rich region.This new nuclear reaction ?ow can change the ?nal heavy-element

abundances by as much as an order of magnitude.

1.Introduction

The r-process is responsible for roughly half of the abundance of nuclei heavier than iron.However,the astrophysical site for this nucleosynthesis process is still a mystery which remains as a major focus of nuclear astrophysics.

Recent detections of the r-process elements in several metal-de?cient halo stars(Sneden et al.1996,1998,2000)have indicated that the observed abundance pattern of heavy elements is very similar to that of the solar r-process abundance(K¨a ppeler et al.1989, Arlandini et al.1999)for the mass region120≤A.This?nding suggests that the r-process occurs in a speci?c environment such that the abundance pattern is completely independent of the metallicity of the progenitor stars.It is generally believed that the r-process occurs under explosive conditions at high neutron density and high temperature.It has been discussed,for sometime,that core-collapse supernovae(type II or type Ib)could provide the most likely environment for such r-process nucleosynthesis.In a supernova explosion, it is now commonly accepted that massive Fe cores do not readily explode in a purely hydrodynamical way,but that they require help from neutrino heating[the so-called delayed explosion(Bethe&Wilson1985)].The r-process occurs in the region between the surface of the newly-formed neutron star and the outward moving shock wave(Meyer et al.1992).In this region,the entropy is so high that the NSE favors abundant free neutrons and alphas rather than heavy nuclei.This is an ideal site in that it naturally satis?es the observed metallicity independence of the r-process yields.

Woosley et al.(1994)have performed an r-process simulation based on a delayed explosion model,from which an excellent?t to the solar r-process abundance pattern was obtained.However,the required high entropy in their supernova simulation has not been duplicated by other numerical calculations(Witti et al.1994,Takahashi et al.1994). Furthermore all those calculations used a limited network for light nuclei and they did not consider neutrino interactions.Since neutrinos can completely dominate the environment just outside a newly born neutron star,their e?ects must be included in nucleosynthesis calculations.Neutrino-nucleus interaction processes during the r-process have been considered by several authors(Meyer et al.1992,Meyer1995,Fuller&Meyer1995, Qian et al.1997,McLaughlin et al.1996).These studies have shown that,among other things,neutrino processes tend to hinder the r-process by decreasing the neutron-to-seed abundance ratio although they can help to smooth the?nal abundance pattern.

These results may restrict the supernova explosion model.However,Cardall&Fuller (1997),Qian&Woosley(1996),Otsuki et al.(1999),and Sumiyoshi et al.(2000)have shown that a short dynamic-time-scale model plus general relativistic e?ects can lead to a successful r-process.This is because the temperature and density decrease very fast.

Therefore,charged-particle reactions to make seed nuclei do not proceed e?ciently,and only a small amount of seed nuclei are produced.Thus,the neutron-to-seed abundance ratio becomes large enough for heavy r-process nuclei to be synthesized.

The seed nuclei in the neutrino-driven wind are produced early in the expansion by alpha-capture processes.When the temperature and density become low and charged-particle reactions almost cease,the r-process starts from these seed nuclei.This is believed to be a general scenario for r-process nucleosynthesis.Thus,in most previous studies of the r-process,interest was paid mainly to heavy nuclei.Although a few thousand heavy nuclei were included in the nuclear reaction network,only a limited number of light-mass nuclei were selected near theβ-stability line.However,the success of wind-models with a short dynamic time scale requires that attention be given to the reactions of light neutron-rich nuclei in a very neutron rich environment.Light-mass nuclei as well as heavy nuclei are expected to play important roles in the production of seed nuclei and r-process elements. Indeed,it has been noted(Cameron2001)that alternate reaction?ow paths involving neutron-rich light nuclei may be important for r-process nucleosynthesis.

In order to study quantitatively the role of light neutron-rich nuclei,we have therefore extended the nuclear reaction network.We have added about40unstable nuclei for Z≤10 to a larger network for Z≥10.We?nd that a new nuclear reaction paths open in the very light neutron-rich region.We also?nd that these new nuclear reaction paths can change the heavy element abundances by as much as an order of magnitude,while still keeping the prominent three peaks of the r-process elements as well as the hill of the rare-earth elements.In the present nucleosynthesis study,we have analyzed the e?ect of this expanded network in the framework of a numerical simulation of the neutrino-driven wind.

2.Reaction Network

The nuclear reaction network used in Meyer et al.(1992)and Woosley et al.(1994)is probably adequate for simulating the nucleosynthesis of intermediate-to-heavy mass nuclei. However,in the mass region Z≤10,this network is limited to only a few neutron-rich unstable nuclei in addition to the stable ones(see Table1).Charged-particle reactions, which assembleα-particles into heavier nuclei(i.e.α-process),are fast at high temperature 2≤T9in the early stage of the expanding neutrino-driven winds.Therefore,the following reactions(and their inverse)linking the light elements up to20Ne were identi?ed to be most important:α(αn,γ)9Be(α,n)12C,α(αα,γ)12C,12C(n,γ)13C,12C(α,γ)16O,13C(α,n)16O, 16O(n,γ)17O,16O(α,γ)20Ne,17O(α,n)20Ne.On the other hand,the onset of the r-process is thought to be delayed until the temperature drops to below T9≈2.By this time many

seed nuclei in the range of70≤A≤120have been produced by charged-particle reactions. This is the reason why the light-mass neutron-rich nuclei were presumed to be unimportant in theα-process as well as the r-process.However,as we will discuss later in more detail, light nuclei can be important in the extremely neutron-rich environment of neutrino-driven winds(Otsuki et al.2000,Sumiyoshi et al.2000,Kajino et al.2001)where they play a signi?cant roles in the production of both seed and r-process nuclei.

We have therefore extended and improved the reaction network so that it covers all radioactive nuclei up to the neutron-drip line for Z<10,as shown in Figure1.Although information is limited on the rates for(2n,γ)reactions,we did consider nuclei which are unbound after an(n,γ)reaction,i.e.6He,8He,11Li,14Be,17,19B,22C,etc.Our extended network thus includes more than63nuclides for Z<10and more than200reactions among them,while the network used in Woosley et al.(1994)includes only27nuclides, most of which are stable.We included all charged-particle reactions for A≤28,in order to study both theα-process and the neutron-capture?ow,as well as their competition in the production of seed nuclei.We take the rates of charged-particle reactions from those of Kajino et al.(1990ab),Orito et al.(1997),and the NACRE compilation(Angulo et al. 1999).Theβ-decay lifetimes are from Tachibana et al.(1990,1995).We also added many heavier,neutron-rich nuclei for Z>10fromβ-stability to the neutron-drip line in addition to our extended network code.The total number of nuclides up to Americium isotopes is 3036.We refer to this hereafter as the”full network”.We believe we have included all possible relevant reactions in this network.

We also used another smaller network which is similar to the ones used in Meyer et al.(1992)and Woosley et al.(1994).This network includes only a few light neutron-rich nuclei.We shall refer to this as the”smallα-network”.Details on the di?erence between the”full network”and the”smallα-network”are shown in Table1and Figure1.

Except for the above modi?cations and extension of our network,the calculation is essentially the same as that of Meyer et al.(1992)and Woosley et al.(1994).Our”full network”includes(α,n)reactions and their inverse up to Z=36.Neutron captures,their inverse reactions,andβ-decays are included for all isotopes.Rates for these reactions are taken from Caughlan&Fowler(1988),Woosley et al.(1978),and OAP-422(Woosley

et al.1975).Neutron capture rates for the heavier nuclei were taken from experiment where known,and otherwise are from Holmes et al.(1976)and Woosley et al.(1978). Theβ-decay rates were taken from Klapdor et al.(1984).We includeβ-delayed neutron emission of up to three neutrons(Thielemann et al.1983).We use the nuclear-mass table from Hilf et al.(1976).

As for neutrino interactions,we include electron-type neutrino capture

(νe+(Z,A)→(Z+1,A)+e?)for all nuclei(Qian et al.1997),and free neutrons (νe+n→p+e?),and electron-type antineutrino capture(ˉνe+p→n+e+)for free protons.These latter two neutrino interactions predominantly control the electron fraction, Y e,during r-process nucleosynthesis.Neutron emission after neutrino-induced excitations can occur.For very neutron-rich nuclei,up to several neutrons can be emitted.We also included these processes following the method of Meyer et al.(1998).

It is noteworthy that the previous r-process calculations of Meyer et al.(1992),and Woosley et al.(1994)had the additional complexity that the seed abundance distribution was?rst calculated by using a smaller network for light-to-intermediate mass elements,and then the result was connected to another r-process network in a di?erent set of calculations. This separation was imposed because it was thought to be numerically more e?cient to run theα-process and the r-process separately.However it was perhaps more di?cult to interpret the whole nucleosynthesis process.Our nucleosynthesis calculation is completely free from this complexity.We have exploited a fully implicit single network code which is applied to the whole sequence from NSE to theα-process to the r-process.

3.Neutrino-Driven Wind Model

3.1.Hydrodynamic Simulation

Our present purpose is to illustrate the di?erences between calculations in our extended network and those of the generally employed smaller network.For purposes of this illustration the details of the wind model employed are not particularly important.We choose a model,however,which is both derived from a”realistic“hydrodynamic simulation and one which exempli?es the possible e?ects.

As a model for the expanding material,we employ results from the numerical simulation of the neutrino-driven winds of Sumiyoshi et al.(2000).After the supernova core bounce, the proto-neutron star emits an intense?ux of neutrinos during a Kelvin-Helmholtz cooling phase.Some of those neutrinos heat the surface material of the proto-neutron star.The surface is gradually ejected from the neutron star,forming a neutrino-driven wind.Qian and Woosley(1996)and Otsuki et al.(2000)have studied such winds above the proto-neutron star by solving the steady-state hydrodynamical equations including neutrino heating and cooling.Otsuki et al.(2000)have included a general-relativistic treatment.They obtained the time evolution of the ejected material for a series of di?erent neutrino luminosities and proto-neutron-star models.They deduced that the wind models with a short dynamic time scale lead to successful r-process nucleosynthesis even for an entropy of S/k B～140.

This is less than that required by Woosley et al.(1994).Sumiyoshi et al.(2000)have con?rmed this?nding in their fully general-relativistic hydrodynamical simulations without the approximation of steady-state?ow for the neutrino-driven wind.

The adopted wind model in the present illustration will consist of a single trajectory which produces signi?cant heavy-element abundances.It thus,has a very short expansion time scale,τdyn=5.1×10?3s,because of the intense neutrino?ux assumed and general relativistic e?ects.The average energy of electron-type neutrinos is set equal to10MeV. For electron-type antineutrinos it is20MeV,and forμ-andτ-neutrinos and anti-neutrinos it is30MeV.This is the same as has been adopted in previous simulations(Qian and Woosley1996;Otsuki et al.2000).The total neutrino luminosity is taken to be6×1052 erg s?1.

Regarding our adopted time scale,it has been proposed,e.g.Meyer&Brown(1997) that for a su?ciently fast time scale in the wind,the neutrons and protons may not completely reassemble to form alpha particles.The r-process might then be facilitated by proton captures instead of beta decay.We note that the time scale considered here is still su?ciently slow that no signi?cant proton abundance contributes to the r-process.A study of this e?ect would require a time scale at least of order5times faster than the one adopted here.Such a study would also require the implementation of many proton capture reactions for intermediate-mass nuclei which is beyond the scope of the present network calculation.

3.2.Neutron-Star Mass

Nucleosynthesis in the r-process is strongly dependent on the gravitational mass of the proto-neutron star(Wanajo et al.2001).Therefore,one can think of the neutron star mass as a parameter to be adjusted to give good r-process yields.A short expansion time is required to obtain a large neutron-to-seed ratio at moderate entropy.In our trajectory this expansion time is obtained by adopting a large neutron-star gravitational mass

(M=2.0M⊙)and a neutron-star radius of10km.Although,this mass is large compared with the”standard”1.4M⊙model,an ideal condition for successful r-process could also have been obtained with relatively rapid expansion time scale being preserved,for example, by altering the outer boundary conditions in the hydrodynamic model.Hence,one should not be too dismayed at this large neutron-star mass.

Furthermore,although a neutron-star mass of2.0M⊙is large,there are established equations of state(Shen et al.1998,Weber1999,Sumiyoshi et al.1995)which can stabilize neutron stars having masses up to M≤2.2M⊙.This is also still consistent with observed

constraints on the maximum neutron star mass ranging0.5M⊙～

Incidentally,a large dispersion in the heavy element abundances of halo stars has been observed(McWilliams et al.1995,Ryan et al.1996).Using an inhomogeneous galactic chemical-evolution model,Ishimaru and Wanajo(1999)have shown that this observed dispersion could be a natural consequence of r-process nucleosynthesis in supernovae of massive M≥30M⊙progenitors.Such progenitors could conceivably have large core masses.

3.3.r-Process Initial Conditions

We start the r-process network calculation at a time when the temperature has dropped to T9=9.0.We display the time variation of the temperature(thin solid curve)and mass density(dashed curve)in the top panel of Figure2.Time t=0s refers to the time at which T9=9.0.From this point the temperature drops very rapidly and then becomes almost constant at around T9～0.62.The initial composition of the material is taken to be free neutrons and protons with an electron fraction of Y e(=Y p)=0.42.This was taken from the hydrodynamical simulation of Sumiyoshi et al.(2000).

4.Results

Once equilibrium between(n,γ)and(γ,n)reactions is obtained,the neutron-capture ?ow path runs through nuclei with almost the same S n-value along the nuclear chart.The r-process path strongly depends on what S n-value is favored by the neutrino-driven winds. The optimal single-neutron separation energies,S n,assuming(n,γ)equilibrium,are given by

S n=

is2～4MeV in the literature.However,in the present wind-model analysis,the calculated S n-value is～1MeV(Figure2).This is because the expansion model has a short dynamic time scale and the material in the neutrino-driven wind is very neutron rich.

Since T9(t),ρ(t)and Y n(t)depend on time,S n also varies with time.S n?rst decreases rapidly due to the expansion of the wind while theα-process operates at high temperatures 2～

We show the calculated seed abundance,Y S,and the neutron-to-seed abundance ratio, Y n/Y S,as a function of time in the lower panel of Figure2.Y S is de?ned as the sum

of the number abundance fractions of intermediate-to-heavy mass elements Y S=ΣY A (70≤A≤120).The solid and dashed curves respectively display the results calculated in the”full network”and the”smallα-network”codes.We also show in Figure3the calculated?nal abundance yields of the r-process elements for this trajectory.Also shown for comparison are the relative solar r-process abundances from K¨a ppeler et al.(1989).

When we use the”full network”code,this particular trajectory happens to more or less reproduces the r-process abundance peaks near A～80,130and195.In the case of the”smallα-network”code,light-mass elements with A～<150are underabundant.The purpose of this illustration,however,is not to argue that this is a better model for the

r-process.Indeed,in most models the challenge has been to provide enough neutrons per seed.Here we see that the?ow to heavier nuclei is considerably diminished in the expanded network.Thus,for most r-process models,this expanded reaction network,as necessary as it may be,may actually make a bad situation worse.

This contrast between the two calculated results can be traced to drastic changes in the seed production.As shown in the lower panel of Figure2,Y S is continuously supplied at10ms～

5.Reaction Flows

Having identi?ed that the production of seed material is quite a bit di?erent in the two network calculations it is important to now analyze the critical reaction?ows in detail as a?uid element expands through the wind.For this analysis we consider two times during the evolution.One corresponding to theαprocess conditions early in the wind,and one corresponding to the later r-process conditions.These two selected times are indicated by arrows and dots on the top panel of Figure2.

Figure4shows the nuclear reaction?ow at t=3.3×10?3s.?From the lower panel on Figure2one can see that the seed abundance Y S is just starting to form at this time in the wind.This is the start ofα-process.This is to be compared with Figure5which shows the?ow at t=0.567s corresponding to near the end of the r-process as identi?ed on the bottom panel of Figure2.

In Figures4and5,the relative abundances for Z≤15in the N-Z plane are shown by circles whose diameters are proportional to the logarithm of the abundance yields

Y A=X A/A as indicated.Small dots denote the network range adapted in the present study:The”full network”is used in the calculated results shown in the upper panels(a) of Figures4and5,and the”smallα-network”in the lower panels(b).The main reaction paths are indicated by arrows.For further clari?cation,the critical reaction?ows to produce carbon isotopes are shown in Figure6.Once formed,these carbon isotopes quickly convert to heavier seed nuclei.Figure7shows the relative abundances of neutrons Y n,protons Y p and alpha particles Yα.

5.1.α-Process

?From Figure2we see that the start of theα-process conditions of Figure4(at t

=3.3×10?3s)corresponds to T9=3.4,andρ=8.0×104g cm?3.At this point the nuclear statistical equilibrium is just shifting to produce a large abundance ofαparticles, as evidenced on Figure7.At early times up to nearly10?2s,the?ow through the large and small networks are quite similar.In both Figures4(a)and4(b)that there are abundant nuclei around theβ-stability line even in our”full network”calculations and the main path resides inside the network range of the”smallα-network”for Z≤15(see Figure4(b)).

Forα-process conditions the main reaction?ow is triggered by the upper path in Figure6,i.e.theα(αn,γ)9Be(α,n)12C orα(αα,γ)12C reaction.The importance of this path was pointed out by Woosley and Ho?man(1992)and Woosley et al.(1994).Although the side?ows ofα(3H,γ)7Li(n,γ)8Li(α,n)11B and7Li(α,γ)11B also involve appreciable

nuclear reaction?ow,the subsequent11B(p,α)8Be reaction returns this?ow back into

α-particles as indicates by the dashed arrows in Figure6.Thus,although there are plenty of protons as well as neutrons andα-particles present at this time(cf.Figure7),three-body and alpha-capture reactions of stable nuclei are more e?cient than neutron captures at this relatively high temperature T9=3.4.This is generally the case in the early epoch of the trajectory because both the temperature and density are still high enough for these reactions to occur.Therefore,we?nd very similar reaction paths in the two di?erent network ranges shown in Figures4(a)and4(b).A few new paths are evident in the full network,e.g.9Be(n,γ)10Be(α,γ)14C,and14C(n,γ)15C(α,n)18O.These,however,make very little di?erence.

We have included the possible three body two-neutron capture reactions(Efros et al.1996)such as4He(2n,γ)6He and6He(2n,γ)8He,and so on as shown in Figure1.

No signi?cant?ow was observed through this channel either at this time or later in

the evolution.However,some two-neutron channels may important in heavier nuclei as described below.

5.2.r-Process

Once seed material has begun to assemble by t≥0.01s(cf.Figure2)some production of r-process nuclei begins.As the temperature and density of a?uid element diminish, charged-particle reactions become progressively slower and eventually neutron capture becomes more important.A classical r-process-like?ow,i.e.(n,γ)and(γ,n)reactions followed by beta decay,starts.

Di?erences in the reaction?ow between the two networks become apparent immediately. One can identify two characteristic features of the light element abundances on Figure7. One is that the alpha abundance is almost the same,but is a little lower in the full network calculation(by about1percent).Since alpha particles are the most abundant nucleus,this small di?erence ofδY～10?3has an in?uence on heavy element production.The other is that the neutrons are exhausted earlier in the full network calculation.The?ow at t =0.567s shown in Figures5(a)and5(b)corresponds to just before freezeout at the end of the r-process when the material in the wind has cooled to T9=0.62,and the density decreasedρ=5.4×102g cm?3.

The di?erences between the two network calculations can be traced to the fact that in the full network the main path runs through very neutron-rich nuclei on the N-Z plane(cf. Figure5a).The added?ow paths mean that both neutrons and alphas are more e?ciently

converted to seed,as shown in Figure7.This process is prohibited in the small network(cf. Figure5b).Hence,their abundance along the trajectory is lower.The presence of more seed material means that neutrons are more quickly exhausted in the wind.

In addition to theα-induced reaction chainsα(αn,γ)9Be(α,n)12C,there are

two main?ow paths to form carbon seed present at this time.They are almost equally important.These are the Be-isotope chainα(αn,γ)9Be(n,γ)10Be(α,γ)14C

and the Li-B chainα(t,γ)7Li(n,γ)8Li(α,n)11B(see Figure6).As for the Be-isotope chain,when the10Be(α,γ)14C reaction is switched o?,the reaction?ow changes to

9Be(n,γ)10Be(n,γ)11Be(n,γ)12Be(β?)12B.We have also studied what happens if the

8Li(α,n)11B reaction is switched o?.In this case,neutron capture on8Li leads to9Li which decays back to9Be.It is to be noted,however,that the results obtained by switching o?either new chain are similar to those calculated in the full network.This is because either the Be-isotope chain or the Li-B chain is still very active even if the other chain is turned o?.When both new chains are turned o?,however,the result is almost the same as that calculated in the small network.The presence of the two new chains in the full network is therefore the main di?erence between two networks.

As a?nal remark we point out that we have also studied the time-integrated nuclear-reaction?ows.This identi?es the most important main?ow paths.We carried out numerical calculations in which the thermonuclear reaction rates times number abundances of interacting nuclei were integrated from time zero to the freezeout time of the r-process. These quantities give the total intensity of the nuclear reaction?ow passing through each nucleus.We?nd that the main?ow paths are almost the same as those indicated by the arrows in Figure5which displays a snap shot at the time t=0.567s.We thus conclude that the main?ow paths identi?ed in Figure5(a)indicate the signi?cance of the new reaction channels for the production of the?nal r-process abundance yields.

5.3.E?ects of Wind Time Scale and Neutrino Interaction

In a realistic supernova simulation one expects that the expansion time in the wind will di?er as the proto neutron star cools and the bubble expands.To identify the conditions at which the expanded full network is important we have run simulations in the full and small network for di?erent expansion time scales.The?nal abundances calculated by using the full network(solid curve)and the smallα-network(dashed curve)are summarized in Figure 8.The di?erent time scales ofτdyn=5.1,53,and100ms correspond to di?erent trajectories obtained in the hydrodynamic supernova model of Sumiyoshi et al(2000).Here we see that there is little di?erence between the two networks when the expansion time scale is

slow.For both networks,the seed build up is too e?cient and there is little production of elements for A>130.

We have also studied the e?ects of neutrino processes(Meyer et al.1992,Meyer 1995,Fuller&Meyer1995,Qian et al.1997,McLaughlin et al.1996)in our full network calculations.Among three wind models which have di?erent time scales(τdyn=5.1,53, and100ms),only the fastest wind withτdyn=5.1ms leads to successful r-process nucleosynthesis.The speci?c neutrino-nucleon collision time scale is given(Qian et al. 1997)by

τν～50×L?1ν,51 Eν 50km 2 10?41cm2

neutron-rich nucleus at Z=10,however in this mass formula it is).17B is known to be stable against particle decay,and19B also is suggested to be stable theoretically,while 16,18B are unstable.29F and some heavier elements beyond the network range adopted in the present studies are predicted to be particle bound in some theoretical nuclear models. Quite recently,the neutron drip line up to?uorine has been studied by the projectile fragmentation experiments and31F has proved to be particle stable,while24,25N,25?28O (except for26O)and30F are unstable(Sakurai et al.1999).Radiative two-neutron capture reactions,i.e.15B(2n,γ)17B(2n,γ)19B,may play a signi?cant role in the r-process.In the present calculations we assumed that the(n,γ)and(γ,n)reactions are in thermal equilibrium for the?rst neutron and then the second neutron is subsequently captured. Although inclusion of the(2n,γ)reactions did not change drastically the?nal result,these rates are presumed to be lower limit.More elaborated theoretical calculation of the(2n,γ) reaction has suggested that the di-neutron correlation may increase these cross sections (Kamimura2001).Photodisintegration reactions of6He and8He and their electromagnetic structure also have been studied experimentally(Aumann et al.1999;Iwata et al.2000). Extensive measurements of the nuclear properties of heavier neutron-rich nuclei near the drip line,including17,19B and29,31F,are yet to be carried out.

At T9=0.62andρ=5.4×102g cm?3in Figure5(a),most of nuclei on the neutron-drip line satisfy Eq.(1)approximately.However,for carbon and magnesium isotopes,two abundant nuclei18C and36Mg are not on the neutron-drip line.The neutron separation energy of19C is smaller than the value of Eq.(1)for sustaining the steady-state?ow at this temperature and neutron density.As a result,even at this low temperature,T9=0.62, the18C(α,n)21O reaction becomes faster than18C(n,γ)19C.Likewise,36Mg(α,n)39Al is faster than36Mg(n,γ)37Mg.For this reason,the(n,γ)A(e?ν)r-process?ow is broken before reaching the neutron-drip line.This approximately satis?es the condition represented by Eq.(1)at this time t=0.567s.Precise experimental studies of the reaction rates for the two competing processes18C(α,n)21O vs.18C(n,γ)19C and36Mg(α,n)29Si vs.36Mg(n,γ)37O would be highly desirable.

Recent progress at radioactive nuclear beam facilities has provided a remarkable opportunity to study the nuclear structure and reactions of extremely neutron-rich radioactive isotopes.The neutron separation energy of19C along with its electric dipole distribution has been measured by using the Coulomb dissociation method(Nakamura

et al.1999).A precise determination of this quantity is important because18C is at a branching point between the(n,γ)19C and(α,n)21O reactions as discussed above.The same technique was applied to?nd a large deformation of32Mg(Motobayashi et al.1995).At t =3.3×10?3s(in Figure4(a)),the main nuclear reaction?ow still stays near the stability line and does not reach32Mg.However,by the time t=0.567s(in Figure5(a)),neutron

captures by magnesium isotopes form a side?ow together with the strongest?ow along the sodium isotopes.(See the abundance of magnesium isotopes).At some intermediate time between3.3×10?3s and0.567s,this side?ow plays an important role.Therefore,a phase transition to large deformation at32Mg may have a non-negligible e?ect on the production of seed abundances.

A new magic number N=16has recently been found near the neutron drip line (Ozawa et al.2000),which may also a?ect strongly the seed abundance distribution because 23N,24O,and25F are some of the most abundant nuclei on the main?ow path in Figure 5(a).The17O(n,α)14C reaction cross section has been precisely measured from thermal to about350keV neutron energies(Wagemans et al.2001).This study indicates a cross section that is lower by a factor of2～3than that of previous measurements.All of these new experimental studies,as well as future work,will be important in order to clarify the role of light neutron-rich nuclei in the high-entropy r-process.

Before closing this section,we should emphasize that our?ndings are based on the r-process nucleosynthesis models in neutrino-driven winds with a short dynamic time scale. For example,if we calculate the r-process abundance patterns using the trajectories with the relatively long dynamic time scale adapted by Woosley et al.(1994),the?nal patterns are almost the same for both the full and small networks.We can understand the reason for this by comparing the dynamic time scale,τdyn,with the time scale for the alpha-capture process,τα.As the value ofτdyn becomes larger thanτα,there is enough time to make seed nuclei by the alpha-process.Thus,seed nuclei are mainly produced byα-captures, and the reactions of light neutron-rich nuclei are unimportant.In the alpha-process,the α(αn,γ)9Be(α,n)12C reaction is the key to make heavy nuclei.Therefore,the value ofταis regulated by theα(αn,γ)9Be reaction and is given(Meyer et al.1992)by

τα≡

the reactions of light neutron-rich nuclei in the r-process,utilizing a wind model of massive SNeII explosions.We?nd that a new path for the r-process opens in these neutron-rich nuclei from the”full network”calculations in models with a short dynamic time scale.In our model calculation,the third r-process peak at A～195is lower,while nuclei around A ～50are more abundant than in the”smallα-network”calculations.This is because the available neutrons to make heavy nuclei are diminished by neutron captures on the light seed nuclei from theβ-stability line to the neutron-drip line.Note that these results are restricted to models with a short dynamic time scale in which the temperature drops rapidly and the alpha-process is relatively unimportant.This result shows that light neutron-rich nuclei could be important in other very neutron-rich situations as well,such as a prompt explosion(Sumiyoshi et al.2001)or binary-neutron-star mergers.

Thus,we?nd that light neutron-rich nuclei can play an important role in r-process nucleosynthesis in models with a short dynamic time scale.The yields of even the most neutron-rich isotopes can be abundant along the r-process path.

One of the authors(MT)wishes to acknowledge the fellowship of RIKEN Junior Research Associate.One of the authors(GJM)also wishes to acknowledge the hospitality of the National Astronomical Observatory of Japan where much of this work was done.This work has been supported in part by the Grant-in-Aid for Scienti?c Research(10640236, 10044103,11127220,12047233)of the Ministry of Education,Science,Sports,and Culture of Japan,and also in part by DoE Nuclear Theory Grant(DE-FG02-95-ER40394at UND). The numerical simulations have been performed on the supercomputers at RIKEN.

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Z A max(smallα?network)

He4

Be9

815

C14

1323

O18

1827

1928

Na35

2338

Al41

2742

··

241279

Table1:Comparison of the”smallα-network”and the”full network”for light-mass nuclear systems.The nuclides(A min≤A≤A max)included in the networks are di?erent from each other only for Z<10.

n 1H 2H

3H 3He

4He 6He 8He 6Li 7Li 8Li 9Li 11

Li 7Be

9Be 10Be 11Be 12Be 14Be 8B 10B 11B 12B 13B 14B 15B 17B 19

B 11

C 12C 13C

14C 15C 16C 17C 18C 19C 20C 22C 13N 14N

15N 16N 17N 18N 19N 20N 21N 22N 23N 15O

16O 17O 18O 19O 20O 21O 22O 23O 24O 18F 19F 20F 21F 22F 23F 24F 25F 27F

29F 19Ne 20Ne 21Ne

22Ne 23Ne 24Ne 25Ne 26Ne 27Ne 28Ne 29Ne 30Ne 22Na

23Na 24Na 25Na 26Na 27Na 28Na 29Na 30Na 31Na 32Na 33Na 34Na 35Na 23Mg

24Mg 25Mg 26Mg 27Mg 28Mg 29Mg 30Mg 31Mg 32Mg 33Mg 34Mg 35Mg 36Mg 26Al 27Al 28Al 29Al 30Al 31Al 32Al 33Al 34Al 35Al 36Al 37

Al 27Si 28Si

29Si 30Si 31Si 32Si 33Si 34Si 35Si 36Si 37Si 38Si F 2631Ne Ne 3231F O 26Fig.1.—Light-mass region of the extended nuclear reaction network (”full network”)used for the present r-process nucleosynthesis calculations.

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院线新片网络防盗有监测426so平台为中国知识产权护航

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有限元网格划分心得

有限元网格划分的基本原则划分网格是建立有限元模型的一个重要环节，它要求考虑的问题较多，需要的工作量较大，所划分的网格形式对计算精度和计算规模将产生直接影响。为建立正确、合理的有限元模型，这里介绍划分网格时应考虑的一些基本原则。 1网格数量网格数量的多少将影响计算结果的精度和计算规模的大小。一般来讲，网格数量增加，计算精度会有所提高，但同时计算规模也会增加，所以在确定网格数量时应权衡两个因数综合考虑。图1中的曲线1表示结构中的位移随网格数量收敛的一般曲线，曲线2代表计算时间随网格数量的变化。可以看出，网格较少时增加网格数量可以使计算精度明显提高，而计算时间不会有大的增加。当网格数量增加到一定程度后，再继续增加网格时精度提高甚微，而计算时间却有大幅度增加。所以应注意增加网格的经济性。实际应用时可以比较两种网格划分的计算结果，如果两次计算结果相差较大，可以继续增加网格，相反则停止计算。图1位移精度和计算时间随网格数量的变化在决定网格数量时应考虑分析数据的类型。在静力分析时，如果仅仅是计算结构的变形，网格数量可以少一些。如果需要计算应力，则在精度要求相同的情况下应取相对较多的网格。同样在响应计算中，计算应力响应所取的网格数应比计算位移响应多。在计算结构固有动力特性时，若仅仅是计算少数低阶模态，可以选择较少的网格，如果计算的模态阶次较高，则应选择较多的网格。在热分析中，结构内部的温度梯度不大，不需要大量的内部单元，这时可划分较少的网格。 2网格疏密网格疏密是指在结构不同部位采用大小不同的网格，这是为了适应计算数据的分布特点。在计算数据变化梯度较大的部位(如应力集中处)，为了较好地反映数据变化规律，需要采用比较密集的网格。而在计算数据变化梯度较小的部位，为减小模型规模，则应划分相对稀疏的网格。这样，整个结构便表现出疏密不同的网格划分形式。图2是中心带圆孔方板的四分之一模型，其网格反映了疏密不同的划分原则。小圆孔附近存在应力集中，采用了比较密的网格。板的四周应力梯度较小，网格分得较稀。其中图b中网格疏密相差更大，它比图a中的网格少48个，但计算出的孔缘最大应力相差1%，而计算时间却减小了36%。由此可见，采用疏密不同的网格划分，既可以保持相当的计算精度，又可使网格数量减小。因此，网格数量应增加到结构的关键部位，在次要部位增加网格是不必要的，也是不经济的。

《小时代4》经典台词

《小时代4》经典台词《小时代4：灵魂尽头》改编自郭敬明小说《小时代》，由郭敬明担任影片的编剧及导演，李力制片，杨幂、郭采洁、陈学冬、郭碧婷、谢依霖主演。该片是《小时代》系列电影的第四部，影片围绕林萧、顾里、南湘、唐宛如四姐妹展开，讲述了顾里癌症、顾源坐牢、姐妹反目及这一群人的友谊方向是如何发展的的各种故事。> 《小时代4》台词欣赏> 血肉横飞只是开始而已。魂飞魄散才是真正的好戏。当然，我们都知道，我们热爱生活中这样刺激有跌宕的drama。> 一个人身边的位置只有那么多,你能给的也只有那么多，在这个狭小的圈子里,有些人要进来,就有一些人不得不离开。> 也许我们最终都会成长为自己曾经最讨厌的模样，但在此之前，面对黑暗无边，让我与你并肩。> 夕阳的光线像是被风吹散一般迅速消失，正如同再也回不去的美好年华。那感觉，像是一个时代最后的剧中。> 生活就是不可抗力，它就是合约里唯一一条、也是永远都会存在的一条无人可以更改的霸王条款。> 旋转着的，五彩缤纷的物质世界。等价交换的，最残酷的也最公平的寒冷人间。> 时间一点一滴地过去，流逝告别。我们慢慢地走向一个被上帝作记号的地点。> 但是生活永远不是连续剧。它不会再应该浪漫的时候，响起煽情的音乐;它不会再男主角深情告白的时候，就让女主角浓烈的回应;它不会再这样需要温柔和甜蜜的时刻，就打翻一杯浓浓的蜂蜜。> 后来，我的梦境总是反复出现这场无声无息的火，无数飞虫朝它飞去，它们仿佛是早就存在于这个世界的记忆碎片，旧时尘埃。> 阴影，都是死神某一个局部的轮廓，当太阳旋转到某一个角度，这些阴影，就会拼成一个完整的高举镰刀的英雄，那是我们就将一起挽手，走向灵魂的尽头。> 我们活在浩瀚的宇宙里，漫天漂浮的宇宙尘埃和星河光尘，我们是比这些还要渺小的存在。你并不知道生活在什么时候突然改变方向，陷入墨水一般浓稠的黑暗里去。你被失望拖进深渊，你被疾病拉进坟墓，你被挫折践踏的体无完肤，你被嘲笑、被讽刺、被讨厌、被怨恨、被放弃。但是我们却总在内心里保留着希望保留着不甘心放弃的心。我们依然在大大的绝望里小小的努力着。这种不想放弃的心情，它们变成无边黑暗的小小星辰。我们都是小小的星辰。> 如果我们的生活充满了以前另一种未知的可能性的话，那么在大学围墙范围内，这一场追逐大战，谁先遇到谁，都可以导致完全不同的结局。这就像有人在转盘里撒下一大把钢珠，在转盘没有停下来之前，谁都不知道最后的赢家会是谁。《小时代》系列经典台词双语版我用青春十年，赴你最后之约。 I bided ten years of my prime for our date with destiny. 这是最好的时代，也是最坏的时代。你会忘记这个时代，但你，会永远记住我们。 This is the best time, but the worst as well. You will eventually forget this day and this age, but you will always remember us.

_基于ANSYS的有限元法网格划分浅析

文章编号:1003-0794(2005)01-0038-02 基于ANSYS的有限元法网格划分浅析杨小兰,刘极峰,陈旋 (南京工程学院,南京210013) 摘要:为提高有限元数值的计算精度和对复杂结构力学分析的准确性,针对不同分析类型采用了不同的网格划分方法,结合实例阐述了ANSYS有限元网格划分的方法和技巧,指出了采用ANSYS有限元软件在网格划分时应注意的技术问题。关键词:ANSYS;有限元;网格;计算精度中图号:O241 82;TP391 7文献标识码:A 1 引言 ANSYS有限元分析程序是著名的C AE供应商美国ANSYS公司的产品,主要用于结构、热、流体和电磁四大物理场独立或耦合分析的CAE应用,功能强大,应用广泛,是一个便于学习和使用的优秀有限元分析程序。在ANSYS得到广泛应用的同时,许多技术人员对ANSYS程序的了解和认识还不够系统全面,在工作和研究中存在许多隐患和障碍,尤为突出的是有限元网格划分技术。本文结合工程实例,就如何合理地进行网格划分作一浅析。 2 网格划分对有限元法求解的影响有限元法的基本思想是把复杂的形体拆分为若干个形状简单的单元,利用单元节点变量对单元内部变量进行插值来实现对总体结构的分析,将连续体进行离散化即称网格划分,离散而成的有限元集合将替代原来的弹性连续体,所有的计算分析都将在这个模型上进行。因此,网格划分将关系到有限元分析的规模、速度和精度以及计算的成败。实验表明:随着网格数量的增加,计算精确度逐渐提高,计算时间增加不多;但当网格数量增加到一定程度后,再继续增加网格数量,计算精确度提高甚微,而计算时间却大大增加。在进行网格划分时,应注意网格划分的有效性和合理性。 3 网格划分的有效性和合理性 (1)根据分析数据的类型选择合理的网格划分数量在决定网格数量时应考虑分析数据的类型。在静力分析时,如果仅仅是计算结构的变形,网格数量可以少一些。如果需要计算应力,则在精度要求相同的情况下取相对较多的网格。同样在响应计算中,计算应力响应所取的网格数应比计算位移响应多。在计算结构固有动力特性时,若仅仅是计算少数低阶模态,可以选择较少的网格。如果计算的模态阶次较高,则应选择较多的网格。在热分析中,结构内部的温度梯度不大,不需要大量的内部单元,可划分较少的网格。 (2)根据分析数据的分布特点选择合理的网格疏密度在决定网格疏密度时应考虑计算数据的分布特点,在计算固有特性时,因为固有频率和振型主要取决于结构质量分布和刚度分布,采用均匀网格可使结构刚度矩阵和质量矩阵的元素不致相差很大,可减小数值计算误差。同样,在结构温度场计算中也趋于采用均匀的网格形式。在计算数据变化梯度较大的部位时,为了更好地反映数据变化规律,需要采用比较密集的网格,而在计算数据变化梯度较小的部位,为了减小模型规模,则应划分相对稀疏的网格,这样整个结构就表现出疏密不同的网格划分形式。以齿轮轮齿的有限元分析模型为例,由于分析的目的是求出齿轮啮合传动过程中齿根部分的弯曲应力,因此,分析计算时并不需要对整个齿轮进行计算,可根据圣文男原理将整个区域缩小到直接参与啮合的轮齿。虽然实际上参与啮合的齿数总大于1,但考虑到真正起作用的是单齿,通常只取一个轮齿作为分析对象,这样作可以大大节省计算机内存。考虑到轮齿应力在齿根过渡圆角和靠近齿面处变化较大,网格可划分得密一些。在进行疏密不同网格划分操作时可采用ANSYS提供的网格细化工具调整网格的疏密,也可采用分块建模法设置网格疏密度。图1所示即为采用分块建模法进行网格划分。图1(a)为内燃机中重要运动零件连杆的有限元应力分析图,由于连杆结构对称于其摆动的中间平面,其厚度方向的尺寸远小于长度方向的尺寸,且载荷沿厚度方向近似均匀分布,故可按平面应力分析处 38 煤矿机械 2005年第1期

CATIA有限元高级划分网格教程

CATIA有限元高级网格划分教程盛选禹李明志 1.1进入高级网格划分工作台（1）打开例题中的文件Sample01.CATPart。（2）点击主菜单中的【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分工具），就进入【Advanced Meshing Tools】（高级网格划分工具）工作台，如图1－1所示。进入工作台后，生成一个新的分析文件，并且显示一个【New Analysis Case】（新分析算题）对话框，如图1－2所示。图1－1【开始】→【分析与模拟】→【Advanced Meshing Tools】(高级网格划分工具）（3）在【New Analysis Case】（新分析算题）对话框内选择【Static Analysis】（静力分析）选项。如果以后打开该对话框的时候均希望是计算静力分析，可以把对话框内的【Keep as default starting analysis case】（在开始时保持为默认选项）勾选。这样，下次进入本工作台时，将自动选择静力分析。（4）点击【新分析算题】对话框内的【确定】按钮，关闭对话框。 1.2定义曲面网格划分参数本节说明如何定义一个曲面零件的网格类型和全局参数。（1）点击【Meshing Method】(网格划分方法)工具栏内的【高级曲面划分】按钮

，如图1－3所示。需要在【Meshing Method】(网格划分方法)工具栏内点击中间按钮的下拉箭头才能够显示出【高级曲面划分】按钮。图1－2【New Analysis Case】（新分析算题）对话框图1－3【高级曲面划分】按钮

2018-2019-咱们结婚吧经典台词-经典语录-对白-句子-语句-片段-桥段-word范文模板 (3页)

本文部分内容来自网络整理，本司不为其真实性负责，如有异议或侵权请及时联系，本司将立即删除！ == 本文为word格式，下载后可方便编辑和修改！ == 咱们结婚吧经典台词|经典语录|对白|句子|语句|片段|桥段《咱们结婚吧》讲述了四对即将举行婚礼的情侣距离婚前结婚一周时间，因为婚前恐惧发生的各种冲突、逆转的故事。那么你知道咱们结婚吧经典台词有哪些吗?下面是咱们结婚吧经典台词，欢迎查阅。《咱们结婚吧》经典台词 1、人生就像减法，吃一顿少一顿，见一面少一面，珍惜你现在拥有的，错过了，就没了!眼前的一切就是最好的安排。 2、“我怕有别的男人，牵起你的手，为你带上婚戒。” 3、知道什么是优胜劣汰么?就是优质的女人剩下了，劣质的女人成为了别人的太太! 4、眼前的安排就是最好的，命运会给人安排个最对的人。 5、“这份感情，会变成在我心中，最珍贵的记忆永存在心底。” 6、”一天当中，最美好的时刻，就是和喜欢的人在一起。” 7、女人一生之中，最美的时刻，就是穿上婚纱那一刻。 8、我怕我一辈子都见不到你。 9、就算有一百匹马，勒住我的脖子，我也会挣脱开，回来陪你去实现梦想。 10、如果你愿意，咱们结婚吧。 11、“我认识你两年的时间，我已暗恋了你七百三十天，我今天站在你面前，我是鼓了一百万次勇气才做到的。” 12、我想要有依靠想要有一个家有错吗? 13、“你是不是从来都没有想过要跟我结婚啊?”

14、婚姻是爱情的坟墓，不结婚，就死无葬身之地。 15、真正有感觉的、能携手终身的人，她不会从天上掉下来的，你不能站在原地等待，你要主动地往前走，主动去寻找。是，这个过程当中，你一定会遇到一些让自己伤心、难过、不开心的事情，你不用太在意，那些都是一个必经的过程。走过它，你就会看见那个你真正有感觉的那个人。 16、“你长这么漂亮，肯定有好多人追你。” “谁追我啊，时间在追我。” 17、小子，为了你不再祸害人间乱搞男女关系，姐姐委屈点收了你吧，从此记得鞍前马后随叫随到，不要挣扎了。 18、小子，你就从了大娘吧!把你的银行卡，信用卡，医疗保险，所有的卡，以及密码统统告诉我，让我帮你好好保管，包括你的人! 19、在这个世界上，任何事物都值得多看两眼。而眼前发生的一切，都是最好的安排。 20、我可能不是最般配你的那个人，但我绝对是最爱你的那个人! 21、你别把自己当空气，你没那么重要，我是离不开空气，但是我离开的了你。 22、北京的九月最美，因为九月的白天像八月，晚上像十月，就像我们三十岁的女人既有二十岁女人的脸蛋，也有四十岁女人的智慧，这是女人一生之中最美的时刻。 23、人的一生啊，总要碰上一些老天爷给的不公平的待遇，那么我们怎么办呢?你不满、你抗议?你都不行的，没有用啊?既然已成事实了，咱们就勇敢的去面对它!过好我们的每一天。把我们这辈子的这日子过好它，你说好不好啊? 24、我知道当别人十分努力的时候，我只有二十分三十分的努力，才有可能和那些十分努力的人一样。 25、我认识你两年的时间，我也暗恋了你七百三十天，我今天站在你面前，我是鼓了一百万次勇气才做到的。 26、女人一生之中最美的时刻，就是穿上婚纱的那一刻。 27、你不能站在原地去等待，你要往前走去寻找，是的，在这个过程中会伤心，难过，这些都是必须经历的，走过它，你才会找到那个人。 28、从我认识你这天起，我恐婚的毛病逐渐治愈了。我现在得了一种不知道什么病，我睁眼闭眼都是你，我现在病入膏肓，濒临死亡。你就是我的解药，我请求你做我的妻子!

有限元网格划分

有限元网格划分摘要:总结近十年有限元网格划分技术发展状况。首先,研究和分析有限元网格划分的基本原则；其次,对当前典型网格划分方法进行科学地分类,结合实例,系统地分析各种网格划分方法的机理、特点及其适用范围,如映射法、基于栅格法、节点连元法、拓扑分解法、几何分解法和扫描法等；再次,阐述当前网格划分的研究热点,综述六面体网格和曲面网格划分技术；最后,展望有限元网格划分的发展趋势。关键词:有限元网格划分；映射法；节点连元法;拓扑分解法；几何分解法；扫描法；六面体网格 1 引言有限元网格划分是进行有限元数值模拟分析至关重要的一步，它直接影响着后续数值计算分析结果的精确性。网格划分涉及单元的形状及其拓扑类型、单元类型、网格生成器的选择、网格的密度、单元的编号以及几何体素。在有限元数值求解中，单元的等效节点力、刚度矩阵、质量矩阵等均用数值积分生成，连续体单元以及壳、板、梁单元的面内均采用高斯（Gauss）积分，而壳、板、梁单元的厚度方向采用辛普生（Simpson）积分。 2 有限元网格划分的基本原则有限元方法的基本思想是将结构离散化,即对连续体进行离散化,利用简化几何单元来近似逼近连续体,然后根据变形协调条件综合求解。所以有限元网格的划分一方面要考虑对各物体几何形状的准确描述,另一方面也要考虑变形梯度的准确描述。为正确、合理地建立有限元模型,这里介绍划分网格时应考虑的一些基本原则。 2.1 网格数量网格数量直接影响计算精度和计算时耗,网格数量增加会提高计

算精度,但同时计算时耗也会增加。当网格数量较少时增加网格，计算精度可明显提高,但计算时耗不会有明显增加;当网格数量增加到一定程度后,再继续增加网格时精度提高就很小,而计算时耗却大幅度增加。所以在确定网格数量时应权衡这两个因素综合考虑。 2.2 网格密度为了适应应力等计算数据的分布特点,在结构不同部位需要采用大小不同的网格。在孔的附近有集中应力,因此网格需要加密;周边应力梯度相对较小,网格划分较稀。由此反映了疏密不同的网格划分原则:在计算数据变化梯度较大的部位,为了较好地反映数据变化规律,需要采用比较密集的网格;而在计算数据变化梯度较小的部位,为减小模型规模,网格则应相对稀疏。 2.3 单元阶次单元阶次与有限元的计算精度有着密切的关联,单元一般具有线性、二次和三次等形式,其中二次和三次形式的单元称为高阶单元。高阶单元的曲线或曲面边界能够更好地逼近结构的曲线和曲面边界,且高次插值函数可更高精度地逼近复杂场函数,所以增加单元阶次可提高计算精度。但增加单元阶次的同时网格的节点数也会随之增加,在网格数量相同的情况下由高阶单元组成的模型规模相对较大,因此在使用时应权衡考虑计算精度和时耗。 2.4 单元形状网格单元形状的好坏对计算精度有着很大的影响,单元形状太差的网格甚至会中止计算。单元形状评价一般有以下几个指标: (1)单元的边长比、面积比或体积比以正三角形、正四面体、正六面体为参考基准。 (2)扭曲度:单元面内的扭转和面外的翘曲程度。 (3)节点编号:节点编号对于求解过程中总刚矩阵的带宽和波前因数有较大的影响,从而影响计算时耗和存储容量的大小 2.5 单元协调性单元协调是指单元上的力和力矩能够通过节点传递给相邻单元。为保证单元协调,必须满足的条件是: (1)一个单元的节点必须同时也是相邻点,而不应是内点或边界

有限元网格划分和收敛性

一、基本有限元网格概念 1．单元概述?几何体划分网格之前需要确定单元类型.单元类型的选择应该根据分析类型、形状特征、计算数据特点、精度要求和计算的硬件条件等因素综合考虑。为适应特殊的分析对象和边界条件,一些问题需要采用多种单元进行组合建模。? 2.单元分类选择单元首先需要明确单元的类型，在结构有限元分析中主要有以下一些单元类型：平面应力单元、平面应变单元、轴对称实体单元、空间实体单元、板单元、壳单元、轴对称壳单元、杆单元、梁单元、弹簧单元、间隙单元、质量单元、摩擦单元、刚体单元和约束单元等。根据不同的分类方法,上述单元可以分成以下不同的形式。?3。按照维度进行单元分类根据单元的维数特征，单元可以分为一维单元、二维单元和三维单元。?一维单元的网格为一条直线或者曲线。直线表示由两个节点确定的线性单元。曲线代表由两个以上的节点确定的高次单元,或者由具有确定形状的线性单元。杆单元、梁单元和轴对称壳单元属于一维单元,如图1~图3所示。 ?二维单元的网格是一个平面或者曲面，它没有厚度方向的尺寸.这类单元包括平面单元、轴对称实体单元、板单元、壳单元和复合材料壳单元等,如图4所示。二维单元的形状通常具有三角形和四边形两种，在使用自动网格剖分时,这类单元要求的几何形状是表面模型或者实体模型的边界面。采用薄壳单元通常具有相当好的计算效率。

??三维单元的网格具有空间三个方向的尺寸，其形状具有四面体、五面体和六面体,这类单元包括空间实体单元和厚壳单元，如图5所示．在自动网格划分时，它要求的是几何模型是实体模型(厚壳单元是曲面也可以）。 ? 4.按照插值函数进行单元分类根据单元插值函数多项式的最高阶数多少，单元可以分为线性单元、二次单元、三次单元和更高次的单元。线性单元具有线性形式的插值函数,其网格通常只具有角节点而无边节点，网格边界为直线或者平面.这类单元的优点是节点数量少，在精度要求不高或者结果数据梯度不太大的情况下，采用线性单元可以得到较小的模型规模.但是由于单元位移函数是线性的，单元内的位移呈线性变化,而应力是常数,因此会造成单元间的应力不连续,单元边界上存在着应力突变，如图6所示。

比较PageRank算法和HITS算法的优缺点

题目：请比较PageRank算法和HITS算法的优缺点，除此之外，请再介绍2种用于搜索引擎检索结果的排序算法，并举例说明。答： 1998年，Sergey Brin和Lawrence Page[1]提出了PageRank算法。该算法基于“从许多优质的网页链接过来的网页，必定还是优质网页”的回归关系，来判定网页的重要性。该算法认为从网页A导向网页B的链接可以看作是页面A对页面B的支持投票，根据这个投票数来判断页面的重要性。当然，不仅仅只看投票数，还要对投票的页面进行重要性分析，越是重要的页面所投票的评价也就越高。根据这样的分析，得到了高评价的重要页面会被给予较高的PageRank值，在检索结果内的名次也会提高。PageRank是基于对“使用复杂的算法而得到的链接构造”的分析，从而得出的各网页本身的特性。 HITS 算法是由康奈尔大学( Cornell University ) 的JonKleinberg 博士于1998 年首先提出。Kleinberg认为既然搜索是开始于用户的检索提问，那么每个页面的重要性也就依赖于用户的检索提问。他将用户检索提问分为如下三种：特指主题检索提问（specific queries，也称窄主题检索提问）、泛指主题检索提问（Broad-topic queries，也称宽主题检索提问）和相似网页检索提问（Similar-page queries）。HITS 算法专注于改善泛指主题检索的结果。 Kleinberg将网页（或网站）分为两类，即hubs和authorities，而且每个页面也有两个级别，即hubs（中心级别）和authorities（权威级别）。Authorities 是具有较高价值的网页，依赖于指向它的页面；hubs为指向较多authorities的网页，依赖于它指向的页面。HITS算法的目标就是通过迭代计算得到针对某个检索提问的排名最高的authority的网页。通常HITS算法是作用在一定范围的，例如一个以程序开发为主题的网页，指向另一个以程序开发为主题的网页，则另一个网页的重要性就可能比较高，但是指向另一个购物类的网页则不一定。在限定范围之后根据网页的出度和入度建立一个矩阵，通过矩阵的迭代运算和定义收敛的阈值不断对两个向量authority 和hub值进行更新直至收敛。从上面的分析可见，PageRank算法和HITS算法都是基于链接分析的搜索引擎排序算法，并且在算法中两者都利用了特征向量作为理论基础和收敛性依据。

ANSYS有限元网格划分的基本要点

ANSYS有限元网格划分的基本要点 1引言 ANSYS有限元网格划分是进行数值模拟分析至关重要的一步，它直接影响着后续数值计算分析结果的精确性。网格划分涉及单元的形状及其拓扑类型、单元类型、网格生成器的选择、网格的密度、单元的编号以及几何体素。从几何表达上讲，梁和杆是相同的，从物理和数值求解上讲则是有区别的。同理，平面应力和平面应变情况设计的单元求解方程也不相同。在有限元数值求解中，单元的等效节点力、刚度矩阵、质量矩阵等均用数值积分生成，连续体单元以及壳、板、梁单元的面内均采用高斯（Gauss）积分，而壳、板、梁单元的厚度方向采用辛普生（Simpson）积分。辛普生积分点的间隔是一定的，沿厚度分成奇数积分点。由于不同单元的刚度矩阵不同，采用数值积分的求解方式不同，因此实际应用中，一定要采用合理的单元来模拟求解。 2ANSYS网格划分的指导思想 ANSYS网格划分的指导思想是首先进行总体模型规划，包括物理模型的构造、单元类型的选择、网格密度的确定等多方面的内容。在网格划分和初步求解时，做到先简单后复杂，先粗后精，2D单元和3D单元合理搭配使用。为提高求解的效率要充分利用重复与对称等特征，由于工程结构一般具有重复对称或轴对称、镜象对称等特点，采用子结构或对称模型可以提高求解的效率和精度。利用轴对称或子结构时要注意场合，如在进行模态分析、屈曲分析整体求解时，则应采用整体模型，同时选择合理的起点并设置合理的坐标系，可以提高求解的精度和效率，例如，轴对称场合多采用柱坐标系。有限元分析的精度和效率与单元的密度和几何形状有着密切的关系，按照相应的误差准则和网格疏密程度，避免网格的畸形。在网格重划分过程中常采用曲率控制、单元尺寸与数量控制、穿透控制等控制准则。在选用单元时要注意剪力自锁、沙漏和网格扭曲、不可压缩材料的体积自锁等问题 ANSYS软件平台提供了网格映射划分和自由适应划分的策略。映射划分用于曲线、曲面、实体的网格划分方法，可使用三角形、四边形、四面体、五面体和六面体，通过指定单元边长、网格数量等参数对网格进行严格控制，映射划分只用于规则的几何图素，对于裁剪曲面或者空间自由曲面等复杂几何体则难以

ANSYS有限元分析中的网格划分

ANSYS有限元分析中的网格划分有限元分析中的网格划分好坏直接关系到模型计算的准确性。本文简述了网格划分应用的基本理论，并以ANSYS限元分析中的网格划分为实例对象，详细讲述了网格划分基本理论及其在工程中的实际应用，具有一定的指导意义。作者: 张洪才关键字: CAE ANSYS 网格划分有限元 1 引言 ANSYS有限元网格划分是进行数值模拟分析至关重要的一步，它直接影响着后续数值计算分析结果的精确性。网格划分涉及单元的形状及其拓扑类型、单元类型、网格生成器的选择、网格的密度、单元的编号以及几何体素。从几何表达上讲，梁和杆是相同的，从物理和数值求解上讲则是有区别的。同理，平面应力和平面应变情况设计的单元求解方程也不相同。在有限元数值求解中，单元的等效节点力、刚度矩阵、质量矩阵等均用数值积分生成，连续体单元以及壳、板、梁单元的面内均采用高斯（Gauss）积分，而壳、板、梁单元的厚度方向采用辛普生（Simpson）积分。辛普生积分点的间隔是一定的，沿厚度分成奇数积分点。由于不同单元的刚度矩阵不同，采用数值积分的求解方式不同，因此实际应用中，一定要采用合理的单元来模拟求解。 2 ANSYS网格划分的指导思想 ANSYS网格划分的指导思想是首先进行总体模型规划，包括物理模型的构造、单元类型的选择、网格密度的确定等多方面的内容。在网格划分和初步求解时，做到先简单后复杂，先粗后精，2D单元和3D单元合理搭配使用。为提高求解的效率要充分利用重复与对称等特征，由于工程结构一般具有重复对称或轴对称、镜象对称等特点，采用子结构或对称模型可以提高求解的效率和精度。利用轴对称或子结构时要注意场合，如在进行模态分析、屈曲分析整体求解时，则应采用整体模型，同时选择合理的起点并设置合理的坐标系，可以提高求解的精度和效率，例如，轴对称场合多采用柱坐标系。有限元分析的精度和效率与单元的密度和几何形状有着密切的关系，按照相应的误差准则和网格疏密程度，避免网格的畸形。在网格重划分过程中常采用曲率控制、单元尺寸与数量控制、穿透控制等控制准则。在选用单元时要注意剪力自锁、沙漏和网格扭曲、不可压缩材料的体积自锁等问题ANSYS软件平台提供了网格映射划分和自由适应划分的策略。映射划分用于曲线、曲面、实体的网格划分方法，可使用三角形、四边形、四面体、五面体和六面体，通过指定单元边长、网格数量等参数对网格进行严格控制，映射划分只用于规则的几何图素，对于裁剪曲面或者空间自由曲面等复杂几何体则难以控制。自由网格划分用于空间自由曲面和复杂实体，采用三角形、四边形、四面体进行划分，采用网格数量、边长及曲率来控制网格的质量。 3 ANSYS网格划分基本原则 3.1 网格数量网格数量的多少将影响计算结果的精度和计算规模的大小。一般来讲，网格数量增加，计算精度会有所提高，但同时计算规模也会增加，所以在确定网格数量时应权衡两个因数综合考虑。图1 位移精度和计算时间随网格数量的变化图1中的曲线1表示结构中的位移随网格数量收敛的一般曲线，曲线2代表计算时间随

31部关于青春的电影

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