Springer’s regular elements over arbitrary fields

SAE Technical Standards Board Rules provide that: “This report is published by SAE to advance the state of technical and engineering sciences. The use of this report is entirely voluntary, and its applicability and suitability for any particular use, including any patent infringement arising therefrom, is the sole responsibility of the user.”SAE reviews each technical report at least every five years at which time it may be reaffirmed, revised, or cancelled. SAE invites your written comments and suggestions.TO PLACE A DOCUMENT ORDER; (724) 776-4970 FAX: (724) 776-0790SAE WEB ADDRESS Copyright 2000 Society of Automotive Engineers, Inc.FIGURE 1A—RETURN FUEL SYSTEMFIGURE 1B—RETURNLESS FUEL SYSTEMFIGURE 2—FUEL CHARGING ASSEMBLY-CFI-CENTRAL FUEL INJECTIONFIGURE 3—FUEL PRESSURE REGULATOR (NON-REFERENCED)FIGURE 4—FUEL PRESSURE REGULATOR (REFERENCED)FIGURE 5A—FUEL PRESSURE RELATIONSHIP TO REFERENCE PRESSUREFIGURE 5B—NON-REFERENCED CONTROLLED FUEL PRESSUREFIGURE 6—FUEL PRESSURE DAMPER (MECHANICAL SPRING)FIGURE 7—FUEL PRESSURE DAMPER (HYDRAULIC SPRING)TABLE OF CONTENTS1Scope (8)2References (8)2.1Applicable Publications (8)2.1.1SAE Publications (8)2.1.2ASTM Publications (8)2.1.3Federal Publications (9)3Performance Characteristics (9)3.1Pressure Regulator (9)3.1.1Set Point Pressure (9)3.1.2Slope (9)3.1.3Operating Flow Range (10)3.1.4Repeatability (11)3.1.5Hysteresis (13)3.1.6Dynamic Response (13)3.1.7Internal Leakage (Leakdown) (14)3.2Pressure Damper Performance (16)3.3External Leakage (17)3.3.1External Leakage–Fuel Side (21)3.3.2External Leakage–Air Side (21)3.4Presentation of Regulator and Damper Data (21)4Application Related Parameters and Tests (22)4.1Structural (22)4.1.1Vibration (22)4.1.2Shock (23)4.1.3Burst (23)4.1.4Proof Pressure (24)4.1.5Physical Tests (24)4.2Durability–Life Cycling (25)4.2.1Flow Cycling (25)4.2.2Pressure Cycling (26)4.2.3Thermal Cycling (27)4.2.4Thermal Soak (28)4.3Material Integrity–External (28)4.3.1Salt Corrosion (28)4.3.2Resistance to Vehicle Fluids (29)4.3.3Surface Corrosion and Blistering (30)4.4Material Integrity–Internal (30)4.4.1Reference Fluids (30)4.4.2Test Procedure (31)4.4.3Assessment of Results (31)5Standard Test Conditions (31)5.1Test Fluid (31)5.2Ambient Test Temperature (32)5.3Test Fluid Temperature (32)5.4Return Line Pressure (32)5.5Mounting Configuration (32)5.6Preparation of Test Specimen (32)6Test Apparatus and Equipment (32)6.1Introduction (32)6.2Test Equipment Description (33)6.2.1Test Fluid Pump (33)6.2.2Vacuum Pump (33)6.2.3Air Pump (33)6.2.4Pressure Gages (33)6.2.5Vacuum Gages (33)6.2.6Vibration Tester (33)6.2.7Salt Spray (34)6.2.8Environmental Test (34)6.2.9Pump Pressure Control (34)6.2.10Shock Test (34)6.3Pressure Regulator Test Equipment Measurement Accuracy (34)7.Notes (34)7.1Marginal Indicia (34)Appendix A Indolene Unleaded Gasoline Specification Code of Federal Regulations, Title 40, Part 86, Section 113 Fuel Specification of Emission Regulations Test Procedure for 1994My andLater Light Duty Vehicles and Light Duty Trucks (35)FIGURES1A Return Fuel System (2)1B Returnless Fuel System (2)2Fuel Charging Assembly–CFI–Central Fuel (3)3Fuel Pressure Regulator (Non-Referenced) (3)4Fuel Pressure Regulator (Referenced) (4)5A Fuel Pressure Relationship to Reference Pressure (4)5B Non-Referenced Controlled Fuel Pressure (5)6Fuel Pressure Damper (Mechanical Spring) (5)7Fuel Pressure Damper (Hydraulic Spring) (8)8Regulator Slope (10)9A Pressure Regulator Flow Range (12)9B Pressure Regulator Population Flow Range (12)10A Example of Dynamic Response (10 kPa Signal Pressure) (15)10B Example of Dynamic Response (50 kPa Signal Pressure) (15)11Internal Leakage Test Stand (16)12Pressure Damper Performance Test Stand (17)13Pressure Damper Overall Acceptance (19)14Example of Pressure Damper Frequency Domain (19)15Pressure Regulator Proof Pressure/Burst Test Stand (24)16Flow Cycle Schedule (25)17Pressure Cycle Schedule (26)18Thermal Cycle Schedule (27)19Pressure Regulator Performance Test Stand (33)1.Scope—This SAE Recommended Practice promotes uniformity in the evaluation and qualification testsconducted on fuel pressure regulators and pressure dampers used in gasoline engine applications. Its scope is limited to fuel pressure regulators and dampers used in automotive port and throttle body fuel injection systems where fuel supply pressure is below 1000 kPa. It is further restricted to bench type tests. More specifically, this document is intended for use as a guide to the following:a.Identify and define those parameters that are used to measure fuel pressure regulator and pressuredamper characteristics of performance. The parameters included in this document are:1.Slope2.Operating Flow Range3.Repeatability4.Hysteresis5.Dynamic Responseb.Establish test procedures and recommend test equipment and methods to measure and quantify theseparameters.c.Establish test procedures and recommend test equipment and methods to quantify simulated fieldreliability over the life of the component.d.Standardize the nomenclature as related to fuel pressure regulation and pressure damping for fuelinjection systems.e.Except where stated, test results are recorded for individual parts. Where population characteristicsare reported, the sample size, selection method, and analysis technique must be explicitly stated.2.References2.1Applicable Publications—The following publications form a part of the specification to the extent specifiedherein. Unless otherwise indicated, the latest revision of SAE publications shall apply.2.1.1SAE P UBLICATIONS—Available from SAE, 400 Commonwealth Drive, Warrendale, PA 15096-0001.SAE J306—Automotive Gear Lubricant Viscosity ClassificationSAE J308—Axle and Manual Transmission LubricantsSAE J310—Automotive Lubricating GreasesSAE J313—Diesel FuelsSAE J814—Engine CoolantsSAE J1541—Fuel Injection Nomenclature—Spark Ignition Engines2.1.2ASTM—Available from ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428-2959.ASTM B 117—Salt Spray (Fog) TestingASTM D 56—Flash Point by Tag Closed TesterASTM D 86—Distillation of Petroleum ProductsASTM D 128—Test Methods for Analysis of Lubricating GreaseASTM D 323—Test Method for Vapor Pressure of Petroleum Products (Reid Method)ASTM D 445—Test Method for Petroleum ProductsASTM D 471-79—Test Method for Rubber Property—Effect of LiquidsASTM D 893—Test for Insolubles in Lubricating OilsASTM D 1266—Test Method for Sulfur in Petroleum Products (Lamp Method)ASTM D 1298—Test for Density and Specific GravityASTM D1319—Test Method for Hydrocarbon Types in Liquid Petroleum Products by Fluorescent Indicator AdsorptionASTM D 1744—Water in Liquid Petroleum ProductsASTM D 2699—Test Method for Research Octane Number of Spark-Ignition Engine FuelASTM D 3231—Test Method for Phosphorus in GasolineASTM D 3237—Test Method for Lead in Gasoline by Atomic Absorption Spectrometry2.1.3F EDERAL P UBLICATIONS—Available from the Superintendent of Documents, U.S. Government Printing Office,Washington, DC 20402.40 CFR 86.113-82—Fuel Specification of Emission Regulations Test Procedure49 CFR 571-116—Motor Vehicle Hydraulic Brake Fluid3.Performance Characteristics3.1Pressure Regulator3.1.1S ET P OINT P RESSUREa.Definition—The test fluid pressure to which the regulator is set at a specified fluid flow rate. It isspecified as pressure (in kPa) at a fluid flow rate (in g/s).b.Background—The regulator set point pressure determines the nominal operating pressure for a fuelinjection system. It is critical to the design of the system that this pressure be known and controlled.The specified regulator set pressure is meaningless without the corresponding flow rate since thepressure will vary as a function of flow rate. The set point specified for different applications may differdue to the differences in required injector flow rates.c.Objective—To measure the regulator set point pressure at the specified set point flow rate.d.Test Apparatus—See 6.2.1, 6.2.4, and 6.2.9.e.Test Conditions—Refer to 5.1 through 5.6.f.Test Method:1.Slowly increase test fluid pressure until fluid just starts to flow through the regulator (approximately1.0 g/s).2.Gradually increase the fluid flow until the specified set point flow rate is achieved (within ±0.5 g/s).Do not allow the flow to exceed the specified flow rate; if so, restart the procedure.3.Allow the flow to stabilize for 15 to 30 s then record the regulator inlet pressure.g.Data Required—Record the regulator inlet pressure, outlet pressure, and actual fluid flow rate. Reportset point pressure as: xxx.x kPa at specified set point flow rate.h.Reported Population Data—Measure the regulator set point pressure for a random sample ofregulators representative of a normal production distribution. Determine the sample mean (X s) andsample standard deviation (s). Report the population set point as the sample mean (X s) ± four timesthe sample standard deviation (4s): xxx.x ± xx.x kPa. Also report the 4 s as a percentage of X s:±xx.x%.3.1.2S LOPEa.Definition—The change in regulator pressure over a linear flow range which results from a change influid flow rate through the regulator, all other parameters remaining constant. It is specified aspressure change per unit flow rate (i.e., kPa/(g/s)) between well defined lower and upper flow rates. Apositive slope represents increasing pressure with increasing flow rate (see Figure 8).b.Background—An ideal regulator maintains the desired fuel system pressure regardless of changes infuel flow rate through the regulator, all other conditions remaining constant. However, with mostproduction regulators the fuel system pressure varies from the desired pressure as a function of fuelflow rate through the regulator. This variation of pressure with respect to flow rate should be veryclose to linear over the operating flow range, and a low value for the slope is generally desired.It is important when designing/specifying a fuel injection system to know the regulator slope. The fuelflow rate through the regulator will vary due to a variety of factors which can include engine load, rpm,battery voltage, and others. Subsequent changes in fuel system pressure, due to flow variation, willchange injector flow, which has to be considered in the fuel injection system design.c.Objective—To measure the change in regulator pressure with change in fluid flow rate, from lower toupper flow rate limit, through the regulator and to establish the regulator slope. The lower and upperflow rate limits are specific to a regulator design and application.d.Test Apparatus—See 6.2.1, 6.2.4, and 6.2.9.e.Test Conditions—See 5.1 through 5.6.FIGURE 8—REGULATOR SLOPEf.Test Method:1.From the set point flow rate, with the regulator air side at ambient pressure, reduce the flow rate to 1to 2 g/s and allow the regulator to stabilize for 15 to 30 s.2.Gradually increase the flow rate to the specified lower flow rate limit, allow the regulator to stabilizefor 15 to 30 s, and record the regulator inlet pressure. Do not allow the flow rate to exceed this flowvalue prior to making the pressure measurement. If it does, restart the procedure.3.Continue with this same procedure, making regulator pressure measurements at specified flow ratesbetween the lower and upper flow rate limits.g.Data Required—A minimum of the previous four measurements should be made from the specifiedlower to upper flow rates inclusive. More intermediate points may be used if desired.Determine the best fit slope for the data using least squares linear regression analysis techniques fordata reduction. If data varies by more than ±2% from linear, the lower and/or upper specified limitsmay have to be reset or the larger error from linear must be noted with the slope.Report slope as: xx.x kPa/(g/s) over the specified flow rate range.h.Reported Population Data—Determine the regulator slopes for a random sample of regulatorsrepresentative of a normal production distribution. Determine the sample mean (X s) and standarddeviation (s) of the slopes. Report the population slope as the sample mean (X s) ± four times thesample standard deviation (4 s): xx.xx ± x.x kPa/(g/s) over the specified flow range.3.1.3O PERATING F LOW R ANGEa.Definition—The range of fluid flow rate over which the pressure regulator response is linear (to within2%) with respect to flow rate Figures 9A and 9B.b.Background—The flow range should equal or exceed the operating flow range the regulator will beexposed to in an actual application.c.Objective—To measure the minimum and maximum regulator fluid flow rates between which theregulator pressure response is linear with respect to flow rate.d.Test Apparatus—See 6.2.1, 6.2.4, and 6.2.9.e.Test Conditions—See 5.1 through 5.6.f.Test Method:1.Slowly increase test fluid pressure until fluid just starts to flow through the regulator (approximately1.0 g/s).2.Gradually increase flow rate, while measuring regulator pressure, until the pressure responsebecomes nonlinear or the test stand capability is reached. Allow regulator to stabilize at eachmeasurement point.g.Data Required—Flow rate measurement increments must be small enough to accurately determinethe flow range, within ±10% at minimum flow rate, ±5% at maximum.Using least squares linear regression analysis for data reduction, for the data points in the specifiedflow rate range, determine the minimum and maximum regulator fluid flow rates between which theregulator pressure response is linear with respect to flow rate to within ±2%. If the maximum flow rateis beyond the capability of the test stand (at least 75 g/s), specify the test stand maximum flow ratefollowed by a plus sign.Report operating flow range of the test specimen as—xx.x g/s to xxx g/s. See Figure 9A.h.Reported Population Data—Measure the regulator pressures at a series of flow points as specified fora random sample of regulators representative of a normal production distribution. Determine thesample mean pressure (X i) and standard deviation (s i) at each flow point. Determine the slope overthe specified flow rate range using least squares linear regression of the mean pressure values (X i).Plot the deviation from the determined slope and the ±4 s; values for each flow point (see Figure 9).The maximum value of the operating flow range is the flow value where the +4 s i pressure deviates by2% from the linear regression slope, and the minimum value of the operating flow range is the flowvalue where the −4 s i pressure deviates by 2% from the linear regression slope. Report the populationoperating flow range as: xx.x g/s to xxx g/s minimum. See Figures 9B.FIGURE 9A—PRESSURE REGULATOR FLOW RANGEFIGURE 9B—PRESSURE REGULATOR POPULATION FLOW RANGE3.1.4R EPEATABILITYa.Definition—The variability in the set point pressure for a given regulator over a number of repeatedpressure on-off cycles.b.Background—An ideal regulator would produce the same set point pressure every time it is cycledfrom zero flow rate to the set point flow rate. Repeatability is important so that the fuel systempressure is predictable for consistent fuel system performance.c.Objective—To determine the variability of the set point pressure for a fuel pressure regulator.d.Test Apparatus—See 6.2.1, 6.2.4, and 6.2.9.e.Test Conditions—See 5.1 through 5.6.f.Test Method:e the test method defined in 3.1 to determine the regulator set point pressure.2.After taking the set point pressure measurement, gradually reduce pressure in the regulator to zeroand allow the regulator to relax for 15 to 30 s.3.Repeat the previous procedure to obtain 20 set point pressure measurements.g.Data Required—Determine the test specimen mean set point pressure and standard deviation for theprevious pressure measurements.The repeatability is defined as four times the standard deviation as a percentage of the test specimenmeasurements mean.Report repeatability as: ±xx.x%h.Reported Population Data—For a random sample size of regulators, determine the sample mean setpoint pressure (X i) and the standard deviations of the individual regulator set pressure measurements.The population repeatability is defined as four times the sample mean standard deviation (mean ofindividual regulator set pressure standard deviations) as a percentage of the mean set point pressure(X i). Report as: ±xx.x%.3.1.5H YSTERESISa.Definition—The change in regulator set point pressure as a result of approaching the set point flowfrom high flow versus approaching from low flow.b.Background—An ideal regulator would maintain the same set point pressure regardless of whether thefluid flow through the regulator has exceeded or has been less than the set point flow. Variability in setpoint pressure due to hysteresis has the same effect as variation due to repeatability.c.Objective—To determine the hysteresis of a fuel pressure regulator by measuring the set pointpressure when approaching the set point flow rate from both directions.d.Test Apparatus—See 6.2.1, 6.2.4, and 6.2.9.e.Test Conditions—See 5.1 through 5.6.f.Test Method:e the method defined in 3.1 to determine the set point pressure as approached from a lower flowrate.2.After taking the set point pressure measurement, gradually increase the fluid flow rate to the upperflow rate limit.3.Gradually reduce the fluid flow rate through the regulator to the set point flow rate and allow theregulator to stabilize for 15 to 30 s. Record the regulator inlet pressure. Do not allow the flow to fallbelow the set point flow rate prior to making this measurement. If so, restart the procedure.g.Data Required—The hysteresis reported is the difference in increasing and decreasing flow set pointpressure measurements as a percent of the set point pressure measured with increasing flow.Report hysteresis as: xx.x%h.Reported Population Data—Measure the regulator set point pressure with increasing and decreasingflow for a random sample of regulators representative of a normal production distribution. Make 20measurements of each regulator and determine the difference between the increasing flow anddecreasing flow set pressures. Determine the maximum difference for each regulator. Next,determine the sample mean and standard deviation of the maximum differences. Report thepopulation hysteresis as ±4 s as a percent of the population set point pressure: ±xx.x% ((4 s/set pointpressure) x 100).3.1.6D YNAMIC R ESPONSEa.Definition—The change in regulator pressure, which results from a change in the reference pressuresignal being applied to the “air side” of the fuel pressure regulator. The response of the pressureregulator to a change in the reference signal consists of both the degree of regulator pressure changeas well as of the time required to respond to changes of the reference signal.b.Background—The fuel system pressure is controlled by the pressure regulator relative to a referencepressure signal applied to the “air side” of the regulator. The reference signal is usually air cleanerpressure (CFI) or intake manifold pressure (MPI). It will be assumed that the desired change insystem pressure will equal the change in reference pressure unless otherwise specified.c.Objective—To measure the response time and accuracy of regulated pressure change of a fuelpressure regulator relative to a change in the reference signal.d.Test Apparatus—See 6.2.1 through 6.2.5, and 6.2.9.e.Test Conditions—See 5.1 through 5.6.f.Test Method:1.Start the test with the regulator operating at set point pressure and flow rate, and ambient pressurefor the reference signal.2.Apply a sudden change in reference signal pressure (not longer than 50 ms to achieve 90% of thefinal value), maintain a constant flow rate, and trace the regulator pressure.g.Data Required—Perform the previous test with reference signal of 10 and 50 kPa vacuum and 10 and50kPa pressure (±1 kPa). Additional reference signal values are optional.Determine the stabilized regulator pressure with the new reference signal pressure. Express thechange of regulator pressure as a percentage relative to the change in signal pressure.Determine the time required for the regulated pressure to reach 90% of its final value as measuredfrom the time that the reference pressure achieves 90% of its final value.Report regulator response as: xx.x% response accuracy, xx.x ms response time, with xx kPa vacuum/pressure signal. Reference Figures 10A and 10B.h.Reported Population Data—Determine the response time and response accuracy for each regulator ina random sample of regulators representative of a normal production distribution. Report the meanresponse time and mean response accuracy of the sample.FIGURE 10A—EXAMPLE OF DYNAMIC RESPONSE (10 KPA SIGNAL PRESSURE)FIGURE 10B—EXAMPLE OF DYNAMIC RESPONSE (50 KPA SIGNAL PRESSURE)3.1.7I NTERNAL L EAKAGE (L EAKDOWN)a.Definition—The fluid leakage from the inlet of the regulator to the lower pressure regulator outlet, at apressure below the regulator set pressure.b.Background—A regulator can be designed to either shut off fuel flow (checking) or allow fuel flowthrough the regulator (bleed-down) when the fuel system pump is de-energized. The purpose,respectively, is to maintain pressure in the system, or bleed-off pressure in the system, after the pumpis de-energized.c.Objective—To measure the internal leakage of a fuel pressure regulator.d.Test Apparatus—See Figure 11.e.Test Conditions—See 5.1 through 5.6.f.Test Method–Fluid (for Shutoff or Bleed-Down Regulators):1.Start the test with the regulator operating at set point pressure and flow rate and ambient pressurefor the reference signal.2.Reduce the regulator inlet pressure to 80% of set pressure, and maintain regulator pressure at thatlevel with the regulator outlet at ambient pressure. Allow regulator to stabilize for 60 s minimum.3.Measure the mass of fluid that leaks out the outlet for a 60 s minimum time period.g.Data Required–Fluid—Report regulator leakdown as: xx.x g/s test fluid at xxx.x kPa.FIGURE 11—INTERNAL LEAKAGE TEST STANDh.Test Method–Air (Optional Method for Shutoff Regulators)—Regulator must be empty of all fluid.1.Apply dry, filtered air at the set pressure for 30 s minimum to dry the regulator interior and sealingsurfaces. The airflow rate or pressure may need to be adjusted to prevent harmful vibration ofinternal components.2.Reduce the regulator pressure to 80% of set pressure to the regulator inlet with the regulator outletat ambient pressure. Allow the regulator to stabilize for 60 s minimum.3.Measure the volume of air that leaks out the outlet for a 60 s minimum time period.i.Data Required–Air—Convert measured volume of air over a measured time period to a standardairflow rate. Scc/min.Report regulator leakdown as—xx.x Scc/min air at xxx.x kPa.j.Reported Population Data—Determine the leak rate (either air or fluid) for each regulator in a random sample representative of a normal production distribution. Using the mean leak rate and standarddeviation for the sample, determine the 4 s upper bound (x + 4 s) and report its value as the maximumpopulation leak rate.3.2Pressure Damper Performancea.Definition—The attenuation effectiveness and characteristics of a pressure damper on a test fixture.b.Background—The ideal pressure damper would remove all pressure pulsations in the fuel system.The pressure trace of the output of the damper would then indicate only the nominal static pressurewith no ripple or noise on the trace. Further, the ideal damper would eliminate all noise over the entirespectral bandwidth of interest, typically DC (static) to 5000 Hz. In actuality, pressure damperperformance might not be equally effective over the entire spectrum of frequencies, thus, the samedamper might appear quite effective in one application and ineffective in another.c.Objective—To define an overall acceptance test of a pressure damper for a particular application, andto suggest techniques for characterizing damper performance over an entire bandwidth.d.Test Apparatus—See Figure 12.FIGURE 12—PRESSURE DAMPER PERFORMANCE TEST STANDe.Test Conditions—See Section 5.f.Test Method:1.Part 1–Overall Acceptance Test—For this test, a good high speed system digital voltmeter isnecessary, which can store multiple readings (at least 1000), average those readings, find maximum/minimum, and read true root mean square voltages (rms).The basic test procedure involves obtaining three pieces of data from the input pressure trace, and the output pressure trace, then comparing the measurements and expressing them in terms of percent differences. This test should be performed at the intended system nominal pressure and at points over the range of fuel system gauge pressures in steps of 10 kPa. (i.e., for a normally aspirated engine at 300 kPa, take readings at 200, 210, 220...300 kPa). The three readings are:a.Average Static Pressure Level—This is determined by taking 1000 readings of the DC equivalent(static) pressure level both at the inlet and outlet, and calculating the average of those pressure readings at each location (x in and x out). The ideal pressure damper has x in and x out equal, which means that no pressure drop has occurred through the device.NOTE—For flow related calculations, the inlet is the side nearest the flow source; for attentuation related calculations, the inlet is the side nearest the acoustic source.b.Random Pressure Noise–Efficiency—This value is the peak-to-peak reading determined from themaximum and minimum values from the same 1000 readings taken in part 1. The value characterizes random pressure excursions present on the trace, and is to be reported as the difference between them (maximum-minimum) for the inlet and outlet traces. The final reported value is the ratio of the outlet to the inlet expressed in percent ((out/in) x 100).c.Root/Mean Square Pressure Ripple—With the pressure traces AC coupled to the digital voltmeter,the true rms value is obtained. This measurement characterizes periodic pressure disturbance with the difference between the in and out values indicating the device effectiveness. Report the ratio of the outlet to the inlet expressed in percent ((out/in) x 100).Figure 13 represents an example of the pressure damper overall acceptance.2.Part 2–Characterizing Damper Performance—This tests attempts to characterize a pressuredamper for any application by measuring the performance characteristics in the frequency domain.A fast Fourier transform (FFT) spectrum analyzer is required for this test, preferably a two channelcondition would be one in which the input pressure trace contains random noise throughout the entire spectral bandwidth of DC to 5000 Hz.The input and output pressure traces are AC coupled to the spectrum analyzer, and a transmissibility trace is obtained. This trace is a good qualitative indication of the performance of the damper over the entire bandwidth. The ideal pressure damper would demonstrate equal performance over the entire display the power spectral density (PSD) of the input and output traces.A visual comparison of the two traces would yield the same basic information as the transmissibilitytrace but with less detail.Figure 14 represents an example of the pressure damper frequency domain.。

长期完整性的分层容错论文翻译长期完整性的分层容错秉健春，佩特罗斯曼尼阿蒂斯英特尔研究大学伯克利分校斯科特 Shenker，约翰 Kubiatowicz 美国加州大学伯克利分校摘要通常，容错服务有关的假设类型和故障，他们可以容忍的最大数量这种故障时，同时提供其正确性的保证;阈值是侵犯性，正确性丢失。

我们重新审视的概念在长期归档存储方面的故障阈值。

我们注意到，故障阈值不可避免地长期违反服务，使得传统的容错不适用长期的。

在这项工作中，我们进行的再分配容错预算的一项长期的服务。

我们分裂投入服务件的服务，每一个都可以容忍不同没有失败的故障数（并不会造成整个服务失败）：每件可在一个关键值得信赖的过错层，它必须永远不会失败，或不可信的故障层，它可以大量经常失败，或其他故障层次之间。

通过仔细工程的一项长期的服务分裂成片，必须服从不同的故障阈值，我们可以延长其必然灭亡。

我们证明了这一点做法 Bonafide，一个长期的 key - value 存储，与所有类似的文献中提出的系统，在保持完整性面对拜占庭故障，无需自我认证的数据。

我们描述了分层容错的概念，设计，实施， Bonafide 和实验1/ 3评价，并主张我们的做法是一个实际仍未显著改善在长期服务的艺术状态。

一，引言当前容错复制服务设计往往不适合长期的应用，如档案，数字文物，这是越来越重要的存储 [42]，为企业的监管[5， 6]，文化[36]原因。

从典型的故障假设这不合适结果这类系统的正确性空调。

例如，在典型的拜占庭容错（BFT）系统[13]，它是假定复制故障副本的数目总是比一些不太固定阈值，如副本人口的 1 / 3。

在典型的短期的应用，这样一个 uniformthreshold 基于故障假设是合理的，可以实现的。

例如，可以说在一个 wellmaintained 人口多元化，高保证副本服务器，由当时总人口的三分之一被打破生长进入或刚刚出现故障，故障副本的运营商可以修复它们。

5.3 State Retention and Restoration Methods5.3 状态保存和恢复方法Given a power switching fabric and an isolation strategy, it is possible to power gate a block of logic. But unless a retention strategy is employed, all state information is lost when the block is powered down. To resume its operation on power up, the block must either have its state restored from an external source or build up its state from the reset condition. In either case, the time and power required can be significant.有了电源开关装置和隔离策略，就可以对逻辑块使用门控电源。

除非使用了状态保持策略，否则所有的状态信息都会在电源关闭的时候丢失。

为了在上电的时候恢复操作，被关闭的块要么从外部源恢复它的状态，要么初始化所有的状态。

这两种情况需要的时间和功率都相当大。

In many cases, an explicit retention strategy for saving and restoring state quickly and efficiently can provide a much faster and power-efficient method of getting the block fully functional after power up.许多情况下，对于保存和恢复状态有一个明确清晰的状态保存策略，可以使模块上电后恢复功能更快，功耗更小。

Tamingfirst order logic:relating the semantic and the syntactic approachMaarten MarxAbstractOut of the joint work of Johan van Benthem and the Hungarian group round HajnalAndr´e ka,Istv´a n N´e meti and Ildik´o Sain and their PhD students,two approaches fortaming a logic evolved.With taming a logic we mean changing the logic in such a waythat it becomes decidable.Forfirst order logic,they took a semantic route using rela-tivisation of models,and a syntactic route focusing on guarded fragments.The purposeof this paper is to show that these two routes are really two sides of the same coin.Wedo this by showing that a certain guarded fragment(called here the packed fragment)offirst order logic forms precisely the set offirst order sentences which are invariant for relativisation with a tolerance relation.Besides this technical contribution we providean intuitive explanation of relativisation in terms of information transmission.Contents1Introduction22Relativisation interpreted as a sceptical information processing strategy2 3Relativised semantics forfirst order logic53.1Admissible assignments (5)3.2Different relativisations (8)3.3Bisimulations (10)4Packed fragment135Conclusion17Dedicated to Johan van Benthem on hisfiftieth birthday.1IntroductionThe main purpose of this paper is to give a semantic characterisation of the guarded fragment(here called the packed fragment)in terms of invariance for relativisation(Theorem4.11).We will argue that relativisation in afirst order setting is the analogue of taking generated submodels in modal logic.Our tentative conclusion will be that the packed fragment is the true modal fragment offirst order logic,because it has the same“localflavour”and is decidable for very similar reasons.Besides this technical result we provide an interpretation of relativisation as a sceptical information processing strategy in section2.We show that for sentences in the packed fragment,this sceptical strategy leads to the same results(in terms of validity and satisfaction)as the classicalfirst order interpretation.This result makes the sceptical strategy an interesting alternative because it has great computational advantages over the classical first order way of interpreting sentences.The technical part of this paper is organised as follows.We introduce relativised semantics forfirst order logic in Section3.We take a modal view onfirst order logic and arrive to the notion of admissible assignments.We show how relativised semantics can be obtained from two primitive concepts:context sets and tolerance relations.Then we introduce our version of the(loosely)guarded fragment,called the packed fragment,and relate it to relativised semantics.We conclude by arguing that the packed fragment is the true modal fragment of first order logic.2Relativisation interpreted as a sceptical informa-tion processing strategyWe employ a very simple model of information transmission:there are two participants of which one does all the talking and the other merely listens and interprets the incoming information.It will be convenient to ascribe gender to the two participants.We assume that the speaker is male and the listener is female.The language used by the speaker is afirst order language,which will be interpreted dynamically,in the style of Dynamic Predicate Logic(DPL)of [3,4].The update–semantics version of DPL provides a model for the interpreta-tion process performed by the listenerIn the dynamic view the interpretation process consists of two components:1.finding antecedents for anaphora(interpreting discourse information)2.building a model of the world described by the speaker(interpretation ofworld–information).The speaker however is not God or some other embodiment of the world just reporting what is the case,the speaker is an observer of the world reportinghow he perceives it.Thus the interpretation of the world–information is more accurately described by2’building a model of the world perceived by the speaker on the basis of his report of it.Once the listener realises that she is building a model of the world relative to the perception of the speaker,a number of information processing strategies are open to her.In this paper,we will focus on one such a strategy:the listener does not accept universal statements made by the speaker unconditionally,but relativises them with the proviso“provided that the elements quantified over are perceived together bythe speaker.”We will make this proviso more precise in due course.First an example Example2.1A company like McDonald’s can be modeled in many different ways.A natural way to think about it is as a collection of databases each containing the employees of one outlet,databases containing the managers of a region,databases containing the whatevers of some larger geographical unit, and so on,all the way up to the database containing the president and his com-panions.In other words,a hierarchical setup of partially overlapping databases.Let’s look at the following situation.The president of McDonald’s gives his yearly address and says:“In this company,everybody loves each other.”A logician who ows his money baking hamburgers raises hisfinger and says that he doesn’t love his neighbour in the audience at all,since that man works in an outlet at the other side of the world.The president answers,rather annoyed, that he meant of course that“everyone within a unit in McDonald’s loves each other”.The president and the logician interpret thefirst sentence differently because they amalgamated the databases in a different way.The logician used the classical way,while the president amalgamated them in a relativised manner.In this way he could keep the natural structure of the company.For the president, the natural way of interpreting his universal statement was using the proviso given above.Before we go into the strategy,let’s look at some more basic questions.Why would a listener employ such a strategy?And,supposing there are good reasons, how does she know when elements of discourse are perceived together by the speaker?In other words,can she practically perform such a strategy at all?To start with thefirst question,why would a listener not want to accept universal statements unconditionally?The answer is that the computational costs connected to universal statements are very high.They can quickly lead to infinite models,moreover it is undecidable tofind out whether a set of sentences has a model at all.Given this it is reasonable to postulate that the listener uses some kind of mechanism to cope with these difficulties of interpretation.What kind of mechanism she uses is of course open to debate.Whatever mechanism she will use,we can expect some natural properties of it:•It should be sound:“whatever can still be deduced,can also be deduced in the“classical setting”.•It shouldn’t lead to too much loss of information:on a large natural frag-ment of natural language,the classical and the adjusted interpretation process should lead to the same results.•It should have definite computational advantages.The strategy we propose satisfies all these.So let’s look at the second question: suppose the listener uses the“proviso–strategy”,how does she implement it? In particular how does she know which elements in the domain of discourse are perceived together by the speaker,and what does that mean?We start with the latter.Given a domain D of individuals,and a subset X⊆D,how can we describe when the elements in X are perceived together by the speaker?One natural way to do this is to postulate that there exists a“dis-tance”function f:D×D−→R,which describes for the speaker the distance between each two elements in D.With distance we mean something inherently vague with many dimensions.It has at least spatial,temporal,conceptual and cultural components,but also cognitive ones,individualised to each speaker. The listener could now state that the elements in X are perceived together by the speaker if for all a,b in X,the distance between a and b is less than some fixed value d.But if the listener cannot know the function f,she also has no idea about d.So what is left for her is just the abstract information that there exists a tolerance relation on D for the speaker.With the assumption that the distance between one object is arbitrarily small,the only thing she knows then is that there exists a binary relationδ(x,y)on D which is reflexive and symmetric.So far so good,but how does she know which elements stand in this rela-tion?She can’t know anything but the discourse of the speaker provides her with clues.That is,from the discourse she gets information which makes it reasonable to assume that indeed the speaker can perceive certain individuals together.The following are examples of such clues:Named individuals all individuals which the speaker gives a name are per-ceived together.Existentially introduced individuals If the speaker introduces two individ-uals existentially in the same discourse,then their denotation can be per-ceived together by him.Primitive relations If the speaker puts certain individuals together in a prim-itive relation,then he can perceive them together.(We can view this as “naming”a group.)Related to named individuals Quite a bit stronger is to postulate that the distance between any individual and any named individual is arbitrarily small.Macho or classicalδis the universal relation on the domain of discourse.That is,the speaker can perceive each two individuals in the domain of discourse together.The last clue brings us back to the classical interpretation of a discourse.Clearlysuch clues can be provided by the speaker.We will assume though that he canonly do this outside the object language that we are studying.So the speakershould make a kind of meta–statement in order to effectuate this information.(Think of a math teacher who starts a class with:“Everything I will say holdsfor all the natural numbers and for them only”.)This concludes our view on relativisation in a discourse setting.We wil nowturn to the technical work.The clues presented above will re-occur there.3Relativised semantics forfirst order logicWe providefirst order logic with a different semantics than the standard seman-tics.We assume we are working with a standardfirst order language withoutfunction symbols:thus the language contains equality,the usualfirst order con-nectives,a countable stock of variables and individual constants,and a count-able stock of n-ary relation symbols,for every n.In addition we will assumethat we have as primitive symbols also∃¯v,where¯v is afinite set of variables.When¯v={v1,...,v k},then∃¯vϕjust means∃v1...∃v kϕin classical logic.For ϕa formula in this language,FV(ϕ)denotes the set of free variables ofϕ,defined in the standard way.In classicalfirst order logic,the interpretation of a formula in a model(D,I)is given relative to an assignment of the variables s.Given a domainD,the set of assignments consists of all functions from the set of variables intoD.The key idea of relativised semantics is that meaning of formulas becomesrelativised to a subset of the set of all assignments.We call such a set theadmissible assignments.In what follows we willfirst define this relativised semantics.Then we seewhat intuitions one can develop about the set of admissible assignments.Wefinish with providing the connection with standardfirst order logic and establisha notion of bisimulation.3.1Admissible assignmentsAssignments.Given a model(D,I),an assignment is a function from the setof variables into D.We assume our language hasωmany variables v0,v1,...,.An assignment g can then be viewed as a sequence fromωD:g(i)then givesthe value of v i according to g.To define the meaning of the existential quantifier,it is handy to create thefollowing relation between assignments:s≡i t ifffor all j=i:s(j)=t(j).(1) That is:two assignments s and t are≡i related iffthey agree on all values of the variables except possibly for v i.We can also define these relations for sets of variables¯v:s≡¯v t ifffor all j∈¯v:s(j)=t(j).(2)Using the relation≡i,we can give an alternative equivalent definition of the meaning of the existential quantifier.Given a model M=(D,I)and an assign-ment s∈ωD,defineM|=∃v iϕ[s]⇐⇒there exists a t≡i s such that M|=ϕ[t].Let us now look at this definition from a modal perspective.We view the as-signments as worlds and≡i as an accessibility relation.Then this definition is just the standard modal truth–definition of the“diamond”∃v i.Given afirst order model(D,I),the set of assignments(worlds)is uniquely determined:it is the setωD.The theory we will develop below abandons this classical rigidness:we will allow other subsets ofωD to be set of“worlds”of ourfirst order models.Before we can start we have to solve a technical difficulty.First order logic satisfies the following appealing locality condition.It says that the meaning of a formula depends only on the model and the variables occurring free in the formula.Fact3.1[Locality]Letϕbe afirst order formula.Let M=(D,I)be a model, and s,t be two assignments such that s(i)=t(i)for all v i∈FV(ϕ).ThenM|=ϕ[s]if and only if M|=ϕ[t].When we givefirst order logic a relativised semantics,locality does not neces-sarily hold,cf[1,6].We will give meaning to the existential quantifier using the dual of the relation≡¯v.This relation was introduced into cylindric alge-bra theory by Y.Venema.On relativised models,this will ensure locality,and on standard models,the meaning of the existential quantifier is just the same. Define for s,t assignments,¯v a set of variables,s≡∂¯v t ifffor all v i∈¯v s(i)=t(i).(3) Admissible assignments.The key idea of relativised semantics forfirst or-der logic is that given a model M=(D,I),only a subset of the set of all assignmentsωD is available for the interpretation of the formulas.We will now provide the truth definition for afirst order language relative to such an admis-sible set of assignments V⊆ωD.First we give meaning to terms:let s be an assignment,and M=(D,I)a model.We define a function i from the set ofterms into D asi(t)=I(t)if t is a constants(i)if t is the variable v i(4)Now for M=(D,I)a model,and V⊆ωD a set of assignments,we define truth of a formula in M relative to assignments in V.For s∈V,M|=V R(t1,...,t n)[s]⇐⇒(i(t1),...,i(t n))∈I(R)M|=V t1=t2[s]⇐⇒i(t1)=i(t2)M|=V¬ϕ[s]⇐⇒M|=Vϕ[s]M|=Vϕ∧ψ[s]⇐⇒M|=Vϕ[s]and M|=Vψ[s]M|=V∃¯vϕ[s]⇐⇒there exists a t∈V such thats≡∂FV(∃¯vϕ)t and M|=Vϕ[t].Note that the only difference with the definition in any textbook onfirst order logic is in the clause for the existential quantifier:the assignment t witnessing ϕmust be admissible,and we use the dual relation≡∂.For comparison,let us define|=c V—for“classical|=”,where all clauses are the same as for|=V above, except the existential quantifier is defined asM|=c V∃¯vϕ[s]⇐⇒there exists a t∈V such that s≡¯v t and M|=c Vϕ[t]. The next fact states that the two definitions are equivalent on classical models. Fact3.2Let M=(D,I)be a model and let V=ωD.Then for any s∈V, for any formulaϕ,M|=Vϕ[s]if and only if M|=c Vϕ[s].Proof.The only difference is in the meaning of the existential quantifier.That case goes through by the facts that locality holds on classical models and s≡¯v t implies that s≡∂FV(∃¯vϕ)t.qed The next fact states that on relativised models,locality holds as well.Fact3.3Let M=(D,I)be a model and V⊆ωD a set of admissible assign-ments.For any formulaϕ,for any s,t∈V such that s≡∂FV(ϕ)t(that is,s and t assign the same values to the free variables inϕ):M|=Vϕ[s]if and only if M|=Vϕ[t].Truth at non–admissible assignments.Given a model M=(D,I)and a set of admissible assignments V⊆ωD,we have defined what it means for a formula to be true in M at assignments in V.But what about the assignments inωD\V?The obvious way to do this for s∈ωD\V is as follows,M|=Vϕ[s]⇐⇒there exists t∈V such that t≡∂FV(ϕ)s and M|=Vϕ[t]. Note that for formulasϕ,M|=Vϕ[s]can still be undefined.If we assume that V is always non–empty,then for sentences however it is always defined.We now have two ways of defining truth in a model for sentences:M|=1Vϕiff(∀s∈ωD):M|=Vϕ[s]M|=2Vϕiff(∀s∈V):M|=Vϕ[s].For sentences,these two definitions give the same result:Fact3.4For every model M,for every non–empty V⊆ωD,for every sentence ϕ,M|=1Vϕif and only if M|=2Vϕ.Since|=2V is more economical,we will use this notion from now on and delete the superscript.Finally we define the notion of validity and of valid consequence.As usual we will overload the meaning of the symbol|=.LetΣbe a set of conditions onsets of admissible assignments.Letϕbe a sentence andΓa set of sentences. We define|=Σϕifffor everyfirst order model M=(D,I),for every V⊆ωD satisfyingΣ,M|=Vϕ.Γ|=Σϕifffor everyfirst order model M=(D,I),for every V⊆ωD satisfyingΣ,M|=VΓimplies M|=Vϕ.3.2Different relativisationsAbove we have defined relativised semantics for any choice of V⊆ωD.In the literature several restrictions on V have been proposed.Here we will show how one can define a set of admissible assignments from a tolerance relation on the domain of the model and from the notion of a context set[9].Tolerances were introduced in the previous section.For M=(D,I) a model,a context is just a subset of D.The intuitive meaning of a context X⊆D isall elements in X can be perceived together by the speaker.We will now define these notions and investigate their effects.Let M=(D,I)be afirst order model.Suppose f:D×D−→R+0is a function associating with every pair of elements in the model a value,which we think of as the distance between the elements.We can then defineδ(x,y)⇐⇒f(x,y)≤d for somefixed positive d.If we assume that f(x,x)=0,thenδsatisfiesδ1(∀x∈D):δ(x,x)δ2(∀xy∈D):(δ(x,y)→δ(y,x)).We call a relationδ⊆D×D satisfying these two requirements a tolerance on D.Let C be a collection of subsets of D.Let C satisfy the following two conditions: C1all singleton sets belong to CC2C is closed under subsets.If in addition C satisfies the following packed dense condition,C PD if for all x,y∈X,{x,y}∈C,then also X∈C,we call C a context set.Tolerances and context sets are closely related.Fact3.5(i)Ifδis a tolerance on D,then the set C defined byX∈C ifffor all x,y∈X,δxy,satisfies C1,C2and C PD.(ii)If C⊆P(D)satisfies C1,thenδdefined byδxy iffthere exists a set X∈C such that{x,y}⊆X,defines a tolerance on D.(iii)Tolerances and context sets are inter definable by the above definitions.We now relate tolerances and context sets to sets of admissible assignments.Definition3.6Let(D,I)be a model,and letδbe a tolerance and C a context set on(D,I).We define two sets of admissible assignments Vδand V C as the smallest subsets ofωD satisfyings∈Vδiff(∀i,j):δ(s(i),s(j))s∈V C iff{s(i)|i∈ω}∈C,respectively.The following fact is immediate by Fact3.5.Fact3.7Let(D,I)be a model,and letδbe a tolerance and C a context set on(D,I).Then Vδ=V C.Until now we have given only minimal requirements on the notions of context sets and tolerances.With our intended interpretation it makes sense to make them language–dependent as well(cf.,the clues provided in Section2).The following extra conditions make sense in a language with constants.¿From the perspective of a distance function,it says that the distance between any element and a named element is arbitrarily small.δ3(∀x∈D):δ(x,I(m))for all constants mC3(∀x∈D):{x,I(m)}∈C for all constants m.A further restriction onδ(and hence C)is to ask that the distance between two elements which stand together in a primitive relation is arbitrarily small. This would lead to the following extra conditions onδand C:δ4(∀x,y∈D):if(∃z1...z k(x=z i∧y=z j∧(z1,...,z k)∈I(R) for some R,i,j,thenδ(x,y).C4(∀x,y∈D):if(∃z1...z k(x=z i∧y=z j∧(z1,...,z k)∈I(R) for some R,i,j,then{x,y}∈C.We now look at the effect of the extra restrictions about constants and primitive relations on the logic.The conditionδ3does have an effect(of course only in languages with constants in the signature),whileδ4does not.Fact3.8There exists a sentence which can be falsified on a model with a set of admissible assignments Vδdefined by a toleranceδ,but which holds on all models where Vδis defined by a tolerance satisfyingδ3.Proof.Consider thefirst order tautology[∀x∃yRxy∧∀xyz((Rxy∧Ryz)→Sxz)]→∃xySxy.The counter model has the natural numbers as its domain,R is interpreted as successor,S=∅andδis generated by the successor relation(that isδ(x,y)iffx=y or Rxy or Ryx holds).Clearlyδis a tolerance and the consequent fails on this model.To see that the antecedent holds,observe that for no assignments∈Vδ,the range of s contains more than two elements.Thus the second conjunct in the antecedent cannot be falsified.On the other hand,consider any model M where Vδis defined from a tol-erance satisfyingδ3.Assume the antecedent holds in M.Let a be the element named by some constant a.Then by thefirst conjunct,there exists a b∈D such that Rab.Let s be the assignment sending every variable to b.Then M|=V∃yRby[s],whence there exists a c such that Rbc andδ(b,c)holds.But byδ3,alsoδ(a,b)andδ(b,c).But then{a,b,c}∈Vδ,whence by the truth of the second conjunct of the antecedent,Sac must hold.qedFor sentences,conditionδ4does not lead to extra validities.Fact3.9For every sentenceϕ,|={δ1,δ2,δ3}ϕif and only if|={δ1,δ2,δ3,δ4}ϕ.The same holds when we disregard conditionδ3.Proof.From left to right is obvious.For the other direction,assume M|=Vϕwhere V is defined from a tolerance not satisfyingδ4.Change the valuation of the relation symbols such thatδ4holds as well,by deleting any tuple¯a containing elements a i,a j which are notδ–related from the interpretation of every relation symbol.Call this model M .But then still M |=Vϕ,since to determine the truth of a sentence at an assignment in Vδone only needs toconsider assignments in Vδ.qedSumming up.We have given several ways of defining relativised semantics. Now it is time to make a choice.In a language without constants this would be easy:we only allow admissible assignments defined from a toleranceδ.Then, just because it is handy,we can ask for conditionδ4as well,since it does not alter the logic anyway.With constants in the language we should make a decision aboutδ3.Since it seems a natural condition and it makes the logic stronger,we have chosen to include that as well.So from now on we only use relativised models where the set of admissible assignments is defined from a toleranceδsatisfyingδ3andδ4(or equivalently,from a context set C which satisfies C3and C4).¿From now on a tolerance means a tolerance satisfying δ3andδ4.Definition3.10Let M=(D,I)be a model,andδ⊆D×D.The relationδis called a tolerance if it satisfiesδ1,δ2,δ3andδ4.3.3BisimulationsExplicit relativisations.Let M=(D,I)be a model and C⊂P(D)a context set on it.Recall that the intuitive meaning of C was as follows: X∈C if and only if all elements in X can be perceived together by thespeaker.The language does not have explicit means to state that two elements are per-ceived together.So we could add constants δij for every i,j ∈ω,and provide them with the following meaning.For any s ∈ωD ,M |=δij [s ]if and only if {s (i ),s (j )}∈C .(5)Following Tarski,we call an operation logical if it’s truth is preserved under automorphisms.Clearly δij is not a logical constant.But intuitively it should not be one in this sense.δij indicates that the elements denoted by s (i )and s (j )are part of a group of elements which can be perceived together.Arbitrary automorphisms can destroy this intuitive meaning of δij .On the other hand,the truth of δij is preserved under automorphisms which respect C .Fact 3.11Let M =(D,I )be a model and C ⊂P (D )a context set on it.Let g be an automorphism of M such that for any set X ⊆D ,X ∈C if and only if {g (x )|x ∈X }∈C .Then for any s ∈ωD ,M |=δij [s ]if and only if M |=δij [g (s )].Note that δij is true on every s ∈V C ,so on admissible assignments it is equiv-alent to .In particular the following equivalence holds.M |=V C ∃¯v ϕ[s ]if and only if M |=V C ∃¯v ( {δij |v i ,v j ∈FV (ϕ)}∧ϕ)[s ].What we just did is to make the implicit relativisation to admissible assignments in the meaning definition of the quantifiers explicit in the object language.This provides us with a translation to ordinary first order logic as follows.Define recursively the following translation function (·)δfrom first order for-mulas to first order formulas.(·)δdoes nothing to atomic formulas,it commutes with the booleans and(∃¯v ϕ)δ=∃¯v ( {δ(v i ,v j )|v i ,v j ∈FV (ϕ)}∧ϕδ).Here δis just a binary predicate.As expected we have,Fact 3.12Let M =(D,I )be a model and δa tolerance on it.Then for every formula ϕ,for all s ∈V δ,M |=V δϕ[s ]if and only if (D,I,δis defined as the set{(x,y )∈D ×D |x,y stand in the tolerance relation δ},and forms the interpretation of the binary predicate δ.Bisimulations and packed sets.Let(D,I)be a model andδa tolerance on it.We call a set X⊆Dδ–packed ifδ(x,y)holds for all x,y∈X.Then Vδ—the set of admissible assignments defined fromδ—is just the set of all sequences whose elements form aδ–packed subset of ing this we can define the appropriate notion of bisimulation for this logic.Note that the definition is very close to the one for the guarded fragment in[2].Two pieces of notation come handy:define for s∈ωD,R(s)={s(i)|i∈ω}.Also for g a function from D to D ,and s∈ωD,defineg(s)=that sequence inωD such that for all i,g(s)(i)=g(i).Definition3.13(Bisimulation)Let M=(D,I)and N=(D ,I )be two models for the same signature.LetδM andδN be tolerances on them respec-tively.A family F offinite partial isomorphisms between D to D is called a δ–bisimulation if F satisfies the following conditions:•if f∈F and g⊆f,then also g∈F•(totality)–for everyδ–packed set X⊆D,there exists an f∈F whose domain is X–similar forδ–packed subsets of D•(forth)if f∈F and dom(f)⊆X for someδ–packed set X,then there exists a g∈F which extends f and whose domain is X•(back)a similar condition in the other direction.Note that bisimulations are always non–empty,by totality and the fact that every singleton set isδ–packed.Of course we have the followingFact3.14For everyϕ,for every M,N,for everyδ–bisimulation F between them,for every f∈F,and for every assignment s such that R(s)=dom(f),M|=Vϕ[s]if and only if N|=Vϕ[f(s)].Proof.The proof is by induction on formulas.We only consider the case for the existential quantifier.So let M|=V∃¯vϕ[s]and f∈F with dom(f)=R(s). The case when∃¯vϕis a sentence is easy and left to the reader(use totality). So suppose otherwise.Then there exists a t∈Vδsuch that t≡∂FV(∃¯vϕ)s and M|=Vϕ[t].Let s be such that R(s )={s(i)|v i∈FV(∃¯vϕ)}.Then by locality also M|=V∃¯vϕ[s ].Since F is closed under subsets,also f R(s )∈F. From t≡∂FV(∃¯vϕ)s it follows that R(s )⊆R(t).Whence by forth,there exists a g⊇f R(s )with dom(g)=R(t).Thus by induction hypothesis,N|=Vϕ[g(t)]. But g⊇f s implies f R(s )(s )≡∂FV(∃¯vϕ)g(t).Thus N|=V∃¯vϕ[f R(s )(s )], whence by locality N|=V∃¯vϕ[f(s)].qed4Packed fragmentIn this section,we look for sentences whose truth in a model is unaffected by adding or deleting a tolerance.The syntactic characterisation of this fragment forms a slight generalisation of van Benthem’s loosely guarded fragment.We first define the fragment.We work in a standardfirst order language with equality with one restriction:terms are variables or constant symbols.We say that a formulaϕpacks a set of variables{x1,...,x k}ifϕis a conjunction of formulas of the form t i=t j or R(t1,...,t n)or∃¯y R(t1,...,t n) such that for every x i=x j,there is a conjunct inϕin which x i and x j both occur free.In the definition of the packed fragment we use generalised quantifiers ∀¯x(ϕ,ψ)where¯x=x1,x2,...,x n is a sequence of variables.The meaning of this quantifier is nothing but the meaning of∀x1...∀x n(ϕ→ψ)infirst order logic.A generalised quantifier∀¯x(ϕ,ψ)is called packed ifϕpacks all free variables ofψ.We callϕthe guard of∀¯x(ϕ,ψ).Note that ifψcontains only one free variable,then thefirst argument of the universal quantifier can be anything: packedness only speaks about pairs of variables.The packed fragment is defined as follows:a packed formula is constructed from atoms using the booleans and packed universal quantification∀¯x(ϕ,ψ), whereψmust be a packed formula.It will be useful to define two more fragments.A packed existential quan-tification is nothing but¬∀¯v(ϕ,¬ψ),where∀¯v(ϕ,¬ψ)is a packed universal quantification(i.e.,it is of the form∃¯v(ϕ∧ψ),whereϕpacks all free variables ofψ).The∀–packed fragment is defined as follows:formulas are constructed from atoms and their negations using∧,∨,∃and packed universal quantification ∀¯x(ϕ,ψ),whereψmust be a∀–packed formula.The∃–packed fragment is defined dually:so we may use unpacked∀,but only packed∃.The three fragments are of course closely relatedFact4.1Afirst order sentenceϕis equivalent to a packed sentence if and only if it is equivalent to a∀–and a∃–packed sentence.We will now related the packed fragment to relativised semantics.One direction is obvious,and observed in[8].Fact4.2The translation(·)δgoes to the packed fragment.Just as allfirst order sentences are invariant forδ–bisimulations when they are interpreted relativised to a set of admissible assignments,all packed sen-tences are invariant forδ–bisimulations when they are classically interpreted. Definition4.3A sentenceϕis invariant forδ–bisimulations if for all models M,N,for all tolerancesδM,δN,and for allδ–bisimulations F:M F N,M|=ϕif and only if N|=ϕ.。

a r X i v :c o n d -m a t /9604012v 1 2 A p r 1996Incomplete melting of the Si(100)surface frommolecular-dynamics simulations using the Effective-MediumTight-Binding model.K.Stokbro,1,2,3K.W.Jacobsen,3J.K.Nørskov,3D.M.Deaven,4C.Z.Wang,4and K.M.Ho 41Mikroelektronik Centret,Danmarks Tekniske Universitet,Bygning 345ø,DK-2800Lyngby,Denmark.2Scuola Internazionale Superiore di Studi Avanzati,via Beirut 4,I-34014Trieste,Italy.3Center for Atomic-scale Materials Physics and Physics Department,Danmarks Tekniske Universitet,DK 2800Lyngby,Denmark 4Ames Laboratory,Ames,Iowa 50011Abstract We present molecular-dynamics simulations of the Si(100)surface in the temperature range 1100-1750K.To describe the total energy and forces we use the Effective-Medium Tight-Binding model.The defect-free surface is found to melt at the bulk melting point,which we determine to be 1650K,but fora surface with dimer vacancies we find a pre-melting of the first two layers100K below the melting point.We show that these findings can rationalizerecent experimental studies of the high temperature Si(100)surface.Typeset using REVT E XI.INTRODUCTIONMost work on the Si(100)surface has focussed on the behaviour at low and intermediate temperatures,while to our knowledge there have to date only been two experimental studies [1,2]of the atomic structure at temperatures near the bulk melting point.Both of these studies indicate that the surface undergoes a phase transition at a temperature below the bulk melting point,however,there is no consistent picture of the nature of the transition, the exact transition temperature,and the atomic structure beyond the phase transition.Metois and Wolf[1]studied the structure of the Si(100)surface with Reflection-Electron Microscopy(REM).In the temperature interval1400-1455K the experiment shows a sudden increase in the number of holes on the surface,and at1455K the surface structure cannot be resolved any longer and it is suggested that the surface at this temperature becomes rough at the atomic scale.In the experimental study by Fraxedas et.al[2]the surface order of Si(100)was studied as a function of the surface temperature by means of X-ray Photo-electron Diffraction(XPD).At1400K the order suddenly decreases,and this is attributed to a phase transition where the twofirst surface layers melt and form a liquid layer.For the Si(100)surface molecular-dynamics simulations based on the Stillinger-Weber potential have been used to study the solid-liquid interface[4],but there are no studies of the surface melting.To our knowledge,the onlyfirst principles molecular-dynamics study of the melting of a semiconductor surface is by Takeuchi et.al.[3].They study a Ge(111) surface at a temperature close to the melting point andfind evidence for the existence of an incomplete melted phase.The experiment by Metois and Wolf shows that the Si(100)surface has an increasing concentration of vacancies as a function of the temperature.Since the vacancy formation takes place on a time scale of seconds,it is not possible to study the vacancy formation in a molecular dynamics simulations,however,given a certain vacancy concentration we can study what is the surface structure attained within a time scale of≈10−100ps.In this report we present a molecular-dynamics study of the Si(100)surface with andwithout vacancies for a number of temperatures in the range1100-1750K.To describe the bonding of Si we have used the Effective-Medium Tight-Binding(EMTB)model[5].We show that this approximate total energy method describes the melting point and the prop-erties of liquid Si very ing the computational efficiency of the method to simulate a slab with a unit cell containing188atoms for up to35ns at a number of differ-ent temperatures and surfaces with and without defects,we show that while the defect-free surface shows no sign of premelting,a surface with25%of dimer vacancies shows melting of thefirst layer100K below the melting point.We show that the results of the simulation can be used to understand the experimental observations for this system.II.TECHNICAL DETAILSThe Effective-Medium Tight-Binding(EMTB)model[5]is based on Effective-Medium Theory[6,7]in which the total energy calculations are simplified by comparisons to a ref-erence system(the effective medium).The so-called one-electron energy sum is calculated using an LMTO tight-binding model[8].The main approximations in the model is the use of afirst-order LMTO tight-binding model and a non-self-consistent electron density [9]and potential[10].In previous studies we have demonstrated the ability of the model to describe the low temperature structure of the Si(100)surface,[5]and generally we have found the accuracy for the total energy and atomic structure of solid Si systems to be cor-rect within10-20percent,which is comparable to plane-wave calculations with a cutoffof 8Ry.However,the EMTB method is in computational efficiency comparable to empirical tight-binding methods and for the system sizes treated here something like three orders of magnitude faster than plane-wave calculations.It is therefore very well suited for large scale molecular-dynamics simulations.The super-cell for the calculation is a12layer thick slab with16atoms in each layer except for the upper layer in which we have introduced a vacancy concentration of25percent by removing four atoms;thus a total of188atoms.Since all atoms are free to move we obtaininformation of a clean surface and a surface with vacancies in the same simulation run.The classical equations of motion for the nuclei are integrated using the Verlet algorithm with a time step of1.08fs and we control the temperature by using Langevin dynamics[11]with a friction coefficient of2ps−1.The lattice constant is determined by scaling the EMTB (T=0K)value(10.167a0)using the experimental thermal-expansion coefficient.III.RESULTSInitially,the atomic structure of the bottom surface consists of two dimer rows with each4dimers,while the vacancies are introduced on the upper surface by removing a dimer from each row.Starting from the c(2x4)reconstruction wefirst increase the temperature of the thermostat to1100K.After thermalization this system is followed over3ps.Thefinal configuration is used as input for four other simulation temperatures:1450K,1550K,1650K, and1750K,which are followed over29ps,35ps,33ps and18ps,respectively.Wefirst summarize the calculations by showing in Fig.1the average structure factor parallel to the dimer rows,|S−11|2,as a function of the simulation temperature.The structure factor is projected onto3intervals of the z axis corresponding to the position of layers1-2, 3-4,and5-6on both the vacancy covered and the clean surface.The structure factor is averaged over the atomic configurations of the last2ps,15ps,15ps,15ps and5ps for each of thefive simulations temperatures,respectively.The calculated variances in the structure factors show that except for the1650K calculation,the simulations seem to have obtained an equilibrium state.In Fig.2we show the corresponding average atomic density perpendicular to the surfaces,and for the1550K calculation a snapshot of the atomic-structure is shown in Fig.3.The observed atomic structures obtained in the simulations are the following:At1100K the two surfaces show no tendency to break up the dimer rows;each dimer stays in a buckling mode moving up and down with a frequency of≈3THz,and the buckling angle reverses independently of the configurations of the other dimers.For the clean surface wefind this tobe the picture up to1750K where the entire slab melts,and Fig.1b shows that the structure factor for this surface is constant up to the melting transition.The vacancy surface shows a much more complex behaviour.At1450K we observe diffusion of one of the vacancies within the simulation time.After the diffusion the dimers remain intact and the structure factor shown in Fig.1a is only slightly affected.At1550K the twofirst layers of the vacancy surface lose all their order and form a disordered state, see Fig.3,and there is a strong interaction with the third layer which forms an interface between the bulk structure and the disordered surface.From Fig.1a we see that now the structure factor has vanished for the two outermost layers,while the structure factor of the third layer is slightly decreased due to the interaction with the disordered state.At1650K thefirst four layers disorders(cf.Fig.1a and Fig.2).At1750K all the layers melt after a short transient period,and Fig.2shows a constant atomic density n=2.54g/cm3in the range−10a0<z<10a0.This is an increase of6percent compared to the crystalline density,and in good agreement with the experimental atomic density n=2.51g/cm3for liquid silicon at1750K.Wefirst extract from the simulations the Si bulk melting point,T m,predicted by the EMTB model.At1750K the whole slab melts,without a partially-melted transient period, and we therefore have T m<1750K.At1550K there are two phases present in the calculation with a large interaction between the two.Since Fig.1a shows that there are only small fluctuations in the structure factor of the layers,this strongly suggests that the simulation has obtained an equilibrium state.Furthermore,if the melting temperature was close to 1550K,the temperature in the1650K simulation would be well above the melting point and the slab should readily melt since a melted phase is already present in the calculation. From these considerations we estimate a melting temperature of T m≈1650±100K,in good agreement with the experimental melting point of1685K.The rather large error bar in the melting temperature comes fromfinite size effects leading to largefluctuations in the temperature.On top of this there may be a systematic overestimate of the melting temperature in the simulation because of thefixed super-cellsize parallel to the slab.However,the possibility of the system to respond with volume changes perpendicular to the slab reduces this problem considerably.The presence of the surface thus allows the liquid to acquire the correct density as noted above.In the simulation with the thermostat set at1650K the temperaturefluctuates around the melting point,and we may have a coexistence of solid and liquid phases at this temperature. This suggests that the melting of the two additional layers in the1650K simulation is rather due to largefluctuations of the temperature around the melting point than the existence of a surface phase transition where four layers disorders.Note also the largefluctuations observed in the structure factor of layer5-6at this temperature,shown in Fig.1a,which makes it questionable whether an equilibrium state has been obtained within the simulation time.The temperature in the1550K simulation,on the other hand,is well below the bulk melting point and the atomic structure in this case must be due to a surface phase transition taking place at a temperature T F<1550.From simulations of metal surfaces it is known[14] that just at the surface melting temperature the defects become mobile,and we therefore estimate T F=1450±100K.Since we have T F<1550<T M we can expect to have obtained equilibrium within our simulation time for this temperature.We now consider the1550K simulation in more detail.In Fig.4we show the radial distribution function averaged over the atoms in layer2and layer6-7.For comparison we also show similar averages obtained for the1750K simulation in which case the whole slab has become liquid.Thefigure clearly shows that the two outermost layers in the1550K simulation have formed a liquid state,with an average coordination very different from the bulk state,but almost identical to that of liquid Si.It is worth noting that the radial distribution function obtained for the liquid Si is in good agreement with experiment[12]and that obtained by Stich et.al.[13]in a Car-Parrinello calculation(the position of thefirst maximum is4.67a0(4.65a0)and integration up to5.90a0gives an average coordination of6.55(6.5),where the values obtained in Ref.[13]are given in parenthesis). The formation of a liquid surface bilayer in the1550K simulation is further illustrated byFig.5,where we from the lateral mean-square displacement of the surface atoms deduce a two dimensional diffusion constant of D=2.1×10−5cm2/s.We also observe a jump in the average potential energy during melting at1550K from which we deduce a latent heat of melting the surface atoms of∆E≈3eV.This must be compensated by an increase in the entropy∆E=Nk B T∆S(where N is the number of atoms taking part in the phase transition),which when attributed to the atoms in the twofirst layers(N=28),gives an increase in the entropy of∆S≈0.8.We now relate our simulations for the Si(100)to those of Ref.[3]for the Ge(111)surface. The outermost layer of the Ge(111)surface consists of fourth a monolayer Ge adatoms, which gives this surface a more open structure than the Si(100)surface.The Ge(111)surface has more resemblance with the vacancy covered Si(100)surface,and there are indications that the adatoms on the Ge(111)surface and the vacancies on the Si(100)surface play similar roles in the surface melting process.Similar to the vacancy diffusion we found in the1450K simulation,Takeuchi et.al.found a metastable state in thefirst part of their simulation where only the adatoms diffuse.Afterwards the Ge(111)surface undergoes a phase transition where the twofirst layers become liquid,and the increase in entropy and the diffusion constant are very similar to the values we have found for the vacancy covered Si(100)surface.(For Ge(111)the values are D Ge=3.5×10−5cm2/s,∆S Ge≈1.0).For the Ge(111)surface it was also found that the electronic structure of the melted layers was metallic,and lately this has been supported by an electron energy loss spectroscopy (EELS)study of the surface conductivity of Ge(111)[15].However,we onlyfind a weak metallic behaviour for the incomplete melted Si(100)surface.This is illustrated in Fig.6, which shows the Projected Density of States(PDOS)of the atoms in the twofirst layers of the vacancy surface(solid line)and the clean surface(dotted line),and we see that the number of states in the gap is only slightly increased for the incomplete melted vacancy surface compared to the non-melted clean surface.We therefore expect that the surface melting of Si(100)will be difficult to detect with EELS.PARISON WITH EXPERIMENTSWe now compare ourfindings to the experimental results of Ref.[1,2].One of our results is that the phase transition only takes place when a high concentration of vacancies is present on the surface,and the question arises what are the vacancy concentrations in the exper-iments.STM studies yield estimates that the surface at room temperature has a vacancy concentration of5-10percent.[16]When the surface is heated up one may observe atoms evaporating,and since they leave a vacancy behind this process may increase the vacancy concentration.Metois and Wolf found that at temperatures below1400K the sublimation process proceeds slowly enough,that the sublimation holes arefilled with atoms diffusing from step edges and the vacancy concentration is relatively stable up to this temperature. In the temperature interval1400-1455K the number of holes increases drastically and at 1455K the surface structure cannot be resolved any longer.The experiment by Metois and Wolf is in a non-equilibrium state and they relate the observed behaviour to kinetic surface roughening.The kinetic roughening is caused by the larger activation energy for sublimation compared to the activation energy for diffusion, which means that the sublimation rate will increase faster with temperature than the dif-fusion rate,and at some temperature the sublimation rate will become too fast for the vacancies to befilled with atoms diffusing from step edges.However,we note that the ob-served behaviour is not necessarily due to kinetic roughening but can alternatively be due to the existence of a roughening phase transition.Both kinetic roughening and a roughening phase transition will give rise to an increased vacancy concentration as a function of the temperature,and the vacancy concentration in the experiments of Ref.[1,2]at1450K will therefore be well above the5-10percent concentration at room temperature.From the avail-able data we cannot determine the exact vacancy concentration in the two experiments,and probably they will not have the same concentrations,since it will depend on experimental details,such as base pressure and the crystal miscut angle.It is not possible to use molecular dynamics simulations to determine the(equilibrium ornon-equilibrium)concentrations of vacancies since the simulation time scales are too short. However,for a given vacancy concentration molecular dynamics can be used to study the attained surface structure within a time scale of≈10−100ps,which is long compared to fundamental surface vibration frequencies.In the simulation we have studied a vacancy concentration of25percent,where wefind a phase transition at1550K to an incomplete melted state with two melted layers.We expect the same phenomena to take place for higher or slightly lower vacancy concentrations.However,the results cannot be generalized to low vacancy concentrations,since in this case the time scale of the formation of an incomplete melted phase may become comparable to the time scale of the processes that alter the vacancy concentration.For instance,on the time scale of desorption of Si dimers(seconds) the simulation for the clean surface will show incomplete melting,since the desorption of Si dimers will give rise to an increased vacancy concentration on the surface.In a real experiment there will also be adatoms diffusing of step edges and adsorption of atoms from the gas phase,and these processes will lower the vacancy concentration.We now make quantitative comparisons with the experimental study of Ref.[2],where the XPD anisotropy(A(hkl)(T))is measured along the(100)and(111)directions as a function of temperature.The anisotropy in a given direction is defined as A(hkl)(T)= (I max(T)−I min(T))/(I max(300K)−I min(300K)),where I max(I min)stands for the maxi-mum(minimum)intensity for a given peak.We have simulated the XPD intensity peaks using Single-Scattering Cluster(SSC)calculations[17],with the same SSC parameters as quoted in Ref.[18].For the electron mean-free path the value is25˚A(corresponding to the energy of the2s photo electrons),in spite of the generally accepted rule of using half of its nominal value.[19]The anisotropy along the(111)direction is extremely sensitive to the actual choice of mean-free path,and we have therefore chosen only to make comparison with the anisotropy along the(100)direction,which we have found to be rather insensitive to the value of the mean-free path.For each simulation temperature we have calculated the anisotropy along the(100)di-rection for20different configurations and Fig.7shows the corresponding averages.Wehave performed calculations using both a mean-free path of12.5˚A and25˚A.In each case the anisotropies have been scaled such that at1100K the anisotropy of the clean surfacefits the experimental measurements,and Fig.7shows that the relative anisotropies are quite insensitive to the choice of mean-free path.We may therefore relate our calculations directly to the experimental measurements.First of all we see that vacancies lead to a drop in the anisotropy,and the observed increase in the vacancy concentration of Ref.[1]should there-fore be measurerable by XPD.Wefind that an increase in the vacancy concentration by10 percent would be consistent with the small decrease in the experimental anisotropy observed in the temperature range≈1390−1425K.Since we must expect[16]that the surface al-ready has a vacancy concentration around10percent at1390K,the vacancy concentration will be20percent at1425K,very similar to the concentration in our simulation.An increase in the vacancy concentration,however,can hardly describe the overall drop in the anisotropy observed in the range1390-1500K.Fig.7indicates strongly that the decrease in the anisotropy after1425K is related to the defect-induced disordering we have described above.This picture is in accordance with Ref.[18],where they found that the experimentally observed drop in anisotropy could be explained by assuming a disordered bilayer at the surface.The experiment also shows that the thickness of the melted layer is constant upto the bulk melting point,again indicating that the melting of the two additional layers in the1650K simulation is related to bulk melting rather than surface melting.V.SUMMARYIn summary,we have performed molecular-dynamics simulations for the Si(100)surface using the EMTB model.Wefind that in the presence of a vacancy concentration of25 percent the surface undergoes a phase transition at1550K,where the twofirst layers melt and form a liquid state.The incompletely melted state shows several similarities to the incomplete melting of Ge(111)[3],except that the electronic surface structure shows a much weaker metallic behaviour.We have also shown that the simulation results are in bothquantitative and qualitative agreement with experimental studies of the surface structure.ACKNOWLEDGMENTSWe thank C.Fadley for providing us the program for the SSC calculations and J.Fraxedas for instructions on how to use the program.Many discussions with P.Stoltze and E.Tosatti are gratefully acknowledged.The Center for Atomic-scale Materials Physics is sponsored by the Danish National Research Foundation.Kurt Stokbro acknowledges eec contract ERBCHBGCT920180and contract ERBCHRXCT930342and CNR project Supaltemp.REFERENCES[1]J.Metois and D.Wolf,Surf.Sci.298,71(93).[2]J.Fraxedas,S.Ferrer,and in,Europhys.Lett.25,119(1994).[3]N.Takeuchi,A.Selloni,and E.Tosatti,Phys.Rev.Lett.72,2227(1994).[4]ndman,W.D.Luedtke,R.N.Barnett,C.L.Clevelannd,M.W.Ribarsky,E.Arnold,S.Ramesh,H.Baumgart,A.Martinez,and B.Kahn,Phys.Rev.Lett.56,155 (1986).[5]K.Stokbro,N.Chetty,K.Jacobsen,and J.Nørskov,Phys.Rev.B50,10727(1994).[6]J.Nørskov and ng,Phys.Rev.B21,2131(1980).[7]K.Jacobsen,J.Nørskov,and M.Puska,Phys.Rev.B35,7423(1987).[8]O.Andersen and O.Jepsen,Phys.Rev.Lett.53,2571(1984).[9]N.Chetty,K.Jacobsen,and J.Nørskov,Lett.J.Phys.Condens.Matter3,5437(1991).[10]K.Stokbro,N.Chetty,K.Jacobsen,and J.Nørskov,J.Phys.Condens.Matter6,5415(1994).[11]D.Heermann,Computer Simulation Methods in Theoretical Physics(Springer-Verlag,Berlin,Heidelberg,1986).[12]Y.Waseda and K.Suzuki,Z.Phys.B.20,339(1975);P.Gabathuler and S.Steeb,Z.Naturforsch.Teil A34,1314(1979).[13]I.Stich,R.Car,and M.Parrinello,Phys.Rev.Lett.60,204(1988).[14]P.Stoltze,J.K.Nørskov,and ndman,Phys.Rev.Lett.61,440(1988).[15]S.Modesti,et.al.,Phys.Rev.Lett.73,1951(1994).[16]R.Hamers,R.Tromp,and J.Demuth,Phys.Rev.B34,5343(1986).[17]C.Fadley,Prog.in Surf.Sci.16,275(1984).[18]J.Fraxedas,S.Ferrer,and in,Surf.Sci.307-309,775(1994).[19]D.Naumovic et al.,Phys.Rev.B47,7462(1993).FIGURESFIG.1.The equilibrium average structure factor parallel to the dimer rows,|S−11|2,as a function of the simulation temperature.The structure factor is projected onto3intervals of the z axis corresponding to the position of layer1-2,3-4,and5-6,for a)The vacancy covered surface, and b)the clean surface.FIG.2.The equilibrium average atomic density perpendicular to the surfaces for each simula-tion temperature.FIG.3.A snapshot of the atomic structure in the1550K simulation at t=30ps.The supercell is repeated twice in the horizontal direction.FIG.4.The radial distribution function projected onto layer2and layer6-7for the1550K simulation(solid line)and the1750K simulation(dashed line).FIG.5.The atomic displacement in the surface plane projected onto layer1-2,3-4,and5-6, as a function of time for the1550K simulation.The2-dimensional diffusion constant,D,for layer 1-2is defined by<r2>=4Dt.FIG.6.The PDOS of layer1-2of the vacancy surface(solid line)and the clean surface(dotted line)in the1550K simulation.The average is taken over the last15ps of the simulation.FIG.7.The XPD anisotropy as obtained from SSC calculations with an electron mean-free path of12.5˚A(solid line)and25˚A(dashed line)as a function of the simulation temperature.The experimental measurements are from Ref.[2].-20-15-10-5051015201100K1450K 1550K 1650K1750Kρ(z ) [a r b i t r a r y u n i t s ]0246810r [a 0]01234g (r )05101520253035time [ps]050100(x −x 0)2+(y −y 0)2[a 02]。

heterogeneous element -回复Heterogeneous Element: A Journey Through Diversity and UnityIntroduction:In a world filled with diverse cultures, belief systems, and perspectives, the concept of a heterogeneous element has become a vital aspect of our society. A heterogeneous element refers to any entity that displays a combination of different elements, whether it be people, objects, or ideas. This article aims to explore the significance and impact of such elements, focusing on their manifestation in various realms of human life and highlighting the ways in which they contribute to the unity of our global community.1. Defining Heterogeneous Elements:To begin, let us delve into the definition of a heterogeneous element. Put simply, it is an entity that possesses different characteristics or qualities within itself. These elements can be found in a wide range of contexts, such as biodiversity, cultural diversity, or even technological advancements. The beauty of these elements lies in their unique ability to blend different componentstogether, offering new perspectives and possibilities.2. Heterogeneous Elements in Nature:One of the most striking manifestations of heterogeneous elements can be observed in nature through biodiversity. From the vast array of plant and animal species to the intricate ecosystems they inhabit, biodiversity encompasses the remarkable variety that exists on our planet. Each species contributes to the overall balance and functionality of the ecosystem, creating a harmonious blend of different life forms. This rich diversity not only enhances the aesthetics of nature but also plays a vital role in maintaining ecological stability.3. Heterogeneous Elements in Culture:Moving on to the cultural realm, heterogeneous elements are prevalent in the diverse traditions, customs, and beliefs that shape our societies. Different cultures bring their own unique sets of values and perspectives, enriching our collective understanding of the world. The celebration of cultural diversity fosters unity by encouraging mutual respect, appreciation, and interculturaldialogue. The exchange of ideas and experiences between cultures can lead to innovation, creativity, and greater social cohesion.4. Heterogeneous Elements in Technology:Advancements in technology also embrace heterogeneous elements. The field of artificial intelligence, for instance, combines various disciplines such as computer science, mathematics, and cognitive psychology, creating a hybrid approach toproblem-solving. By amalgamating different components, breakthroughs in technology become possible, leading to novel solutions and improvements in various aspects of human life. The power of collaboration between diverse fields of expertise is exemplified by the development of the internet, which has revolutionized communication, commerce, and the exchange of information globally.5. Heterogeneous Elements in Society:In society, heterogeneous elements can be observed in the composition of communities and organizations. A diverse workforce, for example, brings together people from differentbackgrounds, ethnicities, and skillsets. This amalgamation of perspectives and experiences enhances problem-solving capabilities, fosters creativity, and promotes inclusion. It has been proven that diverse communities and organizations perform better when it comes to innovation and adaptability, thanks to the diversity of ideas and insights brought to the table.6. Nurturing Unity Through Heterogeneous Elements:While heterogeneous elements thrive on diversity, they also have the potential to foster unity. By recognizing the value and unique contributions of each individual or component, we can build a stronger, more harmonious society. Embracing the concept of unity in diversity requires active efforts to promote inclusive practices, educate others about different perspectives, and provide platforms for open dialogue. Organizations and communities that successfully embrace heterogeneous elements are likely to experience increased creativity, productivity, and satisfaction among their members.7. Conclusion:In conclusion, the concept of a heterogeneous element serves as a powerful reminder of the richness, depth, and complexity of the world we live in. From nature's biodiversity to cultural exchanges and technological advancements, these elements symbolize the endless possibilities that arise when diversity is embraced and celebrated. By valuing and actively promoting heterogeneous elements, we can harness collective strengths, build bridges between different communities, and create a more united and inclusive global society. Let us embrace the heterogeneous element in all its forms, for it is through diversity that we truly flourish.。

A fractal analysis of permeability for fracturedrocksTongjun Miao a ,b ,Boming Yu a ,⇑,Yonggang Duan c ,Quantang Fang caSchool of Physics,Huazhong University of Science and Technology,1037Luoyu Road,Wuhan 430074,Hubei,PR ChinabDepartment of Electrical and Mechanical Engineering,Xinxiang Vocational and Technical College,Xinxiang 453007,Henan,PR China cState Key of Oil and Gas Reservoir Geology and Exploitation,Southwest Petroleum University,8Xindu Road,Chengdu 610500,Sichuan,PR Chinaa r t i c l e i n f o Article history:Received 6September 2013Received in revised form 28March 2014Accepted 5October 2014Keywords:Permeability Rock Fractal FracturesFracture networksa b s t r a c tRocks with shear fractures or faults widely exist in nature such as oil/gas reservoirs,and hot dry rocks,etc.In this work,the fractal scaling law for length distribution of fractures and the relationship among the fractal dimension for fracture length distribution,fracture area porosity and the ratio of the maxi-mum length to the minimum length of fractures are proposed.Then,a fractal model for permeability for fractured rocks is derived based on the fractal geometry theory and the famous cubic law for laminar flow in fractures.It is found that the analytical expression for permeability of fractured rocks is a function of the fractal dimension D f for fracture area,area porosity /,fracture density D ,the maximum fracture length l max ,aperture a ,the facture azimuth a and facture dip angle h .Furthermore,a novel analytical expression for the fracture density is also proposed based on the fractal geometry theory for porous media.The validity of the fractal model is verified by comparing the model predictions with the available numerical simulations.Ó2014Elsevier Ltd.All rights reserved.1.IntroductionFractured media and rocks with shear fractures or faults widely exist in nature such as oil/gas reservoirs,and hot dry rocks,ually,the fractures are embedded in porous matrix with micro pores,which play negligible effect on the seepage characteristic,and randomly distributed fractures dominate the seepage charac-teristic in the media.The randomly distributed fractures are often connected to form irregular networks,and the seepage character-istic of the fracture networks has the significant influence on nuclear waste disposal [1],oil or gas exploitation [2],and geother-mal energy extraction [3].In this work,we focus our attention on the seepage characteristics of fracture networks in fractured rocks and ignore the seepage performance from micro pores in porous matrix.Over the past four decades,many investigators studied the seepage characteristics of fracture networks/rocks and proposed several models.Snow [4]developed an analytical method for per-meability of fracture networks according to parallel plane model.Kranzz et al.[5]studied the permeability of whole jointed granite and tested the parallel plane model by experiments.Koudina et al.[6]investigated the permeability of fracture networks with numer-ical simulation method in the three-dimensional space,they assumed that fracture network consists of polygonal shape frac-tures and fluid flow in each fracture meets the Darcy’s law.Dreuzy et al.[7]studied the permeability of randomly fractured networks by numerical and theoretical methods in two dimensions,and they verified the validity of the model by comparing to naturally frac-tured networks.Klimczak et al.[8]obtained the permeability of a single fracture by parallel plate model with the fracture length and aperture satisfying power-law and verified by the numerical simulation.However,these models did not provide a quantitative relationship among the permeability of fracture networks,poros-ity,fracture density and microstructure parameters of fractures,such as fracture length,aperture,inclination,orientation etc.Fractures in rocks are usually random and disorder and they have been shown to have the statistically self-similar and fractal characteristic [3,9–13].Chang and Yortsos [10]studied the single phase fluid flow in the fractal fracture networks.Watanabe and Takahashi [3]investigated the permeability of fracture networks and heat extraction in hot dry rock by using fractal method.But,they did not propose an expression of permeability with micro-scopic parameters included.Jafari and Babadagli [14]obtained the permeability expression with multiple regression analysis of random fractures by the fractal geometry theory according to observed data in the well logging.In addition,their expression with several empirical constants does not include the orientation factor and microstructure parameters of fracture networks.The tree-like fractal branching networks were often considered as/10.1016/j.ijheatmasstransfer.2014.10.0100017-9310/Ó2014Elsevier Ltd.All rights reserved.⇑Corresponding author.E-mail address:yubm_2012@ (B.Yu).fracture networks by many investigators.Xu et al.[15,16]studied the seepage and heat transfer characteristics of fractal-like tree networks.Recently,Wang et al.[17]studied the starting pressure gradient for Binghamfluid in a special dual porosity medium with randomly distributed fractal-like tree network embedded in matrix porous media.Most recently,Zheng and Yu[18]investigated gas flow characteristics in the dual porosity medium with randomly distributed fractal-like tree networks.However,the fractal-like tree network is a kind of ideal and symmetrical network.The purpose of the present work is to derive an analytical expression and establish a model for permeability of fracture rocks/media based on the parallel plane model(cubic law)and frac-tal geometry theory.The proposed permeability and the predicted fracture density will be compared with the numerical simulations.2.Fractal characteristics for fracture networksMany investigators[3,9–13,19–23]reported that the relation-ships between the length and the number of fractures exhibit the power-law,exponential and log-normal types.Torabi and Berg [19]made a comprehensive review on fault dimensions and their scaling laws,and they summarized several types of scaling laws such as the length distributions for faults and fractures in siliciclas-tic rocks from different scales and tectonic settings.The power-law exponents of the scaling-law between the fault length and the number of faults were found to be in the range of1.02–2.04and are probably influenced by factors such as stress regime,linkage of faults,sampling bias,and size of the dataset.Interested readers may consult Refs.[3,9–13,19–23]for detail.In addition,the self-similar fractal structures of fracture net-works were extensively studied[22,23],and the application in complex rock structures with the fractal technique was recently reviewed by Kruhl[24].Velde et al.[25]and Vignes-Adler et al.[26]studied the data at several length scales with fractal method and found that the fracture networks are fractal.Barton and Zoback [27]analyzed the2D maps of the trace length of fractures spanning ten orders,ranging from micro to large scale fractures and found that D f=1.3–1.7.The width between two plates/walls of a fracture,i.e.the paral-lel plate model is used to represent the effective aperture of a frac-ture.Generally,the relationship between the effective aperture a and the fracture length l is given by[28,29]a¼b l nð1Þwhere b and n are the proportionality coefficient and a constant according to fracture scales,respectively.The value of n=1is important,which indicates a linear scaling law,and the fracture network is self-similarity and fractal[19,29].Thus,in the current work the value of n=1is chosen for fractures with fractal characteristic.Thus,Eq.(1)can be rewritten asa¼b lð2ÞEq.(2)will be used in this work.It is well-known that the cumulative size distribution of islands on the Earth’s surface obeys the fractal scaling law[30]NðS>sÞ/sÀD=2ð3aÞwhere N is the total number of island of area S greater than s,and D is the fractal dimension for the size distribution of islands.The equality in Eq.(3a)can be invoked by using s max to represent the largest island on Earth to yield[31]NðS>sÞ¼s maxsD=2ð3bÞEq.(3b)implies that there is only one largest island on the Earth’s surface,and Majumdar and Bhushan[31]used this power-law equation to describe the contact spots on engineering surfaces,where s max¼g k2max(the maximum spot area)and s¼g k2(a spot area),with k being the diameter of a spot and g being a geometry factor.It has been shown that the length distribution of fractures sat-isfies the fractal scaling law[3,9–13,19,22,23,32],hence,Eq.(3b) for description of islands on the Earth’s surface and spots on engi-neering surfaces can be extended to describe the area distribution of fractures on a fractured surface,i.e.NðS!sÞ¼a max l maxalD f=2ð3cÞwhere a max l max represents the maximum fracture area with a max and l max respectively being the maximum aperture and maximum fracture length,and al refers to a fracture area with the aperture and length being a and l,respectively.Inserting Eq.(2)into Eq.(3c),we obtainNðS!sÞ¼b l2maxb l!D f=2ð3dÞThen,from Eq.(3d),the cumulative number of fractures whose length are greater than or equal to l can be expressed by the follow-ing scaling law:NðL!lÞ¼l maxD fð4Þwhere D f is the fractal dimension for fracture lengths,0<D f<2(or 3)in two(or three)dimensions;and Eq.(4)implies that there is only one fracture with the maximum length.Some investigators [3,9–13,19,32]reported that the length distribution of fractures in rocks has the self-similarity and the fractal scaling law can be described by N/ClÀD f,where C is afitting constant,D f is the fractal dimension for the length(l)distribution of fractures and N is the number of fractures,and this fractal scaling law is similar to Eq.(4).Eq.(4)is also the base of the box-counting method[33]for mea-suring the fractal dimension of fracture lengths in fracture net-works,and Chelidze and Guguen[9]applied the box-counting method and found that the fractal dimension of fracture network (described by Nolen-Hoeksema and Gordon[34])in a2D cross sec-tion is1.6.Since there usually are numerous fractures in fracture net-works,Eq.(4)can be considered as a continuous and differentiable function.So,differentiating Eq.(4)with respect to l,we can get the number of fractures whose lengths are in the infinitesimal rang l to l+dl:ÀdNðlÞ¼D f l D fmaxlÀðD fþ1Þdlð5ÞEq.(5)indicates that the number of fractures decreases with the increase of fracture length andÀdN(l)>0.The relationship among the fractal dimension,porosity and the ratio k max=k min for porous media was derived based on the assump-tion that pores in porous media are in the form of squares with self-similarity in sizes in the self similarity range from the mini-mum size k min to the maximum size k max,i.e.[35]D f¼d Eþln emax minð6Þwhere e is the effective porosity of a fractal porous medium,d E is the Euclid dimension,and d E=2and3respectively in two and three dimensions.It has been shown that Eq.(6)is valid not only for exactly self-similar fractals such as Sierpinski carpet and Sierpinski gasket but also for statistically self-similar fractal porous media.Fractures in rocks or in fractured media are analogous to pores in porous media.Therefore,Eq.(6)can be extended to describe the76T.Miao et al./International Journal of Heat and Mass Transfer81(2015)75–80relationship among the fractal dimension for length distribution, porosity of fractures and the ratio l max/l min of fractures in rocks,i.e.D f¼d Eþln/lnðl max=l minÞð7Þwhere l max and l min are the maximum and the minimum fracture lengths,respectively,and/is the effective porosity of fractures in a rock.The area porosity/of fractures is defined as/¼A PAð8Þwhere A is the area of a unit cell,A P is the total area of all fractures in the unite cell.Based on Eq.(5),the total area of all fractures in the unite cell can be obtained asA p¼ÀZ l maxl min aÁlÁdNðlÞ¼b D f l2max2ÀD f1Àl minl max2ÀD f"#ð9ÞInserting Eq.(7)into Eq.(9)yieldsA p¼b D f l2max2ÀD f1À/ðÞð10Þwhere porosity/is applied in Eq.(7)in two dimensions,i.e.d E=2is used.3.Relationship between fracture density and fractal dimensionThe total fracture lengths in a unit cell of area A can be obtained byl total¼ÀZ l maxl min lÁdNðlÞ¼D f l max1ÀD f1Àl minl max1ÀD f"#ð11ÞThe fracture density is defined by[36]D¼l totalð12Þwhere l total is the total length of all fractures(not a single fracture) which may be connected to form a network in the unit cell.Inserting Eqs.(7),(8)and(11)into Eq.(12)results in the fracture densityD¼ð2ÀD fÞ/1Àl minl max1ÀD fð1ÀD fÞb l max1Àl minmax2ÀD fð13aÞInserting Eq.(7)into Eq.(13a),the fracture density can also bewritten asD¼ð2ÀD fÞ1À/ðÞ1ÀD ff"#/ð1ÀD fÞb l max1À/ðÞð13bÞIt is evident that the fracture density D of fractures is a functionof the fractal dimension D f for fracture area,area porosity/,proportionality coefficient b and l max.Fig.2compares the predictions by the present fractal model(Eq.(13a))with numerical simulations of four groups of randomfracture networks by Zhang and Sanderson[36],who proposed anew numerical method for producing the self-avoiding randomgenerations,and the parameters such as the lengths of fracturescan be controlled.In their simulations,the lengths of fractures liefrom0.0005to1.5m,and the averaged fractal dimension D f is1.3.So,in this work we take the maximum length and minimumlength of fractures are1.5m and0.0005m,respectively,and theaveraged fractal dimension D f=1.3.The average porosity/is0.018calculated by Eq.(7).It can be seen from Fig.2that the pre-dictions are in good agreement with the numerical simulations.Fig.2clearly indicates that the fracture density increases withthe increase of the fractal dimension,and this is consistent withpractical situation.Fig.3presents the fracture density versus porosity of fracturenetworks as l max=1.5m,b=0.01.It can be seen from Fig.3thatthe fracture density increases with porosity.This can be explainedthat the pore area of fractures increasing with porosity means thatthe fracture density increases with porosity.This result is in agree-ment with the Monte Carlo simulations by Yazdi et al.[37].4.Fractal model for permeability of fractured rocksThe orientation of each fracture in fracture networks is definedby two angles,the fracture azimuth and fracture dip angle,whichsignificantly affect theflow and transport properties.The orienta-tions of fractures in a fracture network are non-uniform,but usu-ally with a preferred orientation[38,39].In general,the numberof fractures in fracture networks is very large.Based on generalpractice,the fracture azimuths of all fractures are taken as aver-aged/mean angle,for instance,Massart et al.[40]showed a meandip angle of70°,mean N–S(North–South)orientation from thetotal number of1878fractures.In this work,the mean dip angleof fractures between fracture orientations andfluidflow direction,and the mean azimuth of fractures perpendicular tofluidflowdirection are assumed to be h and a,respectively(see Fig.1(a)).(a)(b)T.Miao et al./International Journal of Heat and Mass Transfer81(2015)75–8077Therefore,the scalar quantity of permeability alongflow direction needs to be calculated.Iffluidflow through fractures is assumed to be laminarflow,the flow rate along theflow direction through a fracture can be described by the famous cubic law[41,42]qðlÞ¼a3l12lD PL0ð14Þwhere L0is the length of the structural unit,l is the fracture trace length,a is the fracture aperture,and D P is the pressure drop across a fracture alongflow direction.If the single fracture forms an angle with theflow direction,due to the projection on theflow direction of the fracture,theflow rate through the fracture can be written by[43,44]qðlÞ¼a3l1Àcos2a sin2h12lD PL0ð15Þwhere a and h are respectively the mean facture azimuth and facture dip angle.When a=0,Eq.(15)is reduced toqðlÞ¼a3l cos2h12lD PL0ð16ÞThis is the famous Parsons’model.See Fig.1(b)[43,44].The totalflow rate through all the fractures can be obtained by integrating Eq.(16)from the minimum length to the maximum length in a unit cross section,i.e.Q¼ÀZ l maxl minqðlÞdNðlÞ¼b3128lD f1Àcos2a sin2h4ÀD fD PL0l4max1Àl minl max4ÀD f"#ð17Þwhere D f represents the fractal dimension for the length distribu-tion of fractures.In general,l min<<l max.Since0<D f<2in two dimensions,andðl min=l maxÞ4ÀD f<<1,so that Eq.(17)can be simplified as:Q¼b3128lD f1Àcos2a sin2h4ÀD fD PL0l4maxð18ÞEq.(18)indicate that the totalflow rate through the fracture net-work is related to the fractal dimension D f of the fracture lengths, the facture azimuth a and facture dip angle h.Eq.(18)also indicates that theflow rate is very sensitive to the maximum fracture length l max.Darcy’s law for Newtonianfluidflow in porous media is given byQ¼KAlD PL0ð19ÞComparing Eq.(18)to Eq.(19),we can obtain the permeability for Newtonianfluidflow through the fracture networks asK¼b3128AD f1Àcos2a sin2h4ÀD fl4maxð20ÞInserting Eqs.(12)and(13b)into Eq.(20),the permeability for Newtonianfluidflow through fracture networks can be written asK¼b3D1281ÀD f4ÀD fl3max1Àcos2a sin2h1À/ðÞ1ÀD ff"#ð21ÞEq.(21)shows that the permeability is a function of the fractal dimension D f for the fracture length distribution,the structural parameters(maximum fracture length l max,fracture density D,fac-ture azimuth a and facture dip angle h)and fracture porosity/of fracture networks.Eq.(21)also reveals that the permeability strongly depends on the maximum fracture length l max,and the longer fracture with wider apertures conduct the higher volume offluid and higher permeability.As a result,the present fractal model can well reveal the mechanisms of seepage characteristics78T.Miao et al./International Journal of Heat and Mass Transfer81(2015)75–80in fracture networks than conventional methods.For example, many investigators proposed fracture network models by assum-ing that the media have ideal structures such as the parallel frac-ture network[4,5,8,45],the orthogonal plane network cracks [46,47],alternate level matrix layer and fractures[48]etc.The frac-ture network permeability was often expressed as K=/a2/12, where/is fracture porosity and a is fracture aperture.Recently, Jafari and Babadagli[14]obtained an expression(with several empirical constants)by fractal geometry for fracture networks according to the well logging and observation data.Therefore,it is clear that Eq.(21)has the obvious advantages over the conven-tional models/methods.5.Results and discussionIn this section,the model predictions will be compared with the simulated data and the effects of model parameters on the perme-ability will be discussed.The procedures for determination of the relevant parameters in Eq.(21)are as follows:(1)Given the fracture network parameters(such as l max,/,a,hand b)based on a real sample.(2)Find the fractal dimension D f of fracture lengths in a fracturenetwork by the box-counting method or by Eq.(7).(3)Determine the fracture density D by Eq.(13b).(4)Finally,calculate the permeability by Eq.(21).Jafari and Babadagli[49]obtained the fractal dimensions D f of2D maps from22different nature fracture networks by box-counting method,and then they calculated the equivalent fracture network permeability by a3D model with a block size of100Â100Â10m simulated/constructed.The maximum fracture length was taken to be2m and dip angle h=0.In comparison,the fracture density D and permeability are calculated by procedures3and4,respec-tively.Fig.4shows that the present model predictions are in good agreement with the simulation results[49].Fig.5depicts the permeability for Newtonianfluid through frac-ture networks against porosity of fracture networks at different dip angles at l max=10mm,b=0.01.It is seen from Fig.5that the per-meability for fracture networks increases with porosity.This is consistent with practical situation.From Fig.5,we can also see that the permeability decreases as the fracture plane dip angle increases.This can be explained that a higher fracture plane dip angle leads to an increase of theflow resistance.Fig.6plots the permeability versus the fracture density of the fracture networks at l max=10mm,a=0,h=p/4and b=0.01.It suggests that the permeability of the fracture networks increases with the increases of fracture density.The reason is that when the fracture density D increases,the area of fracture networks increases and thus results in increasing the permeability.This result agrees with the numerical simulation results in Ref.[50]. 6.ConclusionsIn this paper,the fractal geometry theory has been applied to describe the fractal fracture system,and the fractal scaling law for length distribution of fractures and the relationship among the fractal dimension for fracture length distribution,fracture area porosity and the ratio of the maximum length to the minimum length of fractures have been proposed.Then,a model for perme-ability of fractured rocks has been derived based on the famous cubic law,fractal geometry theory and technique.A novel expres-sion for the fracture density has also been proposed based on the fractal scaling law of length distribution of fractures.The present results show that the permeability of fracture networks increases with the increases of porosity and fracture density.Our results agree well the available numerical simulations.This verifies the validity of the proposed models.It should be point out that the percolation and critical behavior are not involved in this work.In this paper,we focus on the perme-ability that all fractures are assumed to be connected to form frac-ture network,which contributes the permeability of the fracture system.This means that we have ignored the interaction between fractures.The permeability after including the interaction and con-nectivity between fractures and critical behavior of fractures near the threshold undoubtedly is an interesting topic,and this may be our next workConflict of interestNone declared.AcknowledgmentThis work was supported by the National Natural Science Foundation of China through Grant Number10932010. 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