Determination of workpiece flow stress and friction at the chip-tool contact for high-speed cutting

International Journal of Machine Tools &Manufacture 40(2000)133–152

Determination of workpiece ?ow stress and friction at the

chip–tool contact for high-speed cutting

Tug

?rul O ¨zel a,*,Taylan Altan b a Department of Industrial and Manufacturing Engineering,Cleveland State University,Cleveland,OH 44115-2425,

USA

b Engineering Research Center for Net Shape Manufacturing,The Ohio State University,Columbus,OH 43210-

1271,USA

Received 12August 1998;accepted 12April 1999

Abstract

This paper presents a methodology to determine simultaneously (a)the ?ow stress at high deformation rates and temperatures that are encountered in the cutting zone,and (b)the friction at the chip–tool interface.This information is necessary to simulate high-speed machining using FEM based programs.A ?ow stress model based on process dependent parameters such as strain,strain-rate and temperature was used together with a friction model based on shear ?ow stress of the workpiece at the chip–tool interface.High-speed cutting experiments and process simulations were utilized to determine the unknown parameters in ?ow stress and friction models.This technique was applied to obtain ?ow stress for P20mold steel at hardness of 30HRC and friction data when using uncoated carbide tooling at high-speed cutting conditions.The average strain,strain-rates and temperatures were computed both in primary (shear plane)and secondary (chip–tool contact)deformation zones.The friction conditions in sticking and sliding regions at the chip–tool interface are estimated using Zorev’s stress distribution model.The shear ?ow stress (k chip )was also determined using computed average strain,strain-rate,and temperatures in secondary deformation zone,while the friction coef?cient (m )was estimated by minimizing the difference between predicted and meas-ured thrust forces.By matching the measured values of the cutting forces with the predicted results from FEM simulations,an expression for workpiece ?ow stress and the unknown friction parameters at the chip–tool contact were determined.?1999Elsevier Science Ltd.All rights reserved.

Keywords:Orthogonal cutting;High-speed cutting;FEM simulation of cutting;Flow stress modeling;Friction at high-speed cutting

*Corresponding author.Tel.:+1-216-523-7251;fax:+1-216-687-9330.E-mail address:t.ozel@https://www.wendangku.net/doc/994680595.html, (T.O

¨zel)0890-6955/00/$-see front matter.?1999Elsevier Science Ltd.All rights reserved.

PII:S 0890-6955(99)00051-6

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Nomenclature

F c Tool cutting force(per unit width)in orthogonal cutting

F t Tool thrust force(per unit width)in orthogonal cutting

F n Normal force(per unit width)on the tool rake face in orthogonal cutting

F f Friction force(per unit width)on the tool rake face in orthogonal cutting

k chip Average shear?ow stress at chip–tool interface

l c Chip–tool contact length on the tool rake face

l p Chip–tool contact length on the sticking region of the tool rake face

m Shear friction factor

t u Uncut/undeformed chip thickness

t c Cut/deformed chip thickness

T ave Average temperature in the deformation zones

V c Cutting speed/velocity

V f Feed rate

a Rake angle

eˉStrain

eˉ·Strain-rate

eˉ·n Nominal average strain rate in the primary deformation zone

eˉ·ave Average strain rate in deformation zones

g Clearance angle

f s Shear angle at the primary deformation zone

r Tool cutting edge hone radius

s n Normal stress on the tool rake face

sˉFlow stress of work material

t f Frictional stress over the entire tool rake face

1.Introduction

High-speed cutting(HSC)of hard alloy steels(up to hardness of62HRC)offers several advan-tages such as reduction of?nishing operations,elimination of part distortion,achievement of high metal removal rates and lower machining costs as well as improved surface integrity[1].However, HSC results in high temperatures and stresses at the tool–workpiece interface.Consequently,cost-effective application of this technology requires a fundamental understanding of the relationships between process variables and cutting conditions.Thus,it is necessary to understand how tempera-tures and stresses,developed during HSC,in?uence tool wear and premature tool failure as well as residual stresses on machined surfaces.

In machining hard materials,continuous(Type II)chip formation is observed at conventional to high cutting speeds and low to moderate feed rates.However,at higher feed rates“saw tooth”or“shear localized”(Type IV)chips are produced[2].The latter type of chip formation can cause cyclic variations of both cutting and thrust forces and can result in high frequency vibrations that

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135 affect tool life and failure[3].Throughout the scope of this work,only the cutting conditions that result in continuous chip formation have been considered.Therefore,the mechanics and simulation of saw tooth(shear localized)chip formation is not discussed in this paper.

2.Background in orthogonal metal cutting

Metal cutting process can be considered as a deformation process where deformation is highly concentrated in a small zone[4].Thus,cutting can be considered as a chip formation process and simulated using Finite Element Method(FEM)techniques.The main advantage of such an approach is to be able to predict chip?ow,cutting forces,tool temperatures and stresses that result in various cutting conditions.However,material?ow characteristics,or?ow stress,at high temperature and deformation rates are required to conduct such predictions.There are very few material?ow stress data available for the deformation conditions that exist in machining.Those data are mainly obtained by using impact compression tests for various materials at the moderate deformation rates[5].However,further data is needed to overcome the uncertainty in the high temperature and strain rate material property data suitable for simulation of high-speed cutting. Classic orthogonal cutting model for continuous chip formation assumes plane-strain defor-mation conditions.Basic representation of the process model is illustrated in Fig.1.Geometric relations in orthogonal cutting model yield the following equations for normal and frictional forces,F n and F f,on the tool rake face as functions of measured cutting force components,F c and F t,and tool rake angle as:

Fig.1.Forces generated during orthogonal cutting process.

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F n?F c cos a?F t sin a(1)

F f?F c sin a?F t cos a(2) In conventional machining at low cutting speeds,the friction mechanism is mostly effective at the tool?ank face.However,in high-speed machining due to tremendous increase in the chip velocity the chip–tool contact friction is much more signi?cant at the tool rake face[6].As a result,the increasing sliding velocity and frictional stress cause signi?cant wear on the tool rake face.Therefore,the rate of the tool wear in high-speed machining depends heavily on the chip–tool contact conditions.

2.1.Stress distributions on the tool rake face

The normal(s n)and frictional shear(t f)stress distributions at the chip–tool interface charac-terize the cutting temperature and tool wear.Those distributions are commonly represented as shown in Fig.2[7].According to Zorev[7],two regions exist simultaneously on the tool rake face when machining under dry conditions.From the tip of the tool up to a point,frictional stress is considered constant in a sticking region.After this point,frictional stress decreases on the tool rake face in a sliding region where Coulomb’s friction law can be applied.

Fig.2.Curves representing normal(s n)and frictional(t f)stress distributions on the tool rake face[7].

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2.2.Friction characteristics on the chip–tool interface

The contact regions and the friction parameters between the chip and the tool are in?uenced by factors such as cutting speed,feed rate,rake angle,etc.[8],mainly because of the very high normal pressure at the surface.The prevalent conditions at the chip–tool interface constrain the use of the empirical values of the coef?cient of friction found from ordinary sliding test conditions. While numerous works have been reported to quantitatively explain the variable and high values of friction at the rake face,none has provided reliable quantitative predictive models that are devoid of experimental testing.

2.3.Flow stress models for workpiece materials

The?ow stress or instantaneous yield stress at which workpiece material starts to?ow is mostly in?uenced by temperature(T),effective strain(eˉ),effective strain-rate(eˉ·),and microstructure(S), i.e.chemical composition,phases,grain size.Thus,the?ow stress(sˉ)can be given as:

sˉ?f(T,eˉ,eˉ·,S)(3) For practical cutting speeds in machining,the average strain-rate in the shear zone lies in the range of103to105sec?1or even higher[4].These values are much higher than the strain rates of10?3to10?1sec?1that are normally encountered in compression and tension tests.Many researchers developed several techniques to determine the?ow stress of metals at high strain-rates and temperatures[9–11].Some assumed that for a particular strain-rate and temperature combination the relationship between the effective?ow stress and effective strain for the work material considered varying with strain-rate(e˙)and temperature[12].Later Oxley used velocity-modi?ed temperature that consist of strain-rate and temperature for carbon steels[4].

Some used an impact compression material testing machine to measure?ow stress at tempera-tures20–1000°C and strain-rates200–2000sec?1[12].Empirical expressions for the?ow stress including the history effects of the temperature and strain-rate without anneal softening and age hardening effects are also developed[5].As an example,an expression for the?ow stress of cold work mold steel(Cr–Mo steel)is here given[13]:

sˉ?A(10?3eˉ·)M e kT(10?3eˉ·)kT[?T,eˉ?(eˉ·)e?kT/N(10?3eˉ·)?m/N d eˉ]N(4) where,

A(T)=1.46exp(?0.0013T)+0.196exp{?0.000015(T?400)2}?0.0392exp{?0.01(T?100)2}

N(T)=0.162exp(?0.001T)+0.092exp{?0.0003(T?380)2}

M(T)=0.047,k=0.000065,m=0.0039

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3.Modeling of high-speed orthogonal cutting process

Earlier models of metal cutting were based on only basic shear plane assumption or slip line ?eld analysis[14,15].Friction conditions at the chip–tool interface in early FEM models of metal cutting were ignored[16]or assumed to be constant with a coef?cient of friction based on Cou-lomb’s law at the chip–tool interface[17].Usui[12]modeled the tool–chip interaction with a frictional force as a function of normal force as boundary conditions using the shear?ow stress of the workpiece at the primary zone obtained from experiments.Iwata[18]incorporated the frictional stress as a function of normal stress at the boundary conditions with an empirical relationship based on a coef?cient of friction obtained from friction tests.Shih[19]used a constant coef?cient of friction in the sticking region at the chip–tool interface and a coef?cient linearly decaying to zero in the sliding https://www.wendangku.net/doc/994680595.html,ter,models that include both chip–tool contact friction and material behavior at high strains,strain-rates and temperatures were proposed[4]and note-worthy attempts for FEM simulation of cutting processes were presented[20–23].The FEM approach proposed by Marusich et al.has led to a commercially available metal cutting simulation software which appears to be successful[22].

Recently,orthogonal cutting was also simulated using a software developed for large plastic deformations,DEFORM?and chip formation for continuous and segmented chips were predicted using a fracture criteria[24].Capabilities in generating a very dense mesh near the tool tip and remeshing adaptively makes this software applicable to simulate the cutting process.Although the assumed input data for material properties and friction were quite approximate,simulation of metal cutting was carried out with relatively little effort[25].These preliminary investigations demonstrated that with reliable input data on material properties it is possible to estimate chip ?ow and cutting forces.In addition,this model was also extended to simulate chip?ow in2-D ?at end milling with straight cutting edges and veri?ed with experiments[26].Work is in progress to extend this approach to estimate temperatures and stresses in cutting operations involving3-

D chip?ow,e.g.turning and end milling with nose radius tools[27].

4.Methodology to determine?ow stress and friction at chip–tool interface

In HSC,extremely high strain-rates(about1.67×105sec?1at500m/min cutting speed and 0.05mm undeformed chip thickness)and temperatures(about1400°C)at the chip–tool interface occur in the primary deformation zone and secondary deformation zone,respectively.To address the issues of?ow stress and friction,a methodology was developed for determining simul-taneously both the?ow stress of workpiece material and the friction conditions at the chip–tool contact interface.

The basic concept of the proposed methodology is the use of orthogonal cutting experiments and FEM simulations in order to determine the?ow stress and friction conditions used for the range of high-speed cutting.Therefore,a limited number of orthogonal end turning experiments on P20mold steel disks(at hardness of30HRC)was conducted using uncoated tungsten carbide (WC)tooling(Fig.3).From the experiments,two components of cutting force(F c and F t),chip thickness(t c),and chip–tool contact length(l c)were measured.In addition,the microscopic pic-

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Fig.3.A schematic for tool and workpiece geometry used in high-speed orthogonal cutting experiments.

tures of chips were collected to identify chip https://www.wendangku.net/doc/994680595.html,ter,FEM simulations of continuous chip?ow in orthogonal cutting process were conducted.

4.1.High-speed orthogonal cutting experiments

The purpose of these experiments was to obtain cutting force data for a process model of orthogonal cutting.

In these orthogonal cutting experiments Fig.3,the following conditions were used:

?Workpiece:P20mold steel disks,3mm thickness,30HRC hardness

?Tool:uncoated carbide(WC)inserts,rake angle a=?7°,edge radius r=0.012mm

?Cutting speed,V c:200,300and550m/min

?Feed rate,V f:0.025,0.051,0.075and0.100m/rev

Experiments were replicated twice at each cutting condition in order to minimize experimental errors.A sudden tool failure occurred during experiments at a cutting speed of550m/min and feed rates of0.075and0.100mm/rev.

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4.2.Chip formation

The shape and geometry of the chips were investigated using the optical-microscope Fig.4.In this analysis,the chip shapes were mostly continuous(Type II).However,saw-tooth shaped chips were observed at conditions with cutting speed of300m/min and undeformed chip thickness of 0.100mm.Nevertheless,it can be stated that the chip formation indicates continuous type of chips at the lower feed rates(?0.100mm/rev)and at the lower cutting speeds(?550m/min)when high-speed cutting of P20mold steel.This observation supports the shear localization behavior of hard steels in the secondary deformation zone due to runaway thermo-mechanical deformation at

Fig.4.Chip geometry measured from the experiments in orthogonal cutting of P20mold steel.

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high feed and cutting speeds postulated by Zhen-Bin et al.[28].Furthermore,it is in direct opposition to the brittle fracture hypothesis postulated by Elbestawi et al.[29].It should be noted that simulation of shear localized chip formation is not studied in this work.

4.3.Cutting force measurements

The cutting force(F c)in the cutting direction(along z-axis)and thrust force(F t)in the feed direction(along y-axis),Fig.3,were measured during orthogonal turning of P20mold steel disks. In each experiment,a fresh part of the cutting tool was used and experiments were replicated twice at each cutting condition in order to reduce experimental error.In Fig.5,measured cutting forces(F c)and thrust forces(F t)per unit width of cut are presented.Although a signi?cant increment in cutting forces with the increase of undeformed chip thickness was found,this behavior was rather insigni?cant for thrust forces.This is due to high and consistent friction contact between chip and tool at the tool rake face.However,it should be noted that additional replication of the experiments may change the average values of cutting forces as opposed to the average of two replications.

4.4.Flow diagram of the methodology to determine?ow stress and friction

For the process simulation of high-speed cutting of P20mold steel(30HRC)with uncoated carbide(WC)tooling,the?ow stress data and friction model,described earlier,were used.At each of the cutting conditions,the chip?ow was simulated,and cutting force(F c),thrust force (F t),deformed chip thickness(t c)and chip–tool contact length(l c)were predicted.The predictions were compared with measurements and the?ow stress data and friction values were modi?ed until an acceptable correlation could be obtained between experimental and predicted values.A detailed?ow chart explaining this procedure is given in Fig.6.

4.5.Material properties for process simulation of orthogonal cutting

Since chemical composition of P20mold steel is similar to the Cr–Mo steel,the?ow stress equation obtained for Cr–Mo steel was assumed to be also valid for P20mold steel.Therefore, the data for Cr–Mo steel,given in Eq.(4),was used as initial?ow stress[13].Other data for input preparation of the process simulations is given in Table1.

4.6.Estimation of friction conditions at chip–tool interface

Friction conditions at the chip–tool contact can be interpreted in terms of two frictional modes, which are represented by shear friction and friction coef?cient relations Fig.2.Constant shear stress law(t f=sˉ/√3=k chip)with Von Mises plastic?ow criterion and friction coef?cient,m=t f/s n need to be properly de?ned on the sticking region(0?x?l p)and the sliding region(l p?x?l c), respectively.

An appropriate initial value for the friction coef?cient(m p,initial)was selected as the ratio of the frictional force(F f)and normal force(F n)acting on the tool rake face.Both forces were calculated from the measured force components at the given rake angle.

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Fig.5.Measured cutting(F c)and thrust(F t)forces per unit width and the in?uence of cutting speed.

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143

Fig.6.Flow diagram representing the procedure for determination of workpiece?ow stress behavior and friction conditions at chip–tool interface.

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Table1

List of cutting conditions,material properties for P20mold steel workpiece and uncoated carbide tool used in the process simulations

Orthogonal cutting conditions

Cutting speed,V c,(m/min)200,300,550

Feed,uncut/undeformed chip thickness,t u,(mm)0.025,0.051,0.075,0.100

Width of cut(mm)1

Properties of workpiece(P20mold steel)

Emissivity(?)0.60(100°C)0.65(500°C)0.75(1000°C)

Coef?cient of thermal expansion(10?6/°C) 1.3(425°C)1.4(650°C)

Speci?c heat(J/kg/°C)470

Thermal conductivity(W/m°C)51.5

Poisson’s ratio(?)0.3

Young’s modulus(GPa)260

Geometry and properties of cutting tool(Tungsten Carbide F Grade)

Tool tip radius,r,(mm)0.012

Rake angle,a,(deg)?7

Clearance angle,g,(deg)15.3

Emissivity(?)0.5

Coef?cient of thermal expansion(10?6/°C) 5.2

Speci?c heat(J/kg/°C)343.3

Thermal conductivity(W/m°C)120

Piosson’s ratio(?)0.22

Young’s modulus(GPa)522(20°C)620(100°C)

F n,estimated?(F c,measured)cos a?(F t,measured)sin a(5)

F f,estimated?(F c,measured)sin a?(F t,measured)cos a(6) Therefore,the initial friction coef?cient was calculated using measured cutting forces as follows:

m p,initial?F f,estimated

F n,estimated

(7)

This average friction coef?cient was found between0.5to0.7except for the undeformed chip thickness of0.025mm which was about1.0.The increase in the mean friction coef?cient was due to so-called size effect of the cutting edge.At the undeformed chip thickness of0.025mm, the ratio of the cutting edge radius(r=0.012mm)to the undeformed chip thickness was0.5.It is well known that the higher this ratio is,the higher the speci?c cutting pressure,i.e.friction force. When using the simulation software,DEFORM-2D?,the most appropriate way of implementing chip–tool contact friction is to use a variable friction coef?cient that is a function of the normal pressure at the tool rake surface(m i=f(s n)).Since the uniformly distributed shear frictional stress in the sticking region is known to be equal to the local shear?ow stress(t f=k chip), the friction coef?cient in the sticking region was de?ned as:

m i?m0?k chip

s max

at x?0(8)

145

T.O zel,T.Altan /International Journal of Machine Tools &Manufacture 40(2000)133–152m i ?k chip s n 0?x ?l p (9)

In the sliding region,a constant friction coef?cient (m p )was de?ned as:

m i ?m p l p ?x ?l c (10)Thus,this friction model includes two major unknowns,the local shear ?ow stress of the chip (k chip )in the sticking region,and a constant friction coef?cient (m p )in the sliding region,used in the process simulations.

https://www.wendangku.net/doc/994680595.html,putation of average strain rate and temperatures in deformation zones

In order to determine ?ow stress data by using proposed methodology,average strain-rates,strains and temperatures for primary and secondary deformation zones are computed after each process simulation for different cutting conditions.Figs.7and 8indicate the primary and second-ary deformation zones.Process simulation predicts the state variables such as strain,strain-rate,temperature and Von Mises stresses.By using the post processor ?les for element connectivity,coordinates of the nodes,de?ned boundary conditions,strain-rates,and node temperatures,the

Fig.7.Predicted strain-rate distribution and identi?ed primary and secondary regions in orthogonal cutting of P20mold steel (V c =550m/min,t u =0.051mm).

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133–152Fig.8.Predicted temperature distribution and the average temperatures in orthogonal cutting of P20mold steel (V c =550m/min,t u =0.051mm).

deformation zones were identi?ed using a computer program developed.In the identi?ed primary and secondary deformation zones,average strain-rates (e ˉ·ave )and temperatures (T ave )were calcu-lated as listed in Table 2.These values were calculated as area-weighted averages by extracting the simulated data and from the following equation:

f ave ??n

i ?1

f i A i ?n

i ?1A i

(11)Table 2

Calculated average temperatures and strain-rates in the primary and the secondary deformation zones

V f =0.051mm/rev

V f =0.075mm/rev Primary zone Secondary zone Primary zone Secondary zone V c (m/min)T ave (°C)e

ˉ·ave (sec ?1)T ave (°C)e ˉ·ave (sec ?1)T ave (°C)e ˉ·ave (sec ?1)T ave (°C)e ˉ·ave (sec ?1)200

53560,74673636,03144452,85775128,497300

544120,44876240,59655088,17583332,961550560157,62578871,190581122,09898985,450

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147

Table3

Determined shear?ow stress and estimated friction coef?cients for friction at chip–tool contact interface

V f=0.051mm/rev V f=0.075mm/rev

V c(m/min)k chip(MPa)m p m0k chip(MPa)m p m0 2009070.70.189150.50.18 3009110.60.189320.50.18 5509230.50.189540.60.18

where f i=predicted value of strain rate(eˉ·i)and temperature(T i),A i=area of the element i,and n=number of elements in primary or secondary deformation zones.

The extent of primary deformation zone for calculation was de?ned as the region in which elemental strain-rate is higher than the value of nominal average strain rate(eˉ·n)determined from cutting conditions as:

eˉ·n?V c/t u(12) At the same cutting conditions,process simulations were conducted in an iterative scheme until predicted thrust and cutting force match with the experiments.The unknowns of the friction model,shear?ow stress in the chip and the friction parameters,were found from iterations as listed in Table3.

By using the determined?ow stress and friction data,process simulations were conducted repeatedly and the predicted cutting forces were compared in Figs.9and10.The prediction error

??0.10).

in each simulation was set to be less than10%for cutting and thrust force(?eˉF

c,F t

https://www.wendangku.net/doc/994680595.html,parison of predicted and measured cutting forces per unit width of cut at cutting speed of200m/min.

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133–https://www.wendangku.net/doc/994680595.html,parison of predicted and measured cutting forces per unit width of cut at cutting speed of 300m/min.

4.8.Identi?cation of coef?cients in ?ow stress expression

An empirical expression of ?ow stress,developed by Maekawa [5],was used in order to rep-resent the ?ow stress data determined for high-speed machining conditions (cutting speed up to 550m/min).

s ˉ?K (e aT ?A e b (T ?T 0)2)?e ˉ·e ˉ·R ?

c (e ˉ)

d (13)In Eq.(13),th

e speci?ed parameter (e ˉ·R )is introduced to neutralize the strain-rate unit.The

unknown coef?cients,a ,b ,c ,d ,A ,K ,and T 0,were identi?ed by using determined ?ow stress data at temperatures of 20,100,500,1000°C,strains 0.01,0.1,1,10mm/mm and strain-rates 0.1,100,1000,10000and 50000sec ?1by applying the least squares method to minimize the sum of the square of the error (see Appendix A).Therefore,the coef?cients of ?ow stress equation for P20mold steel were found as;K =1339.4,T 0=400,A =0.18268,a =?0.0013,b =?0.00001,c =0.02964,and d =0.0363.The determined ?ow stress graph for P20mold steel is presented in Fig.11.

5.Conclusions

In this paper,a methodology is presented to determine simultaneously the ?ow stress of the workpiece material,and the friction at the chip–too1contact interface at high deformation rates,

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149

Fig.11.Flow stress detemined for P20mold steel for high-speed cutting conditions.

temperatures encountered in the cutting zone.This methodology uses the cutting and thrust force data measured from high-speed orthogonal cutting experiments as reference in order to calibrate a simulated process model.The methodology was applied to obtain?ow stress data for P20mold steel(at hardness of30HRC)and friction data in machining with uncoated tungsten carbide (WC)cutting tool.The experimental cutting conditions of200–550m/min cutting speed(V c),and 0.025–0.100mm/rev feed rate(V f)were used.From the process simulations,boundaries of the primary deformation zone were identi?ed and the average strain-rates(eˉ·ave)and temperatures (T ave)in this region were computed.Flow stress of the workpiece material at computed average strain-rates and temperatures was determined using an iterative scheme until the difference between predicted and measured cutting forces becomes less than10%.In addition,the average strain-rate and temperature were computed at the chip–tool contact site and the average shear friction(k chip)was estimated.The unknown friction coef?cients were determined in another iterat-ive scheme until the difference between predicted and measured thrust forces was less than10%. Finally,cutting force predictions from process simulations,using the determined?ow stress and friction models,were compared with experiments and good agreements were obtained(Figs.10 and11).

Data for?ow stress of the workpiece material and friction at the chip–tool interface,necessary as input into the FEM codes,are usually not available for the deformation rate and temperature regimes that exist in high-speed cutting conditions.Thus,it is possible to apply the methodology,

150T.O

zel,T.Altan /International Journal of Machine Tools &Manufacture 40(2000)133–152developed here,to other cutting conditions and workpiece/tool materials than those investigated in this paper.At present,this FEM based simulation technique is being extended to predict the tool temperature and stress distributions in turning and end milling processes with nose radius cutting tools.

Appendix A.Least squares estimation of coef?cients for ?ow stress equation

Least squares method was used in estimating the unknown coef?cients in the ?ow stress equ-ation given as:

s ˉ?K (e aT ?A e b (T ?T 0)2)?e ˉ·e ·R ?

c (e ˉ)

d (A1)Whil

e this ?ow stress model is not linear when it is used,by taking a log transform o

f Eq.(A1),the model becomes:

log(s ˉ)?log K ?log (e aT ?A e b (T ?T 0)2)?c log ?e ˉ·e

ˉ·R ?

?d log(e ˉ)(A2)where,T 0=400°C is used from the initial ?ow stress model,and e ˉ·R =10000sec ?1is used for cleaning units.The linear model with the error associated with each determined ?ow stress data

d i included results in:Y i ?Y 0?xZ i ?d i

(A3)

The linear least squares method is aimed at ?nding estimates of Y ?0and x ?that minimize:?n i ?1d 2

i ??n

i ?1

(Y i ?Y ?0?x ?Z i )2(A4)The solution can be written using a matrix notation.

Y ??Y 1Y 2..Y n ?,X ??1X 11X 2....1X n ?,and ???d 1d 2d 3

.

d n ?(A5)

The solution is found by differentiating the squared error in Eq.(A4)with respect to the model parameters.Setting the resulting linear equations equal to zero leads to ?nal solution:

b ???Y ?0x ???(X T X )?1(X T Y )(A6)

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151 References

[1]H.K.To¨nshoff,W.Bussmann,C.Stanske,Requirements on tools and machines when machining hard materials,

in:Proceedings of the26th International Machine Tool and Research Conference,1986,pp.349–357.

[2]W.Ko¨nig,A.Berktold,K-F.Koch,Turning versus grinding—A comparison of surface integrity aspects and

attainable accuracies,Annals of the CIRP42(1)(1993)39–43.

[3]M.A.Davies,Y.Chou,C.J.Evans,On chip morphology,tool wear and cutting mechanics in?nish hard turning,

Annals of the CIRP45(1)(1996)77–82.

[4]P.L.B.Oxley,Mechanics of Machining,An Analytical Approach to Assessing Machinability,Halsted Press,John

Wiley&Sons Limited,New York,1989.

[5]K.Maekawa,T.Shirakashi,https://www.wendangku.net/doc/994680595.html,ui,Flow stress of low carbon steel at high temperature and strain rate(Part2),

Bulletin of Japan Society of Precision Engineering17(3)(1983)167–172.

[6]H.Schulz,T.Moriwaki,High-speed machining,Annals of the CIRP41(2)(1992)637–643.

[7]N.N.Zorev,Inter-relationship between shear processes occurring along tool face and shear plane in metal cutting,

in:International Research in Production Engineering ASME,New York,1963,pp.42–49.

[8]T.H.Childs,M.H.Dirikolu,M.D.S.Sammons,K.Maekawa,T.Kitagawa,Experiments on and?nite element

modeling of turning free-cutting steels at cutting speeds up to250m/min,in:Proceedings of1st French and German Conference on High-speed Machining,1997,pp.325–331.

[9]S.Kobayashi,E.G.Thomsen,Metal-cutting analysis—I and II,ASME Journal of Engineering for Industry84

(1962)63–80.

[10]F.Lira,E.G.Thomsen,Metal cutting as a property test,ASME Journal of Engineering for Industry89(1967)

489–493.

[11]R.G.Fenton,P.L.B.Oxley,Mechanics of orthogonal machining:predicting chip geometry and cutting forces from

work-material properties and cutting conditions,Proceeding of Institute of Mechanical Engineering184(1970) 927–942.

[12]https://www.wendangku.net/doc/994680595.html,ui,T.Shirakashi,Mechanics of machining—from descriptive to predictive theory,in:On the Art of Cutting

Metals—75Years Later,ASME Publication PED7,1982,pp.13–35.

[13]K.Maekawa,H.Ohhata,T.Kitigawa,T.H.C.Childs,Simulation analysis of machinability of leaded Cr–Mo and

Mn–B structural steels,Journal of Materials Processing Technology62(1996)363–369.

[14]H.Ernst,M.E.Merchant,Chip formation,friction and high quality machined surfaces,Transactions of American

Society for Metals29(1941)299–378.

[15]E.H.Lee,B.W.Shaffer,The theory of plasticity applied to a problem of machining,Journal of Applied Mechanics

18(1951)405–413.

[16]A.O.Tay,M.G.Stevenson,G.de Vahl Davis,Using the?nite element method to determine temperature distri-

butions in orthogonal machining,Proceedings of Institution for Mechanical Engineers188(1974)627–638. [17]J.T.Carroll,J.Strenkowski,Finite element models of orthogonal cutting with application to single point diamond

turning,International Journal of Mechanical Sciences30(12)(1988)889–920.

[18]K.Iwata,K.Osakada,Y.Terasaka,Process modeling of orthogonal cutting by the rigid-plastic?nite element

method,ASME Journal of Engineering for industry106(1984)132–138.

[19]A.J.Shih,S.Chandrasekar,H.T.Yang,Finite element simulation of metal cutting process with strain-rate and

temperatures effects,ASME Publication PED43(1990)11–24.

[20]J.S.Strenkowski,J.T.Carroll,A?nite element model of orthogonal metal cutting,ASME Journal of Engineering

for Industry107(1985)346–354.

[21]K.Komvopoulos,S.A.Erpenbeck,Finite element modeling of orthogonal metal cutting,ASME Journal of Engin-

eering for Industry113(1991)253–267.

[22]T.D.Marusich,M.Ortiz,Modeling and simulation of high-speed machining,International Journal for Numerical

Methods in Engineering38(1995)3675–3694.

[23]K.Ueda,K.Watanabe,Rigid-plastic FEM analysis of three-dimensional deformation?eld in chip formation

process,Annals of the CIRP42(1)(1993)35–38.

[24]E.Ceretti,P.Fallbo¨hmer,W.T.Wu,T.Altan,Application of2-D FEM to chip formation in orthogonal cutting,

Journal of Materials Processing Technology59(1996)169–181.

[25]S.Kumar,P.Fallbo¨hmer,T.Altan,Computer simulation of orthogonal metal cutting processes:determination of

152T.O zel,T.Altan/International Journal of Machine Tools&Manufacture40(2000)133–152 material properties and effects of tool geometry on chip?ow,Technical Papers of NAMRI/SME25(177)(1997) 1–6.

[26]T.O¨zel,M.Lucchi,C.Rodriguez,T.Altan,Prediction of chip formation and cutting forces in?at end milling:

comparison of process simulations with experiments,Transactions of NAMRI/SME26(1998)231–236. [27]T.O¨zel,T.Altan,Modeling of high speed machining processes for predicted tool forces stresses and temperatures

using FEM simulations,in:Proceedings of CIRP International Workshop on Modeling of Machining Operations, Atlanta,GA,2B(5),1998,pp.1–10.

[28]H.Zhen-Bin,R.Komandwi,On a thermo-mechanical model of shear instability in machining,Annals of the CIRP

44(1)(1995)69–73.

[29]M.A.Elbestawi,A.K.Srivastava,T.I.El-Wardany,A model for chip formation during machining of hardened

steel,Annals of the CIRP45(1)(1996)71–76.

BP神经网络实验——【机器学习与算法分析 精品资源池】

实验算法BP神经网络实验 【实验名称】 BP神经网络实验 【实验要求】 掌握BP神经网络模型应用过程，根据模型要求进行数据预处理，建模，评价与应用； 【背景描述】 神经网络：是一种应用类似于大脑神经突触联接的结构进行信息处理的数学模型。BP神经网络是一种按照误差逆向传播算法训练的多层前馈神经网络，是目前应用最广泛的神经网络。其基本组成单元是感知器神经元。 【知识准备】 了解BP神经网络模型的使用场景，数据标准。掌握Python/TensorFlow数据处理一般方法。了解keras神经网络模型搭建，训练以及应用方法 【实验设备】 Windows或Linux操作系统的计算机。部署TensorFlow，Python。本实验提供centos6.8环境。 【实验说明】 采用UCI机器学习库中的wine数据集作为算法数据，把数据集随机划分为训练集和测试集，分别对模型进行训练和测试。 【实验环境】 Pyrhon3.X，实验在命令行python中进行，或者把代码写在py脚本，由于本次为实验，以学习模型为主，所以在命令行中逐步执行代码，以便更加清晰地了解整个建模流程。 【实验步骤】 第一步：启动python： 1

命令行中键入python。 第二步：导入用到的包，并读取数据： (1).导入所需第三方包 import pandas as pd import numpy as np from keras.models import Sequential from https://www.wendangku.net/doc/994680595.html,yers import Dense import keras (2).导入数据源,数据源地址:/opt/algorithm/BPNet/wine.txt df_wine = pd.read_csv("/opt/algorithm/BPNet/wine.txt", header=None).sample(frac=1) (3).查看数据 df_wine.head() 1

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

数据挖掘常用资源及工具

资源Github，kaggle Python工具库：Numpy，Pandas，Matplotlib，Scikit-Learn，tensorflow Numpy支持大量维度数组与矩阵运算，也针对数组提供大量的数学函数库 Numpy : 1.aaa = Numpy.genfromtxt(“文件路径”,delimiter = “,”,dtype = str)delimiter以指定字符分割，dtype 指定类型该函数能读取文件所以内容 aaa.dtype 返回aaa的类型 2.aaa = numpy.array([5,6,7,8]) 创建一个一维数组里面的东西都是同一个类型的 bbb = numpy.array([[1,2,3,4,5],[6,7,8,9,0],[11,22,33,44,55]]) 创建一个二维数组aaa.shape 返回数组的维度print(bbb[:,2]) 输出第二列 3.bbb = aaa.astype(int) 类型转换 4.aaa.min() 返回最小值 5.常见函数 aaa = numpy.arange(20) bbb = aaa.reshape(4,5)

numpy.arange(20) 生成0到19 aaa.reshape(4,5) 把数组转换成矩阵aaa.reshape(4,-1)自动计算列用-1 aaa.ravel()把矩阵转化成数组 bbb.ndim 返回bbb的维度 bbb.size 返回里面有多少元素 aaa = numpy.zeros((5,5)) 初始化一个全为0 的矩阵需要传进一个元组的格式默认是float aaa = numpy.ones((3,3,3),dtype = numpy.int) 需要指定dtype 为numpy.int aaa = np 随机函数aaa = numpy.random.random((3,3)) 生成三行三列 linspace 等差数列创建函数linspace(起始值，终止值，数量) 矩阵乘法： aaa = numpy.array([[1,2],[3,4]]) bbb = numpy.array([[5,6],[7,8]]) print(aaa*bbb) *是对应位置相乘 print(aaa.dot(bbb)) .dot是矩阵乘法行乘以列 print(numpy.dot(aaa,bbb)) 同上 6.矩阵常见操作

_基于ANSYS的有限元法网格划分浅析

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

题库深度学习面试题型介绍及解析--第7期

1.简述激活函数的作用 使用激活函数的目的是为了向网络中加入非线性因素；加强网络的表示能力，解决线性模型无法解决的问题 2.那为什么要使用非线性激活函数？ 为什么加入非线性因素能够加强网络的表示能力？——神经网络的万能近似定理 ?神经网络的万能近似定理认为主要神经网络具有至少一个非线性隐藏层，那么只要给予网络足够数量的隐藏单元，它就可以以任意的精度来近似任何从一个有限维空间到另一个有限维空间的函数。 ?如果不使用非线性激活函数，那么每一层输出都是上层输入的线性组合；此时无论网络有多少层，其整体也将是线性的，这会导致失去万能近似的性质 ?但仅部分层是纯线性是可以接受的，这有助于减少网络中的参数。3.如何解决训练样本少的问题？ 1.利用预训练模型进行迁移微调（fine-tuning），预训练模型通常在特征上拥有很好的语义表达。此时，只需将模型在小数据集上进行微调就能取得不错的效果。CV 有 ImageNet，NLP 有 BERT 等。 2.数据集进行下采样操作，使得符合数据同分布。

3.数据集增强、正则或者半监督学习等方式来解决小样本数据集的训练问题。 4.如何提升模型的稳定性？ 1.正则化（L2, L1, dropout）：模型方差大，很可能来自于过拟合。正则化能有效的降低模型的复杂度，增加对更多分布的适应性。 2.前停止训练：提前停止是指模型在验证集上取得不错的性能时停止训练。这种方式本质和正则化是一个道理，能减少方差的同时增加的偏差。目的为了平衡训练集和未知数据之间在模型的表现差异。 3.扩充训练集：正则化通过控制模型复杂度，来增加更多样本的适应性。 4.特征选择：过高的特征维度会使模型过拟合，减少特征维度和正则一样可能会处理好方差问题，但是同时会增大偏差。 5.你有哪些改善模型的思路？ 1.数据角度 增强数据集。无论是有监督还是无监督学习，数据永远是最重要的驱动力。更多的类型数据对良好的模型能带来更好的稳定性和对未知数据的可预见性。对模型来说，“看到过的总比没看到的更具有判别的信心”。 2.模型角度

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

人工智能实践：Tensorflow笔记 北京大学 7 第七讲卷积网络基础 (7.3.1) 助教的Tenso

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例：先将此图进行多次特征提取，再把提取后的计算机可读特征喂给全连接网络。 √卷积Convolutional 卷积是一种有效提取图片特征的方法。一般用一个正方形卷积核，遍历图片上的每一个像素点。图片与卷积核重合区域内相对应的每一个像素值乘卷积核内相对应点的权重，然后求和，再加上偏置后，最后得到输出图片中的一个像素值。 例：上面是5x5x1的灰度图片，1表示单通道，5x5表示分辨率，共有5行5列个灰度值。若用一个3x3x1的卷积核对此5x5x1的灰度图片进行卷积，偏置项

b=1，则求卷积的计算是：(-1)x1+0x0+1x2+(-1)x5+0x4+1x2+(-1)x3+0x4+1x5+1=1（注意不要忘记加偏置1）。 输出图片边长=（输入图片边长–卷积核长+1）/步长，此图为：（5 – 3 + 1）/ 1 = 3，输出图片是3x3的分辨率，用了1个卷积核，输出深度是1，最后输出的是3x3x1的图片。 √全零填充Padding 有时会在输入图片周围进行全零填充，这样可以保证输出图片的尺寸和输入图片一致。 例：在前面5x5x1的图片周围进行全零填充，可使输出图片仍保持5x5x1的维度。这个全零填充的过程叫做padding。 输出数据体的尺寸=(W?F+2P)/S+1 W：输入数据体尺寸，F：卷积层中神经元感知域，S：步长，P：零填充的数量。 例：输入是7×7，滤波器是3×3，步长为1，填充为0，那么就能得到一个5×5的输出。如果步长为2，输出就是3×3。 如果输入量是32x32x3，核是5x5x3，不用全零填充，输出是（32-5+1）/1=28，如果要让输出量保持在32x32x3，可以对该层加一个大小为2的零填充。可以根据需求计算出需要填充几层零。32=（32-5+2P）/1 +1，计算出P=2，即需填充2

常用函数 类参考

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dedecms通用消息提示框 function ShowMsg($msg,$gourl,$onlymsg=0,$limittime=0) 保存一个cookie function PutCookie($key,$value,$kptime=0,$pa="/") 删除一个cookie function DropCookie($key) 获取cookie function GetCookie($key) 获取验证码 function GetCkVdValue() 过滤前台用户输入的文本内容 // $rptype = 0 表示仅替换html标记 // $rptype = 1 表示替换html标记同时去除连续空白字符// $rptype = 2 表示替换html标记同时去除所有空白字符// $rptype = -1 表示仅替换html危险的标记 function HtmlReplace($str,$rptype=0) 获得某文档的所有tag function GetTags($aid) 过滤用于搜索的字符串 function FilterSearch($keyword) 处理禁用HTML但允许换行的内容 function TrimMsg($msg) 获取单篇文档信息 function GetOneArchive($aid)

有限元网格划分

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

人工智能实践：Tensorflow笔记 北京大学 4 第四讲神经网络优化 (4.6.1) 助教的Tenso

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√神经网路待优化的参数：神经网络中所有参数w 的个数 + 所有参数b 的个数 例如： 输入层 隐藏层 输出层 在该神经网络中，包含1个输入层、1个隐藏层和1个输出层，该神经网络的层数为2层。 在该神经网络中，参数的个数是所有参数w 的个数加上所有参数b 的总数，第一层参数用三行四列的二阶张量表示（即12个线上的权重w ）再加上4个偏置b ；第二层参数是四行两列的二阶张量（）即8个线上的权重w ）再加上2个偏置b 。总参数 = 3*4+4 + 4*2+2 = 26。 √损失函数（loss ）：用来表示预测值（y ）与已知答案（y_）的差距。在训练神经网络时，通过不断改变神经网络中所有参数，使损失函数不断减小，从而训练出更高准确率的神经网络模型。 √常用的损失函数有均方误差、自定义和交叉熵等。 √均方误差mse ：n 个样本的预测值y 与已知答案y_之差的平方和，再求平均值。 MSE(y_, y) = ?i=1n (y?y_) 2n 在Tensorflow 中用loss_mse = tf.reduce_mean(tf.square(y_ - y)) 例如： 预测酸奶日销量y ，x1和x2是影响日销量的两个因素。 应提前采集的数据有：一段时间内，每日的x1因素、x2因素和销量y_。采集的数据尽量多。 在本例中用销量预测产量，最优的产量应该等于销量。由于目前没有数据集，所以拟造了一套数据集。利用Tensorflow 中函数随机生成 x1、 x2，制造标准答案y_ = x1 + x2，为了更真实，求和后还加了正负0.05的随机噪声。 我们把这套自制的数据集喂入神经网络，构建一个一层的神经网络，拟合预测酸奶日销量的函数。

有限元网格划分和收敛性

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

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

比较PageRank算法和HITS算法的优缺点

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

数据库常用函数

数据库常用函数

一、基础 1、说明：创建数据库 CREATE DATABASE database-name 2、说明：删除数据库 drop database dbname 3、说明：备份和还原 备份：exp dsscount/sa@dsscount owner=dsscount file=C:\dsscount_data_backup\dsscount.dmp log=C:\dsscount_data_backup\outputa.log 还原：imp dsscount/sa@dsscount file=C:\dsscount_data_backup\dsscount.dmp full=y ignore=y log=C:\dsscount_data_backup\dsscount.log statistics=none 4、说明：创建新表 create table tabname(col1 type1 [not null] [primary key],col2 type2 [not null],..) CREATE TABLE ceshi(id INT not null identity(1,1) PRIMARY KEY,NAME VARCHAR(50),age INT) id为主键，不为空，自增长 根据已有的表创建新表： A：create table tab_new like tab_old (使用旧表创建新表) B：create table tab_new as select col1,col2… from tab_old definition only 5、说明：删除新表 drop table tabname 6、说明：增加一个列 Alter table tabname add column col type 注：列增加后将不能删除。DB2中列加上后数据类型也不能改变，唯一能改变的是增加varchar类型的长度。 7、说明：添加主键： Alter table tabname add primary key(col) 说明：删除主键： Alter table tabname drop primary key(col) 8、说明：创建索引：create [unique] index idxname on tabname(col….) 删除索引：drop index idxname 注：索引是不可更改的，想更改必须删除重新建。 9、说明：创建视图：create view viewname as select statement 删除视图：drop view viewname 10、说明：几个简单的基本的sql语句 选择：select * from table1 where 范围 插入：insert into table1(field1,field2) values(value1,value2) 删除：delete from table1 where 范围 更新：update table1 set field1=value1 where 范围

ANSYS有限元网格划分的基本要点

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

人工智能tensorflow实验报告

一、软件下载 为了更好的达到预期的效果，本次tensorflow开源框架实验在Linux环境下进行，所需的软件及相关下载信息如下： 1.CentOS 软件介绍： CentOS 是一个基于Red Hat Linux 提供的可自由使用源代码的企业级Linux 发行版本。每个版本的CentOS都会获得十年的支持（通过安全更新方式）。新版本的CentOS 大约每两年发行一次，而每个版本的CentOS 会定期（大概每六个月）更新一次，以便支持新的硬件。这样，建立一个安全、低维护、稳定、高预测性、高重复性的Linux 环境。CentOS是Community Enterprise Operating System的缩写。CentOS 是RHEL（Red Hat Enterprise Linux）源代码再编译的产物，而且在RHEL的基础上修正了不少已知的Bug ，相对于其他Linux 发行版，其稳定性值得信赖。 软件下载： 本次实验所用的CentOS版本为CentOS7,可在CentOS官网上直接下载DVD ISO镜像文件。 下载链接： https://www.wendangku.net/doc/994680595.html,/centos/7/isos/x86_64/CentOS-7-x86_64-DVD-1611.i so. 2.Tensorflow 软件介绍： TensorFlow是谷歌基于DistBelief进行研发的第二代人工智能学习系统，其命名来源于本身的运行原理。Tensor（张量）意味着N维数组，Flow（流）意味着基于数据流图的计算，TensorFlow为张量从流图的一端流动到另一端计算过程。TensorFlow是将复杂的数据结构传输至人工智能神经网中进行分析和处理过程的系统。TensorFlow可被用于语音识别或图像识别等多项机器深度学习领域，对2011年开发的深度学习基础架构DistBelief进行了各方面的改进，它可在小到一部智能手机、大到数千台数据中心服务器的各种设备上运行。TensorFlow将完全开源，任何人都可以用。

ANSYS有限元分析中的网格划分

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