Engineering Computation in

Transcription

1 Engineering Computation in Dr. Kyle Horne Department of Mechanical Engineering Spring, 2018

2 What is It? 140 Explicit Implicit Temperature T [C] Position y [px] Position y [px] Position x [px] Position x [px] Fit Data Radial Position r [um] Position y [µm] 240 Probe Laser Position x [µm] 180 r =44.80 [µm] r =32.30 [µm] r =18.80 [µm] 170 Engineering Computation in Python; Dr. Kyle Horne 2 / 44

3 Where to Learn More? Python Language Numpy (Array Support) Scipy (Numerical Methods) Sympy (Symbolic Math) Matplotlib (Plotting) Engineering Computation in Python; Dr. Kyle Horne 3 / 44

4 Basic Usage Procedural Execution, Comments and Text Formatting g = 9.81 h = # h = 0.5*g*t**2 -> t = sqrt(2*h/g) import math t = math.sqrt(2*h/g) print('t = %f [s]'%t) Results t = Engineering Computation in Python; Dr. Kyle Horne 4 / 44

5 Basic Usage Lists and Arrays import pylab as pl A = [1.0,2.0,3.0] A.append(4.0) B = pl.array(a) C = 2*B print(a) print(b) print(c) Results [1.0, 2.0, 3.0, 4.0] [ ] [ ] Engineering Computation in Python; Dr. Kyle Horne 5 / 44

6 Flow Control If-Then-Else a = 1 if a==1: print('a=1') elif a==2: print('a=2') else: print('a!=1') Results a =1 Engineering Computation in Python; Dr. Kyle Horne 6 / 44

7 Flow Control For-Loops s = 0 for k in range(4): x = 2.0*k-1.0 s = s+x print('k = %5d; x = %10f'%(k,x)) print('s = %f'%s) Results k = 0; x = k = 1; x = k = 2; x = k = 3; x = s = Engineering Computation in Python; Dr. Kyle Horne 7 / 44

8 Flow Control Functions def f(x): y = x**2-1.0 return y g = lambda x: x**2+1.0 x = 2.0 y = f(x) print('x = %f; y = %f; g(x) = %f'%(x,y,g(x))) Results x = ; y = ; g(x) = Engineering Computation in Python; Dr. Kyle Horne 8 / 44

9 Plotting Line Plots and Boilerplate import pylab as pl N = 100 x = pl.linspace(0,1,n) f = lambda x: x**2-1 y = f(x) fig = pl.figure( tight_layout=true, figsize=(5,4)) ax = fig.add_subplot(1,1,1) ax.plot(x,y,'b-',lw=2) ax.plot(x[::10],y[::10],'rs',mew=0) ax.set_xlabel('abscissa $x$ [-]') ax.set_ylabel('ordinate $y$ [-]') pl.savefig('exampleplot.pdf') Ordinate y [-] Abscissa x [-] Engineering Computation in Python; Dr. Kyle Horne 9 / 44

10 Calculation of π (Series) Theory Leibniz s series formula: [ ] ( 1) k π = 4 2k + 1 k= ; N = 50 Term Magnitude vk Terms Sum vk Term Number k Term Count N Engineering Computation in Python; Dr. Kyle Horne 10 / 44

11 Calculation of π (Series) I Example Solution 1 import numpy 2 import pylab 3 4 N = terms = numpy.empty(n) 7 for k in range(n): 8 terms[k] = 4.0*(-1)**k/(2.0*k+1.0) 9 10 sums = numpy.empty(n) 11 for k in range(n): 12 sums[k] = terms[:k+1].sum() numbers = numpy.linspace(0,n-1,n) fig = pylab.figure(tight_layout=true,figsize=(5,4)) 17 ax = fig.add_subplot(1,1,1) 18 ax.plot(numbers,abs(terms),'-',lw=2) 19 ax.set_yscale('log') 20 ax.set_xlabel('term Number $k$') Engineering Computation in Python; Dr. Kyle Horne 11 / 44

12 Calculation of π (Series) II Example Solution 21 ax.set_ylabel('term Magnitude $v_k$') 22 ax.set_title(r' ') 23 fig.savefig('pitermsplot.pdf') fig = pylab.figure(tight_layout=true,figsize=(5,4)) 26 ax = fig.add_subplot(1,1,1) 27 ax.axhline(numpy.pi,color='k',linestyle='--',linewidth=2) 28 ax.plot(numbers,sums,'.',lw=2) 29 ax.set_yticks([2.6,3.0,numpy.pi,3.4,3.8]) 30 ax.set_yticklabels(['2.6','3.0','$\pi$','3.4','3.8']) 31 ax.set_xlabel(r'term Count $N$') 32 ax.set_ylabel(r'terms Sum $\sum v_k$') 33 ax.text(5,3.8,r'$\pi \approx %f;\,n = %d$'%(terms.sum(),n)) 34 fig.savefig('pisumsplot.pdf') Engineering Computation in Python; Dr. Kyle Horne 12 / 44

13 Calculation of π (Monte Carlo) A c = πr 2 A s = (2R) 2 A c = π A s 4 r i = xi 2 + yi 2 π 4 = lim N count i [1,N] (r i < 1) N y-coordinate ; N = x-coordinate Engineering Computation in Python; Dr. Kyle Horne 13 / 44

14 Calculation of π (Monte Carlo) I Example Solution 1 import pylab 2 import numpy 3 from numpy.random import rand 4 5 def calculatepi(n): 6 x = rand(n,2) 7 r = numpy.sqrt( (x**2).sum(1) ) 8 c = (r<1.0).sum() 9 10 pi = 4.0*c/float(N) th = numpy.linspace(0,numpy.pi/2,100) 13 xt = numpy.cos(th) 14 yt = numpy.sin(th) xp = x[:,0] 17 yp = x[:,1] fig = pylab.figure(tight_layout=true,figsize=(4,4)) 20 ax = fig.add_subplot(1,1,1,aspect=1.0) Engineering Computation in Python; Dr. Kyle Horne 14 / 44

15 Calculation of π (Monte Carlo) II Example Solution 21 ax.plot(xt,yt,'k--',lw=2) 22 ax.plot(xp[r<1],yp[r<1],'x',ms=7,mew=1) 23 ax.plot(xp[r>1],yp[r>1],'+',ms=7,mew=1) 24 ax.set_xlabel('$x$-coordinate') 25 ax.set_ylabel('$y$-coordinate') 26 ax.set_title( r'$\pi \approx %.5f; N=%d$'%(pi,N) ) 27 fig.savefig('randompi.pdf') calculatepi(500) Engineering Computation in Python; Dr. Kyle Horne 15 / 44

16 Sieve of Eratosthenes Theory Sift the Twos and sift the Threes, The Sieve of Eratosthenes! When the multiples sublime, The numbers that remain are Prime. Algorithm 1 P = {1, 2,..., N} 2 For i [2, N] 1 if P [i] is 0, then skip i 2 For j [2i, N] 1 P [j] = 0 Number of Primes [#] (n) f(n) = n/log(n) f(n)/ (n) Ratio [1] Number n [#] Engineering Computation in Python; Dr. Kyle Horne 16 /

17 Sieve of Eratosthenes I Example Solution 1 import numpy 2 import pylab 3 4 N = P = numpy.empty(n,dtype=int) 6 7 for i in range(n): 8 P[i] = i 9 for i in range(2,n): 10 if P[i]==0: 11 continue 12 for j in range(2*i,n,p[i]): 13 P[j] = s = 2 16 px = (P[P!=0])[s:] 17 py = numpy.linspace(s,px.size,px.size) 18 fy = px/numpy.log(px) fig = pylab.figure(tight_layout=true,figsize=(5,4)) Engineering Computation in Python; Dr. Kyle Horne 17 / 44

18 Sieve of Eratosthenes II Example Solution 21 ax = fig.add_subplot(1,1,1) 22 ax.set_xscale('log') 23 ax.set_yscale('log') 24 lp, = ax.plot(px,py,'.') 25 lf, = ax.plot(px,fy,'-',lw=2) 26 ax.set_xlabel('number $n$ [#]') 27 ax.set_ylabel('number of Primes $\pi$ [#]') 28 ax2 = ax.twinx() 29 lr, = ax2.plot(px,fy/py,'c2_',mew=2) 30 ax2.set_ylabel(r'ratio [1]') 31 lines = [lp,lf,lr] 32 labels = [r'$\pi(n)$',r'$f(n)=n/log(n)$',r'$f(n)/\pi(n)$'] 33 ax2.legend(lines,labels,loc='upper center') 34 fig.savefig('seiveplot.pdf') Engineering Computation in Python; Dr. Kyle Horne 18 / 44

19 Orbital Simulation Theory 1 au Engineering Computation in Python; Dr. Kyle Horne 19 / 44

20 Orbital Simulation Solver Compute the acceleration on the Earth using Newton s law of gravitation. Use this to update velocity at each time step. Use velocity to update position at each time step. x (t) = x 0 + v (t) = v 0 + t 0 t a ( x) = G M x x 3 0 v (τ) dτ a (τ) dτ x 0 = r 0 î v 0 = ĵ G M r 0 r 0 = 1 au The integrals can be converted into much simpler expressions by using Euler s explicit method: x+ x f (x + x) = f (x) + f (ξ) dξ f (x) + x f (x) x Engineering Computation in Python; Dr. Kyle Horne 20 / 44

21 Orbital Simulation Algorithm Orbit Sun Earth 1 x R 2 = (r 0 ) ( î ) 2 v R 2 = G M r ĵ 0 3 t R = 0 4 While t < t end : 1 x = x + t v 2 r = x 3 a = G M x r 3 4 v = v + t a 5 t = t + t Position y [au] Position x [au] Engineering Computation in Python; Dr. Kyle Horne 21 / 44

22 Orbital Simulation I Example Solution 1 import pylab 2 import numpy 3 4 MS = e30 5 au = e9 6 G = 6.67E N = U0 = numpy.sqrt(g*ms/au) 11 Y0 = dt = Y0/ xh = numpy.empty( (N,2) ) x = numpy.array([au,0.0]) 17 v = numpy.array([0.0,u0]) xh[0,:] = x 20 Engineering Computation in Python; Dr. Kyle Horne 22 / 44

23 Orbital Simulation II Example Solution 21 for k in range(1,n): 22 x = x+v*dt 23 r = numpy.sqrt( (x**2).sum() ) 24 a = -G*MS*x/r**3 25 v = v+a*dt xh[k] = x fig = pylab.figure(figsize=(4,4), 30 tight_layout={'rect':(0,0,1,0.85)}) 31 ax = fig.add_subplot(1,1,1,aspect=1.0) 32 ax.plot(xh[:,0]/au,xh[:,1]/au,'k--',lw=2,label='orbit') 33 ax.plot([0],[0],'o',color='orange', 34 ms=30,mew=1.5,mec='yellow',label='sun') 35 ax.plot([1],[0],'o',color='green', 36 ms=10,mew=1.5,mec='cyan',label='earth') 37 ax.set_xlim(-1.1,1.1) 38 ax.set_ylim(-1.1,1.1) 39 ax.set_xlabel('position $x$ [au]') 40 ax.set_ylabel('position $y$ [au]') Engineering Computation in Python; Dr. Kyle Horne 23 / 44

24 Orbital Simulation III Example Solution 41 ax.legend(frameon=false,ncol=3,numpoints=1,handletextpad=0.5, 42 bbox_to_anchor=(0,1.05),loc='lower left') 43 fig.savefig('orbitplot.pdf') Engineering Computation in Python; Dr. Kyle Horne 24 / 44

25 Curve Fitting Theory Find Γ, α such that R (Γ, α) is minimized V θ (r) = Γ ( r exp αr 2) [ R (Γ, α) = i Tangential Velocity V [m/s] Fit Experiment (V i V θ (r i )) 2] Position r [m] from scipy.optimize import minimize Engineering Computation in Python; Dr. Kyle Horne 25 / 44

26 Curve Fitting I Example Solver 1 import numpy 2 import pylab 3 from scipy.optimize import minimize 4 5 re,ve = numpy.loadtxt('vortex.txt',unpack=true) 6 7 def candidate(g,a): 8 g = lambda r: G/r*(1-numpy.exp(-a*r**2)) 9 return g def residual(x): 12 G,a = x 13 g = candidate(g,a) 14 R = numpy.sqrt( ( (Ve-g(re))**2 ).sum() ) 15 return R x0 = [1.0,200.0] 18 opt = minimize(residual,x0) 19 G,a = opt.x 20 print(g,a) Engineering Computation in Python; Dr. Kyle Horne 26 / 44

27 Curve Fitting II Example Solver N = rf = numpy.linspace(re.min(),re.max(),n) 24 gf = candidate(g,a) 25 Vf = gf(rf) fig = pylab.figure(tight_layout=true,figsize=(5,4)) 28 ax = fig.add_subplot(1,1,1) 29 ax.plot(rf,vf,'--',lw=2,label='fit') 30 ax.plot(re,ve,'.',label='experiment') 31 ax.set_xlabel(r'position $r$ [m]') 32 ax.set_ylabel(r'tangential Velocity $V_\theta$ [m/s]') 33 ax.legend() 34 fig.savefig('vortexplot.pdf') Engineering Computation in Python; Dr. Kyle Horne 27 / 44

28 Artillery Fire Theory x (t) = v 0 t cos (θ) x t (x) = v 0 cos (θ) y (x) = v 0 t (x) sin (θ) 1 gt (x)2 2 ( x 2 g (x) = 50) g (x h ) = y (x h ) Position y [m] Position x [m] from scipy.optimize import newton Engineering Computation in Python; Dr. Kyle Horne 28 / 44

29 Artillery Fire I Example Solver 1 import pylab 2 import numpy 3 from scipy.optimize import newton 4 5 def py(x,th,v0): 6 g = t = x/(v0*numpy.cos(th)) 8 y = t*(v0*numpy.sin(th)-0.5*g*t) 9 return y 10 def gy(x): 11 y = (x/50)**2 12 return y 13 def hit(th,v0): 14 f = lambda x: py(x,th,v0)-gy(x) 15 x = newton(f,1000.0) 16 return x def plottrajectory(th,v0): 19 N = xh = hit(th,v0) Engineering Computation in Python; Dr. Kyle Horne 29 / 44

30 Artillery Fire II Example Solver 21 yh = gy(xh) 22 xp = numpy.linspace(0,xh,n) 23 xg = numpy.linspace(-0.1*xh,1.2*xh,n) 24 p = py(xp,th,v0) 25 g = gy(xg) 26 fig = pylab.figure(tight_layout=true,figsize=(5,4)) 27 ax = fig.add_subplot(1,1,1,aspect=1.0) 28 lg = ax.fill_between(xg,g,-30,color='g') 29 lp = ax.plot(xp,p,'--',lw=2) 30 lh = ax.plot([xh],[yh],'kx',ms=10,mew=4) 31 ax.set_xlabel('position $x$ [m]') 32 ax.set_xlim(xg.min(),xg.max()) 33 ax.set_ylim(-30,1.1*p.max()) 34 ax.set_ylabel('position $y$ [m]') 35 fig.savefig('artilleryplot.pdf') th = 75.0*numpy.pi/ v0 = plottrajectory(th,v0) Engineering Computation in Python; Dr. Kyle Horne 30 / 44

31 Heated Wall Theory import sympy k d2 T dx 2 + Q = 0 k dt dx + Qx = C 0 kt + Q 2 x2 = C 0 x + C 1 Temperature T [C] Exact Numerical Position x [m] 0 x L k i T i+1 2T i + T i 1 x 2 + Q i = 0 0 ξ (i 1) i (i + 1) N + 1 Engineering Computation in Python; Dr. Kyle Horne 31 / 44

32 Heated Wall I Example Solver 1 import pylab 2 import numpy 3 import scipy.sparse as sparse 4 import scipy.sparse.linalg as linalg 5 import sympy 6 7 Q = # W 8 k = 50.0 # W/m.K 9 L = 1.0 # m T0 = 25.0 # K 12 TL = 25.0 # K def plotsolution(xe,te,xn,tn): 15 fig = pylab.figure(tight_layout=true,figsize=(5,4)) 16 ax = fig.add_subplot(1,1,1) 17 eline = ax.plot(xe,te,'-',lw=2,label='exact') 18 nline = ax.plot(xn,tn,'v',lw=2,label='numerical') 19 ax.legend(loc='best') 20 ax.set_xlabel('position $x$ [m]') Engineering Computation in Python; Dr. Kyle Horne 32 / 44

33 Heated Wall II Example Solver 21 ax.set_ylabel('temperature $T$ [C]') 22 fig.savefig('wallplot.pdf') def finitedifferences(n): 25 dx = L/(N+1) DP = -2.0*numpy.ones(N) 28 DL = numpy.ones(n-1) 29 DU = numpy.ones(n-1) 30 A = sparse.diags([dl,dp,du],(-1,0,1)) 31 A = A.tocsc() S = (-Q*dx**2/k)*numpy.ones(N) 34 S[0] = S[0]-T0 35 S[N-1] = S[N-1]-TL T = linalg.spsolve(a,s) x = numpy.linspace(0,l,n+2) 40 T = numpy.insert(t,0,t0) Engineering Computation in Python; Dr. Kyle Horne 33 / 44

34 Heated Wall III Example Solver 41 T = numpy.append(t,tl) return x,t def differentialequation(n): 46 x = sympy.symbols('x') 47 T = sympy.symbols('t',cls=sympy.function) 48 deq = sympy.eq( k*t(x).diff(x,x)+q, 0 ) 49 sol = sympy.dsolve(deq,t(x)) 50 C1,C2 = sympy.symbols('c1 C2') system = [] 53 system.append( sol.subs(x,0).subs(t(0),t0) ) 54 system.append( sol.subs(x,l).subs(t(l),tl) ) 55 coeffs = sympy.solve(system,[c1,c2]) 56 Ts = sympy.solve( sol.subs(coeffs), T(x) )[0] 57 Tf = sympy.lambdify(x,ts,'numpy') x = numpy.linspace(0,l,n+2) 60 T = Tf(x) Engineering Computation in Python; Dr. Kyle Horne 34 / 44

35 Heated Wall IV Example Solver return x,t xn,tn = finitedifferences(6) 65 xe,te = differentialequation(100) 66 plotsolution(xe,te,xn,tn) Engineering Computation in Python; Dr. Kyle Horne 35 / 44

36 Duct Flow Theory from scipy.integrate import ode w t = 1 ρ ( ) P z + ν 2 w x + 2 w 2 y Max Velocity w [m/s] Position y [cm] Velocity w [m/s] Time t [ms] Position x [cm] 0.0 Engineering Computation in Python; Dr. Kyle Horne 36 / 44

37 Duct Flow I Example Solver 1 import pylab 2 import numpy 3 from scipy.integrate import ode 4 5 #=============# 6 #= Constants =# 7 #=============# 8 9 dpdz = -10.0e3 10 mu = rho = nu = mu/rho 13 L = 10.0e-2 14 D = L/ Dc = 4*(L**2-numpy.pi*D**2/4)/(4*L) 17 Vc = numpy.sqrt(2*abs(dpdz)*dc/rho) 18 gre = rho*vc*dc/mu N = 75 Engineering Computation in Python; Dr. Kyle Horne 37 / 44

38 Duct Flow II Example Solver 21 x = numpy.linspace(-l/2,l/2,n) 22 y = numpy.linspace(-l/2,l/2,n) 23 dx = L/(N-1) 24 dy = L/(N-1) X,Y = numpy.meshgrid(x,y) 27 R = numpy.sqrt(x**2+y**2) mask = numpy.logical_and(y<l/15,y>-l/15) 30 mask = numpy.logical_and(x>0.0,mask) 31 mask = numpy.logical_or(r<d/2.0,mask) #==============# 34 #= Derivative =# 35 #==============# def dwdt(t,w): 38 wl = w.reshape( (N,N) ) 39 o = numpy.zeros( (N,N) ) 40 Engineering Computation in Python; Dr. Kyle Horne 38 / 44

39 Duct Flow III Example Solver 41 o[1:-1,1:-1] = -dpdz/rho+nu*( 42 (wl[2:,1:-1]-2*wl[1:-1,1:-1]+wl[:-2,1:-1])/dx** (wl[1:-1,2:]-2*wl[1:-1,1:-1]+wl[1:-1,:-2])/dy**2 44 ) o[mask] = return o.reshape(-1) #==================# 50 #= Initialization =# 51 #==================# w0 = numpy.zeros( (N,N) ).reshape(-1) 54 t = numpy.linspace(0.0,0.007,100) 55 wm = numpy.zeros(t.size) 56 wm[0] = w0.max() I = ode( dwdt ) 59 I.set_initial_value(w0,0.0) 60 Engineering Computation in Python; Dr. Kyle Horne 39 / 44

40 Duct Flow IV Example Solver 61 #============# 62 #= Solution =# 63 #============# for k in range(1,t.size): 66 I.integrate(t[k]) 67 wm[k] = I.y.max() 68 print('%10d %20.5e %20.5e'%(k,t[k],wm[k])) #============# 71 #= Plotting =# 72 #============# w = I.y.reshape( (N,N) ) fig = pylab.figure(tight_layout=true,figsize=(5,4)) 77 ax = fig.add_subplot(1,1,1) 78 ax.plot(1e3*t,wm,'-',lw=2) 79 ax.set_ylim(0,1.1*wm.max()) 80 ax.set_xlabel('time $t$ [ms]') Engineering Computation in Python; Dr. Kyle Horne 40 / 44

41 Duct Flow V Example Solver 81 ax.set_ylabel('max Velocity $w$ [m/s]') 82 fig.savefig('velocitymaxplot.pdf') fig = pylab.figure(tight_layout=true,figsize=(5,4)) 85 ax = fig.add_subplot(1,1,1,aspect=1.0) 86 cf = ax.contourf(1e2*x,1e2*y,w,20,cmap=pylab.get_cmap('viridis')) 87 cl = ax.contour(1e2*x,1e2*y,w,20,cmap=pylab.get_cmap('viridis')) 88 ax.set_xlabel('position $x$ [cm]') 89 ax.set_ylabel('position $y$ [cm]') 90 fig.colorbar(cf,ax=ax,label='velocity $w$ [m/s]') 91 fig.savefig('velocityprofileplot.pdf') V = w.max() 94 Re = rho*v*dc/mu print('') 97 print(gre) 98 print( Re) Engineering Computation in Python; Dr. Kyle Horne 41 / 44

42 Group Work Theory from glog import glob Doubled Fall Height, 2 h [m] z = 1 2 g 0t 2 + ż 0 t + z 0 2h = g 0 t 2 f Data 1 Data 2 Data 3 Data 4 g 0 = 9.27 m/s Fall Time Squared, t 2 f [s 2 ] Data # Height [m], Time [s] Engineering Computation in Python; Dr. Kyle Horne 42 / 44

43 Group Work I Example Solver 1 import pylab as pl 2 import numpy as np 3 from glob import glob 4 5 def getdata(): 6 fns = sorted(glob('*.txt')) 7 D = [] 8 for fn in fns: 9 D.append( np.loadtxt(fn) ) 10 return np.array(d) def doplot(d,c): 13 fig = pl.figure(figsize=(5,4),tight_layout=true) 14 ax = fig.add_subplot(1,1,1) ms = 'os^v' 17 for k in range(d.shape[0]): 18 h = D[k,:,0] 19 t = D[k,:,1] 20 ax.plot(t**2,h,ms[k],label='data %d'%(k+1)) Engineering Computation in Python; Dr. Kyle Horne 43 / 44

44 Group Work II Example Solver x = np.linspace((d[:,:,1]**2).min(),(d[:,:,1]**2).max()) 23 y = np.polyval(c,x)/ ax.plot(x,y,'k--',label='$g_0=%.2f$ $m/s^2$'%c[0],zorder=1) ax.set_xlabel('fall Time Squared, $t_f^2$ [$s^2$] ') 27 ax.set_ylabel('doubled Fall Height, $2 \cdot h$ [$m$]') 28 ax.legend() fig.savefig('group.pdf') def dofit(d): 33 x = D[:,:,1].flatten()**2 34 y = 2.0*D[:,:,0].flatten() 35 C = np.polyfit(x,y,1) 36 return C D = getdata() 39 C = dofit(d) 40 doplot(d,c) Engineering Computation in Python; Dr. Kyle Horne 44 / 44