Simulation def sphSim(profile, diffsys, time, output=True, name=''): """ Single-Phase Diffusion Simulation Parameters ---------- profile : DiffProfile Initial diffusion profile before simulation. diffsys : DiffSystem Diffusion coefficients. time : float time in seconds. output : boolean, optional Print simulation progress, default = True. name : str, optional Name the output DiffProfile. Returns ------- profile : DiffProfile Simulated diffusion profile """ if name == '': name = diffsys.name+'_%.1fh' % (time/3600) dis, Xs = profile.dis.copy()/1e6, profile.X.copy() fD = diffsys.Dfunc d = dis[1:]-dis[:-1] dj = 0.5*(d[1:]+d[:-1]) t, m = 0.0, 0 while t < time: Xm = 0.5*(Xs[1:]+Xs[:-1]) DCs = np.exp(splev(Xm, fD[0])) dt = min(d**2/DCs/2) dt = time-t if t+dt > time else dt*0.95 t += dt m += 1 Jf = -DCs*(Xs[1:]-Xs[:-1])/d Xs[1:-1] = Xs[1:-1]-dt*(Jf[1:]-Jf[:-1])/dj Xs[0] -= Jf[0]/d[0]*dt*2 Xs[-1] += Jf[-1]/d[-1]*dt*2 if output and np.mod(m, 3e4) == 0: print('%.3f/%.3f hrs simulated' % (t/3600, time/3600)) if output: print('Simulation Complete') return DiffProfile(dis*1e6, Xs, name=name) def mphSim(profile, diffsys, time, liquid=0, output=True, name=''): """ Single/Multiple-Phase Diffusion Simulation. Liquid phase can be attached at left or right end. For thin film simulation, please set up the interface nearby the end in the initial profile. Parameters ---------- profile : DiffProfile Initial profile before simulation. diffsys : DiffSystem Diffusion coefficients. time : float time in seconds. liquid : 1, 0 or -1, optional Liquid phase provide infinite mass flux during simulation, can only be attached at left or right end. 0 : No liquid phase attached. -1 : Liquid phase attached at left. 1 : Liquid phase attached at right. output : boolean, optional Print simulation progress, default = True. name : str, optional Name the output DiffProfile. Returns ------- profile : DiffProfile Simulated diffusion profile Examples -------- With known diffusion coefficients (dsys), simulate profile of 100 hours of diffusion for a diffusion couple experiment (initial profile is a step) on a 600 micron grids: >>> dis = mesh(0, 600) >>> init_profile = step(dis, 300, dsys) >>> final_profile = mphSim(init_profile, dsys, 3600*100) If liquid phase is attached to the left: >>> final_profile = mphSim(init_profile, dsys, 3600*100, liquid=-1) """ dis, Xs = profile.dis.copy()/1e6, profile.X.copy() Ip, If = profile.Ip.copy(), profile.If.copy()/1e6 Np, Xr = diffsys.Np, diffsys.Xr.copy() fD = [f for f in diffsys.Dfunc] # Ascending or descending profile if (Xs[-1]-Xs[0]) * (Xr[-1, 1]-Xr[0, 0]) < 0: Xr = Xr.flatten()[::-1].reshape((Np, 2)) fD = fD[::-1] if name == '': name = diffsys.name+'_%.1fh' % (time/3600) if len(If) != Np+1: raise ValueError('Number of phases mismatches between profile and diffusion system') if liquid not in [-1, 0, 1]: raise ValueError('liquid can only be 0 1 or -1') try: time = float(time) except TypeError: print('Wrong Type of time') d = dis[1:]-dis[:-1] dj = 0.5*(d[1:]+d[:-1]) Jf, DCs = np.zeros(len(dis)-1), np.zeros(len(dis)-1) JIf = np.zeros([Np+1, 2]) vIf = np.zeros(Np+1) t, m = 0.0, 0 dt0 = time/1e3 while t < time: Xm = (Xs[:-1]+Xs[1:])/2 dt = dt0 # Jf JIf calculation # dt limited by simulation stability inside each phases for i in range(Np): if Ip[i+1] > Ip[i]+1: Ipr = np.arange(Ip[i], Ip[i+1]-1) DCs[Ipr] = np.exp(splev(Xm[Ipr], fD[i])) Jf[Ipr] = -DCs[Ipr]*(Xs[Ipr+1]-Xs[Ipr])/d[Ipr] dtn = np.abs(d[Ipr]**2/DCs[Ipr]/2).min() dt = dtn if dtn < dt else dt if Ip[i+1] == Ip[i]: X0 = np.mean(Xr[i]) JIf[i, 1] = JIf[i+1, 0] = -(Xr[i, 1]-Xr[i, 0])/(If[i+1]-If[i])*np.exp(splev(X0, fD[i])) else: if i < Np-1: X0 = 0.5*(Xr[i, 1]+Xs[Ip[i+1]-1]) JIf[i+1, 0] = -(Xr[i, 1]-Xs[Ip[i+1]-1])/(If[i+1]-dis[Ip[i+1]-1])*np.exp(splev(X0, fD[i])) if i > 0: X0 = 0.5*(Xr[i, 0]+Xs[Ip[i]]) JIf[i, 1] = -(Xs[Ip[i]]-Xr[i, 0])/(dis[Ip[i]]-If[i])*np.exp(splev(X0, fD[i])) # vIf calculation, dt limited by 'No interface passing by' for i in range(1, Np): vIf[i] = (JIf[i, 1]-JIf[i, 0])/(Xr[i, 0]-Xr[i-1, 1]) vid = [Ip[i]-2] if Ip[i] > 1 else [] vid += [Ip[i]] if Ip[i] < len(dis) else [] dtn = np.abs(d[vid]/vIf[i]).min() dt = dtn if dtn < dt else dt if i > 1 and vIf[i-1] > vIf[i]: dtn = (If[i]-If[i-1])/(vIf[i-1]-vIf[i])/2 dt = dtn if dtn < dt else dt # dt limited by grid nearby interfaces cannot exceed solubility limit if Xr[0, 0] < Xr[-1, 1]: for i in range(Np): if Ip[i+1] == Ip[i]: continue elif Ip[i+1] == Ip[i]+1: if JIf[i, 1] > JIf[i+1, 0]: dtn = (Xr[i, 1]-Xs[Ip[i]])*dj[Ip[i]-1]/(JIf[i, 1]-JIf[i+1, 0]) dt = dtn if dtn < dt else dt elif JIf[i, 1] < JIf[i+1, 0]: dtn = (Xs[Ip[i]]-Xr[i, 0])*dj[Ip[i]-1]/(JIf[i+1, 0]-JIf[i, 1]) dt = dtn if dtn < dt else dt else: if i < Np-1 and JIf[i+1, 0] < Jf[Ip[i+1]-2]: dtn = -(Xr[i, 1]-Xs[Ip[i+1]-1])/(JIf[i+1, 0]-Jf[Ip[i+1]-2])*dj[Ip[i+1]-2] dt = dtn if dtn < dt else dt if i > 0 and Jf[Ip[i]] > JIf[i, 1]: dtn = (Xs[Ip[i]]-Xr[i, 0])/(Jf[Ip[i]]-JIf[i, 1])*dj[Ip[i]-1] dt = dtn if dtn < dt else dt else: for i in range(Np): if Ip[i+1] == Ip[i]: continue elif Ip[i+1] == Ip[i]+1: if JIf[i, 1] < JIf[i+1, 0]: dtn = (Xr[i, 1]-Xs[Ip[i]])*dj[Ip[i]-1]/(JIf[i, 1]-JIf[i+1, 0]) dt = dtn if dtn < dt else dt elif JIf[i, 1] > JIf[i+1, 0]: dtn = (Xs[Ip[i]]-Xr[i, 0])*dj[Ip[i]-1]/(JIf[i+1, 0]-JIf[i, 1]) dt = dtn if dtn < dt else dt else: if i < Np-1 and JIf[i+1, 0] > Jf[Ip[i+1]-2]: dtn = -(Xr[i, 1]-Xs[Ip[i+1]-1])/(JIf[i+1, 0]-Jf[Ip[i+1]-2])*dj[Ip[i+1]-2] dt = dtn if dtn < dt else dt if i > 0 and Jf[Ip[i]] < JIf[i, 1]: dtn = (Xs[Ip[i]]-Xr[i, 0])/(Jf[Ip[i]]-JIf[i, 1])*dj[Ip[i]-1] dt = dtn if dtn < dt else dt dt = time-t if t+dt > time else dt*0.95 # If first or last phase will be consumed if If[1] < dis[1] and vIf[1] < 0: dtn = (dis[0]-If[1])/vIf[1] dt = dtn if dtn < dt else dt elif If[-2] > dis[-2] and vIf[-2] > 0: dtn = (dis[-1]-If[-2])/vIf[-2] dt = dtn if dtn < dt else dt t += dt m += 1 # Ficks 2nd law inside each phase for i in range(Np): if Ip[i+1] == Ip[i]: continue elif Ip[i+1] == Ip[i]+1: Xs[Ip[i]] -= dt*(JIf[i+1, 0]-JIf[i, 1])/dj[Ip[i]-1] else: if i > 0: Xs[Ip[i]] -= dt*(Jf[Ip[i]]-JIf[i, 1])/dj[Ip[i]-1] if i < Np-1: Xs[Ip[i+1]-1] -= dt*(JIf[i+1, 0]-Jf[Ip[i+1]-2])/dj[Ip[i+1]-2] if Ip[i+1] > Ip[i]+2: Ipr = np.arange(Ip[i]+1, Ip[i+1]-1) Xs[Ipr] -= dt*(Jf[Ipr]-Jf[Ipr-1])/dj[Ipr-1] # Composition changes at first & last grid # If there is liquid phase attached, composition unchanged. if liquid != -1: Xs[0] -= Jf[0]/d[0]*dt if liquid != 1: Xs[-1] += Jf[-1]/d[-1]*dt # If one phase consumed, delete this phase if If[1]+vIf[1]*dt <= dis[0]: Xs[0] = Xr[1, 0] Np -= 1 Xr, If, Ip, fD = Xr[1:], If[1:], Ip[1:], fD[1:] Ip[0] = 0 JIf = np.zeros([Np+1, 2]) vIf = np.zeros(Np+1) if output: print('First phase consumed, %i phase(s) left, time = %.3f hrs' % (Np, t/3600)) elif If[-2]+vIf[-2]*dt >= dis[-1]: Xs[-1] = Xr[-2, 1] Np -= 1 Xr, If, Ip, fD = Xr[:-1], If[:-1], Ip[:-1], fD[:-1] Ip[-1] = len(dis) JIf = np.zeros([Np+1, 2]) vIf = np.zeros(Np+1) if output: print('Last phase consumed, %i phase(s) left, time = %.3f hrs' % (Np, t/3600)) # Interface move across one grid, passed grid has value change for i in range(1, Np): If[i] += vIf[i]*dt if If[i] < dis[Ip[i]-1]: Ip[i] -= 1 if If[i+1] < dis[Ip[i]+1]: Xs[Ip[i]] = splev(dis[Ip[i]], splrep([If[i], If[i+1]], Xr[i], k=1)) else: Xs[Ip[i]] = splev(dis[Ip[i]], splrep([If[i], dis[Ip[i]+1]], [Xr[i, 0], Xs[Ip[i]+1]], k=1)) elif If[i] > dis[Ip[i]]: Ip[i] += 1 if If[i-1] > dis[Ip[i]-2]: Xs[Ip[i]-1] = splev(dis[Ip[i]-1], splrep([If[i-1], If[i]], Xr[i-1], k=1)) else: Xs[Ip[i]-1] = splev(dis[Ip[i]-1], splrep([dis[Ip[i]-2], If[i]], [Xs[Ip[i]-2], Xr[i-1, 1]], k=1)) if output and np.mod(m, 3e4) == 0: print('%.3f/%.3f hrs simulated' % (t/3600, time/3600)) if output: print('Simulation Complete') # Insert interface informations for i in list(range(Np-1, 0, -1)): dis = np.insert(dis, Ip[i], [If[i], If[i]]) Xs = np.insert(Xs, Ip[i], [Xr[i-1, 1], Xr[i, 0]]) return DiffProfile(dis*1e6, Xs, If[1:-1]*1e6, name=name) def ErrorAnalysis(profile_exp, profile_init, diffsys, time, loc=10, w=None, r=0.3, efunc=None, accuracy=1e-3, output=False): """ Error analysis of diffusion coefficients through comparison with experimental data. Parameters ---------- profile_exp : DiffProfile Experiment measured diffusion profile, which is used to compared against each simulation result. profile_init : DiffProfile The initial profile for diffusion simulation, usually a step profile. diffsys : DiffSystem Reference diffusion coefficients. Simulation based on this datasets will be the reference for the error analysis. Bias will then be applied to this diffusivity datasets before each simulation. time : float Diffusion time in seconds. loc : list or int loc indicates the locations to perform error analysis. If loc is an integer, loc points are selected inside each phase. Each point has both positive and negative error to be calculated. w : list, optional Weights of each phase to calculate error. Passed to 'pydiffusion.utils.error_profile'. r : float, optional The concentration range of bias. Passed to 'pydiffusion.utils.DCbias' efunc : function, optional Function to create bias. Passed to 'pydiffusion.utils.DCbias' accuracy : float Stop criterion of each simulation: within error_cap * (1+-accuracy). Low accuracy value may increase simulation times for each point. output : boolean, optional Print each simulation results, default = False. Returns ------- differror : DiffError Diffusion error object """ if len(profile_init.If) != diffsys.Np+1: raise ValueError('Number of phases mismatches between profile and diffusion system') try: time = float(time) except TypeError: print('Wrong type for time variable') profile_ref = mphSim(profile_init, diffsys, time, output=output) error_ref = error_profile(profile_ref, profile_exp, w) ipt = input('Reference error= % .6f. Input cap error: [% .6f]' % (error_ref, error_ref*1.01)) error_cap = error_ref*1.01 if ipt == '' else float(ipt) print('Cap error = % .6f' % error_cap) if isinstance(loc, int): n = loc loc = np.array([]) for i in range(diffsys.Np): loc = np.append(loc, np.linspace(diffsys.Xr[i, 0], diffsys.Xr[i, 1], n)) dis_compare = np.linspace(profile_init.dis[0], profile_init.dis[-1], 1e4) profile_compare = np.zeros((3, len(dis_compare))) profile_compare[0] = splev(dis_compare, profilefunc(profile_ref)) profile_compare[2] = splev(dis_compare, profilefunc(profile_ref)) errors = np.zeros((len(loc), 2)) deltaD_init = [0.5, -0.5] for i in range(len(loc)): X = loc[i] for p in range(2): n_sim = 0 deltaD = deltaD_init[p]*1.1 De, Xe = [0], [error_ref] while True: n_sim += 1 diffsys_error = DCbias(diffsys, X, deltaD, r, efunc) profile_error = mphSim(profile_init, diffsys_error, time, output=output) error_sim = error_profile(profile_error, profile_exp, w) if output: print('At %.3f, simulation #%i, deltaD = %f, profile difference = %f(%f)' % (X, n_sim, deltaD, error_sim, error_cap)) if error_sim <= error_cap: profile_new = profilefunc(profile_error) profile_compare[1] = splev(dis_compare, profile_new) profile_compare[0] = profile_compare[:2].min(0) profile_compare[2] = profile_compare[1:].max(0) if error_sim > error_cap*(1-accuracy) and error_sim <= error_cap: break if len(De) == 1: if error_sim > error_cap: De += [deltaD] Xe += [error_sim] fe = splrep(Xe, De, k=1) deltaD = float(splev(error_cap, fe)) else: De, Xe = [deltaD], [error_sim] deltaD *= 2 else: j = 1 if error_sim > error_cap else 0 De[j], Xe[j] = deltaD, error_sim if (Xe[1]-Xe[0])/abs(De[1]-De[0]) > 10: deltaD, error_sim = De[0], Xe[0] if output: print('Jump between %f and %f' % (De[0], De[1])) break elif n_sim > 6: deltaD = np.mean(De) else: fe = splrep(Xe, De, k=1) deltaD = float(splev(error_cap, fe)) direction = 'positive' if p == 0 else 'negative' print('Error (%s) at %.3f = %f, %i simulations performed, profile difference = %f' % (direction, X, deltaD, n_sim, error_sim)) errors[i, p] = deltaD deltaD_init[p] = deltaD profiles = (splrep(dis_compare, profile_compare[0], k=1), splrep(dis_compare, profile_compare[2], k=1)) data = {} data['exp'] = profile_exp data['ref'] = profile_ref data['error'] = profiles print('Error analysis complete') differror = DiffError(diffsys, loc, errors, data) return differror def T1DsphSim(profile, tsys, time, name=''): """ 1D diffusion simulation of single-phase ternary system. Parameters ---------- profile : Profile1D Initial profile before simulation. tsys : TSystem Diffusion coefficients. time : float Diffusion time in seconds. name : str Name the output profile. Returns ------- Profile1D Final diffusion profile. """ if name == '': name = tsys.name+'_%.1fh' % (time/3600) dis = profile.dis X1, X2 = np.copy(profile.X1), np.copy(profile.X2) fD11, fD12, fD21, fD22 = tsys.fD t, m = 0, 0 d = (dis[-1]-dis[0])/(dis.size-1) while t < time: X1m = (X1[1:]+X1[:-1])/2 X2m = (X2[1:]+X2[:-1])/2 D11 = np.exp(polyval2d(X1m, X2m, fD11)) D12 = np.exp(polyval2d(X1m, X2m, fD12)) D21 = np.exp(polyval2d(X1m, X2m, fD21)) D22 = np.exp(polyval2d(X1m, X2m, fD22)) dt = d**2/max(D11.max(), D12.max(), D21.max(), D22.max())/4 dt = time-t if t+dt > time else dt*0.95 t += dt m += 1 g1 = (X1[1:]-X1[:-1])/d g2 = (X2[1:]-X2[:-1])/d J1 = -D11*g1-D12*g2 J2 = -D21*g1-D22*g2 X1[1:-1] -= dt*(J1[1:]-J1[:-1])/d X2[1:-1] -= dt*(J2[1:]-J2[:-1])/d X1[X1 < 0] = 0 X2[X2 < 0] = 0 Xsum = X1+X2 X1[Xsum > 1] = X1[Xsum > 1]/Xsum[Xsum > 1] X2[Xsum > 1] = X2[Xsum > 1]/Xsum[Xsum > 1] X1[0] -= J1[0]/d*dt*2 X1[-1] += J1[-1]/d*dt*2 X2[0] -= J2[0]/d*dt*2 X2[-1] += J2[-1]/d*dt*2 if np.mod(m, 3e4) == 0: print('1D Simulation %.3f out of %.3f hrs complete' % (t/3600, time/3600)) return Profile1D(dis, X1, X2, name=name) def T2DsphSim(profile, tsys, time, name=''): """ 2D diffusion simulation of single-phase ternary system. Parameters ---------- profile : Profile2D Initial profile before simulation. tsys : TSystem Diffusion coefficients. time : float Diffusion time in seconds. name : str Name the output profile. Returns ------- Profile2D Final diffusion profile. """ if name == '': name = tsys.name+'_%.1fh' % (time/3600) disx, disy = profile.disx, profile.disy nx, ny = profile.nx, profile.ny X1, X2 = np.copy(profile.X1), np.copy(profile.X2) fD11, fD12, fD21, fD22 = tsys.fD t, m = 0, 0 dx = (disx[-1]-disx[0])/(nx-1) dy = (disy[-1]-disy[0])/(ny-1) d = min(dx, dy) while t < time: X1xm = (X1[:, 1:]+X1[:, :-1])/2 X1ym = (X1[1:, :]+X1[:-1, :])/2 X2xm = (X2[:, 1:]+X2[:, :-1])/2 X2ym = (X2[1:, :]+X2[:-1, :])/2 g1x = (X1[:, 1:]-X1[:, :-1])/dx g1y = (X1[1:, :]-X1[:-1, :])/dy g2x = (X2[:, 1:]-X2[:, :-1])/dx g2y = (X2[1:, :]-X2[:-1, :])/dy D11x = np.exp(polyval2d(X1xm, X2xm, fD11)) D11y = np.exp(polyval2d(X1ym, X2ym, fD11)) D12x = np.exp(polyval2d(X1xm, X2xm, fD12)) D12y = np.exp(polyval2d(X1ym, X2ym, fD12)) D21x = np.exp(polyval2d(X1xm, X2xm, fD21)) D21y = np.exp(polyval2d(X1ym, X2ym, fD21)) D22x = np.exp(polyval2d(X1xm, X2xm, fD22)) D22y = np.exp(polyval2d(X1ym, X2ym, fD22)) dt = d**2/max(D11x.max(), D12x.max(), D21x.max(), D22x.max(), D11y.max(), D12y.max(), D21y.max(), D22y.max())/4 dt = time-t if t+dt > time else dt*0.95 t += dt m += 1 J1x = -D11x*g1x-D12x*g2x J1y = -D11y*g1y-D12y*g2y J2x = -D21x*g1x-D22x*g2x J2y = -D21y*g1y-D22y*g2y X1[:, 1:-1] -= dt*(J1x[:, 1:]-J1x[:, :-1])/dx X1[1:-1, :] -= dt*(J1y[1:, :]-J1y[:-1, :])/dy X2[:, 1:-1] -= dt*(J2x[:, 1:]-J2x[:, :-1])/dx X2[1:-1, :] -= dt*(J2y[1:, :]-J2y[:-1, :])/dy X1[X1 < 0] = 0 X2[X2 < 0] = 0 Xsum = X1+X2 X1[Xsum > 1] = X1[Xsum > 1]/Xsum[Xsum > 1] X2[Xsum > 1] = X2[Xsum > 1]/Xsum[Xsum > 1] X1[:, 0] -= J1x[:, 0]/dx*dt*2 X1[:, -1] += J1x[:, -1]/dx*dt*2 X1[0, :] -= J1y[0, :]/dy*dt*2 X1[-1, :] += J1y[-1, :]/dy*dt*2 X2[:, 0] -= J2x[:, 0]/dx*dt*2 X2[:, -1] += J2x[:, -1]/dx*dt*2 X2[0, :] -= J2y[0, :]/dy*dt*2 X2[-1, :] += J2y[-1, :]/dy*dt*2 if np.mod(m, 3e3) == 0: print('2D Simulation %.3f out of %.3f hrs complete' % (t/3600, time/3600)) return Profile2D(disx, disy, X1, X2, name=name)