escript basics Lutz Gross School of Earth Sciences The University of Queensland 1/46 13/03/13

Transcription

1 escript basics Lutz Gross School of Earth Sciences The University of Queensland 1/46 13/03/13

2 escript Sponsors AuScope National Collaborative Research Infrastructure Strategy Australian Geophysical Observing System (AGOS) (EIF) School of Earth Sciences at the University of Queensland. 2/46 13/03/13

3 Escript? Software for solving partial differential equations Coupled, time-dependent, non-linear Easy to use uses PDE terminology programming in python based on the finite element method parallelized for multi-cores and clusters 3/46 13/03/13

4 Porous Media Flow: Perth Basin 4/46 13/03/13

5 Stress Localization 5/46 13/03/13

6 Mantel Convection 6/46 13/03/13

7 How to get escript Distributed via launchpad: visit FAQ current version some binary versions available Documentation Cookbook (for beginner) User's guide (the bible) Inversion cookbook Reference guide 7/46 13/03/13

8 Python? Programming language Interactive or scripts Object oriented Open source Standard in Linux Tons of useful modules numpy: linear algebra SciPy, obspy, matplotlib, sympy Tutorial: 8/46 13/03/13

9 Python example $ python Python (#62, Sep , 19:20:46) [MSC v bit (Intel)] on win32 Type "help", "copyright", "credits" or "license" for more information. >>> print `Hello World` Hello World >> x=6 >> print x+9 15 >> s=`hello`+` `+`World` # concatenate strings >> print s Hello World >> q=[ 1, `H`, x, s ] # a list of objects >> print q[2] 6 >> print q[2:] [6, 'Hello World'] >> CRTL^D # get out of here $ 9/46 13/03/13

10 escript Data Handling: esys.escript PDEs: esys.escript.linearpdes Finite Element solver: Unstructured: esys.finley Grids: esys.ripley Inversion: esys.downunder Data port: esys.weipa Geometries: esys.pycad 10/46 13/03/13

11 Problem: Temperature Distribution T=0 (l 0,l 1 ) heat flux=0 heat flux=0 x 1 (0,0) T=T bot heat source x 0 11/46 13/03/13

12 1D Temperature Diffusion x 0 ( κ T x 0 ) =Q T temperature κ thermal conductivity (may depend on x) Q heat source 12/46 13/03/13

13 Escript's Notation for derivatives With Z, 0 = Z x 0 (κ T,0 ), 0 =Q 13/46 13/03/13

14 2D Temperature Diffusion ( κ T )=Q or x 0 ( κ T x 0 ) x 1 ( κ T x 1 ) =Q 14/46 13/03/13

15 2D Temperature Diffusion (cont.) x 0 ( κ T x 0 ) x 1 ( κ T x 1 ) =Q or (κ T,0 ), 0 (κ T, 1 ),1 =Q 15/46 13/03/13

16 Insulation Boundary Condition heat flux=0 heat flux=0 boundary heat flux=κ T n =κ n T =κ n T i, i (summation over i) 16/46 13/03/13

17 Flux Boundary Conditions (l 0,l 1 ) outer normal field: n= (n i ) (0,0) Ω n Apply at face x 0 =0 and x 0 =l 0 heat flux=κ n i T, i =0 Called Neuman or natural boundary condition 17/46 13/03/13

18 Temperature Boundary Condition T=0 T=T bot (known value) Also called Dirichlet boundary condition or constraint 18/46 13/03/13

19 How to define a Constraint T=T D =0 appropriate T D over entire domain (l 0,l 1 ) (0,0) Ω T=T D =T bot Constraint is given as: T=T D at x 1 =0 and x 1 =l 1 For instance : T D (x 0, x 1 )= T bot l 1 (l 1 x 1 ) 19/46 13/03/13

20 PDE for Temperature Diffusion (κ T,i ), i =Q κ n i T,i =0 at faces x 0 =0 and x 0 =l 0 T =T D at faces x 1 =0 and x 1 =l 1 (summation over i) 20/46 13/03/13

21 Domain in escript - region where to solve the PDE - a discretization method (0,l 1 ) (l 0,l 1 ) (0,0) (l 0,0) 21/46 13/03/13

22 Finite Element Mesh in finley Node Element Quadrature Point Temperature by values on Nodes Gradients by value on quadrature points. 22/46 13/03/13

23 Create a finley domain # get the tools we want to use from esys.escript import * from esys.finley import Rectangle # generate n0 x n1 elements over # [0,l0] x [0,l1] L0=10. L1=5. mydomain=rectangle(l0=l0,l1=l1,n0=80,n1=20) print dimension =,mydomain.getdim() x=mydomain.getx() print x # coordinates of FEM nodes as list or summary 23/46 13/03/13

24 How do I run this? 1. create a file myprog.py, eg. >> gedit myprog.py & 2. enter statements into file and save 3. run python >> run-escript myprog.py 24/46 13/03/13

25 Write to file from esys.escript import * from esys.finley import Rectangle from esys.weipa import * L0=10. L1=5. T_bot=100. mydomain=rectangle(l0=l0,l1=l1,n0=80,n1=40) x=mydomain.getx() T_D=T_bot/L1*(L1-x[1]) # save T_D for visualization with a VTK tool savevtk('u.vtu',t=t_d) 25/46 13/03/13

26 Run & Visualization >> run-escript myprog.py >> ls myprog.py u.vtu >> visit & # fire up visualization Or any tool supporting VTK files: VisIt, paraview, mayavi2 26/46 13/03/13

27 Visualization with VisIt Click Select data file Click 27/46 13/03/13

28 VisIt (cont.) 1. Click 3. Click to draw 2. Select data set 'T' 28/46 13/03/13

29 Visualization with VisIt (cont.) 29/46 13/03/13

30 Tell me more about T_D! escript data objects: T_D, x Values at data points >> print T_D.getShape(), x.getshape() () (2,) >>print T_D.getRank(), x.getrank() 0 1 >> print x.getfunctionspace() Finley_Nodes >> print T_D.getFunctionSpace() # no surprise! Finley_Nodes >> print grad(t_d).getfunctionspace() Finley_Elements >> print grad(t_d).getshape(), grad(x).getshape() (2,), (2,2) 30/46 13/03/13

31 PDE for Temperature Diffusion (κ T,i ), i =Q κ n i T,i =0 at x 0 =0 and x 0 =l 0 T =T D at x 1 =0 and x 1 =l 1 31/46 13/03/13

32 Steady PDEs in Escript from esys.escript.linearpdes import LinearPDE from esys.finley import Rectangle mydomain=rectangle(l0=l0,l1=l1,n0=80,n1=40) # create a PDE: mypde=linearpde(mydomain) # set the coefficients of the PDE mypde.setvalue(?????) # solve the PDE T = mypde.getsolution() 32/46 13/03/13

33 LinearPDE Interface (simplified) Interface to a single linear PDE for u ( A ij u, j ),i (B i u),i +C j u, j +D u= X j, j +Y natural BC: n i A ij u, j +n i B i u=n j X j + y contraints:u=r where q>0 Summation over i and j 33/46 13/03/13

34 LinearPDE for Diffusion Identify terms: ( A ij u,j ),i ( B i u ),i +C j u,j +Du= ( X,j ),j +Y (κt,i ),i =Q 34/46 13/03/13

35 LinearPDE for Diffusion (cont.) Natural boundary condition: n i A ij u,j +n i B i u=n j X,j +y n i κt,i =0 35/46 13/03/13

36 LinearPDE for Diffusion u=r where q> 0 T=T D where x 1 =0 or x 1 =l 1 36/46 13/03/13

37 LinearPDE for Diffusion (cont.) from esys.escript.linearpdes import LinearPDE from esys.finley import Rectangle mydomain=rectangle(l0=l0,l1=l1,n0=80,n1=40) # create a PDE: mypde=linearpde(mydomain) # set the coefficients of the PDE mypde.setvalue(a=?, Y=?, q=?, r=?, y=0) # solve the PDE T=mypde.getSolution() Not required! 37/46 13/03/13

38 Diffusion Term A (κt, i ),i =(κ δ ij T, j ),i =( A ij T, j ), i Kronecker symbol δ ij =[ ] = { A ij =κ δ ij =[ 1 if i= j } 0 else κ 0 ] 0 κ sum over i and j Check natural boundary conditions!!! 38/46 13/03/13

39 LinearPDE for Diffusion (cont.) from esys.escript.linearpdes import LinearPDE from esys.finley import Rectangle mydomain=rectangle(l0=l0,l1=l1,n0=80,n1=40) # create a PDE: mypde=linearpde(mydomain) # set the coefficients of the PDE mypde.setvalue(a=k*kronecker(mydomain)) Anisotropy is easy. 39/46 13/03/13

40 Source Q Ω Qc Q ( x )={ Q c for x x c <r } 0 otherwise 40/46 13/03/13

41 Source Q (continued) y = y y y 2 2 =esys.escript.length ( y ) from esys.escript import * xc=[7.,2.] r=0.8 Qc=100. x=mydomain.getx() Q=Qc*whereNegative(length(x-xc)-r) mypde.setvalue(y=q) 41/46 13/03/13

42 How to set q Make q one at top and bottom and zero elsewhere. # get the coordinates of points in the domain x=mydomain.getx() # f1=1 where x[1]==0 f1=wherezero(x[1]) f2==1 # f2=1 where x[1]==l1 x[1]==l1 f2=wherezero(x[1]-l1) q=f2+f1 mypde.setvalue(r=t_d,\ Ω q=q) f1==1 x[1]==0 42/46 13/03/13

43 Put it all together from esys.escript import * from esys.escript.linearpdes import LinearPDE from esys.finley import Rectangle from esys.weipa import savevtk L0=10.;L1=5. T_bot=100.; k=1. xc=[7.,2.]; r=0.8; Qc=100. mydomain=rectangle(l0=l0,l1=l1,n0=80,n1=40) x=mydomain.getx() T_D=T_bot/L1*(L1-x[1]) Q=Qc*whereNegative(length(x-xc)-r) mypde=linearpde(mydomain) mypde.setvalue(a=k*kronecker(mydomain),y=q, r=t_d, \ q=wherezero(x[1])+wherezero(x[1]-l1)) T=mypde.getSolution() savevtk('u.vtu', T=T, flux=-k*grad(t)) 43/46 13/03/13

44 Result 44/46 13/03/13

45 Some Remarks This PDE problem is symmetric: use mypde.setsymmetryon() Speeds up calculation Use mypde.settolerance() to control accuracy of linear solver for piecewise constant PDE coefficients use tagging ( mesh generation esys.pycad) 45/46 13/03/13

46 The End (for now) Questions? 46/46 13/03/13