_-PhE+2dZddlZd dZdZdZ d d ZdS) zGenerate a diffusion stencil. Supports isotropic diffusion (FE,FD), anisotropic diffusion (FE, FD), and rotated anisotropic diffusion (FD). The stencils include redundancy to maintain readability for simple cases (e.g. isotropic diffusion). N?FEct|}t|}tj|}tj|}||z}|dz}|dz}|dkrd|zdz |zd|zdz |zzd|zdz |zz} d|zdz |zd|zdz|zz} d|zdz |zd|zdz |zzd|zdz|zz} d|zdz|zd|zdz |zz} d |zd z|zd |zd z|zz} tj| | | g| | | g| | | ggd z }n^|d krId |dz z|z} ||z|z } | } ||z|z } d |dzz} tj| | | g| | | g| | | gg}nt d|S)a-Rotated Anisotropic diffusion in 2d of the form. -div Q A Q^T grad u Q = [cos(theta) -sin(theta)] [sin(theta) cos(theta)] A = [1 0 ] [0 eps ] Parameters ---------- epsilon : float, optional Anisotropic diffusion coefficient: -div A grad u, where A = [1 0; 0 epsilon]. The default is isotropic, epsilon=1.0 theta : float, optional Rotation angle `theta` in radians defines -div Q A Q^T grad, where Q = [cos(`theta`) -sin(`theta`); sin(`theta`) cos(`theta`)]. type : {'FE','FD'} Specifies the discretization as Q1 finite element (FE) or 2nd order finite difference (FD) The default is `theta` = 0.0 Returns ------- stencil : numpy array A 3x3 diffusion stencil See Also -------- stencil_grid, poisson Notes ----- Not all combinations are supported. Examples -------- >>> import scipy as sp >>> from pyamg.gallery.diffusion import diffusion_stencil_2d >>> sten = diffusion_stencil_2d(epsilon=0.0001,theta=sp.pi/6,type='FD') >>> print(sten) [[-0.2164847 -0.750025 0.2164847] [-0.250075 2.0002 -0.250075 ] [ 0.2164847 -0.750025 -0.2164847]] Consider a 2 x 4 grid ([x0, x1] x [y0, y1, y2, y3]). The first dimension of the stencil defines x. >>> nx, ny = (2, 4) >>> sten = pyamg.gallery.diffusion_stencil_2d(epsilon=0.1, type='FD') >>> A = pyamg.gallery.stencil_grid(sten, (nx, ny)).toarray() >>> print(sten) [[-0. -1. 0. ] [-0.1 2.2 -0.1] [ 0. -1. -0. ]] >>> print(A) [[ 2.2 -0.1 0. 0. -1. 0. 0. 0. ] [-0.1 2.2 -0.1 0. 0. -1. 0. 0. ] [ 0. -0.1 2.2 -0.1 0. 0. -1. 0. ] [ 0. 0. -0.1 2.2 0. 0. 0. -1. ] [-1. 0. 0. 0. 2.2 -0.1 0. 0. ] [ 0. -1. 0. 0. -0.1 2.2 -0.1 0. ] [ 0. 0. -1. 0. 0. -0.1 2.2 -0.1] [ 0. 0. 0. -1. 0. 0. -0.1 2.2]] rg@FDg?g@z.only stencil types "FE" and "FD" are supported)floatnpcossinarray ValueError)epsilonthetatypeepsCSCSCCSSabcdestencils W/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/gallery/diffusion.pydiffusion_stencil_2dr&sH ..C %LLE u A u A 1B AB AB t||.VaZOr#vz2o -3B > sUQYNbfqj"_ , VaZOr#vz2o -C!R ? VaZOquqy"n , sUQYNaeai^ +(Q1I1I1I'((*-. " qM" "frkN B"frkN qM(Q1I1I1I'((IJJJ Nc ddlm}m}|d\}}|d\}}|d\}}|||dg| |dggdg}|gdd||gd| |gg} |||dg| |dggdg} || z| z} |d\} } |gdd| dgdd| gg}| |z| jz}t d D]4}t d D]"}t d |d |d |||f#5d S)z?Use SymPy to generate the 3D matrices for diffusion_stencil_3d.rsymbolsMatrixz cpsi, spsizcth, sthz cphi, sphi)rrr )r rrz epsy, epszr zD[z, z] = N)sympyr*r+Trangeprint)r*r+cpsispsicthsthcphisphiRpsiRthRphiQepsyepszADijs r%_symbolic_rotation_helperr@s&%%%%%%%&&JD$wz""HC&&JD$ 6D$?dUD!$4iii@ A AD &)))ac]QcN; < >0Cc3S#sC c3#sC3S/BCCA 6%%%& ' ' )D &C a$A !A$KKKKKr'rc t|}t|}t|}t|}t|}tjd}tj|} tj|} tj|} tj|} tj|} tj|}|| | z|z| | zzdzz||dzz| dzzz| | z| | z|zz dzz|d<| |z|z| dzz|| | z| z| |zz z| | z|z| | zzzz| | z| | z|zz | |z| | z| zz zz|d<| |z| z| | z|z| | zzz| |z|z| zz| | z| | z| | z|zz zz|d<| |z|z| dzz|| | z| z| |zz z| | z|z| | zzzz| | z| | z|zz | |z| | z| zz zz|d<| dz|z| dzz|| | z| z| |zz dzzz| |z| | z| zz dzz|d<| |z| z| | z| z| |zz z| | z|z| zz| | z| |z| | z| zz zz|d<| |z| z| | z|z| | zzz| |z|z| zz| | z| | z| | z|zz zz|d <| |z| z| | z| z| |zz z| | z|z| zz| | z| |z| | z| zz zz|d <| dz|z| dzz| dz|zz| dz| dzzz|d <tjd }|d krt d|dkrd\}}}||dz ||dzg||fxxtjgd|dzz cc<|||dz ||dzg|fxxtjgd|dzz cc<||||dz ||dzgfxxtjgd|d zz cc<tjgd}tjgd}|||z||z|fxxdtjgdz|d|dzzz cc<|||z|||zfxxdtjgdz|d |dzzz cc<||||z||zfxxdtjgdz|d |dzzz cc<|S)a; Rotated Anisotropic diffusion in 3d of the form. -div Q A Q^T grad u Q = Rpsi Rtheta Rphi Rpsi = [ c s 0 ] [-s c 0 ] [ 0 0 1 ] c = cos(psi) s = sin(psi) Rtheta = [ 1 0 0 ] [ 0 c s ] [ 0 -s c ] c = cos(theta) s = sin(theta) Rphi = [ c s 0 ] [-s c 0 ] [ 0 0 1 ] c = cos(phi) s = sin(phi) Here Euler Angles are used: http://en.wikipedia.org/wiki/Euler_angles This results in Q = [ cphi*cpsi - cth*sphi*spsi, cpsi*sphi + cphi*cth*spsi, spsi*sth] [ - cphi*spsi - cpsi*cth*sphi, cphi*cpsi*cth - sphi*spsi, cpsi*sth] [ sphi*sth, -cphi*sth, cth] A = [1 0 ] [0 epsy ] [0 0 epsz] D = Q A Q^T Parameters ---------- epsilony : float, optional Anisotropic diffusion coefficient in the y-direction where A = [1 0 0; 0 epsilon_y 0; 0 0 epsilon_z]. The default is isotropic, epsilon=1.0 epsilonz : float, optional Anisotropic diffusion coefficient in the z-direction where A = [1 0 0; 0 epsilon_y 0; 0 0 epsilon_z]. The default is isotropic, epsilon=1.0 theta : float, optional Euler rotation angle `theta` in radians. The default is 0.0. phi : float, optional Euler rotation angle `phi` in radians. The default is 0.0. psi : float, optional Euler rotation angle `psi` in radians. The default is 0.0. type : {'FE','FD'} Specifies the discretization as Q1 finite element (FE) or 2nd order finite difference (FD) Returns ------- stencil : numpy array A 3x3 diffusion stencil See Also -------- stencil_grid, poisson, _symbolic_rotation_helper, _symbolic_product_helper Notes ----- Not all combinations are supported. Examples -------- >>> import scipy as sp >>> from pyamg.gallery.diffusion import diffusion_stencil_2d >>> sten = diffusion_stencil_2d(epsilon=0.0001,theta=sp.pi/6,type='FD') >>> print(sten) [[-0.2164847 -0.750025 0.2164847] [-0.250075 2.0002 -0.250075 ] [ 0.2164847 -0.750025 -0.2164847]] )r r r)rr)rr )rr)r r)r r )r r)rr)rr )rr)r r r rzFE not implemented yetr)r r r r )rrr)rrr r )rr rr g?)r rrr )rrzerosrrNotImplementedErrorr)epsilonyepsilonzrphipsirr:r;r=r4r5r2r3r0r1r$r>r?kLMs r%diffusion_stencil_3dr[sPj ??D ??D %LLE **C **C A 6#;;D 6#;;D &--C &--C 6#;;D 6#;;DDHTMDI-11DqLa4GG dSXd] "Q&'AdG4inS!V# d4imd4i'($s(4-$t)*CDE dSXd] "dU4Z$s(4-%?@AAdGeDjnd3htmd4i78 D c Hd4i#d(4-&?@AAdG4inS!V# d4imd4i'($s(4-$t)*CDE dSXd] "dU4Z$s(4-%?@AAdGAgdl36!D$t)C-$t)*Ca)G$GG td3htm #a'(AdGeDjnd4imd4i78 S c HteDj48D=&@ABAdGeDjnd3htmd4i783t8D=;LL S$t)c$htm+,-AdGeDjnd4imd4i7848D=;LL S4%*tCx},-.AdGAgdl36!CF4K/$'#q&.@AdGhy!!G t||!":;;; t||1a 1a1 q!#$$$(=(=$(GG$$$ AaCAaC=!#$$$(=(=$(GG$$$ 1qsAqsm#$$$(=(=$(GG$$$ H^^^ $ $ H^^^ $ $Aq1ua   bh~~~...!D'AdG2CD E    Aq!a%   bh~~~...!D'AdG2CD E    1q5!a%   bh~~~...!D'AdG2CD E    Nr')rrr)rrrrrr)__doc__numpyrr&r@rPr[r'r%r_sKKKK\...0"EH'+ccccccr'