_-PhcpdZddlZddlmZdZddZdZddZd Z Gd d Z d Z ddZ dZ dZdZddZdS)z%Poisson problem with finite elements.N)sparsectj||dddfdf||dddfdfz ||dddfdf||dddfdfz tj|jdfj}tj||dddfdf||dddfdfz ||dddfdf||dddfdfz tj|jdfj}tjtj||dddfdk}|S)z6Check the ccw orientation of each simplex in the mesh.Nr)npvstackzerosshapeTallcross)VEE01E12 orientations Q/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/gallery/fem.py check_meshrsU )QqAwz]QqAwz]2qAwz]QqAwz]2Xagaj))+ , ,,- )QqAwz]QqAwz]2qAwz]QqAwz]2Xagaj))+ , ,,-&#s++AAAqD1A566K Fc(t|tjrt|tjstd|jddks|jddkrtd|jd}tjtjd|tjdt}tj tj|dzft| |ff}||j z}tj |d }tj|j|jfj }||d d dfd d f||d d dfd d fzd z } tj|| f}|dz} | tj|jdz} | |_||j z}tj|tj|jddftf}||d d df|d d dff|d d df<||d d df|d d dff|d d d f<||d d df|d d dff|d d d f<|r|||fS||fS)awGenerate a quadratic element list by adding midpoints to each edge. Parameters ---------- V : ndarray nv x 2 list of coordinates E : ndarray ne x 3 list of vertices return_edges : bool indicate whether list of the refined edges is returned Returns ------- V2 : ndarray nv2 x 2 list of coordinates E2 : ndarray ne2 x 6 list of vertices Edges : ndarray nedge x 2 list of edges where the midpoint is generated Notes ----- - midpoints are introduced and globally numbered at the end of the vertex list - the element list includes the new list between v0-v1, v1-v2, and v2-v0 Examples -------- >>> import numpy as np >>> from pyamg.gallery import fem >>> V = np.array([[0.,0.], [1.,0.], [0.,1.], [1.,1.]]) >>> E = np.array([[0,1,2], [2,3,1]]) >>> V2, E2 = fem.generate_quadratic(V, E) >>> print(V2) [[0. 0. ] [1. 0. ] [0. 1. ] [1. 1. ] [0.5 0. ] [0.5 0.5] [0. 0.5] [0.5 1. ] [1. 0.5]] >>> print(E2) [[0 1 2 4 5 6] [2 3 1 7 8 5]] V and E must be ndarrayrr)V should be nv x 2 and E should be ne x 3rrdtypeN@) isinstancerndarray ValueErrorr kronarangeonesintr coo_matrixravelr trilrrowcolmaxdatahstackr ) rr return_edgesneIDGV2VV2VmidEdgesVmidmaxindexnewIDs rgenerate_quadraticr:sf a $ $4Jq"*,E,E42333wqzQ!'!*//DEEE B 1b!!274s#;#;#; < >> import numpy as np >>> dx = 1 >>> V = np.array([[0,0], [dx,0], [0,dx], [dx,dx]]) >>> E = np.array([[0,1,2], [2,3,1]]) >>> h = diameter(V, E) >>> print(h) 1.4142135623730951 rrrrrr)rrrr)r!rr"r#r sqrtdiffr-)rrhIehss rdiameterrBls8 a $ $4Jq"*,E,E42333wqzQ!'!*//DEEE A A  WRWQqtQwZ((!+bga!aj.A.A1.DD E E 26688   Hrc |jd}|jd}|tjd|}tj|}|}tjtjd|gd}tj|dzf}t j|||ff||f}|j|z}tj|j j|_ t j |d } t| j } tjd| | _ tj | j| jfj} | jd} | } tj |ddddgf|ddddgf|ddddgff} | d| | dddf| dddffd|fj} tj| ft( }d || |ddf<ttj|dkd}|| d}tj|dkd}d || |ddf<ttj|dkd}tj|dkd}d || |dfdfzd || |dfdfzz}d || |dfdfzd || |dfdfzz}tj ||fj}tj ||f}tj| ft. }|tjd|z||<tj|ft( }d ||<d ||<tj||d}||df}||df}||df}|| |df}|| |df}|| |df}tj |||fj}tj |||fj} tj |||fj}!tj |||fj}"tj ||| |!|"f}||fS) aRefine a triangular mesh. Parameters ---------- V : ndarray nv x 2 list of coordinates E : ndarray ne x 3 list of vertices marked_elements : array list of marked elements for refinement. None means uniform. Returns ------- Vref : ndarray nv x 2 list of coordinates Eref : ndarray ne x 3 list of vertices Notes ----- - Peforms quad-section in the following where n0, n1, and n2 are the original vertices n2 / | / | / | n5-------n4 / \ /| / \ / | / \ / | n0 --------n3-- n1 rN)rrrrr rraxisrT?F)r rr%r)r$r&rr(r r.triutocoolenrr+r,tocsrsortreshaper boolwheresumr'delete)#rrmarked_elementsNelNvr,r+r.r4V2VupperNedgesEdgeListedges ElementToEdge marked_edgesnsplitedge_numedges3edges1x_newy_newV_newnew_ididsE_newn0n1n2n3n4n5t1t2t3t4s# r refine2dtrirosJ '!*C B)As++h//O ''))C '")As##YYY / /C 7CE8  D  TC:.sBi @ @ @C %#+Cwsx~&&CH{3""((**H  FIa((HMy(, 5668H ^A F~~H IqQF|QF|QF|% & &E JJAJU111a4[%1+56>>3xHHJM8VIT222L>BL1288::;,!+,,Q/ 0 0FM*..A.66H Xh!m $ $Q 'F-1Lvqqqy)* ,!+,,Q/ 0 0F Xh!m $ $Q 'F 8L!O,a/0 1 q,/*A-. / 0E 8L!O,a/0 1 q,/*A-. / 0E Iuen % % 'E 1e*A Xvis + + +F !V 4 44F< '3& % % %CCKCK Iaq 1 1 1E 619B 619B 619B  fai( ) / / 1 1B  fai( ) / / 1 1B  fai( ) / / 1 1B BB< "B BB< "B BB< "B BB< "B Iub"b"- . .E e8Orc &|jdkr|j}|j}n)|jdkr|j}|j}nt dt |tjst d|j ddkrt d|jdkr |j ddkrt d|jdkr |j ddkrt d |jd vrt d d }tj gd }tj ddgddgddgddgddgddgg}|ddd fdzdz |dddfdzdz }}|dz}|jdkrtj d} d} | } n4|jdkrtj d} d} | } nt d|D]} || d f}|| df}tj |d|d z |d|d z z|d|d z |d|d z zz }t|||D]@\}}}||dz |ztj|| | | ||dzzz }Atj|S)aCalculate the L2 norm of a function on mesh (V,E). Parameters ---------- u : array (nv,) list of function values mesh : object mesh object Returns ------- val : float the value of the L2 norm of u, ||u||_2,V Notes ----- - modepy is used to generate the quadrature points q = modepy.XiaoGimbutasSimplexQuadrature(4,2) Examples -------- >>> import numpy as np >>> from pyamg.gallery import fem >>> V = np.array([[0,0], [1,0], [0,1], [1,1]]) >>> E = np.array([[0,1,2], [1,3,2]]) >>> mesh = Mesh(V, E, degree=1) >>> X, Y = mesh.V[:, 0], mesh.V[:, 1] >>> u = X + Y >>> unorm = fem.l2norm(u, mesh) >>> print(f'{unorm:2.6}') 1.08012 >>> mesh = Mesh(V, E, degree=2) >>> X, Y = mesh.V2[:, 0], mesh.V2[:, 1] >>> u = X + Y >>> unorm = fem.l2norm(u, mesh) >>> print(f'{unorm:2.6}') 1.08012 rr!only mesh.degree 1 or 2 supportedzu must be ndarrayzV should be nv x 2rzE should be nv x 3zE should be nv x 6rrdegree = 1 or 2 supportedrė?rvrvE#'%?rwrw&y緡yLy纡xēx#ԓx#' ;G?ʓx#Гx#' ;G?NrGc<tjd|z |z ||gS)Nrrarrayxys rbasis1zl2norm..basis1hs+8QqSU !! !rc tjd|z |z dd|zz d|zz z|d|zdz z|d|zdz zd|zd|z |z zd|z|zd|zd|z |z zgS)Nrrrrrs rbasis2zl2norm..basis2ps8ac!ea!eAaCi0!AY!AYqS!A#a%[qSUqS!A#a%[ *++ +r)degreerrV2E2r#r!rr"r rr%abszipdotr<)umeshrrvalwwxyxxyyr?rbasisrr@rrjacwvxvyvs rl2normrsP {a F F    G G<=== a $ $.,---wqzQ-... {aAGAJ!OO-... {aAGAJ!OO-... {&  4555 C RRR S SB (*>?(*>?(*>?(*>?')=>(*=> @ A AB Ahqj!^bAhqj!^B#IB {a IaLL ! ! !    IaLL + + +<===  F F adG adGfad1Q4i!A$qt),!QqT AaD1I/FFGGb"b// F FJBB C!Gr>BF1QqT7EE"bMM$B$BA$EE ECC F 73<<rc.eZdZdZd dZdZdZd dZd S) Meshz:Simple mesh object that holds vertices and mesh functions.rcFtj|dzfd}d||<tj|}|jd|kr}t dtj|d}tj|dz}tj|||<||}||ddf}t||std||_ ||_ |dddf|_ |dddf|_||_||_|jd|_t%|||_d|_d|_d|_d|_|dkr|dSdS) zInitialize mesh. Parameters ---------- V : ndarray nv x 2 list of coordinates E : ndarray ne x 3 list of vertices rFTrzfixing V and ENz#triangles must be counter clockwiser)rfullr-r)rPr printrOr%rr#rrXYrnvr1rBr>rrr6r9r:)selfrrrrcrr?Js r__init__z Mesh.__init__s}gquuwwqylE**AGGII VC[[ 71:   " # # # a A !%%''!)$$A9R==AaD!A!QQQ$A!Q DBCC C111a4111a4 '!*!Q  Q;;  # # % % % % % ;rc0|jt|j|jd\|_|_|_|jdddf|_|jdddf|_|jtj |jj dz|_ dSdS)zGenerate a quadratic mesh.NT)r0rr) rr:rrrr6X2Y2rrr%r r9)rs rr:zMesh.generate_quadratics 7?+=dfdfKO,Q,Q,Q (DGTWdjgaaadmDGgaaadmDG29TZ-=a-@#A#AADJJJ ?rcd|_d|_d|_d|_t |D])}t |j|j\|_|_*|jjd|_ |jjd|_ t|j|j|_ |jdddf|_ |jdddf|_|jdkr|dSdS)z~Refine the mesh. Parameters ---------- levels : int Number of refinement levels. Nrrr)rrr6r9rangerorrr rr1rBr>rrrr:)rlevels_s rrefinez Mesh.refines  v 9 9A(88NDFDFF&,q/&,q/$&$&))11 ;!    # # % % % % %  r {Gz?c|j}|jddgdf}|jddgdf}tj|jdft }tj|||ff||f}| | tj |j dkd}tj |j|}d|j dd<tj|d} |j} | |df| |dfz d z| |df| |dfz d zz} d }t)|D]} || z} | | dddfz} |j|ddf| |ddf<tj| |df| |dfz d z| |df| |dfz d zz} tjtj| | z | z }| } ||krn| |_| S) a Constrained Laplacian Smoothing. Parameters ---------- maxit : int Iterations tol : float Convergence toleratnce measured in the maximum absolute distance the mesh moves (in one iteration). N)rrrrrr)rrrrrrrrrDrrErd)rrr)rr&r r'rr(sum_duplicateseliminate_zerosrOr.uniquer+rrPflattenrcopyrr<r-r)rmaxittolredge0edge1r.r3bidWVnew edgelength_it newedgelengthmoves rsmoothz Mesh.smoothsnWqqq,,,,-3355qqq,,,,-3355w A(444  teU^4RH E E E  hqv{##A&ic ##qqq HQUUU]] # # + + - -v{{}}5!8ntE1H~595!8ntE1H~59: <<  Ct8D AaaagJ D6#qqq&>DaaaLGT%(^d5!8n%Dq$H'+E1H~UAX'F&J%KLLM6"&!;<<}LMMD&Jczz rN)r)rr)__name__ __module__ __qualname____doc__rr:rrrrrrsdDD +&+&+&+&ZBBB&&&.//////rrc:|||}t|ttfrtjd|zSt|tjr$|jdkr|Std|jtdt|)zkStandardize diffusion tensor/scalar. This will return an ndarray or a scalar, depending on input. r)rrzPkappa must return a scalar or ndarray of shape (2,2), received ndarray of shape zDkappa must return a scalar or ndarray of shape (2,2), received type ) r!r'floatreyer"r r#type) kappa_lmbdarrkappas r_compute_diffusion_matrixrs K1  E%#u&&!vayy5  %$$E ;& LD6;kDDEE E 4&*5kk44 5 55rrc  |dvrtd|d}|d}t|rt|std|j}|dkr|j}|j}|j}n+|dkr|j}|j}|j}ntd |dkrd nd }tj ||dzf} tj ||dzft } tj ||dzft } tj ||f} tj ||ft } tj ||ft }td |D]}||ddf}||d ||d }}||d||d}}||d||d}}tj ||z ||z g||z ||z gg}tj|j}tj|}|dkrtj gdgdg}||}t'|||||}|dz |jz|z|z}||||||dz ztjdz}|dkrtj gd}tj ddgddgddgddgddgddgg} | ddd fdzdz | dddfdzdz }"}!|d z}tj ||f}tj |f}t-||!|"D]m\}#}$}%tj d|$z |%z dd|$zz d|%zz z|$d|$zdz z|%d|%zdz zd!|$zd|$z |%z zd!|$z|%zd!|%zd|$z |%z zg}&tj d!|$zd!|%zzd z d!|$zdz d d"|$zd!|%zz d!zd!|%zd#|%zgd!|$zd!|%zzd z d d!|%zdz d#|$zd!|$zd#|$zd$|%zz d!zgg}||}|tj |$|%gtj ||gz\}'}(t'||'|(}||dz |#z|jz|z|zz }||dz |#z||'|(z|&zz }o|| |ddf<tj|tj||| |ddf<tj|tj||| |ddf<|| |ddf<|tj|| |ddf<d ||ddf<t7j| | | ff})|)t7j| | |ff}*|)|*fS)%a Finite element discretization of a Poisson problem. - div . kappa(x,y) grad u = f(x,y) Parameters ---------- V : ndarray nv x 2 list of coordinates E : ndarray ne x 3 or 6 list of vertices kappa : function diffusion coefficient, kappa(x,y) with vector input can either return a scalar value or a 2x2 matrix that transforms f : function right hand side, f(x,y) with vector input degree : 1 or 2 polynomial degree of the bases (assumed to be Lagrange locally) Returns ------- A : sparse matrix finite element matrix where A_ij = b : array finite element rhs where b_ij = Notes ----- - modepy is used to generate the quadrature points q = modepy.XiaoGimbutasSimplexQuadrature(4,2) Examples -------- >>> import numpy as np >>> from pyamg.gallery import fem >>> import scipy.sparse.linalg as sla >>> V = np.array( ... [[ 0, 0], ... [ 1, 0], ... [2*1, 0], ... [ 0, 1], ... [ 1, 1], ... [2*1, 1], ... [ 0,2*1], ... [ 1,2*1], ... [2*1,2*1]]) >>> E = np.array( ... [[0, 1, 3], ... [1, 2, 4], ... [1, 4, 3], ... [2, 5, 4], ... [3, 4, 6], ... [4, 5, 7], ... [4, 7, 6], ... [5, 8, 7]]) >>> mesh = Mesh(V, E) >>> A, b = fem.gradgradform(mesh) >>> print(A.toarray()) [[ 1. -0.5 0. -0.5 0. 0. 0. 0. 0. ] [-0.5 2. -0.5 0. -1. 0. 0. 0. 0. ] [ 0. -0.5 1. 0. 0. -0.5 0. 0. 0. ] [-0.5 0. 0. 2. -1. 0. -0.5 0. 0. ] [ 0. -1. 0. -1. 4. -1. 0. -1. 0. ] [ 0. 0. -0.5 0. -1. 2. 0. 0. -0.5] [ 0. 0. 0. -0.5 0. 0. 1. -0.5 0. ] [ 0. 0. 0. 0. -1. 0. -0.5 2. -0.5] [ 0. 0. 0. 0. 0. -0.5 0. -0.5 1. ]] >>> print(b) [0. 0. 0. 0. 0. 0. 0. 0. 0.] >>> f = lambda x, y : 0*x + 1.0 >>> g = lambda x, y : 0*x + 0.0 >>> g1 = lambda x, y : 0*x + 1.0 >>> tol = 1e-12 >>> X, Y = V[:,0], V[:,1] >>> id1 = np.where(abs(Y) < tol)[0] # north >>> id2 = np.where(abs(Y-2) < tol)[0] # south >>> bc = [{'id': id1, 'g': g}, ... {'id': id2, 'g': g}] >>> A, b = fem.gradgradform(mesh, f=f) >>> A, b = fem.applybc(A, b, mesh, bc) >>> A = A.tocsr() >>> u = sla.spsolve(A, b) >>> print(A.toarray()) [[ 1. 0. 0. 0. 0. 0. 0. 0. 0.] [ 0. 1. 0. 0. 0. 0. 0. 0. 0.] [ 0. 0. 1. 0. 0. 0. 0. 0. 0.] [ 0. 0. 0. 2. -1. 0. 0. 0. 0.] [ 0. 0. 0. -1. 4. -1. 0. 0. 0.] [ 0. 0. 0. 0. -1. 2. 0. 0. 0.] [ 0. 0. 0. 0. 0. 0. 1. 0. 0.] [ 0. 0. 0. 0. 0. 0. 0. 1. 0.] [ 0. 0. 0. 0. 0. 0. 0. 0. 1.]] >>> print(b) [0. 0. 0. 0.5 1. 0.5 0. 0. 0. ] >>> print(u) [0. 0. 0. 0.5 0.5 0.5 0. 0. 0. ] rsrtNcdS)Nr_x_ys rfzgradgradform..f3rcdS)N?rrs rrzgradgradform..kapparrz#f, kappa must be callable functionsrrrqrrrrr)rrr)rrrrg@rrurxryrzr{r|r}r~rrrrrrGr)r#callabler1rrrrrrrr r'rrlinalginvr detrrmeanr&rr)repeatr%tilerr(rtoarray)+rrrrr1rrrmAAIAJAbbibjbeiKx0y0x1y1x2y2rinvJdetJdbasisdphi kappaelemAelembelemrrrrwrrrxtytAbs+ r gradgradformr"s"LV4555y    }    A;;@huoo@>??? B {{ F F F 1 G G G<===q[[aA 2q!t*  B 2q!t*C ( ( (B 2q!t*C ( ( (B 2q'  B 2q' % % %B 2q' % % %BArllJJ b!!!eH1Q4!AaD'B1Q4!AaD'B1Q4!AaD'B HrBwR(BwR(* + +y}}QS!!y}}Q Q;;Xzzz)zz+,,F88F##D2%1adiikkRRICZ46)I5NE Q;;ZZZ[[B02FG02FG02FG02FG/1EF02EF HIIB Ahqj!^bAhqj!^B #IBHaV$$EHaTNNEr2r?? < <1a1Q3q51QqS519"5"#QqSU)"#QqSU)"#A#qs1u+"#A#a%"#A#qs1u+ "/00qS1Q3Y]AaCEq"Q$1*q.!A#QSTUQUVqS1Q3Y]1Q3q5BqD!A#r!taPQczTU~V# xx''rxA//0028RH3E3EEB5eRDD $(a$&09RF 88F##D tax1nad(D(DE EF tax1nad(D(DE EF FJJt    LLNN2qqq5 Yq2/44BE WQry}}-r22BE LLNN2qqq5 Yq2/44BE WQry}}-r22BE   BHHJJciikk(BC D DB  BHHJJciikk(BC D DB r6Mrcz|D]\}t|dstdd|vrd|d<d|vrd|d<]|D]z}|ddkrl|d}g}t|j|jD].\}}|d|vr|d|vr||/tj||f|d<{tj |j df} |D]y}|d|dz}|ddkr|j } |j } n*|ddkr|j } |j} ntd |d| || || |<z||| zz }|D]}|d|dz}| |||<tj|j dfd } |D]}|d|dz}d | |<t!t|j|jD]3\} \}}| |s| |r||kr d |j| <)d |j| <4||fS)a Apply boundary conditions. Parameters ---------- A : sparse matrix Fully assembled sparse matrix b : ndarray Fully assembled right-hand side mesh : Mesh Mesh object bc : list list of boundary conditions bc = [bc1, bc2, ..., bck] where bck = {'id': id, a list of vertices for boundary "k" 'g': g, g = g(x,y) is a function for the vertices on boundary "k" 'var': var the variable, given as a start in the dof list 'degree': degree degree of the variable, either 1 or 2 } Returns ------- A : sparse matrix Modified, assembled sparse matrix b : ndarray Modified, assembled right-hand side gz$each bc g must be callable functionsrrvarrridrqFTrr)rr#keysrr9r6appendrr/r r rrrrr enumerater+r,r.)rrrbccidxnewidxjedu0rrDflagkis rapplybcr%as6# ECDD D 16688 # #AhK  AeH// X;!  D'CFTZ44 % %2a5C<r6s?++   VVVVr' ' ' T~~~~BpppfBBBBBBBBJ555&]]]]@\\\~UUUp   EEEEEEr