_-Ph3&XdZddlZddlmZddlmZd dZ ddZ d Z dd Z d Z d Z dS)zLConstructs linear elasticity problems for first-order elements in 2D and 3D.N)sparsej@333333?ctt|dkrt|||||Std|)aLinear elasticity problem with Q1 finite elements on a regular rectangular grid. Parameters ---------- grid : tuple length 2 tuple of grid sizes, e.g. (10, 10) spacing : tuple length 2 tuple of grid spacings, e.g. (1.0, 0.1) E : float Young's modulus nu : float Poisson's ratio format : string Format of the returned sparse matrix (eg. 'csr', 'bsr', etc.) Returns ------- A : csr_matrix FE Q1 stiffness matrix B : array rigid body modes See Also -------- linear_elasticity_p1 Notes ----- - only 2d for now Examples -------- >>> from pyamg.gallery import linear_elasticity >>> A, B = linear_elasticity((4, 4)) References ---------- .. [1] J. Alberty, C. Carstensen, S. A. Funken, and R. KloseDOI "Matlab implementation of the finite element method in elasticity" Computing, Volume 69, Issue 3 (November 2002) Pages: 239 - 263 http://www.math.hu-berlin.de/~cc/ )spacingEnuformatzNo support for grid=)lenq12dNotImplementedError)gridrr r r s X/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/gallery/elasticity.pylinear_elasticityr sDZ 4yyA~~D'Q2fEEEE ;T;; < <<Tc t|\}}|dks|dkrtd|r |dz }|dz }tjd|dzd|dzf}tj|djdd|dz z |djdd|dz z f}|d\} } nt|\} } || | gz}||zd|zdd|zz zz } |dd|zzz } tjddg| dg| | gd| gg} t| | | }tj |dz|dzz|dz|dz}|ddddf}d|z |j dd d }| }|tj dddd d|zd zd|zd zd|zdzd|zd zgd z }|tj dddd d|zd zd|zd zd|zdzd|zd zgd jz }tj |||zdf}tj|}tj|}tj|}tj|||ff|j |j f}|d}~~~~~tjd|dzz|dzzd f}d|ddddf<d|ddddf<|dddf |ddddf<|dddf|ddddf<|rtj|dz|dzfd}d|ddddf<tj|}tj|dz |dz zddf}d|ddddf<d|ddddf<tj |dz |dz z}tjtjdgtj|f}tj|||fd|dzz|dzzd|dz z|dz zf}|j}|j|z|z}||z}|||fS)zSQ1 elements in 2 dimensions. See Also -------- linear_elasticity zinvalid grid shaper@Nrrr)rrshape)rr blocksizebooldtypeT)tuple ValueErrornpmgridhstackTreshapearray q12d_localarangerepeatsizecopytileravelr coo_matrixtocsrtobsrzeros concatenatecumsum bsr_matrixasformat)rrr r dirichlet_boundaryr XYptsDXDYlamemuverticesKnodesLLIdJVABmaskdataindicesindptrPPts rr r <s ;;DAq1uuA-... Q Q (1QqS5!AaC%< C )SVX%%b!,,q3w6VX%%b!,,q3w68 9 9CBBwB Bx r6a"fQV, -D a!b&j Bx!Q"a2r(QG<==H8T2&&A IqsQqSk " " * *1Q3! 4 4E ssCRCxB B$qv   & &r1a 0 0B  A"'1aAqsQw!a1q!A#'BF K KKB!Q1acAgqsQw!a1qA6 J J LLA AaC8A "B  A  A 1r1g,sx.BCCCIIKKA &!!A Aq"e !qs)QqS/1%&&AAaddAgJAaddAgJaaad)AaddAgJQQQTAaddAgJx1ac &111QrT1R4Zx~~x!A#!a+,,QQQ1W QQQ1W )QqS1Q3K((1# $ @AA  tWf5%&!Wac]AqsGQqSM$B D D D S C!GaK F ::f  q  rc"|d|zz}tjgdgdgdgdgdz }tjgdgdgdgdgd z }tjgd gd gd gd gdz }tjtj|d|dz |d|dz f}tjd}|jtj|dgd|gg|} | d|z| d|zz| d|jzz| d|zz|ddddddf<|jtj|dgd|gg|} | d|z| d|zz| d|jzz| d|zz|ddddddf<|jtjd|g|dgg|} | d|z| d|zz| d|jzz| d|zz|ddddddf<|ddddddfj|ddddddf<|tj|z}|S)a9Local stiffness matrix for two dimensional elasticity on a square element. Parameters ---------- lame : Float Lame's first parameter mu : Float shear modulus See Also -------- linear_elasticity Notes ----- Vertices should be listed in counter-clockwise order:: [3]----[2] | | | | [0]----[1] Degrees of freedom are enumerated as follows:: [x=6,y=7]----[x=4,y=5] | | | | [x=0,y=1]----[x=2,y=3] r)rrr)rRrrr)rrrrR)rrrRrg@)rrrr)rrrrg@)rrrrR)rrrRr)rrRrr)rRrrrrrr)rr)rr)rr)rrrN) r%r*slainvvstackr5r(dotdet) rBr@rAMR_11R_12R_22FrCr s rr+r+s> qt A 8^^^#^^#^^#^^% & &), ,D 8^^^#^^#^^#^^% & &), ,D 8^^^#^^#^^#^^% & &), ,D  8A;!4hqkHQK6OPQQRRA A 1a&1b'*++,,0033AdGdNQtWt^3 $$&T7T>*AaddADqDjM 2q'Aq6*++,,0033AdGdNQtWt^3 $$&T7T>*AaddADqDjM 1b'D!9-..//33A66AdGdNQtWt^3 $$&T7T>*AaddADqDjMaddADqDjMOAaddADqDjMOA Hrc||zd|zdd|zz zz }|dd|zzz }tj|}tj|}|jd}||jdz}|jd} |jd|dzkrtd|dkrt} n|dkrt } ntd||d|} | |z} | tj|z } | d||dzz||dzzd} | d||dzz||dzz} | dd} tj | ||dzz||dzzft } t| D]*}||d d f}||d d f}| |||| |<+| } | } | } tj| | | ff||f }|||f }|dkrTtj|df}d|dd ddf<d|dd ddf<|d d df |dd ddf<|d d df|dd ddf<ntj|d f}d|dd ddf<d|dd ddf<d|dd ddf<|d d df |dd ddf<|d d df|dd ddf<|d d df |dd ddf<|d d df|dd ddf<|d d df |dd ddf<|d d df|dd ddf<|||fS)aP1 elements in 2 or 3 dimensions. Parameters ---------- vertices : array_like array of vertices of a triangle or tets elements : array_like array of vertex indices for tri or tet elements E : float Young's modulus nu : float Poisson's ratio format : string 'csr', 'csc', 'coo', 'bsr' Returns ------- A : csr_matrix FE Q1 stiffness matrix Notes ----- - works in both 2d and in 3d Examples -------- >>> from pyamg.gallery import linear_elasticity_p1 >>> import numpy as np >>> E = np.array([[0, 1, 2],[1, 3, 2]]) >>> V = np.array([[0.0, 0.0],[1.0, 0.0],[0.0, 1.0],[1.0, 1.0]]) >>> A, B = linear_elasticity_p1(V, E) References ---------- .. [1] J. Alberty, C. Carstensen, S. A. Funken, and R. KloseDOI "Matlab implementation of the finite element method in elasticity" Computing, Volume 69, Issue 3 (November 2002) Pages: 239 - 263 http://www.math.hu-berlin.de/~cc/ rrrzdimension mismatchrz$only dimension 2 and 3 are supportedr)axisr!Nrrrr)r%asarrayrr$ p12d_local p13d_localr-r)r,swapaxesemptyfloatranger1rr2r3r4r5r9)rBelementsr r r r@rADDoFNElocal_KrowcolrLielement_indiceselement_verticesrIrJs rlinear_elasticity_p1rqsT r6a"fQrT* +D a!B$hBz(##Hz(##HqA HN1  C  B~aAE!!-...Avv a?@@@ //!   $ $R + +C1HC29Q<<C ++b!QqS' " " ) )!QqS' ) : :C ++b!QqS'1ac7 + +C ,,q!  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