UFL Specification and User Manual 0.3 - FEniCS Project

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UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg2.5.2 Taking components of tensorsBasic fixed indexing of a vector valued expression v or matrix valued expressionA:• v[0]: component access, representing the scalar value of the first componentof v• A[0,1]: component access, representing the scalar value of the firstrow, second column of ABasic indexing:• v[i]: component access, representing the scalar value of some componentof v• A[i,j]: component access, representing the scalar value of some componenti,j of AMore advanced indexing:• A[i,0]: component access, representing the scalar value of some componenti of the first column of A• A[i,:]: row access, representing some row i of A, i.e. rank(A[i,:]) ==1• A[:,j]: columnaccess,representingsomecolumnjofA,i.e. rank(A[:,j])== 1• C[...,0]: subtensor access, representing the subtensor of A with thelast axis fixed, e.g., A[...,0] == A[:,0]• C[j,...]: subtensor access, representing the subtensor of A with thelast axis fixed, e.g., A[j,...] == A[j,:]32
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg2.5.3 Making tensors from componentsIf you have expressions for scalar components of a tensor and wish to convertthem to a tensor, there are two ways to do it. If you have a single expressionwith free indices that should map to tensor axes, like mapping v k to v or A ijto A, the following examples show how this is done.vk = Identity(cell.d)[0,k]v = as_tensor(vk, (k,))Aij = v[i]*u[j]A = as_tensor(Aij, (i,j))Here v will represent unit vector i 0 , and A will represent the outer productof v and u.If you have multiple expressions without indices, you can build tensors fromthem just as easily, as illustrated here:v = as_vector([1.0, 2.0, 3.0])A = as_matrix([[u[0], 0], [0, u[1]]])B = as_matrix([[a+b for b in range(2)] for a in range(2)])Here v, A and B will represent the expressionsv = i 0 +2i 1 +3i 2 , (2.12)[ ]u0 0A = , (2.13)0 u 1[ ] 0 1B = . (2.14)1 2Note that the function as tensor generalizes from vectors to tensors of arbitraryrank, while the alternative functions as vector and as matrix workthe same way but are only for constructing vectors and matrices. They areincluded for readability and convenience only.33
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Page 3 and 4: ContentsAbout this manual 111 Intro
Page 5 and 6: 2.8.10 skew . . . . . . . . . . . .
Page 7 and 8: 4 Internal Representation Details 7
Page 9: B Installation 97B.1 Installing fro
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Bibliography[1] M. Alnæs and K.-A.
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