UFL Specification and User Manual 0.3 - FEniCS Project

More documents

Recommendations

Info

UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg• ln(f)• cos(f)• sin(f)These functions do not accept non-scalar operands or operands with freeindices or BasisFunction dependencies.2.8 Tensor Algebra Operators2.8.1 transposeThe transpose of a matrix A can be written asAT = transpose(A)AT = A.TAT = as_matrix(A[i,j], (j,i))The definition of the transpose isAT[i,j] ↔ (A ⊤ ) ij = A ji . (2.15)For transposing higher order tensor expressions, index notation can be used:AT = as_tensor(A[i,j,k,l], (l,k,j,i))2.8.2 trThe trace of a matrix A is the sum of the diagonal entries. This can bewritten as36
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggt = tr(A)t = A[i,i]The definition of the trace is∑n−1tr(A) ↔ trA = A ii = A ii . (2.16)i=02.8.3 dotThe dot product of two tensors a and b can be written# General tensorsf = dot(a, b)# Vectors a and bf = a[i]*b[i]# Matrices a and bf = as_matrix(a[i,k]*b[k,j], (i,j))The definition of the dot product of unit vectors is 3i i ·i j = δ ij (2.17)where δ ij is the Kronecker delta as explained earlier. The dot product ofhigher order tensors follow from this, as illustrated with the following examples.An example with two vectorsv·u = (v i i i )·(u j i j ) = v i u j (i i ·i j ) = v i u j δ ij = v i u i (2.18)3 Assuming an orthonormal basis for a Euclidean space.37
Page 1 and 2: UFL Specification and User Manual 0
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
Page 86 and 87:
UFL Specification and User Manual 0
Page 88 and 89:
Page 90 and 91:
Page 93:
Bibliography[1] M. Alnæs and K.-A.
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118:
show all

UFL Specification and User Manual 0.3 - FEniCS Project

Create successful ePaper yourself

Delete template?

Save as template?