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deal.II version 9.7.0
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deal.II version 9.7.0
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#include <deal.II/base/config.h>#include <deal.II/base/exceptions.h>#include <deal.II/base/kokkos.h>#include <deal.II/base/numbers.h>#include <deal.II/base/table_indices.h>#include <deal.II/base/template_constraints.h>#include <deal.II/base/tensor_accessors.h>#include <Kokkos_Array.hpp>#include <adolc/adouble.h>#include <cmath>#include <complex>#include <ostream>#include <type_traits>Go to the source code of this file.
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| class | Tensor< 0, dim, Number > |
| class | Tensor< rank_, dim, Number > |
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Multiplication of a tensor of general rank with a scalar number from the right.
Only multiplication with a scalar number type (i.e., a floating point number, a complex floating point number, etc.) is allowed, see the documentation of EnableIfScalar for details.
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Multiplication of a tensor of general rank with a scalar number from the left.
Only multiplication with a scalar number type (i.e., a floating point number, a complex floating point number, etc.) is allowed, see the documentation of EnableIfScalar for details.
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Division of a tensor of general rank with a scalar number. See the discussion on operator*() above for more information about template arguments and the return type.
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Entrywise multiplication of two tensor objects of general rank.
This multiplication is also called "Hadamard-product" (c.f. https://en.wikipedia.org/wiki/Hadamard_product_(matrices)), and generates a new tensor of size <rank, dim>:
\[ \text{result}_{i, j} = \text{left}_{i, j}\circ \text{right}_{i, j} \]
| rank | The rank of both tensors. |
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The dot product (single contraction) for tensors. This function return a tensor of rank \((\text{rank}_1 + \text{rank}_2 - 2)\) that is the contraction of the last index of a tensor src1 of rank rank_1 with the first index of a tensor src2 of rank rank_2:
\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \sum_{k} \text{left}_{i_1,\ldots,i_{r1}, k} \text{right}_{k, j_1,\ldots,j_{r2}} \]
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Generic contraction of a pair of indices of two tensors of arbitrary rank: Return a tensor of rank \((\text{rank}_1 + \text{rank}_2 - 2)\) that is the contraction of index index_1 of a tensor src1 of rank rank_1 with the index index_2 of a tensor src2 of rank rank_2:
\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \sum_{k} \text{left}_{i_1,\ldots,k,\ldots,i_{r1}} \text{right}_{j_1,\ldots,k,\ldots,j_{r2}} \]
If for example the first index (index_1==0) of a tensor t1 shall be contracted with the third index (index_2==2) of a tensor t2, this function should be invoked as
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Generic contraction of two pairs of indices of two tensors of arbitrary rank: Return a tensor of rank \((\text{rank}_1 + \text{rank}_2 - 4)\) that is the contraction of index index_1 with index index_2, and index index_3 with index index_4 of a tensor src1 of rank rank_1 and a tensor src2 of rank rank_2:
\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \sum_{k, l} \text{left}_{i_1,\ldots,k,\ldots,l,\ldots,i_{r1}} \text{right}_{j_1,\ldots,k,\ldots,l\ldots,j_{r2}} \]
If for example the first index (index_1==0) shall be contracted with the third index (index_2==2), and the second index (index_3==1) with the first index (index_4==0) of a tensor t2, this function should be invoked as
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The scalar product, or (generalized) Frobenius inner product of two tensors of equal rank: Return a scalar number that is the result of a full contraction of a tensor left and right:
\[ \sum_{i_1,\ldots,i_r} \text{left}_{i_1,\ldots,i_r} \text{right}_{i_1,\ldots,i_r} \]
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Full contraction of three tensors: Return a scalar number that is the result of a full contraction of a tensor left of rank rank_1, a tensor middle of rank \((\text{rank}_1+\text{rank}_2)\) and a tensor right of rank rank_2:
\[ \sum_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} \text{left}_{i_1,\ldots,i_{r1}} \text{middle}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} \text{right}_{j_1,\ldots,j_{r2}} \]
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Return the cross product in 2d. This is just a rotation by 90 degrees clockwise to compute the outer normal from a tangential vector. This function is defined for all space dimensions to allow for dimension independent programming (e.g. within switches over the space dimension), but may only be called if the actual dimension of the arguments is two (e.g. from the dim==2 case in the switch).
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Return the cross product of 2 vectors in 3d. This function is defined for all space dimensions to allow for dimension independent programming (e.g. within switches over the space dimension), but may only be called if the actual dimension of the arguments is three (e.g. from the dim==3 case in the switch).
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Return the cofactor of the given tensor of rank 2. The cofactor of a tensor \(\mathbf A\) is defined as
\[ \textrm{cof}\mathbf A \dealcoloneq \textrm{det}\mathbf A \; \mathbf{A}^{-T} = \left[ \textrm{adj}\mathbf A \right]^{T} \; . \]
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Return the nearest orthogonal matrix \(\hat {\mathbf A}=\mathbf U \mathbf{V}^T\) by combining the products of the singular value decomposition (SVD) \({\mathbf A}=\mathbf U \mathbf S \mathbf V^T\) for a given input \({\mathbf A}\), effectively replacing \(\mathbf S\) with the identity matrix.
This is a (nonlinear) projection operation since when applied twice, we have \(\hat{\hat{\mathbf A}}=\hat{\mathbf A}\) as is easy to see. (That is because the SVD of \(\hat {\mathbf A}\) is simply \(\mathbf U \mathbf I \mathbf{V}^T\).) Furthermore, \(\hat {\mathbf A}\) is really an orthogonal matrix because orthogonal matrices have to satisfy \({\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}\), which here implies that
\begin{align*} {\hat {\mathbf A}}^T \hat {\mathbf A} &= \left(\mathbf U \mathbf{V}^T\right)^T\left(\mathbf U \mathbf{V}^T\right) \\ &= \mathbf V \mathbf{U}^T \mathbf U \mathbf{V}^T \\ &= \mathbf V \left(\mathbf{U}^T \mathbf U\right) \mathbf{V}^T \\ &= \mathbf V \mathbf I \mathbf{V}^T \\ &= \mathbf V \mathbf{V}^T \\ &= \mathbf I \end{align*}
due to the fact that the \(\mathbf U\) and \(\mathbf V\) factors that come out of the SVD are themselves orthogonal matrices.
| A | The tensor for which to find the closest orthogonal tensor. |
| Number | The type used to store the entries of the tensor. Must be either float or double. |
A must not be singular. This is because, conceptually, the problem to be solved here is trying to find a matrix \(\hat{\mathbf A}\) that minimizes some kind of distance from \(\mathbf A\) while satisfying the quadratic constraint \({\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}\). This is not so dissimilar to the kind of problem where one wants to find a vector \(\hat{\mathbf x}\in{\mathbb R}^n\) that minimizes the quadratic objective function \(\|\hat {\mathbf x} - \mathbf x\|^2\) for a given \(\mathbf x\) subject to the constraint \(\|\mathbf x\|^2=1\) – in other words, we are seeking the point \(\hat{\mathbf x}\) on the unit sphere that is closest to \(\mathbf x\). This problem has a solution for all \(\mathbf x\) except if \(\mathbf x=0\). The corresponding condition for the problem we are considering here is that \(\mathbf A\) must not have a zero eigenvalue. 
1.14.0