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Reference documentation for deal.II version 9.3.3
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Reference documentation for deal.II version 9.3.3
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#include <deal.II/base/symmetric_tensor.h>
Public Member Functions | |
| constexpr | SymmetricTensor ()=default |
| template<typename OtherNumber > | |
| SymmetricTensor (const Tensor< 2, dim, OtherNumber > &t) | |
| constexpr | SymmetricTensor (const Number(&array)[n_independent_components]) |
| template<typename OtherNumber > | |
| constexpr | SymmetricTensor (const SymmetricTensor< rank_, dim, OtherNumber > &initializer) |
| Number * | begin_raw () |
| const Number * | begin_raw () const |
| Number * | end_raw () |
| const Number * | end_raw () const |
| template<typename OtherNumber > | |
| constexpr SymmetricTensor & | operator= (const SymmetricTensor< rank_, dim, OtherNumber > &rhs) |
| constexpr SymmetricTensor & | operator= (const Number &d) |
| constexpr | operator Tensor< rank_, dim, Number > () const |
| constexpr bool | operator== (const SymmetricTensor &) const |
| constexpr bool | operator!= (const SymmetricTensor &) const |
| template<typename OtherNumber > | |
| constexpr SymmetricTensor & | operator+= (const SymmetricTensor< rank_, dim, OtherNumber > &) |
| template<typename OtherNumber > | |
| constexpr SymmetricTensor & | operator-= (const SymmetricTensor< rank_, dim, OtherNumber > &) |
| template<typename OtherNumber > | |
| constexpr SymmetricTensor & | operator*= (const OtherNumber &factor) |
| template<typename OtherNumber > | |
| constexpr SymmetricTensor & | operator/= (const OtherNumber &factor) |
| constexpr SymmetricTensor | operator- () const |
| template<typename OtherNumber > | |
| constexpr internal::SymmetricTensorAccessors::double_contraction_result< rank_, 2, dim, Number, OtherNumber >::type | operator* (const SymmetricTensor< 2, dim, OtherNumber > &s) const |
| template<typename OtherNumber > | |
| constexpr internal::SymmetricTensorAccessors::double_contraction_result< rank_, 4, dim, Number, OtherNumber >::type | operator* (const SymmetricTensor< 4, dim, OtherNumber > &s) const |
| constexpr Number & | operator() (const TableIndices< rank_ > &indices) |
| constexpr const Number & | operator() (const TableIndices< rank_ > &indices) const |
| constexpr internal::SymmetricTensorAccessors::Accessor< rank_, dim, true, rank_ - 1, Number > | operator[] (const unsigned int row) const |
| constexpr internal::SymmetricTensorAccessors::Accessor< rank_, dim, false, rank_ - 1, Number > | operator[] (const unsigned int row) |
| constexpr const Number & | operator[] (const TableIndices< rank_ > &indices) const |
| constexpr Number & | operator[] (const TableIndices< rank_ > &indices) |
| constexpr const Number & | access_raw_entry (const unsigned int unrolled_index) const |
| constexpr Number & | access_raw_entry (const unsigned int unrolled_index) |
| constexpr numbers::NumberTraits< Number >::real_type | norm () const |
| constexpr void | clear () |
| template<class Archive > | |
| void | serialize (Archive &ar, const unsigned int version) |
Static Public Member Functions | |
| static constexpr unsigned int | component_to_unrolled_index (const TableIndices< rank_ > &indices) |
| static constexpr TableIndices< rank_ > | unrolled_to_component_indices (const unsigned int i) |
| static constexpr std::size_t | memory_consumption () |
Static Public Attributes | |
| static const unsigned int | dimension = dim |
| static const unsigned int | rank = rank_ |
| static constexpr unsigned int | n_independent_components |
Private Types | |
| using | base_tensor_descriptor = internal::SymmetricTensorAccessors::StorageType< rank_, dim, Number > |
| using | base_tensor_type = typename base_tensor_descriptor::base_tensor_type |
Private Attributes | |
| base_tensor_type | data |
Friends | |
| template<int , int , typename > | |
| class | SymmetricTensor |
| struct | internal::SymmetricTensorImplementation::Inverse< 2, dim, Number > |
| struct | internal::SymmetricTensorImplementation::Inverse< 4, dim, Number > |
| template<int dim2, typename Number2 > | |
| constexpr Number2 | trace (const SymmetricTensor< 2, dim2, Number2 > &d) |
| template<int dim2, typename Number2 > | |
| constexpr Number2 | determinant (const SymmetricTensor< 2, dim2, Number2 > &t) |
| template<int dim2, typename Number2 > | |
| constexpr SymmetricTensor< 2, dim2, Number2 > | deviator (const SymmetricTensor< 2, dim2, Number2 > &t) |
| template<int dim2, typename Number2 > | |
| constexpr SymmetricTensor< 2, dim2, Number2 > | unit_symmetric_tensor () |
| template<int dim2, typename Number2 > | |
| constexpr SymmetricTensor< 4, dim2, Number2 > | deviator_tensor () |
| template<int dim2, typename Number2 > | |
| constexpr SymmetricTensor< 4, dim2, Number2 > | identity_tensor () |
Related Functions | |
(Note that these are not member functions.) | |
| template<int rank, int dim, typename Number > | |
| SymmetricTensor< rank, dim, Number > | sum (const SymmetricTensor< rank, dim, Number > &local, const MPI_Comm &mpi_communicator) |
| template<int rank_, int dim, typename Number , typename OtherNumber > | |
| constexpr SymmetricTensor< rank_, dim, typename ProductType< Number, OtherNumber >::type > | operator+ (const SymmetricTensor< rank_, dim, Number > &left, const SymmetricTensor< rank_, dim, OtherNumber > &right) |
| template<int rank_, int dim, typename Number , typename OtherNumber > | |
| constexpr SymmetricTensor< rank_, dim, typename ProductType< Number, OtherNumber >::type > | operator- (const SymmetricTensor< rank_, dim, Number > &left, const SymmetricTensor< rank_, dim, OtherNumber > &right) |
| template<int rank_, int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< rank_, dim, typename ProductType< Number, OtherNumber >::type > | operator+ (const SymmetricTensor< rank_, dim, Number > &left, const Tensor< rank_, dim, OtherNumber > &right) |
| template<int rank_, int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< rank_, dim, typename ProductType< Number, OtherNumber >::type > | operator+ (const Tensor< rank_, dim, Number > &left, const SymmetricTensor< rank_, dim, OtherNumber > &right) |
| template<int rank_, int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< rank_, dim, typename ProductType< Number, OtherNumber >::type > | operator- (const SymmetricTensor< rank_, dim, Number > &left, const Tensor< rank_, dim, OtherNumber > &right) |
| template<int rank_, int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< rank_, dim, typename ProductType< Number, OtherNumber >::type > | operator- (const Tensor< rank_, dim, Number > &left, const SymmetricTensor< rank_, dim, OtherNumber > &right) |
| template<int dim, typename Number > | |
| constexpr Number | determinant (const SymmetricTensor< 2, dim, Number > &t) |
| template<int dim, typename Number > | |
| constexpr Number | third_invariant (const SymmetricTensor< 2, dim, Number > &t) |
| template<int dim, typename Number > | |
| constexpr Number | trace (const SymmetricTensor< 2, dim, Number > &d) |
| template<int dim, typename Number > | |
| constexpr Number | first_invariant (const SymmetricTensor< 2, dim, Number > &t) |
| template<typename Number > | |
| constexpr Number | second_invariant (const SymmetricTensor< 2, 1, Number > &) |
| template<typename Number > | |
| constexpr Number | second_invariant (const SymmetricTensor< 2, 2, Number > &t) |
| template<typename Number > | |
| constexpr Number | second_invariant (const SymmetricTensor< 2, 3, Number > &t) |
| template<typename Number > | |
| std::array< Number, 1 > | eigenvalues (const SymmetricTensor< 2, 1, Number > &T) |
| template<typename Number > | |
| std::array< Number, 2 > | eigenvalues (const SymmetricTensor< 2, 2, Number > &T) |
| template<typename Number > | |
| std::array< Number, 3 > | eigenvalues (const SymmetricTensor< 2, 3, Number > &T) |
| template<int dim, typename Number > | |
| std::array< std::pair< Number, Tensor< 1, dim, Number > >, std::integral_constant< int, dim >::value > | eigenvectors (const SymmetricTensor< 2, dim, Number > &T, const SymmetricTensorEigenvectorMethod method=SymmetricTensorEigenvectorMethod::ql_implicit_shifts) |
| template<int rank_, int dim, typename Number > | |
| constexpr SymmetricTensor< rank_, dim, Number > | transpose (const SymmetricTensor< rank_, dim, Number > &t) |
| template<int dim, typename Number > | |
| constexpr SymmetricTensor< 2, dim, Number > | deviator (const SymmetricTensor< 2, dim, Number > &t) |
| template<int dim, typename Number > | |
| constexpr SymmetricTensor< 2, dim, Number > | invert (const SymmetricTensor< 2, dim, Number > &t) |
| template<int dim, typename Number > | |
| constexpr SymmetricTensor< 4, dim, Number > | invert (const SymmetricTensor< 4, dim, Number > &t) |
| template<int dim, typename Number > | |
| constexpr SymmetricTensor< 4, dim, Number > | outer_product (const SymmetricTensor< 2, dim, Number > &t1, const SymmetricTensor< 2, dim, Number > &t2) |
| template<int dim, typename Number > | |
| constexpr SymmetricTensor< 2, dim, Number > | symmetrize (const Tensor< 2, dim, Number > &t) |
| template<int rank_, int dim, typename Number > | |
| constexpr SymmetricTensor< rank_, dim, Number > | operator* (const SymmetricTensor< rank_, dim, Number > &t, const Number &factor) |
| template<int rank_, int dim, typename Number > | |
| constexpr SymmetricTensor< rank_, dim, Number > | operator* (const Number &factor, const SymmetricTensor< rank_, dim, Number > &t) |
| template<int rank_, int dim, typename Number , typename OtherNumber > | |
| constexpr SymmetricTensor< rank_, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > | operator/ (const SymmetricTensor< rank_, dim, Number > &t, const OtherNumber &factor) |
| template<int rank_, int dim> | |
| constexpr SymmetricTensor< rank_, dim > | operator* (const SymmetricTensor< rank_, dim > &t, const double factor) |
| template<int rank_, int dim> | |
| constexpr SymmetricTensor< rank_, dim > | operator* (const double factor, const SymmetricTensor< rank_, dim > &t) |
| template<int rank_, int dim> | |
| constexpr SymmetricTensor< rank_, dim > | operator/ (const SymmetricTensor< rank_, dim > &t, const double factor) |
| template<int dim, typename Number , typename OtherNumber > | |
| constexpr ProductType< Number, OtherNumber >::type | scalar_product (const SymmetricTensor< 2, dim, Number > &t1, const SymmetricTensor< 2, dim, OtherNumber > &t2) |
| template<int dim, typename Number , typename OtherNumber > | |
| constexpr ProductType< Number, OtherNumber >::type | scalar_product (const SymmetricTensor< 2, dim, Number > &t1, const Tensor< 2, dim, OtherNumber > &t2) |
| template<int dim, typename Number , typename OtherNumber > | |
| constexpr ProductType< Number, OtherNumber >::type | scalar_product (const Tensor< 2, dim, Number > &t1, const SymmetricTensor< 2, dim, OtherNumber > &t2) |
| template<typename Number , typename OtherNumber > | |
| constexpr void | double_contract (SymmetricTensor< 2, 1, typename ProductType< Number, OtherNumber >::type > &tmp, const SymmetricTensor< 4, 1, Number > &t, const SymmetricTensor< 2, 1, OtherNumber > &s) |
| template<typename Number , typename OtherNumber > | |
| constexpr void | double_contract (SymmetricTensor< 2, 1, typename ProductType< Number, OtherNumber >::type > &tmp, const SymmetricTensor< 2, 1, Number > &s, const SymmetricTensor< 4, 1, OtherNumber > &t) |
| template<typename Number , typename OtherNumber > | |
| constexpr void | double_contract (SymmetricTensor< 2, 2, typename ProductType< Number, OtherNumber >::type > &tmp, const SymmetricTensor< 4, 2, Number > &t, const SymmetricTensor< 2, 2, OtherNumber > &s) |
| template<typename Number , typename OtherNumber > | |
| constexpr void | double_contract (SymmetricTensor< 2, 2, typename ProductType< Number, OtherNumber >::type > &tmp, const SymmetricTensor< 2, 2, Number > &s, const SymmetricTensor< 4, 2, OtherNumber > &t) |
| template<typename Number , typename OtherNumber > | |
| constexpr void | double_contract (SymmetricTensor< 2, 3, typename ProductType< Number, OtherNumber >::type > &tmp, const SymmetricTensor< 4, 3, Number > &t, const SymmetricTensor< 2, 3, OtherNumber > &s) |
| template<typename Number , typename OtherNumber > | |
| constexpr void | double_contract (SymmetricTensor< 2, 3, typename ProductType< Number, OtherNumber >::type > &tmp, const SymmetricTensor< 2, 3, Number > &s, const SymmetricTensor< 4, 3, OtherNumber > &t) |
| template<int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< 1, dim, typename ProductType< Number, OtherNumber >::type > | operator* (const SymmetricTensor< 2, dim, Number > &src1, const Tensor< 1, dim, OtherNumber > &src2) |
| template<int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< 1, dim, typename ProductType< Number, OtherNumber >::type > | operator* (const Tensor< 1, dim, Number > &src1, const SymmetricTensor< 2, dim, OtherNumber > &src2) |
| template<int rank_1, int rank_2, int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< rank_1+rank_2-2, dim, typenameProductType< Number, OtherNumber >::type >::tensor_type | operator* (const Tensor< rank_1, dim, Number > &src1, const SymmetricTensor< rank_2, dim, OtherNumber > &src2) |
| template<int rank_1, int rank_2, int dim, typename Number , typename OtherNumber > | |
| constexpr Tensor< rank_1+rank_2-2, dim, typenameProductType< Number, OtherNumber >::type >::tensor_type | operator* (const SymmetricTensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2) |
| template<int dim, typename Number > | |
| std::ostream & | operator<< (std::ostream &out, const SymmetricTensor< 2, dim, Number > &t) |
| template<int dim, typename Number > | |
| std::ostream & | operator<< (std::ostream &out, const SymmetricTensor< 4, dim, Number > &t) |
Provide a class that stores symmetric tensors of rank 2,4,... efficiently, i.e. only store those off-diagonal elements of the full tensor that are not redundant. For example, for symmetric \(2\times 2\) tensors, this would be the elements 11, 22, and 12, while the element 21 is equal to the 12 element. Within this documentation, second order symmetric tensors are denoted as bold-faced upper-case Latin letters such as \(\mathbf A, \mathbf B, \dots\) or bold-faced Greek letters such as \(\boldsymbol{\varepsilon}\), \(\boldsymbol{\sigma}\). The Cartesian coordinates of a second-order tensor such as \(\mathbf A\) are represented as \(A_{ij}\) where \(i,j\) are indices ranging from 0 to dim-1.
Using this class for symmetric tensors of rank 2 has advantages over matrices in many cases since the dimension is known to the compiler as well as the location of the data. It is therefore possible to produce far more efficient code than for matrices with runtime-dependent dimension. It is also more efficient than using the more general Tensor class, since fewer elements are stored, and the class automatically makes sure that the tensor represents a symmetric object.
For tensors of higher rank, the savings in storage are even higher. For example for the \(3 \times 3 \times 3 \times 3\) tensors of rank 4, only 36 instead of the full 81 entries have to be stored. These rank 4 tensors are denoted by blackboard-style upper-case Latin letters such as \(\mathbb A\) with components \(\mathcal{A}_{ijkl}\).
While the definition of a symmetric rank-2 tensor is obvious, tensors of rank 4 are considered symmetric if they are operators mapping symmetric rank-2 tensors onto symmetric rank-2 tensors. This so-called minor symmetry of the rank 4 tensor requires that for every set of four indices \(i, j, k, l\), the identity \(\mathcal{C}_{ijkl} = \mathcal{C}_{jikl} = \mathcal{C}_{ijlk}\) holds. However, it does not imply the relation \(\mathcal{C}_{ijkl} = \mathcal{C}_{klij}\). Consequently, symmetric tensors of rank 4 as understood here are only tensors that map symmetric tensors onto symmetric tensors, but they do not necessarily induce a symmetric scalar product \(\mathbf A : \mathbb C : \mathbf B = \mathbf B : \mathbb C : \mathbf A\) or even a positive (semi-)definite form \(\mathbf A : \mathbb C : \mathbf A\), where \(\mathbf A, \mathbf B\) are symmetric rank-2 tensors and the colon indicates the common double-index contraction that acts as a scalar product for symmetric tensors.
Symmetric tensors are most often used in structural and fluid mechanics, where strains and stresses are usually symmetric tensors, and the stress-strain relationship is given by a symmetric rank-4 tensor.
SymmetricTensor<2,dim>, SymmetricTensor<4,dim>, etc, but SymmetricTensor<1,dim> and SymmetricTensor<3,dim> do not exist and their use will most likely lead to compiler errors.The elements of a tensor \(\mathbb C\) can be accessed using the bracket operator, i.e. for a tensor of rank 4, C[0][1][0][1] accesses the element \(\mathcal{C}_{0101}\). This access can be used for both reading and writing (if the tensor is non-constant at least). You may also perform other operations on it, although that may lead to confusing situations because several elements of the tensor are stored at the same location. For example, for a rank-2 tensor that is assumed to be zero at the beginning, writing A[0][1]+=1; A[1][0]+=1; will lead to the same element being increased by one twice, because even though the accesses use different indices, the elements that are accessed are symmetric and therefore stored at the same location. It may therefore be useful in application programs to restrict operations on individual elements to simple reads or writes.
Definition at line 597 of file symmetric_tensor.h.
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A structure that describes properties of the base tensor.
Definition at line 936 of file symmetric_tensor.h.
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Data storage type for a symmetric tensor.
Definition at line 942 of file symmetric_tensor.h.
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Default constructor. Creates a tensor with all entries equal to zero.
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Constructor. Generate a symmetric tensor from a general one. Assumes that t is already symmetric, and in debug mode this is in fact checked. Note that no provision is made to assure that the tensor is symmetric only up to round-off error: if the incoming tensor is not exactly symmetric, then an exception is thrown. If you know that incoming tensor is symmetric only up to round-off, then you may want to call the symmetrize() function first. If you aren't sure, it is good practice to check before calling symmetrize().
Because we check for symmetry via a non-constexpr function call, you will have to use the symmetrize() function in constexpr contexts instead.
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A constructor that creates a symmetric tensor from an array holding its independent elements. Using this constructor assumes that the caller knows the order in which elements are stored in symmetric tensors; its use is therefore discouraged, but if you think you want to use it anyway you can query the order of elements using the unrolled_index() function.
This constructor is currently only implemented for symmetric tensors of rank 2.
The size of the array passed is equal to SymmetricTensor<rank_,dim>::n_independent_components; the reason for using the object from the internal namespace is to work around bugs in some older compilers.
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Copy constructor from tensors with different underlying scalar type. This obviously requires that the OtherNumber type is convertible to Number.
| Number * SymmetricTensor< rank_, dim, Number >::begin_raw | ( | ) |
Return a pointer to the first element of the underlying storage.
| const Number * SymmetricTensor< rank_, dim, Number >::begin_raw | ( | ) | const |
Return a const pointer to the first element of the underlying storage.
| Number * SymmetricTensor< rank_, dim, Number >::end_raw | ( | ) |
Return a pointer to the element past the end of the underlying storage.
| const Number * SymmetricTensor< rank_, dim, Number >::end_raw | ( | ) | const |
Return a const pointer to the element past the end of the underlying storage.
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Assignment operator from symmetric tensors with different underlying scalar type. This obviously requires that the OtherNumber type is convertible to Number.
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This operator assigns a scalar to a tensor. To avoid confusion with what exactly it means to assign a scalar value to a tensor, zero is the only value allowed for d, allowing the intuitive notation \(\mathbf A = 0\) to reset all elements of the tensor to zero.
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Convert the present symmetric tensor into a full tensor with the same elements, but using the different storage scheme of full tensors.
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Test for equality of two tensors.
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Test for inequality of two tensors.
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Add another tensor.
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Subtract another tensor.
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Scale the tensor by factor, i.e. multiply all components by factor.
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Scale the tensor by 1/factor.
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Unary minus operator. Negate all entries of a tensor.
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Double contraction product between the present symmetric tensor and a tensor of rank 2. For example, if the present object is the symmetric rank-2 tensor \(\mathbf{A}\) and it is multiplied by another symmetric rank-2 tensor \(\mathbf{B}\), then the result is the scalar-product double contraction \(\mathbf A : \mathbf B = \sum_{i,j} A_{ij} B_{ij}\). In this case, the return value evaluates to a single scalar. While it is possible to define other scalar products (and associated induced norms), this one seems to be the most appropriate one.
If the present object is a rank-4 tensor such as \(\mathbb A\), then the result is a rank-2 tensor \(\mathbf C = \mathbb A : \mathbf B\), i.e., the operation contracts over the last two indices of the present object and the indices of the argument, and the result is a tensor of rank 2 ( \(C_{ij} = \sum_{k,l} \mathcal{A}_{ijkl} B_{kl}\)).
Note that the multiplication operator for symmetric tensors is defined to be a double contraction over two indices, while it is defined as a single contraction over only one index for regular Tensor objects. For symmetric tensors it therefore acts in a way that is commonly denoted by a "colon multiplication" in the mathematical literature.
There are global functions double_contract that do the same work as this operator, but rather than returning the result as a return value, they write it into the first argument to the function.
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Contraction over two indices of the present object with the rank-4 symmetric tensor given as argument.
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Return a read-write reference to the indicated element.
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Return a const reference to the value referred to by the argument.
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Access the elements of a row of this symmetric tensor. This function is called for constant tensors.
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Access the elements of a row of this symmetric tensor. This function is called for non-constant tensors.
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Return a const reference to the value referred to by the argument.
Exactly the same as operator().
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Return a read-write reference to the indicated element.
Exactly the same as operator().
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Access to an element according to unrolled index. The function s.access_raw_entry(unrolled_index) does the same as s[s.unrolled_to_component_indices(unrolled_index)], but more efficiently.
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Access to an element according to unrolled index. The function s.access_raw_entry(unrolled_index) does the same as s[s.unrolled_to_component_indices(unrolled_index)], but more efficiently.
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Return the Frobenius-norm of a tensor, i.e. the square root of the sum of squares of all entries. This norm is induced by the scalar product defined above for two symmetric tensors. Note that it includes all entries of the tensor, counting symmetry, not only the unique ones (for example, for rank-2 tensors, this norm includes adding up the squares of upper right as well as lower left entries, not just one of them, although they are equal for symmetric tensors).
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Tensor objects can be unrolled by simply pasting all elements into one long vector, but for this an order of elements has to be defined. For symmetric tensors, this function returns which index within the range [0,n_independent_components) the given entry in a symmetric tensor has.
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The opposite of the previous function: given an index \(i\) in the unrolled form of the tensor, return what set of indices \((k,l)\) (for rank-2 tensors) or \((k,l,m,n)\) (for rank-4 tensors) corresponds to it.
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Reset all values to zero.
Note that this is partly inconsistent with the semantics of the clear() member functions of the standard library containers and of several other classes within deal.II, which not only reset the values of stored elements to zero, but release all memory and return the object into a virginial state. However, since the size of objects of the present type is determined by its template parameters, resizing is not an option, and indeed the state where all elements have a zero value is the state right after construction of such an object.
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Determine an estimate for the memory consumption (in bytes) of this object.
| void SymmetricTensor< rank_, dim, Number >::serialize | ( | Archive & | ar, |
| const unsigned int | version | ||
| ) |
Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.
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Definition at line 951 of file symmetric_tensor.h.
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Definition at line 976 of file symmetric_tensor.h.
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Definition at line 976 of file symmetric_tensor.h.
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Return a unit symmetric tensor of rank 2, i.e., the \(\text{dim}\times\text{dim}\) identity matrix \(\mathbf I\).
unit_symmetric_tensor<dim>() is the specialization of the function unit_symmetric_tensor<dim,Number>() which uses double as the data type for the elements.
Definition at line 3211 of file symmetric_tensor.h.
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Return the tensor of rank 4 that, when multiplied by a symmetric rank 2 tensor \(\mathbf T\) returns the deviator \(\text{dev}\ \mathbf T\). It is the operator representation of the linear deviator operator \(\mathbb P\), also known as the volumetric projection tensor, calculated as:
\begin{align*} \mathbb{P} &=\mathbb{I} -\frac{1}{\text{dim}} \mathbf I \otimes \mathbf I \\ \mathcal{P}_{ijkl} &= \frac 12 \left(\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} \right) - \frac{1}{\text{dim}} \delta_{ij} \delta_{kl} \end{align*}
For every tensor T, there holds the identity deviator<dim,Number>(T) == deviator_tensor<dim,Number>() * T, up to numerical round-off.
\[ \text{dev}\mathbf T = \mathbb P : \mathbf T \]
\[ \frac{\partial \text{dev}\mathbf{T}}{\partial \mathbf T} = \mathbb P. \]
This version of the deviator_tensor<dim>() function is a specialization of deviator_tensor<dim,Number>() that uses double as the data type for the elements of the tensor.
Definition at line 3283 of file symmetric_tensor.h.
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Return the fourth-order symmetric identity tensor \(\mathbb I\) which maps symmetric second-order tensors, such as \(\mathbf A\), to themselves.
\[ \mathbb I : \mathbf A = \mathbf A \]
Note that this tensor, even though it is the identity, has a somewhat funny form, and in particular does not only consist of zeros and ones. For example, for dim=2, the identity tensor has all zero entries except for
\[ \mathcal{I}_{0000} = \mathcal{I}_{1111} = 1 \]
\[ \mathcal{I}_{0101} = \mathcal{I}_{0110} = \mathcal{I}_{1001} = \mathcal{I}_{1010} = \frac 12. \]
In index notation, we can write the general form
\[ \mathcal{I}_{ijkl} = \frac 12 \left( \delta_{ik} \delta_{jl} + \delta_{il} \delta_{jl} \right). \]
To see why this factor of \(1 / 2\) is necessary, consider computing \(\mathbf A= \mathbb I : \mathbf B\). For the element \(A_{01}\) we have \(A_{01} = \mathcal{I}_{0100} B_{00} + \mathcal{I}_{0111} B_{11} + \mathcal{I}_{0101} B_{01} + \mathcal{I}_{0110} B_{10}\). On the other hand, we need to have \(A_{01} = B_{01}\), and symmetry implies \(B_{01}=B_{10}\), leading to \(A_{01} = (\mathcal{I}_{0101} + \mathcal{I}_{0110}) B_{01}\), or, again by symmetry, \(\mathcal{I}_{0101} = \mathcal{I}_{0110} = \frac 12\). Similar considerations hold for the three-dimensional case.
This issue is also explained in the introduction to step-44.
This version of the identity_tensor<dim>() function is the specialization of identity_tensor<dim,Number>() which uses double as the data type for the elements of the tensor.
Definition at line 3364 of file symmetric_tensor.h.
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Perform an MPI sum of the entries of a symmetric tensor.
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Addition of two symmetric tensors of equal rank. The result is another SymmetricTensor that has a number type that is compatible with the operation.
If possible (e.g. when Number and OtherNumber are of the same type, or if the result of Number() + OtherNumber() is another Number), you should use operator+= instead since this does not require the creation of a temporary variable.
Definition at line 2503 of file symmetric_tensor.h.
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Subtraction of two symmetric tensors of equal rank. The result is another SymmetricTensor that has a number type that is compatible with the operation.
If possible (e.g. when Number and OtherNumber are of the same type, or if the result of Number() - OtherNumber() is another Number), you should use operator-= instead since this does not require the creation of a temporary variable.
Definition at line 2528 of file symmetric_tensor.h.
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Addition of a SymmetricTensor and a general Tensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2548 of file symmetric_tensor.h.
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Addition of a general Tensor with a SymmetricTensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2565 of file symmetric_tensor.h.
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Subtraction of a general Tensor from a SymmetricTensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2582 of file symmetric_tensor.h.
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Subtraction of a SymmetricTensor from a general Tensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2599 of file symmetric_tensor.h.
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Compute the determinant of a rank 2 symmetric tensor. The determinant is also commonly referred to as the third invariant of rank-2 tensors.
For a one-dimensional tensor, the determinant equals the only element and is therefore equivalent to the trace.
For greater notational simplicity, there is also a third_invariant() function that returns the determinant of a tensor.
Definition at line 2622 of file symmetric_tensor.h.
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Compute the determinant of a rank 2 symmetric tensor. This function therefore computes the same value as the determinant() functions and is only provided for greater notational simplicity (since there are also functions first_invariant() and second_invariant()).
\[ I_3 (\mathbf A) = III (\mathbf A) = \det (\mathbf A) \]
Definition at line 2662 of file symmetric_tensor.h.
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Compute and return the trace of a tensor of rank 2, i.e. the sum of its diagonal entries. The trace is the first invariant of a rank-2 tensor.
\[ \text{tr} \mathbf A = \sum_i A_{ii} \]
Definition at line 2680 of file symmetric_tensor.h.
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Compute the trace of a rank 2 symmetric tensor. This function therefore computes the same value as the trace() functions and is only provided for greater notational simplicity (since there are also functions second_invariant() and third_invariant()).
\[ I_1 (\mathbf A) = I (\mathbf A) = \text{tr} \mathbf A = \sum_i A_{ii} \]
Definition at line 2702 of file symmetric_tensor.h.
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Compute the second invariant of a tensor of rank 2. The second invariant of a tensor \(\mathbf A\) is defined as \(I_2 (\mathbf A) = II(\mathbf A) = \frac 12 \left[ (\text{tr} \mathbf A)^2 - \text{tr} (\mathbf{A}^2) \right]\).
For the kind of arguments to this function, i.e., a rank-2 tensor of size 1, the result is simply zero.
Definition at line 2721 of file symmetric_tensor.h.
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Compute the second invariant of a tensor of rank 2. The second invariant of a tensor \(\mathbf A\) is defined as \(I_2 (\mathbf A) = II(\mathbf A) = \frac 12 \left[ (\text{tr} \mathbf A)^2 - \text{tr} (\mathbf{A}^2) \right]\).
For the kind of arguments to this function, i.e., a symmetric rank-2 tensor of size 2, the result is (counting indices starting at one) \(I_2(\mathbf A) = II(\mathbf A) = \frac 12 \left[ (A_{11} + A_{22})^2 - (A_{11}^2+2 A_{12}^2+ A_{22}^2) \right] = A_{11} A_{22} - A_{12}^2\). As expected, for the \(2\times 2\) symmetric tensors this function handles, this equals the determinant of the tensor. (This is so because for \(2\times 2\) symmetric tensors, there really are only two invariants, so the second and third invariant are the same; the determinant is the third invariant.)
Definition at line 2748 of file symmetric_tensor.h.
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Compute the second invariant of a tensor of rank 2. The second invariant of a tensor \(\mathbf A\) is defined as \(I_2 (\mathbf A) = II(\mathbf A) = \frac 12 \left[ (\text{tr} \mathbf A)^2 - \text{tr} (\mathbf{A}^2) \right]\).
Definition at line 2765 of file symmetric_tensor.h.
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Return the eigenvalues of a symmetric \(1 \times 1\) tensor. The (single) entry of the tensor is, of course, equal to the (single) eigenvalue.
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Return the eigenvalues of a symmetric \(2\times 2\) tensor. The array of eigenvalues is sorted in descending order.
For \(2\times 2\) tensors, the eigenvalues of tensor \(\mathbf T\) are the roots of the characteristic polynomial \(0 = \lambda^2 - \lambda\;\text{tr}\mathbf{T} + \det \mathbf{T}\) as given by \(\lambda_1, \lambda_2 = \frac{1}{2} \left[ \text{tr} \mathbf{T} \pm \sqrt{(\text{tr} \mathbf{T})^2 - 4 \det \mathbf{T}} \right]\).
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Return the eigenvalues of a symmetric \(3\times 3\) tensor. The array of eigenvalues is sorted in descending order.
For \(3\times 3\) tensors, the eigenvalues of tensor \(\mathbf T\) are the roots of the characteristic polynomial \(0 = \lambda^3 - \lambda^2\;\text{tr}\mathbf T - \frac{1}{2} \lambda \left[\text{tr}(\mathbf{T}^2) - (\text{tr}\mathbf T)^2\right] - \det \mathbf T\).
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Return the eigenvalues and eigenvectors of a real-valued rank-2 symmetric tensor \(\mathbf T\). The array of matched eigenvalue and eigenvector pairs is sorted in descending order (determined by the eigenvalues).
The specialized algorithms utilized in computing the eigenvectors are presented in
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Return the transpose of the given symmetric tensor. Since we are working with symmetric objects, the transpose is of course the same as the original tensor. This function mainly exists for compatibility with the Tensor class.
Definition at line 3170 of file symmetric_tensor.h.
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Compute the deviator of a symmetric tensor, which is defined as \(\text{dev} \mathbf T = \mathbf T - \frac{1}{\text{dim}} \text{tr}\mathbf T \; \mathbf I\), where \(\mathbf I\) is the identity operator. This quantity equals the original tensor minus its contractive or dilative component and refers to the shear in, for example, elasticity.
Definition at line 3189 of file symmetric_tensor.h.
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Invert a symmetric rank-2 tensor.
Definition at line 3415 of file symmetric_tensor.h.
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Invert a symmetric rank-4 tensor. Since symmetric rank-4 tensors are mappings from and to symmetric rank-2 tensors, they can have an inverse.
If a tensor is not invertible, then the result is unspecified, but will likely contain the results of a division by zero or a very small number at the very least.
Definition at line 3435 of file symmetric_tensor.h.
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Return the tensor of rank 4 that is the outer product of the two tensors given as arguments, i.e. the result \(\mathbb A = \mathbf{T}_1 \otimes \mathbf{T}_2\) satisfies \(\mathbb A : \mathbf B = (\mathbf{T}_2 : \mathbf B) \mathbf{T}_1\) for all symmetric tensors \(\mathbf B\). In index notation
\[ \mathcal{A}_{ijkl} = (T_1)_{ij} (T_2)_{kl} \]
For example, the deviator tensor \(\mathbb P = \mathbb I - \frac{1}{\text{dim}} \mathbf I \otimes \mathbf I\) can be computed as identity_tensor<dim>() - 1/d * outer_product (unit_symmetric_tensor<dim>(), unit_symmetric_tensor<dim>()), since the (double) contraction with the unit tensor yields the trace of a symmetric tensor ( \(\mathbf I : \mathbf B = \text{tr} \mathbf B\)).
Definition at line 3466 of file symmetric_tensor.h.
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Return the symmetrized version of a full rank-2 tensor, i.e. \(\text{sym}\mathbf A = \frac 12 \left(\mathbf A + \mathbf{A}^T\right)\), as a symmetric rank-2 tensor. This is the version for general dimensions.
Definition at line 3492 of file symmetric_tensor.h.
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Multiplication of a symmetric tensor of general rank with a scalar from the right. This version of the operator is used if the scalar has the same data type as is used to store the elements of the symmetric tensor.
Definition at line 3516 of file symmetric_tensor.h.
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Multiplication of a symmetric tensor of general rank with a scalar from the left. This version of the operator is used if the scalar has the same data type as is used to store the elements of the symmetric tensor.
Definition at line 3534 of file symmetric_tensor.h.
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Division of a symmetric tensor of general rank by a scalar.
Definition at line 3623 of file symmetric_tensor.h.
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Multiplication of a symmetric tensor of general rank with a scalar from the right.
Definition at line 3642 of file symmetric_tensor.h.
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Multiplication of a symmetric tensor of general rank with a scalar from the left.
Definition at line 3659 of file symmetric_tensor.h.
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Division of a symmetric tensor of general rank by a scalar.
Definition at line 3675 of file symmetric_tensor.h.
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Compute the scalar product \(\mathbf A: \mathbf B=\sum_{i,j} A_{ij}B_{ij}\) between two tensors \(\mathbf A, \mathbf B\) of rank 2. In the current case where both arguments are symmetric tensors, this is equivalent to calling the expression A*B which uses SymmetricTensor::operator*().
Definition at line 3693 of file symmetric_tensor.h.
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Compute the scalar product \(\mathbf A: \mathbf B=\sum_{i,j} A_{ij}B_{ij}\) between two tensors \(\mathbf A, \mathbf B\) of rank 2. We don't use operator* for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor. For example, if B is a Tensor, calling A*B (instead of scalar_product(A,B)) provides \((\mathbf A \cdot\mathbf B)_{ij}=\sum_k A_{ik}B_{kj}\).
Definition at line 3716 of file symmetric_tensor.h.
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Compute the scalar product \(\mathbf A:\mathbf B=\sum_{i,j} A_{ij}B_{ij}\) between two tensors \(\mathbf A, \mathbf B\) of rank 2. We don't use operator* for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor. For example, if A is a Tensor, calling A*B (instead of scalar_product(A,B)) provides \((\mathbf A \cdot\mathbf B)_{ij}=\sum_k A_{ik}B_{kj}\).
Definition at line 3743 of file symmetric_tensor.h.
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Double contraction between a rank-4 and a rank-2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrix-vector multiplication.
This function does the same as SymmetricTensor::operator*(). It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general Tensor class.
Definition at line 3765 of file symmetric_tensor.h.
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Double contraction between a rank-4 and a rank-2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrix-vector multiplication.
This function does the same as SymmetricTensor::operator*(). It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general Tensor class.
Definition at line 3790 of file symmetric_tensor.h.
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Double contraction between a rank-4 and a rank-2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrix-vector multiplication.
This function does the same as SymmetricTensor::operator*(). It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general Tensor class.
Definition at line 3815 of file symmetric_tensor.h.
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Double contraction between a rank-4 and a rank-2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrix-vector multiplication.
This function does the same as SymmetricTensor::operator*(). It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general Tensor class.
Definition at line 3845 of file symmetric_tensor.h.
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Double contraction between a rank-4 and a rank-2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrix-vector multiplication.
This function does the same as SymmetricTensor::operator*(). It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general Tensor class.
Definition at line 3875 of file symmetric_tensor.h.
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Double contraction between a rank-4 and a rank-2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrix-vector multiplication.
This function does the same as SymmetricTensor::operator*(). It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general Tensor class.
Definition at line 3906 of file symmetric_tensor.h.
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Multiply a symmetric rank-2 tensor (i.e., a matrix) by a rank-1 tensor (i.e., a vector). The result is a rank-1 tensor (i.e., a vector).
Definition at line 3930 of file symmetric_tensor.h.
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Multiply a rank-1 tensor (i.e., a vector) by a symmetric rank-2 tensor (i.e., a matrix). The result is a rank-1 tensor (i.e., a vector).
Definition at line 3949 of file symmetric_tensor.h.
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The dot product (single contraction) for tensors: 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:
\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \sum_{k} \text{left}_{i_1,\ldots,i_{r1}, k} \text{right}_{k, j_1,\ldots,j_{r2}} \]
Definition at line 3986 of file symmetric_tensor.h.
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The dot product (single contraction) for tensors: 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:
\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \sum_{k} \text{left}_{i_1,\ldots,i_{r1}, k} \text{right}_{k, j_1,\ldots,j_{r2}} \]
Definition at line 4022 of file symmetric_tensor.h.
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Output operator for symmetric tensors of rank 2. Print the elements consecutively, with a space in between, two spaces between rank 1 subtensors, three between rank 2 and so on. No special amends are made to represents the symmetry in the output, for example by outputting only the unique entries.
Definition at line 4040 of file symmetric_tensor.h.
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Output operator for symmetric tensors of rank 4. Print the elements consecutively, with a space in between, two spaces between rank 1 subtensors, three between rank 2 and so on. No special amends are made to represents the symmetry in the output, for example by outputting only the unique entries.
Definition at line 4067 of file symmetric_tensor.h.
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Provide a way to get the dimension of an object without explicit knowledge of it's data type. Implementation is this way instead of providing a function dimension() because now it is possible to get the dimension at compile time without the expansion and preevaluation of an inlined function; the compiler may therefore produce more efficient code and you may use this value to declare other data types.
Definition at line 610 of file symmetric_tensor.h.
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Publish the rank of this tensor to the outside world.
Definition at line 615 of file symmetric_tensor.h.
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An integer denoting the number of independent components that fully describe a symmetric tensor. In \(d\) space dimensions, this number equals \(\frac 12 (d^2+d)\) for symmetric tensors of rank 2.
Definition at line 622 of file symmetric_tensor.h.
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The place where we store the data of the tensor.
Definition at line 947 of file symmetric_tensor.h.
1.9.4