deal.II version GIT relicensing-2149-g6c46669274 2024-11-13 14:40:00+00:00
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#include <deal.II/base/polynomials_bdm.h>
Public Member Functions | |
PolynomialsBDM (const unsigned int k) | |
void | evaluate (const Point< dim > &unit_point, std::vector< Tensor< 1, dim > > &values, std::vector< Tensor< 2, dim > > &grads, std::vector< Tensor< 3, dim > > &grad_grads, std::vector< Tensor< 4, dim > > &third_derivatives, std::vector< Tensor< 5, dim > > &fourth_derivatives) const override |
std::string | name () const override |
virtual std::unique_ptr< TensorPolynomialsBase< dim > > | clone () const override |
unsigned int | n () const |
unsigned int | degree () const |
Static Public Member Functions | |
static unsigned int | n_polynomials (const unsigned int degree) |
Private Attributes | |
const PolynomialSpace< dim > | polynomial_space |
std::vector< Polynomials::Polynomial< double > > | monomials |
Threads::Mutex | mutex |
std::vector< double > | p_values |
std::vector< Tensor< 1, dim > > | p_grads |
std::vector< Tensor< 2, dim > > | p_grad_grads |
std::vector< Tensor< 3, dim > > | p_third_derivatives |
std::vector< Tensor< 4, dim > > | p_fourth_derivatives |
const unsigned int | polynomial_degree |
const unsigned int | n_pols |
This class implements the Hdiv-conforming, vector-valued Brezzi-Douglas-Marini ( BDM ) polynomials described in Brezzi and Fortin's Mixed and Hybrid Finite Element Methods (refer to pages 119 - 124).
The BDM polynomial space contain the entire \((P_{k})^{n}\) space (constructed with PolynomialSpace Legendre polynomials) as well as part of \((P_{k+1})^{n}\) (ie. \((P_{k})^{n} \subset BDM_{k} \subset (P_{k+1})^{n}\)). Furthermore, \(BDM_{k}\) elements are designed so that \(\nabla \cdot q \in P_{k-1} (K)\) and \(q \cdot n |_{e_{i}} \in P_{k}(e_{i})\). More details of two and three dimensional \(BDM_{k}\) elements are given below.
\( BDM_{k} = \{\mathbf{q} | \mathbf{q} = p_{k} (x,y) + r \; \text{curl} (x^{k+1}y) + s \; \text{curl} (xy^{k+1}), p_{k} \in (P_{k})^{2} \}\).
Note: the curl of a scalar function is given by \(\text{curl}(f(x,y)) = \begin{pmatrix} f_{y}(x,y) \\ -f_{x}(x,y) \end{pmatrix}\).
The basis used to construct the \(BDM_{1}\) shape functions is
\begin{align*} \phi_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \phi_1 = \begin{pmatrix} -\sqrt{3}+2\sqrt{3}x \\ 0 \end{pmatrix}, \phi_2 = \begin{pmatrix} -\sqrt{3}+2\sqrt{3}y \\ 0 \end{pmatrix}, \phi_3 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \phi_4 = \begin{pmatrix} 0 \\ -\sqrt{3}+2\sqrt{3}x \end{pmatrix}, \phi_5 = \begin{pmatrix} 0 \\ -\sqrt{3}+2\sqrt{3}y \end{pmatrix}, \phi_6 = \begin{pmatrix} x^2 \\ -2xy \end{pmatrix}, \phi_7 = \begin{pmatrix} 2xy \\ -y^2 \end{pmatrix}. \end{align*}
The dimension of the \(BDM_{k}\) space is \((k+1)(k+2)+2\), with \(k+1\) unknowns per edge and \(k(k-1)\) interior unknowns.
\( BDM_{k} = \{\mathbf{q} | \mathbf{q} = p_{k} (x,y,z) + \sum_{i=0}^{k} ( r_{i} \; \text{curl} \begin{pmatrix} 0\\0\\xy^{i+1}z^{k-i} \end{pmatrix} + s_{i} \; \text{curl} \begin{pmatrix} yz^{i+1}x^{k-i}\\0\\0 \end{pmatrix} + t_{i} \; \text{curl} \begin{pmatrix}0\\zx^{i+1}y^{k-i}\\0\end{pmatrix}) , p_{k} \in (P_{k})^{3} \}\).
Note: the 3d description of \(BDM_{k}\) is not unique. See Mixed and Hybrid Finite Element Methods page 122 for an alternative definition.
The dimension of the \(BDM_{k}\) space is \(\dfrac{(k+1)(k+2)(k+3)}{2}+3(k+1)\), with \(\dfrac{(k+1)(k+2)}{2}\) unknowns per face and \(\dfrac{(k-1)k(k+1)}{2}\) interior unknowns.
Definition at line 98 of file polynomials_bdm.h.
PolynomialsBDM< dim >::PolynomialsBDM | ( | const unsigned int | k | ) |
Constructor. Creates all basis functions for BDM polynomials of given degree.
Definition at line 29 of file polynomials_bdm.cc.
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overridevirtual |
Compute the value and the first and second derivatives of each BDM polynomial at unit_point
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The size of the vectors must either be zero or equal n()
. In the first case, the function will not compute these values.
Implements TensorPolynomialsBase< dim >.
Definition at line 55 of file polynomials_bdm.cc.
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inlineoverridevirtual |
Return the name of the space, which is BDM
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Implements TensorPolynomialsBase< dim >.
Definition at line 193 of file polynomials_bdm.h.
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static |
Return the number of polynomials in the space BDM(degree)
without requiring to build an object of PolynomialsBDM. This is required by the FiniteElement classes.
Definition at line 361 of file polynomials_bdm.cc.
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overridevirtual |
A sort of virtual copy constructor, this function returns a copy of the polynomial space object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.
Some places in the library, for example the constructors of FE_PolyTensor, need to make copies of polynomial spaces without knowing their exact type. They do so through this function.
Implements TensorPolynomialsBase< dim >.
Definition at line 376 of file polynomials_bdm.cc.
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inlineinherited |
Return the number of polynomials.
Definition at line 151 of file tensor_polynomials_base.h.
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inlineinherited |
Return the highest polynomial degree of polynomials represented by this class. A derived class may override this if its value is different from my_degree
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Definition at line 160 of file tensor_polynomials_base.h.
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private |
An object representing the polynomial space used here. The constructor fills this with the monomial basis.
Definition at line 151 of file polynomials_bdm.h.
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private |
Storage for monomials. In 2d, this is just the polynomial of order k. In 3d, we need all polynomials from degree zero to k.
Definition at line 157 of file polynomials_bdm.h.
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mutableprivate |
A mutex that guards the following scratch arrays.
Definition at line 162 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 167 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 172 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 177 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 182 of file polynomials_bdm.h.
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mutableprivate |
Auxiliary memory.
Definition at line 187 of file polynomials_bdm.h.
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privateinherited |
The highest polynomial degree of this functions represented by this object.
Definition at line 139 of file tensor_polynomials_base.h.
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privateinherited |
The number of polynomials represented by this object.
Definition at line 144 of file tensor_polynomials_base.h.