Hellan–Herrmann–Johnson
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Orders | \(0\leqslant k\) |
Reference elements | triangle |
Polynomial set | \(\mathcal{Z}^{(2)}_{k}\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑\(\mathcal{Z}^{(2)}_k=\left\{\mathbf{M}\in\mathcal{P}_{k}^{d\times d}\middle|\mathbf{M}^t=\mathbf{M}\right\}\)
\(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i\leqslant k\right\}\) |
DOFs | On each edge: integral moments of inner products of normal to edge with an order \(k\) Lagrange space
On each face: point evaluations of tensor products with symmetric matrices whose entries are in (lagrange,k-1) |
Number of DOFs | triangle: \(3(k+1)(k+2)/2\) (A045943) |
Mapping | double contravariant Piola |
continuity | Inner products with normals to facets are continuous |
Categories | Matrix-valued elements |
Implementations
Symfem | "HHJ" ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑Before trying this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem
# Create Hellan-Herrmann-Johnson order 1 on a triangle element = symfem.create_element("triangle", "HHJ", 1)
# Create Hellan-Herrmann-Johnson order 2 on a triangle element = symfem.create_element("triangle", "HHJ", 2) |
UFL | "HHJ" ↓ Show UFL examples ↓↑ Hide UFL examples ↑Before trying this example, you must install UFL: pip3 install UFL This element can then be created with the following lines of Python: import ufl
# Create Hellan-Herrmann-Johnson order 1 on a triangle element = ufl.FiniteElement("HHJ", "triangle", 1)
# Create Hellan-Herrmann-Johnson order 2 on a triangle element = ufl.FiniteElement("HHJ", "triangle", 2) |
Examples
References
- Arnold, Douglas N. and Walker, Shawn W. The Hellan–Herrmann–Johnson method with curved elements, SIAM Journal on Numberical Analysis 58(5), 2829–2855, 2020. [DOI: 10.1137/19M1288723] [BibTeX]
DefElement stats
Element added | 06 February 2021 |
Element last updated | 02 August 2022 |