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# Hellan–Herrmann–Johnson

 Orders $$0\leqslant k$$ Reference elements triangle Polynomial set $$\mathcal{Z}^{(4)}_{k}$$↓ Show polynomial set definitions ↓ 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) Categories Matrix-valued elements

## Implementations

 Symfem "HHJ"↓ Show Symfem examples ↓ UFL "HHJ"↓ Show UFL examples ↓

## Examples

triangle
order 1
triangle
order 2
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{matrix}1 & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & 1\end{matrix}\right)$$, $$\left(\begin{matrix}x & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & x\\x & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & x\end{matrix}\right)$$, $$\left(\begin{matrix}y & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & y\\y & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & y\end{matrix}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{8}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & 3 x - 1\\3 x - 1 & 0\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & 3 y - 1\\3 y - 1 & 0\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}- 6 x - 6 y + 4 & 3 x + 3 y - 2\\3 x + 3 y - 2 & 0\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}6 y - 2 & 1 - 3 y\\1 - 3 y & 0\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}0 & 3 x + 3 y - 2\\3 x + 3 y - 2 & - 6 x - 6 y + 4\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}0 & 1 - 3 x\\1 - 3 x & 6 x - 2\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}1&0\\0&0\end{array}\right)$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}6 x & - 6 x - 3 y + 3\\- 6 x - 3 y + 3 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}0 & - 3 x - 3 y + 3\\- 3 x - 3 y + 3 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&1\end{array}\right)$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}0 & - 3 x - 6 y + 3\\- 3 x - 6 y + 3 & 6 y\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{matrix}1 & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & 1\end{matrix}\right)$$, $$\left(\begin{matrix}x & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & x\\x & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & x\end{matrix}\right)$$, $$\left(\begin{matrix}x^{2} & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & x^{2}\\x^{2} & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & x^{2}\end{matrix}\right)$$, $$\left(\begin{matrix}y & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & y\\y & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & y\end{matrix}\right)$$, $$\left(\begin{matrix}x y & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & x y\\x y & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & x y\end{matrix}\right)$$, $$\left(\begin{matrix}y^{2} & 0\\0 & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & y^{2}\\y^{2} & 0\end{matrix}\right)$$, $$\left(\begin{matrix}0 & 0\\0 & y^{2}\end{matrix}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{17}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & 15 x^{2} - 12 x + \frac{3}{2}\\15 x^{2} - 12 x + \frac{3}{2} & 0\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(s_{0} \left(2 s_{0} - 1\right))|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & 15 y^{2} - 12 y + \frac{3}{2}\\15 y^{2} - 12 y + \frac{3}{2} & 0\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(4 s_{0} \left(1 - s_{0}\right))|{e_{0}}|\hat{\boldsymbol{n}}^{\text{t}}_{0}\mathbf{V}\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}0 & \frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2}\\\frac{15 x^{2}}{4} + 15 x y - 6 x + \frac{15 y^{2}}{4} - 6 y + \frac{3}{2} & 0\end{matrix}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9 & - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\- 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2} & 0\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(s_{0} \left(2 s_{0} - 1\right))|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}30 y^{2} - 24 y + 3 & - 15 y^{2} + 12 y - \frac{3}{2}\\- 15 y^{2} + 12 y - \frac{3}{2} & 0\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(4 s_{0} \left(1 - s_{0}\right))|{e_{1}}|\hat{\boldsymbol{n}}^{\text{t}}_{1}\mathbf{V}\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}\frac{15 x^{2}}{2} - 15 x y - 3 x - 15 y^{2} + 15 y - \frac{3}{2} & - \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4}\\- \frac{15 x^{2}}{4} + \frac{15 x y}{2} + \frac{3 x}{2} + \frac{15 y^{2}}{2} - \frac{15 y}{2} + \frac{3}{4} & 0\end{matrix}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(2 s_{0}^{2} - 3 s_{0} + 1)|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}0 & - 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2}\\- 15 x^{2} - 30 x y + 18 x - 15 y^{2} + 18 y - \frac{9}{2} & 30 x^{2} + 60 x y - 36 x + 30 y^{2} - 36 y + 9\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(s_{0} \left(2 s_{0} - 1\right))|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}0 & - 15 x^{2} + 12 x - \frac{3}{2}\\- 15 x^{2} + 12 x - \frac{3}{2} & 30 x^{2} - 24 x + 3\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(4 s_{0} \left(1 - s_{0}\right))|{e_{2}}|\hat{\boldsymbol{n}}^{\text{t}}_{2}\mathbf{V}\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}0 & \frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4}\\\frac{15 x^{2}}{2} + \frac{15 x y}{2} - \frac{15 x}{2} - \frac{15 y^{2}}{4} + \frac{3 y}{2} + \frac{3}{4} & - 15 x^{2} - 15 x y + 15 x + \frac{15 y^{2}}{2} - 3 y - \frac{3}{2}\end{matrix}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}- s_{0} - s_{1} + 1&0\\0&0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{9} = \left(\begin{matrix}24 x \left(- 5 x - 5 y + 4\right) & 90 x^{2} + 120 x y - 96 x + 30 y^{2} - 48 y + 18\\90 x^{2} + 120 x y - 96 x + 30 y^{2} - 48 y + 18 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&- s_{0} - s_{1} + 1\\- s_{0} - s_{1} + 1&0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{10} = \left(\begin{matrix}0 & 30 x^{2} + 60 x y - 48 x + 30 y^{2} - 48 y + 18\\30 x^{2} + 60 x y - 48 x + 30 y^{2} - 48 y + 18 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&- s_{0} - s_{1} + 1\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{11} = \left(\begin{matrix}0 & 30 x^{2} + 120 x y - 48 x + 90 y^{2} - 96 y + 18\\30 x^{2} + 120 x y - 48 x + 90 y^{2} - 96 y + 18 & 24 y \left(- 5 x - 5 y + 4\right)\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{12}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}s_{0}&0\\0&0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{12} = \left(\begin{matrix}12 x \left(5 x - 2\right) & - 90 x^{2} - 60 x y + 84 x + 12 y - 12\\- 90 x^{2} - 60 x y + 84 x + 12 y - 12 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&s_{0}\\s_{0}&0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{13} = \left(\begin{matrix}0 & - 60 x^{2} - 60 x y + 72 x + 12 y - 12\\- 60 x^{2} - 60 x y + 72 x + 12 y - 12 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&s_{0}\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{14} = \left(\begin{matrix}0 & - 60 x^{2} - 120 x y + 72 x + 24 y - 12\\- 60 x^{2} - 120 x y + 72 x + 24 y - 12 & 24 y \left(5 x - 1\right)\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{15}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}s_{1}&0\\0&0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{15} = \left(\begin{matrix}24 x \left(5 y - 1\right) & - 120 x y + 24 x - 60 y^{2} + 72 y - 12\\- 120 x y + 24 x - 60 y^{2} + 72 y - 12 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{16}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&s_{1}\\s_{1}&0\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{16} = \left(\begin{matrix}0 & - 60 x y + 12 x - 60 y^{2} + 72 y - 12\\- 60 x y + 12 x - 60 y^{2} + 72 y - 12 & 0\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\mathbf{V}\mapsto\displaystyle\int_{R}\mathbf{V}:\left(\begin{array}{cc}0&0\\0&s_{1}\end{array}\right)$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \mathbf{\Phi}_{17} = \left(\begin{matrix}0 & - 60 x y + 12 x - 90 y^{2} + 84 y - 12\\- 60 x y + 12 x - 90 y^{2} + 84 y - 12 & 12 y \left(5 y - 2\right)\end{matrix}\right)$$

This DOF is associated with face 0 of the reference element.

## References

• Arnold, D. N. and Walker, S. W. The Hellan–Herrmann–Johnson method with curved elements, arΧiv: 1909.09687, 2020. [BibTeX]

## DefElement stats

 Element added 06 February 2021 Element last updated 13 June 2021