Series: Finite Element Method › Shape Functions on Reference Elements

The Hex8 Element

The Hex8 element is the simplest hexahedral element: 8 nodes, one at each corner of the reference cube, with trilinear shape functions. The reference domain is H^=[0,1]3\hat{H} = [0,1]^3, matching the hexahedron reference domain. The 8 nodes are ordered bottom face first, then top face, each traversed in the same cyclic order:

x^0=(0,0,0),x^1=(1,0,0),x^2=(1,1,0),x^3=(0,1,0),\hat{x}_0 = (0,0,0), \quad \hat{x}_1 = (1,0,0), \quad \hat{x}_2 = (1,1,0), \quad \hat{x}_3 = (0,1,0),

x^4=(0,0,1),x^5=(1,0,1),x^6=(1,1,1),x^7=(0,1,1).\hat{x}_4 = (0,0,1), \quad \hat{x}_5 = (1,0,1), \quad \hat{x}_6 = (1,1,1), \quad \hat{x}_7 = (0,1,1).

In the library, it is implemented in elements/hex8.py by the class Hex8.

Nodes of the Hex8 element

The shape functions extend the pattern of Quad4 from two to three dimensions. Writing ℓ0(t)=1−t\ell_0(t) = 1 - t and ℓ1(t)=t\ell_1(t) = t for the linear Lagrange basis on [0,1][0,1], each Hex8 shape function is a triple tensor product

Ni(x^,y^,z^)=ℓa(x^) ℓb(y^) ℓc(z^),N_i(\hat{x}, \hat{y}, \hat{z}) = \ell_a(\hat{x})\,\ell_b(\hat{y})\,\ell_c(\hat{z}),

where (a,b,c)∈{0,1}3(a, b, c) \in \{0,1\}^3 is the index triple that places node ii at (x^a,y^b,z^c)(\hat{x}_a, \hat{y}_b, \hat{z}_c). Written out explicitly:

N0=(1−x^)(1−y^)(1−z^),N1=x^(1−y^)(1−z^),N2=x^ y^(1−z^),N3=(1−x^)y^(1−z^),N_0 = (1-\hat{x})(1-\hat{y})(1-\hat{z}), \quad N_1 = \hat{x}(1-\hat{y})(1-\hat{z}), \quad N_2 = \hat{x}\,\hat{y}(1-\hat{z}), \quad N_3 = (1-\hat{x})\hat{y}(1-\hat{z}),

N4=(1−x^)(1−y^)z^,N5=x^(1−y^)z^,N6=x^ y^ z^,N7=(1−x^)y^ z^.N_4 = (1-\hat{x})(1-\hat{y})\hat{z}, \quad N_5 = \hat{x}(1-\hat{y})\hat{z}, \quad N_6 = \hat{x}\,\hat{y}\,\hat{z}, \quad N_7 = (1-\hat{x})\hat{y}\,\hat{z}.

Each NiN_i equals one at its own corner and zero at all other seven. The eight functions form a partition of unity. The gradients follow from the product rule; for node 0 for example,

∇N0=(−(1−y^)(1−z^)−(1−x^)(1−z^)−(1−x^)(1−y^)),\nabla N_0 = \begin{pmatrix} -(1-\hat{y})(1-\hat{z})\\ -(1-\hat{x})(1-\hat{z})\\ -(1-\hat{x})(1-\hat{y}) \end{pmatrix},

and similarly for the remaining nodes with appropriate sign changes on each factor. As with Quad4, the gradients depend on position because of the cross terms in the trilinear functions.

Hex8 spans the trilinear space Q1\mathbb{Q}_1 — all polynomials of degree at most one in each variable separately. This includes all linear functions, so the element achieves first-order accuracy. The face element is Quad4, since each of the six square faces carries four corner nodes forming a bilinear quadrilateral element.