Fig. 3

\(L_2\) errors of a single element internal forces (a) and stiffnesses (b) as a function of the number of Gauss integration points. The accuracy is measured relative to the force or stiffness matrix obtained for the maximum number of integration points used (\(n_\mathrm {g} = 13\) for triangles and \(n_\mathrm {g} = 49\) for quadrilaterals). Solid lines correspond to \({\underline{f}}_0^e\) or \({\varvec{\mathsf {K}}}_{00}^e\), dashed lines to \({\underline{f}}_1^e\) or \({\varvec{\mathsf {K}}}_{11}^e\), and dash-dot lines to \({\varvec{\mathsf {K}}}_{01}^e = ({\varvec{\mathsf {K}}}_{10}^e)^{\mathsf {T}}\) sub-matrices of \({\underline{f}}_\mathrm {M}^e\) or \({\varvec{\mathsf {K}}}^e_\mathrm {M}\) in Eqs. (18) and (21); elements from Table 1 are used. Configuration of the element corresponds to homogeneous overall deformation \(\vec {\nabla }\vec {v}_0 = - 0.05\vec {e}_2\vec {e}_2\) with an affine micromorphic field \(v_1(\vec {X}) = (X_1+X_2+3)/2\), \(X_i \in [-0.5,0.5]\), \(i = 1,2\) (microstructure with a square stacking of holes used, i.e., \(n=1\))