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Fig. 2 | Advanced Modeling and Simulation in Engineering Sciences

Fig. 2

From: Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials

Fig. 2

\(L_2\) errors of a single element stiffness matrix for triangular (a) and quadrilateral (b) elements as a function of the number of Gauss integration points. The accuracy is measured relative to the 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 \({\varvec{\mathsf {K}}}_{00}^e\) and dashed lines to \({\varvec{\mathsf {K}}}_{11}^e\) sub-matrices of \({\varvec{\mathsf {K}}}^e_\mathrm {M}\) in Eq. (21), whereas \({\varvec{\mathsf {K}}}_{01}^e = ({\varvec{\mathsf {K}}}_{10}^e)^{\mathsf {T}} = {\varvec{\mathsf {0}}}\); elements from Table 1 are used. Configuration of elements correspond to the reference undeformed state (microstructure with a square stacking of holes used, i.e., \(n=1\)). Integration rules for which spurious singular modes occur are denoted with red circles

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