## 4 Computational Features and CPU Parallelisation
### Comment on the Computational Attractiveness of the (Discrete) Macro-Element  
It has been seen that the macro-scale behaviour arises from the deformation of in- and out-of-plane FE trusses that carry the material information (from experimentation or numerical homogenisation). Restricting the macro-deformation to linear elements brings direct advantages due to its simplicity and one-dimensionality of the constitutive equations, cyclic behaviour, strength domain, inelastic strain evolution laws, and damage evolution and tracking (closure-opening crack states). The use of advanced structural analysis software, such as ABAQUS, is of utmost importance. In cases where a given instability renders a not purely positive definite stiffness matrix, the traditional equation solvers (as the Newton-Raphson, the modified Newton-Raphson, or secant methods) are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Node Renumbering Algorithm** & **None (Reference)** & **Geometric Algorithm** & **AMD Algorithm** \\ \hline CPU total time (s) & 14.95 & 5.523 (\(-\)63.1\%) & 6.006 (\(-\)59.8\%) \\ Optimum physical memory RAM (Mbytes) & 137.1 & 69.93 (\(-\)49.0\%) & 76.23 (\(-\)44.4\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance test on an out-of-plane loaded masonry wall (\(35\times 65\) elements) for two–node renumbering algorithms and within a direct sparse solver. unable to give an adequate solution. In such situations, the software is robust because a modified Riks method ([115; 116; 117; 118]) is at its disposal to overcome snap-back and snap-through issues. This is especially important when conducting quasi-static analysis in which material (and of other kinds) non-linearities are active. A quasi-static type of analysis is performed on the so-called LNEC brick house benchmark to demonstrate the efficiency of the macro-element. Figure 12a depicts the brick structure, which is composed of three walls in a U-shaped plan arrangement. The main facade presents a gable wall (dimensions 3.50 \(\times\) 2.75 m\({}^{2}\)) and is linked with two transversal walls (dimensions 2.50 \(\times\) 2.25 m\({}^{2}\)). Walls are constructed with clay brickwork in an English-bond arrangement and have 235 mm of thickness. A mass proportional pushover analysis has been performed following the load direction indicated in Figure 12a. Results from the macro-element (discrete) are compared against a smeared (fixed) crack FE model. The assumed material properties (see Table 4) are based on experimental and literature evidence [75; 119], specifically the tension (\(f_{i}\)), compression (\(f_{c}\)), and shear (\(f_{shear}\)) strength values, and corresponding fracture energy terms. A mesh size equal to 200 mm has been assumed for both models to guarantee the objectivity of the comparison. Likewise, the following have been considered (see [4]): (i) an exponential softening in tension [4] governed by the tensile fracture energy, \(G_{\textit{fension}}\); and (ii) a parabolic softening in compression [4] governed by the compressive fracture energy, \(G_{\textit{f compression}}\). An exponential softening has been also assumed for the shear regime within the macro-element model, albeit it is disregarded for the smeared crack model. Analyses have been computed on a 64-bit Windows 7 computer with 16 GB RAM and Intel i7 CPU running at 3.4 GHz. The obtained capacity curves and the required computational times are given in Figure 12b. Differences in the peak (around 12%) and post-peak response were expected due to the differences that both formulations contain. Nonetheless, the aim of this comparison resorts especially on the computational time rather than on the capacity curves. Herein, the macro-element allows reducing the computational cost by 41%. This is especially relevant if one states that the FE smeared crack models are typically the option for large-scale structures due to computational attractiveness [2; 4; 5].