## 3 Calibration: Compact Tension Testing
### Genetic Algorithm  
The objective here is to identify a specific set of input parameters for simulations that can accurately replicate the results observed in corresponding physical experiments. In a simulation model, the input parameters affecting numerical results are denoted as design variables \(\mathbf{x}=[x_{1},x_{2},\cdots,x_{n}]^{\mathrm{T}}\) where \(\mathbf{x}\) is the vector of \(n\) design variables, \(x_{1},x_{2},\cdots,x_{n}\). The mean squared error (MSE) between the experimental and simulation outcomes can be used to determine the extent to which simulations can replicate experimental measurements accurately. The ordinate-based MSE given by \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Modulus Along the Grain \(E_{1}\) (MPa)** & **Modulus Perpendicular to Grain \(E_{2}\) (MPa)** & **Shear Modulus (MPa)** & **Poisson’s Ratio \(\
u_{12}\) (-)** \\ \hline
14,000 & 2280 & 1080 & 0.073 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elastic properties of uni-directional beech veneers [1]. Figure 3: Dimensions, finite element mesh, and boundary conditions of open-hole tension test sample. Figure 2: Dimensions, finite element mesh, and boundary conditions of double-tapered compact tension test sample. \[f_{\text{MSE}}(\mathbf{x})=\frac{1}{P}\sum_{p=1}^{P}\left[f_{p}^{\text{num}}( \mathbf{x})-G_{p}^{\text{exp}}\right]^{2} \tag{5}\]  
evaluates \(P\) number of points to compare numerical results \(f_{p}^{\text{num}}(\mathbf{x})\) with target experimental measurements \(G_{p}^{\text{exp}}\). In order to identify the most suitable input parameters, the following optimisation problem is formulated aiming at minimising the MSE value related to Equation (5). The general optimisation problem with (in)equality constraints is expressed by  
\[ \begin{split}\min f_{\text{MSE}}(\mathbf{x})\\ \text{subject to}\\ \mathcal{L}\leq\mathbf{x}\leq\mathcal{U}\\ g_{m}(\mathbf{x})=0,& m\in\Xi\\ g_{m}(\mathbf{x})\geq 0,& m\in\zeta\end{split}  \tag{6}\]  
where \(\mathcal{L}\) and \(\mathcal{U}\) are the lower and upper bounds of the design variables \(\mathbf{x}\), respectively; \(\Xi\) and \(\zeta\) are sets of indices for equality and inequality constraints, respectively; and \(g_{m}(\mathbf{x})\) represent constraint functions. The optimisation software LS-OPT [31] is employed to perform data-driven calibration of the CODAM2 material parameters. The direct coupling of LS-OPT to CODAM2 simulations enables the determination of the most suitable design variables (input parameters) by solving the optimisation problem described in Equation (6) in conjunction with experimentally measured load vs. displacement data obtained from CT tests [6]. To solve the optimisation problem, a genetic algorithm (GA) is selected, a metaheuristic inspired by the process of natural selection [31]. In GA, selection, crossover, and mutation are the primary genetic operators utilised to achieve optimisation. For more specific details about GA parameters and implementation, please refer to [24]. As visualised in the stress-strain curve in Figure 1, the CODAM2 material model requires a total of four input parameters related to damage initiation and progression in the two principal directions: two damage initiation strains \(\varepsilon_{1}^{i}\) and \(\varepsilon_{2}^{i}\) and two damage saturation strains \(\varepsilon_{1}^{s}\) and \(\varepsilon_{2}^{s}\) in the direction of the grains and perpendicular to it, respectively. Therefore, the design variables \([x_{1},x_{2},x_{3},x_{4}]^{\text{T}}\) in GA are defined as \(\mathbf{x}=[\varepsilon_{1}^{i},\varepsilon_{2}^{i},\varepsilon_{1}^{s}, \varepsilon_{2}^{s}]^{\text{T}}\). The lower \(\mathcal{L}\) and upper bounds \(\mathcal{U}\) are set as follows:  
* \(0.0001\leq\varepsilon_{1}^{i}\leq 0.15\),
* \(0.0001\leq\varepsilon_{2}^{i}\leq 0.15\),
* \(0.001\leq\varepsilon_{1}^{s}\leq 0.3\),
* \(0.001\leq\varepsilon_{2}^{s}\leq 0.3\),  
enforcing that \(\varepsilon_{a}^{s}>\varepsilon_{a}^{i}\) for \(\alpha=1,2\).