# From Stress to Shape: Equilibrium of Cloister and Cross Vaults
## 3 Outline of the Method
### Numerical Discretization and Algorithm  
The transverse equilibrium Equation (6) is solved numerically by discretizing the domain into a finite number of triangular elements, and adopting a Lumped Stress Method [28] approximation of the Hessian of both the stress potential \(F\) and of the membrane shape \(f\). With the LSM approximation, the two functions are described by simplicial surfaces, defined by the nodal values \(\hat{F}_{i}\) and \(\hat{f}_{i}\), with \(i=1,2,\ldots,N\), \(N\) being the total number of nodes of the given triangulation. The vectors collecting the nodal values of the shape function \(f\) and of the stress potential \(F\) are denoted \(\hat{\mathbf{f}}\) and \(\hat{\mathbf{F}}\). \(\hat{\mathbf{f}}\) is divided into two vectors \(\hat{\mathbf{f}}_{l}\) and \(\hat{\mathbf{f}}_{B}\), of length \(N_{l}\) and \(N_{B}\), distinguishing between internal and boundary nodes, respectively. The search of a statically admissible stress regime for the vault is carried out through an algorithm divided in several steps. Given the geometry of the vault, defined by its intrados and extrados, we start on assigning an appropriate functional form for the stress potential, depending on a limited number \(n\) of parameters, such that  
\[F=F(x_{1},x_{2};a_{1},a_{2},\ldots,a_{n}). \tag{11}\]A first step is the parametric solution of the discretized version of Equation (6), with suitable boundary conditions, in terms of the membrane shape defined by \(f\); therefore, \(f=f(x_{1},x_{2};a_{1},a_{2},\ldots,a_{n})\). The second step consists in the optimization of the parameters \(a_{1},a_{2},\ldots,a_{n}\), performed by minimizing an error function \(I\) defined as the integral of the quadratic approximation of the characteristic function of the vault domain. Given a point \(\mathbf{x}\in\Omega\), the characteristic function indicates whether the corresponding values of the rise of the membrane is comprised between the volume of the vault, or not. In particular  
\[C(f(\mathbf{x}))=\left\{ \begin{aligned} 0&\text{if}f_{intr}( \mathbf{x})<f(\mathbf{x})<f_{extr}(\mathbf{x})\\ +\infty&\text{if}f(\mathbf{x})<f_{intr}(\mathbf{x}) \quad\text{or}\quad f(\mathbf{x})>f_{extr}(\mathbf{x})\end{aligned} \right.. \tag{12}\]  
Its quadratic approximation reads  
\[C(f(\mathbf{x}))=\left\{ \begin{aligned} 0&\text{if}f_{intr}( \mathbf{x})<f(\mathbf{x})<f_{extr}(\mathbf{x})\\ \left(f(\mathbf{x})-f_{intr}(\mathbf{x})\right)^{2}& \text{$f(\mathbf{x})<f_{intr}(\mathbf{x})$}\\ \left(f(\mathbf{x})-f_{extr}(\mathbf{x})\right)^{2}& \text{$f(\mathbf{x})>f_{extr}(\mathbf{x})$}\end{aligned} \right.. \tag{13}\]  
For a generic point \(\mathbf{x}\) of the domain, given \(f_{intr}(\mathbf{x})\) and \(f_{extr}(\mathbf{x})\), the approximated characteristic function is represented in Figure 2  
We denote \(\Omega_{c}\) the subregion of the entire planform \(\Omega\) obtained by removing from the domain a boundary strip of size \(c\), defined as a fraction of the diameter \(L\) of \(\Omega\) (in the applications we set \(c=L/5\)). The rationale behind this procedure is linked to the nature of the governing equation: due to ellipticity, the solution is only weakly dependent on the boundary data, that is, altering the boundary values has a detectable effect only in a narrow band close to the boundary. So, we can split the optimization into an interior and an exterior part, postponing the optimization of the boundary data. The error function \(I\) is then defined as follows:  
\[I=\int_{\partial\Omega_{a}}{(f-f_{intr})^{2}d\Omega}+\int_{\partial\Omega_{b}} {(f-f_{extr})^{2}d\Omega} \tag{14}\]  
\(\Omega_{a}\) and \(\Omega_{b}\) being the parts of the domain \(\Omega_{c}\), for which the membrane surface is below the intrados, or above the extrados, respectively. In the third step, the values of \(a_{1},a_{2},\ldots,a_{n}\), found in the previous step, are fixed and the boundary values of the discretized membrane shape are considered as variables.