On the numerical solution of constrained eigenvalue problems in structural engineering

The poster is devoted to the analysis of the numerical linear algebra issues arising in the modal analysis of structures with application to masonry construction of historical interest. Although the constitutive equation adopted for masonry is nonlinear, modal analysis gives important qualitative information on the dynamic behavior of masonry structures and allows for assessing their seismic vulnerability, while taking Italian regulations into account. Modal analysis consists in the solution of a constrained eigenvalue problem arising from the solution of the free vibration equilibrium equations in a finite-element setting and involves the mass and stiffness matrices and a set of constraints which enforce relationships between degrees of freedom. A simple example of a constraint, is the imposition of the Dirichlet boundary conditions which usually consists in setting certain degrees of freedom to zero (single-point or fixed constraints). A further example is given by the so called master-slave constraints which impose that the displacement of a node (called the slave) depends linearly on the displacements of other nodes (called the masters). These constraints are crucial, e.g., in modeling the contact interaction between masonry and reinforcement. We propose an efficient implementation of numerical methods for constrained eigenvalue problems, specialized for the modal analysis of structures taking into account both the sparsity of the matrices and the features of master-slave constraints. The implementation will be based on open-source packages embedded in the finite-element code NOSA-ITACA developed in the framework of a research project funded by the Region of Tuscany (www.nosaitaca.it/en/). Numerical examples will be shown on the Project case study "Voltone" - a large vaulted masonry structure located beneath Piazza della Repubblica in Livorno, Italy.