Poster Recent Advances on Optimization, Toulouse (France) July 24-26 2013


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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 [1] 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 (

Numerical examples will be shown on the Project case study "Voltone" - a large vaulted masonry
structure located beneath Piazza della Repubblica in Livorno, Italy.

[1] Lucchesi M., Padovani C., Pasquinelli G., Zani N., "Masonry constructions: mechanical models and numerical applications",
Series: Lecture Notes in Applied and Computational Mechanics, 39, Berlin Heidelberg, Springer-Verlag, 2008.

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Poster Recent Advances on Optimization, Toulouse (France) July 24-26 2013

  1. 1. Geometric survey Recent Advances on Optimization, Toulouse, July 24-26 2013 On the numerical solution of constrained eigenvalue problems in structural engineering Margherita Porcelli Istituto di Scienza e Tecnologie dell’Informazione “A. Faedo” – CNR Vincenzo Binante, Maria Girardi, Cristina Padovani, Giuseppe Pasquinelli Istituto di Scienza e Tecnologie dell’Informazione “A. Faedo” – CNR Daniele Pellegrini Dipartimento di Ingegneria Civile e Ambientale - UNIFI Finite element mesh Modal analysis Visualization of results N O S A I T A C A N O S A I T A C A SPARSEKIT A tool for sparse matrix computations Saad, 1990 ARPACK Implicity restarted Lanczos method Lehoucq, Sorensen, 1998 Mat-vec products Storage Mat-vec products Eigensolver ICFS An incomplete Cholesky factorization with limited memory Lin, Moré, 1999 PCG Form the condensed problem Solve the sdp eigenvalue problem Case study: the “Voltone”, Piazza della Repubblica, Livorno, Italy The “Voltone” is a large underground vaulted masonry structure (thick 0,43 meters) located beneath Piazza della Repubblica in Livorno, Italy; it extends for a total length of 220 meters and is constituted by two longitudinal walls (thick 2,32 meters) with variable height, on which the vault is placed. The aim of the research is the application of the NOSA-ITACA code to this real case for assessing its static safety and seismic vulnerability; in particular, the modal analysis is used for a preliminary study on the dynamic response of the structure, as recommended by the Italian Technical Regulations. The structure’s geometry was obtained by a 3D digital survey. The finite element mesh is constituted by 43084 thick shell elements and by 45379 nodes. The modal frequencies calculated by NOSA-ITACA mainly involve the vaulted structure, which is much more slender than the lateral walls. The activity is carried-out within the project “Tools for the modelling and assessment of the structural behaviour of ancient construction: the NOSA-ITACA code”, funded by Region of Tuscany, 2011-2013. 3D digital acquisition Mechanical characterization of materials FE mesh K Zoom K ZTKZ