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 (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. [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.

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
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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.
www.nosaitaca.it
3D digital acquisition
Mechanical
characterization of
materials
FE mesh
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ZTKZ