256 U. Rauf et al. where K and M are the stiffness and mass matrices of the model, respectively, ˆare the eigenvectors or mode shapes associated with the matrices, and 2 is the eigenvalue matrix, which contains the undamped natural frequencies. The FEMUP aims to leverage experimentally derived estimates of ˆand to update the structural matrices, KandM, under constraints that ensure that the results are physically meaningful. As shown in the equation, the analytical eigenvector and eigenvalue matrices are square matrices, where n is the number of degrees of freedom in the model. However, it is often logistically unfeasible or impossible to experimentally identify more than a small subset of the total number of mode shapes and corresponding natural frequencies. Furthermore, it is also not practical to install a large enough array of sensors to measure all corresponding degrees of freedom in the analytical model. These limitations have significant practical consequences in the application of FEMUP, as the governing eigenvalue equation is reduced to: Kn n " ˆs m ˆ .n s/ m# DMn n " ˆs m ˆ .n s/ m# 2 m m (27.2) wheres is the number of measured degrees of freedom or sensors andmis the number of experimentally reconstructed mode shapes. In this statement of the eigenvalue problem, the ˆ terms indicate the unmeasured components of the eigenvectors, which can represent a significant number of additional unknowns, particularly if the number of degrees of freedom in the analytical model is much larger than the number of sensors used in the experimental test. To ensure physically meaningful results, FEMUP approaches generally prescribe the connectivity structure of the mass and stiffness matrices by assembling them by parameterization of a finite element model. In this way, the matrices are formed by elemental contributions that satisfy meaningful structural constraints: 2 4 pX jD1 ˛jK j n n3 5 " ˆs m ˆ .n s/ m# D2 4 pX jD1 ˇjMj n n3 5 " ˆs m ˆ .n s/ m# 2 m m (27.3) where ˛ and ˇ are unknown scalars that are used to identify structural stiffness or mass of elements or groups of elements in the model that are uncertain or likely to be affected by structural damage. In this parameterized formulation, Kj is the stiffness contribution of the jth element normalized to ˛j and Mj is the mass contribution of the jth element normalized toˇj. The resulting form of the inverse eigenvalue problem presents a challenging nonlinear equation that cannot be directly solved. Numerous optimization schemes, including gradient-based methods [6], simulated annealing [7], genetic algorithms [8], and many other computational intelligence techniques [9], have been explored for producing solutions to the FEMUP problem. However, significant challenges remain before structural identification can serve as a reliable and attractive alternative to data-driven approaches to structural health monitoring. Predominantly, the challenges are associated with computational speed and the generation and evaluation of multiple or alternative solutions in the presence of measurement noise. Additional issues related to effective parameterization of the model for damage detection and uncorrected discretization and idealization errors inherent to the finite element model are also persistent challenges. In this paper, we first explore a novel approach for structural identification based upon formalizing a set of nonlinear constraint equations from the governing eigenvalue problem and using the paradigm of Constraint Logic Programming (CLP) to solve for uncertain parameters in the structural model. By casting the FEMUP as a constraint satisfaction problem rather than an optimization problem, we seek to explore a potential paradigm shift in the structural identification method that may lead to an effective means of generating multiple or alternative solutions in the presence of measurement noise. An overview of the basic ideas essential to our approach to structural identification by nonlinear constraint satisfaction programming is provided followed by a preliminary application of the proposed methodology on a numerical model of a truss. 27.2 Methodology In the domain of engineering sciences, many applications require finding all possible and isolated solutions satisfiable to a set of constraints over real numbers. The system may be non polynomial and the computational complexity to inherently solve such systems is NP-hard. These set of problems are called Constraint Satisfaction Problems (CSPs) [10–12]. Our aim is to model the FEMUP problem as a CSP and later to intersect mechanics-based constraints with advanced computer science
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