We consider two finite element model updating problems, which incorporate the measured modal data into the analytical finite element model, producing an adjusted model on the (mass) damping and stiffness, that closely matches the experimental modal data. We develop two efficient numerical algorithms for solving these problems.The new algorithms are direct methods that require O O(nk 2 ) and O O(nk 2 + k 6 ) flops, respectively, and employ sparse matrix techniques when the analytic model is sparse. Here n is the dimension of the coefficient matrices defining the analytical model, and k is the number of measured eigenpairs.
Nomenclature= adjusted stiffness matrix, n × n K a = stiffness matrix of the original FEM model, n × n k = number of measured eigenvalues or eigenvectors M = adjusted mass matrix, n × n M a = mass matrix of the original FEM model, n × n Q = orthogonal matrix, n × n R = upper triangular matrix, k × k= trace operator vec(·) = vectorization operator x = solution of linear system, k × 1 j = see Eq. (18); k × k δ = see Eq. (41) = diagonal block eigenvalue matrix, k × k λ = eigenvalues of quadratic eigenvalue problem μ = weight factor ν = weight factor = eigenvector matrix, n × k
Finite element model correction of quadratic eigenvalue problems (QEPs) using a symmetric eigenstructure assignment technique is proposed by Zimmerman and Widengren 1989, which incorporates the measured model data into the finite element model to produce an adjusted finite element model on the damping and stiffness matrices that matches the experimental model data, and minimizes the distance between the analytical and corrected models. In this paper, we mainly develop an efficient algorithm to solve the corresponding optimization problem in a least-squares sense. The resulting matrices obtained by the new method are necessary and sufficient to the optimization problem. Furthermore, the proposed algorithm only needs to solve a linear system and totally requires O(nm 2 ) flops, where n is the size of coefficient matrices of the QEP and m is the number of the measured modes. The numerical results show that the new method is reliable and attractive.
In this paper, we analyze phase separation of multi-component Bose-Einstein condensates (BECs) in the presence of strong optical lattices. This paper is in threefold. We first prove that when the inter-component scattering lengths go to infinity, phase separation of a multi-component BEC occurs. Furthermore, particles repel each other and form segregated nodal domains. Secondly, we show that the union of these segregated nodal domains equal to the entire domain. Thirdly, we show that if the intra-component scattering lengths are bounded by some finite number, each nodal domain is connected. For large intracomponent scattering lengths, however, the third result is not true and a counter example of non-connected nodal domains is given.
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