Free and Forced Vibration Analysis in Abaqus Based on the Polygonal Scaled Boundary Finite Element Method

The polygonal scaled boundary finite element method (PSBFEM) is a novel approach integrating the standard scaled boundary finite element method and the polygonal mesh technique. In this work, a user-defined element (UEL) for dynamic analysis based on the PSBFEM is developed in the general finite element software ABAQUS. We present the main procedures of interacting with Abaqus, updating AMATRX and RHS, defining the UEL element, and solving the stiffness and mass matrices through eigenvalue decomposition. Several benchmark problems of free and forced vibration are solved to validate the proposed implementation. The results show that the PSBFEM is more accurate than the FEM with mesh refinement. Moreover, the PSBFEM avoids the occurrence of hanging nodes by constructing a polygonal mesh. Thus, the PSBFEM can choose an appropriate mesh resolution for different structures ensuring accuracy and reducing calculation costs.

Reconstruction of Phase Space and Eigenvalue Decomposition from a Biological Time Series: A Malayalam Speech Signal Case Study

Our objective is to describe the speech production system from a non-linear physiological system perspective and reconstruct the attractor from the experimental speech data. Mutual information method is utilized to find out the time delay for embedding. The False Nearest Neighbour (FNN) method and Principal Component Analysis (PCA) method are used for optimizing the embedding dimension of time series. The time series obtained from the typical non-linear systems, Lorenz system and Rössler system, is used to standardize the methods and the Malayalam speech vowel time series of both genders of different age groups, sampled at three sampling frequencies (16[Formula: see text]kHz, 32[Formula: see text]kHz, 44.1[Formula: see text]kHz), are taken for analysis. It was observed that time delay varies from sample to sample and, it ought to be better to figure out the time delay with the embedding dimension analysis. The embedding dimension is shown to be independent of gender, age and sampling frequency and can be projected as five. Hence a five-dimensional hyperspace will probably be adequate for reconstructing attractor of speech time series.

Bregman primal–dual first-order method and application to sparse semidefinite programming

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

Subspace Based Adaptive Beamforming Algorithm with Interference Plus Noise Covariance Matrix Reconstruction

As we all know, the model mismatch, primarily when the desired signal exists in the training data, or when the sample data is used for training, will seriously affect algorithm performance. This paper combines the subspace algorithm based on direction of arrival (DOA) estimation with the adaptive beamforming. It proposes a reconstruction algorithm based on the interference plus noise covariance matrix (INCM). Firstly, the eigenvector of the desired signal is obtained according to the eigenvalue decomposition of the subspace algorithm, and the eigenvector is used as the estimated value of the desired signal steering vector (SV). Then the INCM is reconstructed according to the estimated parameters to remove the adverse effect of the desired signal component on the beamformer. Finally, the estimated desired signal SV and the reconstructed INCM are used to calculate the weight. Compared with the previous work, the proposed algorithm not only improves the performance of the adaptive beamformer but also dramatically reduces the complexity. Simulation experiment results show the effectiveness and robustness of the proposed beamforming algorithm.

Time-Domain Sound Field Reproduction with Pressure and Particle Velocity Jointly Controlled

Particle velocity has been introduced to improve the performance of spatial sound field reproduction systems with an irregular loudspeaker array setup. However, existing systems have only been developed in the frequency domain. In this work, we propose a time-domain sound field reproduction method with both sound pressure and particle velocity components jointly controlled. To solve the computational complexity problem associated with the multi-channel setup and the long-length filter design, we adopt the general eigenvalue decomposition-based approach and the conjugate gradient method. The performance of the proposed method is evaluated through numerical simulations with both a regular loudspeaker array layout and an irregular loudspeaker array layout in a room environment.

Low rank approximation of positive semi-definite symmetric matrices using Gaussian elimination and volume sampling

Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve systems with such a matrix can be very costly. A core idea to reduce computational complexity is to approximate the matrix by one with a low rank. The optimal and well understood choice is based on the eigenvalue decomposition of the matrix. Unfortunately, this is computationally very expensive. Cheaper methods are based on Gaussian elimination but they require pivoting. We show how invariant matrix theory provides explicit error formulas for an averaged error based on volume sampling. The formula leads to ratios of elementary symmetric polynomials on the eigenvalues. We discuss several bounds for the expected norm of the approximation error and include examples where this expected error norm can be computed exactly. References A. Dax. “On extremum properties of orthogonal quotients matrices”. In: Lin. Alg. Appl. 432.5 (2010), pp. 1234–1257. doi: 10.1016/j.laa.2009.10.034. M. Dereziński and M. W. Mahoney. Determinantal Point Processes in Randomized Numerical Linear Algebra. 2020. url: https://arxiv.org/abs/2005.03185. A. Deshpande, L. Rademacher, S. Vempala, and G. Wang. “Matrix approximation and projective clustering via volume sampling”. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm. SODA ’06. Miami, Florida: Society for Industrial and Applied Mathematics, 2006, pp. 1117–1126. url: https://dl.acm.org/doi/abs/10.5555/1109557.1109681. S. A. Goreinov, E. E. Tyrtyshnikov, and N. L. Zamarashkin. “A theory of pseudoskeleton approximations”. In: Lin. Alg. Appl. 261.1 (1997), pp. 1–21. doi: 10.1016/S0024-3795(96)00301-1. M. W. Mahoney and P. Drineas. “CUR matrix decompositions for improved data analysis”. In: Proc. Nat. Acad. Sci. 106.3 (Jan. 20, 2009), pp. 697–702. doi: 10.1073/pnas.0803205106. M. Marcus and L. Lopes. “Inequalities for symmetric functions and Hermitian matrices”. In: Can. J. Math. 9 (1957), pp. 305–312. doi: 10.4153/CJM-1957-037-9.