Dynamic Finite Element Modelling and Vibration Analysis of Prestressed Layered Bending–Torsion Coupled Beams

Free vibration analysis of prestressed, homogenous, Fiber-Metal Laminated (FML) and composite beams subjected to axial force and end moment is revisited. Finite Element Method (FEM) and frequency-dependent Dynamic Finite Element (DFE) models are developed and presented. The frequency results are compared with those obtained from the conventional FEM (ANSYS, Canonsburg, PA, USA) as well as the Homogenization Method (HM). Unlike the FEM, the application of the DFE formulation leads to a nonlinear eigenvalue problem, which is solved to determine the system’s natural frequencies and modes. The governing differential equations of coupled flexural–torsional vibrations, resulting from the end moment, are developed using Euler–Bernoulli bending and St. Venant torsion beam theories and assuming linear harmonic motion and linearly elastic materials. Illustrative examples of prestressed layered, FML, and unidirectional composite beam configurations, exhibiting geometric bending-torsion coupling, are studied. The presented DFE and FEM results show excellent agreement with the homogenization method and ANSYS modeling results, with the DFE’s rates of convergence surpassing all. An investigation is also carried out to examine the effects of various combined axial loads and end moments on the stiffness and fundamental frequencies of the structure. An illustrative example, demonstrating the application of the presented methods to the buckling analysis of layered beams is also presented.

Berry–Esseen Bounds of the Quasi Maximum Likelihood Estimators for the Discretely Observed Diffusions

For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, this paper obtains the Berry–Esseen bounds on the rates of convergence to normality of the distributions of the quasi maximum likelihood estimators based on stochastic Taylor approximation, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval, the high-frequency regime. It shows that the higher-order stochastic Taylor approximation-based estimators perform better than the basic Euler approximation in the sense of having smaller asymptotic variance.

Variational eigenvalue approximation of non-coercive operators with application to mixed formulations in elasticity

We present an abstract framework for the eigenvalue approximation of a class of non-coercive operators. We provide sufficient conditions to guarantee the spectral correctness of the Galerkin scheme and to obtain optimal rates of convergence. The theory is applied to the convergence analysis of mixed finite element approximations of the elasticity and Stokes eigensystems.

A stabilizer free spatial weak Galerkin finite element methods for time-dependent convection-diffusion equations

In this paper, we propose a stabilizer free spatial weak Galerkin (SFSWG) finite element method for solving time-dependent convection diffusion equations based on weak form Eq. (4). SFSWG method in spatial direction and Euler difference operator Eq. (37) in temporal direction are used. The main reason for using the SFSWG method is because of its simple formulation that makes this algorithm more efficient and its implementation easier. The optimal rates of convergence of 𝒪⁢(hk) and 𝒪⁢(hk+1) in a discrete H1 and L2 norms, respectively, are obtained under certain conditions if polynomial spaces (Pk⁢(K),Pk⁢(e),[Pj⁢(K)]2) are used in the SFSWG finite element method. Numerical experiments are performed to verify the effectiveness and accuracy of the SFSWG method.

Penalized wavelet estimation and robust denoising for irregular spaced data

Nonparametric univariate regression via wavelets is usually implemented under the assumptions of dyadic sample size, equally spaced fixed sample points, and i.i.d. normal errors. In this work, we propose, study and compare some wavelet based nonparametric estimation methods designed to recover a one-dimensional regression function for data that not necessary possess the above requirements. These methods use appropriate regularizations by penalizing the decomposition of the unknown regression function on a wavelet basis of functions evaluated on the sampling design. Exploiting the sparsity of wavelet decompositions for signals belonging to homogeneous Besov spaces, we use some efficient proximal gradient descent algorithms, available in recent literature, for computing the estimates with fast computation times. Our wavelet based procedures, in both the standard and the robust regression case have favorable theoretical properties, thanks in large part to the separability nature of the (non convex) regularization they are based on. We establish asymptotic global optimal rates of convergence under weak conditions. It is known that such rates are, in general, unattainable by smoothing splines or other linear nonparametric smoothers. Lastly, we present several experiments to examine the empirical performance of our procedures and their comparisons with other proposals available in the literature. An interesting regression analysis of some real data applications using these procedures unambiguously demonstrate their effectiveness.