Least-squares finite element approximations to the Timoshenko beam problem

In this paper a least-squares finite element method for the Timoshenko beam problem is proposed and analyzed. The method is shown to be convergent and stable without requiring extra smoothness of the exact solutions. For sufficiently regular exact solutions, the method achieves optimal order of convergence in the H 1 -norm for all the unknowns (displacement, rotation, shear, moment), uniformly in the small parameter which is generally proportional to the ratio of thickness to length. Thus the locking phenomenon disappears as the parameter tends to zero. A sharp a posteriori error estimator which is exact in the energy norm and equivalent in the H 1 -norm is also briefly discussed

A numerical scheme for the identification of hybrid systems describing the vibration of flexible beams with tip bodies

A cubic spline based Galerkin-like method is developed for the identification of a class of hybrid systems which describe the transverse vibration to flexible beams with attached tip bodies. The identification problem is formulated as a least squares fit to data subject to the system dynamics given by a coupled system of ordnary and partial differential equations recast as an abstract evolution equation (AEE) in an appropriate infinite dimensional Hilbert space. Projecting the AEE into spline-based subspaces leads naturally to a sequence of approximating finite dimensional identification problems. The solutions to these problems are shown to exist, are relatively easily computed, and are shown to, in some sense, converge to solutions to the original identification problem. Numerical results for a variety of examples are discussed

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

Studies on knot placement techniques for the geometry construction and the accurate simulation of isogeometric spatial curved beams

The present paper investigates the use of different knot placement techniques for isogeometric analysis of spatial curved beams, to enhance analysis results in cases when geometries are given in terms of data points. Focusing on analysis-aware modeling for structural static and vibration simulations of spatial free-form curved beams, the knot placement techniques based on uniformly spaced knots as well as on De Boor’s and Piegl and Tiller’s algorithms are studied. For this purpose, an isogeometric formulation for linear Euler–Bernoulli beams based on the Euler–Rodriguez transformation rule is implemented. Different case studies and numerical examples are presented and the results are validated against “overkill” solutions computed with a commercial finite element software. The results show that the De Boor’s knot placement algorithm typically leads to better approximation errors and is therefore the suggested strategy for this kind of problems

Methods for the identification of material parameters in distributed models for flexible structures

Theoretical and numerical results are presented for inverse problems involving estimation of spatially varying parameters such as stiffness and damping in distributed models for elastic structures such as Euler-Bernoulli beams. An outline of algorithms used and a summary of computational experiences are presented