Nonlinear Vibrations of 3D Laminated Composite Beams

A model for 3D laminated composite beams, that is, beams that can vibrate in space and experience longitudinal and torsional deformations, is derived. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body but can deform longitudinally due to warping. The warping function, which is essential for correct torsional deformations, is computed preliminarily by the finite element method. Geometrical nonlinearity is taken into account by considering Green’s strain tensor. The equation of motion is derived by the principle of virtual work and discretized by the p-version finite element method. The laminates are assumed to be of orthotropic materials. The influence of the angle of orientation of the laminates on the natural frequencies and on the nonlinear modes of vibration is presented. It is shown that, due to asymmetric laminates, there exist bending-longitudinal and bending-torsional coupling in linear analysis. Dynamic responses in time domain are presented and couplings between transverse displacements and torsion are investigated

Two-level algorithms for Rannacher-Turek FEM

In this paper a multiplicative two-level preconditioning algorithm for second order elliptic boundary value problems is considered, where the discretization is done using Rannacher-Turek non-conforming rotated bilinear finite elements on quadrilaterals. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners originally introduced for conforming finite elements. The proposed variant of hierarchical two-level splitting is first defined in a rather general setting. Then, the involved parameters are studied and optimized. The major contribution of the paper is the derived uniform estimates of the constant in the strengthened CBS inequality which allow the efficient multilevel extension of the related two-level preconditioners

An optimal order multilevel preconditioner with respect to problem and discretization parameters

Contains fulltext : 18885.pdf ( ) (Open Access)Report No. 001518 p

Finite volume discretization of equations describing nonlinear diffusion in Li-Ion batteries

Numerical modeling of electrochemical process in Li-Ion battery is an emerging topic of great practical interest. In this work we present a Finite Volume discretization of electrochemical diffusive processes occurring during the operation of Li-Ion batteries. The system of equations is a nonlinear, time-dependent diffusive system, coupling the Li concentration and the electric potential. The system is formulated at length-scale at which two different types of domains are distinguished, one for the electrolyte and one for the active solid particles in the electrode. The domains can be of highly irregular shape, with electrolyte occupying the pore space of a porous electrode. The material parameters in each domain differ by several orders of magnitude and can be non-linear functions of Li ions concentration and/or the electrical potential. Moreover, special interface conditions are imposed at the boundary separating the electrolyte from the active solid particles. The field variables are discontinuous across such an interface and the coupling is highly non- linear, rendering direct iteration methods ineffective for such problems. We formulate a Newton iteration for an purely implicit Finite Volume discretization of the coupled system. A series of numerical examples are presented for different type of electrolyte/electrode configurations and material parameters. The convergence of the Newton method is characterized both as function of nonlinear material parameters as well as the nonlinearity in the interface conditions

Preconditioning of voxel FEM elliptic systems

The presented comparative analysis concerns two iterative solvers for large-scale linear systems related to žFEM simulation of human bones. The considered scalar elliptic problems represent the strongly heterogeneous structure of real bone specimens. The voxel data are obtained with high resolution computer tomography. Non-conforming Rannacher-Turek finite elements are used to discretize of the considered elliptic problem. The preconditioned conjugate gradient method is known to be the best tool for efficient solution of large-scale symmetric systems with sparse positive definite matrices. Here, the performance of two preconditioners is studied, namely modified incomplete Cholesky factorization, MIC(0), and algebraic multigrid. The comparative analysis is mostly based on the computing times to run the sequential codes. The number of iterations for both preconditioners is also discussed. Finally, numerical tests of a novel parallel MIC(0) code are presented. The obtained parallel speed-ups and efficiencies illustrate the scope of efficient applications for real-life large-scale problems

CBS constants for graph-Laplacians and application to multilevel methods for discontinuous Galerkin systems

Abstract. The goal of this work is to derive and justify a multilevel preconditioner for symmetric discontinuous approximations of second order elliptic problems. Our approach is based on the following simple idea. The finite element space V of piece-wise polynomials of certain degree that are discontinuous on the partition T0 is projected onto the space of piece-wise constants on the same partition. This will constitute the finest space in the multilevel method. The projection of the discontinuous Galerkin system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a very simple M-matrix with −1 as off diagonal entries and current diagonal entries equal to the number of the neighbours through the interfaces of the current finite element. Then after consecutive aggregation of the finite elements we produce a sequence of spaces of piece-wise constant functions. We develop the concept of hierarchical splitting of the unknowns and using local analysis we derive uniform estimates for the constant in the strengthen Cauchy-Bunyakowski-Schwarz (CBS) inequality. As a measure of the angle between the spaces of the splitting, this further is used to justify a multilevel preconditioner of the discontinuous Galerkin system in spirit of the work [4] of Axelsson and Vassilevski. key words: discontinuous Galerkin, second order elliptic equation, graph-Laplacian, multilevel preconditioning, CBS constant (1.1) 1