Application of the Finite Element Method to Rotary Wing Aeroelasticity

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method

An Introduction to Zooarchaeology

zooarchaeology is a self-reproducing field taught in many university departments of anthropology or archaeology. As archaeologists have literally taken faunal analysis into their own hands, they have debated how best to use animal remains to study everything from early hominin hunting or scavenging to animal production in ancient market economies. Animal remains from archaeological sites have been used to infer three kinds of information: the age of deposits (chronology); paleoenvironment and paleoecological relations among humans and other species; human choices and actions related to use of animals as food and raw materials. Methods for reconstructing human diet and behavior have undergone the greatest growth over the last four decades, and most of this book addresses the second and third areas. This book deals with what I know best: vertebrate zooarchaeology, and within that, analysis of mammalian bones and teeth

On the application of the Reduced Basis Method to Fluid-Structure Interaction problems

With this thesis the author aims at giving an extensive overview on the application of the Reduced Basis Method to Fluid\u2013Structure Interaction (FSI) problems. The work exposed is divided into three main research directions: the First two methods presented are based on a standard Finite Element discretization of the problem of interest, whereas the third method presented differs from the other two because it is based on an embedded Finite Element discretization. In this way the author wants to show the advantages of pursuing a model order reduction with either a standard Finite Element method or with a Cut Finite Element method, depending on the particular problem of interest: throughout the Chapters it will be shown that a reduction method based on a classical Finite Element discretization is well suited for multiphysics problems where the geometry of the domain does not change significantly; on the contrary, a Cut Finite Element approach shows its full potentiality in situations where, for example, the structure undergoes a large deformation. The algorithms presented in this thesis are: a partitioned (or segregated) Reduced Basis Method that is based on a Chorin\u2013Temam projection scheme with semi\u2013implicit coupling of the solid and the fluid problem, a Reduced Basis Method enriched with a preprocessing of the snapshots during the offline phase, and lastly a Reduced Order Method in a Cut Finite Element framework. According to the approach adopted to adress the particular problem of interest, the thesis proposes a modification and an improvement of the Reduced Basis Method in order to obtain a complete model order reduction procedure. Several test cases are considered throughout the work: a toy problem that describes the deformation of two leaflets under the influence of the jet of a fluid; a Fluid\u2013 Structure Interaction problem whose solution exhibits a transport dominated behaviour, and, in addition, some Computational Fluid Dynamics toy problems, also in the case of parameter dependence. For each one of the test cases considered, first there is an introduction to the problem formulation, and then the proposed model order reduction procedure follows

Variational approach to probabilistic finite elements

Probabilistic finite element method (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties, and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes

An effective multigrid method for high-speed flows

The use is considered of a multigrid method with central differencing to solve the Navier-Stokes equations for high speed flows. The time dependent form of the equations is integrated with a Runge-Kutta scheme accelerated by local time stepping and variable coefficient implicit residual smoothing. Of particular importance are the details of the numerical dissipation formulation, especially the switch between the second and fourth difference terms. Solutions are given for 2-D laminar flow over a circular cylinder and a 15 deg compression ramp

Recovery methods for evolution and nonlinear problems

Functions in finite dimensional spaces are, in general, not smooth enough to be differentiable in the classical sense and “recovered” versions of their first and second derivatives must be sought for certain applications. In this work we make use of recovered derivatives for applications in finite element schemes for two different purposes. We thus split this Thesis into two distinct parts. In the first part we derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error for fully discrete schemes of the linear heat equation. To our knowledge this is the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique introduced as an aposteriori analog to the elliptic (Ritz) projection. Our theoretical results are backed up with extensive numerical experimentation aimed at (1) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (2) deriving an adaptive method based on our estimators. An extra novelty is an implementation of a coarsening error “preindicator”, with a complete implementation guide in ALBERTA (versions 1.0–2.0). In the second part of this Thesis we propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galërkin type using conforming finite elements and applied directly to the nonvariational(or nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of the “finite element Hessian” based on a Hessian recovery and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on linear PDEs in nonvariational form. We then use the nonvariational finite element method to build a numerical method for fully nonlinear elliptic equations. We linearise the problem via Newton’s method resulting in a sequence of nonvariational elliptic problems which are then approximated with the nonvariational finite element method. This method is applicable to general fully nonlinear PDEs who admit a unique solution without constraint. We also study fully nonlinear PDEs when they are only uniformly elliptic on a certain class of functions. We construct a numerical method for the Monge–Ampère equation based on using “finite element convexity” as a constraint for the aforementioned nonvariational finite element method. This method is backed up with numerical experimentation