This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Recent progress in flapping wing aerodynamics and aeroelasticity

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.

A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry

Scattered Data Approximation

An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries

In the present study, a computational investigation was carried out to understand the influence of flexibility on the aerodynamic performance of a hovering wing. A flexible, two-dimensional, two-link model moving within a viscous fluid was considered. The Navier-Stokes equations governing the fluid dynamics were solved together with the equations governing the structural dynamics by using a strongly coupled fluid-structure interaction scheme. Harmonic kinematics was used to prescribe the motions of one of the links, thus effectively reducing the wing to a single degree-of-freedom oscillator. The wing's flexibility was characterized by the ratio of the flapping frequency to the natural frequency of the structure. Apart from the rigid case, different values of this frequency ratio (only in the range of 1/2 to 1/6) were considered at the Reynolds numbers of 75, 250 and 1000. It was found that flexibility can enhance aerodynamic performance and that the best performance is realized when the wing is excited by a non-linear resonance at 1/3 of the natural frequency. Specifically, at Reynolds numbers of 75, 250 and 1000, the aerodynamic performance that is characterized by the ratio of lift coefficient to drag coefficient is respectively increased by 28%, 23% and 21% when compared with the corresponding ratios of a rigid wing driven with the same kinematics. For all Reynolds numbers, the lift generated per unit driving power is also enhanced in a similar manner. The wake capture mechanism is enhanced, due to a stronger flow around the wing at stroke reversal, resulting from a stronger end of stroke vortex at the trailing edge. The present study provides some clues about how flexibility affects the aerodynamic performance in low Reynolds number flapping flight. In addition, it points to the importance of considering non-linear resonances for enhancing aerodynamic performance.

/pdf/influence-of-flexibility-on-the-aerodynamic-performance-of-a-14uxidf285.pdf

Influence of flexibility on the aerodynamic performance of a hovering wing.

A strongly coupled, embedded-boundary method for fluid–structure interactions of elastically mounted rigid bodies

Free and forced oscillations of piezoelectric, microelectromechanical resonators fabricated as clamped?clamped composite structures are studied in this effort. Piezoelectric actuation is used to excite these structures on the input side and piezoelectric sensing is carried out on the output side. A refined integro-partial differential model is developed for a clamped?clamped composite beam structure and used for studying the nonlinear transverse vibrations of these resonators. This model accounts for the longitudinal extension due to transverse vibrations, distributed actuation and axially varying properties across the length of the structure. Free oscillations about a post-buckled position are studied, and for weak damping and weak forcing, the method of multiple scales is used to obtain an approximate solution for the response to a harmonic forcing. Analytical predictions are also compared with experimental observations. The model development and the analysis can serve as a basis for analysing the responses of other composite microresonators.

http://iopscience.iop.org/article/10.1088/0960-1317/16/2/021/pdf

Nonlinear free and forced oscillations of piezoelectric microresonators

Numerical Simulations of Interactions Among Aerodynamics, Structural Dynamics, and Control Systems

A new strategy, based on the nonlinear phenomenon of saturation, is proposed for controlling the flutter of a wing. The concept is illustrated by means of an example with a rather flexible, high-aspect wing of the type found on such vehicles as high-altitude long-endurance aircraft and sailplanes. The wing is modeled structurally as an Euler-Bernoulli beam with coupled bending and twisting motions. A general unsteady nonlinear vortex-lattice technique is used to model the flow around the wing and provide the aerodynamic loads. The structure, the flowing air, and the controller are considered the elements of a single dynamic system, and all of the coupled equations of motion are simultaneously and interactively integrated numerically in the time domain. The results indicate that the aerodynamic nonlinearities alone can be responsible for limit-cycle oscillations and that the saturation controller can effectively suppress the flutter oscillations of the wing when the controller frequency is actively tuned.

Sergio Preidikman

Papers

Influence of flexibility on the aerodynamic performance of a hovering wing.

A strongly coupled, embedded-boundary method for fluid–structure interactions of elastically mounted rigid bodies

Nonlinear free and forced oscillations of piezoelectric microresonators

Numerical Simulations of Interactions Among Aerodynamics, Structural Dynamics, and Control Systems

Novel Strategy for Suppressing the Flutter Oscillations of Aircraft Wings