Force Density Term Research Articles

Concept Design of a Novel Superconducting PTO Actuator for Wave Energy Extraction

The role of the Power Take-Off (PTO) as part of modern Wave Energy Converters is becoming more and more relevant and many efforts have been done or are ongoing to improve its performance especially in terms of force density and efficiency. Electric Linear PTOs are inherently the most efficient category, since they are really direct drives with no intermediate stages of energy conversion. Nevertheless, conventional electric machines are usually limited in force while their efficiency is better than other type of drives but still does not allow an intense energy capture in a broad band of wave periods. In this regard, superconductivity may become a very helpful alternative that allows improving both: efficiency and force density in spite of the technological difficulties that are introduced in the Wave Energy Converter. This paper, after justifying the need for better PTO performances, presents a new concept of superconducting PTO in which both, the stator and the translator work at cold temperature, performing a reciprocating displacement inside a flexible cryostat. The concept is later applied to a Cylindrical Switched Reluctance machine whose global design is also presented in the paper. This activity has been performed as one of the work packages of the EU H2020 Sea Titan Project in which also a resistive PTO with a novel configuration has been developed.

Closed-Form Solutions for the Form-Finding of Regular Tensegrity Structures by Group Elements

An analytical form-finding method for regular tensegrity structures based on the concept of force density is presented. The self-equilibrated state can be deduced linearly in terms of force densities, and then we apply eigenvalue decomposition to the force density matrix to calculate its eigenvalues. The eigenvalues are enforced to satisfy the non-degeneracy condition to fulfill the self-equilibrium condition. So the relationship between force densities can also be obtained, which is followed by the super-stability examination. The method has been developed to deal with planar tensegrity structure, prismatic tensegrity structure (triangular prism, quadrangular prism, and pentagonal prism) and star-shaped tensegrity structure by group elements to get closed-form solutions in terms of force densities, which satisfies the super stable conditions.

Hybrid modeling of flows over submerged prismatic vegetation with different areal densities

In the modeling of vegetated flows an efficient approach is to solve the Double Averaged Navier Stokes (DANS) equations, which are obtained from the spatial and temporal averaging of the Navier Stokes equations. The resistance effects of vegetation are modeled by a drag force density term with an empirical bulk drag coefficient, as well as by turbulence terms characterized by a length scale parameter. These empirical parameters are dependent on the areal density of vegetation and the spatial distribution pattern of vegetation elements and have seldom been studied. In this work the effect of spatial distribution of vegetation elements on the bulk drag coefficient is investigated by computing explicitly the flows around the vegetation elements. The results show that the bulk drag coefficient increases with the longitudinal vegetation element spacing, decreases with the lateral vegetation element spacing and can have multiple values for a given areal density of vegetation. In the DANS model the vegetation induced turbulence is simulated by a novel k-ε type model embracing an empirical length scale parameter. The length scale parameters are calibrated against previous experiments and a new set of experiments with high areal density of vegetation. The DANS model is subsequently verified by two cases with the same vegetation density and different distribution patterns.

Self-equilibrium and super-stability of truncated regular hexahedral and octahedral tensegrity structures

This paper presents conditions for self-equilibrium as well as super-stability of the truncated regular hexahedral and octahedral tensegrity structures. Their symmetry can be described by octahedral group in group representation theory, and furthermore, their force density matrix is analytically rewritten in the symmetry-adapted form. The condition for self-equilibrium, in terms of force densities, is found by satisfying the non-degeneracy condition for a tensegrity structure. The condition for super-stability, also in terms of force densities, is further presented by guaranteeing positive semi-definiteness of the force density matrix.

Form-finding of tensegrity structures based on the Levenberg–Marquardt method

This paper presents a novel and versatile form-finding method for tensegrity structures that is based on solving nonlinear equilibrium equations. Linear equilibrium equations in terms of force densities are transformed into nonlinear equilibrium equations in which the nodal coordinates vector is variable. The input parameters for the form-finding method are the topology, initial configuration of the structure, rest lengths, and axial stiffness of elements. The form-finding process is performed by solving nonlinear least-squares problems converted from nonlinear equilibrium equations, and the Levenberg–Marquardt method is used to deal with the singularity of the stiffness matrix. Several numerical examples are given to demonstrate the accuracy and efficacy of the proposed method.

Particle Sedimentation in a Constricted Passage Using a Flexible Forcing IB-LBM Scheme

A novel flexible forcing hybrid immersed boundary-lattice Boltzmann model (IB-LBM) is introduced in the present paper for solving moving boundary problems. In conventional IB-LBM schemes, explicit formulations of force density term may not ensure no-slip boundary condition accurately, which leads to inaccurate force and torque calculations for moving object. Following an implicit force density calculation, a single Lagrangian velocity correction term is advised in this study. The formula for this correction term is much simpler and with the help of flexible number of sub-iteration/forcing, the computational time is significantly saved. The no-slip boundary condition is achieved accurately within a convergence limit. The proposed algorithm shows advantages for unsteady and moving boundary problem, where boundary convergence is satisfied consistently at every time step. In particular, a 2D particulate flow case is simulated in a constricted channel. Interesting observations and results are discussed in this article.

A NOVEL FLEXIBLE FORCING HYBRID IB-LBM SCHEME TO SIMULATE FLOW PAST CIRCULAR CYLINDER

A novel flexible forcing immersed boundary and lattice Boltzmann method (IB-LBM) is introduced in this paper. In the conventional IB-LBM scheme, explicit calculation of the force density term may not guarantee exact satisfaction of the no-slip boundary condition. This produces unphysical streamline penetration into the solid object. In this study, an implicit approach is followed where the force density is obtained from the unknown velocity correction with a unique single Lagrangian velocity correction term and flexible number of forcing steps. The proposed algorithm is computationally efficient and easy for new code development. Also the algorithm is found advantageous for unsteady flow case where no-slip condition is accurately satisfied. Flow past circular cylinder is studied for both steady and unsteady flow cases to validate the proposed scheme.

Self-equilibrium and stability of regular truncated tetrahedral tensegrity structures

This paper presents analytical conditions of self-equilibrium and super-stability for the regular truncated tetrahedral tensegrity structures, nodes of which have one-to-one correspondence to the tetrahedral group. These conditions are presented in terms of force densities, by investigating the block-diagonalized force density matrix. The block-diagonalized force density matrix, with independent sub-matrices lying on its leading diagonal, is derived by making use of the tetrahedral symmetry via group representation theory. The condition for self-equilibrium is found by enforcing the force density matrix to have the necessary number of nullities, which is four for three-dimensional structures. The condition for super-stability is further presented by guaranteeing positive semi-definiteness of the force density matrix.

Review of Form-Finding Methods for Tensegrity Structures

Seven form-finding methods for tensegrity structures are reviewed and classified. The three kinematical methods include an analytical approach, a non-linear optimisation, and a pseudo-dynamic iteration. The four statical methods include an analytical method, the formulation of linear equations of equilibrium in terms of force densities, an energy minimisation, and a search for the equilibrium configurations of the struts of the structure connected by cables whose lengths are to be determined, using a reduced set of equilibrium equations. It is concluded that the kinematical methods are best suited to obtaining only configuration details of structures that are already essentially known. The force density method is best suited to searching for new configurations, but affords no control over the lengths of the elements of the structure. The reduced coordinates method offers a greater control on elements lengths, but requires more extensive symbolic manipulations.

Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

AbstractA variant of immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this paper to simulate incompressible viscous flows around moving objects. As compared with the conventional IB‐LBM where the force density is computed explicitly by Hook's law or the direct forcing method and the non‐slip condition is only approximately satisfied, in the present work, the force density term is considered as the velocity correction which is determined by enforcing the non‐slip condition at the boundary. The lift and drag forces on the moving object can be easily calculated via the velocity correction on the boundary points. The capability of the present method for moving objects is well demonstrated through its application to simulate flows around a moving circular cylinder, a rotationally oscillating cylinder, and an elliptic flapping wing. Furthermore, the simulation of flows around a flapping flexible airfoil is carried out to exhibit the ability of the present method for implementing the elastic boundary condition. It was found that under certain conditions, the flapping flexible airfoil can generate larger propulsive force than the flapping rigid airfoil. Copyright © 2009 John Wiley & Sons, Ltd.

Numerical simulation of two-dimensional complex flows around bluff bodies using the immersed boundary method

This paper presents a two-dimensional numerical simulation of flows around different bluff bodies, at Re = 100 and 200, using the Immersed Boundary (IB) method, as a sequence of a previous work. The force density term required by the IB method is obtained with the Virtual Physical Model (VPM). Simulations were carried out for two circular cylinders of different diameter in tandem, two cylinders of the same diameter in tandem and two cylinders placed in side by side arrangement. The configurations of seven cylinders in a 'V' arrangement, for angles of 40o < a < 180o, were also simulated. A configuration of 23 different bluff bodies, representing a transverse cut in a central tower of an offshore structure, has been also simulated and the results were compared with a single compact square, of equivalent size. The Strouhal number, the drag and the lift coefficients were also calculated. The Strouhal number is calculated using the Fast Fourier Transform (FFT) of the lift coefficient temporal distribution. Visualization of the vorticity and pressure fields and the streamlines are presented for each simulation showing the flow dynamics and patterns. It was possible to verify that the IB method with VPM is a powerful methodology to simulate flows in the presence of complex geometries.

The immersed boundary-lattice Boltzmann method for solving fluid–particles interaction problems

A new computational method, the immersed boundary-lattice Boltzmann method, is presented. This method is a combination and utilizes the most desirable features of the lattice Boltzmann and the immersed boundary methods. The method uses a regular Eulerian grid for the flow domain and a Lagrangian grid to follow particles that are contained in the flow field. The rigid body conditions for the fluid and the particles are enforced by a penalty method, which assumes that the particle boundary is deformable with a high stiffness constant. The velocity field of the fluid and particles is solved by adding a force density term into the lattice Boltzmann equation. This novel method preserves the advantages of LBM in tracking a group of particles and, at the same time, provides an alternative and better approach to treating the solid–fluid boundary conditions. The method also solves the problems of fluctuation of the forces and velocities on the particles when the “bounce-back” boundary conditions are applied. This method enables one to simulate problems with particle deformation and fluid–structure deformation. Its results are validated by comparison with results from other methods.

Review of Form-Finding Methods for Tensegrity Structures

Seven form-finding methods for tensegrity structures are reviewed and classified. The three kinematical methods include an analytical approach, a non-linear optimisation, and a pseudo-dynamic iteration. The four statical methods include an analytical method, the formulation of linear equations of equilibrium in terms of force densities, an energy minimisation, and a search for the equilibrium configurations of the struts of the structure connected by cables whose lengths are to be determined, using a reduced set of equilibrium equations. It is concluded that the kinematical methods are best suited to obtaining only configuration details of structures that are already essentially known. The force density method is best suited to searching for new configurations, but affords no control over the lengths of the elements of the structure. The reduced coordinates method offers a greater control on elements lengths, but requires more extensive symbolic manipulations.

Numerical simulations of xylophones. I. Time-domain modeling of the vibrating bars

A time-domain modeling of xylophone bars excited by the blow of a mallet is presented. The flexural vibrations of the bar, with nonuniform cross section, are modeled by a one-dimensional Euler–Bernoulli equation, modified by the addition of two damping terms for the modeling of losses and a restoring force for the modeling of the stiffness of the suspending cord. The action of the mallet against the bar is described by Hertz’s law of contact for linear elastic bodies. This action appears as a force density term on the right-hand side of the bending wave equation. The model is completed by the equation of motion for the mallet, and by free–free boundary conditions for the bar. The bending wave equation of the bar is put into a numerical form by means of an implicit finite-difference scheme, which ensures a sufficient spatial resolution for an accurate tuning of the bar. The geometrical, elastic, and damping parameters of the model are derived from experiments carried out on actual xylophones and mallets. The validity of the numerical model is confirmed by three different procedures: First, a comparison is made between numerical results and analytical solutions. The second series of tests consist of examining the effects of the number of spatial steps on the convergence of the solution. Finally, various comparisons are made between measured and simulated impact forces and bar accelerations. The present model reproduces adequately the main features of a real instrument. The most significant physical parameters of bars, mallets and players’ actions can be controlled independently for producing a remarkable variety of tones.

A numerical method for suspension flow

Peskin's (J. Comput. Phys.25, 220, 1977) immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles. As before, the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations. As a result, one set of equations holds in the entire computational domain, eliminating the need to apply boundary conditions on the surface of suspended objects. The new method computes the force density by discretizing the stress-strain constitutive equations for an elastic solid on a grid, using data provided by clusters of Lagrangian points. This approach clearly specifies the material properties of the suspended objects. A simple data structure for the Lagrangian points makes it easy to model suspended solids with arbitrary shape and size. The method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids, with theory and analysis. The capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems.

A discrete model for synthesizing guitar sounds

In an attempt to simulate guitar sounds, a discrete model was developed that consisted of real strings with stiffness and internal damping, the player's finger, the resonance box, and the modelization of the sound pressure. The vibrating string equation was solved in the time domain using the finite‐difference method. The player's figure was modeled as a force density term in the string equation, which accounted for the initial plucking conditions. The resonance box was modeled as impedancelike boundary conditions at one end of the string. These end conditions were obtained under the assumption that the driving force exerted by the string at its ends excites a set of second‐order resonances in parallel, the first resonance of the plate being coupled with the air resonance of the box. Finally, the sound pressure was modeled as the contribution of a set of monopoles, where each monopole corresponds to one given resonance. The radiation efficiency of each monopole was derived from a curve fitting on a guitar sound‐pressure spectrum measured in an anechoic chamber. The waveforms obtained are in good agreement with experimental waveforms measured on real guitars. Sound examples will be presented high‐lighting the realism of the simulation.