A rheological interface model and its space–time finite element formulation for fluid–structure interaction

This contribution discusses extended physical interface models for fluid–structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical simulation. Besides the various types of friction at the fluid–structure interface the most interesting phenomena are related to effects due to additional interface stiffness and damping. The paper introduces extended models at the fluid–structure interface on the basis of rheological devices (Hooke, Newton, Kelvin, Maxwell, Zener). The interface is decomposed into a Lagrangian layer for the solid‐like part and an Eulerian layer for the fluid‐like part. The mechanical model for fluid–structure interaction is based on the equations of rigid body dynamics for the structural part and the incompressible Navier–Stokes equations for viscous flow. The resulting weighted residual form uses the interface velocity and interface tractions in both layers in addition to the field variables for fluid and structure. The weak formulation of the whole coupled system is discretized using space–time finite elements with a discontinuous Galerkin method for time‐integration leading to a monolithic algebraic system. The deforming fluid domain is taken into account by deformable space–time finite elements and a pseudo‐structure approach for mesh motion. The sensitivity of coupled systems to modification of the interface model and its parameters is investigated by numerical simulation of flow induced vibrations of a spring supported fluid‐immersed cylinder. It is shown that the presented rheological interface model allows to influence flow‐induced vibrations. Copyright © 2010 John Wiley & Sons, Ltd.     

高温下混凝土中热-湿-气-力学耦合过程数值模拟

对高温下混凝土中热-湿-气-力学耦合过程分析提出了一个多孔多相介质的非混溶-混溶两级数学模型。数学模型基于控制干空气、湿份及基质溶解物的质量守恒、混凝土介质混合体的动量守恒和焓(能量)守恒的耦合偏微分方程组。给出了模型的控制方程、状态方程与所采用的本构定律。发展了相应的非线性耦合问题的有限元数值分析过程,以数值模拟火灾和热辐射等热荷载作用下的热-湿-气-力学耦合行为,并进而分析所发生的破坏现象。数值结果例题显示所发展的数学模型和数值方法在重现高温下混凝土中热-湿-气-力学耦合行为的有效性。     

Mathematical modeling of sway, roll and yaw motions: order-wise analysis to determine coupled characteristics and numerical simulation for restoring moment’s sensitivity analysis

This paper investigates sway, roll and yaw motions of a floating body with the aim to determine coupled motion characteristics based on the order-wise analysis of hydrodynamic coefficients. To compute the hydrodynamic coefficients and wave force exerted on the floating body, we employ speed-dependent strip theory. The governing equations are solved analytically for linear restoring moment. For nonlinear restoring moment which is expressed as an odd-order polynomial of fifth degree in roll angle, we apply the Runge–Kutta–Gill method to solve the coupled equations. To investigate the effect of initial disturbances on sway, roll, yaw and speed of the body, numerical experiments have been carried out for a Panamax Container ship under the action of a sinusoidal wave of periodicity 11.2 s with varying wave height and speed. For the linear restoring moment, we first derive associated motion equations for various cases based on the relative magnitude of the hydrodynamic coefficients. The order-wise analysis leads to the classification of coupled characteristics exhibiting the nature of coupling. For the nonlinear restoring moment, we notice that the initial disturbance plays an important role in the ship’s stability. The effects of forward speed and variation in wave heights are illustrated through typical numerical experiments.     

Three‐dimensional coupled heat,moisture, and air transfer in a deformable unsaturated soil

A three‐dimensional numerical model is presented for three‐phase flow (moisture, air, and heat) in a deformable partly saturated soil with deformation calculated via a non‐linear elastic theory. The present work is an extension of a two‐dimensional analysis presented by Thomas and He. The objective of this work is the solution of problems of greater geometric complexity. The mathematical formulation of this coupled problem consists of four governing equations, developed from the principles of mass and energy conservations as well as the stress equilibrium equation. Darcy's flow law is used to describe the motion of liquid and air in the porous medium, and a Philip and de Vries type vapour flow approach is employed in the formulation. A Galerkin finite element method coupled with a finite difference recurrence relationship is used to obtain simultaneous solutions to the governing equations where pore liquid, pore air pressures, temperature and displacements are the primary variables. The method allows the non‐linear nature of the soil parameters to be modelled. Three‐dimensional 20‐noded isoparametric elements are used to simulate different types of cases for the verification of the work. Results are presented of the application of the new model to four problems, two of which are isothermal and two heating simulations. The three‐dimensional nature of the results achieved is highlighted. Copyright © 1999 John Wiley & Sons, Ltd.     

Nonlinear free vibrations of thin-walled beams in torsion

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional–axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped–clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin’s method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the modulation of the amplitudes and phases of the vibration modes. The obtained results are compared with the available results in the literature that are obtained from boundary element and finite element methods, which reveals an excellent agreement between different solution methodologies. Finally, the internal resonance and the stability of coupled and uncoupled nonlinear modes are investigated. This study can be a preliminary step in the understanding of complex dynamics of such systems in internal resonance excited by external resonant excitations.     

Dusty Plasma Studies in the Gaseous Electronics Conference Reference Cell

Particle “dust” in processing plasmas is of critical concern to the semiconductor industry because of the threat particles pose to device yield. A number of important investigations into the formation, growth, charging, transport and consequences of particulate dust in plasmas have been made using the Gaseous Electronics Conference Reference Cell as the reactor test-bed. The greatest amount of work to date has been directed toward a better understanding of the role that electrostatic, ion drag, neutral fluid drag and gravitational forces play in governing the dynamic behavior of particle cloud motion. This has been accomplished by using laser light scattering (LLS) techniques to track the motion of suspended particle clouds in rf discharges. Also, statistical correlation’s in the fluctuation of scattered laser light intensity [dynamic laser light scattering (DLLS)] can be used to determine information about particle size, motion, and growth dynamics. These results are reviewed, along with recent work demonstrating that charged dust particles in a plasma can form a strongly coupled Coulomb liquid or solid. New results from DLSS experiments performed in the Reference Cell are presented that show process-induced dust particles confined in an electrostatic trap exhibit low-frequency oscillatory motion consistent with charge density wave (CDW) motion predicted for strongly coupled Coulomb liquids.     

Hygrothermomechanical evaluation of porous media under finite deformation. Part I - finite element formulations

The coupled thermomechanical responses of fluid-saturated porous continua subjected to finite deformation are investigated. Field equations governing the transient response of the media are derived from a continuum thermodynamics mixture theory based on mass balance, momentum balance and energy balance laws as well as the Clausius-Duhem inequality. Finite element procedures for the two-dimensional response, employing updated Lagrangian formulations for the solid skeleton deformation and the weak formulations for fluid and thermal transport equations, are implemented in a fully implicit form. Temperature-dependent mechanical properties for the non-linear solid matrix, characterized by Perzyna's viscoplastic model, are assumed. An iterative scheme based on the full Newton-Raphson method is presented for simultaneously solving the coupled non-linear equations.     

A general formulation in rigid multibody dynamics

The dynamics of rigid multibodies is traditionally formulated by means of either minimal or redundant co-ordinates methods. An alternative approach is here proposed whereby a highly redundant set of coordinates is adopted. As a result, the equations of motion of the constrained bodies are decoupled. Several meaningful parameters are directly available and the constraint conditions are enforced in a very natural way. The first part of the paper presents the basic meanings and the theoretical developments of the formulation. The second develops a numerical approximation for the methodology proposed in the first part. The non-linear system of differential-algebraic equations governing the motion of the multibody is reduced to its weak form. It is linearized by applying a Newton–Raphson procedure and approximated through the method of finite elements in time. The details of the numerical application of this method are discussed and a solution procedure is presented. Finally, some numerical examples involving tree and closed loop topologies prove the capability of the present formulation in handling multibody dynamics.     

Flexible multibody systems–fluid interaction

Modelling the dynamics of a flexible multibody system coupled to a rigid container carrying a fluid with a free surface is addressed. The proposed methodology allows the analyst to implement all sort of non-linearities inherent in the dynamics of the structure. Potential flow with modified Rayleigh damping is used to model the fluid. Non-linear sloshing effects are considered and no simplifications are made on the field equations and boundary conditions. A set of first-order differential equations for the motion of both the structure and the fluid are presented. Emphasis is placed on the point that the motion of the flexible multibody system is not prescribed but is found as part of the solution procedure. Some improvements are presented with respect to a previous introductory work by the authors. Detailed derivations and two numerical examples are presented: a flexible column supporting a rigid water tank (with a comparison using an approximate method) and a double flexible-link pendulum coupled to a rigid container. © 1998 This paper was produced under the auspices of the U.S. Government and it is therefore not subjected to copyright in the U.S.     

Non-liner dynamic coupled equations on controlling patterns of the solid/liquid interface

In non-equilibrium non-linear region, assuming that there is a local equilibrium at the solid/liquid interface, the non-linear dynamics coupled equations of the time dependence of the perturbation amplitude and wave number of the solid/liquid interface during the solidification of a dilute binary alloy are established.     

Non-linear dynamic coupled equations on controlling patterns of the solid/liquid interface

In non-equilibrium non-linear region, assuming that there is a local equilibrium at the solid/liquid interface, the non-linear dynamics coupled equations of the time dependence of the perturbation amplitude and wave number of the solid/liquid interface during the solidification of a dilute binary alloy are established.     

A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution

The governing equations for the interaction of two immiscible fluids within a deforming porous medium are formulated on the basis of generalized Biot theory. The displacement of the solid skeleton, the pressure and saturation of wetting fluid are taken as primary unknowns of the model. The finite element method is applied to discretize the governing eqations in space. The time domain numerical solution to the coupled problem is achieved by using an unconditionally stable direct integration procedure. Examples are presented to illustrate the performance and capability of the approach.     

Free vibration analysis of sandwich beams using improved dynamic stiffness method

In this paper, free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness and finite element methods. To determine the governing equations of motion by the present theory, the core density has been taken into consideration. The governing partial differential equations of motion for one element contained three layers are derived using Hamilton’s principle. This formulation leads to two partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are combined to form one ordinary differential equation. Closed form analytical solution for this equation is determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. They are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the known Wittrick–Williams algorithm. After validation of the present model, the effect of various parameters such as density, thickness and shear modulus of the core for various boundary conditions on the first natural frequency is studied.     

Surfactant-driven motion and splitting of droplets on a substrate

A theoretical and computational model is presented to predict the motion of a small sessile liquid droplet, lying on a solid substrate including surfactant effects. The model, as formulated, consists of coupled partial differential equations in space and time, and several auxilliary relationships. The validity of the long-wave, or ‘lubrication’ approximation is assumed. It is shown that there are circumstances where surfactant injection or production will cause the droplet to split into two daughter droplets. It is conjectured that the results are relevant to basic mechanisms involved in biological cell division (cytokinesis). It is also demonstrated that motion of a droplet, analogous to the motility of a cell, can be produced by surfactant addition. Computed examples are given here, in both two and three space dimensions. Approximate energy requirements are also calculated for these processes. These are found to be suitably small.     

LQG controller designs from reduced order models for a launch vehicle

The suppression of liquid fuel slosh motion is critical in a launch vehicle (LV). In particular, during certain stages of the launch, the dynamics of the fuel interacts adversely with the rigid body dynamics of the LV and the feedback controller must attentuate these effects. This paper describes the effort of a multivariable control approach applied to the Geosynchronous Satellite Launch Vehicle (GSLV) of the Indian Space Research Organization (ISRO) during a certain stage of its launch. The fuel slosh dynamics are modelled using a pendulum model analogy. We describe two design methodologies using the Linear-Quadratic Gaussian (LQG) technique. The novelty of the technique is that we apply the LQG design for models that are reduced in order through inspection alone. This is possible from a perspective that the LV could be viewed as many small systems attached to a main body and the interactions of some of these smaller systems could be neglected at the controller design stage provided sufficient robustness is ensured by the controller. The first LQG design is carried out without the actuator dynamics incorporated at the design stage and for the second design we neglect the slosh dynamics as well.     

Mathematical and Numerical Modelling of Microconvection in the Long Rectangular Domains

The thermal gravitational convection of liquids under conditions of microgravity is studied in a 2D long rectangular domain elongated in the direction of the gravity force. The liquid is located between two solid regions of equal thickness. The solid parts are heat conducting. The mathematical modelling of the coupled problem is presented. Two mathematical models of convection are used to describe a motion of liquid: the classical Oberbeck-Boussinesq model and the microconvection model of isothermally incompressible liquid. The numerical experiments of convection are performed and demonstrate the qualitative and quantitative differences in the flow characteristics.     

Study on the global chaotic dynamics and control of liquid-filled spacecraft with flexible appendage

This paper investigates the global chaotic attitude dynamics and control of completely viscous liquid-filled spacecraft with flexible appendage. The focus in this paper is on the way in which the dynamics of the liquid and flexible appendage vibration are coupled. The equations of motion are derived and then transformed into a form suitable for the application of Melnikov’s method. Melnikov’s integral is used to predict the transversal intersections of the stable and unstable manifolds for the perturbed system. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of spacecraft. In addition, the Melnikov criterion is compared with numerical simulations of the system. Numerical solutions to these equations show that the attitude dynamics of liquid-filled flexible spacecraft possesses characteristics common to random, non-periodic solutions and chaos. This paper demonstrated that the desired final polarity control is guaranteed by using a pair of thruster impulses. The control strategy for a reorientation maneuver is designed and the numerical simulation results are presented for both the uncontrolled and controlled spin transition.     

Elasticity Theories with Higher-order Gradients of Inertia and Stiffness for the Modelling of Wave Dispersion in Laminates

Dispersive wave propagation is simulated with a continuum elasticity theory that incorporates gradients of strain and inertia. The additional parameters are the Representative Volume Element (RVE) sizes in statics and dynamics, respectively. For the special case of a periodic laminate, expressions for these two RVE sizes can be provided based on the properties of the two components. The fourth-order governing equations are rewritten in two sets of coupled second-order equations, whereby the two sets of unknowns are the macroscopic displacements and the microscopic displacements. The resulting formulation is thus a true multi-scale continuum. In a numerical wave propagation example it is shown that the higher-order continuum model provides an excellent approximation of an explicit model of the heterogeneous laminate.     

Analysis of transient flow in pipelines with fluid–structure interaction using method of lines

A mathematical model is presented for transient flow in a pipeline with fluid–structure interaction. Water hammer theory and equations of axial motion for the pipeline are employed and the Poisson, junction and transient shear stress couplings are taken into account, which give rise to four coupled non‐linear, first‐order hyperbolic partial differential equations governing the fluid flow and pipe motion. These equations are discretized in space using the Keller box scheme and the method of lines is employed to reduce the partial differential equations to a system of ordinary differential equations. The resulting system is solved using a backward differentiation formulation method. The effect of transient shear stress on transient flow is investigated and the mechanisms underlying this effect are explored. The results revealed that the influence of transient shear stress can be significant and varies considerably, depending on the boundary conditions, viz, valve closure time. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical simulation of breakup and detachment of an axially stretching Newtonian liquid bridge with a moving contact line phase field method

The extensional, breakup and detachment dynamics of an axially stretching Newtonian liquid bridge are investigated numerically with a dynamic domain multiphase incompressible flow solver. The multiphase flow solver employs a Cahn–Hilliard phase field model to describe the evolution of the diffuse interface separating the liquid bridge fluid from the surrounding medium. The governing axisymmetric Navier–Stokes and Cahn–Hilliard phase field equations are discretized on a continuously expanding domain, the boundaries of which coincide with the planar solid surfaces containing the liquid bridge. The entire formulation, including the fast pressure correction for high density ratios and the semi-implicit discretization that overcomes the numerical stiffness of the fourth-order spatial operators, is performed on a fixed simplified computational domain using time-dependent transformation. Simulations reveal that the dynamic domain interface capturing technique effectively captures the deformation dynamics of the stretching liquid bridge, including the capillary wave formation, necking and interface evolution post breakup and detachment. It is found that the liquid bridge detachment is strongly influenced by the contact angle prescribed at the stationary and moving solid surfaces. At relatively small pulling speeds, the entire liquid is found to preferentially adhere to the less hydrophobic surface. When the prescribed contact angles are equal, however, the liquid bridge undergoes complete detachment so that no liquid resides either on the stationary or on the moving solid surface.