Computational algorithms for tracking dynamic fluid–structure interfaces in embedded boundary methods

A robust, accurate, and computationally efficient interface tracking algorithm is a key component of an embedded computational framework for the solution of fluid–structure interaction problems with complex and deformable geometries. To a large extent, the design of such an algorithm has focused on the case of a closed embedded interface and a Cartesian computational fluid dynamics grid. Here, two robust and efficient interface tracking computational algorithms capable of operating on structured as well as unstructured three‐dimensional computational fluid dynamics grids are presented. The first one is based on a projection approach, whereas the second one is based on a collision approach. The first algorithm is faster. However, it is restricted to closed interfaces and resolved enclosed volumes. The second algorithm is therefore slower. However, it can handle open shell surfaces and underresolved enclosed volumes. Both computational algorithms exploit the bounding box hierarchy technique and its parallel distributed implementation to efficiently store and retrieve the elements of the discretized embedded interface. They are illustrated, and their respective performances are assessed and contrasted, with the solution of three‐dimensional, nonlinear, dynamic fluid–structure interaction problems pertaining to aeroelastic and underwater implosion applications. Copyright © 2012 John Wiley & Sons, Ltd.     

A fluid–structure interaction solver for the study on a passively deformed fish fin with non-uniformly distributed stiffness

Research on fish locomotion has made extensive progress towards a better understanding of how fish control their flexible body and fin for propulsion and maneuvering. Although the biologically flexible fish fins are believed to be one of the most important features to achieve optimal swimming performance, due to the limitations of the existing numerical modeling tool, studies on a deformable fin with a non-uniformly distributed stiffness are rare. In this work, we present a fully coupled fluid–structure interaction solver which can cope with the dynamic interplay between flexible aquatic animal and the ambient medium. In this tool, the fluid is resolved by solving Navier–Stokes equations based on the finite volume method with a multi-block grid system. The solid dynamics is solved by a nonlinear finite element method. A sophisticated improved IQN-ILS coupling algorithm is employed to stabilize solution and accelerate convergence. To demonstrate the capability of the developed Fluid–Structure-Interaction solver, we investigated the effect of five different stiffness distributions on the propulsive performance of a caudal peduncle-fin model. It is shown that with a non-uniformly distributed stiffness along the surface of the caudal fin, we are able to replicate similar real fish fin deformation. Consistent with the experimental observations, our numerical results also indicate that the fin with a cupping stiffness profile generates the largest thrust and efficiency whereas a heterocercal flexible fin yields the least propulsion performance but has the best maneuverability.     

Simulation of self‐propelled anguilliform swimming by local domain‐free discretization method

The local domain‐free discretization method is extended in this work to simulate fluid–structure interaction problems, the class of which is exemplified by the self‐propelled anguilliform swimming of deforming bodies in a fluid medium. Given the deformation of the fish body in its own reference frame, the translational and rotational motions of the body governed by Newton's Law are solved together with the surrounding flow field governed by Navier–Stokes equations. When the body is deforming and moving, no mesh regeneration is required in the computation. The loose coupling strategy is employed to simulate the fluid–structure interaction involved in the self‐propelled swimming. The local domain‐free discretization method and an efficient algorithm for classifying the Eulerian mesh points are described in brief. To validate the fluid–structure interaction solver, we simulate the ‘lock‐in’ phenomena associated with the vortex‐induced vibrations of an elastically mounted cylinder. Finally, we demonstrate applications of the method to two‐dimensional and three‐dimensional anguilliform‐swimming fish. The kinematics and dynamics associated with the center of mass are shown and the rotational movement is also presented via the angular position of the body axis. The wake structure is visualized in terms of vorticity contours. All the obtained numerical results show good agreement with available data in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Swimming in an inviscid fluid

We present a set of equations governing the motion of a body due to prescribed shape changes in an inviscid, planar fluid with nonzero vorticity. The derived equations, when neglecting vorticity, reduce to the model developed in Kanso et al. (J Nonlinear Sci 15:255–289, 2005) for swimming in potential flow, and are also consistent with the models developed in Borisov et al. (J Math Phys 48:1–9, 2007), Kanso and Oskouei (J Fluid Mech 800:77–94, 2008), Shasikanth et al. (Phys Fluids 14(3):1214–1227, 2002) for a rigid body interacting dynamically with point vortices. The effects of cyclic shape changes and the presence of vorticity on the locomotion of a submerged body are discussed through examples.     

Simulation of flow‐flexible body interactions with large deformation

A modified front‐tracking method was proposed for the simulation of fluid‐flexible body interactions with large deformations. A large deformable body was modeled by restructuring the body using a grid adaptation. Discontinuities in the viscosity at the fluid‐structure interface were incorporated by distributing the viscosity across the interface using an indicator function. A viscosity gradient field was created near the interface, and a smooth transition occurred between the structure and the fluid. The fluid motion was defined on the Eulerian domain and was solved using the fractional step method on a staggered Cartesian grid system. The solid motion was described by Lagrangian variables and was solved by the finite element method on an unstructured triangular mesh. The fluid motion and the structure motion were independently solved, and their interaction force was calculated using a feedback law. The interaction force was the restoring force of a stiff spring with damping, and spread from the Lagrangian coordinates to the Eulerian grid by a smoothed approximation of the Dirac delta function. In the numerical simulations, we validated the effect of the grid adaptation on the solid solver using a vibrating circular ring. The effects of the viscosity gradient field were verified by solving the deformation of a circular disk in a linear shear flow, including an elastic ring moving through a channel with constriction, deformation of a suspended catenary, and a swimming jellyfish. A comparison of the numerical results with the theoretical solutions was presented. Copyright © 2011 John Wiley & Sons, Ltd.     

自主泳动弹性绳的轨迹模拟

研究柔性结构与流体间耦合作用,可以促进软体机器人的发展.通过速度快、精度高的数值模拟方法模拟水下机器人的实时运动轨迹,可以为真实实验提供测试方向与理论牵引,增大实验成功的可能性.本文研究有自主运动趋势的弹性绳在二维流场中的运动轨迹.首先,对弹性绳离散化建模并同时考虑拉压与扭转弹性力,从能量角度建立动力学方程,此模型可以较为真实地反映弹性绳内力对其运动产生的作用.然后基于半拉格朗日法建立流体求解器. 最后,提出简化的基于动量方程的浸入边界法作为耦合算法,通过直接修正网格速度代替浸入边界力法中力源项的作用.使用这种算法求解耦合作用兼具简便性与快速性.对弹性绳模型、流体模型与简化耦合模型依次解算,模拟了正弦形式波动弹性绳在水中的运动轨迹.结果显示,弹性绳在弹性内力与流固相互作用力共同影响下,该种新的浸入边界法可以实现对水下弹性绳运动轨迹的模拟.数值实验显示弹性绳的自主运动参考模型的初相位改变时,其前进方向会发生改变.该仿真模拟算法与平台可以为细长形软体水生机器人的研发提供参考.      

Finite element computations for unsteady fluid and elastic membrane interaction problems

The interaction between the hydrodynamic forces of a flow field and the elastic forces of adjacent deformable boundaries is described by elastohydrodynamics, a coupled fluid–elastic membrane problem. Direct numerical solution of the unsteady, highly non-linear equations requires that the dynamic evolution of both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. This paper describes a numerical algorithm based on the deformable spatial domain space–time (DSD/ST) finite element method for the unsteady motion of an incompressible, viscous fluid with elastic membrane interaction. The unsteady Navier–Stoke and elastic membrane equations are solved separately using an iterative procedure by the GMRES technique with an incomplete lower-upper (ILU) decomposition at every time instant. One-dimensional, two-dimensional and deformable domain model problems are used to demonstrate the capabilities and accuracy of the present algorithm. Both steady state and transient problems are studied. © 1997 John Wiley & Sons, Ltd.     

A time splitting fictitious domain algorithm for fluid–structure interaction problems (A fictitious domain algorithm for FSI)

A Finite Element Method in mixed Eulerian and Lagrangian formulation is developed to allow direct numerical simulations of dynamical interaction between an incompressible fluid and a hyper-elastic incompressible solid. A Fictitious Domain Method is applied so that the fluid is extended inside the deformable solid volume and the velocity field in the entire computational domain is resolved in an Eulerian framework. Solid motion, which is tracked in a Lagrangian framework, is imposed through the body force acting on the fluid within the solid boundaries. Solid stress smoothing on the Lagrangian mesh is performed with the Zienkiewicz–Zhu patch recovery method. High-order Gaussian integration quadratures over cut elements are used in order to avoid sub-meshing within elements in the Eulerian mesh that are intersected by the Lagrangian grid. The algorithm is implemented and verified in two spatial dimensions by comparing with the well validated simulations of solid deformation in a lid driven cavity and periodic elastic wall deformation driven by a time-dependent flow. It shows good agreement with the numerical results reported in the literature. In 3-D the method is validated against previously reported numerical simulations of 3-D rhythmically contracting alveolated ducts.     

Unified one‐fluid formulation for incompressible flexible solids and multiphase flows: Application to hydrodynamics using the immersed structural potential method (ISPM)

In this paper, we present a two‐dimensional computational framework for the simulation of fluid‐structure interaction problems involving incompressible flexible solids and multiphase flows, further extending the application range of classical immersed computational approaches to the context of hydrodynamics. The proposed method aims to overcome shortcomings such as the restriction of having to deal with similar density ratios among different phases or the restriction to solve single‐phase flows. First, a variation of classical immersed techniques, pioneered with the immersed boundary method (IBM), is presented by rearranging the governing equations, which define the behaviour of the multiple physics involved. The formulation is compatible with the “one‐fluid” formulation for two‐phase flows and can deal with large density ratios with the help of an anisotropic Poisson solver. Second, immersed deformable structures and fluid phases are modelled in an identical manner except for the computation of the deviatoric stresses. The numerical technique followed in this paper builds upon the immersed structural potential method developed by the authors, by adding a level set–based method for the capturing of the fluid‐fluid interfaces and an interface Lagrangian‐based meshless technique for the tracking of the fluid‐structure interface. The spatial discretisation is based on the standard marker‐and‐cell method used in conjunction with a fractional step approach for the pressure/velocity decoupling, a second‐order time integrator, and a fixed‐point iterative scheme. The paper presents a wide d range of two‐dimensional applications involving multiphase flows interacting with immersed deformable solids, including benchmarking against both experimental and alternative numerical schemes.     

A potential-flow,deformable-body model for fluid–structure interactions with compact vorticity: application to animal swimming measurements

This paper presents an approach to quantify the unsteady fluid forces, moments and mass transport generated by swimming animals, based on measurements of the surrounding flow field. These goals are accomplished within a framework that is independent of the vorticity field, making it unnecessary to directly resolve boundary layers on the animal, body–vortex interactions, or interactions among vortex lines in the wake. Instead, the method identifies Lagrangian coherent structures in the flow, whose dynamics in flows with compact vorticity are shown to be well approximated by potential flow concepts, especially the Kirchhoff and deformation potentials from deformable body theory. Examples of the application of these methods are given for pectoral fin locomotion of the bluegill sunfish and undulatory swimming of jellyfish, and the methods are validated by analysis of a canonical starting vortex ring flow. The transition to a Lagrangian approach toward animal swimming measurements suggests the possibility of implementing recently developed particle tracking (vis-à-vis DPIV) techniques for fully three-dimensional measurements of animal swimming.     

A volume-tracking method for incompressible multifluid flows with large density variations

A numerical technique (FGVT) for solving the time-dependent incompressible Navier–Stokes equations in fluid flows with large density variations is presented for staggered grids. Mass conservation is based on a volume tracking method and incorporates a piecewise-linear interface reconstruction on a grid twice as fine as the velocity–pressure grid. It also uses a special flux-corrected transport algorithm for momentum advection, a multigrid algorithm for solving a pressure-correction equation and a surface tension algorithm that is robust and stable. In principle, the method conserves both mass and momentum exactly, and maintains extremely sharp fluid interfaces. Applications of the numerical method to prediction of two-dimensional bubble rise in an inclined channel and a bubble bursting through an interface are presented. © 1998 John Wiley & Sons, Ltd.     

A hybrid immersed boundary and material point method for simulating 3D fluid–structure interaction problems

A numerical method is developed for solving the 3D, unsteady, incompressible Navier–Stokes equations in curvilinear coordinates containing immersed boundaries (IBs) of arbitrary geometrical complexity moving and deforming under forces acting on the body. Since simulations of flow in complex geometries with deformable surfaces require special treatment, the present approach combines a hybrid immersed boundary method (HIBM) for handling complex moving boundaries and a material point method (MPM) for resolving structural stresses and movement. This combined HIBM & MPM approach is presented as an effective approach for solving fluid–structure interaction (FSI) problems. In the HIBM, a curvilinear grid is defined and the variable values at grid points adjacent to a boundary are forced or interpolated to satisfy the boundary conditions. The MPM is used for solving the equations of solid structure and communicates with the fluid through appropriate interface‐boundary conditions. The governing flow equations are discretized on a non‐staggered grid layout using second‐order accurate finite‐difference formulas. The discrete equations are integrated in time via a second‐order accurate dual time stepping, artificial compressibility scheme. Unstructured, triangular meshes are employed to discretize the complex surface of the IBs. The nodes of the surface mesh constitute a set of Lagrangian control points used for tracking the motion of the flexible body. The equations of the solid body are integrated in time via the MPM. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at stationary curvilinear grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The influence of the fluid on the body is defined through pressure and shear stresses acting on the surface of the body. The HIBM & MPM approach is validated for FSI problems by solving for a falling rigid and flexible sphere in a fluid‐filled channel. The behavior of a capsule in a shear flow was also examined. Agreement with the published results is excellent. Copyright © 2007 John Wiley & Sons, Ltd.     

A combined level set/ghost cell immersed boundary representation for floating body simulations

A six degrees of freedom (6DOF) algorithm is implemented in the open‐source CFD code REEF3D. The model solves the incompressible Navier–Stokes equations. Complex free surface dynamics are modeled with the level set method based on a two‐phase flow approach. The convection terms of the velocities and the level set method are treated with a high‐order weighted essentially non‐oscillatory discretization scheme. Together with the level set method for the free surface capturing, this algorithm can model the movement of rigid floating bodies and their interaction with the fluid. The 6DOF algorithm is implemented on a fixed grid. The solid‐fluid interface is represented with a combination of the level set method and ghost cell immersed boundary method. As a result, re‐meshing or overset grids are not necessary. The capability, accuracy, and numerical stability of the new algorithm is shown through benchmark applications for the fluid‐body interaction problem. Copyright © 2016 John Wiley & Sons, Ltd.     

斑马鱼C型起动中动力学特性的活体实验研究

提出了一种从鱼类自主游动的运动学实验数据出发推算其动力学特性的实验研究方法。该方法基于变形体动力学方程,不仅可以计算出变形鱼体的整体转动角速度以完善其运动学数据,还可以计算出作用在自主游动的鱼体上的流体合力和流体合力矩,进而分析鱼体的力能学特征。本文运用此方法研究了斑马鱼的两种典型C型起动,对比分析了它们的运动学及力能学特征。结果表明,在相仿的C型弯曲变形下,逃逸型C型起动与非逃逸型C型起动相比,前者推力较大,导致其在前进方向上的质心速率较大;前者侧向力较大,导致其转弯半径较小;前者转矩较大,但因在起动中后期的S型摆动产生了反向转矩,最终导致转动角度小于后者。通过对两种典型C型起动的实验研究发现,斑马鱼会因不同的运动需求而表现出不同的机动性能。     

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

In the present paper, an embedded-boundary formulation that is applicable to fluid–structure interaction problems is presented. The Navier–Stokes equations for incompressible flow are solved on a Cartesian grid which is not aligned with the boundaries of a body that undergoes large-angle/large-displacement rigid body motions through the fixed grid. A strong-coupling scheme is adopted, where the fluid and the structure are treated as elements of a single dynamical system, and all of the governing equations are integrated simultaneously and interactively in the time domain. A demonstration of the accuracy and efficiency of the method is given for a variety of fluid–structure interaction problems.     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

Motions of elastic solids in fluids under vibration

Motion of a rigid or deformable solid in a viscous incompressible fluid and corresponding fluid–solid interactions are considered. Different cases of applying high frequency vibrations to the solid or to the surrounding fluid are treated. Simple formulas for the mean velocity of the solid are derived, under the assumption that the regime of the fluid flow induced by its motion is turbulent and the fluid resistance force is nonlinearly dependent on its velocity. It is shown that vibrations of a fluid’s volume slow down the motion of a submerged solid. This effect is much pronounced in the case of a deformable solid (i.e., gas bubble) exposed to near-resonant excitation. The results are relevant to the theory of gravitational enrichment of raw materials, and also contribute to the theory of controlled locomotion of a body with an internal oscillator in continuous deformable (solid or fluid) media.     

不同网格扭曲率下压力修正全隐算法——IDEAL求解性能研究

2008年,本文作者和陶文铨等提出了一种用于速度和压力耦合求解的高效稳定压力修正全隐算法IDEAL,该算法通过在每个迭代层次上对压力方程进行两次内迭代计算,完全克服了SIMPLE算法的两个假设,充分满足了速度和压力之间的耦合,从而大大提高了计算的收敛性和健壮性.为了进一步实现IDEAL算法的推广应用,本文基于三维倾斜方腔顶盖驱动流动,研究了IDEAL算法在不同网格扭曲率下的求解特性.研究发现,在不同网格扭曲率下,IDEAL算法的健壮性和收敛性均优于SIMPLE算法,特别在高网格扭曲率情况下,IDEAL算法求解性能更加优于SIMPLE算法.在不同网格扭曲率下,IDEAL算法健壮性保持不变,几乎可以在任意速度亚松弛因子下获得收敛的解,同时IDEAL算法最短计算耗时较SIMPLE算法减少了56%~89%,验证了IDEAL算法的优越性.     

Numerical modelling of liquid infusion into fibrous media undergoing compaction

In the present paper, an overall model for the study of a non isothermal fluid flow across a highly compressible porous medium is proposed, in order to be included into a finite element software. This model can be applied to a wide range of activities, and as an application it is used here to model dry route composite manufacturing processes. Indeed, it is to be noted that the spreading of these promising processes fails due to the absence of a numerical model able to capture the resin infusion across the compressed preform thickness. The main difficulty being that this infusion results from a pressure prescribed over the deformable preform/resin stacking involved. From the modelling point of view, problems of this multi-physical analysis are two fold. First the coupling of liquid regions, ruled by Stokes equations, with the fibrous preform regions governed by a Darcy's law, yield badly posed boundary conditions. Second, the interaction phenomena due to the resin flow in the highly compressible preform are not classical. The model developed here includes a modified Beaver–Shaffman–Joseph condition to couple Stokes and Darcy zones, and is based on an ALE formulation of the liquid flow across the deformable porous medium in which finite strains are accounted for with an updated Lagrangian scheme. These mechanical models are also coupled with thermo-chemical models, accounting for resin reticulation under the temperature cycle prescribed by the processing equipment.