An extended finite element method for the simulation of particulate viscoelastic flows

We present an extended finite element method (XFEM) for the direct numerical simulation of the flow of viscoelastic fluids with suspended particles. For moving particle problems, we devise a temporary arbitrary Lagrangian–Eulerian (ALE) scheme which defines the mapping of field variables at previous time levels onto the computational mesh at the current time level. In this method, a regular mesh is used for the whole computational domain including both fluid and particles. A temporary ALE mesh is constructed separately and the computational mesh is kept unchanged throughout the whole computations. Particles are moving on a fixed Eulerian mesh without any need of re-meshing. For mesh refinements around the interface, we combine XFEM with the grid deformation method, in which nodal points are redistributed close to the interface while preserving the mesh topology. Our method is verified by comparing with the results of boundary fitted mesh problems combined with the conventional ALE scheme. The proposed method shows similar accuracy compared with boundary fitted mesh problems and superior accuracy compared with the fictitious domain method. If the grid deformation method is combined with XFEM, the required computational time is reduced significantly compared to uniform mesh refinements, while providing mesh convergent solutions. We apply the proposed method to the particle migration in rotating Couette flow of a Giesekus fluid. We investigate the effect of initial particle positions, the Weissenberg number, the mobility parameter of the Giesekus model and the particle size on the particle migration. We also show two-particle interactions in confined shear flow of a viscoelastic fluid. We find three different regimes of particle motions according to initial separations of particles.     

A particle finite element method applied to long wave run‐up

This paper presents a Lagrangian–Eulerian finite element formulation for solving fluid dynamics problems with moving boundaries and employs the method to long wave run‐up. The method is based on a set of Lagrangian particles which serve as moving nodes for the finite element mesh. Nodes at the moving shoreline are identified by the alpha shape concept which utilizes the distance from neighbouring nodes in different directions. An efficient triangulation technique is then used for the mesh generation at each time step. In order to validate the numerical method the code has been compared with analytical solutions and a preexisting finite difference model. The main focus of our investigation is to assess the numerical method through simulations of three‐dimensional dam break and long wave run‐up on curved beaches. Particularly the method is put to test for cases where different shoreline segments connect and produce a computational domain surrounding dry regions. Copyright © 2006 John Wiley & Sons, Ltd.     

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

A novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. 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.     

A fictitious domain/mortar element method for fluid–structure interaction

A new method for the computational analysis of fluid–structure interaction of a Newtonian fluid with slender bodies is developed. It combines ideas of the fictitious domain and the mortar element method by imposing continuity of the velocity field along an interface by means of Lagrange multipliers. The key advantage of the method is that it circumvents the need for complicated mesh movement strategies common in arbitrary Lagrangian–Eulerian (ALE) methods, usually used for this purpose. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Fluid–structure interactions of photo-responsive polymer cantilevers

A new class of photomechanical liquid crystal networks (LCNs) has emerged, which generate large bending deformation and fast response times that scale with the resonance of the polymer films. Here, a numerical study is presented that describes the photomechanical structural dynamic behavior of an LCN in a fluid medium; however, the methodology is also applicable to fluid–structure interactions of a broader range of adaptive structures. Here, we simulate the oscillation of photomechanical cantilevers excited by light while simultaneously modeling the effect of the surrounding fluid at different ambient pressures. The photoactuated LCN is modeled as an elastic thin cantilever plate, and gradients in photostrain from the external light are computed from the assumptions of light absorption and photoisomerization through the film thickness. Numerical approximations of the equations governing the plate are based on cubic B-spline shape functions and a second order implicit Newmark central scheme for time integration. For the fluid, three dimensional unsteady incompressible Navier–Stokes equations are solved using the arbitrary Lagrangian–Eulerian (ALE) method, which employs a structured body-fitted curvilinear coordinate system where the solid–fluid interface is a mesh line of the system, and the complicated interface boundary conditions are accommodated in a conventional finite-volume formulation. Numerical examples are given which provide new insight into material behavior in a fluid medium as a function of ambient pressure.     

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

This paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

The Fixed‐Mesh ALE approach for the numerical simulation of floating solids

In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. We pay special attention to the tracking of the various interfaces and their intersections, and to the approximate imposition of coupling conditions between the solid and the fluid. Copyright © 2010 John Wiley & Sons, Ltd.     

An immersed boundary method for unstructured meshes in depth averaged shallow water models

The representation of geometries as buildings, flood barriers or dikes in free surface flow models implies tedious and time‐consuming operations in order to define accurately the shape of these objects when using a body fitted numerical mesh. The immersed boundary method is an alternative way to define solid bodies inside the computational domain without the need of fitting the mesh boundaries to the shape of the object. In the direct forcing immersed boundary method, a solid body is represented by a grid of Lagrangian markers, which define its shape and which are independent from the fluid Eulerian mesh. This paper presents a new implementation of the immersed boundary method in an unstructured finite volume solver for the 2D shallow water equations. Moving least‐squares is used to transmit information between the grid of Lagrangian markers and the fluid Eulerian mesh. The performance of the proposed implementation is analysed in three test cases involving different flow conditions: the flow around a spur dike, a dam break flow with an isolated obstacle and the flow around an array of obstacles. A very good agreement between the classic body fitted approach and the immersed boundary method was found. The differences between the results obtained with both methods are less relevant than the errors because of the intrinsic shallow water assumptions. Copyright © 2015 John Wiley & Sons, Ltd.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

Fluid–solid interaction problems with thermal convection using the immersed element‐free Galerkin method

In this work, the immersed element‐free Galerkin method (IEFGM) is proposed for the solution of fluid–structure interaction (FSI) problems. In this technique, the FSI is represented as a volumetric force in the momentum equations. In IEFGM, a Lagrangian solid domain moves on top of an Eulerian fluid domain that spans over the entire computational region. The fluid domain is modeled using the finite element method and the solid domain is modeled using the element‐free Galerkin method. The continuity between the solid and fluid domains is satisfied by means of a local approximation, in the vicinity of the solid domain, of the velocity field and the FSI force. Such an approximation is achieved using the moving least‐squares technique. The method was applied to simulate the motion of a deformable disk moving in a viscous fluid due to the action of the gravitational force and the thermal convection of the fluid. An analysis of the main factors affecting the shape and trajectory of the solid body is presented. The method shows a distinct advantage for simulating FSI problems with highly deformable solids. Copyright © 2009 John Wiley & Sons, Ltd.     

粘性流体中弹性板振动的有限元耦合问题

流体－结构耦合作用问题是工程中比较常见的问题，具有重要意义，由于流体计算的复杂性，迄今为止，大部分的流体－结构耦合分析都是建立在对流体充分简化的基础上，尤其是将流体视为无粘、无旋的理想流体。该文在近年来前人工作的基础上，发展了一种流体－弹性结构耦合计算模式。流体视为不可压、有粘的介质，流场没有自由表面。该文采用ＳＵ／ＰＧ方法形成流体的有限元方程，采用ＡＬＥ格式处理流体和结构之间的移动界面。采用预估     

An Eulerian–Lagrangian hybrid model for the coarse-grid simulation of turbulent liquid jet breakup

In this paper we present a numerical model for the coarse-grid simulation of turbulent liquid jet breakup using an Eulerian–Lagrangian coupling. To picture the unresolved droplet formation near the liquid jet interface in the case of coarse grids we considered a theoretical model to describe the unresolved flow instabilities leading to turbulent breakup. These entrained droplets are then represented by an Eulerian–Lagrangian hybrid concept. On the one hand, we used a volume of fluid method (VOF) to characterize the global spreading and the initiation of droplet formation; one the other hand, Lagrangian droplets are released at the liquid–gas interface according to the theoretical model balancing consolidating and disruptive energies. Here, a numerical coupling was required between Eulerian liquid core and Lagrangian droplets using mass and momentum source terms. The presented methodology was tested for different liquid jets in Rayleigh, wind-induced and atomization regimes and validated against literature data. This comparison reveals fairly good qualitative agreement in the cases of jet spreading, jet instability and jet breakup as well as relatively accurate size distribution and Sauter mean diameter (SMD) of the droplets. Furthermore, the model was able to capture the regime transitions from Rayleigh instability to atomization appropriately. Finally, the presented sub-grid model predicts the effect of the gas-phase pressure on the droplet sizes very well.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

Mesh regularization for an ALE code based on the limitation of the Lagrangian mesh velocity

The Lagrangian approach is usually used for the simulation of flow with strong shock waves. Moreover, this approach is particularly well suited to treatment of material interfaces in the case of multimaterial flows.Unfortunately, this formulation leads to very large deformations in the mesh. The arbitrary Lagrangian‐Eulerian method overcomes this drawback by using a mesh regularization that is based on an analysis of cell geometry. The regularization step may be considered as a method used to correct the nonconvex and potentially tangled cells that constitute the mesh. In this paper, we present a new approach to mesh regularization. Instead of using a purely geometric criterion, we propose that the mesh evolution is computed on the basis of the flow vorticity. This approach is called the large Eddy limitation method, and it is aimed here to be used in finite volume direct arbitrary Lagrangian‐Eulerian methods. The large Eddy limitation method is general, which means that it is not restricted to applications in the finite volume framework dedicated to fluid flow simulation; for instance, it could also be naturally applied to the finite element framework.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.