A Parallel Implementation of ALE Moving Mesh Technique for FSI Problems using OpenMP

This paper investigates a high performance implementation of an Arbitrary Lagrangian Eulerian moving mesh technique on shared memory systems using OpenMP environment. Moving mesh techniques are considered an integral part of a wider class of fluid mechanics problems that involve moving and deforming spatial domains, namely, free-surface flows and Fluid Structure Interaction (FSI). The moving mesh technique adopted in this work is based on the notion of nodes relocation, subjected to a certain evolution as well as constraint conditions. A conjugate gradient method augmented with preconditioning is employed for solution of the resulting system of equations. The proposed algorithm, initially, reorders the mesh using an efficient divide and conquer approach and then parallelizes the ALE moving mesh scheme. Numerical simulations are conducted on the multicore AMD Opteron and Intel Xeon processors, and unstructured triangular and tetrahedral meshes are used for the 2D and 3D problems. The quality of generated meshes is checked by comparing the element Jacobians in the reference and current meshes, and by keeping track of the change in the interior angles in triangles and tetrahedrons. Overall, 51 and 72% efficiencies in terms of speedup are achieved for both the parallel mesh reordering and ALE moving mesh algorithms, respectively.     

Alternating Schwarz methods for partial differential equation-based mesh generation

To solve boundary value problems with moving fronts or sharp variations, moving mesh methods can be used to achieve reasonable solution resolution with a fixed, moderate number of mesh points. Such meshes are obtained by solving a nonlinear elliptic differential equation in the steady case, and a nonlinear parabolic equation in the time-dependent case. To reduce the potential overhead of adaptive partial differential equation-(PDE) based mesh generation, we consider solving the mesh PDE by various alternating Schwarz domain decomposition methods. Convergence results are established for alternating iterations with classical and optimal transmission conditions on an arbitrary number of subdomains. An analysis of a colouring algorithm is given which allows the subdomains to be grouped for parallel computation. A first result is provided for the generation of time-dependent meshes by an alternating Schwarz algorithm on an arbitrary number of subdomains. The paper concludes with numerical experiments illustrating the relative contraction rates of the iterations discussed.     

Arbitrary Lagrangian–Eulerian methods for analysis of regressing solid domains and interface tracking

We present the details of the formulation and implementation of the arbitrary Lagrangian–Eulerian (ALE) finite element method for three-dimensional problems involving regressing solid domains and moving boundaries. An example of such problems is the simulation of solid-propellant rockets in which the evolution of a fluid–solid interface is governed by a combustion law and the transfer of mass and momentum across it. The ALE method, while providing a means to track the location of the interface, allows the adaptation of the finite element mesh to the constantly changing solid domain. In this study, the mesh adaptation is achieved via a novel smoothing technique in which the shape of finite elements with smaller volumes, which are more susceptible to mesh-entanglement, are better preserved compared to those with larger volumes. An analysis of the stability of the ALE computations, under certain simplifying assumptions, is also performed. The stability limits determined from this analysis can be utilized as constraints for adjusting mesh velocities or time increments in the convective mesh-motion phase of the ALE computations. In addition, a method is provided for generating verification problems with moving interfaces from those with known solutions on stationary material domains. A problem in which the prescribed growth of a cavity in an infinite medium under a time-varying pressure loading is used to verify the implementation and to demonstrate the verification technique.     

一种物理量重映方法的研究

§1.引 言 求解流体力学问题,依照采用的坐标可分为Lagrange方法和Euler方法两大类[1].用拉氏方法局部图像可以算得比较精细,物质界面清晰,但是由于二维流体运动中可能出现严重的扭曲现象,可能造成拉氏网格相交,以致于计算不能继续下去.欧拉方法当然没有网格相交的问题,但当系统中含有多种介质时,不加特殊处理,会使物质界面逐渐模糊,得不到正确的结果.为了避免拉氏方法和欧氏方法的缺点,Frank.Lazarus(1964)提出了一种混合欧拉拉格朗日方法,Noh(1964)的耦合欧拉拉格朗日方法(CEL方法),则是将求     

A comprehensive generalized mesh system for CFD applications

The numerical approaches used for the solution of governing equations of fluid flow are dictated highly by the topology of the domain discretization. Two of the most commonly used discretization approaches are the structured and unstructured topologies. This paper describes the discretization of the domain using generalized elements with an arbitrary number of nodes to combine the advantages of both the structured and unstructured methodologies. Numerical algorithms for the solution of the governing equations for generalized mesh, an approach for handling mesh movement applicable to rotating machineries, and the application of this framework for overset meshes to handle moving body problems are discussed. A library-based approach has been adopted for the implementation of overset capability for the framework. The results from the application of this framework for various applications are presented.     

Implicit Lagrangian-Eulerian Tvd Method For Solving Two-Dimensional Hydrodynamic Equations On Unstructured Meshes

An ALE method for solving hydrodynamic equations on unstructured meshes is presented. It is based on an implicit finite-volume scheme derived in Godunov’s approach. The basic quantities (the density, temperature, and velocity) are defined at cell centers. For relations between pressure and velocities at the centers and their analogs at the nodes, we use the relations proposed by P.H. Maire et al. A piecewise-linear TVD reconstruction of the pressure and velocity in the cell is used to achieve the second-order approximation of smooth solutions, preserving their monotonicity. Mesh rezonings are implemented during the calculation. To recalculate the values, the old mesh is covered onto the new one so that a bounded piecewise-linear representation is used for the values in the cells of the old mesh, while the interface in the mixed cells is reconstructed by the VOF method. The mass, momentum, and total energy are conserved under the recalculation.     

Fluid-structure interactions using different mesh motion techniques

In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure interactions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solving an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examine an implementation of time discretization that is designed with finite differences. Spatial discretization is based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newton method with exact Jacobian matrix is used. Our results show that the biharmonic model produces the smoothest meshes but has increased computational cost compared to the other two approaches.     

The simulation of 3D unsteady incompressible flows with moving boundaries on unstructured meshes

The development of a computational model for the simulation of three-dimensional unsteady incompressible viscous fluid flows with moving boundaries is presented. The numerical model is based upon the solution of the Navier–Stokes equations on unstructured meshes using the artificial compressibility approach. An ALE formulation is adopted and the equations are discretized using a cell vertex finite volume method. The formulation ensures the satisfaction of the geometric conservation law when the mesh is allowed to move. An implicit time discretization is adopted and a dual time approach is employed. Explicit relaxation is used for the sub-iterations, with multigrid acceleration. For moving geometries, the mesh is deformed by adopting a spring analogy, combined with a wall distance function approach. The numerical procedure is validated on a standard problem and is then used for the simulation of flow over a flexible fish-like body.     

Fluid simulation with adaptively sharpening and embedded boundary conditions

In this paper, we present a physically based technique for simulating inviscid fluids. Our contribution is concerned with two issues. First, for solving the advection equation, we introduce a hybrid scheme that couples the FLIP scheme with the semi-Lagrangian scheme by adaptively distributing implicit particles and using a transition layer to propagate information. Secondly, for solving pressure, we develop a flux based scheme that can embed arbitrary solid boundaries into a Poisson equation. And based on this scheme we make further improvement to achieve two-way fluid/solid coupling on an octree structure with second-order accuracy. Finally, the experimental results demonstrate that our hybrid scheme for advection can preserve relatively fine surface details with less computation expenditure; and simultaneously our robust pressure solver can handle both stationary and moving obstacles more efficiently compared with unstructured meshes.     

ReALE: A Reconnection Arbitrary-Lagrangian–Eulerian method in cylindrical geometry

This paper deals with the extension to the cylindrical geometry of the recently introduced Reconnection algorithm for Arbitrary-Lagrangian–Eulerian (ReALE) framework. The main elements in standard ALE methods are an explicit Lagrangian phase, a rezoning phase, and a remapping phase. Usually the new mesh provided by the rezone phase is obtained by moving grid nodes without changing connectivity of the underlying mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In ReALE we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and permits to follow Lagrangian features much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tesselation machinery. In this work we focus on the extension of each phase of ReALE to cylindrical geometry. The Lagrangian, rezone with reconnection and remap phases are revamped to take into account the cylindrical geometry. We demonstrate the efficiency of our ReALE in cylindrical geometry on series of numerical examples.     

Finite elements in fluids: Stabilized formulations and moving boundaries and interfaces

We provide an overview of the finite element methods we developed for fluid dynamics problems. We focus on stabilized formulations and moving boundaries and interfaces. The stabilized formulations are the streamline-upwind/Petrov-Galerkin (SUPG) formulations for compressible and incompressible flows and the pressure-stabilizing/Petrov-Galerkin (PSPG) formulation for incompressible flows. These are supplemented with the discontinuity-capturing directional dissipation (DCDD) for incompressible flows and the shock-capturing terms for compressible flows. Determination of the stabilization and shock-capturing parameters used in these formulations is highlighted. Moving boundaries and interfaces include free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed for this class of problems can be classified into two main categories: interface-tracking and interface-capturing techniques. The interface-tracking techniques are based on the deforming-spatial-domain/stabilized space-time (DSD/SST) formulation, where the mesh moves to track the interface. The interface-capturing techniques were developed for two-fluid flows. They are based on the stabilized formulation, over typically non-moving meshes, of both the flow equations and an advection equation. The advection equation governs the time-evolution of an interface function marking the interface location. We also describe some of the additional methods and ideas we introduced to increase the scope and accuracy of these two classes of techniques. Among them is the enhanced-discretization interface-capturing technique (EDICT), which was developed to increase the accuracy in capturing the interface. Also among them is the mixed interface-tracking/interface-capturing technique (MITICT), which was introduced for problems that involve both interfaces that can be accurately tracked with a moving-mesh method and interfaces that call for an interface-capturing technique.     

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

Numerical techniques for moving meshes are many and varied. In this paper we present a novel application of a moving grid finite element method applied to biological problems related to pattern formation where the mesh movement is prescribed through a specific definition to mimic the growth that is observed in nature. Through the use of a moving grid finite element technique, we present numerical computational results illustrating how period doubling behaviour occurs as the domain doubles in size.     

Euler–Lagrange coupling with damping effects: Application to slamming problems

During a high velocity impact of a structure on a nearly incompressible fluid, impulse loads with high-pressure peaks occur. This physical phenomenon called ‘slamming’ is a concern in shipbuilding industry because of the possibility of hull damage. Shipbuilding companies have carried out several studies on slamming modeling using FEM software with added mass techniques to represent fluid effects. In the added mass method inertia effects of the fluid are not taken into account and are only valid when the deadrise angle is small. This paper presents the prediction of the local high pressure load on a rigid wedge impacting a free surface, where the fluid is represented by solving Navier–Stokes equations with an Eulerian or ALE formulation. The fluid–structure interaction is simulated using a coupling algorithm; the fluid is treated on a fixed or moving mesh using an ALE formulation and the structure on a deformable mesh using a Lagrangian formulation.A new coupling algorithm is developed in the paper. The coupling algorithm computes the coupling forces at the fluid–structure interface. These forces are added to the fluid and structure nodal forces, where fluid and structure are solved using an explicit finite element formulation. Predicting the local pressure peak on the structure requires an accurate fluid–structure interaction algorithm. The Euler–Lagrange coupling algorithm presented in this paper uses a penalty based formulation similar to penalty contact in Lagrangian analyses. Both penalty coupling and penalty contact can generate high frequency oscillations due to the nearly incompressible nature of the fluid. In this paper, a damping force based on the relative velocity of the fluid and the structure is introduced to smooth out non-physical high frequency oscillations induced by the penalty springs in the coupling algorithm.     

A global arbitrary Lagrangian–Eulerian method for stratified Richtmyer–Meshkov instability

Richtmyer–Meshkov (RM) instability arises when a material interface is accelerated impulsively by shock waves. In this work, an arbitrary Lagrangian–Eulerian method, global ALE method, was proposed for the simulation of stratified RM instability. In the global ALE method, an Eulerian diffusion interface model was implemented based on mass fraction function. Thus all the meshes can be remeshed arbitrarily no matter whether they are material interface or not. Some benchmark problems, such as shock tube problem with different specific ratio, RM instability with small initial perturbation, were computed with the global ALE method, and the numerical results agree well with exact solution or theoretical model. Also, we proposed some stratified RM instability model problems with two or more material interfaces in planar, cylindrical and spherical geometries. Then the stratified RM instabilities were simulated with global ALE method. The interface evolution process was studied and compared in different geometry cases based on simulation results. To overcome the spurious mesh distortion, a sub-zonal Riemann solver method was proposed in appendix part of the paper based on the analysis of the error source of 2D Lagrangian computation due to non-uniform multi-dimensional mesh.     

Modelling and simulation of moving contact line problems with wetting effects

This paper presents a numerical scheme for computing moving contact line flows with wetting effects. The numerical scheme is based on Arbitrary Lagrangian Eulerian (ALE) finite elements on moving meshes. In the computations, the wetting effects are taken into account through a weak enforcement of the prescribed equilibrium contact angle into the model equations. The equilibrium contact angle is included in the variational form of the model by replacing the curvature with Laplace Beltrami operator and integration by parts. This weak implementation allows that the contact angle determined by the numerical scheme differs from the equilibrium value and develops a certain dynamics. The Laplace Beltrami operator technique with an interface/boundary resolved mesh is well-suited for describing the dynamic contact angle observed in experiments. We consider the spreading and the pendant liquid droplets to investigate this implementation of the contact angle. It is shown that the dynamic contact angle tends to the prescribed equilibrium contact angle when time goes to infinity. However, the dynamics of the contact angle is influenced by the slip at the moving contact line. This work has been partially supported by the German Research Foundation (DFG) through the grant To143/9.     

Finite volume flow simulations on arbitrary domains

We present a novel method for solving the incompressible Navier–Stokes equations that more accurately handles arbitrary boundary conditions and sharp geometric features in the fluid domain. It uses a space filling tetrahedral mesh, which can be created using many well-known methods, to represent the fluid domain. Examples of the method’s strengths are illustrated by free surface fluid simulations and smoke simulations of flows around objects with complex geometry.     

Variable time-step method with coordinate transformation

The variable time-step methods for solving moving boundary problems are presented by transforming the variable space domain. This results in dissociating the mode of advancement of the boundary from the size of the space mesh. That is, a small movement of the moving boundary may be chosen for computing the time interval while the space domain is subdivided into larger space meshes. As a consequence, an enormous amount of saving in computer time may be achieved by using the proposed method. Two sample problems are selected for the illustration of the method.     

变形法移动网格求解双曲型守恒律方程研究

在现有格式的基础上要提高偏微分方程数值解的分辨率,自适应移动网格技术是一种有效而且可行的方法。文中将文献[1]提出的自适应移动网格技术推广到三角形网格,并将该方法用于求解双曲型守恒量方程。用网格自适应技术求解守恒律问题时,当生成新网格之后,需要将旧网格上的函数值更新到新的网格,并保持物理量的守恒性。针对这个问题,文中提出了函数值更新过程中守恒型插值公式的具体形式,并针对二维双曲型守恒律方程进行了仿真实验,取得了满意的结果。     

Effect of substrate flexibility on dynamic wetting: a finite element model

Dynamic wetting plays an important role in coating processes. In this paper, we present a new finite element formulation that can predict the effect of substrate deformation on the location of the dynamic contact line. Our model solves for the fluid-structural interactions between an elastic solid and a viscous liquid with a dynamic contact line that can move across the deformable solid surface. Surface tension forces acting at the dynamic contact line pull outwards on the substrate and cause the formation of capillary ridge. To predict the shape of the capillary ridge and motion of dynamic contact line, our model uses arbitrary Lagrangian Eulerian (ALE) mesh motion in both fluid and solid phases, because ALE decouples the motion of solid and mesh points. In dynamic wetting of rigid solids it is known that a singularity arises at the dynamic wetting line due to a double-valued velocity. The singularity is often relieved by allowing a slip in a small contact region near the dynamic contact line. Dynamic wetting on flexible substrates involves a second singularity, which arises in the solid domain due to a line force acting at the contact line. The line force singularity is relieved by distributing the force over a finite contact region. Two ALE methods of mesh motion are implemented and compared. The variation of dynamic contact line position with respect to various parameters such as downstream pressure, contact angle, capillary number and elasticity number are presented for the finite element model and compared with an analytical model.     

An arbitrary Lagrangian Eulerian formulation for residual distribution schemes on moving grids

The arbitrary Lagrangian Eulerian formulation is derived for the residual distribution method on moving meshes. The system of Euler equations is discretized on moving meshes and in case of deforming meshes a geometrical source term has to be taken into account. A conservative linearization guarantees the conservation property of the discretized equations.From the geometric conservation law we obtain the appropriate integration points in time for the cell fluctuation and a guideline for how to distribute the geometrical source term.Testcases include the flow around a transonic oscillating airfoil and a convected vortex. In the first case a rigidly moving mesh is employed, while in the other testcase a deforming mesh is used to investigate the influence of the geometrical source term on the solution.