Arbitrary Lagrangian Eulerian Coordinates Research Articles

A regularity result for the incompressible inviscid magnetohydrodynamics equations in the Arbitrary Lagrangian-Eulerian coordinates

Consider herein the incompressible inviscid magnetohydrodynamics equations in a moving domain with free surface boundary conditions. Without smallness assumption in the fluid volume, a priori estimates for solutions of this model are established when the initial data in H2.5+δ (δ>0) upon the Arbitrary Lagrangian-Eulerian coordinates under the boundary condition p+12|B|2=0. Indeed, this is achieved by reformulating the system into a new formulation with the Arbitrary Lagrangian-Eulerian variables, presenting the uniform estimates for the pressure, the tangential estimates for the system, as well as the curl and divergence estimates.

Multigoal-oriented error estimation and mesh adaptivity for fluid–structure interaction

In this work, we consider multigoal-oriented error estimation for stationary fluid–structure interaction. The problem is formulated within a variational-monolithic setting using arbitrary Lagrangian–Eulerian coordinates. Employing the dual-weighted residual method for goal-oriented a posteriori error estimation, adjoint sensitivities are required. For multigoal-oriented error estimation, a combined functional is formulated such that several quantities of interest are controlled simultaneously. As localization technique for mesh refinement we employ a partition-of-unity. Our algorithmic developments are substantiated with several numerical tests such as an elastic lid-driven cavity with two goal functionals, an elastic bar in a chamber with two goal functionals, and the FSI-1 benchmark with three goal functionals.

An energy method for flutter analysis of wing using one-way fluid structure coupling

In this paper, an energy method for flutter analysis of wing using one-way fluid structure coupling was developed. To consider the effect of wing vibration, Reynolds-averaged Navier–Stokes equations based on the arbitrary Lagrangian Eulerian coordinates were employed to model the flow. The flow mesh was updated using a fast dynamic mesh technology proposed by our research group. The pressure was calculated by solving the Reynolds-averaged Navier–Stokes equations through the SIMPLE algorithm with the updated flow mesh. The aerodynamic force for the wing was computed using the pressure on the wing surface. Then the aerodynamic damping of the wing vibration was computed. Finally, the flutter stability for the wing was decided according to whether the aerodynamic damping was positive or not. Considering the first four modes, the aerodynamic damping for wing 445.6 was calculated using the present method. The results show that the aerodynamic damping of the first mode is lower than the aerodynamic damping of higher order modes. The aerodynamic damping increases with the increase of the mode order. The flutter boundary for wing 445.6 was computed using the aerodynamic damping of the first mode in this paper. The calculated flutter boundary is consistent well with the experimental data.

Updated Lagrangian/Arbitrary Lagrangian–Eulerian framework for interaction between a compressible neo‐Hookean structure and an incompressible fluid

SummaryWe propose a numerical method for a fluid–structure interaction problem. The material of the structure is homogeneous, isotropic, and it can be described by the compressible neo‐Hookean constitutive equation, while the fluid is governed by the Navier–Stokes equations. Our study does not use turbulence model. Updated Lagrangian method is used for the structure and fluid equations are written in Arbitrary Lagrangian–Eulerian coordinates. One global moving mesh is employed for the fluid–structure domain, where the fluid–structure interface is an ‘interior boundary’ of the global mesh. At each time step, we solve a monolithic system of unknown velocity and pressure defined on the global mesh. The continuity of velocity at the interface is automatically satisfied, while the continuity of stress does not appear explicitly in the monolithic fluid–structure system. This method is very fast because at each time step, we solve only one linear system. This linear system was obtained by the linearization of the structure around the previous position in the updated Lagrangian formulation and by the employment of a linear convection term for the fluid. Numerical results are presented. Copyright © 2016 John Wiley & Sons, Ltd.

A monolithic geometric multigrid solver for fluid-structure interactions in ALE formulation

We present a monolithic geometric multigrid solver for fluid-structure interaction problems in Arbitrary Lagrangian Eulerian coordinates. The coupled dynamics of an incompressible fluid with nonlinear hyperelastic solids gives rise to very large and ill-conditioned systems of algebraic equations. Direct solvers usually are out of question because of memory limitations, and standard coupled iterative solvers are seriously affected by the bad condition number of the system matrices. The use of partitioned preconditioners in Krylov subspace iterations is an option, but the convergence will be limited by the outer partitioning. Our proposed solver is based on a Newton linearization of the fully monolithic system of equations, discretized by a Galerkin finite element method. Approximation of the linearized systems is based on a monolithic generalized minimal residual method iteration, preconditioned by a geometric multigrid solver. The special character of fluid-structure interactions is accounted for by a partitioned scheme within the multigrid smoother only. Here, fluid and solid field are segregated as Dirichlet–Neumann coupling. We demonstrate the efficiency of the multigrid iteration by analyzing 2d and 3d benchmark problems. While 2d problems are well manageable with available direct solvers, challenging 3d problems highly benefit from the resulting multigrid solver. Copyright © 2015 John Wiley & Sons, Ltd.

A Fully Eulerian formulation for fluid–structure-interaction problems

In this work, we present a Fully Eulerian framework for fluid–structure interaction (fsi) problems coupling the incompressible Navier–Stokes equations with a hyperelastic solid.The Fully Eulerian framework is a monolithic implicit variational formulation for the coupled problem. In contrast to the well-established Arbitrary Lagrangian Eulerian (ALE) coordinates, the Fully Eulerian framework formulates both subproblems, fluid and solid, in Eulerian coordinates. This concept circumvents various difficulties connected to ALE coordinates since no artificial domain mapping is used. The formulation is an interface-capturing method and uses an extension of the solid’s deformation, the Initial Point Set, to detect the interface location.By construction, very large deformation as well as topology changes like contact of the solid to the domain boundary or other solid parts are possible.

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.

Investigation into the fluid—structure interaction of a hydraulically damped rubber mount on the basis of finite element analysis

This paper investigates the working process simulation of a hydraulically damped rubber mount (HDM) based on finite element (FE) analysis of the fluid—structure interaction (FSI). The direct computing method of coupled FSI formulation in arbitrary Lagrangian—Eulerian coordinates and the mixed displacement—pressure FE formulation for the incompressible medium, rubber, are adopted. The constitutive law of rubber is determined by FE analysis of incompressible material. A proportion mesh control algorithm helps to overcome fluid mesh distortion. Comparison of the predicted and experimental static elasticities verifies the effectiveness of the presented modelling approach of an HDM. Analysis of fluid pressure—velocity fields and deformation—stress fields of rubber components under typical working conditions clarifies the working process of the HDM, which can help to evaluate the carrying capacity, to identify the chamber volumetric characteristics, and to determine the fluid field distribution and stress distribution of rubber parts for the structural design of the HDM.

Numerical simulation of the flow around oscillating wind turbine airfoils Part 2: Free vibrating airfoil

The present paper is Part 2 of a two parts paper on flow around vibrating wind turbine airfoils. The first part of the paper dealt with a forced oscillating airfoil. Part 2 focuses on free vibrating airfoils. The flow induced vibrations on two airfoils used for wind turbine blades are investigated by applying a fluid structure interaction approach. A commercial Computational Fluid Dynamics (CFD) code is coupled to a computational structural program that solves the dynamic equations of the airfoil oscillations. The fluid governing equations are described in the Arbitrary Lagrangian Eulerian coordinates and solved with a moving mesh. A straightforward meshing technique is implemented in a subroutine called by the CFD code at each time step for updating the grid. The method is applied to a free pitch oscillating airfoil and to combined pitch and vertical oscillations known as the flutter instability

Numerical simulation of a pulsatile flow through a flexible channel

An algorithm for approximation of an unsteady fluid-structure interaction problem is proposed. The fluid is governed by the Navier-Stokes equations with boundary conditions on pressure, while for the structure a particular plate model is used. The algorithm is based on the modal decomposition and the Newmark Method for the structure and on the Arbitrary Lagrangian Eulerian coordinates and the Finite Element Method for the fluid. In this paper, the continuity of the stresses at the interface was treated by the Least Squares Method. At each time step we have to solve an optimization problem which permits us to use moderate time step. This is the main advantage of this approach. In order to solve the optimization problem, we have employed the Broyden, Fletcher, Goldforb, Shano Method where the gradient of the cost function was approached by the Finite Difference Method. Numerical results are presented.

The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries

Using the Arbitrary Lagrangian Eulerian coordinates and the least squares method, a two-dimensional steady fluid structure interaction problem is transformed in an optimal control problem. Sensitivity analysis is presented. The BFGS algorithm gives satisfactory numerical results even when we use a reduced number of discrete controls.

A stabilized finite element procedure for turbulent fluid–structure interaction using adaptive time–space refinement

AbstractThis paper presents our effort to addressing fluid–structure interaction (FSI) problems by means of computational mechanics. A stabilized finite element formulation is used to solve the incompressible Navier–Stokes equations written in primitive variables. The structure is modelled using rigid‐body dynamic equations solved using a Runge–Kutta method. The distinctive feature of our approach is the combination of large eddy simulation (LES)—based on implicit turbulence modelling—with time–space adaptive techniques in arbitrary Lagrangian Eulerian co‐ordinates (ALE). Three representative numerical examples are presented. The first one is the simulation of turbulent vortex shedding around a fixed obstacle, comparing our two‐dimensional (2D) LES results with experiments and more refined three‐dimensional (3D) numerical solutions. The second example presents an ALE computation with moving boundaries, where we were able to detect the lock‐in phenomenon for an oscillating cylinder driven by periodic vortex shedding. Our final example is the FSI problem associated to the flow around a dominant central span section of the Rio‐Niterói bridge, where we compare our results with experiments performed in a wind tunnel. Copyright © 2004 John Wiley & Sons, Ltd.

Numerical Investigation of Unsteady Transitional Flow Over Oscillating Airfoil

A numerical investigation of unsteady transitional flow over an oscillating NACA 0012 airfoil at a Reynolds number of 44,000 and reduced frequency of 0.2 is carried out and the results are compared with experimental data. The Navier-Stokes equations defined on arbitrary Lagrangian-Eulerian coordinates are solved by a time-accurate finite volume method that incorporates an incremental pressure equation for the conservation of mass. The transitional turbulence field is described by multiple-time-scale turbulence equations. The numerical method successfully predicts the large dynamic stall vortex (DSV) and the trailing edge vortex (TEV) that are periodically generated by the oscillating airfoil. The numerical results show that the transition from laminar to turbulent state and relaminarization occur widely in time and in space. The calculated streaklines and the ensemble-averaged velocity profiles are in good agreement with the measured data.