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An overset grid methodology is developed for the fully coupled analysis of fluid-structure interaction (FSI) problems. The overset grid approach alleviates some of the computational geometry difficulties traditionally associated with Arbitrary-Lagrangian-Eulerian (ALE) based, moving mesh methods for FSI. Our partitioned solution algorithm uses separate solvers for the fluid (finite volume method) and the structure (finite element method), with mesh motion computed only on a subset of component grids of our overset grid assembly. Our results indicate a significant reduction in computational cost for the mesh motion, and element quality is improved. Numerical studies of the benchmark test demonstrate the benefits of our overset mesh method over traditional approaches.

We present a fully-coupled, arbitrary Lagrangian-Eulerian (ALE) fluid-structure interaction (FSI) algorithm that uses overset meshes [

Fluid-structure interaction modeling describes the complex interactions between a deformable solid structure with external and/or internal fluid flow. The fluid flow exerts forces on the wetted solid surface to cause defor- mation, and the resulting structural motion impacts the fluid flow. For such interaction, a two-way coupled model must be used, wherein the fluid flow and the solid deformations mutually affect each other. Moreover, for time-accurate transient models, a fully-coupled approach must be used such that all governing equations are si- multaneously satisfied (at least in the appropriate approximate sense) at each discrete time level.

Numerical simulations of transient, fully coupled^{1} FSI phenomena have been performed by many authors, us- ing a variety of approaches. The common problem that all approaches must overcome is the disparity in the “natural” coordinate frames of the materials: the equations of solid mechanics are cast in the Lagrangian (refer- ence) frame, while fluid dynamics equations are traditionally written in a fixed Eulerian (spatial) configuration. All computational schemes for coupled FSI problems must bridge these two coordinate frames.

One approach to resolving this mathematical discrepancy in the descriptions is to cast all of the governing equations into an Eulerian framework [

Fixed grid methods for FSI have a fixed Eulerian grid, and the solid-fluid interface position is solved for as part of the solution procedure. Gerstenberger et al. [

Examples of interface-capturing methods include the immersed boundary method [

Particle methods, such as smoothed particle hydrodynamics (SPH) or the immersed particle method, have also been applied to coupled FSI problems; see, e.g., [

Arbitrary Lagrangian-Eulerian methods, first studied in [

The ALE approach has been used for problems with large structural displacements [

Our overset grid approach to FSI problems shares similarities with both ALE and EL approaches. We utilize a fixed Eulerian background mesh for our fluid domain as in EL methods. We compute the structural solution on the fixed Lagrangian mesh. The fluid flow on the moving, boundary-conforming composite meshes is computed in the ALE frame. Solution coupling between the fluid and solid regions is done via the ALE approach; however, the flow coupling between the fixed background mesh and the composite meshes surrounding the solids is han- dled through the use of overset interpolation operators, which is somewhat analogous to the use of the coupling schemes that EL methods use to couple the Eulerian and Lagrangian frames. The difference is that in our me- thod we implicitly couple the Eulerian and the ALE frames rather than the Eulerian and Lagrangian frames.

The overset grid method for fully-coupled FSI simulations has only previously been studied by Wall et al. [

Overset grid methods coupling flow fields to rigid solids or reduced order models of solids have been rela- tively popular in the computational fluid dynamics community. Small deformation transonic flutter using an overset, implicit aeroelastic solver were studied in [

Our approach is similar to [

Another drawback of moving mesh ALE methods is the need to solve for “arbitrary” mesh geometry to accommodate the structural displacements. No physical equation governs the mesh motion, and the only driving parameter is the known displacements of the fluid-structure boundary. There have been numerous proposed methods on what equations will yield quasi-optimal mesh topology (see, e.g., [

The objective of the present work is to implement and demonstrate the advantages of coupling the overset mesh technology and an ALE, partitioned FSI solution algorithm. The governing equations, which are also outlined by Campbell and Paterson [

One of the main challenges for coupling fluid and solid domains arises from the “natural” mathematical descriptions of the physics. The governing equations for a fluid flow are cast in an Eulerian (spatial) description, while the equations for a solid are written in Lagrangian (referential) form. Aside from that, the balance laws governing the behavior of both the fluid and solid domains are identical. We introduce the general continuum mechanics balance laws in an arbitrary Lagrangian-Eulerian (ALE) frame which provides the framework to describe the Eulerian, Lagrangian, or an arbitrary frame of reference.

We first consider the balance of mass:

where

Performing a force balance and making use of the continuity equation leads to the momentum equation:

where

An additional constraint for the ALE approach is that the mesh velocity satisfy the Geometric Conservation Law (GCL) [

where

Equations (1) and (2) are referred to as the governing equations. Application of constitutive relationships provides the necessary closure of the governing equations. The constitutive relationships and resulting equations for the fluid and solid domains are provided next, followed by the procedure used to couple these domains at the fluid/solid interface

We consider only incompressible Newtonian fluids in this work. The constitutive equation for the Cauchy stress tensor is

where

Substitution of (4) into the momentum Equation (2) and utilizing (5) yields the Navier-Stokes equations:

where

We deal with three types of boundary conditions for our simulations: inflow

Equations (5) and (6) will be solved for pressure and velocity using an ALE formulation in the present work. The ALE formulation is required to accommodate the structural deformations, thus imparting a non-trivial mesh velocity into the fluid region. We describe the numerical method used and its implementation in Section 3.1.

The Lagrangian frame-of-reference specifies the material velocity is equal to the frame velocity

where

The numerical benchmarks published in Turek and Hron [

where

is the Green-Lagrange strain. The Piola-Kirchhoff stresses are related by

Boundary conditions for (10) on the Dirichlet

In this work, it happens that the Neumann partition of the boundary is coincident with the fluid-structure boundary

The fluid-structure interaction is accomplished by imposing fluid stresses on the solid (i.e., the tractions in (14)) and imparting the solid displacements and velocities to the fluid. The requirements for compatibility and the no-slip condition require the following:

where

Our FSI approach requires changes to the spatial discretization (“mesh”) to accommodate the structural displacements in the fluid domain. The solution of the governing equations for the solid provides the interfacial displacement; however, there is no physical specification of the manner in which the mesh should deform away from the interface. The only requirement is that the mesh points on

There have been numerous methods proposed to solve for the mesh motion; see the reviews in [

where

Equation (17) is similar to Equation (10) for the solid, but without the inertial or body force terms and with Cauchy stress instead of the first Piola-Kirchhoff stress. The mesh is represented by a linear-elastic solid

and uses a small-strain approximation

In either choice, the mesh motion equation governs the fluid mesh position, and the fluid solver approximates the mesh velocity based on the solution time step and the mesh displacement. For instances where the interface motion is a small fraction of the smallest characteristic length of the interface cells, only the boundary points of the fluid mesh are moved and the remainder of the mesh remains stationary.

Commercial software packages for Computer Aided Engineering (CAE) have recently been developing “multi- physics” capabilities, one of which is FSI. Most, if not all, of the available packages implement an ALE ap- proach. The ADINA System [

We utilize a finite volume method (FVM) for the fluid equations and a finite element method (FEM) for the solid equations. The overset capability is added through the use of the foamed Over library [

The governing fluid mechanics equations are solved using the open source software OpenFOAM [

The flow problems considered herein are transient and incompressible, and on moving meshes; therefore OpenFOAM’s icoDyMFoam solver provides a baseline for development. The only modifications necessary for our overset-FSI method were adding the overset grid capability (see Section 3.4 for details). The icoDyMFoam solver uses the PISO (Pressure Implicit Splitting of Operators) [

The structural mechanics governing equations are solved using the finite element approach, implemented in an author-written C++ program called feanl (finite element analysis non-linear). The governing Equation (10) is discretized via a Bubnov-Galerkin weighted residual method to obtain the weak form [

The governing mesh motion equations are solved using either OpenFOAM or feanl, depending on the type of mesh motion chosen at run time. Both approaches require a Dirichlet condition on the fluid/solid interface and the fluid mesh motions are computed. The Laplace mesh motion, represented by (16), uses a finite volume ap- proach within OpenFOAM to determine mesh deformation. Either a constant or a variable diffusivity is chosen and mesh motion is computed based on the motion of the interface vertices. Equation (16) is discretized using the fluid mesh and thus the mesh motion solution requires a separate equation solution but does not require a separate mesh.

The linear-elastic solid analogy, described by Equations (17)-(19), employs a finite element approach and is solved using feanl. This approach requires a finite element mesh that overlays the fluid mesh in the moving region, which can be the entire fluid domain, or a subset of the domain. This mesh in general is substantially coarser than the fluid mesh, and is limited only by 1) maintaining sufficient resolution at the fluid-solid interface to preserve the physical boundary, and by 2) producing sufficient quality of the morphed fluid mesh. An inter- polation scheme is employed within the finite element solver to prescibe motion of all fluid verticies that fall within the elements. The ability to employ a vastly coarser mesh reduces solution times. All aspects of the general-purpose finite element solver are applicable to the mesh motion solver, including variation of material properties to mimmic the variable diffusivity option of the Laplace mesh motion approach.

In the present study, the grid deformation that results from solving the coupled fluid-structure interaction is facilitated through the use of overset grids. The capability in this case is provided by an overset grid library named foamedOver that was previously added to OpenFOAM by one of the current authors (Boger) [

Several hole-cutting methods and interpolation schemes are available in Suggar++. The domain connectivity information for the cases presented here result from an octree-based hole cutting method and use a least-squares method for interpolation.

We employ the partitioned FSI approach of Campbell et al. [

We present numerical results for a single FSI validation problem defined by Turek and Hron [

Partitioned approach to FSI showing a fixed-point iteration with under-relaxation for tightly coupled solutions

The results presented herein demonstrate the behavior of three distinct mesh motion assemblies, as listed below. The name in bold type is used from this point forward to reference a mesh motion scheme.

1. Non-overset: A single body-fitted fluid mesh which is deformed throughout the entire fluid domain. This is the legacy approach.

2. Full Overset: A body-fitted overset mesh assembly where the motion of the entire assembly is computed over the entire fluid domain. This is the naive implementation of overset mesh technology into the existing algorithm.

3. Subset Overset: A body-fitted overset mesh assembly where the background mesh is assumed static and only the motion of the overset grid components attached to the deformable structure is computed.

In all of the above cases, the mesh motion displacement is computed by solving a vector-valued Laplace equation with boundary conditions prescribed by the exterior of the fluid domain and the deformation of the solid structure. As described in Section 3.3, we have implemented an additional solver option that can be used with any of the aforementioned mesh assemblies. Therefore, the two mesh motion solver options are:

1. Standard isotropic vector Laplace equation solver with variable diffusivity.

2. An overlay solver (described in Section 3.3) that uses an auxiliary finite element mesh that overlays the moving fluid region and is solved using feanl; variable mesh stiffness is accomplished through mesh property variation.

The reference test case proposed by Turek and Hron [

. Dimensions of the Turek benchmark test case [3] .

Dimension | Value [m] | |
---|---|---|

Channel Width | . Dimensions of the Turek benchmark test case [3] . | 0.41 |

Channel Length | . Dimensions of the Turek benchmark test case [3] . | 2.5 |

Cylinder Radius | . Dimensions of the Turek benchmark test case [3] . | 0.05 |

Flag Length | . Dimensions of the Turek benchmark test case [3] . | 0.35 |

Flag Width | . Dimensions of the Turek benchmark test case [3] . | 0.02 |

Cylinder Center | . Dimensions of the Turek benchmark test case [3] . | (0.2, 0.2) |

Diagram of the Turek and Hron FSI benchmark test case geometry [3]

(21). Physical constants, such as fluid velocity and material properties, were chosen to support simple periodic oscillations of the tail. We will compare our calculations with those of Turek and Hron [

The implementation of an overset grid FSI case requires the generation of three separate mesh components: the background fluid mesh, the dynamic overset fluid mesh attached to the deformable structure, and the solid mesh, as seen in

Decomposition of Overset FSI meshes: static background (gray), dynamic overset (blue), solid (red). Note the disparate number of cells between the solid and fluid meshes along the fluid-structure interface. The lower images demonstrate the ease with which the solid geometry can be changed

. Physical Properties of the Turek benchmark test case [3] .

Parameter | Value | Units |
---|---|---|

Fluid | ||

. Physical Properties of the Turek benchmark test case [3] . | . Physical Properties of the Turek benchmark test case [3] . | . Physical Properties of the Turek benchmark test case [3] . |

. Physical Properties of the Turek benchmark test case [3] . | . Physical Properties of the Turek benchmark test case [3] . | . Physical Properties of the Turek benchmark test case [3] . |

. Physical Properties of the Turek benchmark test case [3] . | 1 | . Physical Properties of the Turek benchmark test case [3] . |

Solid | ||

. Physical Properties of the Turek benchmark test case [3] . | . Physical Properties of the Turek benchmark test case [3] . | . Physical Properties of the Turek benchmark test case [3] . |

. Physical Properties of the Turek benchmark test case [3] . | 0.4 | [ ] |

. Physical Properties of the Turek benchmark test case [3] . | . Physical Properties of the Turek benchmark test case [3] . | [Pa] |

overset fluid meshes is performed by Suggar++, as described in Section 3.4. The result of the overset approach is a solver with the ability to easily add and modify solid features in a given fluid domain, as illustrated by

Before delving into the formal study, we present in

To address the issue of verification and validation, we extract from our simulations the displacement of the vertical midpoint of the the trailing end of the flexible tail and compare it to the values reported by Turek and Hron [

In

The right image in

This section contains the primary result of this research effort, namely the demonstration of significant improvement in mesh quality during large deformation FSI simulations via the implementation of overset grid methods. We compare three reproductions of the FSI2 test case in this section: the legacy non-overset mesh approach, the overset mesh approach, and the overset mesh approach with the overlay mesh motion solver. We note that there is no difference in the solution between the full and subset overset techniques, and as such we only report results of the subset overset approach here.

Still images of our reproduction of the FSI2 case from Turek and Hron [3] , starting at the upper left and moving right then down and left and so on. The grey scale represents velocity magnitude, with contours showing re- gions of equal value

Verification of the overset FSI algorithms. We compare the cal- culated tip displacement between all three mesh assembly approaches (left) and our calculated displacements to those from Turek and Hron [3] (right). All of the approaches produce the same results and match the previously re- ported results

notably, cell deformations are distributed through the domain in the non-overset case. This leads to stretched cells at the top of the channel and compressed cells at the bottom. Concerns in cell skewness and aspect ratio occur in regions that would be important for boundary layers in this case, but in general the motion of the mesh in the entire domain is undesirable for preserving accuracy. However, in the overset case, background mesh quality is maintained due to the fact that the overset mesh is moved independently. This leads to a predictable background mesh in FSI problems, which is important for preserving mesh quality in areas of importance away from the fluid-structure interface.

In the overset fluid mesh (middle,

One way to resolve this involves stiffening the mesh around the tail, which results in the mesh remaining tangent to the tip of the tail. This is precisely what is done in the overlay mesh motion solver. As shown in the

Comparison of mesh motion strategies: the legacy non-overset approach (top), the overset approach with the same mesh motion solver as the legacy approach (middle), and the overset approach with the overlay mesh motion solver (bottom). While all three approaches find the same deflection, the mesh quality clearly improves from top to bottom. The introduction of overset mitigates much of the mesh distortion throughout the domain, and the improved non-uniform stiffness mesh motion solver greatly improves the mesh quality in the deformed overset mesh

Zoomed view of the fluid mesh surrounding the tail in Figure 6. The non-overset case (left) and overset case (middle) show similar mesh distortion at the tip of the tail, as is expected since they were computed with the same standard mesh motion solver. The overset case with the non-uniform stiffness overlay mesh motion solver (right) corrects this issue and creates a large displacement mesh with very good cell quality

bottom image of

In summary, all of the mesh assembly methods we have proposed are capable of reproducing the results in the FSI2 case from Turek and Hron [

In the previous section, we address and demonstrate the improvement in mesh quality during large deformation simulations that is enabled via overset mesh techniques. In the present section we explore the computational cost of this added capability. To do so, we track the time required to run the FSI2 case from the Turek and Hron [

In order to judge the net effect of reducing the mesh motion solver time on the entire simulation time, we include in

This is an important observation, given that we have added functionality, i.e., improved mesh quality for large deformations, while retaining or reducing the run-time of the problem. This benefit is expected to be amplified for three dimensional domains, providing further advantage of the overset approach.

Average wall-time for mesh motion solution per timestep plotted as a func- tion of simulation time. The naive addition of overset mesh technology into the solver increases the mesh motion wall-time, while the optimization for moving only the sub- set of the overset assembly attached to the deforming structure results in reduced wall- time

Average wall-time for the fluid and structural solutions per timestep (a) plotted as a function of simulation time, and the total solver average wall- time (b). The fluid solver dominates the run-time of this problem, but the over- ball simulation time does not increase with the addition of the overset meshing technology

We have developed and demonstrated an FSI simulation method based on the ALE formulation of the governing equations. Our partitioned algorithm allows for the modular introduction of separate, optimized fluid and structural solvers that are joined by a custom framework to pass information between them and perform the necessary mesh motion. The focus of this work is on the mesh motion component of the problem, as it is the source of many difficulties in FSI simulations where large deformation occurs.

To this end, we have proposed to use overset grid technology to alleviate some of the mesh motion difficulties. We have implemented overset technology into the FSI algorithm, verified it against the legacy mesh motion technique, and validated it against previously published computational results. When only the motions of the overset mesh components attached to the deforming immersed structure are calculated, we are able to simulta- neously preserve mesh quality without expensive re-meshing, but also fundamentally maintain or reduce the runtime of the algorithm. A key component of this achievement is the use of the non-uniform stiffness overlay solver to compute the mesh motion in a manner that optimizes deformation of the mesh away from the fluid- structure interface.

While the authors are excited about the results contained in this work, we are also optimistic that when this algorithm is applied to more complex geometries and three dimensional domains, where mesh motion can be a large fraction of the computation time, we will see an even more pronounced improvement in mesh quality and performance. We plan to extend this solver and apply it to three dimensional RANS simulations in the near fu- ture.

The authors would like to acknowledge support for this research from the Applied Research Laboratory at The Pennsylvania State University, especially the student support from the ARL Undergraduate Honors Program.