Handling of Rotating Geometries in the Context of

Finite Element Methods CCES-Seminar WS 12/13

Maximilian Harmel

RWTH Aachen

Supervisor: Dr.-Ing. Stefanie Elgeti

Chair for Computational Analysis of Technical Systems

RWTH Aachen

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Contents:

1 Introduction ........................................................................................................................ 2

2 The Shear-Slip Mesh Update Method ................................................................................. 2

2.1 The Deformable-Spatial-Domain/Stabilized Space-Time Formulation ....................... 2

2.2 The Idea of the SSMUM ............................................................................................... 4

2.3 Numerical Examples .................................................................................................... 5

2.4 Space-Time Shear-Slip Mesh Update Method ............................................................ 8

3 Isogeometric Analysis for the Computation of Flow about Rotating Components............ 9

3.1 Numerical Example: Two Propellers Rotating in Opposite Directions ...................... 11

4 Comparison and Conclusion ............................................................................................. 13

References ................................................................................................................................ 14

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1 Introduction

This elaboration was developed during the CES-seminar in winter semester 2012/13 at

RWTH Aachen University. It deals with the handling of rotating geometries in the context of

finite element methods. One possible approach, the shear-slip mesh update method

developed by Behr and Tezduyar, is presented in Section 2. A NURBS-based method,

established through Bazilevs and Hughes, is presented in Section 3. A comparison of both

methods concludes this elaboration.

Rotating geometries play an important role in a large field of simulation applications. Thus,

for example, flow past a propeller is a very common problem in aviation technology. Due to

the energy transition the simulation and computation of wind turbines becomes more and

more important [6].

2 The Shear-Slip Mesh Update Method (SSMUM)

In this section we consider flow problems with large but regular moving boundaries and

interfaces, such as straight-line transformations, rotations or a combination of those. In

section 2.1 the deformable-spatial-domain/stabilized space-time (DSD/SST), which covers

the case of small and medium deformations, is presented. The shear-slip mesh update

method (SSMUM) [1], which is developed to apply DSD/SST formulation to larger

deformations, is presented in section 2.2 in detail.

2.1 The Deformable-Spatial-Domain/Stabilized Space-Time Formulation

The DSD/SST formulation was introduced to model flow problems with moving boundaries

and interfaces. If there is a single translating or rotating object considered separately, the

coordinate system can be attached to the moving object. This simple approach does not

require a movement of the finite element mesh and models rotating geometries thus in an

effective way. However, if the model contains another object, which is fixed or in relative

motion to the first one, an attached coordinate frame will not lead to success.[1]

To obtain a practicable handling of multiple rotating or translating objects the mesh has to

be moved with respect to time. Thus, a remeshing of the objects is required, which leads to

a high computational effort. The projection from the old mesh to the new one also results in

a smearing of the solution. In the case of small or medium changes in shape, the number of

remeshing processes can be reduced or remeshing can be even prevented completely. In the

case of larger regular deformations, the DSD/SST approach is not practicable, because it

causes very high computational costs. To avoid this problem, the shear-slip mesh update

method (SSMUM) was developed. The basic idea behind this method is to use a thin layer of

shearable elements between two objects with relative motion to each other. We first

introduce the mathematical background of the DSD/SST formulation, which is also used in

the SSMUM.

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Differential Equation

The considered flow problems are described by a system of time-dependent Navier-Stokes

equations. This system of partial differential equations can be obtained by applying the law

of conservation of mass and momentum to a continuum. We assume the fluid to be

incompressible and viscous.

Due to the moving mesh in the DSD/SST formulation, the spatial domain may change with

respect to time. For the time-dependence of the bounded region with

boundary is denoted by the subscript , where is the number of space dimensions.

The unknowns of these equations are the velocity and the pressure . The

external forces acting on the flow (e.g. the gravity) are represented by . The

assumptions and denotations made above lead to the following form of the Navier-Stokes

equation:

(

)

Due to the incompressibility of the fluid, the density is constant. describes the stress

tensor which can be decomposed into its isotropic and deviatoric parts. For Newtonian fluids

the deviatoric stress is related linearly to the strain rate tensor:

⏟

⏟

where is the dynamic viscosity. The Dirichlet and Neumann boundary conditions are

represented as

where and are complementary subsets of the boundary . The initial

conditions at consist of a velocity field without divergence over the entire domain:

To take turbulence effects into account at high Reynolds numbers (turbulent flow), we have

to introduce a turbulence model. Here, the kinematic viscosity

is augmented by an

eddy viscosity .

Due to the moving boundaries and interfaces, as well as the computational costs, we have to

divide the time interval into subintervals with corresponding space-time slabs. The weak

formulation with test functions for the velocity and pressure at each space-time slab leads to

a nonlinear system of equation.

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This system is for example solved with the Newton-Raphson method. An iterative solution

technique is used to solve the resulting large linear system at each Newton-Raphson

iteration.

2.2 The Idea of the SSMUM

The basic idea of the shear-slip mesh update method is to divide the model into regions of

rigid non-deforming elements, which are connected with thin layers of deforming elements

that can absorb shearing. This reduces the number of elements that have to be remeshed

and thus the computational cost drastically. To illustrate this domain decomposition, two

simple examples for translation and rotation are given:

Figure 1: Sketches of domain decomposition [1]

In the left hand side of Figure 1 a translating object is embedded in a strip. The elements in

the strip move “glued” to the non-deforming elements of the moving object. The exterior

boundaries also consist of non-deforming elements. The regions of non-deforming elements,

which have a relative motion to each other, are connected by a layer of shear-absorbing

elements (gray in Figure 1).

The right hand side of Figure 1 shows a sketch of a rotating object that is embedded in a disc.

Analogous to the translating case, the elements of the disk rotate “glued” to the rotating

object in the center. Between the fixed boundary and the disk, there is a thin layer of shear-

absorbing elements (grey).

The shear-absorbing elements in the layer may deform during one time step. These

deformed elements have to be rebuilt in a proper way. To illustrate this process, a two-

dimensional translating object is shown in Figure 2.

Figure 2 shows a non-moving object (grey) and one object that translates upwards (pink).

The objects are connected by a single-element layer of shear-absorbing elements. In this

example these deformed elements all behave the same, so we can focus on one, highlighted

in red. After the first time step the red element is sheared, due to the moving of the pink

elements. To obtain again a shear-absorbing element with comparable quality, is has to be

reconnected to new pink nodes, which moved into proper position. In this example, the red

element is initially connected to the boundary nodes of element C.

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After the first time step the red element is sheared, due to the moving of the pink elements.

To obtain again a shear-absorbing element with comparable quality, it has to be

reconnected to new pink nodes, which moved into proper position. In this example, the red

element is initially connected to the boundary nodes of element C. After one time step it is

connected to these of Element E. After the reconnection of the deformed elements, the next

time step can be performed without projection of the solution.

Figure 2: Single-element shear-slip layer under translation, [1]

In general the thickness of the layer with the shear-absorbing elements can span more than

one element. Multi-element layers lead to a greater flexibility, but also to an increase of

computational costs.

2.3 Numerical Examples

2D Flow past Two Counter-Rotating Square Cylinders

The first numerical experiment to illustrate the capability of the SSMUM is a two-

dimensional model, which contains two square cylinders (which are modeled as squares in

2D) with equal dimensions (for all dimensions, see Figure 3). Both squares rotate in different

direction (lower square: clockwise, upper square: counterclockwise) but with the same

magnitude of rotation velocity (rotational velocity ). A free-stream horizontal

velocity of 1.0 is assumed on the left boundary (x=0). The Reynolds number based on the

upstream velocity and the size of the squares is 400.

The shear-slip layers that connect the rotating square cylinder with the fixed boundary are

located in a thin ring around the squares. Each of the layers is one element thick in radial

direction and divided in 160 elements in circumferential direction. The initial mesh consists

of 31,928 space-time nodes and 31,492 triangular elements, which are concentrated in the

vicinity of the squares.

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Figure 3: 2D flow past two counter-rotating square cylinders and close-up [1]

The computation is done on the parallel computer IBM SP2. The nonlinear equations at

every time step are solved with 4 Newton-Raphson iterations. The linear system that arises

at each iteration of the Newton-Raphson algorithm is solved in turn iteratively with the

GMRES update techniques.

Figure 4: Velocity field in the vicinity of two counter-rotating squares at t=125.0, 126.0,..., 133.0 [1]

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Figure 4 shows the velocity field in the vicinity of the squares. One complete revolution is

shown in nine equally spaced instants.

Moving opposite to the flow direction and entering the gap, the squares cause a regular high

frequency vortex shedding at their corners. We observe a lower frequency shedding

corresponding to flow past a compound object. The layer of shear-absorbing elements

seems not to affect strongly the results of the simulation.

3D Flow Past Helicopter

In this experiment [2] the flow past a helicopter with its main rotor in motion is considered,

but with omitted tail rotor. The fuselage (length: 13.22 m) of the helicopter is designed

according to the Boeing Sikorsky Comanche prototype. The five-blade rotor has a diameter

of 11.90 m. The helicopter is assumed to fly horizontally with a velocity of 10.0 m/s while its

main rotor is rotating counter-clockwise with a tip velocity of 200 m/s. The Reynolds number

based on the translation velocity and the diameter of the rotor is approximately .

Figure 5: Boeing Sikorsky helicopter: photo (left) and FE-model (right) [2]

To realize the SSMUM, the rotor is separated from the fuselage of the helicopter. The

rotating (interior) elements of the rotor are connected via shear-absorbing elements to the

stationary (exterior) elements. The shear-slip layer is an axisymmetric shell with interior

radius of 12.00 m and a thickness of 0.10 m. Only in the area of the close spacing between

the top of the fuselage and the rotor blades, the thickness of the layer is reduced to 0.04 m.

The shell is closed, except for the opening at the base of the rotor, so that small clearances

between the rotating and the stationary objects can be accommodated. The regular shear-

slip layer consists of one element in the axial, 80 elements in the circumferential and 100

elements in the radial direction. The layer is filled by a manually generated structured mesh.

There is a one-to-one correspondence between the surface elements on the interior and the

exterior surface. As the rotor rotates, new surface nodes on the inner surface become

aligned with the surface node on the outer surface. Due to the structuredness of the shear-

slip layer it is possible to recreate all shear-elements at the same time. The intervals of node-

reconnecting are specified by the circumferential resolution, the time step and the rotation

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velocity. The unstructured mesh in both rigid regions (the rotating and the stationary) is

generated by using an automatic mesh generator.

All in all the mesh of the whole helicopter (fuselage and main rotor) consists of 361,434

space-time nodes and 1,096,225 tetrahedral elements. The computation is carried out on

CRAY T3E-1200. The coupled nonlinear equations at every time step are again solved with 4

Newton-Raphson iterations. The linear system that arises at each Newton-Raphson step is

solved iteratively again.

Figure 1 shows the air pressure on the fuselage surface and on the rotor surface separately.

The result gives the highest pressures at the tips of the rotor blades where the highest

absolute velocity is located. The highest air pressure on the fuselage surface is located at the

tail of the helicopter, where the influence of the rotor is a maximum. Overall the tip air

pressure on the rotor is much higher than on the fuselage. This is a realistic result according

to the simulated load case.

Figure 6: air pressure at t= 0.625 s on the fuselage surface (left, limits: )

and on rotor surface (right, limits:

) [2]

The SSMUM, combined with efficient parallel implementation for distributed-memory

parallel computing, is a powerful technique for computation of complex flow-problems with

fast rotating mechanical components in 3D.

2.4 Space-Time Shear-Slip Mesh Update Method

After deforming of the shear-slip elements, the node connectivity might have to be changed.

In the original SSMUM there is no projection calculation performed. The sudden change in

the node connectivity from one time stab to the next leads to an instantaneous change in

the representation of the solution field. This discontinuous change decreases the

conservation property of the numerical method.

To solve this problem, the Space-Time Shear-Slip Mesh Update Method (ST-SSMUM) [3],

changes the spatial node connectivity continuously in the space-time domain. In contrary to

SSMUM, the spatial node connectivity is not changed from a previous time slab to the next

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one. Instead, the space-time connectivity takes the spatial movement into account to

decrease the deformation of the elements over space-time slabs.

In the following, the reconnection procedure of ST-SSMUM is illustrated by an example. A spatially two-dimensional space-time element deforms during one space-time slab due to the movement of its upper nodes (Figure 7, red and green) in positive x-direction while its lower nodes (Figure 7, red and green) are rigid. The node connectivity remains unchanged to the beginning of the second space-time slab. To avoid a complete deterioration of the element until the end of the second slab, we have to alter the spatial node connectivity. The movement of the upper nodes is taken into account to create new nodes (Figure 7, brown) at the end of this space-time slab in a way that the element is less deformed than in the beginning of this time slab. Thus, there is no projection error implicitly introduced. In the ST-SSMUM an element changes its shape during a space-time slab, but there is no connectivity alteration in the space-time framework.

Figure 7: Reconnecting Procedure of ST-SSMUM [3]

3 Isogeometric Analysis for the Computation of Flow about Rotating

Components

Handling of rotating components via NURBS-based isogeometric analysis as described in [4]

tries to overcome the decrease of the conservation properties of the classical SSMUM, due

to the instantaneous change in its solution field. So the motivation is similar to that of the

ST-SSMUM. In contrary to the SSMUM this approach is restricted on rotating components.

The case of translating components is not considered. The following considerations are

based on Bazilevs and Hughes [4].

The central idea of this approach is using non-rational uniform B-Splines (NURBS)-based

isogeometric analysis instead of the classical finite element method for spatial discretization.

For a detailed view on NURBS-based isogeometrc analysis, see [5]. NURBS are capable to

represent geometries exactly. We use this fact to embed a rotating body in a circular (2D) or

cylindrical (3D) domain with a unique interface between the rotating and the stationary

subdomains. That means that the rotating subdomain remains circular or cylindrical over

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time. There is no need of special procedures to ensure the geometric compatibility between

the subdomains with relative motion (see Figure 8).

Figure 8: Stationary domain contains a rotating domain: FE (left) and isogeometric analysis (right) [4]

The representation of geometries with NURBS supports is exact, but a compatibility of the

solution in the solution space is not imposed. Therefore we have to device a numerical

technique that imposes the continuity of the discrete solution weakly.

We want to model similar problems as with the SSMUM, so we again use incompressible

Navier-Stokes equations. The domain is subdivided into a stationary domain and a

rotating domain ( . The rotating domain rotates rigidly

inside with the rotating speed of its containing objects. The boundary between both

subdomains is defined as ̅ ̅ . The circular (2D) or cylindrical (3D) surface

will keep its shape, even under a rotation of . Discretization over both subdomains

leads to a variational equation that causes difficulties in solving. To deal with this problem,

we need a discrete formulation that imposes the continutity of the solution over the

interface weakly. This formulation is presented in detail below.

Let both subdomains be decomposed into NURBS elements. The discretization of that

is variable over time, is obtained by applying a rotation to the rotating subdomain at initial

time . It is sufficient to apply the rotation only to the control points of ), due to

the affine covariance property of NURBS. Discretization of lead to two

separate discretizations of the interface . These discretizations have to be made identical

by means of h-refinement (knot insertions). The refinement is performed by insertion of

knots of the discretized rotating boundary into the mesh of the discretized boundary

of the initial rotating boundary . This creation of a sliding interface is illustrated in

Figure 9. For details of h-refinement by knot insertion in NURBS-based isogeometric analysis

see [5].

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Figure 9: Creating a sliding interface by knot insertion [4]

3.1 Numerical Example: Two Propellers Rotating in Opposite Directions

In the following example the isogeometric handling is applied to a typical flow problem past

rotating objects. Two four-blade propellers are rotating in opposite directions with a

constant rotation velocity ( , with cyclic frequency ). Due to the high Taylor

number ( consecutive instabilities are expected to set in and create complex

vortex structures known as Taylor vortices. The Reynolds number of the flow is 196 in this

experiment.

The computational mesh in the initial configuration is shown in Figure 10 . The rotating

domain (propellers, middle) and the stationary domain (remaining, border) are connected by

a sliding interface. One can see that the mesh is non-matching at this interface even for the

initial configuration. To transfer information through the sliding interface the rotating

domain is rotated from its initial position. The rotation needs to be applied only on the

control points. The lengths of element at both sides of the interface have to be equal, which

is realized by h-refinement. The mesh consists of 4,360 quadratic NURBS-elements.

Figure 10: Computational mesh in initial configuration [4]

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The result of the computation is given for six different instants in Figure 11. After the abrupt start of the flow the propellers have to perform several revolutions to let the vertical structure appear. The first subfigures of Figure 11 are snapshots of the flow field after the flow loses its symmetry. The latter subfigures illustrate the flow field at later stages where smaller vortices are predominant. It is note-worthy that there is a continuity of the flow vectors over the sliding interface, although they were discontinuously discretized. Furthermore, the method produces a nearly continuous pressure field at the interface without having defined this condition explicitly.

In conclusion we can say that the NURBS-based handling of the two rotating propellers leads

to physically comprehensible results. Consistency, stability and adjoint consistency are built

in the method formulation, resulting in a robust and accurate procedure.

Figure 11: Velocity field at various times (t=60, t=124, t=242, t=460, t=604, t=874) [4]

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4 Comparison and Conclusion

Both approaches, the SSMUM and the NURBS-based, are capable of solving flow problems

with multiple rotating objects. In both cases, mentioned numerical experiments give realistic

results. The loss of continuity of the standard SSMUM is avoided in the ST-SSMUM by

changing the spatial node connectivity of a deformed element continuously in the space-

time domain. The practicability of the SSMUM even to handle more complex geometries is

illustrated.

The isogeometric approach of handling rotating objects considers the geometry exactly. This

is a big advantage of NURBS-elements in comparison to standard finite elements. The

continuity of the flow field over the interface is attributed to the geometric compatibility

engaged by the NURBS-based discretization. Only a rather simple example (rotating

propellers in 2D) is performed by Bazilevs and Hughes. Further works may show the

feasibility of NURBS-based handling of more complex rotating geometries, also in three

dimensions. In contrary to the (ST-)SSMUM, the NURBS-based formulation is restricted to

rotations, it does not consider translations.

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References

[1] Behr, M.; Tezduyar, T.;

The Shear-Slip Mesh Update Method,

Computer Methods in Applied Mechanics and Engineering 174: 174 - 261, 1999.

[2] Behr, M.; Tezduyar, T. E.;

Shear-Slip Mesh Update in 3D Computations of Complex Flow Problems with Rotating

Mechanical Components,

Computer Methods in Applied Mechanics and Engineering 190: 3189 - 3200, 2001.

[3] Schippke, H.; Zilian, A.; Modification of the Shear-Slip Mesh Update Method with Respect to Space-Time Finite Element Discretisation of Fluid Flows First ECCOMAS Young Investigators Conference (YIC 2012). ECCOMAS, April 2012. Aveiro, Portugal.

[4] Bazilevs, Y.; Hughes, T.J.R.;

NURBS-based isogeometric analysis for the computation of flows about rotating components Computational Mechanics 43(1): 143-150, 2008

[5] Cotrell, J.A.; Hughes, T.J.R.; Bazilevs, Y.;

Isogeometric Analysis - Toward Integration of CAD and FEA John Wiley & Sons, Chichester (UK), 2009 [6] Achmus, M.; Abdel-Rahman, K;

Numerische Untersuchung zum Tragverhalten horizontal belasteter Monopile-

Gründungen für Offshore-Windenergieanlagen Universität Hannover, 2004