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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