A coupled meshfree-mesh-based solution scheme on hybrid grid for flow-induced vibrations

TLDR: In this article, a coupled mesh-free-mesh-based fluid solver is employed for flow-induced vibration problems, where the fluid domain comprises of a hybrid grid which is formed by generating a body conformal mesh free nodal cloud around the solid object and a static Cartesian grid which surrounds the mesh free cloud in the far field.

Abstract: In this paper, a coupled meshfree-mesh-based fluid solver is employed for flow-induced vibration problems. The fluid domain comprises of a hybrid grid which is formed by generating a body conformal meshfree nodal cloud around the solid object and a static Cartesian grid which surrounds the meshfree cloud in the far field. The meshfree nodal cloud provides flexibility in dealing with solid motion by moving and morphing along with the solid boundary without necessitating re-meshing. The Cartesian grid, on the other hand, provides improved performance by allowing the use of a computationally efficient mesh-based method. The flow equations, in arbitrary Lagrangian–Eulerian formulation, are solved by a local radial basis function in finite difference mode on moving meshfree nodes. Conventional finite differencing is used over the static Cartesian grid for flow equations in Eulerian formulation. The equations for solid motion are solved using a classic Runge–Kutta method. Closed coupling is introduced between fluid and structural solvers by using a sub-iterative prediction–correction algorithm. In order to reduce computational overhead due to sub-iterations, only near-field flow (in the meshfree zone) is solved during the inner iterations. The full fluid domain is solved during outer (time step) iterations only when the convergence at the solid–fluid interface has already been reached. In the meshfree zone, adaptive sizing of the influence domain is introduced to maintain suitable number of neighbouring particles. The use of a hybrid grid has been found to be useful in improving the computational performance by faster computing over the Cartesian grid as well as by reducing the number of computations in the fluid domain during fluid–solid coupling. The solution scheme was tested for problems relating to flow-induced cylindrical and airfoil vibration with one and two degrees of freedom. The results are found to be in good agreement with previous work and experimental results.

Figures

Fig. 19 1-DoF cylindrical vibration at Re=100: Variation of parameters with effective elasticity (k∗eff ) ( —•—Shiels et al. [50], O Present work (Ordered grid), 4 Present work (Randomized grid)

Fig. 20 Comparison of vorticity plots for ’lock-in’ and ’un-locked’ configurations

Fig. 7 Activation and deactivation of meshfree nodes during movement of meshfree grid

Fig. 13 Error Plots for static test case at t=0.5 ( *-RMS error, o-Max error)

Fig. 25 Variation in system response with changing frequency ratio (fNx/fNy) for cylindrical vibration with 2-DoF at = m∗ = 1.25 and Re = 150. −o− fNx/fNy = 2.0, − ∗ − fNx/fNy = 1.5, − − fNx/fNy = 1.0

Fig. 1 Hybrid grid configuration in fluid domain

Full text

Acta Mechanica
A coupled meshfree-mesh based solution scheme on hybrid grid for flow induced
vibrations
--Manuscript Draft--
Manuscript Number: ACME-D-15-00558R1
Full Title: A coupled meshfree-mesh based solution scheme on hybrid grid for flow induced
vibrations
Article Type: Original Paper
Keywords: Fluid Structure Interaction; Meshfree Methods; Hybrid grid; RBF-FD; Partitioned FSI;
Aeroelasticity
Corresponding Author: Ali Javed, PhD
Southampton, Hampshire UNITED KINGDOM
Corresponding Author Secondary
Information:
Corresponding Author's Institution:
Corresponding Author's Secondary
Institution:
First Author: Ali Javed, PhD
First Author Secondary Information:
Order of Authors: Ali Javed, PhD
Kamal Djijdeli, PhD
Jing T. Xing, PhD
Order of Authors Secondary Information:
Funding Information:
Abstract: In this paper, a coupled meshfree-mesh based fluid solver is employed for flow
induced vibration problems. Fluid domain comprises of a hybrid grid which is formed by
generating a body conformal meshfree nodal cloud around the solid object and a static
Cartesian grid which surrounds the meshfree cloud in the far field. The meshfree nodal
cloud provides flexibility in dealing with solid motion by moving and morphing along
with the solid boundary without necessitating re-meshing, and the Cartesian grid, on
the other hand, provides improved performance by allowing the use of computationally
efficient mesh based method. Flow equations, in Arbitrary Lagrangian-Eulerian (ALE)
formulation, are solved by local Radial Basis Function in Finite Difference mode (RBF-
FD) on moving meshfree nodes. Conventional finite differencing is used over static
Cartesian grid for flow equations in Eulerian formulation. The equations for solid motion
are solved using classical Runge Kutta method. Closed coupling is introduced between
fluid and structural solvers by using a sub-iterative prediction-correction algorithm. In
order to reduce computational overhead due to sub-iterations, only near field flow (in
meshfree zone) is solved during inner iterations, and the full fluid domain is solved
during outer (time step) iterations only when the convergence at solid-fluid interface
has already been reached. In meshfree zone, adaptive sizing of influence domain has
been introduced to maintain suitable number of neighbouring particles. The use of
hybrid grid has been found to be useful in improving the computational performance by
faster computing over Cartesian grid as well as by reducing the number of
computations in the fluid domain during fluid-solid coupling. The solution scheme was
tested for problems relating to flow induced cylindrical and airfoil vibration with one and
two degrees of freedom. The results are found to be in good agreement with previous
work and experimental results.
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The Editor and Reviewers,
Acta Mechanica
18 January 2016
Dear Sir/Madam,
Revised Manuscript: A coupled meshfree-mesh based solution scheme on hybrid
grid for flow induced vibrations
Manuscript Reference No: ACME-D-15-00558
Reference: Your email on 04 January, 2016
We gratefully acknowledge the feedback about the manuscript. The comments from
anonymous reviewers are found to be quite useful. Necessary amendments have been
incorporated in the manuscript to account for the remarks by the reviewers. The comments
from the reviewers are reproduced below, accompanied with point-by-point responses
illustrating how the manuscript has been changed based on the reviews comments rovided
therein.
It is believed that the attached submission fulfils the journal requirements you will consider
it ready for publication in Acta Mechanica.
Sincerely,
Sincerely,
(Ali Javed)
Corresponding Author
Click here to download Authors' response to Reviewers'
comments Letter_ACME-D-15-00558_Corrections.docx

Comments
Reply
Remarks of Reviewer-1
Comment-1
I do not understand how the
pressure problem is solved across
the different domains. Being the
pressure problem an elliptic problem,
it should be solved directly in the
whole domain considered otherwise
some errors are introduced.
It seems that in the outer region,
where the Cartesian mesh is used, a
Dirichlet velocity condition is
prescribed on the inner boundary
(Cat-III points). Which conditions are
imposed for the pressure on these
points? How is solved the pressure
problem in the mesh-less domain?
Where are imposed the Boundary
Conditions for the pressure in the
mesh-less domain? Can the authors
provide a detailed description of
these important aspects?
The pressure problem is set up separately in
each domain (meshfree and Cartesian zones).
In each time step, de-coupled momentum
equations and pressure poison equation (Eqs.
(5) to (7)) are first solved in meshfree zone. The
governing equations are then solved in
Cartesian zone. During solution of Eqs. (5) to (7)
on Cartesian zone, the values of pressure and
velocity at Cat-III nodes are used as boundary
conditions on inner boundary (interface of the
two zones) of Cartesian zone. In order to further
clarify this, the words ‘field parameters’ are
replaced by the words ‘field parameters
(pressure and velocity values)’ in line 272, 298
and 308 respectively.
Nevertheless, the idea coined by the reviewer is
quite interesting. Instead of solving separate
pressure problems in meshfree and Cartesian
zones, a single problems can be setup, for
pressure, in the entire domain. The proposed
scheme is similar to what was used by Peng and
Street in 1991 (A coupled multigrid domain-
splitting technique for simulating incompressible
flows in geometrically complex domains). It is
likely that this will improve the accuracy of the
solution. However, simultaneous solution for
pressure will compromise the modular
characteristics of the solution scheme. Currently,
the meshfree and mesh based solvers run
independent to each other exchanging data at
the interface nodes. Simultaneous solution of
pressure equations over the entre domain would
require the derivation of a separate pressure
equation applicable to all the zones. Further
research can however be conducted in this
direction.
For this purpose, remarks have been included in
the conclusion section of the manuscript (line
819 - 831).

Comment-2
In the final part of the Introduction
there is a detailed description of the
proposed method (more than 1
page). I think that it could be
reduced since it is a repetition of
what shown in the successive
sections. On the other hand, I think
that other methods recently adopted
in simulating moving objects need to
be discussed in the introduction. E.g.
The Immersed Boundary Method,
see e.g. Uhlmann JCP 2005, Picano
et al. JFM2015.
As suggested by the reviewer, the description of
current method in the introduction section is
reduced. Moreover, a description of various grid
generation techniques has been (including
immersed boundary methods suggested by the
reviewer) is included in the introduction (line 33 -
47).
Comment-3
In order to be easier to read, several
figures need to be regenerated
making larger plots and labels. E.g.
fig 13, 14 (color codes are missing),
15, 19, 24
All the figures have been enlarged to improve
readability.
Comment-4
There are some typos. E.g. pg 5.
l.98 Arbitrary, pg.10 ln 202
Multiquadratic
The typos have been corrected. Moreover, a
review of the manuscript has been made to
ensure that it is free from such mistakes.
Remarks of Reviewer-2
Comment-1
The only minor points concern the
quality of figures 5-8 where it's hard
to discern graphically the different
categories of points.
The figures have been enlarged in order to
improve their readability.

Noname manuscript No.
(will be inserted by the editor)
A. Javed · K. Djijdeli · J. T. Xing
A coupled meshfree-mesh based solution scheme on
hybrid grid for flow induced vibrations
Received: date / Accepted: date
Abstract In this paper, a coupled meshfree-mesh based fluid solver is employed for flow induced
vibration problems. Fluid domain comprises of a hybrid grid which is formed by generating a body
conformal meshfree nodal cloud around the solid object and a static Cartesian grid which surrounds
the meshfree cloud in the far field. The meshfree nodal cloud provides flexibility in dealing with solid
motion by moving and morphing along with the solid boundary without necessitating re-meshing.
The Cartesian grid, on the other hand, provides improved performance by allowing the use of com-
putationally efficient mesh based method. Flow equations, in Arbitrary Lagrangian-Eulerian (ALE)
formulation, are solved by local Radial Basis Function in Finite Difference mode (RBF-FD) on moving
meshfree nodes. Conventional finite differencing is used over static Cartesian grid for flow equations in
Eulerian formulation. The equations for solid motion are solved using classical Runge Kutta method.
Closed coupling is introduced between fluid and structural solvers by using a sub-iterative prediction-
correction algorithm. In order to reduce computational overhead due to sub-iterations, only near field
flow (in meshfree zone) is solved during inner iterations. The full fluid domain is solved during outer
(time step) iterations only when the convergence at solid-fluid interface has already been reached. In
meshfree zone, adaptive sizing of influence domain is introduced to maintain suitable number of neigh-
bouring particles. The use of hybrid grid has been found to be useful in improving the computational
performance by faster computing over Cartesian grid as well as by reducing the number of compu-
tations in the fluid domain during fluid-solid coupling. The solution scheme was tested for problems
relating to flow induced cylindrical and airfoil vibration with one and two degrees of freedom. The
results are found to be in good agreement with previous work and experimental results.
Keywords Fluid Structure Interaction · Meshfree Methods · Hybrid grid · RBF-FD · Partitioned
FSI · Aeroelasticity
1 Introduction1
Meshfree methods refer to the class of computational techniques in which, at least, the structure of2
mesh is eliminated and the solution is approximated over a set of arbitrarily distributed data points3
(or nodes). In the absence of pre-specified grid connectivity constraint, computational nodes can be4
moved, added or removed more flexibly, from computational domain, during the simulation. Owing5
to these features, meshfree methods are considered to be better suited for problems involving large6
deformation, moving boundaries and complex geometries [6]. However, meshfree methods developed7
so far, are in general, computationally more expensive than conventional mesh based methods.8
Radial Basis Functions (RBF) are primarily used for multivariate data interpolation over scattered9
data points. They are ’truly’ meshfree in nature and can be also be used for solution of differential10
A. Javed · K. Djijdeli · J. T. Xing
Faculty of Engineering and Environment, University of Southampton, UK
E-mail: salijav@hotmail.com
Click here to download Manuscript Manuscript Acta
Mechanica2.tex
Click here to view linked References

Frequently asked questions

Q1. What are the contributions mentioned in the paper "A coupled meshfree-mesh based solution scheme on hybrid grid for flow induced vibrations" ?

In this paper, a coupled meshfree-mesh based fluid solver is employed for flow induced vibration problems. The meshfree nodal cloud provides flexibility in dealing with solid motion by moving and morphing along with the solid boundary without necessitating re-meshing. The Cartesian grid, on the other hand, provides improved performance by allowing the use of computationally efficient mesh based method. 

Q2. How is the movement of mesh accomplished?

The movement of mesh is accomplished by displacing594 the grid nodes (of meshfree zone) only according to prescribed pitching motion. 

Q3. What makes the cylinder an attractive choice to test current FSI solution scheme?

Simple geometry and well established results437 available in literature make it an attractive choice to test current FSI solution scheme. 

Q4. What is the use of adaptive shape parameters for RBFs?

Adaptive shape86 parameters are used for RBFs, as suggested by [31], to ensure well conditioning coefficient matrices over87 a grid with variable nodal density. 

Q5. What is the order of convergence in time for pressure and velocity field?

Order of convergence in time is found to be around 1.0 for both pressure355 and velocity field with known pressure boundary conditions. 

Q6. What is the simplest way to test the spatial order of convergence of a mesh?

In order to study spatial order of convergence, static test cases are run for time step value of 10−5346 and for varying grid sizes. 

Q7. How is the deficiency of partitioned methods overcome?

This deficiency of partitioned methods is overcome67 by the use of closely coupled systems, in which several inner or sub-iterations of fluid and structure68 solvers are run, within a single time step, to reach convergence at the interface before moving on to the69 next time step [14,20,12]. 

Q8. How are the flow parameters assigned to the newly activated nodes?

in order to244 set these newly activated nodes for next time step calculations, field parameter (pressure and velocity)245 values are assigned by interpolating the data from surrounding nodes. 

Q9. What is the effect of the condition number of interpolation matrix for RBFs?

In fact, condition number of interpolation matrix for RBFs grows with increasing the number of com-261 putational nodes participating in derivative approximation at a certain point [49]. 

Q10. How many collocation data points in the influence domain can be subdued?

At the same time, requirement of sufficient number of collocation data points266 in the influence domain to ensure accurate derivative approximation using local RBF [12] cannot be267 subdued. 

Q11. how much slope of the error curve decreases with increasing grid speeds?

slope of the error curve reduces at higher speeds making error to stabilize and not to367 increase with further increase in grid velocities. 

Q12. What are the advantages of using meshfree methods?

meshfree methods developed7 so far, are in general, computationally more expensive than conventional mesh based methods. 

Q13. What is the heave velocity in a single oscillation period?

Variation of heave and pitch displacements in a single oscillation period of airfoil, for θ0 = 76.33 o, ω =599 0.28π case, is shown in Fig. 28(a).