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Radial Basis Functions for Interface Interpolation and Mesh Deformation

3.5 3D mesh deformation

To show the capability of the new method for 3D applications we consider in this

section a test case similar to test case 1, only this time in a 3D domain and a real

FSI computation considering the AGARD 445.6 test case.

3.6 Rotation and translation of 3D block

We consider a test case in a 3D domain similar to test case 1 (section 3.3.1). A cross-

section showing the initial mesh and location of the block is given in Figure 38. The

block is translated 2.5 times the thickness of the block and rotated 15 degrees in

all three directions. A cross-section of the ﬁnal mesh and location of the block is

shown in Figure 39. Visually it is quite hard to judge the quality of the mesh and

Fig. 38 Cross-section of initial mesh.

Fig. 39 Cross-section of ﬁnal mesh.

therefore the values of the mesh quality metric in the cross-sections is displayed

in Figures 40 and 41 for the CP C

and TPS function, respectively. It can be seen

that the mesh quality is everywhere larger than 0.5 and therefore the meshes are

suitable for computational analysis. The main difference between the two ﬁgures

is that with CP C

the mesh quality close to the moving structure remains a little

higher than with TPS.

173

A. de Boer, A.H. van Zuijlen, and H. Bijl

Fig. 40 Mesh quality in cross-section using

CP C

Fig. 41 Mesh quality in cross-section using

TPS.

3.6.1 Flutter of the AGARD 445.6 wing

To demonstrate the practical applicability for real-world three-dimensional ﬂuid-

structure interaction the AGARD 445.6 test case [37, 43] is used. The consistent

approach is used for the interpolation of data at the ﬂuid-structure interface. For the

investigation into the performance of the third order IMEX scheme [39] with the

RBF mesh deformation a time step convergence study is performed. The solution

obtained with

∆

t = 0.001 is taken as the temporally exact solution. In Fig. 42 the

results for the pressure ﬁeld, velocity in vertical direction w, structural displacement

and lift at the end of the simulation are shown. The ﬁgures show a clear third order

-3 -2.8

-2.6

-2.4 -2.2 -2 -1.8

log ∆t

-6

-5

-4

-3

-2

-1

log ε

| p |

|| p ||

max

(a) Pressure.

-3 -2.8

-2.6

-2.4 -2.2 -2 -1.8

log ∆t

-6

-5

-4

-3

-2

-1

log ε

| w |

|| w ||

max

(b) w-velocity.

-3 -2.8

-2.6

-2.4 -2.2 -2 -1.8

log ∆t

-6

-5

-4

-3

-2

-1

log ε

| Q |

|| Q ||

max

-3 -2.8

-2.6

-2.4 -2.2 -2 -1.8

log ∆t

-5

-4

-3

-2

-1

log ε

| L |

(d) Lift.

Fig. 42 Time step convergence for the third order IMEX scheme.

convergence for all the properties in all the norms, which shows that the combination

of the third order partitioned IMEX scheme with RBF mesh deformation retains the

order of the time integration scheme without the necessity to sub-iterate.

The displacements generated in the AGARD 445.6 test case are rather small

and do not place a large demand on the mesh deformation method. To investigate

the performance of the mesh deformation method for large displacements we also

enforced a large deformation on a 3D beam. The resulting mesh on the beam and

the mirror plane is shown in Figure 43. It can be seen that the mesh is still regular

174

Radial Basis Functions for Interface Interpolation and Mesh Deformation

Fig. 43 Mesh obtained with

RBF interpolation around a

heavily deformed 3D beam.

and also the mesh quality metrics indicated a high mesh quality, such that accurate

ﬂow calculations are possible.

4 Conclusions

In this chapter we investigated the use of radial basis functions for application to

interpolation of data at non-matching interfaces and for application to mesh defor-

mation.

RBFI for non-matching meshes

In Section 2 the difference in accuracy between a conservative and a consistent

approach for different RBFI coupling methods is investigated. The performance is

analysed for two analytical test problems as well as a simple quasi-1D FSI problem.

For RBFI coupling methods the conservative approach results in large unphysical

oscillations in the pressure received by the structure. When the structure is ﬂexible

enough these oscillations can result in deviations in the displacement of the ﬂow

interface. Therefore the consistent approach provides the best accuracy.

RBFI for mesh deformation

In Section 3 a point-by-point mesh movement algorithm is presented for the de-

formation of unstructured grids. Radial basis functions (RBF’s) are used to inter-

polate the displacements of the boundary nodes of the mesh to the inner domain.

The method requires solving a small system of equations, only involving the nodes

on the boundary of the ﬂow domain. The implementation of the method is rela-

tively simple, even for 3D applications, because no grid-connectivity information is

needed. Also the implementation for partitioned meshes, occurring in parallel ﬂow

computations, is straightforward.

175

A. de Boer, A.H. van Zuijlen, and H. Bijl

The new algorithm is tested with fourteen RBF’s for a variety of problems. The

method can handle large deformations of a mesh caused by translation, rotation

and deformation of a structure both on 2D and 3D meshes. The performance of the

method is not the same for all RBF’s. Five RBF’s generate meshes of high quality

after deformation. However, when efﬁciency is more important, the CP C

RBF with

compact support is the best choice, closely followed by the thin plate spline.

In a ﬁrst comparison the RBF-method produces meshes of higher quality than the

popular semi-torsional spring analogy for very large deformations. The quality of

the meshes after deformation is high enough to perform accurate ﬂow calculations.

It is shown that working with higher quality meshes can increase the efﬁciency

of the computation. This is due to the fact that the new method preserves the design

order of higher order time integration methods, due to its smooth deformation in

time. The capability of the method to deform 3D meshes is shown by rotating and

translating a 3D block and performing a case study on ﬂutter of the AGARD 445.6

wing.

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178

Least-Squares Spectral Element Methods in

Computational Fluid Dynamics

Marc Gerritsma and Bart De Maerschalck

Abstract The least-squares spectral element method (LSQSEM) is a relatively novel

technique for the numerical approximation of the solution of partial differential

equations. The method combines the weak formulation based on the minimization

of a residual norm, the least-squares formulation, with the higher-order spectral el-

ement discretization. A well-posed least-squares formulation leads to a symmetric,

positive-deﬁnite system of algebraic equations which are highly amenable to well-

established solvers such as the preconditioned conjugate gradient method. Further-

more, the formulation is very robust in the sense that no stabilization operators are

required to acquire convergent solutions. The spectral element discretization ren-

ders high order accuracy to the scheme. This new numerical scheme is applied to

incompressible, compressible and non-Newtonian ﬂow problems.

1 Introduction

The least-squares spectral element method (LSQSEM) combines the weak formu-

lation obtained from a least-squares minimization problem and a spectral element

discretization. In the early 70’s the least-squares technique was introduced for the

solution of partial differential equations by Bramble, [10, 11]. The method was al-

ready described in Russian literature by Dˇziˇskariani, [30] and Luˇcka, [55]. In its

initial form it was less competitive than the Galerkin formulation due to the high

condition numbers and the more stringent regularity requirements. It was only when

it was recognized that partial differential equations need to be rewritten in terms of

Marc Gerritsma

Dept. of Aerospace Engineering, TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands e-mail:

M.I.Gerritsma@TUDelft.nl

Bart De Maerschalck

Flemish Institute for Technological Research, Boeretang 200, BE-2400 Mol, Belgium e-mail:

bart.demaerschalck@vito.be

Lecture Notes in Computational Science and Engineering 71,

179

DOI 10.1007/978-3-642-03344-5_7,

B. Koren and C. Vuik (eds.), Advanced Computational Methods in Science and Engineering,

Marc Gerritsma and Bart De Maerschalck

equivalent ﬁrst order partial differential equations that the method gained renewed

interest.

In 2002/2003 two papers appeared shortly after another in the Journal of Com-

putational Physics: [70] by Proot and Gerritsma, and [61] by Pontaza and Reddy. In

both papers the least-squares formulation was combined with higher order spectral

element methods. Since then the number of practitioners and publications in this

ﬁeld has grown.

The outline of this chapter is as follows: In Section 2 the Rayleigh-Ritz prin-

ciple, the Galerkin formulation and the least-squares formulation are presented. In

Section 3 a brief overview of the spectral element methods is given. In Section 4

convergence rates for well-posed problems are discussed. Applications of the nu-

merical scheme are presented in the Sections 5 and 6, in which incompressible and

compressible ﬂows are discussed, respectively. In Section 7 new developments are

brieﬂy described. The ﬁnal section, Section 8, gives an overview of the relevant

literature.

2 The least-squares formulation

Weak formulations of differential equations consist of functionals over function

spaces whose stationary points solve the original differential equation. These sta-

tionary points can be distinguished in (global) minima or saddle points. If a partial

differential equation can be rephrased in a minimization problem, very robust nu-

merical methods can be developed. A well-known form is the Rayleigh-Ritz formu-

lation.

If, on the other hand, no equivalent minimization formulation can be established

a more general formulation in terms of stationary points can be devised. The disad-

vantage of this approach is that the analysis of well-posedness of the weak formula-

tion is much harder to perform and in many instances the well-posedness properties

on the continuous level are not inherited by a discrete approximation.

In order to position the least-squares formulation within the framework of weighted

residual methods and to introduce some notation and concepts we will brieﬂy de-

scribe the Rayleigh-Ritz method, the Galerkin formulation and eventually the least-

squares formulation.

2.1 Rayleigh-Ritz method

Consider the Poisson problem

∆φ

(x) = f (x) , x ∈

Ω

⊂ R

, (1)

with

180

Least-Squares Spectral Element Methods in Computational Fluid Dynamics

(x) = 0 for x ∈

⊂

∂Ω

and

∂φ

∂

= 0 for x ∈

⊂

∂Ω

∩

= /0 . (2)

This partial differential equation with homogeneous boundary conditions can be

converted in the following minimization problem:

Find

∈H which minimizes J (

) =

Ω



(∇

)

+ f



Ω

. (3)

The set H contains all admissible functions for the minimization problem to make

sense. Firstly, all functions in H must satisfy the essential boundary condition

(x) = 0 for x ∈

. Furthermore, the integral in (3) should exist. The set of func-

tions for which the gradient of a scalar is square integrable, i.e. ∇

∈



(

Ω

)



and

∈ L

(

Ω

), ensures that

J (

) ≤Ck

, (4)

where H is deﬁned as

H =

∈ L

(

Ω

) and ∇

∈



(

Ω

)



and

(x) = 0 for x ∈

, (5)

with norm

= k

(

Ω

)

+ k

[

(

Ω

)

]

, ∀

∈ H . (6)

minimizes the functional J (

) we need to have that

DJ (

)

= 0 , ∀

∈ H , (7)

where the linear operator DJ (

) is deﬁned by, [79]

lim

→0

J (

) −J (

) −

DJ (

)

= 0 . (8)

For the functional given by (3) this gives

DJ (

)

= lim

→0

Ω

(∇

∇

)

−2 f (

) −(∇

)

+ 2 f

Ω

(9)

= lim

→0

Ω

∇

(∇

)

−2

Ω

(10)

Ω

[∇

∇

− f

] d

Ω

(11)

= 0 , ∀

∈ H . (12)

This condition for a minimizer can be abstractly written as

) = ( f ,

) , ∀

∈ H . (13)

181

Marc Gerritsma and Bart De Maerschalck

The bi-linear form a(·,·) is in this case symmetric, i.e. a(u,v) = a(v,u).

Deﬁnition 1. A bi-linear form a(·,·) on a normed linear space H is said to be

bounded (or continuous) if ∃C < ∞ such that

|a(u,v)| ≤Ckuk

kvk

, ∀u, v ∈H , (14)

and coercive on H if ∃

> 0 such that

a(u,u) ≥

kuk

, ∀u ∈H . (15)

⊓⊔

If a partial differential equation can be converted into a minimization problem over

a function space H, where H is a Hilbert space and the bilinear symmetric form

a(·,·) is bounded and coercive, then the minimization possesses a unique solution.

Furthermore, if the bilinear form is only coercive on a proper (closed) subspace

V ⊂ H, then the minimization problem possesses a unique solution in V.

This last property, the fact that one may look for a minimizer on a closed subspace

of H, allows one to construct ﬁnite-dimensional subspaces (ﬁnite elements) and look

for an approximate solution in such a subspace V .

Furthermore, existence and uniqueness of a solution for the whole function space

H are inherited by proper closed subsets V ⊂ H.

An alternative way to derive the conditions for a minimizer (12) is to take the dif-

ferential equation and multiply it with an arbitrary function

∈H. After integration

over the domain

Ω

this gives

Ω

(−

∆φ

+ f )

Ω

= 0 . (16)

Applying integration by parts and using the boundary conditions gives

0 =

Ω

(−

∆φ

+ f )

Ω

(17)

Ω

[−∇·(

∇

) + ∇

∇

+ f

] d

Ω

(18)

Ω

[∇

∇

+ f

] d

Ω

−

∂Ω

(∇

,n) , d

(19)

Ω

[∇

∇

+ f

] d

Ω

, (20)

where n denotes the outward unit normal at the boundary

∂Ω

. The boundary can be

decomposed as

∂Ω

. On

= 0 and on

∂φ

∂

n = 0, so the boundary

integral vanishes. The ﬁnal result of the exercise shows that if

solves the PDE, the

weak formulation (20) is equal to the condition for a minimizer of the functional

J , (12).

The variational approach described above was already used by Rayleigh in 1870,

[74], Ritz in 1908, [75] and Galerkin in 1915, [32]. But it was only until the 1960’s

182