Application of Numerical Manifold Method in Fluid-Solid Interaction Harmonic Analysis
SU HAIDONG1 and HUANG YUYING2
1
DDA Center, Yangtze River Scientific Research Institute, Wuhan, China
2
School of Civil Eng. & Mechanic, HUST, Wuhan, China
Numerical Manifold Method (NMM) is first applied to fluid-solid interaction analysis in this
paper. Based on rectangular mathematical meshes and simplex integration, high-order NMM
equations are presented for two-dimensional fluid-solid interaction harmonic analysis,
concerning inviscid, irrotational, incompressable potential flow and undamped structures.
Known value of fluid pressure is introduced via Lagrange multiplier method. It is also proposed
that cover of analytical series is applied to simulate infinite fluid field, which can greatly
decrease unknowns to be solved. Given examples of computing frequencies and harmonic
responses prove the validity and high efficiency of the approach, suggesting that NMM is very
suitable for combination of numerical solutions and analytical solutions.
Key words: high-order numerical manifold method; fluid-solid interaction; combination of
numerical solutions and analytical solutions
1. Introduction
structures. Readers can refer to Wang et al (2003) and Li
et al (2004). This paper applies NMM to fluid-solid
Based on mathematical manifold of modern mathematics,
interaction
(coupling)
numerical manifold method (NMM) was invented by Shi
considered that NMM has following advantages in the
(1992). NMM is capable of solving continuum and
new region:
discontinuum problems and has a bright future in
(1) By virtue of the independence of mathematical
computation mechanics.
meshes from physical meshes, it is not requested that
In NMM, the entire material volume is divided into many
mathematical meshes should strictly satisfy the geometric
finite covers overlapped each other which are also named
boundaries of structures and fluid field, so element
physical covers. They are formed by two independent
subdivision is convenient to be carried out.
cover systems: one is mathematical mesh defining only
(2)
the fine or rough approximations; the other is physical
computational precision can be improved by adding
mesh defining the boundaries of the material volume and
number of series terms of the local cover functions
the interfaces of different material zones. On each cover,
without subdividing mathematical meshes.
an independent local cover function is defined, which can
(3) In some special area of fluid field, such as infinite
be constants, polynomials or other series (including
field, we can use analytical series as cover functions to
series of analytical solution). These functions are
improve computational efficiency because number of
connected together to form a global function on the entire
unknowns to be solved is greatly reduced.
material volume by means of weighted average via
We shall make some remarks on (3). As we all know, for
weight functions. The common part of these covers is
infinite field problems or singular point problems,
named manifold element whose shape can be assumed
classical approach of using analytical series converges
arbitrarily. Element matrices are usually calculated
much quickly than numerical approach, but it is limited
exactly by means of simplex integration.
to the domains of regular shape and simple boundary
By now, researches on NMM are almost about solid
conditions. Then people have the idea of combination of
Benefited
from
analysis.
the
It
is
property
of
preliminary
p-version,
numerical solutions and analytical solutions: numerical
approach is used in domains of complicated shape, while
Boundary conditions are:
p  0 on free surface SF , (gravitational wave is
in some regions, such as infinite field, analytical
neglected);
solutions can be adopted. However, a new difficulty is
encountered: because the types of the unknowns are
different (the unknowns of numerical solutions are field
freedoms, but the unknowns of analytical solutions are
p
0
n
on fixed boundary S B and infinite boundary S R ;
..
p
  un
n
on fluid-solid interface S I ( un is normal
series coefficients), it is not easy to deal with the
displacement of the structure on it), reflecting coupling
compatibility problem on their interface. For example,
action of structure and fluid.
Huang (1988) computed frequencies of a dam and water
FEM expression of the coupling vibration system is
interaction system involving a regular infinite fluid field,
M s 

  G 
using finite element method (FEM) to simulate near field
and a series solution to represent the far infinite field.
0{D} Ks   GT {D}  {F0} 


0{P.. }  0 H  {P}  {Q0}
..

(2)

Although the method was more effective than pure FEM,
in which, D  is displacement vector of the structure;
the procedure was very complicated: a local variational
P is pressure vector of the fluid field;
principle was introduced, and a functional was derived to
{F0}
is dynamic
satisfy both governing equations and all boundary
vector loading on the structure (not including fluid
pressure); {Q0} is known exciting vector or given
conditions (including the compatibility of the FEM
boundary moving vector in fluid field; M s  and K s 
freedoms and the series coefficients on their interface).
Now we can solve this problem conveniently by NMM in
are mass matrix and stiffness matrix of the structure,
respectively; H  is fluid matrix and G  is coupling
the way that covers of numerical solutions and cover of
matrix. Expression of all these matrices can be seen in
analytical solutions are linked together simply by means
Huang (1998).
of weight functions.
Now we discuss the harmonic vibration of the system.
In this paper, high-order NMM equations are presented
Let {F0}  { f0}eit , {Q0}  {q0}eit , {P}  { p}eit , {D}  {d}eit
for two-dimensional fluid-solid interaction harmonic
analysis. Preliminary research is also done on using
cover of analytical solution to simulate infinite fluid
field.
(  denotes the circular frequency of the system, i   1 ),
then we have
K    2 M   G T  {d }  { f0} 
s
 s
   
.
2
H   { p}  {q0}
   G 
2. Governing equations and boundary conditions
(3)
In the case of free vibration, let { f 0}  {q0}  {0} and
eliminate {p} , we obtain the equation for modal analysis
K    M   M d   0
2
s
s
w
(4)
in which M w    G T H 1G  is called added mass.
Fig. 1: Fluid-solid interaction system.
3. Expressions in manifold method
3.1 Description of displacement and fluid pressure
A fluid-solid coupling system is shown in Fig.1, in which,
solid structure A is an undumped elastic body, and fluid
field  is inviscid, irrotational, incompressible. m, 
At present, FEM meshes are often employed to define
are mass density of solid and fluid, respectively.
FEM node form a mathematical cover. The FEM shape
Governing equation of fluid pressure p is
function is the weight function here. A rectangular
2 p  0 in  .
(1)
finite covers of NMM. All elements connected with any
mathematical mesh with its four covers is shown in Fig.2,
whose weight function is
wi 
stiffness matrix [ K s (e) ] and mass matrix [ M s e ] for the
1
1   0 1  0 
4
(5)
with  0   i  ,  0  i in which ( i ,i ) and ( , )
are local coordinates of each node and any point in the
mesh, respectively,
structure in NMM can be referred to Shi (1992) or Su et
al (2005a). Sub-matrix of [ H (e ) ] for fluid element is
T
T


H ij     Twix  Txwj   Twiy  Tywj d ,

i =1,2,3,4.
(7)

in which i,j=1,2,3,4. Expression of the coupling matrix
will be given later on.
3.3 Introduction of known boundary conditions
Hard springs are used to introduce fixed boundary
conditions on structure (Shi, 1992). We shall discuss how
Fig. 2: A rectangular mathematical mesh and four covers.
The polynomial series is in most common use as local
cover functions. Computational accuracy is improved
when terms and order of the series increase. In two
dimensions, polynomials are given as xn-kyk in which k is
repeated from 0 to n order. The total number of terms is
m=(1+n)(n+2)/2. For example, 0 to 2 order polynomial
cover functions are combined to
{t}={1
x
x2
y
in fluid field, that is, p  0 on free surface SF .
In NMM, pre-process is simplified because mathematical
meshes need not satisfy physical boundaries strictly.
However, known freedom can not be eliminated directly
as what is usually done in FEM. One approach is using
the third boundary condition of Laplace equation,
p
  ( p  pa )  0
n
y2 }.
xy
to introduce the known boundary conditions to matrix [H]
(on q ), to simulate the first condition
Matrices of displacement and fluid pressure in a
p  pa  0
rectangular mesh are
[Ts]=[Ts1
[Tw]=[Tw1
Ts2
Ts3
Tw2
Tw4],
(6)
respectively, where Tsi, Twi are covers of i-th node as
[Tsi]=[Ti1
Ti2
...
Tim]s, [Twi]=[Ti1
t ( j ) 0 

 0 t ( j )
with [Tij ]  wi 

q
are known,  is a
coefficient). That is, p  pa when    . This is called
Ts4],
Tw3
(on p ), ( pa and
Ti2
... Tim]w,
penalty function method, similar to the hard spring of the
solid. However, artificial large number  will have a
bad effect on the inversion of [H], resulting in an unture
frequency computed by Eq. (4). So Lagrange multiplier
for displacement field, and
method is used in this paper.
Define functional as
[Tij ]  wi t ( j ) for fluid field ( i=1,2,3,4, j=1, ...,m).
u 
(e)
(e)
   [Ts ]{d } and p  [Tw ]{ p }
v 
represent displacement
and fluid pressure in the element, respectively,
where {d (e ) } , { p(e ) } are unknowns (coefficients of
polynomial series).
3.2 System equations in the form of manifold method
 1 p
1 p 
   ( )2  ( )2 d 
2 y 
 2 x



 p qd   ( p  p )d
a
q
in which  p  S F , q  S B  S R  S I ,  is the Lagrange
multiplier. After applying variation to Eq.(8), governing
equation of Eq.(1) and all boundary conditions are all
satisfied automatically due to the arbitrary of p and
 , and we get

p . Now substituting
p

n
n
With respect to manifold method using in this paper, only
description of physical field is different from FEM. Thus,
we can use the same system Eqs. (2) to (4). Element
(8)
p
into Eq. (8), we obtain a new functional

 1 p
1 p 
p
    ( )2  ( )2 d   p qd   ( p  pa )d
2 x
2 y 
n

q
p
(9)
Fluid field  shown in Fig.3, including an infinite
Variations of this functional lead to a matrix form
H p  q1  q2 
(10)
in which H   [ H 0 ]  [ H p ] , [ H 0 ] is calculated by Eq. (7),
H    H      T 

(e)
p
T
p
e
q1  
e  (e)
p

e  (e)
p
q2   
 

 T T p a d
 x

 T
 T 

(e) 

e 
q

T    T T T d ,
n
n

(11)
(12)

q d .


(13)
e

 ..  .
ny Ts dS I d 
 
SI (e)
(14)
T
(e)
w
x

cos n ( y  y0 )
,
(16)
in which,  n  (2n  1) , n=1,2,..., An are unknowns to
2h
n y Ts dSI
.
we still adopt numerical solutions based on polynomial
cover functions in 1 .
We shall discuss the treatment of the transition
domain  2 (rectangular meshes of  2 are shown in
Fig.3). Two schemes are considered in this paper: One is
numerical covers on the 1,2 nodes, while analytical
covers on 3,4 nodes, and weight functions are the same
as Eq.(5); The other is three covers shown in Fig.4 where
Then we obtain the element coupling matrix
G     T  n
nx
the traditional four covers shown in Fig.2, in which
according to Eq. (13) we have
x

equal to zero and can be ignored in 3 . At the same time,
n
T
n
governing equation and all the boundary conditions
in 3 . Thus, after variations all integrals in Eq. (9) are
..
w
A e 
be solved. It is easy to check that Eq. (16) satisfies the
and q  p    un . In the element linked to the interface,
 T  n
p ( x, y ) 
n
are direction cosines of the outward normal n,
{q2}   
respectively. In 3 , analytical solution for Laplace

,
On fluid-solid interface S I shown in Fig. 1, n x and

numerical solutions, transition domain  2 (x0<x<x1),
and domain  3 (x>x1) for analytical solution,
equation is
3.4 Coupling matrix G
ny
domain of equal depth h, is divided into three
parts,   1  2  3 , which are domain 1 (x<x0) for
(15)
S I (e)
Simplex integration is used in the integration of the
the third cover represents analytical series, and the
weight function w3 becomes w3  12 (1   ) .
above equations, including one and two dimensional
Then the three covers in the element are
simplex.
[T]=[T1
T2
T3] ,
where, T1,T2 are the same as those in Eq. (6), while
4. Analytical solution cover of infinite fluid field

T3  w3 p 
A w e 
n 3

nx
cos n ( y  y0 ) .
(17)
n
Fig. 4: Three covers in a rectangular mathematical mesh.
Fig. 3: Fluid field including an infinite domain of equal
depth, rectangles are manifold meshes of  2 .
Computation shows that these two schemes achieve very
close results. However, the three covers scheme is more
reasonable theoretically and can save computation
amount.
Because there is not formula of simplex integration for
5.1 Compute base frequency of the system
Eq. (17) by now, numerical integration is performed. In
addition, the value of the function e n x goes little when
We apply one order polynomial cover functions to the
x goes large, leading to bad condition of the matrix when
and only 0 order to the fluid to consider the added mass.
solving the equations. So we rewrite Eq. (16) as
Tab.1 shows the results. It can be seen that frequencies

p ( x, y ) 
A e 
n

n ( x  x0 ) cos
 n ( y  y0 )
(18)
structure to improve the precision of the displacements,
(with or without fluid) are very exact.
n
5.2 Harmonic responses analysis
5. Examples
Compute harmonic responses according to Eq. (3). A
We select the example in Huang (1988), concerning a
horizontal exciting force acts on the top of the beam,
coupling system of a cantilever and fluid shown in Fig.5.
with the amplitude of 10kN and frequency of 10rad/s.
The cantilever beam has a width of 1m. The length of the
Density of solid m = 1kN/m3. One order polynomial
beam is the same as the depth of the fluid field h=10m.
cover functions are used in structure and fluid field. The
Young’s modulus E of solid is 3  10 kN/m , Poisson’s
deformation of the beam is the same as the results with
ratio u is 0.2. Density of solid m is 20, 10, 5, 3.33 times
of the density of fluid  (=0.1kN/m3), respectively.
FEM. Maximum displacement on the top is 0.0345m,
Rectangles in the figure are manifold meshes. One can
of finite elements is at least 240, while number of
see that the meshes do not agree with the boundaries of
manifold meshes is only 48. It can be seen that high
the structure or the fluid. For example, the meshes of the
order NMM with less number of meshes can obtain the
first column comprise solid and fluid material. Fluid field
same precision as FEM with much more elements.
6
2
close to 0.0364m of FEM. Via trial calculation, number
is discretized to meshes at a distance of 30m from the
beam in x direction. Via trial calculation, it is considered
5.3 Compute frequency considering analytical solution
Shown in Fig.6, Discretized domain for numerical
solutions of the fluid field is at a distance of only 3m
from the beam in x direction. Considering analytical
cover to simulate the infinite field, we obtain wet
frequency 1 listed in Tab.2, in which some results
show the effect of length hx of the transition domain and
number of terms n for analytical series.
that the fluid field has been discretized far enough.
Fig. 5: Fluid-beam interaction system and NMM meshes.
Wet frequency 1
h / m
numerical solutions
Tab.1 Base frequency of the beam (rad/s)
frequency of the structure without fluid 01
1 / 01
numerical
analytical
relative error
numerical
series
relative error
solutions
solutions*
(%)
solutions
solutions*
(%)
0.5
1.175
1.238
1.243
-0.40
0.949
0.947
0.21
1.0
1.585
1.750
1.758
-0.46
0.906
0.900
0.66
2.0
2.063
2.475
2.486
-0.44
0.834
0.825
1.09
3.0
2.352
3.032
3.045
-0.43
0.776
0.766
1.31
*series solutions in Huang (1988) adopt 6 terms of series
analytical solutions can play a great role.
With respect to nonlinear analysis concerning large
displacement of structure and large disturbance of fluid,
arbitrary shape of manifold elements presents a
possibility of using fixed mathematical meshes to
calculate large deformation of the structure (Su et al,
2005b). Thus, solid and fluid can be conveniently solved
Fig. 6: NMM meshes considering analytical solutions.
Tab.2 Wet frequency 1 (considering analytical cover)
hx  2 m
h / m
hx  4m
hx  10m
n=5
n=20
n=5
n=20
n=5
n=20
0.5
1.172
1.165
1.174
1.175
1.178
1.177
1.0
1.579
1.560
1.584
1.584
1.592
1.590
2.0
2.049
2.008
2.060
2.061
2.078
2.074
3.0
hx
2.331 2.271 2.347 2.350 2.374 2.369
is the length of the transition domain, and n is the
number of terms of analytical series
with the same fixed background meshes to settle the
problem of incompatibility on the interface of solid and
fluid. Therefore, it is very important to study NMM in
the region of fluid-solid coupling analysis.
The authors are grateful to Dr. Shi for his valuable
suggestions on the research.
References
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analytical cover, wet frequency 1 are close to those in
Tab.1. For example, when h / m  3 , 1 is between
2.271 and 2.374, 1 / 01 is between 0.7490 and 0.7830.
Comparing with the series solution result 0.7661, relative
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