Coupling Finite Element and Meshless Local Petrov-Galerkin Methods for Two-Dimensional Potential Problems

28 Pages·2002·1.3 MB·English

by  Chen, T.

i
AIAA-2002-1659
Coupling Finite Element and Meshless Local
Petrov-Galerkin Methods for Two-Dimensional
Potential Problems
T. Chen
Army Research Laboratory
NASA Langley Research Center
Hampton, VA 23681
I.S. Raju
NASA Langley Research Center
Hampton, VA 23681
43 rdAIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics, and
Materials Conference
22-25 April, 2002
Denver, Colorado
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AIAA-2002-1659
COUPLING FINITE ELEMENT AND MESHLESS LOCAL PETROV-GALERKIN
METHODS FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS
T. Chent and I.S. Raju*
NASA Langley Research Center, Hampton, Virginia 23681-0001 ,USA
Abstract for remeshing because of heavily distorted elements in
large deformation. Meshless Methods (MM), such as
A coupled finite element (FE) method and meshless
the diffusion element method [3], the Element-Free
local Petrov-Galerkin (MLPG) method for analyzing
Galerkin Method (EFGM) [4-6], h-p clouds [7],
two-dimensional potential problems is presented in this
Partition of Unity (PU) [8], the Reproducing Kernel
paper. The analysis domain is subdivided into two
Particle Method (RKPM) [9], the Local Boundary
regions, a finite element (FE) region and a meshless
Integral Equation (LBIE) method [10]_ and the
(MM) region. A single weighted residual form is
Meshless Local Petrov-Galerkin (MLPG) method [11]
written for the entire domain. Independent trial and test
are developed to avoid these deficiencies. Of these
functions are assumed in the FE and MM regions. A
methods, the MLPG method is a very promising
transition region is created between the two regions.
numerical method and has been successfully used on
The transition region blends the trial and test functions
both potential and elasticity problems to obtain very
of the FE and MM regions. The trial function blending
accurate results [12-14]. The MLPG method is based on
is achieved using a technique similar to the 'Coons
a local weak form and a moving least squares (MLS)
patch' method that is widely used in computer-aided
approximation. This method is a "'true" meshless
geometric design. The test function blending is
method and does not need any "element" or "mesh" but
achieved by using either FE or MM test functions on uses a distributed set of nodes for both field
the nodes in the transition element.
interpolation and background integration. Although
The technique was evaluated by applying the coupled the MLPG method is attractive, the method is more
method to two potential problems governed by the computationally expensive than the FE method, and it
Poisson equation. The coupled method passed all the is not as mature and comprehensive as the FE method.
patch test problems and gave accurate solutions for the
There is a great interest in combining two different
problems studied.
numerical methods, because such coupling would
exploit the potential of each method while avoiding
their deficiencies. Several techniques have been
Keywords: Finite element method, Meshless Local
proposed in the literature [15-22] to couple different
Petrov-Galerkin (MLPG), Coupling method, Potential
numerical methods, such as combining the FEM with
problem
the Boundary Element Method (BEM) [15, 16], the
Introduction FEM with the EFGM [17, 20-22], the EFGM with the
BEM [18], the FEM with the MLPG method [19], etc.
The finite element (FE) method [I, 2] is a widely used
In coupling methods, the analysis domain is divided
method, because of its accuracy, convenience, and
into two regions and each method is used in a region
flexibility. There are, however, deficiencies, such as where it is most efficient. At the interface between the
element locking, discontinuous derivatives of the
regions, the continuity of the primary variable and the
primary variable at element boundaries, and the need
balance of the secondary variables need to be
maintained. One technique that is commonly used in
the coupling methods [15-19] defines a transition
"_ Army Research LaboratoD'. MS-240. Anablical and
Computational Methods Branch. NASA Langley Research Center. region between the two regions. In the transition region,
a ramp transition function is chosen to combine the trial
* Head. Analytical and Computational Methods Branch, Associate
functions (shape functions) of the two numerical
Fellow AIAA.
methods. This technique is straighttbrward and widely
Copynghl © 2002 by the American Institute of Aeronautics and Astronautics, used in coupling different numerical methods. Using
Inc No copyrighl isasserted in the United Stales under Title 17, U.S. Code.
this technique, the continuity condition of the primary
The US. Government has aroyalty-free licen_ to exercise all rights under the
copyright claimed hereto for Govenunental Purposes All olher rights arc variable is satisfied. However, the reciprocity condition
reserved b_tile copyright owner of the secondary variables is not guaranteed. Other
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techniques, such as Lagrange multipliers method [20], a
hierarchical mixed approximation method [21], and a , Ou &' Ou __).d_+ c3u Ou
method that comprises both FE and EFGM shape
functions in the interface region [22] have also been -_p. v. dD = 0
presented recently.
(4)
In this paper, a formulation that couples the FE and
MLPG methods is presented for two-dimensional (2D) where n_and n,.are the direction cosines of the outward
potential problems based on a single variational form. normal n of the boundary in the x and y directions,
In this method, the analysis domain is divided into two respectively (see Figure 1).
regions (FEM region and MLPG region). Several
In the coupling method, the domain f2 is divided into an
computational issues related to the coupling method are
discussed. Two numerical examples for the potential FE region (DF') and an MM region (_M) as shown in
Figure 2. The subscripts F and M are used to describe
problems governed by Poisson equation are presented
the variables in the FE and MM regions, respectively.
to evaluate the accuracy and efficiency of the proposed
technique. The interface between the two regions is denoted as _.
Variational Coupling Procedure In the coupling method, the weighted residual equation
(Eq. (3)) and subsequently the weak form (Eq. (4)) are
In this section the variational coupling procedure is written for the FE and MM regions separately as
described for coupling two disjoint regions (sub-
domains) that are governed by apotential function.
Consider a Poisson's equation for a 2D problem
<3u Ov
governing the potential u (primary variable) in a two- _u _v. da +I (Ou.__ _u
dimensional (2D) domain f2 hounded by contour F (see
_1.' FI , G_; "
Figure l)
+ - I p.v.an:0
V2u =p (1) _Ou _u
_1,"
(5)
The right hand sidep in Eq. (1) is, in general, a function
ofx andy. Inthe general case, the boundary F can have
IM =
mixed boundary conditions. On one part of the
Ou Ov Ou Ov ,Ou Ou
boundary, F,, the potential u is prescribed, and on the -°I (g '- +- ' +I '- '",o+x-- .,,).v.dr v
remaining part Fq, the secondary variables, flux, M FM "
(q = du /dn )are prescribed, i.e.
u= u on Fu and q= q on Fq (2) L_M
(6)
where F = F U Fq. For example, in a steady-state heat
Notice that the outward normals of interface boundary
transfer problem, which is a typical Poisson's problem,
from FE and MM domains are in opposite directions
u is a temperature function, q is the heat flux function
(see Figure 2), i.e.
and p is internal heat generation.
(7)
The solution for Eq. (1) is sought in a weighted residual nF :-n M
manner as
Therefore, the terms along the interface boundary
should meet the requirement
2u 02u
(_x2+-_,:-p).v.dx.dy =0 (3)
I (_O-'un,.:, +_._.nl,). v.dF
FI ' '.'
where v=v(x,y) is an arbitrary weight function. (8)
+ f(a,, o.
Using the divergence theorem, one can rewrite Eq. (3) Ox'n'" +_ "n'+_')'v'dr=O
as
The balance of the secondary variables represented by
Eq. (8) along the interface needs to be satisfied. This
issue will be discussed in detail later inthis paper.
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Combing Eqs. (5) and (6) and using Eq. (8), a single gl;
Jyv';'
variational equation that is valid for the entire domain
v_,= (I 3)
can be written as
l"=lO x _ _M
where v_l_'_arechosen to be the variations on u'[_(i,e.
3u &, c_u &'). an +r! _3u c_u du_ )as is customary in thetraditional FE method and
_u _ 0u &
- I ,vdn - I dn
0 x_ _v
_2/," _At " = ,vM (14)
0ll "" l___t(v,, ), X_DM
+ f (-._n-,x&_ +_-7.n,).v.dF- _.,,_p.v.dD =0
[-M
where (vA¢),are chosen to be the Petrov-Galerkin weight
(9)
function on a compact support sub-domain of each node
The next step in the coupling method is to choose the [13]. The weight function of node i equals unity at x,,
trial and test functions. The approximations for u in the
and monotonically decreases as I"-',11 increases and
variational form (Eq. (9)) are called the trial functions
while the weight functions vin the variational form are is equal to zero at the boundary and outside of the sub-
called the test functions. The trial and test functions of domain.
the FEand MM methods along the interface (F0 arenot
Trial functions in the transition region
the same. The coupling between the FE and MM
regions can be achieved by defining atransition region The trial functions in the transition region need special
attention. Different numerical methods (MM, BEM,
_T (see in Figure 3.) The region _r' is split into two
and different types of elements of FEM) use distinct
parts, f2r and _T, with an interface FT.The region f2x is
trial functions. Directly coupling two numerical
now between the FE region _v and the meshless region
methods by only matching the interface nodal values
_M. may not satisfy the compatibility condition along the
Trial and test functions interface. In other words, the function values produced
in different regions along the interface may be different
The trial and test functions for the FE and MM regions
although a few nodal values are the same. Therefore, a
are presented first and then the functions in the transition element between the two regions that
transition region are discussed. The trial and test
smoothly transforms the values of the trial functions
functions are chosen such that
from one region to another ispreferred.
u= uj. +u,_ and v =vj +v_ (I 0) A blending technique is developed to generate the
interpolation function in a 4-sided region (quadrilateral
The trial functions of a point x in the FE region are
element) in terms of its edge (boundary) functions (see
Figure 4). The edges of the element can be curved and
/, /. X_E ['2F the number of the nodes on each of the edges need not
u, = ,,=l (1I)
be the same. This technique is similar to a technique,
NI.
0 x_ f2M 'Coons patch' [23], that is used in computer-aided
geometric design and FEM mesh generation. The
where u"F_ are the interpolation functions used in the Coons patch technique is used to construct a smooth
traditional FE method, and N_, is the number of the surface using the bounded curves. By using the Coons
elements in the FE region. Similarly, the trial functions patch technique in an analogous fashion, the various
in the MM region are trial functions on the edges of the transition elements
are blended smoothly. This technique is also similar to
the method used to generate serendipity elements.
(12) However, instead of using the boundary nodal values to
generate the serendipity element, the boundary
functions are used in this blending technique to
construct the transition element. These boundary
where NMis the number of nodes inthe MM region and
functions may be explicit functions of the boundary
(u_t), are the moving least squares (MLS)
nodal values, such as FEM, or implicit functions of the
approximations at each of the nodes in the MM region. nodal values, such as MM.
The test functions are chosen as
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Abilineabrlendinigsused
in this study to construct the example, the side lengths are a and b in type C of
trial function of the element. Although the blending is Figure 5 for node g). For a node that connects to the
bilinear, the trial functions can be an arbitrary function MM region, the MM test function with a support
that depends on the edge functions of the element [23]. domain of 1,,is used (for example nodes q and kof type
To be compatible with the FE method, the trial B in Figure 5).
functions of the transition element are developed in a
One additional concern about the test function of the
square element (parent element) in the '_, r/' domain,
transition region is the size of the support domain of a
where -I _<(_, q)_<1. The trial function can be written
node using an MM test function. The size of the support
as
domain of an MM node can be defined by a process
similar to that used in references [11-14]. Although
u = 2l--(] - _:) •F3 (/7) + _ (1 + _=)•F4 (/7) there is no restriction on the size of the support domain
by the weighted residual method, the radius (lo) of the
support domain of a node is defined to be less than the
+l(I-/'7)-Fl(_)+/(l+r/).F2(_ =)
(15) size of the element to which that node belongs (see for
example, nodes q and k of type B in Figure 5). This
-!(14 -4)(l-v).u -¼0 +4)0-,7).u,,
limitation on the size of the support domain eliminates
complicated numerical integration that may be needed
-¼(I+_)(,+rl).u v- ¼(I-_)(l+rl).U, to integrate accurately the weak form of the transition
element.
In Figure 5, three types of possible transition elements,
where F, (i = !,4) are the boundary functions of the A, B, and C, are identified. In the type B element, the
quadrilateral element and uj (j"= n, p, q, and m) are the nodes k and q are on the interface F_. Therefore, the
values of the primary variable at the corner nodes. MM trial function is applied on edge kq, and the MM
test functions should be applied on both nodes k and q.
The derivatives of the primary variable with respect to x
Note the sizes of the support domain (l,) of the test
and y can be obtained by calculating the derivatives
function for nodes k and q need not be the same. Since
with respect to _ and r/, then forming the Jacobian
node q is shared by elements of type A and B, in order
matrix of the element and following standard FE
to satisfy the reciprocity conditions of the secondary
procedures.
variables on the common edge pq, the MM test function
Typical transition elements between the FE region (DF) should be applied on node q of the element of type A,
and MM region (DM) are shown in Figure 5. The open although no MM trial function is applied on any edge in
circles and solid circles represent the nodes in the FE element of type A.
and MM regions, respectively. The shaded elements are System equations
the transition elements. In these transition elements, the
solid lines represent the boundaries that are connected The trial and test functions outlined in Eqs. (10-15) are
to finite elements. On these boundaries, the edge substituted into the weak form in Eq. (9) and the
functions can be generated by FE trial functions. The integrations are performed. This leads to a system of
dashed lines in Figure 5 are the boundaries that connect equations similar to the FE and MLPG methods
to the MM region, and the edge functions are generated
[K].{u}= {f} (16)
by the MM trial functions. The next step is to form the
test function for the transition elements. where [K] contains the [KF], [KM] and [KT] matrices
corresponding to the FE, MM and transition regions,
Test functions in the transition region
respectively. Similarly, {f} contains contributions from
Although the test function can be arbitrarily selected in the three regions. The solution of the system of
the weighted residual method, to satisfy the weak form equations leads to the potentials {u} at all the nodes in
requirement of the reciprocity conditions of the the model. Note that instead of working with fictitious
secondary variables with constant derivatives (see Eq. nodal potentials {t_}in the MM method the nodal
(8)), the test functions v must (a) be the same function
potentials {u} are used by making use of the procedures
and (b) have the same integration length along the
contained inreference [I4].
interface of both of the regions. Therefore, the test
function of a node in the transition element depends on Boundary conditions
the adjacent regions of that node. if the node is adjacent
There are no special considerations necessary to apply
only to the FE region, and not to the MM region, the FE
essential boundary conditions on the model. However,
test function is used and the integration length of the
the values of the natural boundary condition ('force')
function is identical to the side of the element (for
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depend on the type of nodes (FE or MM) on the The second type of interface forms a closed loop and
transition element. Based on Eq.(4), the 'force' value does not intersect the domain boundary (see Figure
on a node is calculated as 2(b)). Two types of models are used to evaluate this
type of interface. The first model uses an FE region that
t'"_u au is embedded in an MM region, and the second model
J_ = j(_x.n +-_.n ).v .dr - _p.v .d_ (17)
uses an MM region that is embedded in an FE region.
In the examples (see Figure 6), 8-node quadratic
where v_ is the test function of the nodes j , _ is the
isoparametric elements and quadratic basis functions in
boundary of the test function where the secondary
the MM region are used. When the coupled method
variables, fluxes, are prescribed, and f_j is the domain
reproduces the exact solution, then the method "passes"
of the test function.
the patch tests.
The formulation presented above for the coupled
Example 1
method is evaluated by performing patch test problems
and by application to problems for which exact A potential problem governed by the Poisson's
solutions are available. equation, Vu2= 2.(q + c2) is considered in a
Numerical Examples rectangular domain (Figure 6(a)). The exact solution for
the problem is u=c_x 2+c2y 2+c3_,, where c, are
To evaluate the algorithm of coupling FE-MM for
potential problems, two example problems governed by arbitrary constants. All four models in Figure 7 are used
Poisson's equation, as shown in Figure 6, are to evaluate the coupling method using quadratic
considered. For the numerical integration in the MM quadrilateral elements and a quadratic basis function in
domain, 8 and 12 Gauss points are used in the radial the FE and MM regions, respectively. A type-i interface
is used in the first model, and a type-II interface is used
and tangential directions, respectively, of the circular
in the models 2, 3 and 4. In these patch tests, the
support domain. A 3x3 quadrature rule is used for the
quadratic elements in the FE region. The Gauss potential u is prescribed on the boundaries of the entire
domain corresponding to the exact values. As expected,
quadrature of a node in the transition element is
dependent on the test function of the node. For the node the exact solutions for both potential and fluxes in the
entire domain are reproduced to machine accuracy.
using the MM test function, the same Gauss quadrature
Similar results are obtained using the mixed boundary
is used as in the MM region. A 4x4 quadrature rule is
conditions.
used for the node using the FE test functions.
The potential and its partial derivatives obtained using Example 2
the coupling method are compared to exact solutions The second example considered is that of the torsion of
wherever available. The error norm (t[e_f]12)is used to a shaft of elliptical cross-section as shown in Figure
evaluate the effectiveness of various parameters. This
6(b). The governing equation is a Poisson's equation in
norm isdefined as terms of Prandtl stress function u with the boundary
condition u= 0 on the boundary of the cross-section.
The shear stresses in the shaft are determined as the
,_.1 2/M derivatives of the stress function. For this problem, the
right-hand side function of Eq. (i) isdefined as
where M is the total number of randomly distributed
internal points in the domain at which the numerical p(x, y) =-2GO = - 2. M. (a 2+ b2)
solution is evaluated and compared to the exact rc'a _'b3 (19)
solution. Note that these internal points are independent
where G is the modulus of rigidity, 0 is the twist angle,
points that are randomly generated and are not
associated with the nodes used in the models. In the M is the torsion moment applied on the shaft, and a and
b arethe semi-major and semi-minor axes of the elliptic
MM region, a value of M = 50 is used, and in the FE
cross-section, respectively. The exact solution of the
region, six randomly distributed points are generated
for each element. The error norm is calculated for the problem [24] is
entire domain. If the exact solution of the problem is
not available, the results of the coupling method are u(x, y) g X2 V2
- _.a.b(_ -+ _ - 1) (20)
compared to the full FE solutions and the full MM
solutions over the analysis domain. Two coupling models are used for the problem, a full
model of the elliptic shaft and a model of one-quarter of
Two types of interfaces are used to model the examples.
the shaft, as shown in Figure 8. As the exact solution is
The first type of interface cuts across the domain and
a quadratic, quadratic elements and quadratic basis
intersects the domain boundary as shown in Figure 2(a).
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functionsareused.Inthefull
model, 16 quadratic [3] Nayroles, B., Touzot, G., and Villon, P.,
elements (8-node quadrilateral) and 102 MM nodes "Generalizing the Finite Element Method:
using a quadratic basis function are used. The essential Diffuse Approximation and Diffuse Elements,"
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[4] Belytschko, T., Lu, Y.Y., and Gu, L., "Element-
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Free Galerkin Methods," International Journal
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[8] Melenk, J.M., and Babuska, I., "The Partition of
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A single variational weighted residual formulation is
used on the entire domain to develop the coupling [9] Liu, W.K., Jun, S., and Zhang, Y.F.,
method. A transition region is defined between the FE "Reproducing Kernel Particle Methods,"
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blending technique is used to construct the trial function Engineering, Vol. 20, 1995, pp. 108!-I106.
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Computational Mechanics," Computational
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F
q
!1
Y
0 X u
Figure 1 Two-dimensional potential problem
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AIAA-2002-1659
F FM F
n n
Y
• _.
X 0 X
(a) Type-I interface (b) Type-II interface
Figure 2 Finite element and meshless method regions
FF f_T
FM
Y
0 x
Figure 3 Finite element (FE) region, transition (TR) region and
meshless method (MM) region
8
American Institute ofAeronautics and Astronautics