a new hybrid meshless-differential order reduction (hm-dor)

8/2/2019 A New Hybrid Meshless-differential Order Reduction (hM-DOR)

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Engineering Structures 25 (2003) 141–154

www.elsevier.com/locate/engstruct

A new hybrid meshless-differential order reduction (hM-DOR)method with applications to shape control of smart structures via

distributed sensors/actuators

T.Y. Ng ∗, Hua Li, J.Q. Cheng, K.Y. Lam

Institute of High Performance Computing, National University of Singapore, 89C Science Park Drive, #02-11/12, The Rutherford, Singapore

Science Park 1, Singapore 118261

Received 17 October 2001; received in revised form 14 June 2002; accepted 20 July 2002

Abstract

This paper presents the development of a new hybrid numerical method, the hM-DOR method, which is based on an order-reduction technique for partial differential equations, and combines the true-meshless collocation technique with a fixed reproducingkernel approximation. The proposed method is able to directly impose overlapping boundary conditions, a procedure difficult toaccomplish with existing collocation-based meshless methods. The present method is applied to examine thin-plate bending forplates of various geometries and different boundary conditions. Its efficiency and accuracy are validated by numerical comparisonswith finite element results. Furthermore, it is also used to simulate the deformation shape control of the uniformly loaded smartbeams and plates integrated with piezoelectric sensors/actuators. The numerical results here again verify the efficiency of this newlydeveloped method.

2002 Elsevier Science Ltd. All rights reserved.

Keywords: Meshless method; Point collocation; Reproducing kernel; Order reduction; Smart structure; Shape control

1. Introduction

Numerical approaches in computational mechanicshave been developed on a continual basis ever since theFEM proved effective for a wide range of engineeringapplications. Over the last few years, the meshlessapproach, often referred to as the next-generationnumerical tool, has attracted much attention amongstresearchers world-wide [1]. In general, meshlessmethods can be classified roughly into two groups,namely, those that require a background mesh such asGalerkin-based techniques [2–16] and those that do notrequire a background mesh such as collocation tech-niques [17–20]. The latter collocation-based techniquesare true meshless methods, and include the finite cloudmethod [21], which combines collocation technique withthe fixed reproducing kernel approximation. One of itsadvantages is that both Dirichlet and Neumann boundaryconditions may be implemented exactly, but this is pro-

∗ Corresponding author.

0141-0296/03/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0141-0296(02)00116-5

vided there are no two boundary conditions, involvingthe same variable, which need to be satisfied concur-rently at a boundary point. This leads us to one of itsdrawbacks, i.e., since only one boundary conditioninvolving one variable can be enforced at each scatteredpoint along the edge, it is difficult to implement the over-lapping boundary conditions, where there are more thanone boundary condition involving the same variable,describing the edge condition. An example of overlap-ping boundary conditions is the bending problem of asimply-supported plate which includes both the essentialzero-displacement and natural zero-moment boundaryconditions.

In order to overcome the difficulty of implementingthe overlapping boundary conditions, we develop here anew hybrid meshless-differential order reductionmethod, which we call the hM-DOR method. Thismethod combines the true-meshless finite cloud methodwith an order-reduction technique for partial differentialequations (PDE) for full exploitation of the merits of collocation methods and the direct imposition of theoverlapping boundary conditions. A simple and efficient



142 T.Y. Ng et al. / Engineering Structures 25 (2003) 141–154

technique validated for the solution of high-order PDEs

with overlapping boundary-value characteristics, the

present hM-DOR method first introduces selected vari-

ables suitable for the reduction of the higher-order PDEs

into the lower-order PDEs. This is followed by recon-struction of the governing PDEs and given boundary

conditions to correspond to the selected variables. Then,for the scattered set of points in the computational

domain, the collocation technique with fixed reproducing

kernel approximation, is imposed on the order-reduced

PDEs and restructured boundary conditions for discretiz-

ation of the boundary-value problem. Finally, the

numerical results are obtained by solving the set of com-

plete linear discrete algebraic equations for point values.

For verification of the convergence characteristics and

accuracy of the developed hM-DOR method, numericalcomparisons with established results are made for the

plate-bending problem, in which various geometric

shapes and different boundary conditions were con-

sidered. Further, the present true-meshless hM-DOR

method will be applied to smart structures, where defor-

mation shape control of one-dimensional beam and two-

dimensional plate structures integrated with arbitrarily

distributed piezoelectric sensors/actuators are simulated

to optimize both the dimension and location of the actu-ators in an active control systems.

2. hM-DOR method—a true meshless method

The presently developed hM-DOR method exploits

the order-reduction technique for PDEs and the collo-cation technique with fixed reproducing kernel approxi-mation, to solve boundary-value problems with overlap-

ping boundary conditions.

2.1. Fixed reproducing kernel approximation

The fixed reproducing kernel approximation is based

on the classical reproducing kernel particle method

(RKPM) [4]. It defines the approximate solution of an

unknown real function f ( x, y) as

f ̃ ( x, y) C ( x, y,u,v)K ( xk u, yk v) f (u,v)dudv (1)

where K ( xk u, yk v) is the kernel function and the point

( xk , yk )is the center point of kernel. C ( x, y,u,v) is a correc-

tion function and is defined as

C ( x, y,u,v) B(u,v)C∗( x, y) (2)

where C∗T ( x, y) {c1,c2,%,cm} is an unknown mth-

order column coef ficient vector of the correction func-

tion. B(u,v) {b1(u,v),b2(u,v),%,bm(u,v)} is a mth-order

row vector of basis functions and its definition usually

depends on the different boundary-value problems. For

example, in the present two-dimensional plate bending

problem with the overlapping boundary conditions, we

introduce the cubature serendipity-based interpolation

polynomial as the basis functions, namely, when taking

m 8, we have

B(u,v)

{b1(u,v),b2(u,v),…,b

m

(u,v)} (3) {1,u,v,u2,uv,v2,u2v,uv2}

Similarly, depending again on the different boundary-value problems, the fixed kernel function K ( xk u, yk

v) may be constructed by different forms of windowfunctions. In this paper, we use a cubic spline function

to construct the fixed kernel function

K ( xk u, yk v) W ∗(( xk u) / x)W ∗(( yk (4)

v) / y) / ( x y)

in which the cubic-spline form of the window function

W ∗

( z) is given as

W ∗( z) 0 | z|2

(2| z|)3 / 6 1| z|2

(2/3) z2(10.5| z|) | z|1

(5)

where z ( xk u) / x or z ( yk v) / y. x and y

denote the cloud size in the x-direction and y-direction

and they need be adjusted according to the point coordi-

nates and accuracy requirement due to the consistency

conditions of the reproducing kernel approach.The coef ficient vector of the correction function

C ∗T ( x, y) {c1,c2,%,cm} in Eq. (2) is unknown and

determined by satisfying the following consistency con-

ditions

bi( x, y)

C ( x, y,u,v)K ( xk u, yk v)bi(u,v)dudv (6)

i 1,2,…,m

and by substitution of Eq. (2) into Eq. (6) above and

then discretizing the resulting equations, we obtain

bi( x, y)

N T

n 1

B(un,vn)C∗( x, y)K ( xk un, yk (7)

vn)bi(un,vn)sn i 1,2,%m

where N T is the total number of points covering the

interior computational domain and edges and sn is

defined as the surface area of the point n.

Eq. (7) is actually a set of linear algebraic equations

with respect to the coef ficient vector C∗T ( x, y)

{c1,c2,%,cm} and can be rewritten simply into the fol-

lowing matrix form



143T.Y. Ng et al. / Engineering Structures 25 (2003) 141 –154

A( xk , yk )C∗( x, y) BT ( x, y) (8)

in which A( xk , yk ) is a symmetric constant matrix related

to the center point ( xk , yk )

Aij( xk , yk )

N T

n 1

bi(un,vn)K ( xk un, yk (9)

vn)b j(un,vn)sn i, j 1,2,%,m

and the unknown coef ficient vector C∗T ( x, y)

{c1,c2,%,cm} can be obtained through some rearrange-

ment of Eq. (8)

C∗( x, y) A1( xk , yk )BT ( x, y) (10)

Substituting Eqs. (2) and (10) into Eq. (1), the

approximate solution of the unknown real function f ( x, y) in Eq. (1) is obtained as

f ˜ ( x, y) B(u,v)A

1

( xk , yk )BT

( x, y)K ( xk u, yk (11)

v) f (u,v)dudv

and in the discretized form

f ̃ ( x, y)

N T

n 1

N n( x, y) f n (12)

where f n is a unknown point value for the point n and

N n( x, y) is defined as the shape function by

N n( x, y) B(un,vn)A1( xk , yk )BT ( x, y)K ( xk un, yk (13)

vn)sn

which is the interpolation function based on the fixed-

kernel window function. It should be noted that the

present shape functions, Eq. (13), are simply the poly-

nomial with respect to x and y, such that any of thederivatives can be obtained directly by the differentiation

of the basic function. Further, it is also validated that

the present shape functions satisfy the consistency con-

ditions.

The above approximation approach with the collo-

cation and fixed reproducing kernel techniques can be

used to solve generic engineering boundary-value prob-

lems. For example,

L f ( x, y) P( x, y)

PDEs in computational domain (14)

f ( x, y) Q( x, y)

Dirichlet boundary condition on D(15)

∂ f ( x, y) / ∂n R( x, y)

Neumann boundary condition on N(16)

where L is a differential operator and f ( x, y) an unknown

real function. By using the point collocation technique

and taking f ̃ ( x, y) as the approximation of f ( x, y), the

discretized approximation forms of the boundary-value

problem are given as

L f ̃ ( xn, yn) P( xn, yn) n 1,2,…, N (17)

f ̃ ( xn, yn) Q( xn, yn) n 1,2,… N D (18)

∂ f ̃ ( xn, yn)

∂n R( xn, yn) n 1,2,… N N (19)

where N , N D and N N are the number of scattered points

in the interior computational domain and along the

Dirichlet and Neumann edges, respectively, and the total

number of scattered points is thus N T ( N

N D N N ).

Substituting the fixed reproducing kernel approxi-

mation of Eq. (12) into the boundary-value problem of Eqs. (17–19) and then rewriting the resulting equation

with respect to the unknown point value f i into matrixform, we have

[ M ij] N T× N T{ f i} N T×1 {d i} N T×1 (20)

where [ M ij] is a N T × N T coef ficient matrix and {d i} N T-

order column coef ficient vector. They are given as

M ij L N j( xi, yi), d i P( xi, yi) (21)

i 1,2,…, N and j 1,2,…, N T

M ij N j( xi, yi), d i Q( xi, yi), i ( N 1),( N (22)

2),…,( N N D) and j 1,2,…, N T

M ij ∂ N j( xi, yi)

∂n, d i R( xi, yi), i ( N N D (23)

1),…,( N N D N N) and j 1,2,…, N T

Solving numerically the set of completed linear

algebraic equations, Eq. (20), the N T-order point-value

vector { f i} is obtained and then the approximation

f ̃ ( x, y) of the boundary-value problem is computedthrough Eq. (12).

Numerical comparisons have shown that the present

meshless approach is suf ficiently accurate and cost

ef ficient. However, there are constrained drawbacks for

its engineering application. For example, it is required

that the intersection between different boundary con-ditions, such as the present Dirichlet and Neumann

boundary conditions, must result in an empty set whena complete solution system is developed by the point

collocation technique. This results in dif ficulty when

applied directly to the partial differential boundary-value

problems with overlapping boundary conditions. In order

to overcome this dif ficulty, the following order-reduction

technique is employed to solve the higher-order PDEs.



144 T.Y. Ng et al. / Engineering Structures 25 (2003) 141 –154

2.2. Order-reduction technique for partial differential

equations

In general, overlapping boundary conditions, where

there are more than one boundary condition involvingthe same variable imposed at an edge, are very generic

in engineering applications, especially in the area of computational mechanics. For example, a simply-sup-

ported plate requires both the essential boundary con-

ditions (zero displacement) and natural boundary con-

ditions (zero moment) to be met, and both these

boundary conditions involve the same transverse deflec-

tion variable w. However, current meshless approaches

usually require an empty intersection set between two

adjacent boundary conditions. For example, the popular

Galerkin-based meshless methods simply use two setsof functions to formulate a variational form constructed

by various techniques including the penalty and Lag-

range multiplier methods. One set consists of trial func-

tions and the other consists of test functions [11],

{ f ( x, y)| f ( x, y) H 1, (24)

f ( x, y) Q( x, y) on Essential} (trial functions)

{g( x, y)|g( x, y) H 1, (25)

g( x, y) 0 on Essential} (test functions)

and the domain boundary is defined by the following

EssentialNatural (boundary condition) (26)

EssentialNatural f (empty set) (27)

whereEssential is the essential boundary and

Natural thenatural boundary. Q( x, y) is the given essential boundary

condition. The similar empty-set is also required for the

collocation-based meshless methods. In other words, it

is dif ficult to employ directly the present true-meshless

point collocation technique with fixed reproducing ker-nel approximation for solution of PDEs with the overlap-

ping boundary conditions as

{w( x, y)|w( x, y) H 3,

hE(w) h̄ E(w) and (28)

hN(w) h̄ N(w) on EssentialNatural}

(non empty set)

where h̄ E(w) and h̄ N(w) are essential and natural bound-

ary conditions respectively. The present boundary con-

ditions are the overlapping, i.e., EssentialNatural

Essential Natural, or EssentialNatural f is a non-

empty set.To overcome this drawback, suitable variables

F ( x, y) are selected and introduced not only to reduce the

higher-order PDEs into lower-order PDEs, but also to

separate the overlapping boundary conditions respect-

ively for the introduced variable F ( x, y) and problem vari-

able w( x, y), such that they correspond to empty sets

{w( x, y)|w( x, y) H 3 and

F ( x, y)|F ( x, y) H 3,

hE(w,F ) h̄ E(w,F ) and

hN(w,F ) h̄ N(w,F ) on EssentialNatural}

(29)

After the reconstruction of both the governing PDEsand given boundary conditions to correspond to the two

variables F ( x, y) and w( x, y), a complete reduced partial

differential boundary-value system is developed. Then,

by scattering a set of points in the computational domainand using the collocation technique with fixed reproduc-

ing kernel approximation, the system is discretized to a

set of completed linear algebraic equations. Finally, the

approximate solutions are obtained by numerically solv-

ing the set of algebraic equations.Thus far we have described the presently developed

true-meshless hM-DOR method, which is based on the

order-reduction technique for partial differential equa-

tions and combines the point collocation technique withfixed reproducing kernel approximation. The two sub-

sequent sections will examine the accuracy of the hM-DOR method and study the computational stability of

the methodology for certain engineering applications.

3. Numerical validation of the hM-DOR method

In this section, the accuracy and convergence charac-

teristics of the developed hM-DOR method are examined

numerically for the bending problem in classical thin

plates. Following a brief presentation of the formulationof the hM-DOR method for thin plates with different

overlapping boundary conditions, numerical compari-

sons are made with either exact solutions or FEM results

obtained by the commercial solver MSC/NASTRAN, for

plates of various geometric shapes (including square, tri-

angular and trapezium) and different boundary con-

ditions.

3.1. hM-DOR formulation for classical thin plates

The governing equation of an isotropic plate subjected

to a static transverse load is written classically in termsof the transverse deflection w w( x, y) as follows

D0(∂4w

∂ x4 2

∂4w

∂ x2∂ y2

∂4w

∂ y4) q( x, y) (30)

where D0 is the bending stiffness of the plate and

q( x, y) the distributed load in the transverse z-direction.

The boundary conditions we will consider in this study

include simply-supported boundary conditions

w 0, M n 0 (31)

and free boundary conditions




M n 0,V n Qn∂ M ns

∂s 0 (32)

where the subscripts n and s denote the normal and tan-

gential directions at the local edge. The moments andshear forces are usually given in the local edge n–s coor-

dinate system as [22] M n M xcos2 b M ysin2 b2 M xysin bcos b (33)

M ns M xy(cos2 bsin2 b) ( M x M y)sin bcos b (34)

Qn Q xcos b Q ysin b (35)

with b being the angle between the vector n and the axis x of the global x – y coordinate system and

M x D0(∂2w

∂ x2 m

∂2w

∂ y2),

M y D0(∂2w

∂ y2 m

∂2w

∂ x2), (36)

M xy D0(1 m)∂2w

∂ x∂ y

Q x D0∂

∂ x2w,Q y D0

∂

∂ y2w (37)

The boundary conditions, Eqs. (31) and (32), of theplate are overlapping boundary conditions, where there

are two boundary conditions, involving the variable w,

required to be imposed on each edge. These include both

essential boundary conditions w 0 on Essential and

natural boundary conditions M n

0 or V n

0 (both of which are functions of w) on Natural. As described above,

it is very dif ficult to impose directly the overlapping

boundary conditions by the point collocation technique.

By the present hM-DOR method however, the problemcan be solved by introducing a new variable F ( x, y) to

reduce the order of the governing PDE equation, as wellas to separate the overlapping boundary conditions

respectively for the introduced variable F ( x, y) and

deflection variable w( x, y). The introduced variableF ( x, y) here is selected as

F ( x, y) ∂2w

∂ x2

∂2w

∂ y2

in the interior computational domain(38)

and the governing equation, Eq. (30), and the boundaryconditions, Eqs. (31) and (32), can be rewritten as

D0(∂2F

∂ x2

∂2F

∂ y2) q( x, y)

in the interior computational domain

(39)

w 0, F ( x, y)(1 m)∂2w

∂ y2 0

when simply supported edges at x constant

(40)

F ( x, y)(1 m)∂2w

∂ x2 0,

∂F

∂ y (1 m)

∂3w

∂ x2∂ y 0

when free edges at y constant

(41)

In this way, the original boundary-value problem

expressed by Eqs. (30)–(32) are replaced by the reduced

partial differential boundary-value problem expressed byEq. (38)–(41). By the point collocation technique with

fixed reproducing kernel approximation, the approximate

solutions of the transverse deflection and introduced

variable are constructed as

w̃ ( xi, yi)

N T

n 1

N n( xi, yi)wn i 1,2,…, N T (42)

F ˜ ( xi, yi)

N T

n 1

N n( xi, yi)F n i 1,2,…, N T (43)

and the reduced PDEs and boundary conditions arediscretized as:

(1) interior of computational domain

D0(

N T

n 1

N n, xx( xi, yi)F n

N T

n 1

N n, yy( xi, yi)F n) (44)

q( xi, yi) i 1,2,…, N

N T

n 1

N n( xi, yi)F n

N T

n 1

N n, xx( xi, yi)wn (45)

N T

n 1 N n, yy( xi, yi)wn i 1,2,…, N

(2) simply-supported edges at x constant

N T

n 1

N n( xi, yi)wn 0 and

N T

n 1

N n( xi, yi)F n(1

m)

N T

n 1

N n, yy( xi, yi)wn 0 i ( N 1),( N (46)

2),…,( N Nss)

(3) free edges at y constant

N T

n 1

N n( xi, yi)F n(1 m)

N T

n 1

N n, xx( xi, yi)wn 0 (47)

i ( N N s-s 1),…, N T

N T

n 1

N n, y( xi, yi)F n (1 m)

N T

n 1

N n, xxy( xi, yi)wn 0 (48)

i ( N N s-s 1),…, N T

where N and N s s represent the numbers of points in

the interior computational domain and along simply-sup-




ported edges respectively. N T is the total number of

points and N T N N s s N f ( N f is the number of

points along free edges). It should be noted that in the

shape function above, N n( xi, yi) is the interpolation func-

tion based on the fixed-kernel window function, and thevariables in the subscript after a comma indicates partial

differentiation with respect to that variable. As shown inEq. (13), the present shape functions are simply poly-

nomials in x and y such that any of its derivatives are

obtained directly by the differentiation of the basic func-

tion.

Eqs. (44)–(48) are a set of discretized linear algebraic

equations with respect to the unknown point values wnand F n. Similar to the form of Eqs. (20)–(23), rewriting

with reference to the two unknown point-value variables

wn and F n, the resulting equations can be obtained inmatrix form as

M11 N T× N T M12 N T× N T

M21 N T× N T M22 N T× N Tw N T×1

F N T×1 {d2 N T×1} (49)

and through standard procedures for linear algebraicequations, the point values wn and F n of Eq. (49) may

be computed and the approximate solution w̃ ( xi, yi)

(i=1,2,…, N T) of plate deflection are obtained through

Eq. (42).

3.2. Numerical plate bending results

Numerical results for the maximum deflection in the

plate bending problem are used in comparisons to inves-

tigate the accuracy of the present hM-DOR method. Asshown in Fig. 1, the geometric shapes considered forthin plates subjected to uniformly distributed loads

include the square, triangular and trapezoidal geometries,

where a regularly scattered set of points are presented

respectively for computational discretization.

The comparisons for the square plates of various

boundary conditions are shown in Table 1. Three caseshave been considered, namely, SSSS, SFSS and SFSF,

where S represents a simply-supported edge while F rep-

resents a free edge. Presently obtained results for the

non-dimensional maximum deflections W max

w̃ max( x, y) D0 / (q0a4) of the square plates subjected to uni-

formly distributed loads q0 are compared with the exactsolutions. It is observed that as the number of regular

Fig. 1. Geometrical shapes of the thin plates and the distribution of the scattered points—(a) square, (b) triangular and (c) trapezoidal.

scattered points is increased, the relative errors decrease

in a monotonic manner, and generally being less than

1.1% for converged solutions. The convergence charac-

teristics in the form of the variation of the relative error

x with the distance between the points are shown in

Fig. 2, which validates the numerical stability of the

developed hM-DOR method.The comparisons for the different triangular plates are

shown respectively in Table 2 (a simply-supported equi-

lateral triangular plate—SSS) and Table 3 (an isosceles

triangular plate with all three edges simply-supported—SSS or with two simply-supported edges and one free

edge—SFS, the free edge being the base). Table 2 com-

pares the computed non-dimensional maximum deflec-

tions W max w̃ max( x, y) D0 / (q0a4) of the equilateral tri-

angular plates subjected to uniformly distributed loads

q0 with the exact solutions, and it is observed that as the

number of scattered points is increased, the relative

errors decrease monotonically, reaching a discrepancy of less than 0.7% when convergence is attained. Table 3

compares the computed maximum deflections w̃ max( x, y)

by the developed hM-DOR method with the FEM results

obtained through the commercial solver

MSC/NASTRAN, for an isosceles triangular aluminum

plate (a 1 m, α 45°) with two different combi-

nations of boundary conditions (SSS and SFS), and the

discrepancy between the two sets of results is less than

2.8%. Also for different densities of the scattered points,

Fig. 3 shows the deflection variation along the sym-

metric axis for a simply-supported isosceles triangular

plate (a 1 m, α 45°). Once again we observe verystable convergence characteristics in the developed hM-

DOR method. Further, the deflection profile shows that

the boundary conditions have been properly enforced.

For trapezoidal plates of various boundary conditions

(SSSS/SFSS/SFSF) and geometric parameters, Table 4

shows the numerical comparisons of the computed

maximum deflections w̃ max( x, y) by the developed hM-

DOR method again with the FEM results by commercial

solver MSC/NASTRAN. It is evident that the numerical

results are in close agreement with the FEM analysis,

with discrepancies of less than 2.5%, which is further

evidence of the numerical accuracy and stability of theproposed hM-DOR method.




Table 1

Convergence characteristics and comparison of non-dimensional maximum deflection W max for square plates subjected to uniformly distributed

loads q0 for different boundary conditions

Boundary condition Point distribution Maximum deflection W max Relative error

5 × 5 0.005254

9 × 9 0.004337SSSS 11 × 11 0.004236

(Exact solution W max 0.004062) 17 × 17 0.004137

21 × 21 0.004110

29 × 29 0.004090 0.69%

5 × 5 0.01456

SFSS 9 × 9 0.01351


17 × 17 0.01329

21 × 21 0.01307

31 × 31 0.01299 1.09%

5 × 5 0.01686

SFSF 9 × 9 0.01582


17 × 17 0.01556

21 × 21 0.0152525 × 25 0.01512 0.73%

Fig. 2. Convergence characteristics of the hM-DOR method for the

bending of square plates with various boundary conditions, x—relative

error, —distance between the points.

Table 2Convergence characteristics and comparison of non-dimensional

maximum deflection W max for equilateral triangular plates with SSS

boundary conditions, and subjected to uniformly distributed loads q0

No. of Maximum deflection W max Relative error (%)

points (exact solution W max=0.0005787)

16 0.0006998 20.9

49 0.0006463 11.7

100 0.0006114 5.65

169 0.0005991 3.53

256 0.0005827 0.69

Table 3

Comparison of maximum deflection w̃ max(m) for isosceles triangular

plates with various boundary conditions, and subjected to uniformly

distributed load q0 10 Pa (a 1m, h 1mm and α 45°)

Boundary hM-DOR FEM Relative

condition MSC/NASTRAN Difference (%)

SSS 2.64 × 104

2.57 × 104

2.72SFS 7.69 × 103 7.52 × 103 2.26

(free edge-base)

Fig. 3. Variation of the deflection profile along the symmetric axis,

for different scattered densities, of a SSS isosceles triangular plate load

under uniformly distributed load q0=10 Pa (a 1 m, h 1 mm and

α 45°).




Table 4

Comparison of maximum deflection w̃ max(m) for trapezoidal plates with various boundary conditions, and subjected to uniformly distributed load

q0 10 Pa (a 4 m, b 2 m, h 1 mm and α 45°)

α Boundary condition hM-DOR FEM MSC/NASTRAN Relative difference (%)

SSSS 1.87 × 102 1.85 × 102 1.08

45° SFSS (free edge-base) 1.25 1.22 2.46SFSF 3.01 3.00 0.333

15° 1.05 × 104 1.04 × 104 0.962

30° SSSS 2.25 × 103 2.23 × 103 0.897

60° 1.21 × 101 1.19 × 101 1.68

75° 7.26 × 101 7.24 × 101 0.276

4. Shape control of smart structures via distributed

sensors/actuators

The accuracy and stability of the developed hM-DOR

method have been examined numerically for the thin-

plate bending problem. In this section, we will apply itto examine the deformation shape control of smart struc-tures, which include the bending control of one-dimen-

sional beams and two-dimensional plates through distrib-

uted piezoelectric sensor and actuator layers. In general,

it can be very useful for a design engineer to employ

cost-effective computational tools in order to simulate

numerically the deformation control performance of a

smart structure by changing the location and geometry

of the actuators. The hM-DOR method is applied here

to study the shape control in bending deformation of simply-supported laminated beams and plates with inte-

grated distributed sensors/actuators, so as to demonstrate

the elegance and ef ficiency of the present method for

analyzing smart structures.

4.1. Shape control of a one-dimensional beam in

bending

The governing equation based on bending equilibrium

for a beam integrated with laminated sensors and actu-

ators, as shown in Fig. 4, can be described by [23]

D0

L3d 4W

dX 4 q( X ) p( X )

1

L

d 2

d X 2(

n

m 1

E md 2m( z2im (50)

Fig. 4. Geometry and cross-sectional view (thickness direction) of a beam with integrated actuators and sensors.

z2 jm) Rm( X )d 31V m

hm

)

where the non-dimensional deflection is defined as

W w / L and the non-dimensional coordinate variable

is X x / L ( L being the length of beam). D0 is the effec-tive bending stiffness of the laminated beam, p( X ) thedistributed mechanical load applied to the beam. E m,

hm and d m are respectively the Young’s modulus, thick-

ness and width for the mth-lamina. d 31 is the piezoe-

lectric strain constant, Rm( X ) the electrode profile func-

tion and V m the applied voltage on mth-actuator. zim and

z jm are the distances to middle plane from the upper and

lower surfaces of the mth actuator of the laminated

beam, respectively. The relations of shear force Qshear,

bending moment M , and displacement W according forthe piezoelectric effect of actuators are

M D0d 2W dX 2

n

m 1

E md 2m( z2im (51)

z2 jm) Rm( X )d 31V m

hm, Qshear

d M

d x

The simply-supported boundary conditions are con-sidered for the beam subjected to a uniform load

p( X ) p0. The beam is bonded symmetrically by a pair

of sensor and actuator (see Fig. 4), which are fully

covered by electrodes ( Rm( X ) R0). The right-hand side

term in Eq. (50) thus reduces to q( X ) p0. Applying the




hM-DOR methodology, a selected variable is intro-

duced as

F ( X ) d2W

d X 2(52)

and Eq. (50) is subsequently reduced to the following

lower-order differential equation

D0

L3d2F

d X 2 p0. (53)

The simply-supported boundary conditions are

W 0 and D0

LF ( X ) d 31

n

m 1

( z2im (54)

z2 jm) E md 2mV m

hm

Rm( X ) 0 at X 0 and X 1

and by the point collocation technique with fixed repro-

ducing kernel approximation, the approximate solutionsof the non-dimensional beam deflection W ( X ) and intro-

duced variable F ( X ) are constructed by

W ˜ ( X i)

N T

n 1

N n( X i)W n i 1,2,…, N T (55)

F ˜ ( X i)

N T

n 1

N n( X i)F n i 1,2,…, N T (56)

The above differential boundary-value problem is

discretized directly as

D0

L3

N T

n 1

N n, XX ( X i)F n p0( X i) i 1,2,…, N (57)

N T

n 1

N n( X i)F n

N T

n 1

N n, XX ( X i)W n i 1,2,…, N (58)

and at X 0 X N 1

N T

n 1

N n( X N +1)W n 0, D0

L

N T

n 1

N n( X N +1)F n (59)

d 31

n

m 1

( z2im z2 jm) E md

2

mV mhm

Rm( X N +1) 0

and at X 1 X N 2

N T

n 1

N n( X N +2)W n 0, D0

L

N T

n 1

N n( X N +2)F n (60)

d 31 n

m 1

( z2im z2 jm) E md 2mV m

hm

Rm( X N +2) 0

where N is the number of points in the interior compu-

tational domain, and N T is the total number of points,

N T N 2. Rewriting the set of discretized linear

algebraic equations, Eqs. (57) to (60), with respect to the

unknown point values W n and F n, the resulting equations

in matrix form results



W N T×1

F N T×1 {p2 N T×1} (61)

Eq. (61) is a set of linear algebraic equations, and

through standard procedures, the point values W n andF n can be easily computed and the approximate solution

W ˜ ( X i) (i=1,2,…, N T) for the beam deflection are finally

obtained through Eq. (55).Employing the material properties of a commercial

piezoelectric material PVDF for the numerical simula-

tions, the control influence of the applied electric field is

examined through the deformation of a uniformly loaded

beam. The mechanical and piezoelectric properties of

PVDF are taken as, E 1 E 2 E 3 2 GPa, G12

G13 G23 1 GPa, m12 m13 m23 0, e31 e32 0.046 C/m2, k 33 0.1063 × 109 F/m, with

other parameters being zero. An electro-elastic coupling

parameter Q is defined by

Q Vd 31

p0 L2

( n

m 1

( z2im z2 jm) E md 2m

hm

) (62)

This infers that the deformation shape of the beam can

be controlled by adjusting the defined parameter Q,

which is associated with both the applied electric fieldV and mechanical load p0. In Fig. 5, the comparison of

the controlled non-dimensional deflections W , for the

different electro-elastic coupling parameters Q between

the numerical results by the present hM-DOR method

and the exact analytical solutions of Lin and Hsu [23],

Fig. 5. Comparison with exact results for the controlled non-dimen-

sional deflections W of a uniformly loaded beam with uniform

actuators/ sensors for different electro-elastic coupling parameters Q,

(W 0 p0 L3 / (24 D0)).




shows that present simulations are coincident with the

exact solutions.

4.2. Shape control of a plate with distributed

sensor/actuator patches

Here we consider a symmetric laminated plate inwhich the smart layers are partially covered by square

electrodes acting as the sensors/actuators, as shown inFig. 6(a,b) and 7(a,b). The uncoupled governing equ-

ation for thin plate bending subjected to the mechanical

loading and electric field is written as [24]

D11∂4w

∂ x4 2( D12 2 D66)

∂4w

∂ x2∂ y2 D22

∂4w

∂ y4

q( x, y) p( x, y) (63)

Fig. 6. Effects of central actuator dimension S and electro-elastic coupling parameter Q on the deformation shape of the simply-supported bi-

layer-type square laminated plate subjected to a uniformly distributed load.

n

m 1

[ lmepV mhm zm(em31

∂2 Rm( x, y)

∂ x2 2em36

∂2 Rm( x, y)

∂ x∂ y

em32∂2 Rm( x, y)

∂ y2)]

where zm represents the z-coordinate of the mth-layermidplane, hm the mth-layer thickness, and Dij(i, j

1,2,6) the mechanical bending stiffnesses. If each layer

of laminated plate is isotropic, D11 D22 ( D12

2 D66) D0. p( x, y) is the distributed mechanical load,

V m the applied electric field to the mth-layer and emij the

piezoelectric parameters. lmep is a tracer that defines

whether the applied electric field direction corresponds

to the poling direction of the smart layer, with lmep 1 when the two directions are the same and lm

ep




Fig. 7. Effects of central actuator dimension S and electro-elastic coupling parameter Q on the deformation shape of the simply-supported sandwich-

type laminated square plate subjected a uniformly distributed load.

1 if they are opposite. Rm( x, y)is a function to define

the shape and position of mth-layer sensor/actuator

Rm( x, y) [ H ( x x1) H ( x x2)]m·[ H ( y y1) H ( y (64)

y2)]m

where H ( x xi) is the Heaviside step function, which is+1 if x xi and 0 if x xi. Based on the definition of

the Dirac delta function d ( x), we can implement the

derivatives of the function Rm( x, y) with respect to x or y. Usually the Heaviside step function, the Dirac func-

tion and the Dirac-function derivatives are explainedmechanically as a distributed force, concentrated force

and concentrated moment, respectively.

Similarly for the present deformation shape control

problem for plates, we employ the hM-DOR method and

introduce the variable F ( x, y) with the same form as Eq.

(38), and the governing equation, Eq. (63), is reduced to

a lower-order PDE. If the approximate solutions w̃ ( x, y)

and F ˜ ( x, y) are constructed in the same form as Eqs. (42)

and (43), followed by direct discretization through the

point collocation technique with fixed reproducing ker-

nel approximation, the set of reduced governing equa-

tions defined for the interior domain, with respect to theplate deflection w( x, y)and introduced variable F ( x, y)

results in the following discretized form

D0(

N T

n 1

N n, xx( xi, yi)F n

N T

n 1

N n, yy( xi, yi)F n) q( xi, yi)

p( xi, yi) n

m 1

[ lmepV mhm zm(em31

∂2 Rm( xi, yi)

∂ x2

2em36∂2 Rm( xi, yi)

∂ x∂ y em32

∂2 Rm( xi, yi)

∂ y2)]

i 1,2,…, N

(65)




N T

n 1

N n( xi, yi)F n

N T

n 1

N n, xx( xi, yi)wn (66)

N T

n 1

N n, yy( xi, yi)wn i 1,2,…, N

The plate boundary conditions considered here to be

simply-supported, thus the discretized boundary con-

ditions are of the same form as Eq. (46). The correspond-

ing set of discretized linear algebraic equations with

respect to the unknown point values wn and F n is then

set up and solved for simulation of the deformation

shape control of the smart plate structure.

As numerical simulation examples, we consider heretwo types of symmetric square laminated plates with

sensors/actuator patches, which are subjected to a uni-formly distributed load: (1) a bi-layer-type laminated

plate consisting of two layers of piezoelectric material

PXE-52; (2) a sandwich-type laminated plate consistingof an aluminum–alloy core layer with two surface layers

of the commercial piezoelectric material PXE-52. The

Fig. 8. Deformation shape control for a simply-supported equilateral triangular sandwich-type laminated plate subjected to a uniformly distributed

load, via a square actuator with centre point O1gr8(a / 2,√3a / 6), for different electro-elastic coupling parameters Q.

material properties of the aluminum alloy are E 1

E 2 E 3 69 GPa, G12 G13 G23 25.94 GPa,

and m12 m13 m23 0.33. Those of the piezoelectric

material PXE-52 are E 1 E 2 E 3 62.5 GPa,G12 G13 G23 24 GPa, m12 m13 m23 0.3,

d 31 d 32 280 × 1012 m/V, d 33 700 × 1012 m/V,

k 33 3.45 × 108F /m, with other parameters being

zero.

The influences of the electro-elastic coupling para-

meter Q and central actuator dimension S on the defor-

mation shape of the smart square plate subjected to the

uniformly distributed mechanical load p0 are shown

respectively in Fig. 6 for the bi-layer-type laminated

plate and Fig. 7 for the sandwich-type laminated plate.

It is observed from both the figures that the maximum

deflection of laminated plates can be controlled easilyby the distributed sensors/actuators. The deflections

decrease with increasing Q values, and it is also

observed that when Q exceeds a critical value, the

maximum deflection is reduced significantly, see Figs.

6(e) and 7(e). The dimensional effect of the electrode

profile is also examined. By respectively comparing




Figs. 6(e) and 7(e) with Fig. 6(f) and 7(f), it is evident

that with increasing electrode profile surface, the defor-

mation shape of the plate can be qualitatively changed

for the same electro-elastic coupling parameter Q.

In order to demonstrate the versatility of the hM-DORmethod, we extend the deformation shape control study

to sandwich-type laminated plates of equilateral triangu-lar (Fig. 8) and trapezoidal (Fig. 9) geometries. In Fig.

8, the deformation shape control of the simply-supported

sandwich-type triangular plate subjected to a uniformly

Fig. 9. Three-dimensional and corresponding contour variations for the deformation shape control of a simply-supported trapezoidal sandwich-

type laminated plate subjected to a uniformly distributed load, via varying the locations of two rectangular actuators ( S1 S2 0.05a × 0.1b)

with centre points O1 and O2, for different electro-elastic coupling parameters Q.

distributed mechanical load is presented for different Q

values. The deformation shape of the plate is controlled

by a single square actuator with centre point O1 at

(a / 2,√3a / 6). Results show intuitively correct character-

istics, and are qualitatively similar to the earlier squareexamples. Fig. 9 shows the corresponding deformation

shape control results for trapezoidal plates. The defor-mation shape of the trapezoidal plate is controlled by

two actuators, both of which have the same rectangular

area S1 S2 0.05a × 0.1b and their centre points O1




and O2 are located at the different local positions, as

shown in Fig. 9. Results here depict the three-dimen-

sional deformation shapes and corresponding contours

respectively for different locations of O1 and O2, and

once again, intuitively correct trends are observed.Further, the deflection mode profiles of the plates,

observed in Fig. 6 to 8, show that the simply-supportedboundary conditions have been properly enforced by thehM-DOR method.

5. Conclusion

Due to dif ficulty in the direct imposition of overlap-

ping boundary conditions encountered by existing collo-

cation-based meshless methods, a new hybrid meshless-

differential order reduction (hM-DOR) method has been

developed here such that these overlapping boundary

conditions can be imposed directly. Based on the order-reduction technique for partial differential equations, thehM-DOR method combines the collocation technique

with a fixed reproducing kernel approximation. Thedeveloped method has been validated for the thin-plate

bending analysis, where thin plates of various geometricshapes and boundary conditions were considered. The

hM-DOR was found to be very accurate and also pos-

sessed high numerical stability. Further, an application

of the hM-DOR method was demonstrated for the simul-

ation of deformation shape control in beams and plates

under uniformly distributed loading and integrated withpiezoelectric sensors/actuators. The numerical results all

point to the newly developed hM-DOR method beingelegant, accurate and numerically stable.

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a new hybrid meshless-differential order reduction (hm-dor)

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