Engineering Analysis with Boundary Elements 106 (2019) 452–461

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Engineering Analysis with Boundary Elements

journal homepage: www.elsevier.com/locate/enganabound

A novel displacement-based Trefftz plate element with high distortion

tolerance for orthotropic thick plates

Yan Shang

a , ∗ , Chen-Feng Li b , Ming-Jue Zhou

c

a State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics,

Nanjing 210016, China b Zienkiewicz Centre for Computational Engineering and Energy Safety Research Institute, College of Engineering, Swansea University, Swansea SA1 8EN, UK c College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310017, China

a r t i c l e i n f o

Keywords:

Displacement-based Trefftz element

Mindlin–Reissner plate

Generalized conforming

Locking free

Mesh distortion

a b s t r a c t

In this work, a simple but robust 4-node displacement-based Trefftz plate element is proposed for efficiently

analysis of orthotropic plate structures. This is achieved via two steps. First, the element’s deflection and rota-

tion fields are directly formulated at the base of the Trefftz functions of the orthotropic Mindlin–Reissner plate.

Second, to ensure the convergence property and further enhance the element performance, the generalized con-

forming theory is employed to satisfy the requirement for interelement compatibility in weak sense. The resulting

displacement-based element belongs to the nonconforming models on coarse meshes and gradually converges into

a conforming one with the mesh refinement. Numerical benchmarks reveal that the new element can produce

high precision results for both displacement and stress resultant. Moreover, it is free of shear locking and has

good tolerances to mesh distortions.

1

r

g

s

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b

t

o

t

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i

t

t

r

t

s

p

R

b

t

p

s

t

fi

i

c

T

a

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a

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w

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0

. Introduction

Due to the superior strength to weight ratio and stiffness to weight

atio, orthotropic plate structures have been widely used in various en-

ineering applications. For well understanding and effectively designing

uch structures, accurate numerical tools are clearly required, in which

he finite element method (FEM) is usually regarded as the most popular

hoice. In the earlier history, most plate finite elements are developed

ased on the classical Kirchhoff thin plate theory [1] . However, because

he transverse shear deformation is neglected in the Kirchhoff plate the-

ry, these Kirchhoff plate elements are only appropriate for modeling

hin structures. In contrast, in Mindlin–Reissner plate theory, the deflec-

ion and rotations are independently defined and the transverse shear

train is assumed to be constant through the plate thickness. Thus, the

indlin–Reissner plate elements can appropriately describe the behav-

ors of thick plates. Another advantage of the Mindlin–Reissner plate

heory is that it is applicable to both thick and thin plates. In addition,

here are also some element models developed based on other plate theo-

ies, such as the higher order shear deformation theory [2] , the zig-zag

heory [3–5] , etc. These plate theories indeed can provide improved

hear deformation responses but at cost of a significant increase in com-

utation expense.

Over the past decades, how to develop high-performance Mindlin–

eissner plate elements has drawn great attentions from scholars in

∗ Corresponding author.

E-mail address: shangyan@nuaa.edu.cn (Y. Shang).

ttps://doi.org/10.1016/j.enganabound.2019.06.002

eceived 12 March 2019; Received in revised form 15 May 2019; Accepted 6 June 2

955-7997/© 2019 Elsevier Ltd. All rights reserved.

oth computation mechanics and computation engineering communi-

ies. Various methodologies and schemes have been successfully pro-

osed, such as the enhanced assumed strain (EAS) method [6–9] , the

moothed FEM [10–15] , the quasi-conforming element method [16–18] ,

he mixed interpolated tensorial components method [19–21] , the re-

ned non-conforming element method [22,23] , the high-order linked

nterpolation method [24,25] , etc. A comprehensive literature review

an be found in [26] . Amongst these available approaches, the hybrid-

refftz finite element method [27–29] , which successfully blends the

ttributes of the Trefftz method [30] with the usual hybrid element

ethod, is a very attractive one. In hybrid-Trefftz element models, the

nternal displacement or stress trial field is assumed as the sum of the

nalytical homogeneous solutions and particular solution, namely the

refftz functions, so that it can a priori satisfy related governing equa-

ions. As a benefit, good numerical accuracy can be expected and the

hallenging shear locking problem can be easily eliminated. Since its

nception, the hybrid-Trefftz finite element method has become increas-

ngly popular and been applied to solve various solid mechanics prob-

ems [31–37] .

However, to our best knowledge, a very few studies have been per-

ormed on the topic of the orthotropic thick plates. Among the existing

orks, Petrolito [38,39] derived the Trefftz functions of the orthotropic

indlin–Reissner plate by transforming all governing differential equa-

ions into one equation, and then used these Trefftz functions to develop

019

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

Fig. 1. The definitions for a Mindlin–Reissner plate.

a

t

3

g

f

f

e

r

s

e

s

m

e

c

d

w

o

r

o

4

T

n

s

t

l

w

e

i

w

w

c

c

h

a

e

d

2

2

i

t

d

t

b

𝐮

b

𝜺

w

𝐋

a

𝝈

w

𝐃

i

𝐃

𝐃

r

𝐷

𝐶

w

s

r

⎧⎪⎪⎪⎨⎪⎪⎪⎩2

R

𝜓

𝜓

i

𝐻

a

3-node triangular hybrid-Trefftz element for analysis of the orthotropic

hick plate structures. Karkon and Rezaiee-Pajand [40] also developed

-node triangular and 4-node quadrilateral plate elements in an analo-

ous manner. But, it is found that their approach for deriving the Trefftz

unctions will be inapplicable when the order of Trefftz function exceeds

our. More recently, Ray [41] proposed a non-classical hybrid-Trefftz

lement formulation in which the Trefftz functions are obtained by di-

ectly solving the simultaneous governing equations and the particular

olution is no longer required.

Comparing with the hybrid element models, the displacement-based

lements may be more preferred in practical applications because of

ome inherent advantages. For instance, the displacement-based ele-

ents can be extended to the nonlinear cases more directly [42] . How-

ver, it should be noted that the Trefftz functions are generally not

ompatible across the interface between two neighboring elements. A

isplacement-only finite element implementation using Trefftz functions

ill produce non-conforming models whose convergence properties are

ften queried. Thus, it delivers a question that: whether an efficient and

obust displacement-based Trefftz element can be developed for analysis

f orthotropic thick plates.

This work aims to propose an answer to the above question. A novel

-node 12-DOF (degree of freedom) quadrilateral displacement-based

refftz plate element will be constructed via two steps. First, the poly-

omial Trefftz solutions of the orthotropic Mindlin–Reissner plate are

olved by converting related simultaneous governing equations into

hree equations. Then, these Trefftz solutions are employed to formu-

ate the element’s deflection and rotation fields in a straightforward

ay. Second, to ensure the convergence property and future improve the

lement’s performance, the generalized conforming theory [43–45] is

ntroduced to meet the requirement for interelement compatibility in

eak sense. The resulting generalized conforming Trefftz plate element

ill perform as a non-conforming model on coarse meshes and gradually

onverge into the conforming one as the mesh refined.

Several benchmark examples are tested to assess the new element’s

apability. Numerical results reveal that this new element can produce

igh precision results for both the displacement and stress resultant in

nalysis of thin and thick plate structures. Moreover, the proposed new

lement is free of shear locking and has exceptional tolerances to mesh

istortion.

. Analytical solutions for orthotropic Mindlin–Reissner plate

.1. Basic governing equations

For orthotropic plates, it is common to firstly describe their behav-

ors by using a local coordinate system, in which x - and y - axes are the

wo material principal directions whilst z - denotes the plate thickness

irection. As illustrated in Fig. 1 , in Mindlin–Reissner plate theory, the

ransverse deflection and rotations are independently defined and can

e expressed in the following vector form:

=

[𝑤 𝜓 𝑥 𝜓 𝑦

]T . (1)

w

453

The bending curvatures and transverse shear strains can be obtained

y

=

[𝜅𝑥 𝜅𝑦 𝜅𝑥𝑦 𝛾𝑥 𝛾𝑦

]T = 𝐋𝐮 , (2)

ith

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 𝜕 ∕ 𝜕 𝑥 0 0 0 𝜕 ∕ 𝜕 𝑦 0 𝜕 ∕ 𝜕 𝑦 𝜕 ∕ 𝜕 𝑥

𝜕 ∕ 𝜕 𝑥 −1 0 𝜕 ∕ 𝜕 𝑦 0 −1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (3)

For linear elastic cases, the constitutive relations between the strain

nd stress can be generally written as

=

[𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 𝑄 𝑥 𝑄 𝑦

]T = 𝐃𝛆 , (4)

ith

=

[ 𝐃 𝑏

𝐃 𝑠

] , (5)

n which

𝑏 =

⎡ ⎢ ⎢ ⎣ 𝐷 𝑥 𝐷 1 0 𝐷 1 𝐷 𝑦 0 0 0 𝐷 𝑥𝑦

⎤ ⎥ ⎥ ⎦ , (6)

𝑠 =

[ 𝐶 𝑥𝑧

𝐶 𝑦𝑧

] . (7)

Particularly for the single-layer plate composed of orthotropic mate-

ial, the components of Eqs. (6) and (7) can be calculated as follows:

𝑥 =

𝐸 𝑥 ℎ 3

12 (1 − 𝜇𝑥𝑦 𝜇𝑦𝑥

) , 𝐷 𝑦 =

𝐸 𝑦 ℎ 3

12 (1 − 𝜇𝑥𝑦 𝜇𝑦𝑥

) , 𝐷 𝑥𝑦 =

𝐺 𝑥𝑦 ℎ 3

12 ,

𝐷 1 = 𝜇𝑥𝑦 𝐷 𝑦 = 𝜇𝑦𝑥 𝐷 𝑥 , (8)

𝑥𝑧 = 𝑘ℎ 𝐺 𝑥𝑧 , 𝐶 𝑦𝑧 = 𝑘 𝑠 ℎ 𝐺 𝑦𝑧 , (9)

here E x , E y , 𝜇xy , 𝜇yx , G xy , G xz , G yz are the engineering material con-

tants; h denotes the plate thickness; k s is the shear deformation cor-

ection coefficient and usually taken as 5/6.

Besides, the equilibrium equations are given by

𝜕 𝑀 𝑥

𝜕𝑥 +

𝜕 𝑀 𝑥𝑦

𝜕𝑦 + 𝑄 𝑥 = 0

𝜕 𝑀 𝑥𝑦

𝜕𝑥 +

𝜕 𝑀 𝑦

𝜕𝑦 + 𝑄 𝑦 = 0

𝜕 𝑄 𝑥

𝜕𝑥 +

𝜕 𝑄 𝑦

𝜕𝑦 + 𝑞 = 0

. (10)

.2. Analytical solutions of Mindlin–Reissner plate

To derive the analytical solutions of the orthotropic Mindlin–

eissner plate, the next assumptions proposed in [38] are introduced:

𝑥 =

𝜕𝑤

𝜕𝑥 +

1 𝐶 𝑥𝑧

(

𝐷 𝑥

𝜕 3 𝑤

𝜕 𝑥 3 + 𝐻

𝜕 3 𝑤

𝜕 𝑥𝜕 𝑦 2

)

, (11)

𝑦 =

𝜕𝑤

𝜕𝑦 +

1 𝐶 𝑦𝑧

(

𝐷 𝑦

𝜕 3 𝑤

𝜕 𝑦 3 + 𝐻

𝜕 3 𝑤

𝜕 𝑥 2 𝜕𝑦

)

, (12)

n which

= 𝐷 1 + 2 𝐷 𝑥𝑦 . (13)

Then, by substituting Eqs. (11) and (12) into the strain-displacement

nd constitutive relationships which have been given by Eqs. (2) and (4) ,

e can obtain

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

Table 1

The homogeneous solutions of Eq. (18) and the deduced rotations and strains.

i 1 2 3 4 5 6 7 8 9 10 11

w i 1 x y x 2 xy y 2 x 3 x 2 y xy 2 y 3 x 3 y

𝜓 𝑖 𝑥

0 1 0 2 x y 0 3 𝑥 2 + 6 𝐷 𝑥 𝐶 𝑥𝑧

2 xy 𝑦 2 + 2 𝐻 𝐶 𝑥𝑧

0 3 𝑥 2 𝑦 + 6 𝐷 𝑥 𝐶 𝑥𝑧

𝑦

𝜓 𝑖 𝑦

0 0 1 0 x 2 y 0 𝑥 2 + 2 𝐻 𝐶 𝑦𝑧

2 xy 3 𝑦 2 + 6 𝐷 𝑦 𝐶 𝑦𝑧

𝑥 3 + 6 𝐻 𝐶 𝑦𝑧

𝑥

𝜅𝑖 𝑥

0 0 0 2 0 0 6 x 2 y 0 0 6 xy

𝜅𝑖 𝑦

0 0 0 0 0 2 0 0 2 x 6 y 0

𝜅𝑖 𝑥𝑦

0 0 0 0 2 0 0 4 x 4 y 0 6 𝑥 2 + 6 𝐷 𝑥 𝐶 𝑥𝑧

+ 6 𝐻 𝐶 𝑦𝑧

𝛾𝑖 𝑥

0 0 0 0 0 0 − 6 𝐷 𝑥 𝐶 𝑥𝑧

0 − 2 𝐻 𝐶 𝑥𝑧

0 − 6 𝐷 𝑥 𝐶 𝑥𝑧

𝑦

𝛾𝑖 𝑦

0 0 0 0 0 0 0 − 2 𝐻 𝐶 𝑦𝑧

0 − 6 𝐷 𝑦 𝐶 𝑦𝑧

− 6 𝐻 𝐶 𝑦𝑧

𝑥

i 12 13 14

w i xy 3 D y x 4 − D x y 4 Hx 4 − 3( D x + D y ) x 2 y 2 + Hy 4

𝜓 𝑖 𝑥

𝑦 3 + 6 𝐻 𝐶 𝑥𝑧

𝑦 4 𝐷 𝑦 𝑥 3 + 24 𝐷 𝑥 𝐷 𝑦

𝐶 𝑥𝑧 𝑥 4 𝐻 𝑥 3 − 6( 𝐷 𝑥 + 𝐷 𝑦 ) 𝑥 𝑦 2 +

12 𝐻 𝐶 𝑥𝑧

( 𝐷 𝑥 − 𝐷 𝑦 ) 𝑥

𝜓 𝑖 𝑦

3 𝑥 𝑦 2 + 6 𝐷 𝑦 𝐶 𝑦𝑧

𝑥 −4 𝐷 𝑥 𝑦 3 − 24 𝐷 𝑥 𝐷 𝑦

𝐶 𝑦𝑧 𝑦 4 𝐻 𝑦 3 − 6( 𝐷 𝑥 + 𝐷 𝑦 ) 𝑥 2 𝑦 +

12 𝐻 𝐶 𝑦𝑧

( 𝐷 𝑦 − 𝐷 𝑥 ) 𝑦

𝜅𝑖 𝑥

0 12 𝐷 𝑦 𝑥 2 + 24 𝐷 𝑥 𝐷 𝑦

𝐶 𝑥𝑧 12 𝐻 𝑥 2 − 6( 𝐷 𝑥 + 𝐷 𝑦 ) 𝑦 2 +

12 𝐻 𝐶 𝑥𝑧

( 𝐷 𝑥 − 𝐷 𝑦 )

𝜅𝑖 𝑦

6 xy −12 𝐷 𝑥 𝑦 2 − 24 𝐷 𝑥 𝐷 𝑦

𝐶 𝑦𝑧 12 𝐻 𝑦 2 − 6( 𝐷 𝑥 + 𝐷 𝑦 ) 𝑥 2 +

12 𝐻 𝐶 𝑦𝑧

( 𝐷 𝑦 − 𝐷 𝑥 )

𝜅𝑖 𝑥𝑦

6 𝑦 2 + 6 𝐷 𝑦 𝐶 𝑦𝑧

+ 6 𝐻 𝐶 𝑥𝑧

0 − 24( D x + D y ) xy

𝛾𝑖 𝑥

− 6 𝐻 𝐶 𝑥𝑧

𝑦 − 24 𝐷 𝑥 𝐷 𝑦 𝐶 𝑥𝑧

𝑥 − 12 𝐻 𝐶 𝑥𝑧

( 𝐷 𝑥 − 𝐷 𝑦 ) 𝑥

𝛾𝑖 𝑦

− 6 𝐷 𝑦 𝐶 𝑦𝑧

𝑥 24 𝐷 𝑥 𝐷 𝑦 𝐶 𝑦𝑧

𝑦 − 12 𝐻 𝐶 𝑦𝑧

( 𝐷 𝑦 − 𝐷 𝑥 ) 𝑦

𝑀

𝑀

𝑀

𝑄

𝑄

t

a

𝐷

s

m

E

a

𝐷

a

𝐷

m

r

u

s

𝑤

a

𝜓

𝜓

𝜅

T

i

m

t

𝑥 = 𝐷 𝑥

[ 𝜕 2 𝑤

𝜕 𝑥 2 +

1 𝐶 𝑥𝑧

(

𝐷 𝑥

𝜕 4 𝑤

𝜕 𝑥 4 + 𝐻

𝜕 4 𝑤

𝜕 𝑥 2 𝜕 𝑦 2

) ]

+ 𝐷 1

[ 𝜕 2 𝑤

𝜕 𝑦 2 +

1 𝐶 𝑦𝑧

(

𝐷 𝑦

𝜕 4 𝑤

𝜕 𝑦 4 + 𝐻

𝜕 4 𝑤

𝜕 𝑥 2 𝜕 𝑦 2

) ] , (14a)

𝑦 = 𝐷 1

[ 𝜕 2 𝑤

𝜕 𝑥 2 +

1 𝐶 𝑥𝑧

(

𝐷 𝑥

𝜕 4 𝑤

𝜕 𝑥 4 + 𝐻

𝜕 4 𝑤

𝜕 𝑥 2 𝜕 𝑦 2

) ]

+ 𝐷 𝑦

[ 𝜕 2 𝑤

𝜕 𝑦 2 +

1 𝐶 𝑦𝑧

(

𝐷 𝑦

𝜕 4 𝑤

𝜕 𝑦 4 + 𝐻

𝜕 4 𝑤

𝜕 𝑥 2 𝜕 𝑦 2

) ] , (14b)

𝑥𝑦 = 𝐷 𝑥𝑦

[ 2 𝜕

2 𝑤

𝜕 𝑥𝜕 𝑦 +

1 𝐶 𝑥𝑧

(

𝐷 𝑥

𝜕 4 𝑤

𝜕 𝑥 3 𝜕𝑦 + 𝐻

𝜕 4 𝑤

𝜕 𝑥𝜕 𝑦 3

)

(14c)

+

1 𝐶 𝑦𝑧

(

𝐷 𝑦

𝜕 4 𝑤

𝜕 𝑥𝜕 𝑦 3 + 𝐻

𝜕 4 𝑤

𝜕 𝑥 3 𝜕𝑦

) ] , (14c)

𝑥 = 𝐶 𝑥𝑧 𝛾𝑥 = −

(

𝐷 𝑥

𝜕 3 𝑤

𝜕 𝑥 3 + 𝐻

𝜕 3 𝑤

𝜕 𝑥𝜕 𝑦 2

)

, (14d)

𝑦 = 𝐶 𝑦𝑧 𝛾𝑦 = −

(

𝐷 𝑦

𝜕 3 𝑤

𝜕 𝑦 3 + 𝐻

𝜕 3 𝑤

𝜕 𝑥 2 𝜕𝑦

)

. (14e)

Next, by substituting above Eqs. (14a )–( 14e ) back into Eq. (10) , the

hree equilibrium equations of the Mindlin–Reissner plate theory can

lso be expressed in terms of the deflection w , as follows:

𝐷 𝑥 2

𝐶 𝑥𝑧

𝜕 5 𝑤

𝜕 𝑥 5 +

(

𝐷 𝑥 𝐻

𝐶 𝑥𝑧

+

𝐷 1 𝐻

𝐶 𝑦𝑧

+

𝐷 𝑥 𝐷 𝑥𝑦

𝐶 𝑥𝑧

+

𝐷 𝑥𝑦 𝐻

𝐶 𝑦𝑧

)

𝜕 5 𝑤

𝜕 𝑥 3 𝜕 𝑦 2

+

(

𝐷 1 𝐷 𝑦

𝐶 𝑦𝑧

+

𝐷 𝑥𝑦 𝐻

𝐶 𝑥𝑧

+

𝐷 𝑥𝑦 𝐷 𝑦

𝐶 𝑦𝑧

)

𝜕 5 𝑤

𝜕 𝑥𝜕 𝑦 4 = 0 , (15)

𝐷 𝑦 2

𝐶 𝑦𝑧

𝜕 5 𝑤

𝜕 𝑦 5 +

(

𝐷 𝑦 𝐻

𝐶 𝑦𝑧

+

𝐷 𝑥𝑦 𝐻

𝐶 𝑥𝑧

+

𝐷 𝑥𝑦 𝐷 𝑦

𝐶 𝑦𝑧

+

𝐷 1 𝐻

𝐶 𝑥𝑧

)

𝜕 5 𝑤

𝜕 𝑥 2 𝜕 𝑦 3

+

(

𝐷 𝑥𝑦 𝐷 𝑥

𝐶 𝑥𝑧

+

𝐷 𝑥𝑦 𝐻

𝐶 𝑦𝑧

+

𝐷 1 𝐷 𝑥

𝐶 𝑥𝑧

)

𝜕 5 𝑤

𝜕 𝑥 4 𝜕𝑦 = 0 , (16)

𝑥

𝜕 4 𝑤

𝜕 𝑥 4 + 2 𝐻

𝜕 4 𝑤

𝜕 𝑥 2 𝜕 𝑦 2 + 𝐷 𝑦

𝜕 4 𝑤

𝜕 𝑦 4 = 𝑞. (17)

c

454

Note that it is very difficult and even impossible to simultaneously

olve above three equations. Considering that Eqs. (15) and (16) are

uch more complicated, we will first solve Eq. (17) . The solution of

q. (17) can be divided into two parts, i.e., the homogeneous part w

0

nd the particular part w

∗ , which should, respectively, satisfy

𝑥

𝜕 4 𝑤

0

𝜕 𝑥 4 + 2 𝐻

𝜕 4 𝑤

0

𝜕 𝑥 2 𝜕 𝑦 2 + 𝐷 𝑦

𝜕 4 𝑤

0

𝜕 𝑦 4 = 0 , (18)

nd

𝑥

𝜕 4 𝑤

∗

𝜕 𝑥 4 + 2 𝐻

𝜕 4 𝑤

∗

𝜕 𝑥 2 𝜕 𝑦 2 + 𝐷 𝑦

𝜕 4 𝑤

∗

𝜕 𝑦 4 = 𝑞. (19)

For the homogeneous part, fourteen linearly independent polyno-

ial solutions are summarized in Table 1 , together with their deduced

otation and strain solutions. Besides, when the plate is subjected to a

niformly distributed transverse load q , the particular solution can be

imply set as

∗ =

𝑞

48

(

𝑥 4

𝐷 𝑥

+

𝑦 4

𝐷 𝑦

)

, (20)

nd accordingly

∗ 𝑥 =

𝑞 𝑥 3

12 𝐷 𝑥

+

𝑞𝑥

2 𝐶 𝑥𝑧

, (21)

∗ 𝑦 =

𝑞 𝑦 3

12 𝐷 𝑦

+

𝑞𝑦

2 𝐶 𝑦𝑧

. (22)

Then substitutions of Eqs. (20 )–( 22 ) back into Eq. (2) yields

∗ 𝑥 =

𝑞 𝑥 2

4 𝐷 𝑥

+

𝑞

2 𝐶 𝑥𝑧

, 𝜅∗ 𝑦 =

𝑞 𝑦 2

4 𝐷 𝑦

+

𝑞

2 𝐶 𝑦𝑧

, 𝜅∗ 𝑥𝑦

= 0 , 𝛾∗ 𝑥 = −

𝑞𝑥

2 𝐶 𝑥𝑧

,

𝛾∗ 𝑦 = −

𝑞𝑦

2 𝐶 𝑦𝑧

. (23)

It can be easily proven that, the polynomial functions proposed in

able 1 and Eq. (20) , whose orders are no greater than four, can also sat-

sfy Eqs. (15) and (16) which are fifth-order differential equations. That

eans, these provided homogeneous and particular solutions are exactly

he analytical solutions of the orthotropic Mindlin–Reissner plate, and

an be employed for constructing finite elements.

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

Fig. 2. The new 4-node quadrilateral plate element.

3

3

n

c

s

g

Π

i

∏w

l

t

c

w

p

i

e

g

c

m

e

a

r

3

q

B

𝜉

e

𝐪

w

M

s

𝐮

w

𝐔

𝜶

p

𝜺

w

𝐄

fi

a

E

𝛼

c

i

n

i

o

i

𝜓

w

i

p

f

m

b

𝝀

b

𝜶

w

𝐋

. Finite element implementation

.1. Variational principle

As previously discussed, a straightforward displacement-based fi-

ite element implementation using Trefftz solutions will produce non-

onforming models. Generally, the sub-region potential energy principle

hould be employed to ensure the non-conforming elements’ conver-

ence properties [45] :

𝑒 𝑚𝑃

= Π𝑒 𝑃 + 𝐻 𝑖𝑐 , (24)

n which Π𝑒 𝑃

is the conventional potential energy functional

𝑒

𝑝 = ∬Ω

1 2 𝜺 𝑇 𝐃𝛆 dΩ − ∬Ω

𝐟 𝑇 𝐮 dΩ − ∫Γ𝜎 𝐑

T 𝐮 dΓ, (25)

here f and R, respectively, are the prescribed body and boundary

oads; H ic is the additional energy item produced by the incompatibili-

ies along element boundary.

Note that the sub-region potential energy principle is much more

omplicated than the conventional potential energy principle, which

ill make the element construction procedure more tedious. For sim-

licity, we will directly use the conventional potential energy functional

nstead of the sub-region potential energy functional in this work. How-

ver, if the simplified functional is improperly employed, the conver-

ence cannot be guaranteed. To overcome this problem, the generalized

onforming theory proposed by Long et al. [45] , which can effectively

ake H ic →0 as the mesh refined, will be introduced. The resulting gen-

ralized conforming element belongs to the non-conforming model in

coarse mesh and will converge into a conforming one with the mesh

efinement.

.2. The new 4-node quadrilateral plate element

Fig. 2 illustrates a schematic representation of the new 4-node

uadrilateral plate element in which 1–4 are the vertex nodes. A i and

i ( i = 1 − 4) are the Gauss points (the local parametric coordinates are

= ± 1∕ √3 ) for imposing the generalized conforming condition. The el-

ment nodal DOF vector can be expressed as

𝑒 = [ 𝑤 1 𝜓 𝑥 1 𝜓 𝑦 1 𝑤 2 𝜓 𝑥 2 𝜓 𝑦 2 𝑤 3 𝜓 𝑥 3 𝜓 𝑦 3 𝑤 4 𝜓 𝑥 4 𝜓 𝑦 4 ] T .

(26)

As previously discussed, this new displacement-based plate element

ill be constructed based on the analytical solutions of the orthotropic

indlin–Reissner plate. Firstly, the deflection and rotation fields are as-

umed as

=

⎧ ⎪ ⎨ ⎪ ⎩ 𝑤

𝜓 𝑥

𝜓 𝑦

⎫ ⎪ ⎬ ⎪ ⎭ = 𝐔𝛂 + 𝐮 ∗ , (27)

𝐮

455

ith

=

⎡ ⎢ ⎢ ⎢ ⎣ 𝑤

0 1 𝑤

0 2 𝑤

0 3 ⋯ 𝑤

0 12

𝜓

0 𝑥 1 𝜓

0 𝑥 2 𝜓

0 𝑥 3 ⋯ 𝜓

0 𝑥 12

𝜓

0 𝑦 1

𝜓

0 𝑦 2

𝜓

0 𝑦 3

⋯ 𝜓

0 𝑦 12

⎤ ⎥ ⎥ ⎥ ⎦ , (28)

=

[𝛼1 𝛼2 ⋯ 𝛼12

]. (29)

Accordingly, the bending curvatures and shear strains can be ex-

ressed as

=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

𝜅𝑥 𝜅𝑦 𝜅𝑥𝑦 𝛾𝑥 𝛾𝑦

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = 𝐄𝛂 + 𝜺 ∗ , (30)

ith

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝜅0 𝑥 1 𝜅0

𝑥 2 𝜅0 𝑥 3 ⋯ 𝜅0

𝑥 12

𝜅0 𝑦 1

𝜅0 𝑦 2

𝜅0 𝑦 3

⋯ 𝜅0 𝑦 12

𝜅0 𝑥𝑦 1

𝜅0 𝑥𝑦 2

𝜅0 𝑥𝑦 3

⋯ 𝜅0 𝑥𝑦 12

𝛾0 𝑥 1 𝛾0

𝑥 2 𝛾0 𝑥 3 ⋯ 𝛾0

𝑥 12

𝛾0 𝑦 1

𝛾0 𝑦 2

𝛾0 𝑦 3

⋯ 𝛾0 𝑦 12

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (31)

In above equations, the components of the matrices U and E are the

rst twelve groups of homogeneous solutions proposed in Table 1 . u

∗

nd 𝜺 ∗ correspond to the particular parts and can be calculated by using

qs. (20 )–( 23 ) for an element-wise constant load q .

Then, in order to determine the introduced unknown coefficients

i , ( i = 1 − 12) as shown in Eq. (29) , the following twelve generalized

onforming conditions will be imposed:

(i) the compatibility conditions for deflections at the four nodes

𝑤

(𝑥 𝑖 , 𝑦 𝑖

)= 𝑤 𝑖 , ( 𝑖 = 1 − 4 ) , (32)

n which w ( x i ,y i ) is calculated by using Eq. (27) while w i is the element

odal deflection DOF shown in Eq. (26) ;

(ii) the compatibility conditions for normal rotations at the Gauss

points along each edge

𝜓 𝑛

(𝑥 𝑘 , 𝑦 𝑘

)= �� 𝑛𝑘 ,

(𝑘 = 𝐴 1 , 𝐵 1 , 𝐴 2 , 𝐵 2 , 𝐴 3 , 𝐵 3 , 𝐴 4 , 𝐵 4

), (33)

n which 𝜓 n ( x k ,y k ) are also calculated by substituting the Cartesian co-

rdinates into the assumed rotation field shown in Eq. (27) , while �� 𝑛𝑘

s interpolated by

𝑛𝑖𝑗 = ( 1 − 𝑠 ) 𝜓 𝑛𝑖 + 𝑠 𝜓 𝑛𝑗 , 𝑠 = ( 𝜉 + 1 ) ∕2 , (34)

here �� 𝑛𝑖 and �� 𝑛𝑗 are the nodal normal rotations along the element edge

j .

It has been demonstrated in [45] that, for a 4-node quadrilateral

late element, above twelve generalized conforming conditions can ef-

ectively ensure H ic →0 as the mesh refined. That means, the new ele-

ent’s convergence can be guaranteed.

Next, by substituting Eq. (27) into Eqs. (32) and ( 33 ), the relations

etween 𝜶 and the element nodal DOF vector q

e are derived:

𝛼 + 𝝌 = 𝚲𝐪 𝑒 . (35)

For brevity, the detailed expressions of the matrices 𝝀, 𝝌 and 𝚲 will

e summarized in Appendix. From Eq. (35) , we can obtain

= 𝐋 Λ𝐪 𝑒 − 𝐋 𝜒 , (36)

ith

Λ = 𝝀−1 𝚲, 𝐋 𝜒 = 𝝀−1 𝝌 . (37)

Substitution of Eq. (36) back into Eqs. (27) and (30) yields

= 𝐍 𝐪 𝑒 + 𝐍

∗ , (38)

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

Fig. 3. The relationship between the global coordinate system ( X, Y ) and the

local coordinate system ( x, y ).

𝜺

i

𝐍

𝐁

t

𝐊

w

𝐊

a

𝐏

e

w

i

n

𝐊

𝐏

w

𝐐

i

𝐐

‘

c

m

r

4

i

s

1

P

Fig. 4. The mesh for the patch test.

Fig. 5. The square plate subjected to a uniformly distributed load.

4

T

r

𝐸

w

𝑤

c

c

s

c

t

t

e

4

t

e

n

𝑤

(

u

a

c

a

i

n

i

p

= 𝐁 𝐪 𝑒 + 𝐁

∗ , (39)

n which

= 𝐔 𝐋 Λ, 𝐍

∗ = 𝐮 ∗ − 𝐔 𝐋 𝜒 , (40)

= 𝐄 𝐋 Λ, 𝐁

∗ = 𝜺 ∗ − 𝐄 𝐋 𝜒 . (41)

Finally, by substituting Eqs. (38) and (39) into Eq. (25) and applying

he principle of minimum potential energy, we can obtain

𝐪 𝑒 = 𝐏 , (42)

here K is the element stiffness matrix

= ∬Ω𝐁

𝑇 𝐃𝐁 dΩ, (43)

nd P is the element nodal equivalent load vector

= ∬Ω𝐍

𝑇 𝐟dΩ + ∫Γ𝜎 𝐍

T 𝐑 dΓ − ∬Ω𝐁

𝑇 𝐃 𝐁

∗ dΩ. (44)

Note that the above element stiffness matrix and the element nodal

quivalent load vector are obtained in the local coordinate system,

hich is in general different with the global coordinate system, as shown

n Fig. 3 . Thus they should be transformed back into the global coordi-

ate system:

global = 𝐐𝐊 𝐐

T , (45)

global = 𝐏 𝐐

T , (46)

ith

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

𝐐 𝜃

𝐐 𝜃

𝐐 𝜃

𝐐 𝜃

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (47)

n which

𝜃 =

⎡ ⎢ ⎢ ⎣ 1 0 0 0 cos 𝜃 − sin 𝜃0 sin 𝜃 cos 𝜃

⎤ ⎥ ⎥ ⎦ . (48)

This new element will be named by GCTP4, to record that it is a

G eneralized C onforming T refftz P late element with 4 nodes’. As pre-

eding discussed, it will perform as a non-conforming model on coarse

eshes and gradually converge into the conforming one as the mesh is

efined.

. Numerical validation

The proposed new element GCTP4 will be applied to several numer-

cal benchmark tests. The results are then analyzed and discussed. Be-

ides, the data of some existing plate element models, including QHT-

1 [40] , Abaqus S4 and S4R [46] , RDKQ-L20 [47] , SQUAD4 [48] and

etrolito’s element [38,39] , are also provided for comparison.

456

.1. The patch test

As shown in Fig. 4 , the small patch is divided into five elements.

wo different thicknesses ( h = 0.02 and 0.2) are considered. The mate-

ial properties are defined as [40] :

𝑥 = 10 𝐸 𝑦 = 10 , 𝐺 𝑥𝑦 = 0 . 5 𝐸 𝑦 , 𝐺 𝑥𝑧 = 0 . 5 𝐸 𝑦 , 𝐺 𝑦𝑧 = 0 . 2 𝐸 𝑦 , 𝜇𝑥𝑦 = 0 . 25 .

(49)

To produce a constant stress state, the following displacement field

ill be employed:

= 1 + 𝑥 + 𝑦 + 𝑥 2 + 𝑦 2 + 𝑥𝑦, 𝜓 𝑥 = 1 + 2 𝑥 + 𝑦, 𝜓 𝑦 = 1 + 2 𝑦 + 𝑥. (50)

The deflections and rotations at the boundary nodes 1–4 which are

alculated by using Eq. (50) are imposed to the patch as the boundary

onditions, whilst the results at the inner nodes 5–8 are monitored and

ummarized in Table 2 . In the thin plate case with h = 0.02, this element

an deliver exact results. In the thick plate case with h = 0.2, there are

iny errors which are mainly caused by the interelement incompatibili-

ies in a coarse distorted mesh. With the refinement of the mesh, these

rrors can be effectively eliminated.

.2. The square plate subjected to uniformly distributed load

Fig. 5 depicts a square plate subjected to a uniformly distributed

ransverse load q . Due to symmetry, only a quarter of the plate is mod-

led. The material parameters given by Eq. (49) are employed and the

on-dimensional deflection and bending moment are calculated:

= 𝑤 ×

(100 𝐸 𝑦 ℎ

3 ∕ 𝑞 𝐿

4 ), �� = 𝑀 ×(10 ∕ 𝑞 𝐿

2 ). (51)

a) Convergence

The convergence of the proposed new element will be investigated

sing the typical mesh shown in Fig. 5 . Two cases with different bound-

ry conditions, i.e., the clamped case and simply supported case, are

onsidered. Figs. 6–9 give the relative errors of the central deflections

nd moments for two different thickness-span ratios ( h / L = 0.01, 0.1),

n which the reference values are provided in [40] . One can see that the

ew element GCTP4 convergences very rapidly. An interesting feature

s that this new displacement-based element can also provide quite good

erformance when predicting stress resultants.

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

Table 2

The results of the patch test.

Node h = 0.02 h = 0.2

Exact GCPT4 Relative error Exact GCPT4 Relative error

5 w 125 125.00 0% 125 125.06 0.048%

𝜓 x 21 21.00 0% 21 21.00 0.000%

𝜓 y 17 17.00 0% 17 17.00 0.000%

6 w 1291 1291.00 0% 1291 1291.08 0.006%

𝜓 x 71 71.00 0% 71 71.00 0.000%

𝜓 y 45 45.00 0% 45 45.01 0.022%

7 w 1715 1715.00 0% 1715 1715.08 0.005%

𝜓 x 79 79.00 0% 79 78.99 − 0.013%

𝜓 y 61 61.00 0% 61 60.99 − 0.016%

8 w 707 707.00 0% 707 707.10 0.014%

𝜓 x 47 47.00 0% 47 46.99 − 0.021%

𝜓 y 45 45.00 0% 45 45.01 0.022%

Fig. 6. The relative errors of the central deflection and bending moment of the

clamped square plate with h / L = 0.01.

Fig. 7. The relative errors of the central deflection and bending moment of the

clamped square plate with h / L = 0.1.

Fig. 8. The relative errors of the central deflection and bending moment of the

simply supported square plate with h / L = 0.01.

Fig. 9. The relative errors of the central deflection and bending moment of the

simply supported square plate with h / L = 0.1.

Fig. 10. Distorted meshes containing 2 ×2 elements for the clamped square

plate.

(

s

t

o

t

t

c

t

e

4

p

T

457

b) Sensitivity to mesh distortion

To assess the element’s sensitivity to mesh distortion, the clamped

quare plate is analyzed again. As shown in Fig. 10 , two different dis-

orted meshes containing only 2 ×2 elements are employed. In the first

ne, the central node is moved along the main diagonal of the plate to

he corner node. In the second one, the central node is vertically moved

o the symmetry edge. Fig. 11 illustrates the variation of the normalized

entral deflection versus the distortion parameter. It can be observed

hat the maximum deviation is no larger than 5%, revealing that the

lement has good resistances to mesh distortion.

.3. The two-span plate

As shown in Fig. 12 , this test involves a two-span plate which is sim-

ly supported along the y -direction at its left, right and middle edges.

he left-span of the plate is subjected to a distributed transverse load

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

Fig. 11. The normalized central deflection of the

clamped plate versus the distortion parameter.

Fig. 12. The two-span rectangular plate.

Fig. 13. The relative errors of the deflection and bending moment of the two-

span rectangular plate with h / L = 0.01.

q

t

p

r

F

c

e

s

g

r

g

4

a

e

Fig. 14. The relative errors of the deflection and bending moment of the two-

span rectangular plate with h / L = 0.1.

B C

(a) Distorted mesh A

B

C

(b) Distorted mesh B

Fig. 15. Two distorted meshes containing 4 ×2 elements for the two-span plate.

a

𝜇

a

𝑤

3

, while the right-span is without loading. Two cases with different

hickness-span ratios h / L = 0.01 and 0.1 are considered. The material

arameters used here are the same with the previous test.

First, the convergence will be investigated. The computations are

epeatedly carried out by using the mesh 2 N ×N with N = 2, 4, 8, 16. In

igs. 13 and 14 , the relative errors of the deflection and bending moment

alculated at the point A are proposed. Second, the coarse mesh 4 ×2 is

mployed to generate distorted meshes, as shown in Fig. 15 , to further

tudy the influences on element performance of mesh distortion. Fig. 16

ives the variation of the deflection versus the distortion parameter. The

esults of this test reveal once again that this new element GCTP4 has

ood numerical accuracy and low susceptibility to mesh distortion.

.4. The rectangular plate with different sizes

In this test, the rectangular plates with different length a , width b

nd thickness h are analyzed to evaluate the rationalities of the present

lement formulation. The material parameters proposed in [49] are

458

dopted here:

𝐸 𝑥 = 20 . 83 , 𝐸 𝑦 = 10 . 94 , 𝐺 𝑥𝑦 = 6 . 10 , 𝐺 𝑥𝑧 = 3 . 71 , 𝐺 𝑦𝑧 = 6 . 19 ,

𝑥𝑦 = 0 . 44 , (52)

nd the non-dimensional deflection are checked:

= 𝑤 𝑄 11 ∕ ℎ𝑞 , 𝑄 11 = 𝐸 𝑥 ∕

(1 − 𝜇𝑥𝑦 𝜇𝑦𝑥

). (53)

In Table 3 the numerical results calculated by using a refined mesh

2 ×32 are summarized. Besides, the reference values which are ob-

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

Fig. 16. The normalized deflection of the two-span

plate versus the distortion parameter.

Table 3

The dimensionless central deflection of the simply supported rectangular plate with dif-

ferent sizes.

b / a h/a CPT [50] RPT [50] FSDT [50] HSDT [49] 3D [51] GCTP4

2.0 0.05 21,201 21,513.5 21,542 21,542 21,542 21,542.9

0.1 1325.1 1402.24 1408.4 1408.5 1408.5 1409.7

0.14 344.93 384.20 387.27 387.5 387.23 387.55

1.0 0.05 10,246 10,413.4 10,442 10,450 10,443 10,444.4

0.1 640.39 681.73 688.37 689.5 688.57 689.32

0.14 166.7 187.75 191.02 191.6 191.07 191.57

0.5 0.05 1988.1 2042.74 2047.9 2051.0 2048.7 2050.61

0.1 124.26 137.82 138.93 139.8 139.08 139.77

0.14 32.345 39.26 39.753 40.21 39.79 40.21

Fig. 17. The typical meshes for the circular plate.

t

r

c

t

t

s

4

t

d

𝐸

𝑤

(

e

Table 4

The dimensionless central deflection of the clamped circular

plate.

Elements 12 48 192 Reference

h/R = 0.001

RDKQ-L20 [47] 0.1269 0.1259 – 0.1250

SQUAD4 [48] 0.1163 0.1231 0.1246

QHT-11 [40] 0.1186 0.1234 –

GCTP4 0.1194 0.1235 0.1246

h/R = 0.01

RDKQ-L20 [47] 0.1245 0.1251 –

SQUAD4 [48] 0.1193 0.1242 0.1249

QHT-11 [40] 0.1187 0.1235 –

GCTP4 0.1195 0.1236 0.1247

h/R = 0.1

RDKQ-L20 [47] 0.1244 0.1251 –

SQUAD4 [48] 0.1355 0.1378 0.1384

QHT-11 [40] 0.1313 0.1362 –

GCTP4 0.1327 0.1363 0.1379

5

e

o

t

r

t

e

a

i

n

ained using the series expansion method based on different plate theo-

ies, including the high order shear deformation plate theory [49] , the

lassical thin plate theory [50] , the first order shear deformation plate

heory [50] , the two-variable refined plate theory [50] and the 3D elas-

ic theory [51] , are also listed. It seems that the results of the high order

hear deformation theory [49] are more close to the new plate element.

.5. The circular plate subjected to uniformly distributed load

In this test, the clamped circular plate is subjected to a uniformly dis-

ributed load. As shown in Fig. 17 , only a quarter of the plate is analyzed

ue to symmetry. The material parameters are described as [40]

𝑥 = 5 . 6 , 𝐸 𝑦 = 1 . 2 , 𝐺 𝑥𝑦 = 𝐺 𝑥𝑧 = 𝐺 𝑦𝑧 = 0 . 6 , 𝜇𝑥𝑦 = 0 . 26 . (54)

Then the non-dimensional central deflection is calculated:

= 𝑤𝐷 ∕ 𝑞 𝑅

4 , 𝐷 = 3 (𝐷 𝑥 + 𝐷 𝑦

)+ 2 𝐻. (55)

In Table 4 , the results for three different thickness-radius ratio cases

h / R = 0.001, 0.01 and 0.1) are summarized. It can be seen that this new

lement has good numerical accuracy and is free of shear locking.

459

. Conclusions

This work proposes a simple but robust 4-node 12-DOF quadrilat-

ral displacement-based Trefftz plate element for efficiently analysis of

rthotropic plate structures. This is achieved via two main steps. First,

he Trefftz functions of the orthotropic Mindlin–Reissner plate are de-

ived and employed for formulating the element’s deflection and rota-

ion fields. Second, to guarantee the convergence property and future

nrich the element behaviors, the generalized conforming conditions

re employed to satisfy the requirements for interelement compatibility

n weak sense. The resulting displacement-based Trefftz plate element,

amed as GCTP4, has the following characteristics:

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

A

F

S

t

b

(

A

E

b

𝝀

i

𝝀

𝝀

w

𝚽

w

e

𝝌

i

𝝌

𝝌

w

𝚽

𝚲

i

𝚲

𝚲

w

𝚲

𝚲

𝚲

𝚲

𝚲

𝚲

𝚲

𝚲 ⎣ 0 −

𝑙 41 𝑠 𝐵4 𝑦 41 𝑙 41

𝑠 𝐵4 𝑥 41 ⎦

(i) Since the element is developed based on the generalized conform-

ing theory, it will behave as a non-conforming model on coarse

meshes and gradually converge into a conforming model with the

mesh refinement.

(ii) Numerical tests reveal that the new element can produce high

precision results for both the displacement and stress resultant

in analysis of thick and thin plates. Moreover, it is free of shear

locking and has good robustness to mesh distortion.

(iii) Compared with its hybrid/mixed counterparts, the displacement-

based Trefftz plate element can be extended to the geometric non-

linear cases more directly. Related topics will be discussed in our

future papers.

cknowledgments

This work is financially supported by the National Natural Science

oundation of China (Grant numbers 11702133 , 11602219 ), the Natural

cience Foundation of Jiangsu Province (Grant number BK20170772 ),

he Natural Science Foundation of Zhejiang Province (Grant num-

er LQ16A020004 ), and the China Scholarships Council fellowship

201806835031).

ppendix

In this section, the components of the matrices 𝝀, 𝝌 and 𝚲 shown in

q. (35) are presented in details. Firstly, the matrix 𝝀 can be obtained

y

=

[ 𝝀𝑤

𝝀𝜙

] , (A1)

n which

𝑤 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝑤

0 1 (𝑥 1 , 𝑦 1

)𝑤

0 2 (𝑥 1 , 𝑦 1

)𝑤

0 3 (𝑥 1 , 𝑦 1

)⋯ 𝑤

0 12 (𝑥 1 , 𝑦 1

)𝑤

0 1 (𝑥 2 , 𝑦 2

)𝑤

0 2 (𝑥 2 , 𝑦 2

)𝑤

0 3 (𝑥 2 , 𝑦 2

)⋯ 𝑤

0 12 (𝑥 2 , 𝑦 2

)𝑤

0 1 (𝑥 3 , 𝑦 3

)𝑤

0 2 (𝑥 3 , 𝑦 3

)𝑤

0 3 (𝑥 3 , 𝑦 3

)⋯ 𝑤

0 12 (𝑥 3 , 𝑦 3

)𝑤

0 1 (𝑥 4 , 𝑦 4

)𝑤

0 2 (𝑥 4 , 𝑦 4

)𝑤

0 3 (𝑥 4 , 𝑦 4

)⋯ 𝑤

0 12 (𝑥 4 , 𝑦 4

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

(A2)

𝜙 = 𝐓 𝑛 𝚽, (A3)

ith

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝜓

0 𝑥 1 (𝑥 𝐴 1 , 𝑦 𝐴 1

)𝜓

0 𝑥 2 (𝑥 𝐴 1 , 𝑦 𝐴 1

)⋯ 𝜓

0 𝑥 12

(𝑥 𝐴 1 , 𝑦 𝐴 1

)𝜓

0 𝑦 1

(𝑥 𝐴 1 , 𝑦 𝐴 1

)𝜓

0 𝑦 2

(𝑥 𝐴 1 , 𝑦 𝐴 1

)⋯ 𝜓

0 𝑦 12

(𝑥 𝐴 1 , 𝑦 𝐴 1

)⋮ ⋮ ⋱ ⋮

𝜓

0 𝑥 1 (𝑥 𝐵4 , 𝑦 𝐵4

)𝜓

0 𝑥 2 (𝑥 𝐵4 , 𝑦 𝐵4

)⋯ 𝜓

0 𝑥 12

(𝑥 𝐵4 , 𝑦 𝐵4

)𝜓

0 𝑦 1

(𝑥 𝐵4 , 𝑦 𝐵4

)𝜓

0 𝑦 2

(𝑥 𝐵4 , 𝑦 𝐵4

)⋯ 𝜓

0 𝑦 12

(𝑥 𝐵4 , 𝑦 𝐵4

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (A4)

𝐓 𝑛 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

𝐓 1 𝐓 2

𝐓 3 𝐓 4

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐓 𝑖 =

[ 𝑙 𝑖𝑗 𝑚 𝑖𝑗

𝑙 𝑖𝑗 𝑚 𝑖𝑗

] ,

( 𝑖𝑗 = 12 , 23 , 34 , 41 ) , (A5)

here l ij and m ij denote the direction cosines of the outer normal of the

lement edge ij .

Secondly, the matrix 𝝌 is obtained by

=

[ 𝝌𝑤

𝝌𝜙

] , (A6)

n which

460

𝑤 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝑤

∗ (𝑥 1 , 𝑦 1 )𝑤

∗ (𝑥 2 , 𝑦 2 )𝑤

∗ (𝑥 3 , 𝑦 3 )𝑤

∗ (𝑥 4 , 𝑦 4 )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (A7)

𝜙 = 𝐓 𝑛 𝚽∗ , (A8)

ith

∗ =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝜓

∗ 𝑥

(𝑥 𝐴 1 , 𝑦 𝐴 1

)𝜓

∗ 𝑦

(𝑥 𝐴 1 , 𝑦 𝐴 1

)⋮

𝜓

∗ 𝑥

(𝑥 𝐵4 , 𝑦 𝐵4

)𝜓

∗ 𝑦

(𝑥 𝐵4 , 𝑦 𝐵4

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (A9)

Finally, the matrix 𝚲 is given by

=

[ 𝚲𝑤

𝚲𝜙

] , (A10)

n which

𝑤 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

𝐈 1×3 𝐈 1×3

𝐈 1×3 𝐈 1×3

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐈 1×3 =

[1 0 0

], (A11)

𝜙 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

𝚲𝜙

11 𝚲𝜙

12 𝚲𝜙

22 𝚲𝜙

23 𝚲𝜙

33 𝚲𝜙

34 𝚲𝜙

41 𝚲𝜙

44

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (A12)

ith

𝜙

11 =

⎡ ⎢ ⎢ ⎣ 0 −

1 𝑙 12

(1 − 𝑠 𝐴 1

)𝑦 12

1 𝑙 12

(1 − 𝑠 𝐴 1

)𝑥 12

0 −

1 𝑙 12

(1 − 𝑠 𝐵1

)𝑦 12

1 𝑙 12

(1 − 𝑠 𝐵1

)𝑥 12

⎤ ⎥ ⎥ ⎦ , 𝜙

12 =

⎡ ⎢ ⎢ ⎣ 0 −

1 𝑙 12

𝑠 𝐴 1 𝑦 12 1 𝑙 12

𝑠 𝐴 1 𝑥 12

0 −

1 𝑙 12

𝑠 𝐵1 𝑦 12 1 𝑙 12

𝑠 𝐵1 𝑥 12

⎤ ⎥ ⎥ ⎦ , (A13a)

𝜙

22 =

⎡ ⎢ ⎢ ⎣ 0 −

1 𝑙 23

(1 − 𝑠 𝐴 2

)𝑦 23

1 𝑙 23

(1 − 𝑠 𝐴 2

)𝑥 23

0 −

1 𝑙 23

(1 − 𝑠 𝐵2

)𝑦 23

1 𝑙 23

(1 − 𝑠 𝐵2

)𝑥 23

⎤ ⎥ ⎥ ⎦ , 𝜙

23 =

⎡ ⎢ ⎢ ⎣ 0 −

1 𝑙 23

𝑠 𝐴 2 𝑦 23 1 𝑙 23

𝑠 𝐴 2 𝑥 23

0 −

1 𝑙 23

𝑠 𝐵2 𝑦 23 1 𝑙 23

𝑠 𝐵2 𝑥 23

⎤ ⎥ ⎥ ⎦ , (A13b)

𝜙

33 =

⎡ ⎢ ⎢ ⎣ 0 −

1 𝑙 34

(1 − 𝑠 𝐴 3

)𝑦 34

1 𝑙 34

(1 − 𝑠 𝐴 3

)𝑥 34

0 −

1 𝑙 34

(1 − 𝑠 𝐵3

)𝑦 34

1 𝑙 34

(1 − 𝑠 𝐵3

)𝑥 34

⎤ ⎥ ⎥ ⎦ , 𝜙

34 =

⎡ ⎢ ⎢ ⎣ 0 −

1 𝑙 34

𝑠 𝐴 3 𝑦 34 1 𝑙 34

𝑠 𝐴 3 𝑥 34

0 −

1 𝑙 34

𝑠 𝐵3 𝑦 34 1 𝑙 34

𝑠 𝐵3 𝑥 34

⎤ ⎥ ⎥ ⎦ , (A13c)

𝜙

44 =

⎡ ⎢ ⎢ ⎣ 0 −

1 𝑙 41

(1 − 𝑠 𝐴 4

)𝑦 41

1 𝑙 41

(1 − 𝑠 𝐴 4

)𝑥 41

0 −

1 𝑙 41

(1 − 𝑠 𝐴 4

)𝑦 41

1 𝑙 41

(1 − 𝑠 𝐴 4

)𝑥 41

⎤ ⎥ ⎥ ⎦ , 𝜙

41 =

⎡ ⎢ ⎢ 0 −

1 𝑙 41

𝑠 𝐴 4 𝑦 41 1 𝑙 41

𝑠 𝐴 4 𝑥 41

1 1

⎤ ⎥ ⎥ . (A13d)

Y. Shang, C.-F. Li and M.-J. Zhou Engineering Analysis with Boundary Elements 106 (2019) 452–461

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

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