Journal Title
Month Year, Volume *, Issue *, PP.*-*
Calculation Method of Non-Linear Stiffness
Matrix in Spline Finite Element Method
Jian Qin1†, Liang Qiao1, Jun Chen1, Jianhua Feng2, Lili Wang3, Yimin Ma1
1. China Electric Power Research Institute, Beijing, 100055, China
2. Hebei Electric Power Transmission and Transformation Company, Shijiazhuang, China
3. Xinxiang Hoisting Equipment Factory Co. Ltd, Xinxiang, China
†Email:
[email protected]
Abstract
According to the analytical characteristics of cubic B spline function, such as compactness and discontinuous smoothness, two
methods, Global integral method and Assembling method are proposed to calculate the non-linear stiffness matrix in the spline
finite element method. The Global integral method transforms the integral of the product of spline functions into the sum of
boundary data of the product by integral by parts. The Assembling method used the compactness characteristics of the spline
function, reduces the n+3 order stiffness matrix in unit interval to the 4 order matrix, and then obtains the global matrix by
assembling the small ones. The two proposed methods are clear in theory and easy to be realized by computer program, which
can effectively improve the efficiency of the algorithm for nonlinear spline element method.
Keywords: non-linear stiffness matrix; spline function; integral by parts; matrix assembly
1 INTRODUCTION
From 1960 the finite element method developed very fast which was generally regarded as the most powerful tool in
structure analysis. According to virtual work principle or potential energy principle, the different interpolation
functions are used to form different kinds of finite element method, such as spline finite element method [1-2], finite
strip method [3-4], wavelet finite element method [5-6], and so on.
The spline finite element method taking cubic B spline function as interpolation function has more advantages [1].
Compared with bi-cubic Hermite rectangular element, the workload of the spline finite element is only a quarter of
the Hermite element with the same element division and the accuracy of the two methods is equal. The first and
second order derivatives of spline function are continuous, so the turning angle, stress, strain and bending torque of
the element nodes are with more accuracy than the one order element.
In the spline finite element, the calculation method of matrix composed by product of two spline functions or its
derivatives has been discussed in detail [2]. And there is no mature integral method for product of more spline
functions in matrix. But the stiffness matrix is mainly composed by three or four spline functions and its derivatives
product in the nonlinear spline finite element method [7-8].
Therefore, the nonlinear stiffness matrix calculation method is put forward to improve the computational efficiency.
2 SPLINE FINITE ELEMENT METHOD
The cubic B spline function is a class of piecewise continuous function, shown in FIG. 1.
0
d3
d2
d1
d4
-2
-1
0
1
2
FIG. 1 THE CUBIC B SPLINE FUNCTION
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The nonzero items of the B spline function can be decomposed as the 4 cubic polynomials d1, d2, d3, d4. The cubic
polynomials in unit interval [0,1] is
d1  (1  3t  3t 2  3t 3 ) / 6 t  [0,1]
d 2  (4  6t 2  3t 3 ) / 6
t  [0,1]
d3  (1  3t  3t 2  3t 3 ) / 6 t  [0,1]
d4  t 3 / 6
t  [0,1]
and the function space is constructed by the B spline functions at difference nodes such as 1 , 0 , …, n 1 (shown
in FIG. 2).
 n 1
 1  0 1
0
n
2
1
FIG. 2 THE BASE OF FUNCTION SPACE
So the arbitrary function in the interval [0, n] can be interpolated by space base and expressed as the product of
spline function and coefficient.
(1)
u( )  ( )  A
T
The spline function   [1 ( ), 0 ( ), , n ( ), n 1 ( )] and coefficient vector A=[a-1, a0,…, an+1] . The calculation of
one dimension problem is discussed here and the method for high dimension is same.
In the calculation of non-linear spline finite element method [7], the non-linear stiffness matrix is generally
expressed as

n
0
( BT 
d T d 
d T d 
 A)
d
d d
d d
(2)
Here, A and B are the known constant vectors, A  [a-1 ,a0 ,a1 , ,an ,an+1 ]T , B  [b-1 ,b0 ,b1 , ,bn ,bn+1 ]T .
Define
n
G( A, B)  G( B, A)   ( BT 
0

n
0
d T d 
d T d 
 A)
d
d d
d d
d T d 
d T d 
 A  BT 
d
d d
d d
(3)
So the important problem of non-linear spline FEM is the calculate method of the matrix G(A,B).
In this paper two methods are presented to calculate G(A,B). One is to integrate the product of spline functions in the
whole interval [0,n] and sum the results(called as Global integral method), the other is to calculate the product in the
unit interval and then assemble in the whole range(Assembling method).
3 GLOBAL INTEGRAL METHOD
The term of the matrix G(A,B) can be given as
n 1 n 1
Gij   (   bk alkl)i j d 
n
0
k 1 l 1
So the non-linear stiffness matrix can be obtained if the

n
0
n 1 n 1
 b a 
k 1 l 1
k
n
l 0
kli j d
(4)
kli j d  has been calculated.
The order of spline function i (i=-1,…,n+1) is 3, then when the order of derivation p is greater than 3, i ( p )  0 ,
when p=3, i ( p ) is a sectional-continuous function and when p<3, i ( p ) is continuous on the interval.
Integral by parts is preformed to the product of four spline function i p  j q k r l s  (the orders of derivative are p,
q, r, and s, respectively) until one of spline functions is equal to zero.
12  p  q  r  s
n
 m1   p   q   r   s   ( m) n
 p q r  s
m




d

(5)

(

1)
   
i
j
k
l

0
 m  1!  i j k l  0
m0
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Through the Leibniz differential formula
i p  j q k r l s  


( m)
m!
i p ia  j q ib k r ic l s id 
i
!
i
!
i
!
i
!
 is  m a b c d


(6)
0  is  m
So the integral of the product of four spline functions can be calculated when the boundary data of
i p i  j q  i k r  i l s i  is gotten.
a
b
c
d
The method needs multiple iteration and continuous discrimination of spline function, but it can be used in the
integral of the spline functions product with arbitrary derivatives.
4 ASSEMBLING METHOD
d
 [1 , 0 ,
d
element method.
Define  
, n , n1 ] where   [1 , 0 ,
, n , n 1 ] in the matrix G(A,B) of non-linear spline finite
The matrix G(A,B) can be seen as the sum of integration in the interval:
n
G ( A, B)   ( BT  T  A) T  d 
0
1
2
3
n 1
    

 M1  M 2 
 Mn
0
1
2
n2
n
  [( BT  T  A) T  ]d 
n 1
(7)
According to the compactness of spline function, the Φ or Φ’ in any unit interval has at most 4 nonzero terms then
the vectors can be simplified. For example, when ξ  [i,i+1], ( )  [0,0, ,0, i 1 ( ), i ( ), i 1 ( ), i  2 ( ),0, ,0,0] ,
as shown in FIG. 3(the Φ’ has the same simplify method).
i 1
i-1
i
i 1 i  2
i  i+1 i+2
FIG. 3 THE NONZERO TERMS OF THE SPLINE FUNCTION IN THE [i,i+1] INTERVAL
In this range, the function vector can be expressed by the interval polynomials d1, d2, d3, d4.
( )  [0,0, ,0, d1(  ), d2 (  ), d3 (  ), d4 (  ),0, ,0,0] ,     i [0,1]
So in every unit interval, there is nonzero items from row i-1 to row i+2 and from column i-1 to column i+2 of the
i 1
n+3 order integral matrix M i   [( BT T A)T ]d  , which compose a 4×4 nonzero submatrix Ki.
i
In order to reduce the amount of calculation and memory space, the 4 order vectors R  [d1, d2 , d3 , d4 ] ,
Ai  [ ai 1 , ai , ai 1 , ai  2 ] and Bi  [bi 1 , bi , bi 1 , bi  2 ] are defined. The 4 order nonzero submatrix Ki can be expressed as:
1
K i   [( B iT RT RAi ) RT R ]d 
0
(8)
When Ki is calculated, it can be assembled into the global matrix according to the order of coefficients, and the
global stiffness matrix G is formed finally, shown in FIG. 4. (The process is similar with the assembling of structure
stiffness matrix in finite element, so this method can be regarded as the stiffness matrix assembly of spline finite
element.)
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K1
K2
K3
G
Kn2
K n 1
Kn
(n+3)×(n+3)
FIG. 4 STIFFNESS MATRIX ASSEMBLY
Because the terms of Ki are the integral of product of polynomials, and the order of the product of polynomials is 8,
the Gaussian integration with 5 integral points can be used in calculation of Ki to reach 9 order of accuracy.
The method of Gaussian integration
1
5
0
m 1
K i   [( B iT RT RAi ) RT R ]d    H m [( BiT R(m )T R(m ) Ai ) R(m )T R(m )]
(9)
The integral weight coefficient Hm and integral point m in the interval of [0, 1] can be found in the related articles.
The values of R  [d1, d2 , d3 , d4 ] at Gaussian integration points can be saved as a file to be available for the
calculation program, thus the calculation speed can be greatly improved.
5 SUMMARY
The composition and calculation of stiffness matrix is the most important part in the finite element method, and the
calculation of the nonlinear stiffness is more difficult.
In the spline finite element, it needs to calculate the integral of the product of multiple spline function or its
derivative in the nonlinear stiffness matrix, and the nonlinear stiffness matrix also changes with the input parameters
(A and B) during the iteration. So the calculation of nonlinear stiffness matrix is more complex and has larger
amount of calculation.
In this paper the two methods Global integral method and assembling method are proposed to calculate the nonlinear stiffness matrix.
The items of stiffness matrix can be calculated by the boundary data of the product of spline functions in Global
integral method.
The Assembling method used the compactness characteristics of the spline function, reduces the n+3 order stiffness
matrix in unit interval to the 4 order matrix, and then obtains the global matrix by assembling the small ones.
These methods fully make use of the advantage of spline function in interpolation, which are effective calculation
methods for the nonlinear stiffness matrix of the spline finite element method.
ACKNOWLEDGMENT
This research was supported by Science and Technology Research Project of State Grid (Study on the calculation
method and selection test of cargo cableway of transmission line).
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AUTHORS
1Jian
2Liang
Qin was born in Shijiazhuang,
Qiao was born in Datong, China,
China, in 1979. He received his doctor's
in 1989. He received his master's degree in
degree in 2009 from Peking University.
2014 from Southwest Jiaotong University.
He is senior engineer of China Electric
He is engineer of China Electric Power
Power Research Institute and research on
Research Institute and research on the
the numerical calculation of construction
numerical analysis and calculation of
technology in transmission line.
mechanical structure.
3Jun
Chen was born in Zhangjiakou,
China, in 1987. He received his master's
degree in 2013 from Beijing University of
Chemical Technology. He is engineer of
China Electric Power Research Institute
and research on the numerical analysis
and mechanical design.
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