MA Calculation Method of Stiffness Matrix in Non-linear Spline Finite Element for Suspension Cable

Abstract: Through the characteristics of cubic B spline function, the stiffness matrix expression is obtained for the non-linear spline finite element of the suspension cable, and the three calculation method is put forward such as the integral by parts method, the matrix assembly method based on spline integral and the matrix assembly method based on Gauss integral, which are applied to calculate the non-linear stiffness matrix. The integral by parts method using the Leibniz formula as foundation can calculate every item of the stiffness matrix directly, but its efficiency is low when the matrix order is high. The matrix assembly method based on spline integral fully uses the local non-zero property of the spline function, converts the calculation of high order matrix into calculation of 4 order matrix and its assembly, with high efficiency and small storage in calculation. The matrix assembly method based on Gauss integral is similar with the method based on spline integral, but its workload is larger than the method based on spline integral when the matrix order is high. These calculation methods can all be processed by parallel algorithm to improve calculation efficiency, which can provide new ideas for high efficient numerical analysis works.


Introduction
Performance of the modern computer with high speed provides strong technical support for solving the non-linear problems, such as the non-elastic material, the complicated contact issue and the large geometrical deformation of the structure etc.Following the enlargement and refinement of the structure, working quantity for non-linear numerical calculation is also increased greatly [1,2].In order to improve calculation efficiency, solve the problems in structure design and analysis appeared during actual engineering, the more efficient and more accurate theoretical algorithm is necessary to be proposed.
In the finite element, the expression formula and calculation of the stiffness matrix is a matter of prime importance [3][4][5].The main method for stiffness matrix calculation is always numerical integral (generally Gauss integral).In order to ensure integral accuracy in non-linear issue, the order of numerical integral shall be improved 1-2 orders accordingly, so the calculated amount is greatly increased, especially in iteration process of Newton-Raphson method where the stiffness matrix shall be generated repeatedly.When free degree is more, time cost is extremely great [6,7].
On basis of the non-linear spline finite element method [8][9][10], the calculation methods of the stiffness matrix for the suspension cable are forwarded for the large geometrical deformation issue, which can effectively improve generation efficiency of the stiffness matrix.

Characteristic of non-linear spline element
The difference between the spline finite element and the common finite element method is that cubic B spline function is taken as the interpolation function.The spline function has high accuracy for calculating the angle, stress, strain and bending moment etc.
Nonzero parts of the cubic B spline basic function is generally expressed as 4 cubic polynomial d 1 , d 2 , d 3 , d 4 in unit sections during calculation, shown as Figure 1.
The polynomial formulas of the spline function in [0, 1] are 3 ) / 6 Cubic B spline base is expressed as tensor base of one dimension base V ξ , ( ) So the arbitrary function in the interval [0, n] can be interpolated by the base and expressed as the product of spline function and coefficient.
The spline function vector is The non-linear spline finite element function forwarded in this paper sufficiently uses the advantage of the spline function in interpolation for calculation the integral function to reduce the calculated amount.

The stiffness matrix of suspension cable
For the suspension cable, the Green strain is dX dU dU dU ds ds ds ds According to the constitutive relation and the principle of minimum potential energy, the potential energy function can be expressed as The stiffness matrix can be expressed as combination of the following integral matrixes: Here, A and B are the known constant vectors, And then So the importance matter of non-linear spline FEM is the calculate method of the matrix G(A, B).
Set the matrix G(A, B), the entry in the i-th row and j-th column of the matrix can be expressed as: It is seen that Gij is the combination of integral of products of four spline functions.Calculated amount for the expansion of combination is great and efficiency is extremely low.Three effective methods are proposed for the calculation of stiffness matrixes.

Integral by parts method
) is calculated, the Gij is obtained and then the stiffness matrix G.
For products of the 4 spline functions φ φ φ φ (p, q, r, s are derivatives respectively), it is carried out by integral by parts until the derivative of the product of the spline functions becomes zero.
( ) ( ) ( ) ( ) From the differential Leibniz formula So ∫ can be calculated.The derivative of the cubic B spline function can be directly obtained at integral end 0 and n, therefore the integral by parts can be directly calculated.And the spline function is local nonzero, the product of the spline functions is zero when the interval between i, j, k and l is greater than 3.
Because the method needs repeatedly judge the continuity of derivative of the spline function and complete the multi-layer summation, it is inefficient in the computer.The matrix G(A, B) can be seen as the sum of integration in the interval:

Matrix assembly method 5.1 The matrix assembly method based on spline integral
i-1 i i+1
In this section, the function vector ′ Φ can be expressed by d 1 , d 2 , d 3 , d 4 .( ) [0, 0, , 0, ( ), ( ), ( ), ( ), 0, , 0, 0] 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 n+3 order integral matrix In order to reduce the amount of calculation and memory space, the 4 order vectors = are defined.The 4 order nonzero submatrix K i can be expressed as: When K i 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 Figure 3 (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).The entry in the matrix Ki can be written as p q k l ∈ , and then ( , , , ) D pqkl can be obtained by analytical calculation.Saving D pqkl as a file and calling the file in the calculation procedure, the calculation efficiency can be extremely improved.

The matrix assembly method based on Gauss integral
Because the integral term in Ki is the product of the polynomials which is 8 orders polynomial, Ki can be accurately calculated by the Gauss integral of 5 integral points which can reach accuracy of 2×5-1=9 order.So Here Hm is integral coefficient and m ξ is integral point in [0, 1] section.
Value of 0 ( ) at every Gauss integral point can be saved as the file and called in the calculation procedure to improve the calculation efficiency.

Example
For known constant vectors A and B with different orders, the stiffness matrix is calculated by integral by parts method, the matrix assembly method based on spline integral and the matrix assembly method based on Gauss integral respectively, the CPU time cost is shown as Figure 4. Results obtained by three methods are the same.It is seen that efficiency of the matrix assembly method based on spline integral is highest whose time cost is minimum with same matrix order, and no obvious change when the order of the matrix increases.This method has the calculation speed twice as high as the other two methods.
The cost time of the matrix assembly method based on Gauss integral is linearly increasing with the order of the matrix, its working quantity is mainly Gauss integral calculation of the function in every section.The time of integral by parts method is square growth with the matrix order, and the judgment of spline function boundary and summation of product of the spline function are the main workload.When the matrix order is small, efficiency is high, and efficiency decreases quickly when order increases.

Conclusion
The three calculation methods for the stiffness matrixes of suspension cable issues are put forward.It can be seen from the given example, the matrix assembly method based on spline integral has the highest efficiency and fastest calculation speed.The cost time of this method is two orders of magnitude larger than other methods when the matrix order is high, and this method can obtain the accurate result.Therefore the matrix assembly method based on spline integral is the best for calculation of the stiffness matrix.And the technique of this method can also be used in calculation of other matrixes, such as the load matrix in spline element method.

Define
G(A, B) of non-linear spline finite element method.

Figure 3 .
Figure 3. Assembly of stiffness matrix

Figure 4 .
Figure 4.The cost time with different matrix order