An Efficient Method to Generate Element Stiffness Matrix of Quadrilateral Elements in Closed Form on Application of Vehicle Analysis

Abstract:

This paper presents about generating elemental stiffness matrix for quadrilateral elements in closed form solution method for application on vehicle analysis which is convenient and simple as long as Jacobian is matrix of constant. The interpolation function of the field variable to be found can integrate explicitly once for all, which gives the constant universal matrices A, B and C. Therefore, stiffness matrix is no longer integration of the given functional, it is simple calculation of universal matrices and local co-ordinates of the element. So time taken for generation of element stiffness can be reduced considerably compared to Gauss numerical integration method. For effective use of quadrilateral elements hybrid grid generation is recommended that contains all interior element edges are parallel to each other (rectangle or square elements) and outer boundary elements are quadrilaterals with distortion. So in the Proposed method, the closed form and Gauss numerical method is used explicitly for interior elements and outer elements respectively. The time efficiency of proposed method is compared with conventional Gauss quadrature that is used for entire domain. It is found that the proposed method is much efficient than Gauss Quadrature.

• Periodical: Applied Mechanics and Materials (Volume 852)
• Pages: 582-587
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.852.582
• Online since: September 2016
• Authors: P.V. Jeyakarthikeyan*, R. Yogeshwaran, Karthikk Sridharan
• Keywords: Closed Form, Gauss Integration, Numerical Integration, Stiffness Matrix, Time Comparison, Universal Matrix

References

[1] G. Suramanian, C. Jeyachandra Bose, Convenient generation of stiffness matrices for the family of plane triangular elements, Computers and Structures, vol. 15, no. 1, p.85–89, (1982).

DOI: 10.1016/0045-7949(82)90036-0

[2] G. Subramanian, C. Jeyachandra Bose, C. Ramesh Babu, A note on universal matrices for triangular elements for the quasi harmonic equation, Numerical Methods in Engineering, vol. 19, p.617–624, (1983).

DOI: 10.1002/nme.1620190410

[3] C.K. Yew, J.T. Boyle, D. Mackenzie Closed form integration of element stiffness matrices using a computer algebra system, Computers and Structures, vol. 56, no. 4, p.529–539, (1994).

DOI: 10.1016/0045-7949(94)00549-i

[4] Jian-Wu Zhang, Wilfried B. Kratzig, A simple four - node quadrilateral finite element for plates, Finite Elements in Analysis and Design, vol. 19, p.195–207, (1995).

DOI: 10.1016/0168-874x(95)00012-i

[5] P.S. Shia kolas , R. V Nambiar , K. L Lawrence , W. A Rogers, Closed form stiffness matrices for the linear strain and quadratic strain tetrahedron finite elements, Computers and Structures, vol. 45, no. 2, p.237–242, (1992).

DOI: 10.1016/0045-7949(92)90407-q

[6] Sara E. McCaslin, Panos.S. Shiakolas, Brian H. Dennis, Kent L. Lawrence, Closed form stiffness matrices for higher order tetrahedral finite elements, Advances in Engineering Software, vol. 44, p.75–79, (2012).

DOI: 10.1016/j.advengsoft.2011.05.035

[7] Tirupathi.R. Chandrupatla, Finite Element in Engineering. Eastern Economy Edition, (2002).

[8] Moser. K, Swoboda. G, Explicit stiffness matrix of the linearly varying strain triangular element, Computers & Structures, p.311–314, (1978).

DOI: 10.1016/0045-7949(78)90038-x

[9] C. Jeyachandra bose, J. Kirkhopes, least squares strain smoothing for the eight- node serendipity plane stress element.

[10] Rathod.H. T, Explicit matrices for axisymmetric triangular elements, Computers and Structures, vol. 30, no. 5, p.1001, (1988).

DOI: 10.1016/0045-7949(88)90152-6

[11] K.L. Lawrence, R.V. Nambiar,B. Bergmann, Closed form stiffness matrices and error estimators for plane hierarchic triangular elements, IntJ. Numer. Meth. Eng, vol. 31, p.879–894, (1991).

DOI: 10.1002/nme.1620310505