Convergence of Finite Element Method in Rheology

HOU Lei; SUN Xian-yan; ZHAO Jun-jie; LI Han-ling

doi:10.3879/j.issn.1000-0887.2014.04.007

Volume 35 Issue 4

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2014 > 35(4): 412-422

HOU Lei, SUN Xian-yan, ZHAO Jun-jie, LI Han-ling. Convergence of Finite Element Method in Rheology[J]. Applied Mathematics and Mechanics, 2014, 35(4): 412-422. doi: 10.3879/j.issn.1000-0887.2014.04.007

Citation:

PDF( 418 KB)

Convergence of Finite Element Method in Rheology

doi: 10.3879/j.issn.1000-0887.2014.04.007

¹Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, P.R.China;²Computational Science E-Institute of Shanghai Universities, Shanghai Jiao Tong University, Shanghai 200030, P.R.China

Funds: The National Natural Science Foundation of China（11271247）

Received Date: 2013-10-18
Rev Recd Date: 2014-03-13
Publish Date: 2014-04-15

Abstract

Abstract

Convergence of the first-order mixed-type hyperbolic parabola partial differential equations in non-Newtonian fluid problems was studied. The coupling partial differential equations (Cauchy fluid equation, P-T/T stress equation) were used to simulate the flow zone generated by the free surface elements or excessively tensile elements. The semi-discrete finite element method was applied to solve these equations coupling with time. The finite element method was used in space. The trilinear functional was employed to solve the nonlinear problems of partial differential equations. In the time domain the Euler scheme was adopted. The convergence order of the equation set reached O(h2+Δt). Numerical results of the equations were obtained through priori and posteriori error estimation of high performance computation. And the deformed sizes of the grids were presented.
- non-Newtonian fluid,
- semi-discrete finite element,
- coupling equations,
- convergence

FullText(HTML)

References(11)

References

[1]	HOU Lei, Nassehi V. Evaluation of stress-effective flow in rubber mixing[J].Nonlinear Analysis,2001,47(3): 1809-1820.
[2]	HOU Lei, Paris R B, Wood A D. The resistive interchange mode in the presence of equilibrium flow[J].Physics of Plasmas,1996,3(2): 473-481.
[3]	林群, 严宁宁. 关于Maxwell方程混合元方法的超收敛[J]. 工程数学学报, 1996,13(supp): 1-10, 66.(LIN Qun, YAN Ning-ning. Superconvergence of mixed methods for the Maxwell’s equations[J].Journal of Engineering Mathematics,1996,13(supp): 1-10, 66.(in Chinese))
[4]	Van Schaftingen J J, Crochet M J. A comparison of mixed methods for solving the flow of a Maxwell fluid[J].International Journal for Numerical Methods in Fluids,1984,4(11): 1065-1081.
[5]	LIN Jia-fu, LIN Qun. Global superconvergence of the mixed finite element methods for 2-D Maxwell equations[J].Journal of Computational Mathematics,2003,21(5): 637-646.
[6]	HOU Lei, ZHAO Jun-jie, LI Han-ling. Finite element convergence analysis of two-scale non-Newtonian flow problems[J].Advanced Materials Research, 2013,718/720: 1723-1728.
[7]	HOU Lei, CAI Li. Nonlinear property of the visco-elastic-plastic material in the impact problem[J].Journal of Shanghai University(English Edition),2009,13(1): 23-28.
[8]	HOU Lei, LIN De-zhi, WANG Bin, LI Han-ling, QIU Lin. Computational modelling on the contact interface with boundary-layer approach[C]// Proceedings of the World Congress on Engineering. VolI. London, UK, 2011: 39-44.
[9]	Sugeng F, Phan-Thien N, Tanner R I. A study of non-isothermal non-Newtonian extrudate swell by a mixed boundary element and finite element method[J].Journal of Rheology,1987,31(1): 37-58.
[10]	林群, 林甲富. 二维Maxwell方程组的混合有限元高精度近似[J]. 数学物理学报, 2003,23A(4): 499-503.(LIN Qun, LIN Jia-fu. High accuracy approximation of mixed finite element for 2-d Maxwell equations[J].Acta Mathematica Scientia,2003,23A(4): 499-503.(in Chinese))
[11]	HOU Lei, Harwood R. Nonlinear properties in Newtonian and non-Newtonian equations[J].Nonlinear Analysis,1997,30(4): 2497-2505.