The Effects of Newmark Method Parameters on Errors in Dynamic Extended Finite Element Method Using Response Surface Method

Authors

Mechanical Engineering Department, Urmia University, Urmia, Iran

Abstract

The Newmark method is an effective method for numerical time integration in dynamic problems. The results of Newmark method are function of its parameters (β, γ and ∆t). In this paper, a stationary mode I dynamic crack problem is coded in extended finite element method )XFEM( framework in Matlab software and results are verified with analytical solution. This paper focuses on effects of main parameters in Newmark method for dynamic XFEM problems. Also use of the response surface method (RSM) a regression model is presented for estimating error of dynamic stress intensity factors (DSIF) with high validity according to results of analysis of variance (ANOVA). This work enables one to understand the effect of Newmark parameters on error of DSIFs and to find optimum β and γ for a determined number of time steps (N). This procedure is highly effective in order to  manage the computational cost and enhance the accuracy at the desired domain. The effect of the considered parameters on error, is investigated using RSM in Minitab software and optimum state for minimization of errors is illustrated.

Keywords


1.     Heidari, A. and Salajegheh, E., "Approximate dynamic analysis of structures for earthquake loading using fwt", International Journal of Engineering Transactions B Applications,  Vol. 20, No. 1, (2007), 37-47.
2.     Kaynia, A. and Dargush, G., "Fundamental solutions of dynamic poroelasticity and generalized termoelasticity", International Journal of Engineering,  Vol. 5, No. 1&21, 1-10.
3.     Belytschko, T. and Black, T., "Elastic crack growth in finite elements with minimal remeshing", International Journal for Numerical Methods in Engineering,  Vol. 45, No. 5, (1999), 601-620.
4.     Armero, F. and Linder, C., "Numerical simulation of dynamic fracture using finite elements with embedded discontinuities", International Journal of Fracture,  Vol. 160, No. 2, (2009), 119-141.
5.     Dolbow, J. and Belytschko, T., "A finite element method for crack growth without remeshing", International Journal for Numerical Methods in Engineering,  Vol. 46, No. 1, (1999), 131-150.
6.     Stolarska, M., Chopp, D., Moës, N. and Belytschko, T., "Modelling crack growth by level sets in the extended finite element method", International Journal for Numerical Methods in Engineering,  Vol. 51, No. 8, (2001), 943-960.
7.     Belytschko, T., Chen, H., Xu, J. and Zi, G., "Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment", International Journal for Numerical Methods in Engineering,  Vol. 58, No. 12, (2003), 1873-1905.
8.     Réthoré, J., Gravouil, A. and Combescure, A., "An energy‐conserving scheme for dynamic crack growth using the extended finite element method", International Journal for Numerical Methods in Engineering,  Vol. 63, No. 5, (2005), 631-659.
9.     Mohammadi, S., "Xfem fracture analysis of composites, Wiley Online Library,  (2012).
10.   Menouillard, T., Rethore, J., Combescure, A. and Bung, H., "Efficient explicit time stepping for the extended finite element method (x‐fem)", International Journal for Numerical Methods in Engineering,  Vol. 68, No. 9, (2006), 911-939.
11.   Zhou, J. and Zhou, Y., "A new simple method of implicit time integration for dynamic problems of engineering structures", Acta Mechanica Sinica,  Vol. 23, No. 1, (2007), 91-99.
12.   Alamatian, J., "A modified multi time step integration for dynamic analysis", International Journal of Engineering-Transactions B: Applications,  Vol. 25, No. 4, (2012), 303-314.
13.   Alamatian, J., "New implicit higher order time integration for dynamic analysis", Structural Engineering and Mechanics,  Vol. 48, No. 5, (2013), 711-736.
14.   Shojaee, S., Rostami, S. and Abbasi, A., "An unconditionally stable implicit time integration algorithm: Modified quartic b-spline method", Computers & Structures,  Vol. 153, (2015), 98-111.
15.   Newmark, N.M., "A method of computation for structural dynamics", Journal of the Engineering Mechanics Division,  Vol. 85, No. 3, (1959), 67-94.
16.   Zienkiewicz, O.C., "A new look at the newmark, houbolt and other time stepping formulas. A weighted residual approach", Earthquake Engineering & Structural Dynamics,  Vol. 5, No. 4, (1977), 413-418.
17.   Liu, P., Bui, T., Zhang, C., Yu, T., Liu, G. and Golub, M., "The singular edge-based smoothed finite element method for stationary dynamic crack problems in 2d elastic solids", Computer Methods in Applied Mechanics and Engineering,  Vol. 233, No., (2012), 68-80.
18.   Okafor, E.C., Ihueze, C.C. and Nwigbo, S., "Optimization of hardness strengths response of plantain fibers reinforced polyester matrix composites (PFRP) applying taguchi robust design", International Journal of Engineering,  Vol. 26, No. 1, (2013), 1-11.
19.   Alimirzaloo, V. and Modanloo, V., "Minimization of the sheet thinning in hydraulic deep drawing process using response surface methodology and finite element method", International Journal of Engineering (IJE), Transactions B: Applications,  Vol. 29, No. 2, (2016), 264-273.
20.   Nejad, S.J.H., Hasanzadeh, R., Doniavi, A. and Modanloo, V., "Finite element simulation analysis of laminated sheets in deep drawing process using response surface method", The International Journal of Advanced Manufacturing Technology,  Vol. 93, No. 9-12, (2017), 3245-3259.
21.   Freund, L.B., "Dynamic fracture mechanics, Cambridge university press,  (1998).
22.   Azdast, T., Hasanzadeh, R. and Moradian, M., "Optimization of process parameters in FSW of polymeric nanocomposites to improve impact strength using step wise tool selection", Materials and Manufacturing Processes, (2017), doi: 10.1080/10426914.2017.1339324.
23.   Eungkee Lee, R., Afsari Ghazi, A., Azdast, T., Hasanzadeh, R. and Mamaghani Shishavan, S., "Tensile and hardness properties of polycarbonate nanocomposites in the presence of styrene maleic anhydride as compatibilizer", Advances in Polymer Technology, doi: 10.1002/adv.21832. http://onlinelibrary.wiley.com/doi/10.1002/adv.21832/full