Error estimation based on stress recovery by adaptivity in nonlinear problems of elasto-plasticity behavior by isogeometric method

Authors

1 Ph.D. Student, Department of Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran.

2 Assistant Professor, Department of Civil Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran.

Abstract

In this research, the efficiency of error estimation based on two methods of recovering the equilibrium stress of patches and superconvergent points in guiding the adaptive solution of nonlinear problems by isogeometric method has been investigated. Also, by analyzing elasto-plastic problems based on material properties, and adaptive solution by temperature gradient method, the stress improvement process has been investigated. The method of the adaptive solution of this research is based on the movement of control points and it is used in stress recovery, taking into account the difference between the exact stress level and the stress level obtained from the isogeometric analysis for each element, as a criterion to determine the amount of error in it. The element is obtained. For this purpose, the modeling of two problems in the non-linear range, which has an exact solution, has been considered. The results have shown that the total norm difference of exact and approximate error in both stress recovery methods used is more than 33% and in the direction of improving the network of control points. Also, the method by equilibrium patches is more effective than the method based on superconvergent points, which can be used as a suitable solution to improve the stress field.

Keywords

Main Subjects

References

[1] Hughes, T. J., Cottrell, J. A., & Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer methods in applied mechanics and engineering, 194(39-41), 4135-4195.
 [2] Turner, M. J., Clough, R. W., Martin, H. C., & Topp, L. (1956). Stiffness and deflection analysis of complex structures. J. Aero. Sci., 23(9), 805-823.
[3] Argyris, J. (1964). Recent advances in matrix methods of structural analysis (Matrix theory of structures for small and large deflections, using high speed digital computers). NEW YORK, MACMILLAN CO., OXFORD, PERGAMON PRESS, LTD., 1964. 187 P.
[4] Argyris, J. (1965). Continua and Discontinua, opening address to the 1-st Conf. Matrix Methods in Structural Mechanics. Wright-Patterson AFB, Dayton, Ohio.
[5] Oden, J. T. (1967). Numerical formulation of nonlinear elasticity problems. J. Struc. Div., 93(3), 235-255.
[6] Mallett, R. H., & Marcal, P. V. (1968). Finite element analysis of nonlinear structures. J. Struct. Division, 94(9), 2081-2106.
[7] Rezaiee-Pajand, M., Masoodi, A. R., & Arabi, E. (2022). Improved shell element for geometrically non-linear analysis of thin-walled structures. Proceedings of the Institution of Civil Engineers-Structures and Buildings, 175(4), 347-356.‏
[8] Pajand, M., Masoodi, A. R., & Arabi, E. (2018). On the shell thickness-stretching effects using seven-parameter triangular element. European J. Comput. Mech., 27(2),163-185.‏
[9] Zienkiewicz, O. C., & Morice, P. (1971). The finite element method in engineering science (Vol. 1977): McGraw-hill London.
[10] C. Brebbia, J. Connor, Geometrically nonlinear finite-element analysis, J. Eng. Mech. Divis., 95(2) (1969) 463-483.
[11] VeisiAra, A., Mohammad-Sedighi, H., & Reza, A. (2021). Computational analysis of the nonlinear vibrational behavior of perforated plates with initial imperfection using NURBS-based isogeometric approach. J. Comput. Design Eng., 8(5), 1307-1331.
[12] Huynh, G., Zhuang, X., Bui, H., Meschke, G., & Nguyen-Xuan, H. (2020). Elasto-plastic large deformation analysis of multi-patch thin shells by isogeometric approach. Finite Elements in Analysis and Design, 173, 103389.
[13] Payen, D. J., & Bathe, K.-J. (2011). Improved stresses for the 4-node tetrahedral element. Computers & Structures, 89(13-14), 1265-1273.
[14] Payen, D. J., & Bathe, K.-J. (2011). The use of nodal point forces to improve element stresses. Computers & Structures, 89(5-6), 485-495.
[15] Wu, Z., Wang, S., Xiao, R., & Yu, L. (2020). A local solution approach for level-set based structural topology optimization in isogeometric analysis. J. Comput. Design Eng., 7(4), 514-526.
[16] CHEN, Y., DENG, Z., & HUANG, Y. (2022). RECOVERY-BASED A POSTERIORI ERROR ESTIMATION FOR ELLIPTIC INTERFACE PROBLEMS BASED ON PARTIALLY PENALIZED IMMERSED FINITE ELEMENT METHODS. Int. J. Numeric. Anal. Model. 19(1).
[17] Cottrell, J., Hughes, T., & Reali, A. (2007). Studies of refinement and continuity in isogeometric structural analysis. Computer methods in applied mechanics and engineering, 196(41-44), 4160-4183.
[18] Xu, G., Li, B., Shu, L., Chen, L., Xu, J., & Khajah, T. (2019). Efficient r-adaptive isogeometric analysis with winslow’s mapping and monitor function approach. J. Comput. Appl. Math., 351, 186-197.
[19] Ji, Y., Wang, M., Yu, Y., & Zhu, C. (2022). Curvature-Based r-Adaptive Isogeometric Analysis with Injectivity-Preserving Multi-Sided Domain Parameterization. J. Sys. Sci. Compl., 1-24.
[20] Ji, Y., Wang, M.-Y., Wang, Y., & Zhu, C.-G. (2022). Curvature-Based R-Adaptive Planar NURBS Parameterization Method for Isogeometric Analysis Using Bi-Level Approach. Computer-Aided Design, 103305.
[21] Mirzakhani, A., Hassani, B., & Ganjali, A. (2015). Adaptive analysis of three-dimensional structures using an isogeometric control net refinement approach. Acta Mechanica, 226(10), 3425-3449.
[22] Zienkiewicz, O. C., & Zhu, J. Z. (1992). The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. International J. Numeric. Methods Eng., 33(7), 1331-1364.
[23] Hassani, B., Ganjali, A., & Tavakkoli, M. (2012). An isogeometrical approach to error estimation and stress recovery. Europ. J. Mech.-A/Solids, 31(1), 101-109.
[24] Boroomand, B., & Zienkiewicz, O. C. (1997). Recovery by equilibrium in patches (REP). Int. J. Numeric. Methods Eng., 40(1), 137-164.
[25] Ganjali, A., & Hassani, B. (2020). Error Estimation and Stress Recovery by Patch Equilibrium in the‎‎ Isogeometric Analysis Method. Advances in Applied Mathematics and Mechanics, 12.
[26] Boroomand, B., & Zienkiewicz, O. (1999). Recovery procedures in error estimation and adaptivity. Part II: Adaptivity in nonlinear problems of elasto-plasticity behaviour. Computer methods in applied mechanics and engineering, 176(1-4), 127-146.
[27] Piegl, L., & Tiller, W. (1996). The NURBS book: Springer Science & Business Media.
[28] Owen, D. R. J., & Hinton, H. (1980). Finite elements in plasticity, theory and practice.
[29] Sheng, D., Sloan, S. W., & Abbo, A. J. (2002). An automatic Newton–Raphson scheme. The Int. J. Geomech., 2(4), 471-502.
[30] Zienkiewicz, O. C., Taylor, R. L., & Zhu, J. Z. (2005). The finite element method: its basis and fundamentals: Elsevier.
[31] Ainsworth, M., & Oden, J. T. (1993). A unified approach to a posteriori error estimation using element residual methods. Numerische Mathematik, 65(1), 23-50.
[32] Chakrabarty, J. (2010). Applied plasticity (Vol. 758): Springer.
[33] de Souza Neto, E. A., Peric, D., & Owen, D. R. (2011). Computational methods for plasticity: theory and applications: John Wiley & Sons.
Journal of Solid and Fluid Mechanics
Volume 13, Issue 3
May and June 2023
Pages 67-84

Files

Share

How to cite

Statistics