An unconstrained stress update algorithm based on hyper-dual step derivative approximation

LU Dechun; SHI Anyu; ZHOU Xin; DU Xiuli

doi:10.11779/CJGE20240138

Volume 47 Issue 6

Turn off MathJax

Article Contents

Abstract

References

Supplements

Chinese Journal of Geotechnical Engineering > 2025 > 47(6): 1113-1122. > DOI: 10.11779/CJGE20240138

LU Dechun, SHI Anyu, ZHOU Xin, DU Xiuli. An unconstrained stress update algorithm based on hyper-dual step derivative approximation[J]. Chinese Journal of Geotechnical Engineering, 2025, 47(6): 1113-1122. DOI: 10.11779/CJGE20240138

Citation:

PDF (8452 KB)

An unconstrained stress update algorithm based on hyper-dual step derivative approximation

1.
Institute of Geotechnical and Underground Engineering, Beijing University of Technology, Beijing 100124, China
2.
Department of Civil Engineering, Tsinghua University, Beijing 100084, China

More Information

Received Date: February 18, 2024
Available Online: July 25, 2024

Graphical Abstract

Abstract

Abstract

The loading/unloading judgment and analytical derivative operations have been the bottlenecks restricting numerical application of elastoplastic models. An unconstrained implicit stress update algorithm is proposed based on the hyper-dual step derivative approximation, which solves the above calculation difficulties. For the problem of loading/unloading judgment, in the new algorithm, the nonlinear stress integral equations with inequality constraints are transformed into an unconstrained minimization problem by using the smooth function to replace the Karush-Kuhn-Tucker conditions. Thus, there is no need for loading/unloading judgement during the calculation. To solve the problem of derivative evaluation, the algorithm uses the hyper-dual step derivative approximation instead of the analytical derivative to obtain the 1st derivative of the smooth function and the 1st and 2nd derivatives of the plastic potential function, which are used to construct iterative formulas for nonlinear calculation, ensuring the quadratic convergence speed of local stress update iterations and global equilibrium iterations. Numerical examples demonstrate that, compared with other numerical differentiation methods, the hyper-dual step derivative approximation is free from truncation errors and subtraction cancellation errors, and its computational results are almost equivalent to the analytical derivations. Finally, based on the proposed algorithm, a UMAT subroutine of smooth Mohr-Coulomb plasticity model is programmed. The effectiveness and convergence speed are verified through numerical analyses of three typical boundary value problems.

FullText(HTML)

References (24)

References

[1]	BORJA R I, LEE S R. Cam-clay plasticity, part 1: implicit integration of elasto-plastic constitutive relations[J]. Computer Methods in Applied Mechanics and Engineering, 1990, 78(1): 49-72. doi: 10.1016/0045-7825(90)90152-C
[2]	BORJA R I. Cam-clay plasticity, part Ⅱ: implicit integration of constitutive equation based on a nonlinear elastic stress predictor[J]. Computer Methods in Applied Mechanics and Engineering, 1991, 88(2): 225-240. doi: 10.1016/0045-7825(91)90256-6
[3]	贾善坡, 陈卫忠, 杨建平, 等. 基于修正Mohr-Coulomb准则的弹塑性本构模型及其数值实施[J]. 岩土力学, 2010, 31(7): 2051-2058. JIA Shanpo, CHEN Weizhong, YANG Jianping, et al. An elastoplastic constitutive model based on modified Mohr-Coulomb criterion and its numerical implementation[J]. Rock and Soil Mechanics, 2010, 31(7): 2051-2058. (in Chinese)
[4]	ZHAO J D, SHENG D C, ROUAINIA M, et al. Explicit stress integration of complex soil models[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2005, 29(12): 1209-1229. doi: 10.1002/nag.456
[5]	HONG H K, LIU C S. Internal symmetry in the constitutive model of perfect elastoplasticity[J]. International Journal of Non-Linear Mechanics, 2000, 35(3): 447-466. doi: 10.1016/S0020-7462(99)00030-X
[6]	REZAIEE-PAJAND M, NASIRAI C. On the integration schemes for Drucker-Prager's elastoplastic models based on exponential maps[J]. International Journal for Numerical Methods in Engineering, 2008, 74(5): 799-826. doi: 10.1002/nme.2178
[7]	SLOAN S W, ABBO A J, SHENG D C. Refined explicit integration of elastoplastic models with automatic error control[J]. Engineering Computations, 2001, 18(1/2): 121-194. doi: 10.1108/02644400110365842
[8]	SIMO J C, TAYLOR R L. A return mapping algorithm for plane stress elastoplasticity[J]. International Journal for Numerical Methods in Engineering, 1986, 22(3): 649-670. doi: 10.1002/nme.1620220310
[9]	MIRA P, TONNI L, PASTOR M, et al. A generalized midpoint algorithm for the integration of a generalized plasticity model for sands[J]. International Journal for Numerical Methods in Engineering, 2009, 77(9): 1201-1223. doi: 10.1002/nme.2445
[10]	SCALET G, AURICCHIO F. Computational methods for elastoplasticity: an overview of conventional and less-conventional approaches[J]. Archives of Computational Methods in Engineering, 2018, 25(3): 545-589. doi: 10.1007/s11831-016-9208-x
[11]	SIMO J C, HUGHES T J R. Computational Inelasticity[M]. New York: Springer Verlag, 1998.
[12]	范庆来, 栾茂田, 杨庆. 修正剑桥模型的隐式积分算法在ABAQUS中的数值实施[J]. 岩土力学, 2008, 29(1): 269-273. FAN Qinglai, LUAN Maotian, YANG Qing. Numerical implementation of implicit integration algorithm for modified Cam-clay model in ABAQUS[J]. Rock and Soil Mechanics, 2008, 29(1): 269-273. (in Chinese)
[13]	ARMERO F, PÉREZ-FOGUET A. On the formulation of closest-point projection algorithms in elastoplasticity: Part I The variational structure[J]. International Journal for Numerical Methods in Engineering, 2002, 53(2): 297-329. doi: 10.1002/nme.278
[14]	STARMAN B, HALILOVIČ M, VRH M, et al. Consistent tangent operator for cutting-plane algorithm of elasto-plasticity[J]. Computer Methods in Applied Mechanics and Engineering, 2014, 272: 214-232. doi: 10.1016/j.cma.2013.12.012
[15]	陈洲泉, 陈湘生, 庞小朝, 等. 角点型非共轴模型的半隐式应力积分算法及应用[J]. 岩土工程学报, 2023, 45(3): 521-529. doi: 10.11779/CJGE20220050 CHEN Zhouquan, CHEN Xiangsheng, PANG Xiaochao, et al. Semi-implicit integration algorithm for non-coaxial model based on vertex theory and its application[J]. Chinese Journal of Geotechnical Engineering, 2023, 45(3): 521-529. (in Chinese) doi: 10.11779/CJGE20220050
[16]	KRABBENHOFT K, LYAMIN A V. Computational cam clay plasticity using second-order cone programming[J]. Computer Methods in Applied Mechanics and Engineering, 2012, 209/210/211/212: 239-249.
[17]	ZHOU X, LU D C, SU C C, et al. An unconstrained stress updating algorithm with the line search method for elastoplastic soil models[J]. Computers and Geotechnics, 2022, 143: 104592. doi: 10.1016/j.compgeo.2021.104592
[18]	PEREZ F A. Numerical differentiation for local and global tangent operators in computational plasticity[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 189(1): 277-296. doi: 10.1016/S0045-7825(99)00296-0
[19]	CHOI H, YOON J W. Stress integration-based on finite difference method and its application for anisotropic plasticity and distortional hardening under associated and non-associated flow rules[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 345: 123-160. doi: 10.1016/j.cma.2018.10.031
[20]	SU C C, LU D C, ZHOU X, et al. An implicit stress update algorithm for the plastic nonlocal damage model of concrete[J]. Computer Methods in Applied Mechanics and Engineering, 2023, 414: 116189. doi: 10.1016/j.cma.2023.116189
[21]	ZHANG N, LI X, WANG D. Smoothed classic yield function for C2 continuities in tensile cutoff, compressive cap, and deviatoric sections[J]. International Journal of Geomechanics, 2021, 21(3): 4021005. doi: 10.1061/(ASCE)GM.1943-5622.0001910
[22]	MENETREY P, WILLAM K J. Triaxial failure criterion for concrete and its generalization[J]. ACI Structural Journal, 1995, 92(3): 311-318.
[23]	FIKE J A. Multi-Objective Optimization Using Hyper-Dual Numbers[D]. Palo Alto: Stanford University, 2013.
[24]	ZHOU X, SHI A Y, LU D C, et al. A return mapping algorithm based on the hyper dual step derivative approximation for elastoplastic models[J]. Computer Methods in Applied Mechanics and Engineering. 2023, 417: 116418. doi: 10.1016/j.cma.2023.116418