A polygonal finite element method for shakedown analysis of structures

Author affiliations

Authors

  • Phuc L. H. Ho University Core Research Center for Disaster-free & Safe Ocean City Construction, Dong-A University, Busan, 49315, South Korea https://orcid.org/0000-0002-5794-805X
  • Canh V. Le \(^1\) School of Civil Engineering and Management, International University, Ho Chi Minh City, Vietnam
    \(^2\) Vietnam National University, Ho Chi Minh City, Vietnam
    https://orcid.org/0000-0003-4028-6659
  • Phuong H. Nguyen University Core Research Center for Disaster-free & Safe Ocean City Construction, Dong-A University, Busan, 49315, South Korea https://orcid.org/0000-0001-7497-9467
  • Dung T. Tran Faculty of Civil Engineering, Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam https://orcid.org/0000-0002-5132-0544

DOI:

https://doi.org/10.15625/0866-7136/19027

Keywords:

polygonal finite element method, shakedown analysis, cyclic load, second-order cone programming

Abstract

This study presents an innovative numerical method that combines the polygonal finite element method (Poly-FEM) with conic optimization techniques within the framework of structural shakedown analysis. The resulting optimization problem is formulated as a second-order cone programming (SOCP) problem and is efficiently solved using the MOSEK primal-dual interior-point solver. Numerical experiments validate the computational efficiency and efficacy of the proposed approach.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

References

W. T. Koiter. General theorems for elastic plastic solids. Progress of Solid Mechanics, (1960),pp. 167–221.

E. Melan. Zur Plastizität des räumlichen Kontinuums. Ingenieur-Archiv, 9, (2), (1938), pp. 116–126. DOI: https://doi.org/10.1007/BF02084409

T. Belytschko. Plane stress shakedown analysis by finite elements. International Journal of Mechanical Sciences, 14, (9), (1972), pp. 619–625. DOI: https://doi.org/10.1016/0020-7403(72)90061-6

V. Carvelli, Z. Z. Cen, Y. Liu, and G. Maier. Shakedown analysis of defective pressure vessels by a kinematic approach. Archive of Applied Mechanics, 69, (1999), pp. 751–764. DOI: https://doi.org/10.1007/s004190050254

A.-M. Yan and H. Nguyen-Dang. Kinematical shakedown analysis with temperaturedependent yield stress. International Journal for Numerical Methods in Engineering, 50, (5), (2001), pp. 1145–1168. DOI: https://doi.org/10.1002/1097-0207(20010220)50:5<1145::AID-NME70>3.0.CO;2-C

D. K. Vu, A. M. Yan, and H. Nguyen-Dang. A primal–dual algorithm for shakedown analysis of structures. Computer Methods in Applied Mechanics and Engineering, 193, (42-44), (2004), pp. 4663–4674. DOI: https://doi.org/10.1016/j.cma.2004.03.011

J.-W. Simon. Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening. International Journal of Plasticity, 42, (2013), pp. 141–167. DOI: https://doi.org/10.1016/j.ijplas.2012.10.008

K. D. Andersen, E. Christiansen, A. R. Conn, and M. L. Overton. An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms. SIAM Journal on Scientific Computing, 22, (1), (2000), pp. 243–262. DOI: https://doi.org/10.1137/S1064827598343954

C. D. Bisbos, A. Makrodimopoulos, and P. M. Pardalos. Second-order cone programming approaches to static shakedown analysis in steel plasticity. Optimization Methods and Software, 20, (1), (2005), pp. 25–52. DOI: https://doi.org/10.1080/1055678042000216003

C. V. Le, T. D. Tran, and D. C. Pham. Rotating plasticity and nonshakedown collapse modes for elastic–plastic bodies under cyclic loads. International Journal of Mechanical Sciences, 111, (2016), pp. 55–64. DOI: https://doi.org/10.1016/j.ijmecsci.2016.04.001

P. L. H. Ho, C. V. Le, and T. Q. Chu. The equilibrium cell-based smooth finite element method for shakedown analysis of structures. International Journal of Computational Methods, 16, (05), (2019). DOI: https://doi.org/10.1142/S0219876218400133

P. L. H. Ho and C. V. Le. A stabilized iRBF mesh-free method for quasi-lower bound shakedown analysis of structures. Computers & Structures, 228, (2020). DOI: https://doi.org/10.1016/j.compstruc.2019.106157

P. H. Nguyen, C. V. Le, and P. L. H. Ho. Numerical evaluation of macroscopic fatigue criterion of anisotropic materials using computational homogenization and conic programming. European Journal of Mechanics-A/Solids, 95, (2022). DOI: https://doi.org/10.1016/j.euromechsol.2022.104654

J. Gro-Weege. On the numerical assessment of the safety factor of elastic-plastic structures under variable loading. International Journal of Mechanical Sciences, 39, (4), (1997), pp. 417–433. DOI: https://doi.org/10.1016/S0020-7403(96)00039-2

N. Zouain, L. Borges, and J. L. Silveira. An algorithm for shakedown analysis with nonlinear yield functions. Computer Methods in Applied Mechanics and Engineering, 191, (23-24), (2002), pp. 2463–2481. DOI: https://doi.org/10.1016/S0045-7825(01)00374-7

G. Garcea, G. Armentano, S. Petrolo, and R. Casciaro. Finite element shakedown analysis of two-dimensional structures. International Journal for Numerical Methods in Engineering, 63, (8), (2005), pp. 1174–1202. DOI: https://doi.org/10.1002/nme.1316

K. Krabbenhøft, A. V. Lyamin, and S. W. Sloan. Bounds to shakedown loads for a class of deviatoric plasticity models. Computational Mechanics, 39, (2007), pp. 879–888. DOI: https://doi.org/10.1007/s00466-006-0076-3

H. V. Do and H. Nguyen-Xuan. Computation of limit and shakedown loads for pressure vessel components using isogeometric analysis based on Lagrange extraction. International Journal of Pressure Vessels and Piping, 169, (2019), pp. 57–70. DOI: https://doi.org/10.1016/j.ijpvp.2018.11.012

T. N. Tran, G. R. Liu, H. Nguyen-Xuan, and T. Nguyen-Thoi. An edge-based smoothed finite element method for primal–dual shakedown analysis of structures. International Journal for Numerical Methods in Engineering, 82, (7), (2010), pp. 917–938. DOI: https://doi.org/10.1002/nme.2804

H. Nguyen-Xuan, T. Rabczuk, T. Nguyen-Thoi, T. N. Tran, and N. Nguyen-Thanh. Computation of limit and shakedown loads using a node-based smoothed finite element method. International Journal for Numerical Methods in Engineering, 90, (3), (2012), pp. 287–310. DOI: https://doi.org/10.1002/nme.3317

S. Zhou, Y. Liu, D. Wang, K. Wang, and S. Yu. Upper bound shakedown analysis with the nodal natural element method. Computational Mechanics, 54, (2014), pp. 1111–1128. DOI: https://doi.org/10.1007/s00466-014-1043-z

D. W. Spring, S. E. Leon, and G. H. Paulino. Unstructured polygonal meshes with adaptive refinement for the numerical simulation of dynamic cohesive fracture. International Journal of Fracture, 189, (2014), pp. 33–57. DOI: https://doi.org/10.1007/s10704-014-9961-5

S. E. Leon, D.W. Spring, and G. H. Paulino. Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements. International Journal for Numerical Methods in Engineering, 100, (8), (2014), pp. 555–576. DOI: https://doi.org/10.1002/nme.4744

H. Chi, C. Talischi, O. Lopez-Pamies, and G. H. Paulino. Polygonal finite elements for finite elasticity. International Journal for Numerical Methods in Engineering, 101, (4), (2015), pp. 305–328. DOI: https://doi.org/10.1002/nme.4802

E. L. Wachspress. A rational basis for function approximation. In Conference on Applications of Numerical Analysis: Held in Dundee/Scotland, March 23–26, 1971, Springer, Springer, (2006), pp. 223–252, DOI: https://doi.org/10.1007/BFb0069458

A. Tabarraei and N. Sukumar. Application of polygonal finite elements in linear elasticity. International Journal of Computational Methods, 3, (2006), pp. 503–520. DOI: https://doi.org/10.1142/S021987620600117X

E. T. Ooi, C. Song, F. Tin-Loi, and Z. Yang. Polygon scaled boundary finite elements for crack propagation modelling. International Journal for Numerical Methods in Engineering, 91, (3), (2012), pp. 319–342. DOI: https://doi.org/10.1002/nme.4284

C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes. PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Structural and Multidisciplinary Optimization, 45, (2012), pp. 329–357. DOI: https://doi.org/10.1007/s00158-011-0696-x

K. N. Chau, K. N. Chau, T. Ngo, K. Hackl, and H. Nguyen-Xuan. A polytreebased adaptive polygonal finite element method for multi-material topology optimization. Computer Methods in Applied Mechanics and Engineering, 332, (2018), pp. 712–739. DOI: https://doi.org/10.1016/j.cma.2017.07.035

J. A. Koenig. Shakedown of elastic-plastic structures. North-Holland, (1987).

N. Sukumar and E. A. Malsch. Recent advances in the construction of polygonal finite element interpolants. Archives of Computational Methods in Engineering, 13, (2006), pp. 129–163. DOI: https://doi.org/10.1007/BF02905933

J. Warren. On the uniqueness of barycentric coordinates. Contemporary Mathematics, 334, (2003), pp. 93–100. DOI: https://doi.org/10.1090/conm/334/05977

M. S. Floater, K. Hormann, and G. Kós. A general construction of barycentric coordinates over convex polygons. Advances in Computational Mathematics, 24, (2006), pp. 311–331. DOI: https://doi.org/10.1007/s10444-004-7611-6

MOSEK ApS. The MOSEK optimization toolbox for MATLAB manual. Version 10.0, (2022).

F. A. Gaydon and A. W. McCrum. A theoretical investigation of the yield point loading of a square plate with a central circular hole. Journal of the Mechanics and Physics of Solids, 2, (3), (1954), pp. 156–169. DOI: https://doi.org/10.1016/0022-5096(54)90022-8

M. V. da Silva and A. N. Antao. A non-linear programming method approach for upper bound limit analysis. International Journal for Numerical Methods in Engineering, 72, (10), (2007), pp. 1192–1218. DOI: https://doi.org/10.1002/nme.2061

S. Chen, Y. Liu, and Z. Cen. Lower-bound limit analysis by using the EFG method and nonlinear programming. International Journal for Numerical Methods in Engineering, 74, (3), (2008), pp. 391–415. DOI: https://doi.org/10.1002/nme.2177

F. Genna. A nonlinear inequality, finite element approach to the direct computation of shakedown load safety factors. International Journal of Mechanical Sciences, 30, (10), (1988), pp. 769–789. DOI: https://doi.org/10.1016/0020-7403(88)90041-0

Downloads

Published

30-09-2023

How to Cite

[1]
P. L. H. Ho, C. V. Le, P. H. Nguyen and D. T. Tran, A polygonal finite element method for shakedown analysis of structures, Vietnam J. Mech. 45 (2023) 259–272. DOI: https://doi.org/10.15625/0866-7136/19027.

Issue

Section

Research Article

Categories