Comparative Study of Peridynamics and Finite Element Method for Practical Modeling of Cracks in Topology Optimization

Recently, topology optimization of structures with cracks becomes an important topic for avoiding manufacturing defects at the design stage. This paper presents a comprehensive comparative study of peridynamics-based topology optimization method (PD-TO) and classical finite element topology optimization approach (FEM-TO) for designing lightweight structures with/without cracks. Peridynamics (PD) is a robust and accurate non-local theory that can overcome various difficulties of classical continuum mechanics for dealing with crack modeling and its propagation analysis. To implement the PD-TO in this study, bond-based approach is coupled with optimality criteria method. This methodology is applicable to topology optimization of structures with any symmetric/asymmetric distribution of cracks under general boundary conditions. For comparison, optimality criteria approach is also employed in the FEM-TO process, and then topology optimization of four different structures with/without cracks are investigated. After that, strain energy and displacement results are compared between PD-TO and FEM-TO methods. For design domain without cracks, it is observed that PD and FEM algorithms provide very close optimum topologies with a negligibly small percent difference in the results. After this validation step, each case study is solved by integrating the cracks in the design domain as well. According to the simulation results, PD-TO always provides a lower strain energy than FEM-TO for optimum topology of cracked structures. In addition, the PD-TO methodology ensures a better design of stiffer supports in the areas of cracks as compared to FEM-TO. Furthermore, in the final case study, an intended crack with a symmetrically designed size and location is embedded in the design domain to minimize the strain energy of optimum topology through PD-TO analysis. It is demonstrated that hot-spot strain/stress regions of the pristine structure are the most effective areas to locate the designed cracks for effective redistribution of strain/stress during topology optimization.

[1]  Selda Oterkus,et al.  Investigating the effect of brittle crack propagation on the strength of ship structures by using peridynamics , 2020, Ocean Engineering.

[2]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .

[3]  Erkan Oterkus,et al.  A computational model of peridynamic theory for deflecting behavior of crack propagation with micro-cracks , 2019, Computational Materials Science.

[4]  Y. Xie,et al.  A simple evolutionary procedure for structural optimization , 1993 .

[5]  D. S. Dugdale.  Yielding of steel sheets containing slits , 1960 .

[6]  Takayuki Yamada,et al.  A topology optimization method based on the level set method for the design of negative permeability dielectric metamaterials , 2012 .

[7]  Mirco Zaccariotto,et al.  Static solution of crack propagation problems in Peridynamics , 2019, Computer Methods in Applied Mechanics and Engineering.

[8]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[9]  Albert C. To,et al.  Proportional Topology Optimization: A New Non-Sensitivity Method for Solving Stress Constrained and Minimum Compliance Problems and Its Implementation in MATLAB , 2014, PloS one.

[10]  Adnan Kefal,et al.  Damage mechanisms in CFRP/HNT laminates under flexural and in-plane shear loadings using experimental and numerical methods , 2020 .

[11]  Shanshan Zhang,et al.  An improved multi-objective topology optimization model based on SIMP method for continuum structures including self-weight , 2020 .

[12]  M. Y. Wang,et al.  An enhanced genetic algorithm for structural topology optimization , 2006 .

[13]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[14]  Afzal Suleman,et al.  Continuous density-based topology optimization of cracked structures using peridynamics , 2020 .

[15]  A. Kefal,et al.  Dynamic fracture analysis of functionally graded materials using ordinary state-based peridynamics , 2020 .

[16]  H. Feng,et al.  Dynamic Fracture Analysis of Functional Gradient Material Coating Based on the Peridynamic Method , 2019, Coatings.

[17]  Youn Doh Ha,et al.  Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites , 2012 .

[18]  L. Xia,et al.  Topology optimization for maximizing the fracture resistance of quasi-brittle composites , 2018 .

[19]  S. Silling.  Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces , 2000 .

[20]  Stewart Andrew Silling,et al.  Dynamic fracture modeling with a meshfree peridynamic code , 2003 .

[21]  Martin Sotola,et al.  Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam , 2021, Symmetry.

[22]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[23]  Afzal Suleman,et al.  Multi-material topology optimization of structures with discontinuities using Peridynamics , 2021, Composite Structures.

[24]  Kyung K. Choi,et al.  Structural Sensitivity Analysis and Optimization 1: Linear Systems , 2005 .

[25]  Peyman Lahe Motlagh,et al.  A spectral Tchebychev solution for electromechanical analysis of thin curved panels with multiple integrated piezo-patches , 2020 .

[26]  Jeehyun Jung,et al.  Fatigue crack growth analysis in layered heterogeneous material systems using peridynamic approach , 2016 .

[27]  Brian Sanders,et al.  Topology Optimization Approach for the Determination of the Multiple-Configuration Morphing Wing Structure , 2008 .

[28]  Jakub Mesicek,et al.  Design Procedure of a Topologically Optimized Scooter Frame Part , 2020, Symmetry.

[29]  Sujin Bureerat,et al.  Structural topology optimisation using simulated annealing with multiresolution design variables , 2008 .

[30]  Zhan Kang,et al.  Topology optimization using material-field series expansion and Kriging-based algorithm: An effective non-gradient method , 2020 .

[31]  S. Silling,et al.  A meshfree method based on the peridynamic model of solid mechanics , 2005 .

[32]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[33]  Stewart Andrew Silling,et al.  Linearized Theory of Peridynamic States , 2010 .

[34]  E. Madenci,et al.  Peridynamic modeling of delamination growth in composite laminates , 2015 .

[35]  Jihong Zhu,et al.  Topology Optimization in Aircraft and Aerospace Structures Design , 2016 .

[36]  Yangjun Luo,et al.  A material-field series-expansion method for topology optimization of continuum structures , 2019 .

[37]  L. Nilsson,et al.  Topology optimization in crashworthiness design , 2006 .

[38]  A. A. Griffith.  The Phenomena of Rupture and Flow in Solids , 1921 .

[39]  G. K. Ananthasuresh,et al.  Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme , 2001 .

[40]  Yong He,et al.  Altered baseline brain activity in children with ADHD revealed by resting-state functional MRI. , 2007, Brain & development.

[41]  Yin Yu,et al.  Peridynamic analytical method for progressive damage in notched composite laminates , 2014 .

[42]  I..,et al.  The Phenomena of Rupture and Flow in Solids , 2011 .

[43]  K. Saitou,et al.  Multi-material topology optimization using ordered SIMP interpolation , 2016, Structural and Multidisciplinary Optimization.

[44]  K. Tai,et al.  Structural topology design optimization using Genetic Algorithms with a bit-array representation , 2005 .

[45]  Ole Sigmund,et al.  A 99 line topology optimization code written in Matlab , 2001 .

[46]  S. Oterkus,et al.  Three-Dimensional Peridynamic Model for Predicting Fracture Evolution during the Lithiation Process , 2018, Energies.

[47]  Y. Xie,et al.  Computational efficiency and validation of bi-directional evolutionary structural optimisation , 2000 .

[48]  G. I. Barenblatt.  The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks , 1959 .

[49]  Mitsuru Kitamura,et al.  Temperature Distribution Design Based on Variable Lattice Density Optimization and Metal Additive Manufacturing , 2021, Symmetry.

[50]  Erkan Oterkus,et al.  Topology optimization of cracked structures using peridynamics , 2019, Continuum Mechanics and Thermodynamics.

[51]  Ming Li,et al.  Topology optimization considering fracture mechanics behaviors at specified locations , 2017 .

[52]  Ren-Jye Yang,et al.  Automotive applications of topology optimization , 1995 .

[53]  Yiming Liu,et al.  Nonlocal Peridynamic Modeling and Simulation on Crack Propagation in Concrete Structures , 2015 .