Fracture resistance via topology optimization

The fracture resistance of structures is optimized using the level-set method. Fracture resistance is assumed to be related to the elastic energy released by a crack propagating in a normal direction from parts of the boundary that are in tension, and is calculated using the virtual crack extension technique. The shape derivative of the fracture-resistance objective function is derived. Two illustrative two-dimensional case studies are presented: a hole in a plate subjected to biaxial strain; and a bridge fixed at both ends subjected to a single load in which the compliance and fracture resistance are jointly optimized. The structures obtained have rounded corners and more material at places where they are in tension. Based on the results, we propose that fracture resistance may be modeled more easily but less directly by including a term proportional to surface area in the objective function, in conjunction with nonlinear elasticity where the Young’s modulus in tension is lower than in compression.

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

[2]  J. Rice.  A path-independent integral and the approximate analysis of strain , 1968 .

[3]  D. M. Parks.  A stiffness derivative finite element technique for determination of crack tip stress intensity factors , 1974 .

[4]  T. K. Hellen.  On the method of virtual crack extensions , 1975 .

[5]  Jacques Simon,et al.  Etude de Problème d'Optimal Design , 1975, Optimization Techniques.

[6]  M. Deighton.  Fracture of Brittle Solids , 1976 .

[7]  J. Simon.  Differentiation with Respect to the Domain in Boundary Value Problems , 1980 .

[8]  J. Cea.  Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût , 1986 .

[9]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

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

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

[12]  N. Kikuchi,et al.  A homogenization method for shape and topology optimization , 1991 .

[13]  J. Zolésio,et al.  Introduction to shape optimization : shape sensitivity analysis , 1992 .

[14]  M. J. Adams.  Fracture of brittle solids (2nd edition): Brian Lawn , 1994 .

[15]  Edward J. Garboczi,et al.  An algorithm for computing the effective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase poisson ratios , 1995 .

[16]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[17]  E. Garboczi,et al.  Finite Element and Finite Difference Programs for Computing the Linear Electric and Elastic Properties of Digital Images of Random Materials | NIST , 1998 .

[18]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[19]  J. Sethian,et al.  Structural Boundary Design via Level Set and Immersed Interface Methods , 2000 .

[20]  S. Osher,et al.  Level Set Methods for Optimization Problems Involving Geometry and Constraints I. Frequencies of a T , 2001 .

[21]  G. Allaire,et al.  A level-set method for shape optimization , 2002 .

[22]  A. ADoefaa,et al.  ? ? ? ? f ? ? ? ? ? , 2003 .

[23]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[24]  Bessem Samet.  The topological asymptotic with respect to a singular boundary perturbation , 2003 .

[25]  M. Chial,et al.  in simple , 2003 .

[26]  J. Sethian,et al.  FRONTS PROPAGATING WITH CURVATURE DEPENDENT SPEED: ALGORITHMS BASED ON HAMILTON-JACOB1 FORMULATIONS , 2003 .

[27]  Xiaoming Wang,et al.  Color level sets: a multi-phase method for structural topology optimization with multiple materials , 2004 .

[28]  Xiaoming Wang,et al.  Structural shape and topology optimization in a level-set-based framework of region representation , 2004 .

[29]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[30]  M. Burger,et al.  Incorporating topological derivatives into level set methods , 2004 .

[31]  Mei Yulin,et al.  A level set method for structural topology optimization and its applications , 2004 .

[32]  M. Hintermueller.  Fast level set based algorithms using shape and topological sensitivity information , 2005 .

[33]  Erik Lund,et al.  Topology Optimization - Broadening the Areas of Application , 2005 .

[34]  Michael Yu Wang,et al.  Design of multimaterial compliant mechanisms using level-set methods , 2005 .

[35]  G. Allaire,et al.  A level-set method for vibration and multiple loads structural optimization , 2005 .

[36]  Michael Yu Wang,et al.  A level-set based variational method for design and optimization of heterogeneous objects , 2005, Comput. Aided Des..

[37]  G. Allaire,et al.  Structural optimization using topological and shape sensitivity via a level set method , 2005 .

[38]  Xu Guo,et al.  A new approach for simultaneous shape and topology optimization based on dynamic implicit surface function , 2005 .

[39]  Frédéric de Gournay,et al.  Velocity Extension for the Level-set Method and Multiple Eigenvalues in Shape Optimization , 2006, SIAM J. Control. Optim..

[40]  M. Wang,et al.  Radial basis functions and level set method for structural topology optimization , 2006 .

[41]  A. Suleman,et al.  Application of spectral level set methodology in topology optimization , 2006 .

[42]  Daniel N. Wilke,et al.  A quadratically convergent unstructured remeshing strategy for shape optimization , 2006 .

[43]  M. Wang,et al.  A moving superimposed finite element method for structural topology optimization , 2006 .

[44]  M. Wang,et al.  Structural Shape and Topology Optimization Using an Implicit Free Boundary Parametrization Method , 2006 .

[45]  Heiko Andrä,et al.  A new algorithm for topology optimization using a level-set method , 2006, J. Comput. Phys..