Structural optimization under internal porosity constraints using topological derivatives

Porosity is a well-known phenomenon occurring during various manufacturing processes (casting, welding, additive manufacturing) of solid structures, which undermines their reliability and mechanical performance. The main purpose of this article is to introduce a new constraint functional of the domain which controls the negative impact of porosity on elastic structures in the framework of shape and topology optimization. The main ingredient of our modelling is the notion of topological derivative, which is used in a slightly unusual way: instead of being an indicator of where to nucleate holes in the course of the optimization process, it is a component of a new constraint functional which assesses the influence of pores on the mechanical performance of structures. The shape derivative of this constraint is calculated and incorporated into a level set based shape optimization algorithm. Our approach is illustrated by several two-and three-dimensional numerical experiments of topology optimization problems constrained by a control on the porosity effect.

[1]  G. Allaire,et al.  Minimum stress optimal design with the level set method. , 2008 .

[2]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[3]  X. L. Deng,et al.  Optimization of structures under technological casting constraints , 1995 .

[4]  P. Peyre,et al.  Reduction of porosity content generated during Nd:YAG laser welding of A356 and AA5083 aluminium alloys , 2003 .

[5]  G. Allaire,et al.  Thickness control in structural optimization via a level set method , 2016, Structural and Multidisciplinary Optimization.

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

[7]  Grégoire Allaire,et al.  A linearized approach to worst-case design in parametric and geometric shape optimization , 2014 .

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

[9]  Antonio André Novotny,et al.  Topological Derivatives in Shape Optimization , 2012 .

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

[11]  Grégoire Allaire.  Conception optimale de structures , 2007 .

[12]  I. Hlaváček,et al.  On Topological Derivatives for Elastic Solids with Uncertain Input Data , 2009 .

[13]  Mattias Schevenels,et al.  Topology optimization with geometric uncertainties by perturbation techniques , 2012 .

[14]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

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

[16]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[17]  Kurt Maute,et al.  Topology Optimization under Uncertainty , 2014 .

[18]  Jess Martnez-Frutos,et al.  GPU acceleration for evolutionary topology optimization of continuum structures using isosurfaces , 2017 .

[19]  Jesús Martínez-Frutos,et al.  Risk-averse structural topology optimization under random fields using stochastic expansion methods , 2018 .

[20]  Jan Sokoƒowski,et al.  TOPOLOGICAL DERIVATIVES OF SHAPE FUNCTIONALS FOR ELASTICITY SYSTEMS* , 2001 .

[21]  Antoine Laurain,et al.  A level set-based structural optimization code using FEniCS , 2018 .

[22]  Vivien J. Challis,et al.  High resolution topology optimization using graphics processing units (GPUs) , 2013, Structural and Multidisciplinary Optimization.

[23]  Grégoire Allaire,et al.  A deterministic approximation method in shape optimization under random uncertainties , 2015 .

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

[25]  Hai-Lung Tsai,et al.  Porosity Formation and Prevention in Pulsed Laser Welding , 2007 .

[26]  Jesús Martínez-Frutos,et al.  Efficient matrix-free GPU implementation of Fixed Grid Finite Element Analysis , 2015 .

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

[28]  P. Marti,et al.  An implementation of level set based topology optimization using GPU , 2013 .

[29]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[30]  Parviz Davami,et al.  Optimal riser design in sand casting process by topology optimization with SIMP method I: Poisson approximation of nonlinear heat transfer equation , 2008 .

[31]  G. Allaire,et al.  MULTI-PHASE STRUCTURAL OPTIMIZATION VIA A LEVEL SET METHOD ∗, ∗∗ , 2014 .

[32]  D. Chopp.  Computing Minimal Surfaces via Level Set Curvature Flow , 1993 .

[33]  P J Withers,et al.  The Influence of Porosity on Fatigue Crack Initiation in Additively Manufactured Titanium Components , 2017, Scientific Reports.

[34]  Helmut Harbrecht,et al.  Computing quantities of interest for random domains with second order shape sensitivity analysis , 2015 .

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

[36]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[37]  Philippe Guillaume,et al.  The Topological Asymptotic for PDE Systems: The Elasticity Case , 2000, SIAM J. Control. Optim..

[38]  L. M. Delves,et al.  Iterative solution of the global element equations , 1982 .

[39]  Antoine Henrot,et al.  Variation et optimisation de formes : une analyse géométrique , 2005 .

[40]  Jesús Martínez-Frutos,et al.  Robust shape optimization of continuous structures via the level set method , 2016 .

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

[42]  R. Monroe,et al.  Porosity in Castings , 2005 .

[43]  Charlie C. L. Wang,et al.  Current and future trends in topology optimization for additive manufacturing , 2018 .

[44]  William Pao,et al.  A medical-axes-based interpolation method for solidification simulation , 2004 .

[45]  F. Murat,et al.  Sur le controle par un domaine géométrique , 1976 .

[46]  John O. Milewski,et al.  Additive Manufacturing of Metals: From Fundamental Technology to Rocket Nozzles, Medical Implants, and Custom Jewelry , 2017 .

[47]  Ole Sigmund,et al.  Manufacturing tolerant topology optimization , 2009 .

[48]  Reinhart Poprawe,et al.  Formation and reduction of hydrogen porosity during selective laser melting of AlSi10Mg , 2015 .

[49]  Alaa Elwany,et al.  Prediction of porosity in metal-based additive manufacturing using spatial Gaussian process models , 2016 .

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