Isogeometric Analysis for Topology Optimization with a Phase Field Model

We consider a phase field model for the formulation and solution of topology optimization problems in the minimum compliance case. In this model, the optimal topology is obtained as the steady state of the phase transition described by the generalized Cahn–Hilliard equation which naturally embeds the volume constraint on the amount of material available for distribution in the design domain. We reformulate the model as a coupled system and we highlight the dependency of the optimal topologies on dimensionless parameters. We consider Isogeometric Analysis for the spatial approximation which facilitates encapsulating the exactness of the representation of the design domain in the topology optimization and is particularly suitable for the analysis of phase field problems. We demonstrate the validity of the approach and numerical approximation by solving two and three-dimensional topology optimization problems.

[1]  Kurt Maute,et al.  Adaptive topology optimization of shell structures , 1996 .

[2]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[3]  James K. Guest,et al.  Topology optimization with multiple phase projection , 2009 .

[4]  R. Haftka,et al.  Structural shape optimization - A survey , 1985 .

[5]  Thomas J. R. Hughes,et al.  n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method , 2009 .

[6]  Charles M. Elliott,et al.  The Cahn-Hilliard Model for the Kinetics of Phase Separation , 1989 .

[7]  Jonathan J. Hu,et al.  ML 3.1 smoothed aggregation user's guide. , 2004 .

[8]  R. Spatschek,et al.  Phase field modeling of fracture and stress-induced phase transitions. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  John A. Evans,et al.  Isogeometric finite element data structures based on Bézier extraction of NURBS , 2011 .

[10]  Christian Miehe,et al.  Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations , 2010 .

[11]  O. Pironneau,et al.  Applied Shape Optimization for Fluids , 2001 .

[12]  Akira Onuki,et al.  Ginzburg-Landau Approach to Elastic Effects in the Phase Separation of Solids , 1989 .

[13]  John W. Cahn,et al.  Thermochemical equilibrium of multiphase solids under stress , 1978 .

[14]  C. S. Jog,et al.  A new approach to variable-topology shape design using a constraint on perimeter , 1996 .

[15]  Ekkehard Ramm,et al.  ADAPTIVE TOPOLOGY AND SHAPE OPTIMIZATION , 1998 .

[16]  Martin Philip Bendsee.  On Obtaining a Solution to Optimization Problems for Solid, Elastic Plates by Restriction of the Design Space* , 1983 .

[17]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[18]  Harald Garcke,et al.  Numerical approximation of the Cahn-Larché equation , 2005, Numerische Mathematik.

[19]  Sung-Kie Youn,et al.  Isogeometric topology optimization using trimmed spline surfaces , 2010 .

[20]  Jonathan J. Hu,et al.  ML 5.0 Smoothed Aggregation Users's Guide , 2006 .

[21]  G. Allaire,et al.  Topology optimization for minimum stress design with the homogenization method , 2004 .

[22]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[23]  J. Petersson.  Some convergence results in perimeter-controlled topology optimization , 1999 .

[24]  Takayuki Yamada,et al.  Topology optimization using a reaction-diffusion equation , 2011 .

[25]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[26]  E. Ramm,et al.  Topology and shape optimization for elastoplastic structural response , 2001 .

[27]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[28]  C. M. Elliott,et al.  On the Cahn-Hilliard equation with degenerate mobility , 1996 .

[29]  Kyung K. Choi,et al.  Numerical method for shape optimization using T-spline based isogeometric method , 2010 .

[30]  W. Craig Carter,et al.  Variational methods for microstructural-evolution theories , 1997 .

[31]  Thomas J. R. Hughes,et al.  Encyclopedia of computational mechanics , 2004 .

[32]  Ole Sigmund,et al.  On the Design of Compliant Mechanisms Using Topology Optimization , 1997 .

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

[34]  Takayuki Yamada,et al.  A topology optimization method based on the level set method incorporating a fictitious interface energy , 2010 .

[35]  Niels Leergaard Pedersen,et al.  Topology optimization of laminated plates with prestress , 2002 .

[36]  T. Hughes,et al.  Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations , 2010 .

[37]  Shiwei Zhou,et al.  Phase Field: A Variational Method for Structural Topology Optimization , 2004 .

[38]  Paul C. Fife,et al.  Models for phase separation and their mathematics. , 2000 .

[39]  John E. Hilliard,et al.  Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 1959 .

[40]  Y. Xie,et al.  3D and multiple load case bi-directional evolutionary structural optimization (BESO) , 1999 .

[41]  Arun L. Gain,et al.  Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation , 2012 .

[42]  M. Bendsøe,et al.  Topology optimization of continuum structures with local stress constraints , 1998 .

[43]  W. H. Zhang,et al.  Combined shape and sizing optimization of truss structures , 2002 .

[44]  E. Ramm,et al.  Models and finite elements for thin-walled structures , 2004 .

[45]  Michael Yu Wang,et al.  A Variational Method for Structural Topology Optimization , 2004 .

[46]  G. Hulbert,et al.  A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method , 2000 .

[47]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[48]  K. Svanberg,et al.  On the trajectories of penalization methods for topology optimization , 2001 .

[49]  Fermín Navarrina,et al.  Topology optimization of structures with local and global stress constraints , 2007 .

[50]  Shiwei Zhou,et al.  Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition , 2006 .

[51]  Robert Lipton,et al.  A saddle-point theorem with application to structural optimization , 1994 .

[52]  Klaus Schittkowski,et al.  Software systems for structural optimization , 1993 .

[53]  C. S. Jog,et al.  Stability of finite element models for distributed-parameter optimization and topology design , 1996 .

[54]  B. Bourdin.  Filters in topology optimization , 2001 .

[55]  José Francisco Rodrigues,et al.  Mathematical Models for Phase Change Problems , 1989 .

[56]  T. Hughes,et al.  Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows , 2007 .

[57]  George I. N. Rozvany,et al.  On singular topologies in exact layout optimization , 1994 .

[58]  J. Petersson,et al.  Slope constrained topology optimization , 1998 .

[59]  F. Navarrina,et al.  Topology optimization of continuum structures with local and global stress constraints , 2009 .

[60]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[61]  Ahmad H. Nasri,et al.  T-splines and T-NURCCs , 2003, ACM Trans. Graph..

[62]  G. Rozvany.  Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics , 2001 .

[63]  J. Tinsley Oden,et al.  Goal‐oriented error estimation for Cahn–Hilliard models of binary phase transition , 2011 .

[64]  Kurt Maute,et al.  TOWARDS GENERALIZED SHAPE AND TOPOLOGY OPTIMIZATION , 1997 .

[65]  W. Wall,et al.  Isogeometric structural shape optimization , 2008 .

[66]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

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

[68]  Ekkehard Ramm,et al.  Shape optimization with program CARAT , 1993 .

[69]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[70]  O. Sigmund.  Morphology-based black and white filters for topology optimization , 2007 .

[71]  M. P. Bendsøe,et al.  The Michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: An asymptotic study , 1993 .

[72]  S. Youn,et al.  Topology optimization of shell structures using adaptive inner-front (AIF) level set method , 2008 .

[73]  Wei Wu,et al.  Topology optimization of a novel stent platform with drug reservoirs. , 2008, Medical engineering & physics.

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

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

[76]  Ole Sigmund,et al.  Design of multiphysics actuators using topology optimization - Part I: One-material structures , 2001 .

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

[78]  G. Allaire,et al.  Shape and topology optimization of the robust compliance via the level set method , 2008 .

[79]  高等学校計算数学学報編輯委員会編.  高等学校計算数学学報 = Numerical mathematics , 1979 .

[80]  Sung-Kie Youn,et al.  Shape optimization and its extension to topological design based on isogeometric analysis , 2010 .

[81]  Thomas J. R. Hughes,et al.  Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models , 2011, J. Comput. Phys..

[82]  Mathias Stolpe,et al.  Modelling topology optimization problems as linear mixed 0–1 programs , 2003 .

[83]  Seung-Hyun Ha,et al.  Isogeometric shape design optimization: exact geometry and enhanced sensitivity , 2009 .

[84]  Shinji Nishiwaki,et al.  Shape and topology optimization based on the phase field method and sensitivity analysis , 2010, J. Comput. Phys..

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

[86]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[87]  Martin Burger,et al.  Phase-Field Relaxation of Topology Optimization with Local Stress Constraints , 2006, SIAM J. Control. Optim..

[88]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

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

[90]  T. Hughes,et al.  Efficient quadrature for NURBS-based isogeometric analysis , 2010 .

[91]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[92]  Antonin Chambolle,et al.  The Phase-Field Method in Optimal Design , 2006 .

[93]  Piotr Rybka,et al.  Convergence of solutions to cahn-hilliard equation , 1999 .

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

[95]  M. Bendsøe,et al.  A topological derivative method for topology optimization , 2007 .

[96]  G. Buttazzo,et al.  An optimal design problem with perimeter penalization , 1993 .

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

[98]  Krister Svanberg,et al.  A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations , 2002, SIAM J. Optim..

[99]  Thomas A. Poulsen.  A new scheme for imposing a minimum length scale in topology optimization , 2003 .

[100]  Ole Sigmund,et al.  Topology optimization of large scale stokes flow problems , 2008 .

[101]  R. Kohn,et al.  Optimal design and relaxation of variational problems, III , 1986 .

[102]  E. Ramm,et al.  Form finding of shells by structural optimization , 1993, Engineering with Computers.

[103]  Michael Yu Wang,et al.  3D Multi-Material Structural Topology Optimization with the Generalized Cahn-Hilliard Equations , 2006 .

[104]  Yi Min Xie,et al.  Evolutionary Structural Optimization , 1997 .

[105]  M. Bendsøe,et al.  Topology optimization of heat conduction problems using the finite volume method , 2006 .

[106]  G. Burton.  Sobolev Spaces , 2013 .

[107]  Byung Man Kwak,et al.  Smooth Boundary Topology Optimization Using B-spline and Hole Generation , 2007 .

[108]  H. Frieboes,et al.  Computer simulation of glioma growth and morphology , 2007, NeuroImage.

[109]  A. Michell.  LVIII. The limits of economy of material in frame-structures , 1904 .

[110]  J. Tinsley Oden,et al.  GENERAL DIFFUSE-INTERFACE THEORIES AND AN APPROACH TO PREDICTIVE TUMOR GROWTH MODELING , 2010 .

[111]  Jakob S. Jensen,et al.  Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide , 2005 .

[112]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[113]  Tony F. Chan,et al.  A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model , 2002, International Journal of Computer Vision.

[114]  Ashok V. Kumar,et al.  Topology optimization using B-spline finite elements , 2011 .

[115]  Cv Clemens Verhoosel,et al.  A phase-field description of dynamic brittle fracture , 2012 .

[116]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[117]  Daniel Weiss,et al.  Feature-based spline optimization in CAD , 2010 .

[118]  H. Garcke.  On a Cahn-Hilliard model for phase separation with elastic misfit , 2005 .

[119]  John W. Cahn,et al.  On Spinodal Decomposition , 1961 .

[120]  Mark Frederick Hoemmen,et al.  An Overview of Trilinos , 2003 .

[121]  Zafer Gürdal,et al.  Isogeometric sizing and shape optimisation of beam structures , 2009 .

[122]  Luis A. Caffarelli,et al.  An L∞ bound for solutions of the Cahn-Hilliard equation , 1995 .

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

[124]  H. Mlejnek,et al.  An engineer's approach to optimal material distribution and shape finding , 1993 .