A fast method for binary programming using first-order derivatives, with application to topology optimization with buckling constraints

SUMMARY We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analyt...

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Published in:	International journal for numerical methods in engineering Vol. 92; no. 12; pp. 1026 - 1043
Main Authors:	Browne, P. A., Budd, C., Gould, N. I. M., Kim, H. A., Scott, J. A.
Format:	Journal Article
Language:	English
Published:	Chichester Blackwell Publishing Ltd 21.12.2012 Wiley Wiley Subscription Services, Inc
Subjects:	binary programming Buckling eigenvalue Exact sciences and technology Fundamental areas of phenomenology (including applications) Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Numerical analysis. Scientific computation Physics Sciences and techniques of general use Solid mechanics Static elasticity (thermoelasticity...) Structural and continuum mechanics structural optimization topology optimization Geometrical shape High performance Stiffness matrix Constraint structural optimization Minimization Topology Distributed computing Modeling Weight Constrained optimization Finite element method Zero one programming binary programming Eigenvalue problem topology optimization Buckling eigenvalue Structural analysis
ISSN:	0029-5981, 1097-0207
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Abstract	SUMMARY We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd.
AbstractList	SUMMARY We present a method for finding solutions of large-scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite-element setting. Results are presented for a number of two-dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd. [PUBLICATION ABSTRACT] We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd. SUMMARY We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd.
Author	Budd, C. Kim, H. A. Browne, P. A. Gould, N. I. M. Scott, J. A.
Author_xml	– sequence: 1 givenname: P. A. surname: Browne fullname: Browne, P. A. email: P. A. Browne, Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, U.K., P.A.Browne@bath.ac.uk organization: Department of Mathematical Sciences, University of Bath, BA2 7AY, Bath, UK – sequence: 2 givenname: C. surname: Budd fullname: Budd, C. organization: Department of Mathematical Sciences, University of Bath, BA2 7AY, Bath, UK – sequence: 3 givenname: N. I. M. surname: Gould fullname: Gould, N. I. M. organization: Numerical Analysis Group, STFC Rutherford Appleton Laboratory, OX11 0QX, Oxfordshire, UK – sequence: 4 givenname: H. A. surname: Kim fullname: Kim, H. A. organization: Department of Mechanical Engineering, University of Bath, BA2 7AY, Bath, UK – sequence: 5 givenname: J. A. surname: Scott fullname: Scott, J. A. organization: Numerical Analysis Group, STFC Rutherford Appleton Laboratory, OX11 0QX, Oxfordshire, UK
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Keywords	Geometrical shape High performance Stiffness matrix Constraint structural optimization Minimization Topology Distributed computing Modeling Weight Constrained optimization Finite element method Zero one programming binary programming Eigenvalue problem topology optimization Buckling eigenvalue Structural analysis
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References	Stolpe M, Svanberg K.Modelling topology optimization problems as linear mixed 0 − − 1 programs. International Journal for Numerical Methods in Engineering June 2003; 57(5):723-739. DOI: 10.1002/nme.700. Available from: http://doi.wiley.com/10.1002/nme.700. Cook RD, Malkus DS, Plesha M.Concepts and Applications of Finite Element Analysis. John Wiley and Sons: London, UK, 1989. Salajegheh E, Vanderplaats G.Optimum design of trusses with discrete sizing and shape variables. Structural Optimization 1993; 6:79-85. Williams J.Algorithm 232: heapsort. Communications of the ACM 1964; 7(6):347-348. Beckers M.Dual methods for discrete structural optimization problems. International Journal for Numerical Methods in Engineering 2000; 48(12):1761-1784. Sandgren E.Nonlinear integer and discrete programming for topological decision making in engineering design. Journal of Mechanical Design 1990; 112(1):118-122. DOI: 10.1115/1.2912568. Available from: http://link.aip.org/link/?JMD/112/118/1 Hunt G, Thompson J.A General Theory of Elastic Stability. Wiley-Interscience: London, UK, 1973. Kočvara M.On the modelling and solving of the truss design problem with global stability constraints. Structural and Multidisciplinary Optimization April 2002; 23(3):189-203. DOI: 10.1007/s00158-002-0177-3. Available from: http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s00158-002-0177-3 Rozvany GIN.Authors reply to a discussion by Gengdong Cheng and Xiaofeng Liu of the review article On symmetry and non-uniqueness in exact topology optimizationİ by George I. N. Rozvany (2011, Struct Multidisc Optim 43: 297317). Structural and Multidisciplinary Optimization September 2011; 44(5):719-721. DOI: 10.1007/s00158-011-0703-2. Available from: http://www.springerlink.com/index/10.1007/s00158-011-0703-2 Kočvara M, Stingl M.Solving nonconvex SDP problems of structural optimization with stability control. Optimization Methods and Software October 2004; 19(5):595-609. DOI: 10.1080/10556780410001682844. Available from: http://www.informaworld.com/openurl?genre=article&doi=10.1080/10556780410001682844&magic=crossref\|\|D404A21C5BB053405B1A640AFFD44AE3 Wolsey L.Integer programming, Wiley-Interscience series in discrete mathematics and optimization, Wiley: New York, USA, 1998. Available from: http://books.google.com/books?id=x7RvQgAACAAJ Pedersen N.Maximization of eigenvalues using topology optimization. Structural and Multidisciplinary Optimization 2000; 20(1):2-11. Neves MM, Sigmund O, Bendsøe MP.Topology optimization of periodic microstructures with a penalization of highly localized buckling modes. International Journal for Numerical Methods in Engineering June 2002; 54(6):809-834. DOI: 10.1002/nme.449. Available from: http://doi.wiley.com/10.1002/nme.449 Bendsøe MP, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer: Berlin, Germany, 2003. Achtziger W, Stolpe M.Truss topology optimization with discrete design variables-guaranteed global optimality and benchmark examples. Structural and Multidisciplinary Optimization December 2007; 34(1):1-20. DOI: 10.1007/s00158-006-0074-2. Available from: http://www.springerlink.com/index/10.1007/s00158-006-0074-2. Achtziger W, Stolpe M.Global optimization of truss topology with discrete bar areas-Part II: implementation and numerical results. Computational Optimization and Applications 2009; 44(2):315-341. DOI: 10.1007/s10589-007-9152-7. Huang X, Xie YM.Evolutionary Topology Optimization of Continuum Structures. John Wiley & Sons Ltd: Chichester, UK, 2010. Svanberg K.The method of moving asymptotes-a new method for structural optimization. International Journal For Numerical Methods in Engineering 1987; 24:359-373. Stolpe M, Bendsøe MP.Global optima for the Zhou-Rozvany problem. Structural and Multidisciplinary Optimization 2010; 43(2):151-164. Tenek LH, Hagiwara I.Eigenfrequency maximization of plates by optimization of topology using homogenization and mathematical programming. JSME International Journal Series C 1994; 37(4):667-677. Stolpe M.On some fundamental properties of structural topology optimization problems. Structural and Multidisciplinary Optimization 2010; 41(5):661-670. Hogg J, Reid J, Scott J.Design of a multicore sparse cholesky factorization using DAGs. SISC 2010; 32:3627-3649. Sandgren E.Nonlinear integer and discrete programming in mechanical design optimization. Journal of Mechanical Design 1990; 112(2):223-229. DOI: 10.1115/1.2912596. Available from: http://link.aip.org/link/?JMD/112/223/1. Farkas J, Szabo L.Optimum design of beams and frames of welded I-sections by means of backtrack programming. Acta Technica Academiae Scientiarum Hungaricae 1980; 91(1):121-135. Arora J, Huang M.Methods for optimization of nonlinear problems with discrete variables: a review. Structural Optimization 1994; 8:69-85. Toakley A,Optimum design using available sections. Journal of the Structural Division: proceedings of the American Society of Civil Engineers 1968; 94:1219-1241. Achtziger W, Stolpe M.Global optimization of truss topology with discrete bar areas-Part I: theory of relaxed problems. Computational Optimization and Applications November 2008; 40(2):247-280. DOI: 10.1007/s10589-007-9138-5. Available from: http://www.springerlink.com/index/10.1007/s10589-007-9138-5. John K, Ramakrishna C, Sharma K.Optimum design of trusses from available sections-use of sequential linear programming with branch and bound algorithm. 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References_xml	– reference: Beckers M.Dual methods for discrete structural optimization problems. International Journal for Numerical Methods in Engineering 2000; 48(12):1761-1784. – reference: Neves MM, Sigmund O, Bendsøe MP.Topology optimization of periodic microstructures with a penalization of highly localized buckling modes. International Journal for Numerical Methods in Engineering June 2002; 54(6):809-834. DOI: 10.1002/nme.449. Available from: http://doi.wiley.com/10.1002/nme.449 – reference: Williams J.Algorithm 232: heapsort. Communications of the ACM 1964; 7(6):347-348. – reference: Arora J, Huang M.Methods for optimization of nonlinear problems with discrete variables: a review. Structural Optimization 1994; 8:69-85. – reference: Kočvara M.On the modelling and solving of the truss design problem with global stability constraints. Structural and Multidisciplinary Optimization April 2002; 23(3):189-203. DOI: 10.1007/s00158-002-0177-3. Available from: http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s00158-002-0177-3 – reference: Wolsey L.Integer programming, Wiley-Interscience series in discrete mathematics and optimization, Wiley: New York, USA, 1998. Available from: http://books.google.com/books?id=x7RvQgAACAAJ – reference: Achtziger W, Stolpe M.Global optimization of truss topology with discrete bar areas-Part I: theory of relaxed problems. Computational Optimization and Applications November 2008; 40(2):247-280. DOI: 10.1007/s10589-007-9138-5. Available from: http://www.springerlink.com/index/10.1007/s10589-007-9138-5. – reference: Cook RD, Malkus DS, Plesha M.Concepts and Applications of Finite Element Analysis. John Wiley and Sons: London, UK, 1989. – reference: Achtziger W, Stolpe M.Global optimization of truss topology with discrete bar areas-Part II: implementation and numerical results. Computational Optimization and Applications 2009; 44(2):315-341. DOI: 10.1007/s10589-007-9152-7. – reference: Bendsøe MP, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer: Berlin, Germany, 2003. – reference: Tenek LH, Hagiwara I.Eigenfrequency maximization of plates by optimization of topology using homogenization and mathematical programming. JSME International Journal Series C 1994; 37(4):667-677. – reference: Sandgren E.Nonlinear integer and discrete programming for topological decision making in engineering design. Journal of Mechanical Design 1990; 112(1):118-122. DOI: 10.1115/1.2912568. Available from: http://link.aip.org/link/?JMD/112/118/1 – reference: Sandgren E.Nonlinear integer and discrete programming in mechanical design optimization. Journal of Mechanical Design 1990; 112(2):223-229. DOI: 10.1115/1.2912596. Available from: http://link.aip.org/link/?JMD/112/223/1. – reference: Hogg J, Reid J, Scott J.Design of a multicore sparse cholesky factorization using DAGs. SISC 2010; 32:3627-3649. – reference: Toakley A,Optimum design using available sections. Journal of the Structural Division: proceedings of the American Society of Civil Engineers 1968; 94:1219-1241. – reference: Farkas J, Szabo L.Optimum design of beams and frames of welded I-sections by means of backtrack programming. Acta Technica Academiae Scientiarum Hungaricae 1980; 91(1):121-135. – reference: Stolpe M, Svanberg K.Modelling topology optimization problems as linear mixed 0 − − 1 programs. International Journal for Numerical Methods in Engineering June 2003; 57(5):723-739. DOI: 10.1002/nme.700. Available from: http://doi.wiley.com/10.1002/nme.700. – reference: Achtziger W, Stolpe M.Truss topology optimization with discrete design variables-guaranteed global optimality and benchmark examples. Structural and Multidisciplinary Optimization December 2007; 34(1):1-20. DOI: 10.1007/s00158-006-0074-2. Available from: http://www.springerlink.com/index/10.1007/s00158-006-0074-2. – reference: Huang X, Xie YM.Evolutionary Topology Optimization of Continuum Structures. John Wiley & Sons Ltd: Chichester, UK, 2010. – reference: Svanberg K.The method of moving asymptotes-a new method for structural optimization. International Journal For Numerical Methods in Engineering 1987; 24:359-373. – reference: Kočvara M, Stingl M.Solving nonconvex SDP problems of structural optimization with stability control. Optimization Methods and Software October 2004; 19(5):595-609. DOI: 10.1080/10556780410001682844. Available from: http://www.informaworld.com/openurl?genre=article&doi=10.1080/10556780410001682844&magic=crossref\|\|D404A21C5BB053405B1A640AFFD44AE3 – reference: Salajegheh E, Vanderplaats G.Optimum design of trusses with discrete sizing and shape variables. Structural Optimization 1993; 6:79-85. – reference: Pedersen N.Maximization of eigenvalues using topology optimization. Structural and Multidisciplinary Optimization 2000; 20(1):2-11. – reference: John K, Ramakrishna C, Sharma K.Optimum design of trusses from available sections-use of sequential linear programming with branch and bound algorithm. Engineering Optimization 1988; 13(2):119-145. – reference: Rozvany GIN.Authors reply to a discussion by Gengdong Cheng and Xiaofeng Liu of the review article On symmetry and non-uniqueness in exact topology optimizationİ by George I. N. Rozvany (2011, Struct Multidisc Optim 43: 297317). Structural and Multidisciplinary Optimization September 2011; 44(5):719-721. DOI: 10.1007/s00158-011-0703-2. Available from: http://www.springerlink.com/index/10.1007/s00158-011-0703-2 – reference: Stolpe M.On some fundamental properties of structural topology optimization problems. Structural and Multidisciplinary Optimization 2010; 41(5):661-670. – reference: Hunt G, Thompson J.A General Theory of Elastic Stability. Wiley-Interscience: London, UK, 1973. – reference: Stolpe M, Bendsøe MP.Global optima for the Zhou-Rozvany problem. 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Snippet	SUMMARY We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then... We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this... SUMMARY We present a method for finding solutions of large-scale binary programming problems where the calculation of derivatives is very expensive. We then...
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SubjectTerms	binary programming Buckling eigenvalue Exact sciences and technology Fundamental areas of phenomenology (including applications) Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Numerical analysis. Scientific computation Physics Sciences and techniques of general use Solid mechanics Static elasticity (thermoelasticity...) Structural and continuum mechanics structural optimization topology optimization
Title	A fast method for binary programming using first-order derivatives, with application to topology optimization with buckling constraints
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