A 99 line code for discretized Michell truss optimization written in Mathematica

The main purpose of the paper is to provide an easy-to-use code for topological optimization of the least weight trusses, written in the Mathematica programming language. The main idea of the presented approach consists in using a fixed ground structure and the linear programming formulation of the optimization problem. The solver is based on the fast interior point method. The strong effort is done to create the effective generator of the computational model utilizing the high regularity of the ground structure and the high sparsity of the geometric matrix. The efficiency and reliability of the algorithm is confirmed in several numerical tests. Due to a linear programming formulation of the optimization problem the method presented in the paper assures finding the global optimum, hence it may be considered as the useful tool for verification of results obtained in other ways. The appended complete Mathematica code of the program developed will be supplied by the Publisher on SpringerLink.

[1]  Cezary Graczykowski,et al.  Part IV: Complete exact solutions of selected optimal designs and their approximations by trusses of finite number of joints , 2007 .

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

[3]  G. Rozvany,et al.  Extended exact least-weight truss layouts—Part II: Unsymmetric cantilevers , 1994 .

[4]  Harvey J. Greenberg,et al.  Automatic design of optimal structures , 1964 .

[5]  David G. Wilson,et al.  2. Constrained Optimization , 2005 .

[6]  W. Achtziger.  On simultaneous optimization of truss geometry and topology , 2007 .

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

[8]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[9]  M. A. Bhatti,et al.  Practical Optimization Methods with Mathematica Applications (& CD-ROM) , 2002, J. Oper. Res. Soc..

[10]  G. Rozvany,et al.  Extended exact solutions for least-weight truss layouts—Part I: Cantilever with a horizontal axis of symmetry , 1994 .

[11]  M. Gilbert,et al.  Layout optimization of large‐scale pin‐jointed frames , 2003 .

[12]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[13]  Tomasz Sokół,et al.  Application of the interior point method to truss topology optimization , 2009 .

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

[15]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains , 2006 .

[16]  Cezary Graczykowski,et al.  Michell cantilevers constructed within a half strip. Tabulation of selected benchmark results , 2010 .

[17]  H. Chan,et al.  HALF-PLANE SLIP-LINE FIELDS AND MICHELL STRUCTURES , 1967 .

[18]  Tomasz Lewiński,et al.  The lightest plane structures of a bounded stress level transmitting a point load to a circular support , 2005 .

[19]  W. Achtziger,et al.  Topology Optimization of Discrete Structures , 1997 .

[20]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[21]  George I. N. Rozvany,et al.  Layout Optimization of Structures , 1995 .

[22]  G. Rozvany.  Some shortcomings in Michell's truss theory , 1996 .

[23]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[24]  M. A. Bhatti,et al.  Practical Optimization Methods , 2000 .