Trusses, NPCompleteness, and Genetic Algorithms

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Trusses, NP-Completeness, and Genetic Algorithms
Gonzaga University Center for Evolutionary
Algorithms - Spokane, Washington
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Introduction

• The optimization of large trusses often leads to
  a nearly optimal solution, rather than a truly
  optimal design.
• In fact, the problem space for truss optimization
  grows exponentially with the size of the truss.
• Using the method of problem reduction, this paper
  demonstrates that truss optimization is in the
  set of NP-complete problems.
• Hence, the only practical techniques for solving
  the truss problem are heuristic in nature.
• Genetic algorithms provide a viable solution for
  large trusses.

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The Traveling Salesman Problem (TSP)

• The salesman visits each city once, and returns
  to the starting point (Hamiltonian Circuit).
• We want to find the shortest route.

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Traveling Salesman Problem - 2

• NP-Complete.
• O((n-1)!) for a problem of size n (n cities), the
  solution requires fewer than c (n-1)!
  operations for some constant c and n gt N0 where
  N0 is a nonnegative integer.
• For a computer running 1 billion operations per
  sec, it would take 9.83 billion years to solve
  the 24 city problem with a brute-force algorithm
  (check every possible tour).

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Genetic Algorithms

• Also called Evolutionary Algorithms
• A form of artificial intelligence based on the
  theory of evolution.
• Best solutions are combined to form better
  solutions.
• An attempt to find optimal or near optimal
  solutions to NP-Complete problems more quickly,
  without checking every possible solution.

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Flowchart

• GA(Integer parameters)
• Population population
• population
  GeneratePop(parameters)
• Sort(population)
• while(population has not
  converged on a good- enough solution)
• Pair(population)
• Mate(population)
• 
  Mutate(population)
• Sort(population)

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Truss Optimization

• Truss Optimization (TP) Find the minimum total
  volume, V, of a truss, T, with n members, and a
  set, A, of m discrete design variables
  (cross-sectional areas) that satisfy a set of
  prescribed constraints.

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Truss Optimization - 2

• A trusss total volume is the sum of the length
  of each member times the members cross sectional
  area
• Total Volume of a Truss (V)
• where
• V is the volume of the truss
• Li is the length of member i
• ai, the cross-sectional area of member i, is
  selected from the set A of cross-sectional areas.

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Truss Optimization - 3

• Typically, constraints are imposed on stresses
  and displacements.
• Stresses should be less than allowable values as
  a function of the material used.
• Displacements are limited to maximum values
  chosen by the designer.

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TP is NP-Complete

• Theorem 1 The Truss Optimization Problem is
  NP-complete.
• This theorem is demonstrated in the paper using
  the method of problem reduction.
• Since TP is an NP-complete problem, structural
  engineers will have to be satisfied with
  near-optimal as opposed to truly optimal trusses.

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GA for Truss Optimization

• One promising heuristic is the genetic algorithm
  (GA).
• GA is a useful heuristic for intractable
  problems. We show results for the 64-bar truss
  with 64 different variables.

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64-Bar Truss Geometry and Loading

• Aluminum
• E 1.0105 psi.
• Displacement constraints nodes 1 (vertical) and
  9 (horizontal) lt 10 in.
• Max stresses 25 ksi (T and C).

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64-Bar Truss Geometry and Loading 2

• Adjacent nodes are 200 in apart in the horizontal
  and vertical directions.
• Nodes 21, 22, 27, and 28 have no degrees of
  freedom but all other nodes have two.

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64-Bar Truss - Design

• Two designs are proposed
• 1) 64 independent variables
• 2) 64 variables are linked and reduced to
  only 19 independent variables.
• In addition, the truss is solved using integer
  values. The design variables, m, are limited to
  the range 1 in2 m 32 in2 for integer values.

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64-Bar Truss Variable Linking

• This picture shows the 19 linked variables.
• Each member in a set will have the same
  cross-sectional area.
• For example, the set of members identified with 1
  will have the same cross-sectional area.

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64-Bar Truss Results

• The volumes obtained are
• V 31,840 in3 for the 64 independent variables.
• V 43,090 in3 for the 19 linked variables.
• 64 independent variables
• - More detailed description of the design.
• - Lower volume than linked variables.
• - However, higher computational cost.

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Applications - GA for Optimal Structural Design
Using Convex Models of Uncertainties

• Uniform Bound Convex Model
• Non-probabilistic method.
• Uncertainties are bounded within a convex set.
• Identifies the worst-case scenario due to the
  uncertainties.

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Introductory Example

• Two-bar truss.
• Uncertainties static loads P1 and P2.
• Two degrees of freedom X1 and X2.

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Convex Domain


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Superposition Method

• If X(P1,P2 ) gt 0 use and vice versa
• Convex forces Fcon SBXcon Where
• S is the member stiffness matrix
• B is the transpose of the statics matrix A.

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Conclusion

• In this paper we have demonstrated that the truss
  optimization is NP-complete.
• Therefore, practitioners must be satisfied with
  heuristic methods that produce solutions that
  cannot be shown to be optimal.
• The genetic algorithm has been successfully
  applied to large trusses.

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Acknowledgments

• Funding for this project has been provided by the
  McDonald Work Award. Robert and Claire McDonald
  have generously established this grant to the
  benefit of undergraduate students at Gonzaga
  University.
• Sean Fitzgerald, alumnus of the Mathematics and
  Computer Science Department at Gonzaga
  University, is acknowledged for helping providing
  solutions to the 64-bar truss.
• Ann Kilzer, junior student of the Mathematics and
  Computer Science Department at Gonzaga
  University, is acknowledged for helping in
  preparing this presentation.
• Katie Dahmen, alumnus of the Mathematics and
  Computer Science Department at Gonzaga
  University, is acknowledged for helping in
  preparing this presentation.