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Linear Programming . Old name for linear optimization Linear objective functions and constraints Optimum always at boundary of feasible domain First solution algorithm, Simplex algorithm developed by George Dantzig , 1947 What is a simplex (e.g. triangle, tetrahedron)?
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Linear Programming • Old name for linear optimization • Linear objective functions and constraints • Optimum always at boundary of feasible domain • First solution algorithm, Simplex algorithm developed by George Dantzig, 1947 • What is a simplex (e.g. triangle, tetrahedron)? • We will study limit design of skeletal structures as an application of LP.
Example (Vanderplaats, Multidiscipline Design Optimization, p. 128)
Solution with Matlablinprog Simplest form solves f=[-4 -1]; A=[1 -1; 1 2; -1 0; 0 -1]; b=[2 8 0 0]‘; [x,obj]=linprog(f,A,b) Optimization terminated. x =4.0000 2.0000 obj =-18.0000 • Matrix form
Problem linprog • Solve the following problem using linprog and also graphically (do not use the equality constraint to reduce the number of variables).
Limit analysis of trusses • Elastic-perfectly plastic behavior • Normally, beyond yield the stress will continue to increase, so the assumption is conservative. • We will see it will simplify estimating the collapse load of a truss.
Beyond yield • Recall • Member B yields first • However, load can be increased until members A and C also yield
Lower bound theorem • The Lower Bound Theorem: If a stress distribution can be found that is in equilibrium internally and balances the external loads, and also does not violate the yield conditions, these loads will be carried safely by the structure. • Leads to an optimization problem with equations of equilibrium as equality constraints, and yield conditions as inequality constraints.
LP formulation of truss collapse load • Example 3.2 • Implication of lower bound theorem: Any p for which we can find n’s that satisfy the equation is safe • LP problem: Find loads to maximize p subject to above constraints • Non-dimensionalize!
Non-dimensional form • LP problem f=[0 0 0 -1]; A=eye(4); b=[1 1 1 1000]'; Aeq=[0.5 1 0.5 -1; sqrt(3)/2 0 -sqrt(3)/2 -1]; beq=zeros(2,1); lb=-[1 1 1 0]; x=linprog(f,A,b,Aeq,beq,lb)’ Optimization terminated. x =1.0000 1.0000 -0.4641 1.2679
Problem limit design • Limit design is to the truss cross sectional areas to minimize the weight of the truss subject to a given collapse load p. Formulate the limit design of the truss in Slide 9 for given loads p as an LP and solve using linprog. • Define a nominal area • The non-dimensional design variables will now be the areas divided by A and the three member loads, divided by A times sigma0.
