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 ID: 332159
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Slide1
Linear Programming
Old name for linear optimizationLinear objective functions and constraintsOptimum always at boundary of feasible domainFirst solution algorithm, Simplex algorithm developed by George Dantzig, 1947What is a simplex (e.g. triangle, tetrahedron)?We will study limit design of skeletal structures as an application of LP.Slide2
Example
(
Vanderplaats
,
Multidiscipline Design Optimization,
p. 128)Slide3
Solution with
Matlab linprogSimplest 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 2.0000obj =-18.0000Matrix formSlide4
Problem
linprogSolve the following problem using linprog and also graphically (do not use the equality constraint to reduce the number of variables).SolutionSlide5
Limit analysis of trusses
Elastic-perfectly plastic behaviorNormally, 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.Slide6
Three bar truss example
3.1.1Slide7
Beyond yield
Recall Member B yields firstHowever, load can be increased until members A and C also yieldSlide8
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.Slide9
LP formulation of truss collapse load
Example 3.2Implication of lower bound theorem: Any p for which we can find n’s that satisfy the equation is safeLP problem: Find loads to maximize
p subject to above constraints
N
on-
dimensionalize!Slide10
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 1.2679Slide11
Problem limit design
Limit design is to select 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 areaThe non-dimensional design variables will now be the areas divided by A and the three member loads, divided bySolution