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Transcript of 67f5bd23 e166-4bf3-8177-9df7689eab27-160907172352
Comparison of robustness of metaheuristic algorithms for
steel frame optimization
RYAN ALBERDI
KAPIL KHANDELWAL
Dipak Prasad
16CE65R10
Algorithmic robustness: -
It is defined as the ability of algorithms to consistently
converge to low cost design irrespective of variable space
and initial starting point.
Diversification: -
It is the ability of algorithms to search new regions of
variable space by incorporating randomness in variable
space.
This helps to escape local optima by thoroughly
searching variable space.
It may cause convergence issue as it fails to converge at
one solution.
Intensification: -
Intensification is the use of memory to seek out regions of the
variable space that have been shown to produce good
solutions.
Intensification encourages convergence by continually
drawing solutions from favorable regions of the variable
space, but will increase the likelihood of converging to local
optima.
The performance of metaheuristic algorithms depends on
a balance between the conflicting characteristics of
diversification and intensification.
These algorithms provide solutions by searching through
discrete variable space of steel sections while incorporating design
codes and other practical constraints of serviceability and
aesthetic aspects.
Problems of steel frame design optimization
1. Large no. of variable space makes search of global optimum
difficult.
2. Different properties inherent to steel sections as well as lack of
knowledge of governing serviceability.
These challenges defined earlier may cause hindrance to
robustness of algorithms.
Robustness of an algorithm may vary with the size of variable
space i.e. an algorithm may loss its robustness with increasing no.
of design variables.
Due to large no. of variable spaces and different governing
metric mean may cause an algorithm to converge to two different
solutions.
Problem formulation
Here we assume design vector x of size ng where ng is total no. of
group in problem.
These groups are formed in such a manner that each member in
group will be assigned same cross-section.
If nk are the possible choices of cross-section for group no. k
than from combinatorics total no. of solution set is given by
N= n1 x n2 x…x nng
Problem formulation
strength constraints from AISC-LRFD
Where
r=1,2,…n, n is total no. of elements in frame
Pur=Required tensile or compressive strength
Pnr=Nominal tensile or compressive strength
φ = 0.9 for tension and 0.85 for compression
Muxr/Muyr= Required flexure strength about major/minor axis
Mnxr/Mnyr=Nominal flexure strength about major/minor axis
Φb= Resistance factor for flexure = 0.9
Inter story drift constraint
Top story drift constraint
Where
r= 1,2,…n , no. of stories
δr= maximum drift at story r
δa= allowable maximum displacement
δT= maximum displacement at top floor
δaT= allowable maximum displacement
δa or δaT may vary with problem depending on materials used in
construction.
Constructability constraints
Where
r= 1,2 ,…n,
n=total no. of constructability constraints
bbf= flange width of beam
bcf= flange width of column
Objective function
minimize fp(x)
These auxiliary functions are as follows
Where
fp(x) is penalized objective function
f(x) is weight of frame
ρ= density of steel
Aj= cross sectional area of member assigned to jth group
Li= length of ith member of group
αs , αd , αt , αc are penalty factor corresponding to strength,
drift , top displacement, constructability violations respectively.
Drift dominated member
These are the members which violates or comes near to violating
drift constraints but does not violate strength constraints.
Strength dominated member
These are the members which violates or comes closer to violating
strength constraints but does not violate drift constraints.
Table shows neighborhood matrix of a
particular steel section based on plastic
section modulus, moment of inertia and
radius of gyration.
Depending on the behavior of member
algorithms can use any of column for
searching optimum cross-section.
Algorithms involved in study of Robustness
1. Ant colony optimization (ACO)
2. Genetic algorithms (GA)
3. Harmony search (HS)
4. Design driven harmony search (DDHS)
5. Adaptive harmony search (AHS)
6. Particle swarm optimization (PSO)
7. Simulated annealing (SA)
8. Improved simulated annealing (ISA)
9. Tabu search (TS)
Benchmark design problems
Five steel frames are created in order to test the optimization algorithms.
Linear static analysis is performed and all frames use ASTM992.
ASTM992 is a structural steel alloy used for wide flanges and I- sections
in USA.
It has following properties: -
Density = 7850 kg/m3
Tensile yield strength = 345MPa (50ksi)
Ultimate tensile strength = 450Mpa (65ksi)
Strain to rupture = 18% for 200 mm long test specimen;
21% for 50 mm long test specimen.
Generally represented as ‘ASTM992(Fy=50ksi,Fu=65ksi)’
Planar moment frame – 3 bay X 3 story
Here 3 design variables had been
used for optimization.
First group consists of all inside
columns.
Second group consists of all outside
columns.
Third group consists of all beams.
The variable space used for column
consists of all 18 W10 and 29 W12
sections.
The variable space for beam
consists of all 267 wide-flange
sections.
To investigate robustness each algorithm is run 500 times for
planar frame and 200 times for space frame.
Result table indicates the following facts
1. Avg. weight is mean of total wt. of structure from different run.
2. Std. weight is standard deviation of total wt. of structure from
different run.
3. Percentage feasible indicates the percentage of run generating
feasible result.
4. Percentage optimal indicates percentage of run generating
exact optimal solution.
The optimal solution for 3bay-3story planar frame consists of
W21X44 section for beam group,W12X120 section for inside columns
and W12X53 section for outside columns. The weight of this globally
optimal design is 107.32 KN.
Essential facts from result: -
DDHS finds the optimal solution in every run.
Only ISA generates some non feasible results.
ISA have highest standard deviation for its solutions.
After DDHS, ISA generates the global optimum solution maximum times.
PSO could not produce actual result for a single run. Also its results’ avg.
weight is much far from actual one.
Planar moment frame- 5 bay X 14 story
Here 20 design variables are
involved in analysis 12 column
group and 8 beam group.
The variable space used for
column group consists of all
W12,W14,W18,W21,W24,W27
sections while all 267 sections
have been used for beam group.
The total variable space consists of
1.34e45 permutations.
Planar braced frame 5 bay X 14 story
Here 27 design variables are
involved. These are 12 column, 8
beam and 7 bracing groups.
The variable group for column
consists of W12, W14, W18, W21,
W24,W27 sections in AISC manual.
The variable group for beam consists
of all 267 sections and for brace
groups only W8,W10,W12,W14
sections are considered.
Total no. of variable space consists of
1.09e59 permutations.
Space moment frame 5 bay X 5 bay X 20 story
16 column and 10 beam group
W12,W14,W18,W21,W24,W27 for column
All 267 sections for beam group
3.58e58 permutations in variable space
Space braced frame 5 bay X 3 bay X 25 story
20 column group, 10 beam
group and 10 brace group
All 267 sections for all groups
1.15e97 permutations for
variable space.
Algorithms comparison for each cases are as follows
Conclusion
Design driven harmony search is the most robust overall obtaining the lowest
average weight and finding 100% feasible design.
Tabu search, harmony search, adaptive harmony search perform well as
variable space is increased.
Genetic algorithm and ant colony optimization consistently find feasible
solution but unable to converge to optimal design.
Particle swarm optimization is not appropriate in its current form for steel
frame optimization but its performance can be improved by using metric
information by appropriately organizing variable space.
Simulated annealing and improved simulated annealing can achieve results
comparable to DDHS but only for small variable space.
In this case only gravity and wind loads are taken into account so further study
with seismic load is necessary.