Post on 12-Jan-2016
1 Part 4: Multidisciplinary Optimization
Example II: Linear truss structure
• Optimization goal is to minimize the mass of the structure • Cross section areas of trusses as design variables• Maximum stress in each element as inequality constraints• Maximum displacement in loading points as inequality constraints• Gradient-based and ARSM optimization perform much better if
constraint equations are formulated separately instead of using total max_stress and max_disp as constraints
2 Part 4: Multidisciplinary Optimization
Example II: Sensitivity analysis
• MOP indicates only a1, a3, a8 as important variables for maximum stress and displacements,but all inputs are important for objective function
3 Part 4: Multidisciplinary Optimization
Example II: Sensitivity analysis
• For single stress values used in constraint equations, each input variable occurs at least twice as important parameter
Reduction of number of inputs seems not possible
max_stress
max_disp
stress10
stress9
stress8
stress8
stress6
stress5
stress4
stress3
stress2
stress1
disp4
disp2
mass
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
MOP filter
4 Part 4: Multidisciplinary Optimization
Example II: Gradient-based optimization
• Best design with valid constraints: mass = 1595 (19% of initial mass)
• Areas of elements 2,5,6 and 10 are set to minimum
• Stresses in remaining elements reach maximum value
• 153 solver calls (+100 from DOE)
5 Part 4: Multidisciplinary Optimization
Example II: Adaptive response surface
• Best design with valid constraints: mass = 1613 (19% of initial mass)
• Areas of elements 2,6 and are set to minimum, 5 and 10 are close to minimum
• 360 solver calls
6 Part 4: Multidisciplinary Optimization
Example II: EA (global search)
• Best design with valid constraints: mass = 2087 (25% of initial mass)
• 392 solver calls
7 Part 4: Multidisciplinary Optimization
Example II: EA (local search)
• Best design with valid constraints: mass = 2049 (24% of initial mass)
• 216 solver calls (+392 from global search)
8 Part 4: Multidisciplinary Optimization
Example II: Overview optimization results
Method Settings Mass Solver callsConstraints
violated
Initial - 8393 - -
DOE LHS 3285 100 75%
NLPQLdiff. interval 0.01%, single sided
1595 153(+100) 42%
ARSM defaults (local) 1613 360 80%
EA global defaults 2087 392 56%
EA local defaults 2049 216(+392) 79%
PSO global defaults 2411 400 36%
GA global defaults 2538 381 25%
SDI local defaults 1899 400 70%
• NLPQL with small differentiation interval with best DOE as start design is most efficient
• Local ARSM gives similar parameter set• EA/GA/PSO with default settings come close to global optimum• GA with adaptive mutation has minimum constraint violation
9 Part 4: Multidisciplinary Optimization
Gradient-based algorithms
• Most efficient method if gradients are accurate enough
• Consider its restrictions like local optima, only continuous variablesand noise
Response surface method
• Attractive method for a small set of continuous variables (<15)
• Adaptive RSM with default settings is the method of choice
Biologic Algorithms
• GA/EA/PSO copy mechanisms of nature to improve individuals
• Method of choice if gradient or ARSM fails
• Very robust against numerical noise, non-linearities, number of variables,…
Start
When to use which optimization algorithms
10 Part 4: Multidisciplinary Optimization
4) Goal: user-friendly procedure provides as much automatism as possible
1) Start with a sensitivity study using the LHS Sampling
Sensitivity Analysis and Optimization
3) Run an ARSM, gradient based or biological based optimization algorithm
Understand the Problem using
CoP/MoP
Search for Optima
Scan the whole Design Space
optiSLang
2) Identify the important parameters and responses
- understand the problem- reduce the problem
11 Part 4: Multidisciplinary Optimization
• Optimization of the total weight of two load cases with constrains (stresses)
• 30.000 discrete Variables • Self regulating evolutionary
strategy• Population of 4, uniform
crossover for reproduction• Active search for dominant
genes with different mutation rates
Solver: ANSYSDesign Evaluations: 3000Design Improvement: > 10 %
Optimization of a Large Ship VesselEVOLUTIONARY ALGORITHM
12 Part 4: Multidisciplinary Optimization
Optimization of passive safety performance US_NCAP & EURO_NCAP
using Adaptive Response Surface Method
- 3 and 11 continuous variables
- weighted objective function
Solver: MADYMO
Optimization of passive safety
Design Evaluations: 75Design Improvement: 10 %
Adaptive Response Surface Methodology
13 Part 4: Multidisciplinary Optimization
Genetic Optimization of Spot Welds
Solver: ANSYS (using automatic spot weld Meshing procedure)Design evaluations: 200Design improvement: 47%
2)( /140cossinsincos mmNMYMXFZFYFXR
• 134 binary variables, torsion loading, stress constrains
• Weak elitism to reach fast design improvement
• Fatigue related stress evaluation in all spot welds
14 Part 4: Multidisciplinary Optimization
Optimization of an Oil Pan
The intention is to optimize beads to increase the first eigenfrequency of an oil pan by more than 40%. Topology optimization indicate possibility
> 40% improvement, but test failed. Sensitivity study and parametric optimization
using parametric CAD design + ANSYS workbench+optiSLang could solve the task.
Initial design
beads design after parameter
optimization
beads design after topology optimization
Design Parameter 50Design Evaluations: 500CAE: ANSYS workbenchCAD: Pro/ENGINEER
[Veiz. A; Will, J.: Parametric optimization of an oil pan; Proceedings Weimarer Optimierung- und Stochastiktage 5.0, 2008]
15 Part 4: Multidisciplinary Optimization
Multi Criteria Optimization Strategies
• Several optimization criteria are formulated in terms of the input variables x
• Strategy A:• Only the most important objective
function is used as optimization goal• Other objectives as constraints
• Strategy B:• Weighting of single objectives
16 Part 4: Multidisciplinary Optimization
Example: damped oscillator
• Objective 1: minimize maximum amplitude after 5s• Objective 2: minimize eigen-frequency • DOE scan with 100 LHS samples gives good problem overview• Weighted objectives require about 1000 solver calls
17 Part 4: Multidisciplinary Optimization
Strategy C: Pareto Optimization
Multi Criteria Optimization Strategies
18 Part 4: Multidisciplinary Optimization
Multi Criteria Optimization Strategies
Design space Objective space
• Only for conflicting objectives a Pareto frontier exists• For positively correlated objective functions only one optimum exists
19 Part 4: Multidisciplinary Optimization
Correlated objectives
Multi Criteria Optimization Strategies
Conflicting objectives
20 Part 4: Multidisciplinary Optimization
Multi Criteria Optimization Strategies
Pareto dominance
• Solution a dominates solution c since a is better in both objectives• Solution a is indifferent to b since each solution is better than
the respective other in one objective
(a dominates c)
(a is indifferent to b)
21 Part 4: Multidisciplinary Optimization
Multi Criteria Optimization Strategies
Pareto optimality• A solution is called Pareto-optimal if there is no decision vector
that would improve one objective without causing a degradation in at least one other objective
• A solution a is called Pareto-optimal in relation to a set of solutions A, if it is not dominated by any other solution c
Requirements for ideal multi-objective optimization• Find a set of solutions close to the Pareto-optimal solutions
(convergence)• Find solutions which are diverse enough to represent the whole
Pareto front (diversity)
22 Part 4: Multidisciplinary Optimization
Pareto Optimization using Evolutionary Algorithms
Multi Criteria Optimization Strategies
• Only in case of conflicting objectives, a Pareto frontier exists and Pareto optimization is recommended (optiSLang post processing supports 2 or 3 conflicting objectives)
• Effort to resolute Pareto frontier is higher than to optimize one weighted optimization function
23 Part 4: Multidisciplinary Optimization
Example: damped oscillator
• Pareto optimization with EA gives good Pareto frontier with 123 solver calls
24 Part 4: Multidisciplinary Optimization
Example II: linear truss structure
• For more complex problems the performance of the Pareto optimization can be improved if a good start population is available
• This can be found in selected designs of a previous DOE or single objective optimization
1.
Pareto frontAnthill plot from ARSM
25 Part 4: Multidisciplinary Optimization
Gradient-based algorithms
Response surface method (RSM)
Biologic Algorithms Genetic algorithms, Evolutionary strategies & Particle Swarm Optimization
Start
Optimization Algorithms
Pareto Optimization
Local adaptive RSM
Global adaptive RSM