example ii: linear truss structure
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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