Example II: Linear truss structure

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


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


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


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)


Example II: Adaptive response surface


• Areas of elements 2,6 and are set to minimum, 5 and 10 are close to minimum

• 360 solver calls


Example II: EA (global search)


• 392 solver calls


Example II: EA (local search)


• 216 solver calls (+392 from global search)


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


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


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


• 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


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


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


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]


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


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


Strategy C: Pareto Optimization




Design space Objective space

• Only for conflicting objectives a Pareto frontier exists• For positively correlated objective functions only one optimum exists


Correlated objectives


Conflicting objectives



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)



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)


Pareto Optimization using Evolutionary Algorithms


• 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


Example: damped oscillator

• Pareto optimization with EA gives good Pareto frontier with 123 solver calls


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


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

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Example II: Linear truss structure

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