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Design Optimization by usingSupport Vector Machines

NAFEMS NORDIC 18, 24-25 April, Gothenburg

Niclas Strömberg, Ph.D., DocentÖrebro University

niclas.stromberg@oru.se

www.fema.se

• Computational contactmechanics

• Thermomechnical stresses• Topology optimization• Metamodel-based design

optimization• Reliability-based design

optimization

• TopoBox• MetaBox• BrakeBox

• MT502G - Mechanics• MT504G - Solid mechanics• MT506G - Machine elements

TEACHING

RESEARCH

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Outline• What is support vector machines• Two SVM-based approaches for TO and RBDO• TopoBox – an inhouse toolbox for TO• Orthotropic elasticity with mortar contact conditions• SVM-based postprocessing• MetaBox – an inhouse toolbox for sampling-based DO• Reliability-based design optimization• A new SQP-based RBDO approach• SVM-based limit surface• SVM-based adaptive sampling

Many years ago, deep learning …

N. Strömberg, Simulation of Rotary Draw Bending using Abaqus and a Neural Network, NAFEMS NORDIC, 24-25 November, Gothenburg, 2005.

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Today, support vector machines …

C. Cortes & V. Vapnik, Support-VectorNetworks, Machine Learning, 20, 273–297, 1995.

The kernel trick

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Examples – 2D and 3D bulls eye

SVM-based approaches

SVM-based limit state surfaceSVM-based postprocessing

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TopoBox - inhouse toolbox

Trade-off curve

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Topology optimization of a cutting tool

Lattice structure

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Topology optimization of a stamping die

SVM

Compliance formulation

1 e

For given density distribution, solve the state problem

Sensitivity analysis

Solve LP-problem

Convergence

Update density distribution

STOP

YES

NO

Filter sensitivities

1

0V)(

0)(

s.t. max

e

V

ρ

FdρK

dFρd

Tc ) ,(

min

?

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Maximizing potential energy

N. Strömberg & A. Klarbring, Topology Optimization of Structures in Unilateral Contact, Structural and Multidisciplinary Optimization 41, 57-64, 2010.

N. Strömberg, Topology Optimization of Structures with Manufacturing and Unilateral Contact Constraints by Minimizing an Adjustable Compliance-Volume Product, Structural and Multidisciplinary Optimization 42, 341-350, 2010.

Topology optimization with contact constraints

F

F

F

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Additive manufacturing

A. Jansson, L. Pejryd / Additive Manufacturing 9 (2016) 7–13 www.eos.info

Orthotropic elasticity

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The mortar integralLobatto rule

N. Strömberg, Topology Optimization of Orthotropic Elastic Design Domains with Mortar Contact Conditions, in the proceedings of 12th World Congress on Structural and Multidisciplinary Optimization, Braunschweig, Germany, June 5-9, 2017.

Michell’s benchmark

Michell, A. G. M. (1904) The limits of economy of material in frame-structures, Philosophical Magazine, Vol. 8(47), p. 589-597.

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Different build directions

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.64800

4900

5000

5100

5200

5300

5400

5500

5600

5700

SVM-based postprocessing

N. Strömberg, Automatic Postprocessing of Topology Optimization Solutions by using support vector machines, submitted, 2018.

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Orthotropic elasticity with mortar contacts

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.62.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85105

SVM-based postprocessing

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Metamodel

Metamodel”Black box” Computer experiments DoE

Metabox 1.4

Design of experiments• Linear Koshal• Full factorial• Face centered cubic• Symmetrical Koshal• Quadratic Koshal• Spherical• Box-Behnken• S-optimal• Latin hypercube sampling• Halton sampling• Hammersley sampling

Metamodels• Linear regression• Quadratic regression• OPRM• Kriging with LRM• Kriging with QRM

Metamodels• A priori RBN with LRM• A priori RBN with QRM• A posteriori RBFN with LRM• A posteriori RBFN with QRM• Analytical model• Hybrid model of analytical

model and RBFN• Polynomial chaos expansion• Support vector machines• Support vector regression• Least square SVM & SVR

Solvers• Genetic algorithm• SLP• SQP• Succesive response surface

methodology• Newton’s method

Distributions• Normal• Lognormal• Gumbel• Gamma• Weibull

Solvers• RBDO with SLP• RBDO with SQP• FORM based RBDO• SORM based RBDO• Crude Monte Carlo• Quasi-Monte Carlo• Importance sampling• Multi-classification

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RBDO - an impact problemUnderrun protection system

RBDO - an impact problem

Deterministic

RBDO

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A new SQP-based RBDO approach

N. Strömberg, Reliability-based Design Optimization using SORM and SQP, Structural and Multidisciplinary Optimization, 56, 631–645, 2017.

A new SQP-based RBDO approach

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A new SQP-based RBDO approach

RBDO benchmark

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SVM-based limit surface

N. Strömberg, Reliability-based Design Optimization by using Support Vector Machines, to appear in the proceedings of the European Safety and Reliability Conference ESREL, Trondheim, Norway, 17-21 June, 2018.

SVM-based adaptive sampling

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“The devil is in the details”