great challenges in engineering design - materials and ......claus mattheck derives, from...

||ETH Zurich, Laboratory of Composite Materials and Adaptive Structures

9/22/2015Gerald Kress 1

Great Challenges in Engineering Design

- Materials and Topology -

Introduction

Design inspiration provided by biology

Automated structural optimization inspired by biologyGrowth of trees and Computer-Aided Optimization by MattheckDarwinism and structural optimization with evolutionary algorithms

Topology optimization with mathematical programming

Applications to practical design problems

9/22/2015Gerald Kress 2

Contents

||ETH Zurich, Laboratory of Composite Materials and Adaptive Structures 9/22/2015Gerald Kress 3

Life has created highly efficient structured materials

The structures can be at the surface in the interiorperiodic or not

The structures are characterized bytopology shape

Introduction


Bionics is the application of biological methods and systems found in nature to the study and design of engineering systems and modern technology.

Esomba, S., Twenty-First Century's Fuel Sufficiency Roadmap (2012)

Introduction: Bionics


The study of biology with the aim of finding solutions to design problems is called bionics

The observation of the

growth behavior of trees the development of species

has inspired automated design optimization methods

Introduction: Imitating biological design and design optimization


Introduction: Biological design examples

Velcro was inspired by the tiny hooks found on the surface of burs.

https://en.wikipedia.org/wiki/Bionics

Lotus leaf surface, rendered: microscopic view.

https://en.wikipedia.org/wiki/Bionics

Wing for flying apparatus source: L. da Vinci

https://de.wikipedia.org/wiki/Bionik


Introduction: Biological design example bone spongiosa

Spongiosa

The interior of the bone is not solid but consists of fine bone trabeculae, the spongy substance. The alignment of the trabeculae follows the femur exactly the course of the lines of force, acting on the thigh with pressure and train.

In 1865 the engineer Culmann visited an anatomy lecture. At the time, he was concerned with how to construct a new, heavy-duty yet lightweight crane. In the human femur he found exactly the model that he needed. It showed the most effective way of how large loads can be handled with minimum amounts of material.

Spongiosa - Architektur, Biologie des Menschen, Mörike et.al. Quelle & Meyer



• Suction cup: octopus, beetle

• Sonar or echo sounding: used by dolphins and bats.

• Airplane slat: bird bastard wing

• Propeller: maple tree samara

• Eiffel tower: bone trabeculae (spongiosa)

• Flying wing: flying semen of

zanonia macrocarpa



• Syringe: poison sting of bees

• Flipper: web of frogs or water birds

• Rocket propulsion: yellyfish or octopus

• Artificial ventilation: termite‘s nest

• Tubular steel pole: corn stalk


Introduction: Function and form

“Function and form are intimately related to one another.

The function describes what the purpose of a technical object is; the form or structure describes how this product will do it.”

Ulman, D.G., The Mechanical Design Process, vol. 2, McGraw-Hill, New York (1997)


Introduction: Structure characterizations and design parameters


Introduction

Ledermann, Ch., Parametric Associative CAE Methods in Preliminary Aircraft Design, Diss. ETH no. 16778 (2006)


Introduction: Structural optimization definitions

Objective and constraining functions can be linear or non-linear in x.

The choice of solution methods depends upon the nature of the functions.

Non-linear optimization requires iterative solution methods.

1

1

1

,1

,10)(

,10)(

x

x

x

nixxx

lkh

mjg

)f(

Oii

Ui

k

j

x

x

x

xminimize:

under:

objective function

inequality constraint

equality constraint

side constraint

design variablesor search space





• direct methods need function evaluations only, indirect methods require derivatives.

• stochastic methods are based on random generated numbers.

• deterministic methods gather information at a point to systematically find better points in its vicinity.

• indirect methods either require first derivatives (gradient) only or first and second derivatives.

• mathematical programming comprises methods of zeroth, first, and second orders. The order corresponds with the required degree of derivative. Mathematical programming is based on model notions of the objective functions. Therefore one also distinguishes model order and method order.

Mathematical Programming

1st order 0th order 2nd order

Parameter Optimization Techniques

Direct Indirect

Stochastic Deterministic Use f, f Use f, f, 2 f

- Evolutionary Computation

- Cauchy's Method- Fletcher Reeves M.

- Simplex Method- Powell's Method- Response-Surface

- Newton's Method

Müller,S.D., Bio-Inspired Optimization Algorithms for Engineering Applications,Diss. ETH no. 14719 (2002)


Automated structural optimization inspired by biology

C. Mattheck:Axiom of constant stress

Computer-Aided Optimization CAO

C. Darwin:“On the Origin of Species”

Evolutionary AlgorithmsEA


Growth of Trees and Computer-Aided Optimization by Mattheck

Combined axial and bending loads caused by the two weights F1 und F2

Claus Mattheck derives, from observations on the growth behavior of tress and particular the adaptation to load changes, optimization methods for maximizing structural strength.

The structural strength increase follows from mitigation of local notch effects.

He calls his method CAO (Computer Aided Optimization)

The present and other sketches are taken from Claus Mattheck, Design in der Natur,Rombach Wissenschaft, 1993

Mattheck‘s Computer-Aided Optimization CAO


Growth of trees and Computer-Aided Optimization by Mattheck

Local optimization method: Axiom of constant stresses.

Trees always grow (change their shape) such that the stresses under long-term loads are as evenly distributed as possible.

His CAO method subdivides the analysis model, analog to the living tree, into a fixed part (old wood) and a surface layer able to grow, the cambium.

region containing stress concentration

surface layer of constant thickness cambium



initial design FEM simulation of structural response to mechanical load

interpretation of equivalentstress as temperature

modify cambium model:- reduce Young's modulus- assign value

calculation of thermal displacements in cambium

convergence?

addition of displacementsto node coordinates

setting of Young's modulusto actual material property

improved design

optimized design

CAO algorithm



A small change of the clip inner contour effects significant maximum-stress reduction.

This increases structural strength.

Sample problem clip


Growth of trees and Computer-Aided Optimization by Mattheck.

Sample problem shaft with cut-out, automotive application.


Darwinism and structural optimization with evolutionary algorithms



Individual: design candidate, a set of gene alleles

Population: a set of individuals

Generation: the population within the evloutionary cycle

Genotype space: the search space X

Phenotype space: the decision space D

Fitness: consists of objective and constraining functions,objective space Y



König, O., Evolutionary Design Optimization: Tools and Applications, Diss. ETH no. 15486 (2004)





• the initial population must be filled with a defined number of individuals

• the initial population must include a high genetic diversity

• Genotypes coding illegal or infeasible solutions may be filtered out

• the initialization can be based random or on known solutions



Exploration: tap the complete search space, strong point of EA(initialization, mutation)

Exploitation: approximate minimum, strong point of MA(cross-over operations)




Topology optimization of continua

Topology optimization redistributing material and voids within the geometric design space.

The material distribution must be mapped onto an analysis model.

A FE mesh assigns to individual elements the states “filled” or “empty”.

u

vk=1

=0

With mesh-dependent parameterization the number of variables equals that of the finite elements.

We consider maximization of structural stiffness under specified boundary conditions.

Without constraining the total amount of material the domain would be completely filled with material.


Topology optimization of continua

Objective is maximum structural stiffness with given mass M0 under specified boundary conditions.

Maximum stiffness is minimum compliance.

With the node displacements calculated with a FE simulation,

rxuxK )(~)(

rxux )(~)( TW

the inner product of the displacement and the external force vectors gives the external work applied to the structur, which is a suitable objective function.


Topology optimization of continua: Combinatoric problem

Without further restrictions a number of nel elements gives nK=2nel combinations.

The model shown on slide 32 with 300 elements gives nK=2300=2.037•1090 combinations, too many to evaluate.

The total-mass constraint requires for every change from “empty” to “full” another change from “full” to “empty” of another parameter.

The average density gives the number of filled elements m. Then the number of combinations is given with

!!!mnm

nnK


Topology optimization of continua: Combinatoric problem

Sample slide 2: At a mean density of = 0.25, or 75 of 300 elements are filled, the formula gives

The total-mass constraint drastically reduces the number of combinations which, however, remains too high evaluate all of them.

7210796.9!225!75

!300Kn

The combinations also contain mechanically meaningless solutions with no load path between load introduction and supported parts of the boundary. .

The number of mechanically meaningful solutions seems to be much smaller.

4 out of 924!



Bendsøe and Kikuchi introduce a continuous element density function which transforms the discrete problem into a continuous one.

The topology optimization problem becomes a sizing problem as the optimum density values within each element must be found.

The density function tends to iterate against either “0” or “1”.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DichteN

orm

iert

er E

-Mod

ul

pEE 0

density function



average density : =0.225

number of elements : N=300

initial design 1 iteration 20 iterations

40 iterations 60 iterations 83 iterations


LENGTH AND HEIGHT OF DESIGN SPACE: XLENGTH YHEIGHT

100.0 50.0

NO. OF ELEMENTS IN X AND Y DIRECTIONS: NELX NELY

80 40

SUPP. EDGES (0=FREE, 1=X, 2=Y, 3=BOTH): 0 0 0 0

SUPP. CORNERS (0=FREE, 1=X, 2=Y, 3=BOTH): 0 0 3 3 0

SUPP. MIDSIDES (0=FREE, 1=X, 2=Y, 3=BOTH): 0 3 0 3 1

EDGE TRACTIONS: 0.000 1.000 1 EDGE TRACTIONS: 0.000 0.000 1 EDGE TRACTIONS: 0.000 1.000 1 EDGE TRACTIONS: 0.000 0.000 1

CORNER FORCES : 0.000 0.000 25 CORNER FORCES : 0.000 0.000 25 CORNER FORCES : 0.000 0.000 25 CORNER FORCES : 0.000 0.000 25

GRAVITY 0.000

AVERAGE DENSITY 0.200 EXPONENT P 4. TOLERANCE RANGE [%] 10.0 FILTER OFF OR ON? 0 OR 1 0NUMBER OF ITERATIONS 1000

READ MOVIE FILE? 0 OR 1 1

The numbers in the box signify by how many finite elements the respective boundary conditions are moved into the interior of the domain.

This control parameter must be set to zero before solving a new problem.

The meanings of most input data are illustrated with the sketch on the next page.




XLENGTH

YHEIGHT

NELX

NELY

EDGE1

EDGE4

EDGE3

EDGE2

CORNER4 CORNER3

CORNER2CORNER1

LENGTH AND HEIGHT OF DESIGN SPACE: XLENGTH YHEIGHT

100.0 50.0

NO. OF ELEMENTS IN X AND Y DIRECTIONS: NELX NELY

80 40

SUPP. EDGES (0=FREE, 1=X, 2=Y, 3=BOTH): 0 0 0 0

SUPP. CORNERS (0=FREE, 1=X, 2=Y, 3=BOTH): 0 0 3 3 0

SUPP. MIDSIDES (0=FREE, 1=X, 2=Y, 3=BOTH): 0 3 0 3 1

EDGE TRACTIONS: 0.000 1.000 1 EDGE TRACTIONS: 0.000 0.000 1 EDGE TRACTIONS: 0.000 1.000 1 EDGE TRACTIONS: 0.000 0.000 1

CORNER FORCES : 0.000 0.000 25 CORNER FORCES : 0.000 0.000 25 CORNER FORCES : 0.000 0.000 25 CORNER FORCES : 0.000 0.000 25

GRAVITY 0.000

AVERAGE DENSITY 0.200 EXPONENT P 4. TOLERANCE RANGE [%] 10.0 FILTER OFF OR ON? 0 OR 1 0NUMBER OF ITERATIONS 1000

READ MOVIE FILE? 0 OR 1 1





Sizing for several load cases, very fine FE mesh (260.000 elements)

Due to the fine mesh, the topology optimization result is very close to the fabricable solution.

Source: M.P. Bendsøe, O. Sigmund, Topology Optimization


Topology optimization with mathematical programmingthematical programming

Maximum strength with topology and shape optimization steps

source: M.P. Bendsøe, O. Sigmund, Topology Optimization

a) Initial FE model

b) Optimized topology

c) CAD model based on topology result

d) FE mesh for shape optimization

e) Equivalent stress in optimal design

f) Designs before and after shape optimization



Magnesium rims are state of the art in motorcycle racing

Improving racing competitiveness by reducing mass and moment of inertia of rear and front rims

Development of lightweight CFRP racing motorcycle rims

Application of proprietary developed optimization tools based on Evolutionary Algorithms

Close collaboration with SRT and OCP in terms of design and manufacturing



27 sections fivefold symmetric

Parameters:

Number of layers

Layer orientation

Material properties

Genotype45° 0.2 UD . . . -15°0.130°



Minimize mass

subject to:stiffness constraint [mm] target stiffness st(p) = 0.18failure criteria (Tsai-Wu index)

m(p) = m(p1, … , pN)

0.16 < s(p) < 0.20 t(p) < 1

Evaluation of two load cases per iteration:

Load case 1:Drive torque to evaluate maximum Tsai-Wu index

Load case 2:Lateral load case to evaluate stiffness value



CAD-model FE-model Parameterization

Layer orientation Layer thicknessMaterial

Genotype45° 0.2 UD . . . 15°0.130°

Initial population1

N

2

Evolutionary Design Optimization Process

Convergence ?

No!

Yes!

Evaluation

FitnessF1, … , FN

Selection

12 3

Recombination

Mutation nth Population1

N

2



Drive torque introduction side Braking side



Hollow spokes are manufactured using press bag molding techniques.

ACG LTM 26-EL: low to medium viscosity prepregs formulated for cure at low initial cure temperatures.

Autoclave curing process:

• cycle time 5h• temperature 70°• pressure 5.5 bar

Final mass: 2400 g


LiteratureErmanni, P., Making matters: Materials, Shape and Function, in S. Konsorski-Lang and M. Hampe, Eds., The Design of Material, Orgamism, and Minds, Springer (2010)

Kress, G., Structural Optimization, Skript to the lecture class, ETH CMAS (2015)

König, O., Evolutionary Design Optimization: Tools and Applications, Diss. ETH no. 15486 (2004)

Wintermantel, M., Design Encoding for Evolutionary Algorithms in the Field of Structural Optimization,Diss. ETH no. 15323 (2004)

Keller, D., Stochastische Verfahren und Evolutionäre Algorithmen, Skript to the lecture class Structural Optimization, ETH CMAS (2015)

C. Mattheck, Design in der Natur . Der Baum als Lehrmeister, Rombach, Freiburg (1993)

Bendsoe, M.P., N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering 71(2): 197-224 (1988

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