COMPUTATIONAL INTELLIGENCE IN MULTISCALE AND … · Numerical homogenization - requirements • Separation of scales • Averaging theorem • Hill’s condition (the equality of

COMPUTATIONAL INTELLIGENCE

IN

MULTISCALE AND BIOMEDICAL

ENGINEERING

TADEUSZ BURCZYŃSKI Institute of Fundamental Technological Research

Polish Academy of Sciences (IPPT PAN)

and

Cracow University of Technology

JUBILEE SCIENTIFIC CONFERENCE „PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Intelligence and Interdependence between macro and micro

http://hunch.net/~yan/solid.mechanics.html

Contents

• Intelligent computing

• Multiscale inverse problems

• Computational Intelligence Systems (CIS)

• Optimal Design on the micro-macro levels

• Identification problems on the micro-macro levels

• Smart design materials in nano-scale

• Concluding remarks

Three important areas of intelligent computing methods, namely:

• Evolutionary Computing based on Evolutionary Algorithms (EA)

• Immune Computing based on Artificial Immune Systems (AIS)

• Swarm Computing based on Particle Swarm Optimizers (PSO)

are presented as intelligent computing (Artificial Intelligence - AI) methods.

Criteria of AI: • Turing test,

• Intelligent actions:

- heuristics,

- learning,

• Rational perpetration.

COMPUTATIONAL INTELLIGENCE

INTELIGENT COMPUTING METHODS

Common features of intelligent

bio-inspired methods

• Formulation based on population (set of problems in each iteration).

• Operators simulate some biological or natural processes.

• Stochastic approach.

• The great probability of finding global solutions (possibility of closing to

the global optimum also when the starting population is in local optimas

basins).

• Impact of the best solutions on next iterations, even the worst solution can

have impact.

• Time consuming but there is possibility to speed up by parallel computing

and grid environment.

Intelligent optimization methods inspired by biological/natural mechanisms – soft computing

Obje

ctive

fun

ctio

n v

alu

e pathogens

Obje

ctive

fun

ctio

n v

alu

e Individuals

Evolutionary

algorithms (EA) Artificial immune

systems (AIS)

The goal of AIS

find the most dangerous pathogen

i.e. the global optimum

of objective function

The goal of EA

find the fittest chromosom



Obje

ctive

fun

ctio

n v

alu

e Locations

The goal of PSO

find the best location



Particle swarm

optimizers (PSO)

Evolutionary algorithm (EA)

Distributed EA

Sequential EA

Artificial Immune System (AIS)

Parameters of AIS:

• the number of memory cells

• the number of the clones

• crowding factor

• Gaussian mutation

B-cell with antibodies

T-cell (non self protein recognition)

Particle Swarm Optimization (PSO)

Parameters of PSO:

• number of the particles,

• number of design variables,

• inertia weight,

• two acceleration coefficients,

• two random numbers with uniform distribution,

Parallel Bioinspired Algorithm

Hybrid Bioinspired Algorithm

The number of subpopulations

The number of chrom.

Simple crossover

Gaussian mutation

1 20 100% 100%

2 10 100% 100%

Comparison for he mathematical function

46

The number of memory

cells

The number of the clones

Crowding factor

Gaussian mutation

2 4 0.45 40%

The Rastrigin function

21

( ) 10 10cos 2n

i i

i

F x n x x

5.12 5.12ix

for n=2

min 0,0, ,0 0.0F x F

The stop condition: F(x)

< 0.1

The optimal parameters of AIS

The optimal parameters of EA

The optimal parameters of PSO

Number of particles

Interia weight w

Acceleration coefficient c1

Acceleration coefficient c2

74 1 1.9 1.9

Multiscale approach in engineering problems

Nano

Multiscale Modelling

10 -9 10 -6 10 -3 10 0

Length, m

10 -15

10 -12

10 -9

10 -6

10 -3

10 0

10 3 T

ime,

s

Atomistic

Dislocations

Substructures

Grain/Phase

Macro-Interface

FEM/BEM

Celular

Automata

Dislocation

Dynamics

Molecular Dynamics

Tight

Binding

Ab-Initio Physical

Chemical

Biological Mechanical

Inverse Problems in

Multiscale Modelling

B. Inverse problems: Optimization

Identification

Optimization: minimization of a given objective function in macro

scale with respect to design variables in micro scale of the structure

Identification: evaluation of some geometrical or material parameters

of the structures in micro scale having measured information in

macro scale.

CIS Computational Intelligent System

Soft computing Hard computing

FEM (Finite Element Method)

BEM (Boundary Element Method)

MM (Meshless Methods)

MD (Molecular Dynamics)

Bio-inspired

Methods

AI

Ansys

Nastran

Marc

Mentat

Lammps

In-house software

Computational Intelligent System - interfaces

EA

AIS

PSO

Evolutionary Computing

Immune Computing

Swarm Computing

Multiobjective Computing

Computational Intelligent System

Optimization

Problems of

Multiscale

Modelling

Macro-Micro

Nano

Numerical homogenization

Numerical homogenization by using RVE

(Representative Volume Element)

Numerical homogenization - requirements

• Separation of scales

• Averaging theorem

• Hill’s condition (the equality of the averaged micro-scale energy density and the macro-scale energy density at the selected point of macro-structure corresponding to the RVE)

l and L are characteristic lengths of body in

macro/micro scales.

average macroscopic value

volume of RVE element

stress nad strain tensors

temperature gradient and heat fluxes

periodic boundary conditions

1l

L

1

RVE

RVE

RVE

d

RVE

ij ij ij ij

, ,i i i iT q T q

ij ij

,i iT q


• Hook’s law

• Fourier’s law

'

ij ijkl ijc

'

,i ij iq k T

• Tensor of effective elastic constants

• Tensor of effective thermal constants

11 12 13

21 22 23

31 32 33'

44

55

66

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

ij

c c c

c c c

c c cc

c

c

c

11

'

22

33

0 0

0 0

0 0

ij

k

k k

k


avg. - average

Macro-stresses

Homogenization

Macro-strains

Localization

BVP

BVP – Boundary Value Problem

Macro

Micro

RVE

Optimal design on macro-micro scales

min oDV

J

1 2, ,..., ,...i n

Ch

DV B cell x x x x

P

Constraints: min max

0, 1,2,..

, ,1,2,..

( , , ), 0,1,2,...

i i i

J m

x x x i n

J J u m

xi – design variables, play the role of geometrical, material

or topologcal parameters in the micro scale

where

DV=design vector

J0 – objective function described

in the macro scale

Meso scale:

Grains Micro scale: Single grain

Nano scale:

Molecular/atomic

level

Macro scale:

Structure

Illustration of optimization in multiscale approach

0 0( , , )J J u 1 2, ,..., ,...i nDV x x x x

Design variables

RVE

Material parameters

Shape parameters

Topology parameters

Evolutionary/immune/swarm optimization in multiscale

in macro scale in micro scale

RVE

DEA parameters: 2 subpopulations

20 chromosomes in each

Rank selection

Gasuss mutation

Simple crossover

g7,

g8

The best solution

in the 1st generation

The best solution

in the last generation

0 0 maxmin ,Ch

J where J u 1 2 3 4 5 6 7 8, , , , , , ,Ch g g g g g g g g

Optimization of Functionally Graded Materials in Multsicale Modelling

The function or composition changes gradually in the material

http://www.unl.edu/emhome/faculty/bobaru/project_shape_optim.htm

FGM in nature – clam shell

Bamboo

Functionally Graded Materials

The function or composition changes gradually in the material

Metal-ceramic FGMs

http://sbir.nasa.gov/SBIR/successes/ss/3-079text.html

Optimization of FGM parameters

macromodel Micromodel - RVE

Minimization of inclusions total volume

z

1

dA

Z

n

z A

f h

maxiu u

6 design parameters - diameters di


Constraint on maximum displacement value

Displacements map for the best solution (umax=4)


the resuts 1 - 0.187501 2 - 0.137236 3 - 0.123124 4 - 0.104760 5 - 0.143142 6 - 0.101725

FGM material for tooth implant

The simplified model of implant-bone systen with FGM material – optimization of porosity

Minimization of porosity p1 (mat1) and p2 (mat2)

Constraints on max eqivalent stress value in the bone area are imposed

Box constraints on prosity [0.0; 0.4]

igl

voidsi

ch

V

Vp

ppch

ppf

],[

min

21

21F

F

Macromodel

FEM MSC.Nastran

Micromodel

FMBEM model

RVE

Optimization of functionally graded materials in multiscale modelling

Distribution of equivalent stresses in the optimal design

Multicriteria Optimal Design of porous microstructures

1min d

u

def

ux

f u

Optimization functionals for termomechanical problems

• minimization of displacement on selected part of the boundary

2min d

q

def

qx

f q

• minimization of heat flux on selected part of the boundary

3max d

q

def

qx

f q

• maximization of heat flux on selected part of the boundary

4

d

maxd

por

RVE

pordef

xRVE

f

• maximization of the porosity of the microstructure

Numerical example

Boundary conditions

P0(total)=100N

T0=0°C

T1=100°C

Constraints (5 design variables)

Z1 – [0.53 – 0.92] Z2 – [0.09 – 0.45] Z3 – [0.09 – 0.45] Z4 – [0.08 – 0.47] Z5 – [0.09 – 0.45]

Macromodel

of aluminium plate 50x50x1 under thermomechanical loadings

RVE model of microstructure with void modeled using NURBS

Variants of multicriteria optimal design of materials

Variant 1

Variant 2

1min d

u

def

ux

f u

minimization of displacement on selected part of the boundary

2min d

q

def

qx

f q

minimization of heat flux on selected part of the boundary

3max d

q

def

qx

f q

• maximization of heat flux on selected part of the boundary

4

d

maxd

por

RVE

pordef

xRVE

f

• maximization of the porosity of the microstructure

Results of multicriteria optimization (variant 1)

1 d

u

def

uf u

2 d

q

def

qf q

Results of multicriteria optimization (variant 2)

3 d

q

def

qf q

4

d

d

por

RVE

pordef

RVE

f

Identification: macro-micro

Goal – find the FEM microscale model parameters on the base of experimental measurements in macro

Real structure

FEM model

What material properties for FEM gives the same results in sensor points as in real structure ?

0

0

1 1

min

ˆˆ

DV

m m

i i i i

j j

J

where

J a u u b

IDENTIFICATION

1 2, ,..., ,...i nDV x x x x

xi – design variables, play the role of material or geometrical

parameters in the micro scale

min max , ,1,2,..i i ix x x i n

ˆ ,

ˆ

i i

i i

u and u computed and measured displacements

and computed and measured strains

http://www.ucc.ie/bluehist/CorePages/Bone/Bone.htm

Identification of material parameters of a bone tissue

The femur bone is build from trabecular and compact bone. The identification of material properties of single trabeculae.

K.Tsubota, T. Adachi, S. Nishiumi , Y. Tomita, ATEM'03, JSME-MMD, 2003

G. M. Kurtzman, 2006

femur trabecular bone RVE

single trabeculae

Identification can be performed in two stages: I) Identification of anisotropic homogenized material properties of RVE on the basis of measurements for femur II) Identification of isotropic material properties of trabeculae on the basis of homogenized RVE anisotropic material properties

Problem formulation for II stage (RVE)

Design parameters:

chi=[Young modulus E, Poisson ratio ] material properties of single trabeculae

chi=[g1, g2]

The objective function: where: - RVE homogenized material properties from macromodel - computed homogenized RVE material properites n - number of coefficients (9) The homogenized anisotropic material properties for RVE: a[i] ={E11 E22 E33 E12 E13 E23 E44 E55 E66}

The constraints on design parameters values:

1

ˆmin ( )

n

i i

i

F a a

chˆia

ia

min maxi i ig g g

11 12 13

22 23

33

44

55

66

0 0 0

0 0 0

0 0 0

0 0

. 0

x x

x x

z z

xy xy

yz yz

zx zx

E E E

E E

E

E

sym E

E

The FEM model for RVE created on the basis of microCT. The bone sample was taken form the femur.

~70,000 DOF

The fitness function value for the best chromosome in subpopulations

iteration

fitness

Actual Found

E [MPa] 3300.0 3305.5

0.330 0.329

Creation of new graphene-like

materials by means of the hybrid

parallel evolutionary algorithm

Nano level optimization of graphene allotropes

by means of a hybrid parallel evolutionary algorithm

Journal Computational Materials Science (in press)

by A.Mrozek, W. Kuś, T. Burczyński

Carbon allotropes

-diamond

- graphite/graphene

-nanotubes/nanorings

etc.

- fullerenes

- amorphous state

Graphene-like 2D materials / hybridization of carbon atoms

acetylenic linkages,

nanowires benzene rings, base of the graphite/

graphene honeycomb lattice

Optimal searching for the new atomic structures:

• proper interaction model

• optimization’s algorithm

• stable configurations of atoms correspond to the minima

on the potential energy surface

Bond Order (BO) potentials for molecular dynamics simulations of carbon/hydrocarbons

• LCBOP (I + II) Long range Carbon Bond Order Potential • REBO (Reactive Empirical Bond Order)

• AIREBO (Adaptive Intermolecular REBO) a variant of the REBO with additional torsion and (Lennard-Jones-like) long-range terms* • ReaxFF (Reactive Force Fields) with equilibration of atomic charge All of them can handle various hybridization states of carbon atoms

*used in this work S.J. Stuart, A.B. Tutein, J.A. Harrison, A reactive potential for hydrocarbons with intermolecular interactions, The Journal of Chemical Physics, 112(14), 2000, pp. 6472–6486

Evolutionary optimization of atomic structure - minimization of the potential energy

• Fitness function - the total potential energy of the considered atomic cluster (sum

over all atomic interactions)

• Design variables/genes: the real-valued Cartesian

Coordinates of each atom in the considered cluster

• Constrains: all atoms can move freely in the triclinic

or rectangular unit cell with imposed periodic boundary

conditions

• Neighborhood-dependent behavior of carbon atoms (i.e. hybridization’s states,

bond’s lengths and angles) is handled by AIREBO potential and conjugated

gradient-based molecular statics solver

• Periodicity of the lattice is guaranteed by the molecular static solver

, , ,

REBO LJ TORSION

ij ij kijl

i j i k i j l i j k

FF E E E

START - generation of initial

population

(creation of randomly-generated

positions of atoms)

minimization of potential energy using

molecular statics/gradient method

fitness function evaluation

selection

modification of genes

using evolutionary operators

Stop condition

?

Proposed hybrid evolutionary-molecular computational system

END

N Y

- LAMMPS

- E.A.

Parallel hybrid gradient/evolutionary algorithm (small, 2D problems)

Multiple instances of LAMMPS

- LAMMPS

- E.A.

Parallel hybrid gradient/evolutionary algorithm (large, 3D problems)

Proposed hybrid evolutionary/gradient algorithm

• modular structure (each part can be replaced with appropriate equivalent,

e.g.

- AIREBO -> ReaxFF (Reactive Force Fields)

- EA -> AIS etc.

• ready for 3D optimization (and not only carbon atoms…)

Validation & Results obtained using prototype version of the algorithm

Dimensions:12x10 Å triclinic unit cell, 25 atoms

4 threads

100 individuals

124800 FF evaluations

10% mutation & crossover

0 200 400 600 800 1000 1200 1400 1600-156

-154

-152

-150

-148

-146

-144

-142

-140

-138

Example of the progress of optimization

Supergraphene, as presented in: Enyashin A.N., Ivanovskii A.L., Graphene allotropes, Physica Status Solidi, 248, 8, 2011, pp. 1879-1883

bond’s lengths computed using classical MD and AIREBO potential: Mrozek A., Burczynski T.,

Examination of mechanical properties of graphene allotropes by means of computer simulation,

CAMES, 20, 4, 2013, pp. 309-323.

Supergraphene found by g-optim algorithm:

(in 34th generation)

Triclinic unit cell: 10x6Å, 8 atoms

(finally relaxed to 10.65x6.08Å)

0 10 20 30 40 50 60 70 80 90 100-50.5

-50

-49.5

-49

-48.5

-48

-47.5

-47

-46.5

-46

-45.5

1.32Å

1.38Å

potential energy (eV) vs. generation

Graphyne, as presented in: Enyashin A.N., Ivanovskii A.L., Graphene allotropes, Physica Status Solidi, 248, 8, 2011, pp. 1879-1883

bond’s lengths computed using classical MD and AIREBO potential: Mrozek A., Burczynski T.,

Examination of mechanical properties of graphene allotropes by means of computer simulation,

CAMES, 20, 4, 2013, pp. 309-323

Graphyne found by g-optim algorithm:


Triclinic unit cell: c.a. 10x6Å, 12 atoms

(finally relaxed to 10.2x5.9Å)

0 10 20 30 40 50 60-81

-80

-79

-78

-77

-76

-75

-74

-73

-72


Structure found by g-optim algorithm:

(in 223 generation)

Orthogonal unit cell: 4x7Å, 8 atoms

2 threads

100 individuals

22300 FF evaluations


0 50 100 150 200 250 300 350 400 450 500-52.2

-52.1

-52

-51.9

-51.8

-51.7


1. Example of the „new” graphene-like materials X

-5.94eV

-6.31eV

-8.01eV

-6.31eV

-5.94eV

-8.01eV

-5.94eV

a) b)

Unit cell close-up X

2. Example of the „new” graphene-like materials Y

Structure found by g-optim algorithm:


Orthogonal unit cell: 6x4Å, 8 atoms

2 threads

100 individuals

5500 FF evaluations


0 20 40 60 80 100 120 140 160 180 200-54.5

-54

-53.5

-53

-52.5

-52

-51.5

-51

-50.5

-50


-6.81eV

-6.87eV

-6.81eV

-6.59eV

-6.59eV

a) b)

Unit cell close-up Y

equilibrate the investigated „nanospecimen”

at the desired temperature

apply a certain, finite deformation the structure (dε, dγ)

equilibrate the structure

compute all the necessary

time-averaged values (displacements/deformations,

components of microstress tensor)

Examination of the mechanical properties (tensile / shear tests at the finite temperature)

1 1

2

N N

i i i ij ij

i j i

m

σ v v r f

Tensile test – strain(%) - stress(N/m) curve

Young moduli (0-5%) vertical axis: 176 N/m horizontal axis: 183,4 N/m

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

5

10

15

20

25

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

5

10

15

20

25

30

Mechanical Properties of X

Tensile test – strain(%) - stress(N/m) curve

Young moduli (0-5%) horizontal axis: 226 N/m vertical axis: 280 N/m

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

5

10

15

20

25

0 0.05 0.1 0.15 0.2 0.250

5

10

15

20

25

30

35

Mechanical Properties of Y

Concluding remarks

• Two-scale macro-micro materials design needs special analytical and

computational techniques and tools.

• Coupled soft and hard computing techniques based on Computational

Intelligent System (CIS) ensure the great probability of finding global

solutions. CIS has the great flexibility.

• Effective CIS is based on the parallel computing and grid environment.

• Optimal material and geometrical parameters on the micro-scale ensure the

extremum for an objective function in the macro-scale.

• Using CSI it is possible to create material on the nano-scale ……

Concluding remarks cont.

• Proposed algorithm gives possibility of finding new flat carbon networks with unique properties

• Newly created structures are „physically” correct: form proper basic elements (benzene rings,

triads, acetylenic groups etc.), without alone atoms or unconnected branches etc.

• The AIREBO potential seems to be reasonable choice for modeling presented flat carbon

structures (except long-range interactions), where time-consuming ab-inito methods are not

suitable

• Proposed optimization algorithm is easy to parallelize, since the most time-consuming step is

molecular statics and FF evaluation

• It is possible to create a new (even 3D) material with predefined density and

properties using this methodology.

^Institute of Computer Science, Cracow University of Technology, Poland

Thank you for your attention!

Documents

COMPUTATIONAL INTELLIGENCE IN MULTISCALE AND … · Numerical homogenization - requirements • Separation of scales • Averaging theorem • Hill’s condition (the equality of