combined domain decomposition and model order reduction … · 2015. 11. 19. · combined domain...

Combined Domain Decomposition and Model Order Reduction methods for the solution of multi-physics

and non-linear problems in MEMS

Martino Dossi, Stefano Mariani, Alberto Corigliano+ Federica Confalonieri, Mauro Terraneo

Department of Civil and Environmental Engineering

www.mems.polimi.it

MOR 4 MEMS17 & 18 November 2015

Karlsruhe

http://www.mems.polimi.it/

A. Corigliano - MOR 4 MEMS – 17.11.15

2

“9-axis module”: 3.5 x 3 x 1 mm fully packaged

Gyroscope + Accelerometer + Magnetometer

1 Euro cent diameter 15 mm = 177 mm 2

> 16 x 9 = 144 axis !

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

9-axis MEMS

3.5 x 3 = 10.5 mm 2

Microsystems: a commercial product


3Microsystems: modelling and simulation

Design issues- Structural dynamics- Fluid-structure interaction- Electro-mechanical problem- Electro-magneto-mechanical problem- Electro-thermo-mechanical problem- …

Reliability issues- Fracture- Fatigue- Stiction- Temperature drift- …

A

A

A

A

A


4

Macro-scale

mm

Meso-scale

µm

Micro-scale

Sub-µm

Package

Die

Sensor

Polysilicon

Multi-scale problems Multi-physics problems

• Fluid-structure interaction• Electro-mechanical coupling• Electro-Thermo-mechanical coupling• Magneto-mechanical coupling• Moisture diffusion• Surface effects and dispersion forces• …

Microsystems: modelling and simulation

Non-linear problems• Damping phenomena• …

• Non-linear coupling• Fracture-fatigue• Elasto-plasticity


5

Accidental drop

Adhesion-stiction

Fracture processes

Thermo-compressive bonding

Microsystems: large scale computing


6Domain Decomposition and Model Order Reduction for coupled problems

𝑟𝑟 ≪ 𝑁𝑁N

r

Domain decomposition and Model Order reduction techniques applied to thesimulation of the electro-mechanical problem in MEMS

F. Confalonieri, A. Corigliano, M. Dossi, M. Gornati. A domain decomposition technique applied to the solution of the coupled electro-mechanical problem. Int. J. Numer. Meth. Eng. 93 (2) , pp. 137-159, (2013).

A. Corigliano, M. Dossi, S. Mariani. Domain decomposition and model order reduction methods applied to the simulation of multi-physicsproblems in MEMS. Computers and Structures, 122, 113-127, (2013).


7

• the voltage difference (V) between the fixed anddeformable electrodes cause the charges toaccumulate on the surfaces between the electricaland the mechanical domains

• electrostatic forces on the mechanical surface causethe deformation of the structure

• the deformation of the structure causes the shapevariation of the electrostatic domain

• non-linear coupling between the mechanical and theelectrostatic fields

Reference problem: electro-mechanical coupling


8

UΓ,Φ )

Effect of the electrostatic domain on the mechanical part: Coulomb forces

Effect of the mechanical domain on the electrical part: change of the shape of the electrical domain

displacements of the nodes of the mechanical domaindisplacements at the interface Γ

Electric boundary condition

Mechanical boundary condition

𝐔𝐔Γ

𝐔𝐔

Semi-discretized electro-mechanical coupled problem

A domain decomposition technique applied to multi-physics problemsElectro-mechanical problem


9

Domain decomposition approach:each physical field can be viewed as a sub-domain

Electrostatic domain:Mechanical domain:

Free problems:Each physics is

solved indipendently

Link problems:The physics are

re-coupled

A domain decomposition technique applied to multi-physics problemsSimple Domain decomposition technique (SD)


10

Decomposition of the solution

Coupling term: electrostatic forces

Free mechanical problem

Link mechanical problem

Mechanical problem

+

Mechanical part

[N. Mahjoubi, A. Gravouil, A. Combescure: Coupling subdomains with heterogeneous time integrators and incompatible time steps, 2009].

𝐌𝐌�̈�𝐔 + 𝐊𝐊𝑚𝑚𝐔𝐔 = 𝐅𝐅𝑒𝑒𝑒𝑒𝑒𝑒 + 𝑭𝑭𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝐔𝐔Γ,𝚽𝚽

𝐔𝐔 = 𝐔𝐔𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 + 𝐔𝐔𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙

𝐌𝐌�̈�𝐔𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 + 𝐊𝐊𝑚𝑚𝐔𝐔𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 = 𝐅𝐅𝑒𝑒𝑒𝑒𝑒𝑒

𝐌𝐌�̈�𝐔𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 + 𝐊𝐊𝑚𝑚𝐔𝐔𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 = 𝐅𝐅𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝐔𝐔Γ,𝚽𝚽



11

Decomposition of the solution

Free electric problem

Link electric problem

Electric problem

+

Electric part“0” : undeformed configuration of electric domain

𝐊𝐊𝜙𝜙0 + Δ𝐊𝐊𝜙𝜙 𝚽𝚽𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 + 𝚽𝚽𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 = 𝐐𝐐0 + Δ𝐐𝐐

𝚽𝚽 = 𝚽𝚽𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 + 𝚽𝚽𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙

𝐊𝐊𝜙𝜙0 + Δ𝐊𝐊𝜙𝜙 𝚽𝚽𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 + 𝐊𝐊𝜙𝜙

0𝚽𝚽𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 = 𝐐𝐐0 + Δ𝐐𝐐 − Δ𝐊𝐊𝜙𝜙𝚽𝚽𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒

𝐊𝐊𝜙𝜙0𝚽𝚽𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 = 𝐐𝐐0

𝐊𝐊𝜙𝜙0 + Δ𝐊𝐊𝜙𝜙 𝚽𝚽𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 = Δ𝐐𝐐 − Δ𝐊𝐊𝜙𝜙𝚽𝚽𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒


𝐊𝐊𝜙𝜙𝚽𝚽 = 𝐐𝐐


12

To drastically reduce the computational burden of the mechanical system in the s-th sub-domain (𝑠𝑠 = 1,2,⋯ , 𝑛𝑛𝑠𝑠𝑠𝑠𝑠𝑠−𝑑𝑑𝑑𝑑𝑚𝑚𝑑𝑑𝑙𝑙𝑙𝑙𝑠𝑠) of the coupled electro-mechanical problem:Model Order Reduction via Proper Orthogonal Decomposition (POD) technique

𝐔𝐔𝑠𝑠 = �𝑙𝑙=1

𝑁𝑁𝑠𝑠

𝜶𝜶𝑙𝑙𝑠𝑠 𝚵𝚵𝑙𝑙𝑠𝑠

Mechanical displacement field in the sub-domain s:

𝐔𝐔𝑓𝑓𝑠𝑠 = �𝑙𝑙=1

𝑓𝑓𝑠𝑠

𝜶𝜶𝑙𝑙𝑠𝑠 𝚵𝚵𝑙𝑙𝑠𝑠 = 𝐀𝐀𝑓𝑓𝑠𝑠𝚵𝚵𝑓𝑓𝑠𝑠

Linear combination of orthonormal basis

Reduced representation of the system

𝑟𝑟𝑠𝑠 ≪ 𝑁𝑁𝑠𝑠

𝑚𝑚𝑚𝑚𝑛𝑛 𝐔𝐔𝑓𝑓𝑠𝑠 − 𝐔𝐔𝑠𝑠

Minimization of the discrepancy between the full and the reduced representation

Reduced mechanical system

N

r

𝐌𝐌𝑓𝑓𝑠𝑠�̈�𝚵𝑓𝑓𝑠𝑠 + 𝐊𝐊m𝑓𝑓𝑠𝑠𝚵𝚵rs = 𝐅𝐅𝑓𝑓𝑠𝑠

𝑒𝑒𝑒𝑒𝑒𝑒 + 𝐅𝐅𝑓𝑓𝑠𝑠𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

A MOR technique coupled with DD method applied to multi-physics problemsModel Order Reduction via POD in the s-th sub-domain


13

Construction of the reduced basis 𝑨𝑨𝑓𝑓𝑠𝑠 Snapshot , training stage and SVD

Singular Value Decomposition: factorization of matrix 𝐒𝐒𝑠𝑠 which gives the POM

𝐒𝐒𝑠𝑠 = 𝐋𝐋s𝚲𝚲𝑠𝑠𝑹𝑹𝑠𝑠T

∑𝑙𝑙=1𝒓𝒓𝒔𝒔 Λ𝑠𝑠𝑖𝑖𝑖𝑖

2

∑𝑙𝑙=1𝑁𝑁 Λ𝑠𝑠𝑖𝑖𝑖𝑖2 ≥ 𝑘𝑘𝑠𝑠

number of POMs to furnish as much insight as possible into the original multi-physics system

𝑘𝑘𝑠𝑠 defines the energy of the snapshots captured by the 𝒓𝒓𝒔𝒔 first POD basis vectors

𝚲𝚲𝑠𝑠 pseudo diagonal matrix of singular values𝐋𝐋s, 𝐑𝐑s orthogonal matrices that gather the so-called left and right singular vectors𝐋𝐋s collects the Proper Orthogonal modes (POM)

Snapshot: response of the system to the actual excitation at a certain time instant

Training stage: collection of a certain amount of snapshots in matrix 𝐒𝐒𝑠𝑠

𝐒𝐒𝑠𝑠 = 𝐔𝐔𝑠𝑠1 𝐔𝐔s2 ⋯ 𝐔𝐔s𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠



14

Singular Value Decomposition updated

This approach introduces on the fly updates of POMs as a new snapshot is collected

Updated singular value matrices

During training stage:

Convergence conditions:

Number of POMs in the sub-domain is not increased. Relevant oriented energy content in the sub-domain is not

increased.

𝐒𝐒𝑠𝑠 𝐔𝐔𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1 = �̂�𝐋𝑠𝑠�𝚲𝚲𝑠𝑠�𝐑𝐑𝑠𝑠T𝐔𝐔s𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1

new snapshots

SVD updated • Speedups the computation• End of the training stage is not decided a priori

𝐒𝐒𝑠𝑠 = 𝐋𝐋s𝚲𝚲𝑠𝑠𝑹𝑹𝑠𝑠T

�̂�𝐋𝑠𝑠�𝚲𝚲𝑠𝑠�𝐑𝐑𝑠𝑠



15

Free mechanical problem

Link mechanical problem

Interface problem

SD with sub-decomposition of the mechanical domain with POD (SD-DD-POD)

�𝐌𝐌𝑠𝑠𝑓𝑓�̈�𝚵𝑠𝑠𝒓𝒓𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 = �𝐅𝐅𝑠𝑠𝑟𝑟

�𝐌𝐌𝑠𝑠𝑓𝑓�̈�𝚵𝑠𝑠𝑟𝑟𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 = 𝚲𝚲𝒔𝒔𝑓𝑓

�𝐌𝐌𝑠𝑠𝑓𝑓 = 𝐀𝐀𝑠𝑠𝒓𝒓T �𝐌𝐌𝑠𝑠𝐀𝐀𝑠𝑠𝒓𝒓

�𝐅𝐅𝑠𝑠𝑓𝑓 = 𝐀𝐀𝑠𝑠𝒓𝒓T �𝐅𝐅𝑠𝑠

�𝐔𝐔𝑠𝑠 Γ= �𝐀𝐀𝑠𝑠𝒓𝒓 Γ

𝚵𝚵𝑠𝑠𝑟𝑟 Reconstruction of the solution on the interface

The interface problem is not reduced.

In each mechanical subdomain the POD is applied to build a reduced model, 𝐀𝐀𝒔𝒔𝒓𝒓.

Mechanical reduced problem (sub-decomposed):

�̈�𝐔s = 𝐀𝐀𝑠𝑠𝒓𝒓�̈�𝚵𝑠𝑠𝑟𝑟 = 𝐀𝐀𝑠𝑠𝒓𝒓 �̈�𝚵𝑠𝑠𝒓𝒓𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 + �̈�𝚵𝑠𝑠𝑟𝑟

𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙

�̇�𝐔𝑠𝑠 = 𝐀𝐀𝑠𝑠𝒓𝒓�̇�𝚵𝑠𝑠𝑟𝑟 = 𝐀𝐀𝑠𝑠𝒓𝒓 �̇�𝚵𝑠𝑠𝒓𝒓𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 + �̇�𝚵𝑠𝑠𝒓𝒓


𝐔𝐔s = 𝐀𝐀𝑠𝑠𝒓𝒓𝚵𝚵𝑠𝑠𝑟𝑟 = 𝐀𝐀𝑠𝑠𝒓𝒓 �̈�𝚵𝑠𝑠𝒓𝒓𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 + �̈�𝚵𝑠𝑠𝒓𝒓


Reconstruction of the solution.

A. Corigliano, M. Dossi, S. Mariani. Recent advances in computational methods for microsystems. Advanced materials research, 745, 13-25 (2013).

𝚲𝚲𝑠𝑠𝑓𝑓 = 𝐀𝐀𝑠𝑠𝒓𝒓T 𝐂𝐂𝑠𝑠T𝚲𝚲

𝐂𝐂𝑠𝑠T𝚲𝚲 = 𝐊𝐊𝑒𝑒𝑒𝑒 �𝐔𝐔𝑠𝑠 Γ→ 𝔸𝔸𝑒𝑒𝑒𝑒=1

𝑙𝑙𝑒𝑒𝑒𝑒𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒓𝒓𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝐇𝐇𝑒𝑒𝑒𝑒𝚲𝚲 𝑒𝑒𝑒𝑒 = 𝐊𝐊𝑒𝑒𝑒𝑒 𝐂𝐂2𝐔𝐔2

𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒− 𝐂𝐂1𝐔𝐔1

𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒

𝑒𝑒𝑒𝑒


a a


16Numerical examplesDoubly clamped beam - Constant potential

Newmark parameters𝛽𝛽 = 1

4; 𝛾𝛾 = 1

2;

Dynamic parameters𝑡𝑡𝑒𝑒𝑑𝑑𝑒𝑒 = 5 � 10−6; Δ𝑡𝑡 = 10−9;



Vertical displacement of central node

time to store snapshots


19Numerical examplesPlane resonator

[Kaajakari V, Mattila T, Oja A, Kiihamäki J, Seppä H. Square-extensional mode single-crystal silicon micromechanical resonator for low-phase-noise oscillator applications. IEEE Electr Device].



Space discretization (a quarter of whole structure)

sd. Mech. 1

The anchor is part of 1st sub-domainsd. Elec. 1

sd. Elec. 2

sd. Mech. 2



Vertical displacement of node A (with decomposition of the mechanical part)


23Domain Decomposition and Model Order Reduction for coupled problems


r

Domain decomposition and Model Order reduction techniques applied to thesimulation of the thermo-elastic problem in MEMS

F. Confalonieri, M. Terraneo, A. Corigliano. Domain decomposition and model order reduction approaches applied to the solution of fullycoupled thermo-mechanical problems in vibrating microsystems To be submitted (2015).


24

Semi-discretized thermo-mechanical coupled problem

A domain decomposition technique applied to multi-physics problemsThermo-mechanical problem

Overlapped mechanical and thermal domains

DD strategy for the solution of the coupled problem

MOR via POD for mechanical and thermal problems


25

Analytical solution(Zener 1938)

Quality factor Computational time Gain

Staggered method 2359 3046 s

MOR 2344 548 s -82 %

Numerical examplesZener’s cantilever beam: free vibration after imposed displacement


26

0 0.05 0.1 0.15 0.2 0.25-1

0

1

2

3

4 x 10-9

Time [s]

Dis

plac

emen

t [µm

]

MORStaggered

Numerical examplesZener’s cantilever beam: free vibration after imposed displacement

No mechanical damping, only thermo-elastic damping


27Numerical examplesDoubly clamped beam: assigned “step function” distributed loading

Point P

𝑞𝑞

𝑡𝑡

𝑞𝑞


28

Displacement Uy

MOR-POD Staggered

Temperature

Numerical examplesDoubly clamped beam: assigned “step function” distributed loading


29

0 20 40 60 80 100 120-50

-40

-30

-20

-10

0

Time (s)

Dis

plac

emen

t (µ m

)

MORStaggered

Computing time Gain

Staggered 76,18

MOR - POD 14,50 -80,97 %

Numerical examplesDoubly clamped beam: assigned “step function” distributed loading


30

Computing time Gain Quality factor

Staggered 106,06 4086677

MOR - POD 25,67 75,79 % 4086499

Displacement (norm) Temperature

Numerical examplesPlane resonator: free vibration after imposed displacement


31Domain Decomposition and Model Order Reduction for irreversible non-linear problems


r

Domain decomposition and Model Order reduction techniques applied to thesimulation of elasto-plastic solid and structural dynamics

elastic sd

plastic sd

A.Corigliano, M. Dossi, S. Mariani. Combined domain decomposition and model order reduction methods for the solution of coupled and non-linear problems. WCCM 11, (2014).

A. Corigliano, M. Dossi, S. Mariani. Model order reduction and domain decomposition strategies for the solution of the dynamic elasto-plastic structural problem Comp. Meth. Appl. Mech. Engng, 290, 127-155, (2015).


32A MOR technique coupled with DD method applied to elasto-plastic problemsModel Order Reduction via POD

elastic sd

plastic sd

Reduced analysisDD-POD algorithm

Full analysisPlastic code (explicit)

�𝐌𝐌𝑠𝑠𝑓𝑓�̈�𝚵𝑠𝑠𝒓𝒓𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 = �𝐅𝐅𝑠𝑠𝑟𝑟

�𝐌𝐌𝑠𝑠𝑓𝑓�̈�𝚵𝑠𝑠𝑟𝑟𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 = 𝚲𝚲𝒔𝒔𝑓𝑓

𝚲𝚲 → 𝔸𝔸𝑒𝑒𝑒𝑒=1𝑙𝑙𝑒𝑒𝑒𝑒𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒓𝒓𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝐇𝐇𝑒𝑒𝑒𝑒𝚲𝚲 𝑒𝑒𝑒𝑒 = 𝐊𝐊𝑒𝑒𝑒𝑒 𝐂𝐂2𝐔𝐔2𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒

− 𝐂𝐂1𝐔𝐔1𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒

𝑒𝑒𝑒𝑒

𝐌𝐌𝑠𝑠(𝑗𝑗+1)�̈�𝐔𝒔𝒔(𝑗𝑗+1)𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒 = �𝐅𝐅𝑠𝑠(𝑗𝑗+1) − 𝐅𝐅𝑠𝑠 𝑗𝑗+1

𝑙𝑙𝑙𝑙𝑒𝑒 (𝐔𝐔𝑠𝑠𝑗𝑗)

𝐌𝐌𝑠𝑠(𝑗𝑗+1)�̈�𝐔𝒔𝒔(𝑗𝑗+1)𝑒𝑒𝑙𝑙𝑙𝑙𝑙𝑙 = 𝐂𝐂𝑠𝑠T𝚲𝚲𝑠𝑠(𝑗𝑗+1)𝐇𝐇𝑒𝑒𝑒𝑒 = 𝐈𝐈 + �

𝑠𝑠

𝑠𝑠𝑖𝑖𝑖𝑖𝑠𝑠

𝛽𝛽𝑠𝑠Δt2𝐊𝐊𝑒𝑒𝑒𝑒 𝐂𝐂2 �𝐌𝐌𝑠𝑠𝐂𝐂𝑠𝑠T 𝒊𝒊𝒆𝒆

𝚲𝚲 → 𝔸𝔸𝑒𝑒𝑒𝑒=1𝑙𝑙𝑒𝑒𝑒𝑒𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒓𝒓𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝐇𝐇𝑒𝑒𝑒𝑒𝚲𝚲 𝑒𝑒𝑒𝑒 = 𝐊𝐊𝑒𝑒𝑒𝑒 𝐂𝐂2𝐔𝐔2𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒

− 𝐂𝐂1𝐔𝐔1𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒

𝑒𝑒𝑒𝑒

Time scales

A material behavior modelled through an elasto-perfectly plastic constitutive law with von Misesyield criterion.

When plastic deformations develop inside a sub-domain, this sub-domain switches to an explicit time scheme

Multi-scale in space and time


33Numerical examplesStructural 2D frame – elasto-plastic case

load P

Newmarkparameters𝛽𝛽 = 1

4, 𝛾𝛾 = 1

2;



Equivalent stress & plastic strain


36Domain Decomposition and Model Order Reduction for irreversible non-linear problems


r

Domain decomposition and Model Order reduction techniques applied to thesimulation of fracture processes in quasi-brittle polycrystalline materials

F. Confalonieri, A. Ghisi, G. Cocchetti, A. Corigliano. A domain decomposition approach for the simulation of fracture phenomena in polycrystalline microsystems. Comp. Meth. Appl. Mech. Engng,277, 180, 218 (2014).

M. Dossi, S. Mariani, A. Corigliano. Combined domain decomposition and model order reduction approaches for the simulation of fracture phenomena in polycrystalline microsystems. To be submitted (2015).


37A MOR technique coupled with DD method applied to fracture problemsModel Order Reduction via POD

( ) [ ]( )( ) ( ) ( )t t tcohes_intint ext MU + F U + F U = F

+ ICMaterial behaviour

Semi-discretized problem

Cohesive interface law for crack modelling

DD strategy

MOR via POD for elastic domains


38Numerical examplesDouble Cantilever Beam – fracture propagation

alumina


39Numerical examplesDouble Cantilever Beam – fracture propagation

σx


40Closing remarks

• New approach for the solution of multi-physics and irreversible non-linear problems in microsystems which combines DD and MOR by means of POD

• Application to the electro-mechanical and thermo-mechanical coupled problemsin MEMS

• Application to the elasto-plastic and cohesive-fracture structural dynamicproblems

Done

In progress

• Application of the proposed approach to other multi-physics problems

• Application of the proposed approach to other mechanically non linear problems

• Full exploitation of parallel computing


41

Thank you for your attention!

MIUR: funded project 2009XWLFKWMulti-scale, multi-physics and domain decompositionmethods in the mechanics of microsystems and nano andmicro-structured materials

Acknowledgements

Fondazione Cariplo: funded project 2009XWLFKW Safer Helmets

combined domain decomposition and model order reduction … · 2015. 11. 19. · combined domain...

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