coupling heterogeneous models with non-matching meshes by localized lagrange multipliers modeling...

Coupling Heterogeneous Models with Non-matching Meshes by Localized Lagrange Multipliers

Modeling for Matching Meshes with Existing Staggered Methods and Silent Boundaries

Holly Lewis & Mike Ross

Center for Aerospace Structures

University of Colorado, Boulder

20 April 2004

Topics of Discussion Refresh Memory PML Lagrange Multipliers for Dam-

Sandstone Interface Future Work Lagrange Multipliers for Fluid-

Structure Interface and associate issues

A Picture is Worth 1,000 WordsMulti-physic system Modular Systems

Connected by Localized Interaction Technique (Black Lines)

Plan of Attack

Generate a benchmark model Use current available methods Matching meshes

Generate a model with localized frames Maintain matching meshes

Generate a model with localized frames With nonmatching meshes

Benchmark Model Refresher

Fluid (Spectral Elements)

Soil (Brick Elements)

•Output: Displacements of Dam & Cavitation Region

•Assume: Plane Strain (constraints reduce DOF)

•Only looking at seismic excitation in the x-direction

•Linear elastic brick elements

Dam (Brick Elements)

Silent Boundary

Silent Boundary

A Quick Note on Perfectly Matched Layers

The main concept is to surround the computational domain at the infinite media boundary with a highly absorbing boundary layer.

Outgoing waves are attenuated.

Wave amplitude

This boundary layer can be made of the same finite elements. Formulation of the matrices are the same method for both computational domain and the

boundary layer There are just different properties

A Quick Note on Perfectly Matched Layers (PML) Going from the frequency domain

to the time domain is a real pain!!! Can be done see “Perfectly

Matched Layers for Transient Elastodynamics of Unbounded Domains.” U. Basu and A. Chopra

Localized Frame Concept

Frames are connected to adjacent partitions by force/flux fields Mathematically: Lagrange

multipliers “gluing” the state variables of the partition models to that of the frame.

Lagrange multipliers at the frame are related by interface constraints and obey Newton’s Third Law.

Localized Lagrange Multipliers

Analysis by three modules Sequance:

Earthquake hits -> structural displacements Interface Solver Fluid & Structural Solver in Parallel

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface

uB

uD

uSD

s

Dam

Sandstone

Variational Principles and Lagrange Multipliers

Lagrange Method to derive the equilibrium equations of a system of constrained rigid bodies in Newtonian Mechanics Formulation.

1- Treat the problem as if all bodies are entirely free and formulate the virtual work by summing up the contributions of each free body.

2- Identify constraint equations and multiply each by an indeterminate coefficient. Then take the variation and add to the virtual work of the free bodies to yield the total virtual work of the system.

3- The sum of all terms which are multiplied by the same variation are equated to zero. These equations will provide all the conditions necessary for equilibrium.


1- Subsystem Energy Expressions (Variational Formulation)

)(:

)(:

ssssssssS

DDDDDDDDD

SandstoneSystem

DamSystem

fuMuCuKu

fuMuCuKu

2- Identify Interface Constraints

)()()()(

)()(

2211

21

BsSBsSBDDBDDB

BsSBDDB

uuBuuBuuBuuB

uuBuuB

3- Total Virtual Work = 0 ( Stationary)

0δΠδΠ δΠδΠ DSB

)()()(

)()(

21

21

DSBBST

SBDT

D

ssssssssSDDDDDDDDD

uuuBuuB

BfuMuCuKuBfuMuCuKu


(25) Eq.

(24) Eq.

(23) Eq.

(22) Eq.

(21) Eq.

f

f

u

u

u

II

IB

IB

BMCK

BMCK

0

0

0

000

000

000

000

000

21

22

11

22

2

12

2

S

D

B

s

D

S

D

T

T

SSS

DDD

dt

d

dt

ddt

d

dt

d

24240

2424

24264

1

0

0

1000

0

0

010

0001

IB

33

241485

241584

2

0

0100

0010

0001

0000

0

0

000

0000

IB


Midpoint Integration Rule:

nnn

nnn

nnnn

t

tt

uuu

uuu

uuuu

2/11

2/12/1

2/12

2/1

2

)(2

1

4

)()(

2

1

Rearrange in terms of velocity and acceleration at half time step

nnnn

nnn

ttt

tt

uuuu

uuu

2

)(

4

)(

4

22

22/1

22/1

2/12/1


Examine Eqn (21) with midpoint rule:

nnD

nD

nD

nDDDD

nD

nD

nD

nDD

nDD

nDD

ttt

tt

uuMuC

BfMCKu

onaccelerati and velocityfor RuleMidpoint Insert

fBuMuCuK

242

]42

[

2

2/11

2/112

2/1

2/12/11

2/12/12/1

Apply Same Concept to Eqn (22) to get a displacement of the sandstone at the half time step


Look at Eqn (23) with the above eqns for the displacements:

nD

nDD

nBD

nDDDD

T

nB

nDDDD

T

nB

nD

T

tt

ttt

tt

uuM

uCfMCKB

uBMCKB

uuB

24

242

42

2

2/1

1

21

2/12/11

1

21

2/12/11

Apply the same concept to Eqn (24). Then use Eqn (25) and the two above equations to input into Matrix form


00

420

042

21

21

21

24242424

24242

1

22

24241

1

21

S

D

nB

nS

nD

SSST

DDDT

tt

tt

g

g

uII

IBMCKB

IBMCKB

nD

nDD

nDD

nDDDD

TD ttttt

uuMuCfMCKBg 242422

21

1

21

nS

nSS

nSS

nSSSS

TS ttttt

uuMuCfMCKBg 242422

21

1

22


Basic Concept:

Step 1: Solve for ’s using previous three equations.

Step 2: In parallel solve for the displacements at the next time step using:

nD

nDD

nDD

nD

nDDDD

nD

ttt

tt

uuMuC

BfMCKu

242

]42

[

2

2/11

2/112

2/1

Step 3: Update the variables and generate the necessary time step-dependent vectors.

nD

nD

nD uuu 2/11 2


Dam Crest Displacement with Lagrange Multipliers

Dam Crest Displacement with Monolithic Model

Set Sail for the Future Find bug in the Dam-Sandstone Interface Develop structure-fluid interaction via localized

interfaces with nonmatching meshes. Develop structure-soil interaction via localized

interfaces spanning a range of soil media. Develop a localized interface for cavitating fluid

and linear fluid. Develop rules for multiplier and connector frame

discretization. Implement and assess the effect of dynamic

model reduction techniques.

Fluid Structure Interaction with Lagrange Multipliers

Same Concept as with Dam-Sandstone Interface

Separate the two systems Apply interface Constraint

)()( BffBDDB uuuu

However, remember the displacement of the fluid is expressed in terms of the gradient of a scalar function

fu


Can the interface constraint be written?

)()()()(

)()(

BfBfBDDBDBB

BfBDDB

uuuuuu

uuu

What happens with the gradient?

Fluid Structure Interaction with Lagrange Multipliers Another concern is the variational

formulation of the fluid system. If you remember we ended up with

fluid equations of the form:

bHQbHQs 22 cc Can the variation of the fluid be written?

bQH 22 ccfluid


b

f

u

u

II

IB

IB

BQH

BMCK

0

0

0

000

000

000

000

000

2

21

22

11

22

22

12

2

c

f

dt

dc

dt

d

dt

dD

B

s

D

D

T

T

DDD

????????????????

Acknowledgments

NSF Grant CMS 0219422 Professor Felippa & Professor Park Mike Sprague (Professor Geer’s Ph.D

student, now a post-doc in APPM) CAS (Center for Aerospace Structures) CU

coupling heterogeneous models with non-matching meshes by localized lagrange multipliers modeling...

Documents

total virtual work

interface constraints

computational domain

free bodies

boundary layerthere

frequency domain

time domain

equilibrium equations