Coupling Heterogeneous Models with Non-matching Meshes

Modeling for Matching Meshes with Existing Staggered Methods and Silent Boundaries

Mike Ross

Center for Aerospace Structures

University of Colorado, Boulder

24 February 2004

Topics of Discussion Introduction of the Research Topic Benefits Plan of Attack for this Research Progress (Benchmark Model) Direction for the Future

Why am I Here??? NSF Grant

PIs: Professor Felippa & Professor Park Crux: Model different physical systems with

non-matching meshes Use existing models of individual physical

systems. Develop interaction techniques for different

models with use of a localized connection frames and

localized multipliers.

Application Drivers MEMS Device (Mechanical Eng.)

Electrical-Mechanical-Thermal Interaction

Dam under seismic excitation (Civil Eng.) Fluid-Structure-Soil Interaction

Reason: NSF granting division is Civil &

Mechanical systems (CMS)

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

Connected by Localized Interaction Technique (Black Lines)

Benefits/Goals: Maintain software modularity through

an interface frame Provides great flexibility for different

FEM Codes, etc. Custom discretizations and solvers

for different physics Simplify the treatment of non-

matching meshes Efficient high fidelity simulation

Meaning: resources can be put on critical parts of the problem

User Friendly

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

Current Progress (Benchmark Model)

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)

Benchmark Concept:Staggered Method

with Non-Reflecting Boundary

Structure (seismic displacement) Fluid Volume Non-Reflecting

Boundary (NRB)

displacements

displacements

pressures

pressures

Structure Equations:

x = relative displacements Acceleration (a) is from Berkeley’s PEER

database Damping modeled with Rayleigh damping Solved with a Central Difference Method

(Explicit). Easy to implement. Also the physics can be represented with small time steps

Silent Boundary on Soil is modeled with a Viscous Damping Boundary Method

Soil and Dam are modeled monolithically with different properties (i.e. Young’s Modulus, etc.)

wfMaKxxCxM

Relative Displacement Concept

Summation of Forces:

Matrix Form:

0)()( 2121111 uukudgkum 0)( 21222 uukum

000

0

0

0 1

2

1

22

221

2

1

2

1 dgk

u

u

kk

kkk

u

u

m

m

Relative Displacement Concept

11 xdgu ; Thus, 11 xagu (where ag = the acceleration of the earthquake)

22 xdgu ; Thus, 22 xagu

Define Relative Displacements

)()()( 2121111 xdgxdgkxdgdgkxagm =0

)()( 21222 xdgxdgkxagm =0

Insert into summation of forces equation

ag

ag

m

m

x

x

kk

kkk

x

x

m

m

2

1

2

1

22

221

2

1

2

1

0

0

0

0

Matrix Form

Key to remember is that total displacement = earthquake displacement - relative displacement

Central DifferenceExpand Xn+1 & Xn-1 in Taylor Series about time n t

(2) ... )X(2

Δt )XΔt(XX

(1) ... )X(2

Δt )XΔt(XX

n

2

nn1n

n

2

nn1n

Ignore higher order terms and add and subtract (1) & (2) to yield.

1nn1n2n1n1nn X2XXΔt

1X ; XX

t2

1X

Insert these two into the EOM wgnnn fMa)K(X)XC()XM(

1n1nn2nwg1n2XC

t2

1X2XM

Δt

1KXfMaXC

t2

1M

Δt

1

The Central Difference is said to be second-order accurate. Halving the time step should approximately quarter the error.

Stability of Uncoupled Structural Time Integration

Central Difference is conditionally stable (General for explicit methods).

By going through a Fourier/Spectral Stability: maxω

2Δt

For propagating waves:

length sticcharacteri L speed; wave ρ

Ec

Matrix Mass Lumped ;L

2cω emax

CFL condition:

Matrix Mass Consistent 3c

LΔt

Matrix Mass Lumped c

LΔt

min

min

;

;

t must be small enough that information does not propagate across more than one element per time step.

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Fluid Equations: Assumptions

Fluid is inviscid & irrotational Displacements are small; thus, density is constant Fluid is compressible with a bulk modulus Bilinear acoustic fluid (bilinear to account for cavitation)

Fluid cannot transmit negative pressures System is initial in static equilibrium Steady body force field (gravity)

Goal is to develop continuum fluid models that are discretized with the spectral-element method.

Fluid: Momentum Equation

Newton’s Law (F = ma). Momentum Equation

Force)(Body ForceGravity dV)g (ρ

Force dV p-

Mass dV ρ

Fluid ofElement VolumedV

Volume Force is conservative with Potential Energy

pressure chydrostati p pressure; total p

nt vectordisplaceme w) v,(u, d

PotentialGravity dV V dV ρgz)( :Example

h

T

g

Definitions for Fluid

dρ p - V

)dVd(ρ dV p)( - dV )V(

g

g

Fluid: Momentum EquationCompare to Euler’s Equation

t

d

t

d

t

dρ p - V

2

2

g

For an acoustic fluid starting at rest one can assume the following in the acceleration term:

t

d

t

d

t

d2

2

Thus,

dρ p - Vg

Fluid: Displacement PotentialBecause fluid motion is assumed irrotational, the displacement field can be expressed in terms of the gradient of a scalar function (x,y,z,t).

d-ψ

Check Irrotational Condition: 0 ...

y

u

x

v 0 )dcurl(

Ok! yx

ψ

yx

ψ

t

ψ v;

t

ψ u

22

Fluid: Equation of MotionReplace acceleration term with this displacement potential in the Momentum Equation.

ψ dρ -

ψ- p - Vg

Spatial Integrate:

nIntegratio ofConstant :C

C p - V ψ- g

Fluid: Equation of MotionC p - V ψ- g

At Static Equilibrium: 0 ψ

h

hg

hg

p - p ψ

0 C Therefore,

p V However,

C p - V 0

EOM

Fluid: Constitutive Equation•For a linear acoustic fluid: dK - p-p h

• K = fluid Bulk Modulus

•Property to Characterize Compressibility

• Increase Pressure -> Decrease Volume

•K = c2; c = speed of sound of the medium.

•Define ‘Densified Relative Condensation’

dρ- s

•Insert: K = c2 & dρ- s

Into Constitutive Eqn.

sc p-p 2h Constitutive Eqn.

Fluid: Governing Equation•Compare EOM and the Constitutive Equations:

hp - p ψ :EOM

sc p-p :veConstituti 2h

•We see the following :

sc ψ 2•Apply the wave equation for a Linear Fluid:

ψc ψ 22

•Get the Governing Equation:

0 ψ - sor ψ s 22

Fluid Equations: (Spectral Elements) Dependent field Variables representation within each element

Essences of Spectral Elements is the choice of & quadrature rule by Lagrangian Interpolants with Gauss-Lobatto-Legendre (GLL)

quadrature points. Element-node locations are coincident with the quadrature points

)()()( functions basis polynomial-order-N D,-1 of vector

)()()()()(),,(),,,(

)()()()()(),,(),,,(

th

0,,

e

0,,

kji

N

kji

eijkkji

T

N

kji

eijkkji

T

ttt

tstts

es

Standard Gauss Quadrature 1-D GLL Quadrature 1-D

Linear Shape functions

1 -1 1-1 -1/(3) 1/(3) Locations -1 1

Fluid Equations: Discretize governing equation with Galerkin approach

Apply Green’s first formula

Insertion of dependent field variable representation, yields the element-level algebraic equations.

0)( 2 dse

e e enddsd

eeeee bHsQ

dnde

eTe eT

Ω

; ;dΩe

bHQ

Fluid Equations: Assemble into global system

Q : Capacitance Matrix with Spectral elements (becomes diagonal matrix)

H : Reactance Matrix b : boundary-interaction vector

Explicit time integration to solve for s then solve for p

bHQbHQs 22 c c

otherwise ,0

/-ps if 2H2 cscpp

H

Fluid: Bilinear (Cavitation)• Cavitation is the spontaneous vaporization of a fluid. It happens when the fluid pressure < vapor pressure.

• Water’s vapor pressure << atmospheric pressure

• Simple Mathematical model is that if the total pressure is negative then it is just zero.

otherwise ,0

/-ps if 2H2 cscpp

H

Explicit Integration for Fluid Add numerical damping to the EOM to reduce frothing

values nodal of vectors column denotes ? ;sct β p - p ψ 2h

= dimensionless damping coefficient (varies from 0 to 1) Modified Solution Advance Process with Cavitation

11

21

1 ,/

nnn

nhnnnnnn

nnn

t

stcpptt

tsss

111 nnn HbsQ

Insert into Main Fluid Equation:

0,max 12

1 nh

n scpp

bn+1 is from predicted structure and NRB displacements Solve Linear system for sn+1, Remember Q is diagonal

Stability of Uncoupled Fluid Time Integration

Fourier Stability Analysis see Felippa, Deruntz, sec. 2.4

Gerschgorin’s theorem can be used to obtain an upper bound on max.

)21(

2

max c

t

is the eigenvalue of (H- Q)z=0.

eii

N

ijj

eij

eii

e QHH /max3)1(

,1max

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Coupling

NRB:

Structure Forcing:

Fluid Forcing:

z}y,{x, i wet

ei

ei dpNf

st nrb

nrbTe dxdxb

c

px nrbnrb

Temporal integration

Time Stepping:S

F F

up

tn tn+1

Euler ;v*dtuu nnp

S

Etc.

Time Stepping:Staggered Method add subcycling

S

F

tn

Example: Structure time = t; Fluid time = 2t; Subcycling = 2

F

tn+2

up

Euler ;v*dt*2 uu nnp

tn+1

S

pb

S

pn+2Etc.

dt*2

pppp n2n

nb

Time Stepping:Staggered Method add subcycling

Structure (seismic displacement, d) Fluid Volume Non-Reflecting

Boundary (NRB)

•Used Subcycling to reduce computational time and have the time increments near the upper bound of the stable time region for the fluid and the structure. Using a Euler Scheme for the prediction.

nn u*t2u

2np

Example: Structure time = t; Fluid time = 2t; Subcycling = 2

New u

pn+2

start pressure

displacements

dt*2

pppp n2n

nb

Subcycling Trying to integrate the different systems with their

critical time for stability Critical time (fluid,3.5e-3 sec.)>(structure,2.5e-4 sec.) We predict fluid displacements with Euler Scheme.

Comparison of deflections with subcycling and without.Dam Crest Movement without subcycling Dam Crest Movement with subcycling

Output for Dam Crest Under Seismic Load no Soil

Without Fluid Interaction

With Fluid Interaction

Output for Dam Crest Under Seismic Load Soil

Total Displacement of Dam Crest Relative Displacement of Dam Crest

CAVITATION REGION

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.

Example of Localized Lagrange multipliers(Two-spring system)

Subsystems

Subsystem Energy Expressions (Variational Formulation)

3)u*m3-u2)-(u3*k2-δu3(f3 2)u*m2-u3)-(u2*k2-δu2(f2δΠ2 :2 System

1)u*m1-u1*k1-δu1(f1δΠ1 :1 System

Example of Localized Lagrange multipliers

Identify Interface Constraints

(Con’t)

δu2)λ2(δufδu1)λ1(δufu2)δλ2(ufu1)δλ1(ufδΠc

u2)λ2(ufu1)λ1(ufΠ

Total Virtual Work = 0 ( Stationary)

0δΠ2δΠ1 δΠcδΠ

0 λ2)δuf(-λ1uf)δλ2(u2uf)δλ1(u1 3)u*m3u2)-(u3*k2δu3(-f3

λ2)2u*m2u3)-(u2*k2δu2(-f2 λ1) 1u*m1u1*k1δu1(-f1δΠ

The physically represent the interface forces

Example of Localized Lagrange multipliers(Con’t)

Equations of Motion

0

0

0

f3

f2

f1

uf

λ2

λ1

u3

u2

u1

011000

100010

100001

000dt

dm3k2k20

010k2dt

dm2k20

00100dt

dm1k1

2

2

2

2

2

2

mequilibriu 2 Structure

mequilibriu 1 Structure

sconstraint interface 1 Structure

sconstraint interface 2 Structure

interfacesat law thirdsNewton'

Comments:

• Notice the localized multipliers (Lagrange)

• u1 = uf & u2 = uf (constraint)

• Last row states that the sum of reaction forces at a node disappear when partitioned nodes are assembled (Newton’s third law)

•This is just the set up for the transient interaction analysis.

Set Sail for the Future 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 asses the effect of dynamic

model reduction techniques.

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