1. Coupling Heterogeneous Models with Non-matching Meshes
3/31/2017
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

2. Topics of Discussion Introduction of the Research Topic Benefits
3/31/2017
Topics of Discussion
Introduction of the Research Topic
Benefits
Plan of Attack for this Research
Progress (Benchmark Model)
Direction for the Future

3. 3/31/2017
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.

4. Application Drivers MEMS Device (Mechanical Eng.)
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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)

5. A Picture is Worth 1,000 Words
Multi-physic system
Modular Systems
Connected by Localized Interaction Technique (Black Lines)

6. 3/31/2017
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

7. Plan of Attack Generate a benchmark model
3/31/2017
Plan of Attack
Generate a benchmark model
Use current available methods
Matching meshes
Generate a model with localized frames
Maintain matching meshes
With nonmatching meshes

8. Current Progress (Benchmark Model)
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Current Progress (Benchmark Model)
Dam (Brick Elements)
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

9. Benchmark Concept: Staggered Method with Non-Reflecting Boundary
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displacements
pressures
Structure (seismic displacement)
Non-Reflecting Boundary (NRB)
Fluid Volume
pressures
displacements

10. Structure Equations: x = relative displacements
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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.)

11. Relative Displacement Concept
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Summation of Forces:
Matrix Form:

12. Relative Displacement Concept
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Relative Displacement Concept
Define Relative Displacements
Insert into summation of forces equation
Matrix Form
Key to remember is that total displacement = earthquake displacement - relative displacement

13. Central Difference Expand Xn+1 & Xn-1 in Taylor Series about time n t
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Expand Xn+1 & Xn-1 in Taylor Series about time n t
Ignore higher order terms and add and subtract (1) & (2) to yield.
Insert these two into the EOM
The Central Difference is said to be second-order accurate. Halving the time step should approximately quarter the error.

14. Stability of Uncoupled Structural Time Integration
Central Difference is conditionally stable (General for explicit methods).
By going through a Fourier/Spectral Stability:
For propagating waves:
CFL condition:
t must be small enough that information does not propagate across more than one element per time step.

15. JOKE TIME
A football player and an engineer are applying for the same job.
The boss said, “Boys, you need to take a test before you can get this job.
So they took the test and the next day they came back to see who the boss chose. “Well,” he said, “Both of you got the same score except I’m going to choose the engineer.”
The football player complained, “Don’t you think that’s prejudice or something?”
“Well,” the boss said,”Let me tell you what happened. Both of your papers were right all the way through until the last question came up, and the engineer answered ‘I don’t know’ and then when I looked at your paper, you answered, ‘Me either’”.

16. 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.

17. Fluid: Momentum Equation
Definitions for Fluid
Volume Force is conservative with Potential Energy
Newton’s Law (F = ma). Momentum Equation

18. Fluid: Momentum Equation
Compare to Euler’s Equation
For an acoustic fluid starting at rest one can assume the following in the acceleration term:
Thus,

19. Fluid: Displacement Potential
Because fluid motion is assumed irrotational, the displacement field can be expressed in terms of the gradient of a scalar function (x,y,z,t).
Check Irrotational Condition:

20. Fluid: Equation of Motion
Replace acceleration term with this displacement potential in the Momentum Equation.
Spatial Integrate:

21. Fluid: Equation of Motion
At Static Equilibrium:
EOM

22. Fluid: Constitutive Equation
For a linear acoustic fluid:
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’
Insert: K = c2 &
Into Constitutive Eqn.
Constitutive Eqn.

23. Fluid: Governing Equation
Compare EOM and the Constitutive Equations:
We see the following :
Apply the wave equation for a Linear Fluid:
Get the Governing Equation:

24. 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
Standard Gauss Quadrature 1-D
GLL Quadrature 1-D
Linear Shape functions -1 1 -1 1  
Locations
-1/(3)
1/(3) -1 1

25. 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.

26. 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

27. 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.

28. Explicit Integration for Fluid
Add numerical damping to the EOM to reduce frothing
 = dimensionless damping coefficient (varies from 0 to 1)
Modified Solution Advance Process with Cavitation
Insert into Main Fluid Equation:
bn+1 is from predicted structure and NRB displacements
Solve Linear system for sn+1, Remember Q is diagonal

29. Stability of Uncoupled Fluid Time Integration
Fourier Stability Analysis
see Felippa, Deruntz, sec. 2.4
 is the eigenvalue of (H- Q)z=0.
Gerschgorin’s theorem can be used to obtain an upper bound on max.

30. Time for a little laughter:
“Outside of the killings, Washington has one of the lowest crime rates in the country,” - Washington DC mayor Marion Barry
“I’m not going to have some reporters pawing through out papers. We are the President,” - Hillary Clinton
“It isn’t the pollution that’s harming the environment. It’s the impurities in our air and water that are doing it.” - Al Gore
“It’s no exaggeration to say that the undecided could go one way or another” - George Bush

31. Coupling
Temporal integration
NRB:
Structure Forcing:
Fluid Forcing:

32. Time Stepping: S S up
Etc. F F tn
tn+1

33. Time Stepping: Staggered Method add subcycling
Example: Structure time = t; Fluid time = 2t; Subcycling = 2 S S S pb
Etc.
pn+2 up F F tn
tn+1
tn+2

34. Time Stepping: Staggered Method add subcycling
Example: Structure time = t; Fluid time = 2t; Subcycling = 2
New u
start
pressure
Structure (seismic displacement, d)
Non-Reflecting Boundary (NRB)
Fluid Volume
displacements
pn+2
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.

35. Subcycling Comparison of deflections with subcycling and without.
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

36. Output for Dam Crest Under Seismic Load no Soil
With Fluid Interaction
Without Fluid Interaction

37. Output for Dam Crest Under Seismic Load Soil
Total Displacement of Dam Crest
Relative Displacement of Dam Crest

38. CAVITATION REGION

39. 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.

40. Example of Localized Lagrange multipliers
(Two-spring system)
Subsystems
Subsystem Energy Expressions (Variational Formulation)

41. Total Virtual Work = 0 ( Stationary)
Example of Localized Lagrange multipliers
(Con’t)
The  physically represent the interface forces
Identify Interface Constraints
Total Virtual Work = 0 ( Stationary)

42. Example of Localized Lagrange multipliers
(Con’t)
Equations of Motion
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.

43. 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.

44. 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