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

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 Words Multi-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 Dam (Brick Elements) Fluid (Spectral Elements) Silent Boundary Soil (Brick Elements) Silent Boundary • 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

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 uD Dam uB D uS s 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.

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface 1- Subsystem Energy Expressions (Variational Formulation) 2- Identify Interface Constraints 3- Total Virtual Work = 0 ( Stationary)

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface Midpoint Integration Rule: Rearrange in terms of velocity and acceleration at half time step

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface Examine Eqn (21) with midpoint rule: Apply Same Concept to Eqn (22) to get a displacement of the sandstone at the half time step

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface Look at Eqn (23) with the above eqns for the displacements: Apply the same concept to Eqn (24). Then use Eqn (25) and the two above equations to input into Matrix form

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface 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: Step 3: Update the variables and generate the necessary time step-dependent vectors.

Localized Lagrange Multipliers Applied to Dam and Sandstone Interface Dam Crest Displacement with Monolithic Model Dam Crest Displacement with Lagrange Multipliers

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

Fluid Structure Interaction with Lagrange Multipliers • Can the interface constraint be written? • 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: • Can the variation of the fluid be written?

Fluid Structure Interaction with Lagrange Multipliers ????????????????

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