University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures CISM Lectures on Computational Aspects of Structural Acoustics and Vibration Udine, June 19-23, 2006

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Presenter: Carlos A. Felippa Department of Aerospace Engineering Sciences and Center for Aerospace Structures University of Colorado at Boulder Boulder, CO 80309, USA

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Carlos A. Felippa, Michael R. Ross, K. C. Park Partitioned FSI Analysis by Localized Lagrange Multipliers Partitioned FSI Analysis by Localized Lagrange Multipliers Computational Aspects of Structural Acoustics and Vibration - Part 4 Udine, June 19-23, 2006

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Lecture Topics Partitioned Analysis of Coupled Systems: Overview 1. Partitioned Analysis of Coupled Systems: Overview 2. Synthesis of Partitioned Methods 3. Mesh Coupling and Interface Treatment + 4. Partitioned FSI by Localized Lagrange Multipliers

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Detour: my Stay at Barcelona Briefly worked on PFEM with Sergio Idelsohn Lagrangian-Lagrangian FSI Convective terms are gone New idea: treat structure as a “stiff fluid” Will it work? [Stay tuned till next World Cup]

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures PFEM-FSI detour Promising for gravity dominated free surface flows. Not so for acoustics

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Going back to a more established subject... This lecture is on interaction of a structure with an acoustic fluid using LLM to treat non-matching meshes

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures OutlineOutline Part 4A: Overview & basic concepts (Felippa’s CIMNE seminar slides) Part 4B: Selective results from Michael Ross’s thesis, defended May 5

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Part 4A Part 4A: Overview & basic concept s

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FSI Is Important in a Range of Applications aircraft rockets turbines marine structures (fixed, floating, submerged) storage tanks dams suspension bridges bio systems airbags, parachutes inkjet printers

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Exterior FSI Example

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interior FSI Example From Roger Ohayon’s WCCM6 Lecture

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Two Prototype Approaches to Computational Multiphysics Monolithic vs. Partitioned

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Focus in this Part Partitioned Approach Structure by FEM Acoustic Fluid by FEM Interaction by Localized Lagrange Multipliers (LLM)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Project Background Work presented here is part of a research project funded by the Civil & Mechanical Systems (CMS) Division of the US National Science Foundation (NSF) Grant topic: model-based multiphysics simulation using Localized Lagrange Multipliers (LLM) Length: 4 years Two driver problems: one Civil, one Mechanical (next slide)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Project Background (Cont’d) Application Coupled New research driver physics aspects _______________________________________________________ Dam under seismic Fluid-structure- Nonmatching meshes action (Felippa)* soil LLM on fluid side _______________________________________________________ Microelectronic Mechanical-thermo- Model reduction devices (Park) electromagnetics Transform spaces * This presentation

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Project Background (Cont’d) Older technologies used for dam problem:  Partitioned analysis of coupled systems (Park, Felippa & DeRuntz, 1977)  Potential-based acoustic fluid elements (Felippa & DeRuntz, 1984)  Variationally based LLM (Park & Felippa, 1998)  Silent Boundaries Well known ones used: PWA (F) & Lysmer (S)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Driver Civil Engrg Problem Figure is a sketch to explain method. Not a cross section of an actual dam

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Actual Dams Used in Ross’ Examples Later Koyko gravity dam (Japan) “2D slice” analysis Extensive results in literature Morrow Point arch dam (Colorado, US) 3D analysis

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Space Discretization break into partitions (3 shown here) & append silent boundaries for mesh truncation

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures 3-Partition Interaction Diagram

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures 2-Partition Interaction Diagram Structure-soil treated monolithically No LLM interface used there since we know (since ~1999) how to make it work

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FSI Interface Treated by LLM

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Models & Discretization Interface: next slide Structure & soil: linear elastic, standard FEM Soil silent boundary: Lysmer dashpots Fluid silent boundary: PWA Fluid: acoustic (irrotational, inviscid, compressible, linear or bilinear). FEM based on displacement potential Small-amplitude surface waves & cavitation allowed

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Matching Meshes: LLM Interface Treatment S F Matching meshes

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Matching Meshes LLM Interface Treatment Details uBuB F S S F

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Nonmatching Meshes: LLM Interface Treatment S F

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Mismatching in Time If different time stepping schemes are used (for example, implicit in the structure and explicit in the fluid) for the dynamic analysis, solutions may not match in time either. This mismatch (e.g., subcycling methods) is better understood than non-matching in space.

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Basic Formulation Rule The governing functional for the coupled system is the sum of three contributions    total (system) functional  F  fluid functional as subsystem  S  structure functional as subsystem  B  interface functional, aka “interface potential”    F  S  B

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Subsystem Functionals: Partitions Acoustic fluid (D’Alembertian form) Structure (D’Alembertian form)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Subsystem Functionals: Interface Making the total functional    F  S  B stationary with respect to displacement and multipliers as master variables, provides (1) structure & fluid field eqs as Euler-Lagrange eqs (2) interface equations as natural BCs Interface Potential

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Semidiscrete FEM Equations

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Time Domain Dynamic Analysis For time stepping treat semidiscrete S & F equations with time integrators (same or different) and form the time-independent interface equations Equations separate into 3 groups (next slide)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Discrete FEM Equations at Timestep n

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Time Stepping Stages

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Time Stepping Diagram

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Important Feature of LLM Treatment Note the absence of predictor (A consequence of the implicit treatment of the interface)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Software: Research Code

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Software: Validation Code

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Fluid Code Validation: Pressure (1) Compare computational pressure on rigid dam face with analytical solution as Bessel function series (Chopra et al, 1967) Assumptions: ground motion given, fluid infinite in x direction, no reflections, no surface wave effects, valid for early time

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Fluid Code Validation: Pressure (2) Fluid model: p e =  F c 2 B vs u F e, u F = D e 

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Fluid Code Validation: Pressure (3) l = 20m l = 10m Applied acceleration: El Centro earthquake

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Fluid Code Validation: Pressure (4) Applied acceleration: El Centro earthquake l = 5m l = 5m with PWA fluid silent boundary

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FSI Validation: Infinite Piston (1) Matching Meshes

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FSI Validation: Infinite Piston (2) Time step of 0.01 sec give ~4 accurate figures with LLM. Note: CAFE (Mike Sprague’s spectral code) requires over hrs because of stable time step limitations. LLM under Matlab takes 3 sec with TR Matching Meshes Results

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FSI Validation: Infinite Piston (3) Nonmatching Meshes Results Response error (measured by Tom Geers’ C-error measure) is about 10  if ZMR used

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Mortar Method Many variations of method. That used here is variationaly based. Lagrange Multipliers are located at interface frame SS FF

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Koyna Dam A gravity dam (right figure) located in Japan. On December 1967 it was shaken by an earthquake that caused cracking on the downstream face (left figure). The dam has been extensively used in the literature since.

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures 2D Model of Gravity Dam Unit width slice in plane strain Both matching & non-matching meshes used. Transient analysis under El Centro earthquake (May 18, 1967)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures El Centro Earthquake Acceleration Record

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Frequency Content of El Centro Earthquake Horizontal accelerationVertical acceleration

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Vibration Analysis Assume unforced undamped response Insert into Semidiscrete Coupled Equations Transform modal to physical coordinates

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Coupled System Vibration Analysis (1)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Coupled System Vibration Analysis (2)

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Selected Vibration Mode Shapes 6 th mode: 0.54 Hz 20 th mode: 0.54 Hz 46 th mode: 10.1 Hz 49 th mode: 11.3 Hz 55 th mode: 12.4 Hz 65 th mode: 13.8 Hz 21 th mode: 2.53 Hz 27 th mode: 4.94 Hz 34 th mode: 6.69 Hz40 th mode: 8.13 Hz

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Kinematic Continuity Violation LLM discretization u  discretization Mortar

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Kinematic Continuity (2) Same continuity problem with LLM

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Gravity Dam FRF Output DOF Input DOF Output DOF Coupled Alone

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FRF Verifications

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Relative Displacement Concept

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Transient Response Benchmark Mesh Portion of very fine matched-mesh model run with CASE and LLM CASE response considered exact. C-errors on next slide Plot Truncated

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Transient Response Benchmark Accuracy LLMCASE

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Gravity Dam Benchmark Transient Compare Refined LLM Transient vs. CASE (spectral code) with Matching Meshes

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Energy Interface Error Use Energy Difference at interface to warn of trouble in LLM:

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Response For Different Meshings

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interface Force Transmission Based on finer mesh Based on coarser mesh

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interface Meshings: CASE, LLM, Mortar LLM CASE Mortar

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures CASE transient with LLM and Coarse mesh as master CASE transient with LLM and ZMR interface LLM transient with LLM and ZMR interface CASE transient with Mortar and finer mesh as master C Errors: CASE, LLM, Mortar

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Gravity Dam Transient Movie Non-matching Mesh Analysis

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Dam Transient under El Centro Quake

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Non-matching Gravity Dam Transient LLM (Zero) Total DOF C-Error CASE (LLM-Zero) Total DOF C-Error CASE (Mortar) Total DOF

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Reduced Order Modeling Gave excellent results Skipped for lack of time

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Nonlinear Analysis: Fluid Cavitation Gave good to fair results For dam problem, our fluid element could not compete with CASE Skipped for lack of time

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Morrow Point Dam (Gunnison River - Colorado) Geometry

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Seismic Excitation Digitized Taft Earthquake (July 1952) Acceleration historyFrequency power spectrum

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Curved Surface Mesh Generation Map Surfaces in Order to Create Connection Matrices: Interface Frame applying Zero Moment Rule along each direction Structure Fluid

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Matching vs Non-Matching Results Structure has 63 nodes on interface, and fluid meshes range from 42 to 108 nodes. Difference from matched mesh results are taken as error measure

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Conclusions (1) LLM Advantages Complete Subsystem Localization & Decoupling subsystem code “sees” only interface forces simplifies software development “by component” Transient Analysis eliminates prediction step, a difficulty of staggered methods; preserves stability and accuracy Automated Treatment of Non-matching Meshes no need to pick master-slaves or workaround cross points

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Conclusions (2) LLM Advantages Vibration Analysis Symmetry and sparseness of coupled system maintained Lanczos eigensolver ran without problems Model Reduction Effective since vibration analysis is easy

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Conclusions (3) LLM Disadvantages Additional interface unknowns more programming Automated construction of interface frame not fully solved in 3D except for regular configurations

University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Stop  End of Part 4