Computational Aspects of Structural Acoustics and Vibration CISM Lectures on Computational Aspects of Structural Acoustics and Vibration Udine, June 19-23, 2006

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

Synthesis of Partitioned Analysis Procedures C. A. Felippa CISM Lecture Series on Computational Methods in Structural Acoustics and Vibration - Part 2 Udine, June 19-23, 2006

General Comment on Lectures Note that in an FSI simulation (say) I won’t talk on how to do the structure how to do the fluid I assume you know how to do each piece by itself, or to get existing software that do them. My focus is how you may couple the pieces and solve the coupled system.

Lecture Topics 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

Learning by Repetition Effective synthesis practice relies on recognizing some common features of coupled systems. Important is to learn about: software components types of interaction model systems

Coupled System Examples Aeroelasticity Underwater Shock MEMS Resonator Dam Under Seismic Action Common Features favor Partitioned Analysis

Aeroelasticity

Aeroelasticity: Interaction Diagram

Lesson: Try Not to Reinvent the ] Structure model benefits from 50 years of FEM Fluid model benefits from 50 years of CFD

Underwater Shock (UWS)- Early 70s

Resonator (Cell Phone Chip Component)

Resonator: Interaction Diagram System entirely modeled in the frequency domain Reduced Models useful

Dam Under Seismic Action Covered in Part 4

Common Features of Examples (1) Dynamic, two-way interaction Similar physical scales - these are not multiscale problems Interfaces and partitions well defined, surface coupling only

Common Features of Examples (2) Isolated components well understood Software (commercial or public) for components often available Treatment benefits from customization (different models & methods for different components)

Common Features of Examples (3) In production projects, new modeling features are often the result of customer requests; e.g. What happens if “unnamed things” inside the submarine go nonlinear under a strong shock? Can you simulate a fast fighter maneuver? Can turbulence be the cause of a recent crash? Often the request can be “localized”

Common Features of Examples (4) Often the request can be “localized”. For instance when Lockheed was asked by the Navy: What happens if “unnamed things” inside a N-sub go nonlinear under a strong shock? The answer was to replace the linear structural analyzer (NASTRAN) by a nonlinear one (DYNA3D) The fluid and interaction software were not changed.

Starting Project from Scratch with Limited Resources? Mine existing codes for software or data components (e.g, get stiffness/mass mtx from NASTRAN or ANSYS, read in Matlab) Synthesize an interfacing method and a time marching scheme (following slides) Use a higher order language (for example Matlab) as “driver-wrapper” and postprocessor Try realistic problems from start and compare with validation codes. Dont waste time on fancy features (e.g. parallel processing) before the “core stuff” works

Partitioned Method Synthesis

Some Experience So Far (1975-date) Structure-structure, undamped / lightly damped: I-I, A-stable, 2nd-order accurate, single pass, schemes possible. Control-structure interaction I-I, C-stable, 1st-order accurate, single pass schemes possible. A-stable for light or moderate damping. 2nd order requires iteration Underwater shock, no cavitation I-I, A-stable, 2nd-order accurate, single pass, schemes possible only by augmentation of either fluid or structure Underwater shock with cavitation I-E-I, C-stable, 2nd-order accurate, single pass, schemes possible Fluid volume done explicitly. No augmentation required.

Augmentation

Important Step Construct a model equation test system The system contains as many differential equations as coupled components Goal: identify primary physics with minimal number of parameters

Test System Example (Developed Later)

How Long Does It Take? Going through a test system synthesis loop can be time consuming, even for an experienced engineer or scientist. The amount of work strongly depends on many design variables are carried along. This is problem dependent (next slide)

Design Parameters in Test System Structure-structure, undamped 4 Structure-structure, Rayleigh damped 7 Control-structure interaction 7 Underwater shock, no cavitation 5 Underwater shock with cavitation 6 Aeroelasticity with dynamic mesh 7-8 Flexible ship hydrodynamics 6-7 Electrothermomechanics 7-9 Counts are for minimum # of partitions

Recommendations to Save Work Try to reduce number of physical parameters by looking at the essential physics & by forming dimensionless combinations Try to reduce the number of integration & predictor parameters by using theory if available

Your Computer Can Help Synthesis loop may take weeks or months if done by hand (or by “potshot” computations) The use of Computer Algebra Systems (CAS) such as Mathematica o Maple can reduce the time to days or hours. Examples in notes Why the gain? Faster algebra, reduction of human errors & integrated graphics facilities

Recent Research: Partition Localization Basic goal: maximally isolate software modules doing component computations so they communicate only by interface forces aka + Lagrange Multipliers Four Possibilities, with last one covered in Part 4: Distributed Global Lagrange Multipliers Distributed Local Lagrange Multipliers Collocated Global Lagrange Multipliers Collocated Local Lagrange Multipliers

Localization Goals Software reuse, including extrating equations from commercial codes Nonmatching meshes Multilevel parallelization

How Long Does It Take? Going through a test system synthesis loop can be time consuming, even for an experienced engineer or scientist. The amount of work strongly depends on many design variables are carried along. This is problem dependent. Some examples follow.

Test System Example (Developed Later)

Design Variables in Test System (1) Structure-structure interaction, undamped: 2 subsystem frequencies 1 frequency coupling parameter 1 stepsize Total: 4 physical parameters + integrator & predictor parameters With Rayleigh damping: add 3

Recommendations to Save Work Try to reduce number of physical parameters by looking at the essential physics & by forming dimensionless combinations Try to reduce the number of integration & predictor parameters by using theory if available

Getting Computer Help Synthesis loop may take weeks or months if done by hand (or by sample computations) The use of Computer Algebra Systems (CAS) such as Mathematica o Maple can reduce the time to days or hours Why the gain? Faster algebra, reduction of human errors & integrated graphics facilities

Auxiliary Mathematica Modules Provided in notes to help with Stability Analysis Derivation of characteristic equation Polynomial stability (Routh criterion, etc.) Accuracy Analysis Derivation of Modified Equation

A Warning For second order coupled systems, stability depends on the computational path The path dictates how auxiliary variables such as velocities and momenta are computed Consequence: a tiny change in the guts of a program may suddenly affect stability Good news: changes in computational path can be compensated by predictor changes (details in article)

Stability Analysis Example: Structure-Structure Interaction

SSI: Staggered Partition

SSI: Test System (1)

SSI: Test System (2)

SSI: Test System (2)

Theoretical Result Result obtained by Park (1980) If both structures are treated by the Trapezoidal Rule, and an optimal predictor used (adjusted for the computational path) then The staggered solution method is unconditionally stable and has second order accuracy

Verified by Mathematica 24 Years Later

Stable Predictors

Mathematica Code

Summary For undamped structure-structure interaction, optimal staggered methods are known, which do not hinder stability or accuracy If the coupled system is Rayleigh damped, the same methods are recommended For generally damped coupled systems, or control-structure interaction, no general theory is available. Problems must be done case by case

Lecture Sources Parts 1 and 2: Material of recent FSI course (Spr 2003) posted at http://caswww.colorado.edu/courses.d/FSI.d/Home.html contains posted student projects and references to journal papers, including those in CISM brochure: (Felippa-Park-Farhat - CMAME 2001) (Park-Ohayon-Felippa - CMAME 2002) Will add these slides sets on return to Boulder Part 3: a potpourri of bits and pieces, mostly unpublished Part 4: two CMAME papers under preparation (Ross’ Thesis)

Stop End of Part 2