© 2012 Autodesk A Fast Modal (Eigenvalue) Solver Based on Subspace and AMG Sam MurgieJames Herzing Research ManagerSimulation Evangelist

© 2012 Autodesk Class Summary  Linear Dynamics and FEA  Subspace with AMG  Demonstration  Modal  Response Spectrum  Frequency Response  Critical Buckling  Summary  Q&A

© 2012 Autodesk Learning Objectives At the end of this class, you will be able to:  Understand the theory behind the latest performance improvements to Autodesk Simulation’s linear dynamics solvers  Decide when to use linear dynamics as opposed to more computationally intensive options  Set up of the base modal analysis  Set up and solve a dynamics response analysis  Set up and solve a random vibration analysis  Set up and solve a buckling analysis

© 2012 Autodesk Linear Dynamics and FEA

© 2012 Autodesk Tacoma Narrows Bridge Disaster

© 2012 Autodesk Linear Dynamics and FEA  Benefits  Physical prototypes are expensive  Setup is time consuming (shaker table)  Nonlinear structural FEA is expensive  Time constraints  Potentially additional software cost  Avoid catastrophic failure  Multiple loads and other BCs considered  Perfect for cloud

© 2012 Autodesk Linear Dynamics and FEA  Types  Modal (Natural Frequency)  Modal w/Load Stiffening  Random Vibration  Frequency Response  Response Spectrum  Transient Stress  Critical Buckling

© 2012 Autodesk Linear Dynamics and FEA  Types  Modal (Natural Frequency)  Modal w/Load Stiffening  Random Vibration  Frequency Response  Response Spectrum  Transient Stress  Critical Buckling

© 2012 Autodesk Linear Dynamics and FEA  Typical inputs  Boundary conditions and simplifications  Point masses  Spectrum data (g vs Period)  Vibration data (Accel Sq/Hz vs. Freq (Hz))

© 2012 Autodesk Theory - Basics Mathematical equations: -where K is a stiffness matrix, M is a mass matrix

© 2012 Autodesk Theory – Power Method  Mathematically simple  Solves for the largest eigenvalue  Start with a guess vector x 0, and iterate:

© 2012 Autodesk Theory – Power Method  x k will eventually converge to an eigenvector corresponding to the largest eigenvalue  Convergence dependent on initial guess vectors But engineers don’t care about the largest eigenvalues!!!

© 2012 Autodesk Theory – Inverse Method  Inverse method  Variational method requires a linear equation solver

© 2012 Autodesk Theory – Subspace Method  Extension of Inverse method  Starts with a bunch of linearly independent guess vectors - which forms an m-dimension subspace

© 2012 Autodesk Theory – Subspace Method  Follows a similar iteration scheme as Inverse method  Converges to the subspace spanned by the first m eigenvectors  Iterate from k = 0,1,2, …, compute subspace with a fast equation solver

© 2012 Autodesk Theory – Subspace Method  Project original K and M matrices to this subspace with a sparse matrix operation

© 2012 Autodesk Theory – Subspace Method  Solve the projected m-dimension eigenvalue problem with a high- fidelity eigenvalue solver

© 2012 Autodesk Theory – Subspace Method  Improve the previous subspace estimation with sparse matrix operation  Re-iterate until X converge or max iteration is reached

© 2012 Autodesk Subspace with AMG

© 2012 Autodesk Subspace Method with AMG  Rationale  Theory  Performance Improvements

© 2012 Autodesk Subspace with AMG  Rationale  Use subspace method to efficiently yield lowest eigenvalues  Need fast equation solver  Fast sparse matrix operations  High-fidelity eigenvalue solver

© 2012 Autodesk Subspace with AMG  Fast equation solver -> Algebraic Multigrid Solver (AMG)  Fast, iterative solver  Great scalability as the size of the equation increases  Theoretically ~O(N)  User-controlled tolerance for on-demand precision

© 2012 Autodesk Subspace with AMG  Fast equation solver -> Algebraic Multigrid Solver (AMG)  No matrix factorization step as used in a direct sparse solver, therefore, much less memory and hard disk usage  Easily to be adopted under distributed processing environment

© 2012 Autodesk Subspace with AMG  Fast sparse matrix operations -> Intel Math Kernel Library (MKL)  High-fidelity eigenvalue solver -> A LAPACK eigenvalue solver from MKL

© 2012 Autodesk Benchmark Example 1 – Car Front

© 2012 Autodesk Benchmark Example 1 – Car Front

© 2012 Autodesk Benchmark Example 2 - Satellite

© 2012 Autodesk Benchmark Example 2 - Satellite

© 2012 Autodesk Subspace with AMG  Heuristic used for automatic eigenvalue solver selection:

© 2012 Autodesk Demonstrations

© 2012 Autodesk Natural Frequency: Modal Must run before running other linear dynamics analyses Possible to calculate as many frequencies as desired Optimize analysis time by defining a range of frequencies of interest For forces to be included, Modal with Load Stiffening must be used

© 2012 Autodesk Response Spectrum Common Applications: Earthquake analysis Shock loads Blast testing Key Analysis Steps: Apply any additional loads Point to proper Modal analysis scenario Input Spectrum

© 2012 Autodesk Frequency Response Common Applications: Rotating imbalance Frequency sweeps Fans and pumps Key Analysis Steps: Apply any additional loads Point to proper Modal analysis scenario Define nodes with applied excitation Define frequencies, damping and amplitudes

© 2012 Autodesk Critical Buckling Common Applications: Column design Buildings / Towers Bridges Thin walled structures Key Analysis Steps: No Modal analysis necessary Apply loads and multipliers * Results provide a “Buckling Load Multiplier,” which totals your loads and defines how many more times your loads the part can handle.

© 2012 Autodesk Summary

© 2012 Autodesk Summary  Linear dynamics can be used to effectively and efficiently solve complex models  Even more efficient with Subspace AMG  Can avoid computationally intensive nonlinear analysis  Subspace AMG is very good for large linear dynamics models  The cost increases almost linearly with # of modes requested  Can be times faster for common workflows

© 2012 Autodesk Summary  Linear dynamics procedure  Decide when to use linear dynamics as opposed to more expensive options  Set up of the base modal analysis  Choose proper analysis for your loading conditions  Set up and solve the proper advanced linear dynamics analysis  Review for logic and accuracy

© 2012 Autodesk Q & A

© 2012 Autodesk Autodesk, AutoCAD* [*if/when mentioned in the pertinent material, followed by an alphabetical list of all other trademarks mentioned in the material] are registered trademarks or trademarks of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries. All other brand names, product names, or trademarks belong to their respective holders. Autodesk reserves the right to alter product and services offerings, and specifications and pricing at any time without notice, and is not responsible for typographical or graphical errors that may appear in this document. © 2012 Autodesk, Inc. All rights reserved.