A Fast Modal (Eigenvalue) Solver Based on Subspace and AMG Sam MurgieJames HerzingResearch ManagerSimulation Evangelist 2012 Autodesk 1Class SummaryLinear Dynamics and FEASubspace with AMGDemonstrationModalResponse SpectrumFrequency ResponseCritical BucklingSummaryQ&A

2012 Autodesk 2Learning ObjectivesAt the end of this class, you will be able to:Understand the theory behind the latest performance improvements to Autodesk Simulations linear dynamics solversDecide when to use linear dynamics as opposed to more computationally intensive optionsSet up of the base modal analysisSet up and solve a dynamics response analysisSet up and solve a random vibration analysisSet up and solve a buckling analysis

2012 Autodesk 3Linear Dynamics and FEA 2012 Autodesk 4Tacoma Narrows Bridge Disaster

2012 Autodesk Linear Dynamics and FEABenefitsPhysical prototypes are expensiveSetup is time consuming (shaker table)Nonlinear structural FEA is expensiveTime constraintsPotentially additional software costAvoid catastrophic failureMultiple loads and other BCs consideredPerfect for cloud

2012 Autodesk Linear Dynamics and FEATypesModal (Natural Frequency)Modal w/Load StiffeningRandom VibrationFrequency ResponseResponse SpectrumTransient StressCritical Buckling

2012 Autodesk Linear Dynamics and FEATypesModal (Natural Frequency)Modal w/Load StiffeningRandom VibrationFrequency ResponseResponse SpectrumTransient StressCritical Buckling

2012 Autodesk Linear Dynamics and FEATypical inputsBoundary conditions and simplificationsPoint massesSpectrum data (g vs Period)Vibration data (Accel Sq/Hz vs. Freq (Hz))

2012 Autodesk Theory - BasicsMathematical equations:

-where K is a stiffness matrix, M is a mass matrix

2012 Autodesk Theory Power MethodMathematically simpleSolves for the largest eigenvalueStart with a guess vector x0, and iterate:

2012 Autodesk Theory Power Methodxk will eventually converge to an eigenvector corresponding to the largest eigenvalueConvergence dependent on initial guess vectors

But engineers dont care about the largest eigenvalues!!!

2012 Autodesk Theory Inverse MethodInverse methodVariational method

requires a linear equation solver

2012 Autodesk Theory Subspace MethodExtension of Inverse methodStarts with a bunch of linearly independent guess vectors

- which forms an m-dimension subspace

2012 Autodesk Theory Subspace MethodFollows 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 MethodProject original K and M matrices to this subspace with a sparse matrix operation

2012 Autodesk Theory Subspace MethodSolve the projected m-dimension eigenvalue problem with a high-fidelity eigenvalue solver

2012 Autodesk Theory Subspace MethodImprove 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 19Subspace Method with AMGRationaleTheoryPerformance Improvements

2012 Autodesk Subspace with AMGRationaleUse subspace method to efficiently yield lowest eigenvaluesNeed fast equation solverFast sparse matrix operationsHigh-fidelity eigenvalue solver

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

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

2012 Autodesk Subspace with AMGFast 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 AMGHeuristic used for automatic eigenvalue solver selection:

2012 Autodesk Demonstrations 2012 Autodesk 30Natural 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 analysisShock loadsBlast testing

Key Analysis Steps:Apply any additional loadsPoint to proper Modal analysis scenarioInput Spectrum

2012 Autodesk Frequency Response

Common Applications:Rotating imbalanceFrequency sweepsFans and pumps

Key Analysis Steps:Apply any additional loadsPoint to proper Modal analysis scenarioDefine nodes with applied excitationDefine frequencies, damping and amplitudes

2012 Autodesk Critical Buckling

Common Applications:Column designBuildings / TowersBridgesThin walled structures

Key Analysis Steps:No Modal analysis necessaryApply 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 35SummaryLinear dynamics can be used to effectively and efficiently solve complex modelsEven more efficient with Subspace AMGCan avoid computationally intensive nonlinear analysis

Subspace AMG is very good for large linear dynamics modelsThe cost increases almost linearly with # of modes requestedCan be 10-50 times faster for common workflows

2012 Autodesk SummaryLinear dynamics procedureDecide when to use linear dynamics as opposed to more expensive optionsSet up of the base modal analysisChoose proper analysis for your loading conditionsSet up and solve the proper advanced linear dynamics analysisReview for logic and accuracy

2012 Autodesk Q & A 2012 Autodesk 38

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