Model Order Reduction of Elastic Multibody Systems with Large Finite - - PowerPoint PPT Presentation

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

Summer School Trogir 2011

Model Order Reduction of Elastic Multibody Systems with Large Finite Element Models

Michael Fischer

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

Principle of Elastic Multibody Systems

multibody system elastic bodies

discretization finite elements, finite difference, ... continuum

elastic multibody system

rigid bodies coupling and constraint elements

C

reduction of elastic degrees

• f freedom

improvement of the simulation process for flexible multibody systems Petrov-Galerkin projection to reduce the elastic degrees of freedom approximation mit

q V q   ) ( ) ( q q

dim dim



Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

                                                  q D q K q q h q q h q q h q α a M (q) C C (q) C J(q) (q) c C (q) c E

e T T

     

e e t a e er et er T et

m m m ) , ( ) , ( ) , ( ~ ~

Reduction of the Elastic Degrees of Freedom

equations of motion of a single body

u B q K q D q M       

e e e e

   q C y  

T e

reduction of second order MIMO system (ODE) with sparse system matrices forces at cut free nodes are inputs to the MIMO-System displacements of this nodes are outputs to the MIMO-System

u B V q V K V q V D V q V M V              

e T e T e T e T

   q V C y   

T e

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

Programs and Data Flow

different finite element programs Permas Abaqus Ansys user input full elastic body Morembs++ / MatMorembs different reduction methods SVD/ Gramian matrix based reduction Krylov based reduction modal reduction simulation of the elastic multibody system Matlab Neweul-M² Simpack

standard FEM programs used for generation of matrices use of alternative reduction methods (Morembs++ / MatMorembs) advanced MOR can easily be used instead of modal approaches standard MBS programs used for the simulation of flexible MBS  important to keep second order structure

reduced elastic body … …

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

Numerical Problems in Reduction Techniques

eigenvalue problem modal reduction moment matching at expansion points modified Gram Schmidt orthogonalization: energy scalar product

) (

2

   

i e e i e i

   K D M

Component Mode Synthesis

        

 bb ib ii c

I K K φ

1 e e k e k e e e k e

s s s K D M K D M D     

2

~ , 2 ~ ) ( colsp ) ~ , ~ , ~ ~ (

1 1 1 1 ) (

V B K M K D K        

   

V Γ

K k e e e e e e k Jb



Krylov subspace constraint modes

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

Numerical Problems in Reduction Techniques

solution of Lyapunov equations Cholesky factorization Singular Value Decomposition calculation of frequency weighted position Gramian matrix Proper Orthogonal Decomposition: SVD and eigenvalue problem SVD / Gramian matrix based modern reduction methods approximate the transfer function

 

 

e e e e e

s s s B K D M C H     

1 2

main problem for large systems: inverting large sparse matrices

 

   

 

max min min max

) ( ) ( 2 1 ) ( ) ( 2 1

    

        d i i d i i

H H p

Q Q Q Q P

e e e e

i i B K D M Q     

1 2

) ( ) (   

T T T T

            C C A Q Q A B B A P P A

Q P /

controllability/ observability Gramian

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

Solving Large Sparse Systems

direct methods (LU factorization) vs. iterative methods usage of central memory computational time with large right sides numerical libraries fill in is memory-intensive small

• nly one factorization

for each right side free

• ften not free

used numerical libraries for LU factorization UMFPACK GNU GPL Lizenz used in Matlab MUMPS (MUltifrontal Massively Parallel sparse direct Solver) in test mode

• ut-of-core solution possible

parallel computing with MPI is possible with both libraries inverting large sparse matrices = solving a sparse linear system

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

car body 2 278 938 dof sparse system matrices 114 million non-zero elements size of system information as a HDF5/Matlab structure: 16 GB approximate usage of memory for one LU factorization: 40 GB

Example of a Large System

Institute of Engineering and Computational Mechanics University of Stuttgart, Germany

• Prof. Dr.-Ing. Prof. E.h. Peter Eberhard

a lot of different numerical algebra in reduction process main problem: LU factorization usage of different numerical libraries to solve large systems efficiently better reduction results for large systems with modern reduction techniques reduction of large models on desktop PCs with out-of-core solver

Results and Outlook

results better understanding of numerical linear algebra speed up by optimizing numerical linear algebra not only for LU decomposition larger models with up to 10 million degrees of freedom parallelization of numerical operations and data access automated problem depending usage of solution method and numerical library future work

thank you for your attention