[PPT] - A unified library of nonlinear solution schemes Sofie E. Leon, PowerPoint Presentation

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A unified library of nonlinear solution schemes

Sofie E. Leon, Glaucio H. Paulino, Anderson Pereira, Ivan F. M. Menezes, Eduardo N. Lages 7/27/2011

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Motivation

Nonlinear problems are prevalent in structural, fluid, continuum, etc. mechanics

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes

Nonlinear wave interaction simulated with libMesh. libMesh wiki. Simulation of the electronic behavior

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performed with PETSc. Antaki, J., et al., Procceedings of SC2000, 2000. Maxwell model used for viscoelasticity in asphalt pavements. Dave et al. IJNME, 2011 Nonlinear fracture process zone in dynamic fracture processes. Park et al. IJNME, 2010

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Motivation

Many algorithms have been developed to solve such problems, but no single

algorithm is capable of solving any and all nonlinear problems

A library of nonlinear solution schemes, defined by unique constraint equations, is

unified into a single space

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon Bergan, P . G., et al. IJNME 1978. | Yang, Y.B. and Sheih, M.S. AIAA Journal 1990. | Rezaiee-Pajand, M., et al. International Journal of Engineering, Transactions B: Applications. 2009. | Salinger, A. G. et. al., "LOCA 1.0 Library of continuation algorithms", Sandia National Laboratories (2002). | Balay, S. et al Modern Software Tools in Scientific Computing (Birkhäuser Press, 1997)

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u

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Presentation Outline

Nonlinearity overview
N+1 dimensional space formulation
Solution schemes
Numerical results
Evaluation and conclusions

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 4

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Geometric

– Large strains and/or rotations – Linear or nonlinear constitutive relation – Loads have an effect on deformed configuration or configuration can have an effect on the load

Material

– i.e. inelasticity, strain-softening

Critical points

– Structure loses stability or bifurcation

ccurs

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Nonlinearity overview

Stable Unstable Stable A B C D E Load nt Displaceme Point Limit Load Point Limit Load Points Limit nt Displaceme O

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N+1 dimensional space formulation

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Load nt Displaceme

j

p δ

1

u ∆

1 −

∆

j

u

j

u ∆

j

u δ

1

p ∆

1 −

∆

j

p

j

p ∆

Decomposition
Load parameter

Batoz JJ and Dhatt G. IJNME, 1979. 6

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Unified scheme – NLS++

Incremental-

iterative scheme is the same for any algorithm implemented in the N+1 space

Only the

computation of the load factor changes for each algorithm

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

p p

i i

λ ← ) (

1 −

←

i i

u K K 1 ← j ) (

i i i j

u q p r − ← 1 + ← j j TOL

1

< ∆ p r

i i j

λ Yes No

i j i j i j 1 1 1)

(

− − −

← r K uII δ ←

i j II

u δ p K uI

1 1)

(

− −

←

i j i j

δ

i j

δλ Compute ) (

i j i j i j i i i j i i II I

u u u u δ δ δλ δλ λ λ + + ← + ← ) ( Compute

i

u q

i j

Compute Κ 1 + ← i i 1 == j Yes No i Step

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Nonlinear solution schemes

7/27/2011

USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon Crisfield MA. Non-linear Finite Element Analysis of Solids and Structures. Volume 1: Essentials. 1991. | Ramm E. Nonlinear Finite Element Analysis in Structural Mechanics. 1981. | Bergan PG, et al. IJNME. 1978. | Riks E. International Journal of Solids and

Structures. 1979. | Wempner GA. International Journal of Solids and Structures. 1971. | Yang Y-B, McGuire W. Proceedings of the

1985 International Conference on Numerical Methods in Engineering: Theory and Applications. | Yang, Y.B. and Sheih, M.S. AIAA Journal 1990. |Krenk S. IJNME 1995. | Kouhia R. CMAME 2008. 8

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Load control method Arc-length control method

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Common nonlinear solution schemes

A B Surface Constraint p λ ∆ u ∆ p λ

u

1

u ∆

u

p

1

λ ∆ p λ Surface Constraint B

i

s ∆ A 1 = j 2 = j 3 = j

2

u ∆

3

u ∆ p

2

λ ∆ p

3

λ ∆

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Generalized displacement control method

Load parameter is based on a physical quantity called the Generalized Stiffness

Parameter (GSP)

The sign of the GSP changes only at load limit points

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Load

nt Displaceme

+ + + +

i

I i I

1 1

1

:

f

Sign u u δ δ ⋅

−

Yang, Y.B. and Sheih, M.S. AIAA Journal 1990. 10

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Internal force given by
Tangent matrix

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Two-dimensional function

LCM = load control method DCM = displacement control method ALCM = arc-length control method WCM = work control method GDCM = generalized displacement control method ORP = orthogonal residual procedure var = variable u1,u2 = control degrees of freedom Ilinca Stanciulescu. Personal communications. 2009. 11

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Systematically select the best control parameter

7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Two-dimensional function and the variable DCM

Fujii F, et al. Computers & Structures. 1992.

Variable displacement control method

Step Control Snap back 1-73 74-216 217-445

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7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Twelve bar truss

NRM = newton raphson method, DCM = displacement control method, ALCM = arc-length control method, WCM = work control method, GDCM = generalized displacement control method, ORP = orthogonal residual procedure Yang Y-B and Leu LJ. CMAME 1991. Krenk S and Hededal O. CMAME 1995.

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x-z view

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u 697 . 1 414 . 1 . 1

y-z view

. 1 P 25 .

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7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon

Lee frame

LCM = load control method DCM = displacement control method ALCM = arc-length control method WCM = work control method GDCM = generalized displacement control method ORP = orthogonal residual procedure

G I A E , ,

1

u

2

u

P 120 120 x y

Lee SL et al. ASCE Journal of Engineering Mechanics. 1968. Schweizerhof KH and Wriggers P . CMAME 1986. Parente E and Vaz LE. IJNME 2001 14

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Evaluation of solution schemes

In general the most robust and powerful solution schemes in the

context of the N+1 space are – Variable displacement control method – Arc-length control method – Generalized displacement control method

Standard displacement control method and work control method have

predictable difficulties at displacement limit points

Orthogonal residual procedure is sensitive to input parameters and

behavior is sometimes unpredictable near limit points

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Conclusions and future work

NLS++ is a powerful computational framework that provides seamless

implementation of solution schemes of interest – Gives user flexibility to select the most appropriate scheme for a particular problem – Portable, effective and extendable

Plan to integrate NLS++ with the new finite element code based on

topological data representation, called TopFEM

Details and code available in M.S. thesis

www.ideals.illinois.edu www.sofieleon.com

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