Performance of time-marching techniques dedicated to nonsmooth - - PowerPoint PPT Presentation

Euromech 516 — Nonsmooth contact and impact laws in mechanics

Performance of time-marching techniques dedicated to nonsmooth systems A nonlinear modal analysis approach

Mathias Legrand & Denis Laxalde

McGill University Structural Dynamics and Vibration Laboratory

July 07, 2011

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Outline

1 Introduction

Context Challenges Motivation

2 Modal analysis

Linear framework Nonlinear framework

3 Non-smooth application

Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds

4 Time-stepping techniques

Central differences – Implicit contact θ-method Comparison

5 Conclusion

Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Outline

1 Introduction

Context Challenges Motivation

2 Modal analysis

Linear framework Nonlinear framework

3 Non-smooth application

Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds

4 Time-stepping techniques

Central differences – Implicit contact θ-method Comparison

5 Conclusion

Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Context

Context

Framework

• Flexible structures

◮ finite-element models, multibody ◮ possibly with many degrees-of-freedom

• Dynamics and vibration

◮ transient ◮ steady-state — periodic, quasi-periodic regimes

• Interface nonlinearities — contact, friction

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Context

Context

Framework

• Flexible structures

◮ finite-element models, multibody ◮ possibly with many degrees-of-freedom

• Dynamics and vibration

◮ transient ◮ steady-state — periodic, quasi-periodic regimes

• Interface nonlinearities — contact, friction

Nonlinear modal analysis

• nonlinear modes: periodic motion of an automonous system, invariant

manifold on which oscillations take place

• extension of linear modal analysis, i.e. provides an essential signature of

the dynamics

• numerical framework
• useful for engineering applications

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Challenges

Challenges

• Large scale engineering systems (e.g. FE models, multibody) call for

(efficient) numerical methods adapted to generic nonlinearities (i.e. not

• nly polynomial)

◮ Space semi-discretization ◮ Time stepping approaches (incl. shooting for BVP) ◮ Galerkin methods in time (e.g. harmonic balance, finite-element, etc.) ◮ Space-time discretization

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Challenges

Challenges

• Large scale engineering systems (e.g. FE models, multibody) call for

(efficient) numerical methods adapted to generic nonlinearities (i.e. not

• nly polynomial)

◮ Space semi-discretization ◮ Time stepping approaches (incl. shooting for BVP) ◮ Galerkin methods in time (e.g. harmonic balance, finite-element, etc.) ◮ Space-time discretization

• Non-smooth systems (e.g. featuring interface interactions) require

specific treatments and methods

◮ Contact interface laws ◮ Optimization strategies

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Challenges

Challenges

• Large scale engineering systems (e.g. FE models, multibody) call for

(efficient) numerical methods adapted to generic nonlinearities (i.e. not

• nly polynomial)

◮ Space semi-discretization ◮ Time stepping approaches (incl. shooting for BVP) ◮ Galerkin methods in time (e.g. harmonic balance, finite-element, etc.) ◮ Space-time discretization

• Non-smooth systems (e.g. featuring interface interactions) require

specific treatments and methods

◮ Contact interface laws ◮ Optimization strategies

• Beyond nonlinear modal analysis, the dynamic regimes of such systems

have to be fully qualified including their stability

◮ equilibria ◮ periodic or quasi-periodic solutions ◮ chaos

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Motivation

Time-integration algorithms

Validation

• Assessment of performances of time-stepping techniques dedicated to

flexible mechanical systems featuring unilateral contact conditions

• How are such techniques capable of finding periodic orbits of

automonous (and conservative) systems with non-smooth nonlinearities?

• Preservation of invariant quantities

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Motivation

Time-integration algorithms

Validation

• Assessment of performances of time-stepping techniques dedicated to

flexible mechanical systems featuring unilateral contact conditions

• How are such techniques capable of finding periodic orbits of

automonous (and conservative) systems with non-smooth nonlinearities?

• Preservation of invariant quantities

Extension

• Description of the dynamics around these invariant quantities
• Stability analysis

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Outline

1 Introduction

Context Challenges Motivation

2 Modal analysis

Linear framework Nonlinear framework

3 Non-smooth application

Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds

4 Time-stepping techniques

Central differences – Implicit contact θ-method Comparison

5 Conclusion

Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Linear framework

Linear framework (1)

• Linear 1st-order ODE with constant coefficients

˙ z + Dz = 0 (1)

• Assumed solution

z(t) = Zeλt ⇒ (D − λI)Z = 0 (2)

◮ generally complex eigen elements (λi,Zi) ◮ uncouped equations ≡ invariant subsets ◮ superposition principle ≡ eigen modes span the configuration space

• Invariant set formulation

◮ Master coordinates (x1,y1) = (u1,v1) ◮ Functional dependence

xi(t) = a1iu1(t) + a2iv1(t) yi(t) = b1iu1(t) + b2iv1(t) i = 1,2,...,N (3)

◮ Substitution into equation of motion (1) to get aik’s and bik’s

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Linear framework

Linear framework (2)

• Equations of motion in real modal space
• ˙

uk ˙ vk

• =

         −ξkωk ωk

• 1 − ξ2

k

ωk

• 1 − ξ2

k

−ξkωk         

• uk

vk

• k = 1,2,...,N

(4)

• Reconstruction: superposition principle
• System with two degrees-of-freedom: first linear mode

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1 1 2 −2 −1 1 2

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Nonlinear framework

Nonlinear framework (1)

• Theoretical tools

◮ Central manifold theorem ◮ Lyapunov theorem ◮ Normal form theorem

• Assumed solution

z(t) =

• n

Znenλt ⇒ F(λ,Z) = 0 (5)

◮ Z = (Z1,Z2,...,Zn) ◮ continuation technique: Z(λ) ◮ geometry of the invariant manifold

• Invariant set formulation

◮ Master coordinates (x1,y1) = (u,v) ◮ (Nonlinear) functional dependence

xi = Xi(u,v) yi = Yi(u,v) i = 1,2,...,N (6)

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Nonlinear framework

Nonlinear framework (2)

◮ Time differentiation

˙ xi = ∂Xi ∂u ˙ u + ∂Xi ∂v ˙ v ˙ yi = ∂Yi ∂u ˙ u + ∂Yi ∂v ˙ v i = 1,2,...,N (7)

◮ Substitution into equation of motion (1) — i=1,2,...,N

Yi = ∂Xi ∂u v + ∂Xi ∂v f1(u,X1,...,Xn,v,Y2,...,YN ) fi(u,X1,...,Xn,v,y2,...,YN ) = ∂Yi ∂u v + ∂Yi ∂v f1(u,X1,...,Xn,v,Y2,...,YN ) (8)

◮ Power series expansions, nonlinear Galerkin technique... ◮ Reduced-dynamics on the invariant manifold

˙ u = v ˙ v = f1(u,X1,...,Xn,v,Y2,...,YN ) (9)

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Nonlinear framework

Nonlinear framework (3)

◮ energy-dependent frequency ◮ perturbation techniques ◮ stability analysis ◮ Cubic nonlinearity: first nonlinear mode

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1 1 2 −5 5

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Outline

1 Introduction

Context Challenges Motivation

2 Modal analysis

Linear framework Nonlinear framework

3 Non-smooth application

Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds

4 Time-stepping techniques

Central differences – Implicit contact θ-method Comparison

5 Conclusion

Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Unilateral contact without friction

Unilateral contact without friction

• Autonomous and conservative system

◮ contact interface : Γ

c

◮ no external forcing ◮ no friction, no structural damping

• Gap function on Γ

c

g(u) = u(x) · n − g0(x)

◮ n: outward normal ◮ g0: initial gap

• Contact conditions — ∀x ∈ Γ

c

τN 0, g (u) 0, g (u) · τN = 0

◮ τN = σ · n: contact pressure

b

u (x) Ω Γd Γc

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Invariant periodic solution

Invariant periodic solution

• Boundary value problem (Dirichlet and Signorini conditions)

ρ¨ u − divσ(u) = 0

• n Ω × R+

∗

u = 0

• n Γ

d × R+ ∗

g(u) 0, τN 0 and g(u) · τN = 0

• n Γ

c × R+ ∗

• Frequency-domain displacement field

u(t) =

• n∈Z

ˆ unejnωt with ˆ un = 1 T

• T

u(t)e−jnωtdt

• Final formulation

Find {ω, ˆ u}, with ˆ u = {ˆ un, n ∈ Z} such as: −divσ(ˆ un) = (nω)2ˆ un

• n Ω × Z

ˆ un = 0

• n Γ

d × Z

g(ˆ u) 0, τN 0 and g(ˆ u) · τN = 0

• n Γ

c × [0,T]

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Linear rod with contact

Linear rod with contact

g0 x u(x, t) L

b b b b b b b b b b b b b

• Local equation

ρ∂2u ∂t2 (x,t) − E ∂2u ∂x2 (x,t) = 0, x ∈]0,L [

• Boundary condition

u(0,t) = 0

• Unilateral contact conditions

u(L,t) − g0 0, σxx(L,t) = EA ∂u ∂x (L,t) 0, (u(L,t) − g0) · σxx(L,t) = 0

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Modal motions

Modal motions

displacement velocity low energy high energy

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Invariant manifolds

Invariant manifolds

• Cross section of the invariant manifold
• From low to high energy
• Energy–frequency dependence

node in contact node close to the clamped condition

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Invariant manifolds

Invariant manifolds

• Cross section of the invariant manifold
• From low to high energy
• Energy–frequency dependence

0.0 0.5 1.0 1.5 2.0 2.5

• norm. energy

1296 1298 1300 1302 1304 1306 1308 1310 frequency

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Invariant manifolds

Invariant manifolds

• Cross section of the invariant manifold
• From low to high energy
• Energy–frequency dependence
• First linear mode

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Invariant manifolds

Invariant manifolds

• Cross section of the invariant manifold
• From low to high energy
• Energy–frequency dependence
• First nonlinear mode

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Outline

1 Introduction

Context Challenges Motivation

2 Modal analysis

Linear framework Nonlinear framework

3 Non-smooth application

Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds

4 Time-stepping techniques

Central differences – Implicit contact θ-method Comparison

5 Conclusion

Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Central differences – Implicit contact

Central differences – Implicit contact

• Problem Find un+1 and λn+1 such that:

M un+1 − 2un + un−1 ∆t2

• + Kun − GTλn+1 = 0

with Gun+1 0, λn+1 0, and λT

n+1Gun+1 = 0

• Algorithm
• 1. Prediction of un+1 by ignoring contact forces λn+1
• 2. If (Gun+1)(i) < 0 then un+1 is modified so that (Gun+1)(i) = 0
• Properties

◮ conditionally stable ◮ proof of convergence in 1D ◮ possibilities of lumped mass ◮ energy losses affect the solution after few impacts

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

θ-method

θ-method (1)

• Integral formulation

t+∆t

t

Md ˙ u + t+∆t

t

Kudt − t+∆t

t

λdt = 0

• Assumptions

◮ ut+∆t − ut = ∆t

• (1 − θ) ˙

ut + θ ˙ ut+∆t

• with 0 θ 1

◮ t+∆t

t

Md ˙ u = M( ˙ ut+∆t − ˙ ut)

◮ t+∆t

t

Kudt = ∆t

• (1 − ξ)Kut + ξKut+∆t
• with 0 ξ 1

◮ t+∆t

t

λdt = ∆tλt+∆t

• Problem Find un+1 and λn+1 such that:

M θ∆t2 + ξK

• ut+∆t + M

θ∆t vt + M θ∆t2 − (1 − ξ)K

• ut − GTλt+∆t = 0

with Gut+∆t 0, λt+∆t 0, and λT

t+∆tGut+∆t = 0

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

θ-method

θ-method (2)

• Algorithm
• 1. Steppest descent in un+1
• 2. Uzawa in λn+1
• Properties

◮ Best parameters: θ = ξ = 0.5 ◮ Inconditonally stable in a linear framework ◮ Energy conservation

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Comparison

Comparison

• Initial conditions on the calculated invariant manifold
• Time-stepping integration
• Comparison of the three numerical approaches: implicit

, explicit , and NLM

End-rod displacement [m] Mid-rod displacement [m] −1 1 −1 1 Time [ms] Amplitude [MN] 3.4 3.8 4.2 4.6 2 4 6

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Outline

1 Introduction

Context Challenges Motivation

2 Modal analysis

Linear framework Nonlinear framework

3 Non-smooth application

Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds

4 Time-stepping techniques

Central differences – Implicit contact θ-method Comparison

5 Conclusion

Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Summary

Summary

• Full unilateral and periodic formulation, ie, no regularization

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Summary

Summary

• Full unilateral and periodic formulation, ie, no regularization
• Strategy dedicated to flexible structures

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Summary

Summary

• Full unilateral and periodic formulation, ie, no regularization
• Strategy dedicated to flexible structures
• Numerical proof of the existence of these periodic motions

◮ Agreement between three different numerical techniques ◮ Need for mathematical proof (already existing?)

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Summary

Summary

• Full unilateral and periodic formulation, ie, no regularization
• Strategy dedicated to flexible structures
• Numerical proof of the existence of these periodic motions

◮ Agreement between three different numerical techniques ◮ Need for mathematical proof (already existing?)

• Capture the well-known stiffening effects observed for systems

undergoing unilateral contact conditions

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Future work

Future work

• Contact interface involving more nodes

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Future work

Future work

• Contact interface involving more nodes
• Study of higher modes of vibration

◮ Comparison with conservation of energy ◮ Are specific modes more affected?

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Future work

Future work

• Contact interface involving more nodes
• Study of higher modes of vibration

◮ Comparison with conservation of energy ◮ Are specific modes more affected?

• Several contacting flexible structures

◮ Periodic or quasi-periodic motions ◮ More complicated formulation

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Future work

Future work

• Contact interface involving more nodes
• Study of higher modes of vibration

◮ Comparison with conservation of energy ◮ Are specific modes more affected?

• Several contacting flexible structures

◮ Periodic or quasi-periodic motions ◮ More complicated formulation

• Development of other existing time-marching techniques

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Future work

Future work

• Contact interface involving more nodes
• Study of higher modes of vibration

◮ Comparison with conservation of energy ◮ Are specific modes more affected?

• Several contacting flexible structures

◮ Periodic or quasi-periodic motions ◮ More complicated formulation

• Development of other existing time-marching techniques
• Finite-element techniques in time vs. Fourier series

◮ Improved description of the contact efforts ◮ More DoF

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Future work

Future work

• Contact interface involving more nodes
• Study of higher modes of vibration

◮ Comparison with conservation of energy ◮ Are specific modes more affected?

• Several contacting flexible structures

◮ Periodic or quasi-periodic motions ◮ More complicated formulation

• Development of other existing time-marching techniques
• Finite-element techniques in time vs. Fourier series

◮ Improved description of the contact efforts ◮ More DoF

• Stability analysis

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion

Future work

Future work

• Contact interface involving more nodes
• Study of higher modes of vibration

◮ Comparison with conservation of energy ◮ Are specific modes more affected?

• Several contacting flexible structures

◮ Periodic or quasi-periodic motions ◮ More complicated formulation

• Development of other existing time-marching techniques
• Finite-element techniques in time vs. Fourier series

◮ Improved description of the contact efforts ◮ More DoF

• Stability analysis
• Friction?