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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
Mikaël Barboteu, 22-23 juin 2006 à Besançon
Laboratoire de Mathématiques Et Physique pour les
Systèmes (MEPS), équipe Equations aux Dérivées
Partielles et leurs Applications (EDPA)
? Plan of the talk
1) Mechanical modelling and variational form
3) Domain decomposition methods Non symmetric
balancing method
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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1) Mechanical modelling and variational form
1) Contact mechanical modelling for nonlinear
impact
? Contact and Friction laws
Friction of Coulomb
Unilateral Contact
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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1) Mechanical modelling and variational form
? Variational formulation
Find
Such that for all
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods
2) General framework of energy-conserving methods
Time discretization
? Integration scheme (Gonzalez midpoint scheme
(2000))
Find
Such that
(S)
and
with
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods
Some works in this field
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods Newton continuation
method
Newton continuation method
? Extended Newton method 2 steps strategy
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods Newton continuation
method
? Continuation Newton method algorithm
for p 0 ..

step (a) for i 0 ..

until convergence
step (b) for i ..

until convergence
until p P
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods Newton continuation
method
Analysis of the proposed method
? Enforcement of the contact and friction
conditions
? Analysis of the energy conservation (we suppose
fg0)
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods Specific
penalization method
Specific penalization method
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods impact of a ring
A representative application impact of a ring
against a rigid surface
? Evolution of the discrete energy (friction
case)
? Evolution of the discrete energy (frictionless
case)
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods impact of a ring
Dependencies of the two proposed methods compared
to the time step (frictionless case)
? Specific penalization method
? Newton continuation method
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods bounces of the ring
A representative application bounces of the ring
against a rigid surface
? Animation of the bounces of the ring
? scheme with steps (a)-(b) (case with friction)
? scheme without step (b) (case without friction)
? scheme with steps (a)-(b) (case without
friction)
? Displacement of the ring center
? Evolution of the discrete energy
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods bounces of the ring
Comparison between the continuation method and
the penalization method
? Displacements of the ring center
? Discrete energy evolution
continuation method
penalization method
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods bounces of the ring
? Normal contact distances during the bounces and
according to several time steps
penalization method
continuation method
Errors made on the normal contact distances are
proportional to the time step
Similar behaviors between the Newton continuation
method and the specific penalization method
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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3) Domain decomposition method nonsymmetric
Balancing method
3) Domain decomposition method non symmetric
Balancing method
? Decomposition in nonoverlapping subdomains
? 2-level Neumann-Neumann preconditioner
? Variational interface problem
? Decomposition of the interface space
Fine space
Coarse space
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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3) Domain decomposition method nonsymmetric
Balancing method
Towards an efficient definition of the coarse
space
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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3) Domain decomposition method nonsymmetric
Balancing method
? Construction of the 2-level Neumann-Neumann
preconditioner
? The 2-level Neumann-Neumann preconditioner is
classically defined by
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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3) Domain decomposition method nonsymmetric
Balancing method
Numerical application impact of a ring against
rigid surface
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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Some prospects
Some prospects
? Energy-conserving methods comparison with the
Equivalent Mass Matrix method (Khenous, Laborde
and Renard 2005)
? Domain decomposition methods improvements by
using the BDDC (Dohrmann and Mandel 2004)
? Extension to impacts between two hyperelastic
bodies.
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Efficient algorithms to solve large nonlinear
elastodynamic frictional contact problems
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2) Energy-conserving methods bounces of the ring
? Normal contact distances during the bounces and
according to several time steps
? Displacement of the ring center during the
bounces and according to several time steps
continuation method
penalization method