Global-Friction-contact problems (2D or 3D)

Global-Friction-contact problems (2D or 3D)#

Problem statement #

Given

a symmetric positive semi definite matrix $M \in {I R}^{n \times n}$
a matrix $H \in {I R}^{n \times d n_{c}}$
a vector $q \in {I R}^{n}$
a vector $b \in {I R}^{d n_{c}}$
a vector of coefficients of friction $μ \in {I R}^{n_{c}}$

the global frictional contact problem denoted by $PFC (M, H, q, b, μ)$ consists in finding three vectors,

the global velocity $v \in {I R}^{n}$ ,
the relative local velocity $u \in {I R}^{d n_{c}}$ ,
the contact forces $r \in {I R}^{d, n_{c}}$ ,

such that :

\begin{array}{r} \begin{matrix} {\begin{cases} M v = q + H r \\ u = H^{⊤} v + b \\ \hat{u} = u + {[{[\begin{array}{c} μ^{α} ‖ u_{T}^{α} ‖ \\ 0 \\ 0 \end{array}]}^{T}, α = 1 \dots n_{c}]}^{T} \\ C_{μ}^{⋆} ∋ \hat{u} ⊥ r \in C_{μ} \end{cases} \end{matrix} \end{array}

where the Coulomb friction cone is defined by $C_{μ} = \prod_{α = 1 \dots n_{c}} C_{μ^{α}}^{α}$

with $C_{μ^{α}}^{α} = {r^{α}, ‖ r_{t} ‖ \leq μ_{α} | r_{n}^{α} |}$ , and the set $C_{μ^{α}}^{α, ⋆}$ its dual.

The modified local velocity $\hat{u}$ is not considered as an unknown since it can be obtained uniquely from the local velocity $u$ .

Coulomb’s friction law with Signorini’s condition for the unilateral contact written in terms of second order complementarity condition

\begin{array}{r} C_{μ}^{⋆} ∋ \hat{u} ⊥ r \in C_{μ} \end{array}

can be interpreted in a more usual form

\begin{array}{r} \begin{matrix} {\begin{cases} 0 \leq u_{N} ⊥ r_{N} \geq 0 Signorini condition \\ u_{T} = 0 \Rightarrow ‖ r_{T} ‖ \leq μ | r_{n} | Sticking mode \\ u_{T} \neq 0 \Rightarrow r_{T} = - μ | r_{n} | \frac{u_{T}}{‖ u_{T} ‖} Sliding mode \end{cases} \end{matrix} \end{array}

This problem models any instance of discretized frictional contact problem obtained from

the time-discretization of dynamics contact problems with event-capturing of event-tracking schemes,
the time-discretization of quasi-static contact problems,
the modeling of static contact problems. In this last case, $u$ plays the role of the relative displacement at contact

Implementation in numerics #

Structure to define the problem: GlobalFrictionContactProblem.

Solvers list : FRICTION_SOLVER, id containing ‘GLOBAL_FRICTION_3D’

The generic drivers for global friction-contact problems is gfc3d_driver().

Error strategy #

To set internal solver tolerance (when it makes sense!) use gfc3d_set_internalsolver_tolerance().

Check details in Error strategy

Global Friction 3D available solvers #

NSGS (`SICONOS_GLOBAL_FRICTION_3D_NSGS`)#

Non-Smooth Gauss Seidel solver with reformulation.

Driver: gfc3d_nsgs()

Parameters: same as :enumerate:`SICONOS_FRICTION_3D_NSGS`.

Warning : default iparam[SICONOS_FRICTION_3D_IPARAM_ERROR_EVALUATION_FREQUENCY] is 0, which may lead to very expensive computation for error checking. Increase this value to improve performances.

Nonsmooth Newton, Alart-Curnier, (`SICONOS_GLOBAL_FRICTION_3D_NSN_AC`)#

Driver: gfc3d_nonsmooth_Newton_AlartCurnier_wr()

Parameters: same as :enumerate:`SICONOS_FRICTION_3D_NSN_AC`.

iparam[SICONOS_IPARAM_MAX_ITER] = 200;
iparam[SICONOS_FRICTION_3D_NSN_FORMULATION]
- SICONOS_FRICTION_3D_NSN_FORMULATION_ALARTCURNIER_STD (default)
- SICONOS_FRICTION_3D_NSN_FORMULATION_JEANMOREAU_STD
- SICONOS_FRICTION_3D_NSN_FORMULATION_ALARTCURNIER_GENERATED,
- SICONOS_FRICTION_3D_NSN_FORMULATION_JEANMOREAU_GENERATED
- SICONOS_FRICTION_3D_NSN_FORMULATION_NULL
iparam[SICONOS_FRICTION_3D_NSN_MEMORY_ALLOCATED] = 0, 0 if memory for internal work arrays must be allocated, else 1.
iparam[SICONOS_FRICTION_3D_NSN_LINESEARCH]
- SICONOS_FRICTION_3D_NSN_LINESEARCH_GOLDSTEINPRICE (default)
- SICONOS_FRICTION_3D_NSN_LINESEARCH_ARMIJO
- SICONOS_FRICTION_3D_NSN_LINESEARCH_NO
iparam[SICONOS_FRICTION_3D_NSN_LINESEARCH_MAX_ITER] = 100 maximum number of iterations allowed for the line search.
iparam[SICONOS_FRICTION_3D_NSN_MPI_COM] = -1
dparam[SICONOS_DPARAM_TOL] = 1e-10
dparam[SICONOS_FRICTION_3D_NSN_RHO] = 1.
iparam[SICONOS_FRICTION_3D_IPARAM_ERROR_EVALUATION_FREQUENCY] = 1

iparam[SICONOS_IPARAM_MAX_ITER] = 1000;
iparam[SICONOS_FRICTION_3D_IPARAM_INTERNAL_ERROR_STRATEGY] = SICONOS_FRICTION_3D_INTERNAL_ERROR_STRATEGY_ADAPTIVE
dparam[SICONOS_DPARAM_TOL] = 1e-4;
dparam[SICONOS_FRICTION_3D_DPARAM_INTERNAL_ERROR_RATIO] = 2.0

Internal solver: SICONOS_CONVEXQP_ADMM, see Available solvers.

ADMM (`SICONOS_GLOBAL_FRICTION_3D_ADMM`)#

Solver based on ADMM method.

Driver: gfc3d_ADMM()

Parameters:

iparam[SICONOS_IPARAM_MAX_ITER] = 20000;
iparam[SICONOS_FRICTION_3D_ADMM_IPARAM_ACCELERATION]
- SICONOS_FRICTION_3D_ADMM_ACCELERATION
- SICONOS_FRICTION_3D_ADMM_ACCELERATION_AND_RESTART (default)
- SICONOS_FRICTION_3D_ADMM_NO_ACCELERATION
iparam[SICONOS_FRICTION_3D_ADMM_IPARAM_SPARSE_STORAGE]
- SICONOS_FRICTION_3D_ADMM_FORCED_SPARSE_STORAGE
- SICONOS_FRICTION_3D_ADMM_KEEP_STORAGE (default)
iparam[SICONOS_FRICTION_3D_ADMM_IPARAM_INITIAL_RHO] =
- SICONOS_FRICTION_3D_ADMM_INITIAL_RHO_GIVEN (default)
- SICONOS_FRICTION_3D_ADMM_INITIAL_RHO_NORM_INF
- SICONOS_FRICTION_3D_ADMM_INITIAL_RHO_EIGENVALUES;
iparam[SICONOS_FRICTION_3D_ADMM_IPARAM_RHO_STRATEGY]
- SICONOS_FRICTION_3D_ADMM_RHO_STRATEGY_RESIDUAL_BALANCING
- SICONOS_FRICTION_3D_ADMM_RHO_STRATEGY_SCALED_RESIDUAL_BALANCING
- SICONOS_FRICTION_3D_ADMM_RHO_STRATEGY_CONSTANT (default)
iparam[SICONOS_FRICTION_3D_ADMM_IPARAM_GET_PROBLEM_INFO]
- SICONOS_FRICTION_3D_ADMM_GET_PROBLEM_INFO_NO (default)
- SICONOS_FRICTION_3D_ADMM_GET_PROBLEM_INFO_YES
iparam[SICONOS_FRICTION_3D_ADMM_IPARAM_FULL_H] = SICONOS_FRICTION_3D_ADMM_FULL_H_NO;
iparam[SICONOS_FRICTION_3D_ADMM_IPARAM_UPDATE_S] = SICONOS_FRICTION_3D_ADMM_UPDATE_S_YES;
iparam[SICONOS_FRICTION_3D_IPARAM_RESCALING]
- SICONOS_FRICTION_3D_RESCALING_NO (default)
- SICONOS_FRICTION_3D_RESCALING_SCALAR,
- SICONOS_FRICTION_3D_RESCALING_BALANCING_M,
- SICONOS_FRICTION_3D_RESCALING_BALANCING_MH
iparam[SICONOS_FRICTION_3D_IPARAM_RESCALING_CONE]=SICONOS_FRICTION_3D_RESCALING_CONE_NO;
dparam[SICONOS_DPARAM_TOL] = 1e-6;
dparam[SICONOS_FRICTION_3D_ADMM_RHO] = 0.1;
dparam[SICONOS_FRICTION_3D_ADMM_RESTART_ETA] = 0.999;
dparam[SICONOS_FRICTION_3D_ADMM_BALANCING_RESIDUAL_TAU] = 2.
dparam[SICONOS_FRICTION_3D_ADMM_BALANCING_RESIDUAL_PHI] = 10.;

Solvers with reformulation #

All solvers with id ending with ‘_WR’ (which stands for With Reformulation) starts with a reformulation of the global problem into a local one, which is solved with one of the fc3d solvers.

NSGS, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSGS_WR`)#

Non-Smooth Gauss Seidel solver with reformulation.

Driver: gfc3d_nsgs_wr()

Parameters: same as :enumerate:`SICONOS_FRICTION_3D_NSGS`.

NSGS, velocity, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSGS_WR`)#

Non-Smooth Gauss Seidel solver with reformulation.

Driver: gfc3d_nsgs_velocity_wr()

Parameters: same as :enumerate:`SICONOS_FRICTION_3D_NSGSV`.

Global-Friction-contact problems (2D or 3D)

Contents

Global-Friction-contact problems (2D or 3D)#

Problem statement #

Implementation in numerics #

Error strategy #

Global Friction 3D available solvers #

NSGS (`SICONOS_GLOBAL_FRICTION_3D_NSGS`)#

Nonsmooth Newton, Alart-Curnier, (`SICONOS_GLOBAL_FRICTION_3D_NSN_AC`)#

PATH (GAMS) (`SICONOS_GLOBAL_FRICTION_3D_GAMS_PATH`)#

PATHVI (GAMS) (`SICONOS_GLOBAL_FRICTION_3D_GAMS_PATHVI`)#

Fixed-Point projection (VI reformulation) (`SICONOS_GLOBAL_FRICTION_3D_VI_FPP`)#

Extra-Gradient (VI reformulation) (`SICONOS_GLOBAL_FRICTION_3D_VI_EG`)#

ACLM Fixed point (`SICONOS_GLOBAL_FRICTION_3D_ACLMFP`)#

ADMM (`SICONOS_GLOBAL_FRICTION_3D_ADMM`)#

Solvers with reformulation #

NSGS, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSGS_WR`)#

NSGS, velocity, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSGS_WR`)#

Proximal point, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_PROX_WR`)#

DeSaxce FixedPoint, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_DSFP_WR`)#

Tresca FixedPoint, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_TFP_WR`)#

Nonsmooth Newton, Alart-Curnier, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSN_AC_WR`)#

ADMM, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_ADMM_WR`)#

Global-Friction-contact problems (2D or 3D)

Contents

Global-Friction-contact problems (2D or 3D)#

Problem statement#

Implementation in numerics#

Error strategy#

Global Friction 3D available solvers#

NSGS (SICONOS_GLOBAL_FRICTION_3D_NSGS)#

Nonsmooth Newton, Alart-Curnier, (SICONOS_GLOBAL_FRICTION_3D_NSN_AC)#

PATH (GAMS) (SICONOS_GLOBAL_FRICTION_3D_GAMS_PATH)#

PATHVI (GAMS) (SICONOS_GLOBAL_FRICTION_3D_GAMS_PATHVI)#

Fixed-Point projection (VI reformulation) (SICONOS_GLOBAL_FRICTION_3D_VI_FPP)#

Extra-Gradient (VI reformulation) (SICONOS_GLOBAL_FRICTION_3D_VI_EG)#

ACLM Fixed point (SICONOS_GLOBAL_FRICTION_3D_ACLMFP)#

ADMM (SICONOS_GLOBAL_FRICTION_3D_ADMM)#

Solvers with reformulation#

NSGS, with reformulation (SICONOS_GLOBAL_FRICTION_3D_NSGS_WR)#

NSGS, velocity, with reformulation (SICONOS_GLOBAL_FRICTION_3D_NSGS_WR)#

Proximal point, with reformulation (SICONOS_GLOBAL_FRICTION_3D_PROX_WR)#

DeSaxce FixedPoint, with reformulation (SICONOS_GLOBAL_FRICTION_3D_DSFP_WR)#

Tresca FixedPoint, with reformulation (SICONOS_GLOBAL_FRICTION_3D_TFP_WR)#

Nonsmooth Newton, Alart-Curnier, with reformulation (SICONOS_GLOBAL_FRICTION_3D_NSN_AC_WR)#

ADMM, with reformulation (SICONOS_GLOBAL_FRICTION_3D_ADMM_WR)#

Problem statement #

Implementation in numerics #

Error strategy #

Global Friction 3D available solvers #

NSGS (`SICONOS_GLOBAL_FRICTION_3D_NSGS`)#

Nonsmooth Newton, Alart-Curnier, (`SICONOS_GLOBAL_FRICTION_3D_NSN_AC`)#

PATH (GAMS) (`SICONOS_GLOBAL_FRICTION_3D_GAMS_PATH`)#

PATHVI (GAMS) (`SICONOS_GLOBAL_FRICTION_3D_GAMS_PATHVI`)#

Fixed-Point projection (VI reformulation) (`SICONOS_GLOBAL_FRICTION_3D_VI_FPP`)#

Extra-Gradient (VI reformulation) (`SICONOS_GLOBAL_FRICTION_3D_VI_EG`)#

ACLM Fixed point (`SICONOS_GLOBAL_FRICTION_3D_ACLMFP`)#

ADMM (`SICONOS_GLOBAL_FRICTION_3D_ADMM`)#

Solvers with reformulation #

NSGS, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSGS_WR`)#

NSGS, velocity, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSGS_WR`)#

Proximal point, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_PROX_WR`)#

DeSaxce FixedPoint, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_DSFP_WR`)#

Tresca FixedPoint, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_TFP_WR`)#

Nonsmooth Newton, Alart-Curnier, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_NSN_AC_WR`)#

ADMM, with reformulation (`SICONOS_GLOBAL_FRICTION_3D_ADMM_WR`)#