On Solving Contact Problems with Coulomb Friction: Formulations and Numerical Comparisons

Abstract

In this chapter, we review several formulations of the discrete frictional contact problem that arises in space and time discretized mechanical systems with unilateral contact and three-dimensional Coulomb’s friction. Most of these formulations are well–known concepts in the optimization community, or more generally, in the mathematical programming community. To cite a few, the discrete frictional contact problem can be formulated as variational inequalities, generalized or semi–smooth equations, second–order cone complementarity problems, or optimization problems, such as quadratic programming problems over second-order cones. Thanks to these multiple formulations, various numerical methods emerge naturally for solving the problem. We review the main numerical techniques that are well-known in the literature, and we also propose new applications of methods such as the fixed point and extra-gradient methods with self-adaptive step rules for variational inequalities or the proximal point algorithm for generalized equations. All these numerical techniques are compared over a large set of test examples using performance profiles. One of the main conclusions is that there is no universal solver. Nevertheless, we are able to give some hints for choosing a solver with respect to the main characteristics of the set of tests.

Appendices

Appendix 1. Basics in Convex Analysis

Definition 1

([103]) Let $X\subseteq \mathrm {I\!R}^n$. A multivalued (or point-to-set) map** $T:X\rightrightarrows X$ is said to be (strictly) monotone if there exists $c (>) \geqslant 0$ such that, for all $\hat{x}, \widetilde{x} \in X$,

$$\begin{aligned} {(\hat{v} - \widetilde{v})}^\top (\hat{x} - \widetilde{x})\geqslant c \Vert \hat{x} - \widetilde{x}\Vert \qquad \text {with}\; \hat{v}\in T(\hat{x}), \widetilde{v}\in T(\widetilde{x}). \end{aligned}$$

(137)

Moreover, T is said to be maximal when it is not possible to add a pair (x, v) to the graph of T without destroying the monotonicity.

The Euclidean projector $P_X$ onto a closed convex set X: for a vector $x\in \mathrm {I\!R}^n$, the projected vector $z = P_X(x)$ is the unique solution to the convex quadratic program

$$\begin{aligned} {\left\{ \begin{array}{ll} \min \, {\displaystyle \frac{\displaystyle 1}{\displaystyle 2}} (y-x)^\top (y-x), \\ \begin{array}{ll} \text {s.t.} &{} y \in X . \end{array} \end{array}\right. } \end{aligned}$$

(138)

The following equivalences are classical:

$$\begin{aligned} y = P_K(x)&\Longleftrightarrow \begin{array}{ll} \min &{}\frac{1}{2} (y-x)^\top (y-x ) \\ \text{ s.t. } &{} y \in K \end{array} \end{aligned}$$

(139)

$$\begin{aligned}&\Longleftrightarrow - (y-x) \in N_K(y) \end{aligned}$$

(140)

$$\begin{aligned}&\Longleftrightarrow (x-y)^\top (y-z) \geqslant 0, \forall z \in K \end{aligned}$$

(141)

$$\begin{aligned} -F(x) \in N_K(x)&\Longleftrightarrow -\rho F(x)^\top (y-x) \geqslant 0, \forall y \in K\end{aligned}$$

(142)

$$\begin{aligned}&\Longleftrightarrow (x-(x - \rho F(x))^\top (y-x) \geqslant 0, \forall y \in K \end{aligned}$$

(143)

$$\begin{aligned}&\Longleftrightarrow x = P_K(x- \rho F(x)) \text { thanks to (141).} \end{aligned}$$

(144)

Sub-differential of the Euclidean Norm

The sub-differential of the Euclidean norm in $\mathrm {I\!R}^n$ is given by

$$\begin{aligned} \partial \Vert z\Vert = \left\{ \begin{array}{lcl} {\displaystyle \frac{\displaystyle z}{\displaystyle \Vert z\Vert }}, &{} &{}z \ne 0\\ \{x, \Vert x\Vert \leqslant 1 \},&{} &{}z = 0. \end{array} \right. \end{aligned}$$

(145)

Euclidean Projection on the Unit Ball

Let $B=\{x \in \mathrm {I\!R}^n, \Vert x\Vert \leqslant 1\}$. The Euclidean projection on the unit ball is given by

$$\begin{aligned} P_{B}(z) =\left\{ \begin{array}{lcl} z &{}\text {if}&{} z \in B \\ {\displaystyle \frac{\displaystyle z}{\displaystyle \Vert z\Vert }}&{} \text {if}&{}z \notin B. \end{array}\right. \end{aligned}$$

(146)

Its subdifferential can be computed as

$$\begin{aligned} \partial P_{B}(z) =\left\{ \begin{array}{lcl} I &{}\text {if }&{} z \in B \setminus \partial B \\ I + (s-1)z z^\top , s\in [0,1] &{}\text {if }&{} z \in \partial B \\ {\displaystyle \frac{\displaystyle I}{\displaystyle \Vert z\Vert }} - {\displaystyle \frac{\displaystyle z z^\top }{\displaystyle \Vert z\Vert ^3}} &{}\text {if }&{} z \notin B. \end{array}\right. \end{aligned}$$

(147)

Euclidean Projection on the Second-Order Cone of $\mathrm {I\!R}^3$

Let $K=\{x = [x_{\text {N}}x_{\text {T}}]^T \in \mathrm {I\!R}^3, x_{\text {N}}\in \mathrm {I\!R}, \Vert x_{{\text {T}}}\Vert \leqslant \mu x_{{\text {N}}}\}$ be the second-order cone in $\mathrm {I\!R}^3$. The Euclidean projection on K is

$$\begin{aligned} P_{K}(z) =\left\{ \begin{array}{lcl} z &{}\text {if}&{} z \in K \\ 0 &{}\text {if}&{} -z \in K^* \\ {\displaystyle \frac{\displaystyle 1}{\displaystyle 1+\mu ^2}}(z_{\text {N}}+ \mu \Vert z_{\text {T}}\Vert ) \left[ \begin{array}{c} 1 \\ \mu {\displaystyle \frac{\displaystyle z_{\text {T}}}{\displaystyle \Vert z_{\text {T}}\Vert }} \end{array}\right]&\text {if}&z \notin K \text { and } -z \notin K^*. \end{array}\right. \end{aligned}$$

(148)

Direct Computation of an Element of the Subdifferential

The computation of the subdifferential of $P_{K}$ is given as follows:

• if $z\in K\setminus \partial K$, $\partial _z P_K(z) =I$,

• if $-z \in K^*\setminus \partial K^*$, $\partial _z P_K(z) =0$,

• if $z \notin K$ and $-z \notin K^*$ and, $\partial _z P_K(z) =0$, we get

  $$\begin{aligned} \partial _{z_{\text {N}}} P_{K}(z) = {\displaystyle \frac{\displaystyle 1}{\displaystyle 1+\mu ^2}} \left[ \begin{array}{c} 1 \\ \mu z_{\text {T}}\end{array}\right] \end{aligned}$$

  (149)

  and

  $$\begin{aligned} \partial _{z_{\text {T}}} [P_{K}(z)]_{\text {N}}= {\displaystyle \frac{\displaystyle \mu }{\displaystyle 1+\mu ^2}} {\displaystyle \frac{\displaystyle z_{\text {T}}}{\displaystyle \Vert z_{\text {T}}\Vert }} \end{aligned}$$

  (150)
  $$\begin{aligned} \partial _{z_{\text {T}}} [P_{K}(z)]_{\text {T}}= {\displaystyle \frac{\displaystyle \mu }{\displaystyle (1+\mu ^2)}} \left[ \mu {\displaystyle \frac{\displaystyle z_{\text {T}}}{\displaystyle \Vert z_{\text {T}}\Vert }} {\displaystyle \frac{\displaystyle z^\top _{\text {T}}}{\displaystyle \Vert z_{\text {T}}\Vert }}+ (z_{\text {N}}+ \mu \Vert z_{\text {T}}\Vert ) \left( {\displaystyle \frac{\displaystyle I_2}{\displaystyle \Vert z_{\text {T}}\Vert }} - {\displaystyle \frac{\displaystyle z_{\text {T}}z_{\text {T}}^\top }{\displaystyle \Vert z_{\text {T}}\Vert ^3}} \right) \right] , \end{aligned}$$

  (151)

  that is,

  $$\begin{aligned} \partial _{z_{\text {T}}} [P_{K}(z)]_{\text {T}}= {\displaystyle \frac{\displaystyle \mu }{\displaystyle (1+\mu ^2)\Vert z_{\text {T}}\Vert }} \left[ (z_{\text {N}}+ \mu \Vert z_{\text {T}}\Vert ) \,I_2 + z_{\text {N}}{\displaystyle \frac{\displaystyle z_{\text {T}}z_{\text {T}}^\top }{\displaystyle \Vert z_{\text {T}}\Vert ^2}} \right] . \end{aligned}$$

  (152)

Computation of the Subdifferential Using the Spectral Decomposition

In [55], the computation of the Clarke subdifferential of the projection operator is also done by inspecting the different cases using the spectral decomposition

$$\begin{aligned} \partial P_K(x) = \left\{ \begin{array}{ll} I &{} \quad (\lambda _1>0, \lambda _2>0 ) \\ {\displaystyle \frac{\displaystyle \lambda _2}{\displaystyle \lambda _1+\lambda _2}} I +Z &{} \quad (\lambda _1< 0, \lambda _2>0 ) \\ 0 &{} \quad (\lambda _1< 0, \lambda _2<0 ) \\ {{\mathrm{co}}}\{I, I+Z\} &{} \quad (\lambda _1=0, \lambda _2 >0 ) \\ {{\mathrm{co}}}\{0, Z\} &{} \quad (\lambda _1<0, \lambda _2 =0 ) \\ {{\mathrm{co}}}\{0 \cup I \cup S\} &{} \quad (\lambda _1=0, \lambda _2 =0 ), \end{array} \right. \end{aligned}$$

(153)

where

$$\begin{aligned} \begin{array}{l} Z = \frac{1}{2} \begin{bmatrix} - y_{\text {N}}&{} y_{\text {T}}^\top \\ y_{\text {T}}&{} - y_{\text {N}}y_{\text {T}}y_{\text {T}}^\top \end{bmatrix},\\ S = \left\{ \frac{1}{2} (1+\beta ) I + \frac{1}{2} \begin{bmatrix} -\beta &{} w^\top \\ w &{} -\beta w w^\top \end{bmatrix}\mid -1 \leqslant \beta \leqslant 1, \Vert w\Vert =1\right\} , \end{array} \end{aligned}$$

(154)

with $y = x / \Vert x_{\text {T}}\Vert$. A simple verification shows that the previous computation is an element of the subdifferential.

Appendix 2. Computation of Generalized Jacobians for Nonsmooth Newton Methods

Computation of Components of a Subgradient of $F_{{\text {vi}}}^\mathsf{{nat}}$

Let us introduce the following notation for an element of the subdifferential:

$$\begin{aligned} \varPhi (u, r) = \left[ \begin{array}{cc} \rho I &{} - \rho W \\ \varPhi _{r u}(u,r) &{} \varPhi _{r r}(u,r) \end{array}\right] \in \partial F_{{\text {vi}}}^\mathsf{{nat}}(u,r), \end{aligned}$$

(155)

where $\varPhi _{x y}(u,r) \in \partial _{x}[F_{{\text {vi}}}^\mathsf{{nat}}]_{y}(u,r)$. Since $\varPhi _{u u}(u,r) = I$, a reduction of the system is performed in practice and Algorithm 4 is applied or $z =r$ with

$$\begin{aligned} {\left\{ \begin{array}{ll} G(z) = [F_{{\text {vi}}}^\mathsf{{nat}}]_{r}(Wr+q,r) \\ \varPhi (z) = \varPhi _{rr}(r,Wr+q) + \varPhi _{ru}(r,Wr+q) W. \end{array}\right. } \end{aligned}$$

(156)

Let us introduce the following notation for an element of the sub–differential with an obvious simplification:

$$\begin{aligned} \varPhi (v, r) = \left[ \begin{array}{ccc} \rho M &{} - \rho H \\ -\rho H^\top &{} \rho I &{} 0 \\ 0 &{} \varPhi _{r u}(v,u,r) &{} \varPhi _{r r}(v,u,r) \end{array}\right] \in \partial F_{{\text {vi}}}^\mathsf{{nat}}(u,r), \end{aligned}$$

(157)

where $\varPhi _{x y}(v,u,r) \in \partial _{x}[F_{{\text {vi-1}}}^\mathsf{{nat}}]_{y}(v,u,r)$. A possible computation of $\varPhi _{r u}(v,u,r)$ and $\varPhi _{r r}(v,u,r)$ is directly given by (159) and (158). In this case, the variable u can also be substituted.

For one contact, a possible computation of the remaining parts in $\varPhi (u, r)$ is given by

$$\begin{aligned} \varPhi _{r u}(u,r) = \left\{ \begin{array}{lcl} 0 &{} \text{ if } &{} r- \rho (u+g(u)) \in K \\ \\ I - \partial _{r}[P_K(r-\rho (u+g(u)))] &{} \text{ if } &{}r- \rho (u+g(u)) \notin K \end{array}\right. \end{aligned}$$

(158)

$$\begin{aligned} \varPhi _{r u}(u,r) = \left\{ \begin{array}{lcl} \rho \left( I + \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ {\displaystyle \frac{\displaystyle u_{\text {T}}}{\displaystyle \Vert u_{\text {T}}\Vert }} &{} 0 &{} 0 \\ \end{array}\right] \right) &{} \text{ if } &{} {\left\{ \begin{array}{ll} r- \rho (u+g(u)) \in K \\ u_{\text {T}}\ne 0 \end{array}\right. } \\ \\ \rho \left( I + \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ s &{} 0 &{} 0 \\ \end{array}\right] \right) , s \in \mathrm {I\!R}^2 , \Vert s\Vert =1 &{} \text{ if } &{} {\left\{ \begin{array}{ll} r- \rho (u+g(u)) \in K \\ u_{\text {T}}= 0 \end{array}\right. } \\ \\ I +\rho \left( I + \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ {\displaystyle \frac{\displaystyle u_{\text {T}}}{\displaystyle \Vert u_{\text {T}}\Vert }} &{} 0 &{} 0 \\ \end{array}\right] \right) \partial _{u}[P_K(r-\rho (u+g(u)))]&\text{ if }&r- \rho (u+g(u)) \notin K. \end{array}\right. \end{aligned}$$

(159)

The computation of an element of $\partial P_K$ is given in Appendix 11.

Alart–Curnier Function and Its Variants

For one contact, a possible computation of the remaining parts in $\varPhi (u, r)$ is given by

$$\begin{aligned} \varPhi _{r_{\text {N}}u_{\text {N}}}(u,r) = \left\{ \begin{array}{ll} \rho _{{\text {N}}} &{} \text { if } r_{{\text {N}}} - \rho _{{\text {N}}} u_{{\text {N}}} > 0 \\ 0 &{} \text { otherwise } \end{array}\right. \end{aligned}$$

(160)

$$\begin{aligned} \varPhi _{r_{\text {N}}r_{\text {N}}}(u,r) = \left\{ \begin{array}{ll} 0 &{} \text { if } r_{{\text {N}}} - \rho _{{\text {N}}} u_{{\text {N}}} > 0 \\ 1 &{} \text { otherwise } \end{array}\right. \end{aligned}$$

(161)

$$\begin{aligned} \varPhi _{r_{\text {T}}u_{\text {N}}}(u,r) = \left\{ \begin{array}{ll} 0 &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert \leqslant \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ 0 &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ r_{{\text {N}}} - \rho _{{\text {N}}} u_n\leqslant 0 \end{array}\right. } \\ \mu \rho _{{\text {N}}} {\displaystyle \frac{\displaystyle r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}} }{\displaystyle \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert }} &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ r_{{\text {N}}} - \rho _{{\text {N}}} u_n > 0 \end{array}\right. } \\ \end{array}\right. \end{aligned}$$

(162)

$$\begin{aligned} \varPhi _{r_{\text {T}}u_{\text {T}}}(u,r) = \left\{ \begin{array}{ll} \rho _{{\text {T}}} &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert \leqslant \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ \mu \rho _{{\text {T}}}(r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}})_+ \varGamma (r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}) &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ r_{{\text {N}}} - \rho _{{\text {N}}} u_n > 0 \end{array}\right. } \\ \end{array}\right. \end{aligned}$$

(163)

$$\begin{aligned} \varPhi _{r_{\text {T}}r_{\text {N}}}(u,r) = \left\{ \begin{array}{ll} 0 &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert \leqslant \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ 0 &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ r_{{\text {N}}} - \rho _{{\text {N}}} u_n\leqslant 0 \end{array}\right. } \\ -\mu {\displaystyle \frac{\displaystyle r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}} }{\displaystyle \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert }} &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ r_{{\text {N}}} - \rho _{{\text {N}}} u_n > 0 \end{array}\right. } \\ \end{array}\right. \end{aligned}$$

(164)

$$\begin{aligned} \varPhi _{r_{\text {T}}r_{\text {T}}}(u,r) = \left\{ \begin{array}{ll} 0 &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert \leqslant \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ I_2-\mu (r_{{\text {N}}} - \rho _{{\text {N}}}u_{\text {N}})_+ \varGamma (r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}) &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu \max (0 ,r_{{\text {N}}} - \rho _{{\text {N}}} u_{\text {N}}) \\ r_{{\text {N}}} - \rho _{{\text {N}}} u_n > 0, \end{array}\right. } \\ \end{array}\right. \end{aligned}$$

(165)

with the function $\varGamma (\cdot )$ defined by

$$\begin{aligned} \varGamma (x) = {\displaystyle \frac{\displaystyle I_{2\times 2}}{\displaystyle \Vert x\Vert }} - {\displaystyle \frac{\displaystyle x\,x^\top }{\displaystyle \Vert x\Vert ^3}}. \end{aligned}$$

(166)

If the variant (60) is chosen, the computation of $\varPhi _{r_{\text {T}}\bullet }$ simplifies to

$$\begin{aligned} \varPhi _{r_{\text {T}}u_{\text {N}}}(u,r) = 0 \end{aligned}$$

(167)

$$\begin{aligned} \varPhi _{r_{\text {T}}u_{\text {T}}}(u,r) = \left\{ \begin{array}{ll} \rho _{{\text {T}}} &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert \leqslant \mu r_{\text {N}}\\ -\mu \rho _{\text {T}}r_{n,+} \varGamma (r_{\text {T}}-\rho _{\text {T}}u_{\text {T}}) &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert > \mu r_{\text {N}}\\ \end{array}\right. \end{aligned}$$

(168)

$$\begin{aligned} \varPhi _{r_{\text {T}}r_{\text {N}}}(u,r) = \left\{ \begin{array}{ll} 0 &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert \leqslant \mu r_{{\text {N}}} \\ 0 &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu r_n \\ r_{{\text {N}}} \leqslant 0 \end{array}\right. } \\ -\mu {\displaystyle \frac{\displaystyle r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}} }{\displaystyle \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert }} &{} \text { if } {\left\{ \begin{array}{ll} \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert> \mu r_n \\ r_{{\text {N}}} > 0 \end{array}\right. } \\ \end{array}\right. \end{aligned}$$

(169)

$$\begin{aligned} \varPhi _{r_{\text {T}}r_{\text {T}}}(u,r) = \left\{ \begin{array}{ll} 0 &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert \leqslant \mu r_{{\text {N}}}\\ I_2-\mu (r_{{\text {N}}})_+ \varGamma (r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}) &{} \text { if } \Vert r_{{\text {T}}} - \rho _{{\text {T}}} u_{{\text {T}}}\Vert > \mu r_{{\text {N}}}. \\ \end{array}\right. \end{aligned}$$

(170)