2. Theoretical formulation of the method

2.1. The contact problem

2.1.1. Strong formulation

Let us consider the equilibrium problem of two solids in equilibrium in the figure, denoted by the index \(l=\mathrm{1,}2\).

Figure 1: Solids in balance

We are looking for the displacement field \(u=({u}_{\mathrm{1,}}{u}_{2})\mathrm{:}\overline{{\mathrm{\Omega }}^{1}}\times \overline{{\mathrm{\Omega }}^{2}}\to {\mathrm{ℝ}}^{d}\) (\(d=2\) or \(d=3\)) such that () - () are satisfied for \(l=\mathrm{1,}2\):

(2.1)\[ \begin{align}\begin{aligned} -\ text {div}\ mathrm {\ sigma} ({u} _ {l}) = {f} _ {l}\\ \ textrm {in}\\ {\ mathrm {\ omega}}} ^ {l}\end{aligned}\end{align} \]

(2.2)\[ \begin{align}\begin{aligned} \ mathrm {\ sigma} ({u} _ {l}) = {A} _ {l}\ mathrm {\ epsilon} ({u} _ {l})\\ \ textrm {in}\\ {\ mathrm {\ omega}}} ^ {l}\end{aligned}\end{align} \]

(2.3)\[ \begin{align}\begin{aligned} \ mathrm {\ sigma} ({u} _ {l}) {n} _ {l} = {F} _ {l}\\ \ textrm {on}\\ {\ mathrm {\ Gamma}} _ {N} ^ {l}\end{aligned}\end{align} \]

\[\]

: label: eq-4

{u} _ {l} =0

textrm {on}

{mathrm {Gamma}} _ {D} ^ {l}

\[\]

: label: eq-5

[{u} _ {N}] ≤0

textrm {on}

{mathrm {Gamma}} _ {C}

(2.4)\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {N} ≤0\\ \ textrm {on}\\ {\ mathrm {\ Gamma}} _ {C}\end{aligned}\end{align} \]

(2.5)\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {N}} _ {N}} _ {N}] =0\\ \ textrm {on}\\ {\ mathrm {\ Gamma}} _ {C}\end{aligned}\end{align} \]

(2.6)\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {\ mathrm {\ tau}} =0\\ \ textrm {on}\\ {\ mathrm {\ Gamma}} _ {C}\end{aligned}\end{align} \]

where \([{v}_{N}]={\sum }_{l=1}^{2}{v}_{l}\cdot {n}_{l}\).

The equations () to () correspond to the mechanical problem under consideration (to simplify, we place ourselves in the framework of a linear elasticity problem) where \(A\) is an elliptical, symmetric fourth-order tensor with components in \({L}^{\mathrm{\infty }}\) coming from the law of behavior associated with the solid \({\mathrm{\Omega }}^{l}\) coming from the law of behavior associated with solid with Dirichlet condition with Dirichlet condition on edge \({\Gamma }_{D}^{l}\) and Neumann condition on edge \({\Gamma }_{N}^{l}\). To these equations, contact conditions are added on \({\Gamma }_{C}\), namely a non-interpenetration condition for the displacement jump \([{u}_{N}]\) (), a sign condition for normal pressure \({\mathrm{\sigma }}_{\mathrm{N}}\) the contact zone () and a complementarity condition between these last two fields (). The last equation () comes from the fact that we consider frictionless contact.

2.1.2. Weak formulation

From equations () to (), we can introduce a weak formulation of the contact problem in the form of the following variational inequality:

(2.7)\[ \begin{align}\begin{aligned} \ textrm {Find}\\ U\\ \ textrm {such as :math:`\forall v\in K`,}\\ {\ sum} _ {l=1} ^ {2} {\ 2} {\ int} _ {\ mathrm {\ omega}} ^ {l}} {a} _ {l}\ mathrm {\ epsilon} ({u} _ {l}) ({u} _ {l}) ({u} _ {l}) ({u} _ {l}) d {\ mathrm {\ Omega}} ^ {l}} {\ sum} _ {\ sum} _ {l=1} ^ {2} {\ int} _ {\ mathrm {\ Omega}} ^ {l}} {f}} {f}} {f} _ {f}} _ {f}} _ {l} _ {l}) d {\ mathrm {\ Omega}} ^ {l}} {f}} {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {l} _ {l}) d {\ mathrm {\ Omega}}} {f}} {f} _ {f} _ {f} _ {l}} {f} _ {f} _ {l}} {f} _ {\ int} _ {{\ mathrm {\ Gamma}}} _ {N} ^ {l}} {F} _ {l}\ cdot {({v} _ {l} - {u} _ {l})} _ {l})} _ {l})} _ {l})} _ {l})} _ {l})} _ {l})} _ {l})} _ {l})} _ {l})} _ {l})\end{aligned}\end{align} \]

where \(K\subset V\) is the convex cone of the allowable displacements (\(K\) contains the non-interpenetration condition). The following spaces are therefore used:

(2.8)\[ \begin{align}\begin{aligned} {V} _ {l} =\ {{v} _ {l} _ {l}\ in {({H} ^ {1} ({\ mathrm {\ Omega}} ^ {l}))} ^ {d}\ mathrm {::} v=0\ text {on} {\ text {on} {\ mathrm {\ Gamma}}} _ {D} ^ {l}))} ^ {l}\}\ mathrm {::} v=0\ text {on} {on} {on} {\ mathrm {\ Gamma}} _ {D} ^ {l}))} ^ {l}\}\\ V= {V} _ {1}\ times {V} _ {2}\\ K=\ {v\ in V\ mathrm {:} [{v} _ {N}]\ le 0\ text {on} {\ mathrm {\ Gamma}}} _ {C}\}\end{aligned}\end{align} \]

The following will be noted:

(2.9)\[ a (u, v) = {\ sum} _ {l=1} ^ {l=1} ^ {2} {\ int} _ {\ mathrm {\ Omega}} ^ {l} _ {l}\ mathrm {\ epsilon} ({\ epsilon}} ({u} epsilon} ({u} epsilon} ({u} epsilon} ({u} epsilon} ({u} epsilon}) ({u} _ {l}) ({u} _ {l}) ({u} _ {l}) ({u} _ {l}) d {\ mathrm {\ omega}}} ^ {l}\]

And:

(2.10)\[ l (v) =\ sum _ {l=1} ^ {1} ^ {2} {\ int}} _ {{\ mathrm {\ Omega}} {f} _ {l}\ cdot {v} _ {l} _ {l} _ {l} _ {l} d {\ mathrm {\ v} _ {l} d {\ mathrm {\ v} _ {l} _ {l} _ {l} d {\ mathrm {\ v} _ {l} _ {l} d {\ mathrm {\ omega}} d {\ mathrm {\ omega}} d {\ mathrm {\ omega}} {l} d {\ mathrm {\ v} _ {l} _ {l} _ {l} _ {l} _ {l} _ {l}} {F} _ {l}\ cdot {v} _ {l} _ {l} d {\ mathrm {\ Gamma}} ^ {l}\]

The equivalent mixed problem is introduced:

(2.11)\[ \begin{align}\begin{aligned} \ textrm {Find}\\ U\ in V\\ \ textrm {and :math:`\lambda \in M` such as:}\\ a (u, v) -b (\ mathrm {\ lambda}, v) =l (v)\ text {}\ forall v\ in V\\ b (\ mathrm {\ mu} -\ mathrm {\ lambda}, u)\ ge 0\ text {}\ forall\ mathrm {\ mu}\ in M\end{aligned}\end{align} \]

With:

(2.12)\[ b (\ mathrm {\ mu}, v) = {\ int} _ {\ mathrm {\ Gamma}} _ {C}}\ mathrm {\ mu} [{v} _ {N}] d\ mathrm {\ Gamma}] [{v} _ {N}] d\ mathrm {\ Gamma}\]

And:

(2.13)\[ M=\ left\ {\ mathrm {\ mu}\ in {H} ^ {-1/2} ({\ mathrm {\ Gamma}} _ {C})\ mathrm {:} {\ int} _ {{\ mathrm {\ mu}} _ {\ Gamma}}} _ {C}}\ mathrm {\ mu}}\ mathrm {\ Psi} d\ mathrm {\ int} _ {int} _ {int} _ {int} _ {int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ mathrm {\ Gamma}} _ {\ Gamma}}\ 0\ forall\ mathrm {\ Psi}\ in {H} ^ {-1/2} ({\ mathrm {\ Gamma}} _ {C}),\ mathrm {\ Psi}\ in {H} ^ {-1/2} ({\ mathrm {\ Gamma}} _ {C}),\ mathrm {\ Psi}\ in {H} ^ {-1/2} ({\ mathrm {\ Gamma}}\]

**Note: The problems* () and () are well posed. So we have the existence and uniqueness of solutions \((u,\mathrm{\lambda })\). Also, the solutions \(u\) of problem * () and \(u\) of problem () * are the same.

2.2. Discreet issues

2.2.1. Discreet variational inequality

We consider the discrete contact problem from the point of view of variational inequality (discretization of the problem):

(2.14)\[ \begin{align}\begin{aligned} \ textrm {Find}\\ {u} ^ {h}\ in {K} ^ {h}\\ \ textrm {such as}\\ \ forall {v} ^ {h}\ in {K} ^ {h}\\ a ({u} ^ {h}, {v} ^ {h} - {u} ^ {h}) ≥l ({v} ^ {h} - {u} ^ {h} - {u} ^ {h})\end{aligned}\end{align} \]

where \({K}^{h}\subset {V}^{h}\) is the convex cone of the allowable displacements, \(a\) and \(l\) the bilinear and linear shapes defined above.

To define a method, you must choose the discrete spaces \({V}_{l}^{h}\) and the set \({K}^{h}\). Let \({T}_{l}^{h}\) be a regular triangulation of \({\mathrm{\Omega }}^{l}\subset {\mathrm{ℝ}}^{d}\), where \(d=\mathrm{2,}3\). We then define:

(2.15)\[ {V} _ {l} ^ {h} =\ {{v} _ {l} _ {l} ^ {h}\ in {(C (\ overline {{\ mathrm {\ Omega}}}} ^ {l}}))}} ^ {l} {l} ^ {l} {\ text {|}}} ^ {l}} ^ {l}}} ^ {l}} ^ {l}} ^ {l}}} ^ {l}} {l}} ^ {l}}} ^ {l}}} ^ {l}} {l}}\ in {P} _ {k} (T),\ forall T\ in {T} _ {l} _ {l} ^ {h}, {v} _ {l} ^ {h} =0\ text {on} {\ mathrm {\ Gamma}} _ {l}}} _ {D} ^ {l}\}\]

where \(k=\mathrm{1,}2\), \({V}^{h}={V}_{1}^{h}\times {V}_{2}^{h}\).

It is proposed to use the following contact condition:

(2.16)\[ {K} ^ {h} =\ left\ {{v} ^ {v} ^ {h} ^ {h}\ in {V} ^ {h}\ mathrm {:} {\ int} _ {m}} [{v} _ {N} _ {N} _ {N} {N}} {N} ^ {n} ^ {h}] d\ mathrm {\ h}] d\ mathrm {\ Gamma}\ le 0\ text {}\ forall {T} ^ {m}} [{v} _ {N} _ {N} _ {N} {N} ^ {n} {N} ^ {h}] d\ mathrm {\ Gamma}\ le 0\ text {}\ forall {T} ^ {m}}\ in {T} ^ {m}} M}\ right\}\]

where \({T}^{M}\) is a macro-mesh of \({T}^{m}\) elements to be defined according to the choice of \({V}^{h}\).

This contact condition is inspired by mortar-type methods while maintaining a local aspect. In fact, the mortar projection matrix is no longer a full matrix because the inverse of the base matrix of space \({P}^{0}({T}^{M})\) is diagonal. This property assures us that the mortar projection matrices (matrix product between the inverse base matrix of space \({P}^{0}({T}^{M})\) and the coupling matrices between the mortar space and the approximation spaces) are local. However, we know that the simple type condition:

(2.17)\[ {\ int} _ {{T} ^ {h}\ cap {\ Gamma} _ {C}} [{v} _ {N} ^ {h}] d\ mathrm {\ Gamma}\ the 0\]

is not stable for all elements commonly used in engineering studies (especially in 3D, where only the 27-node hexahedron is naturally stable), and therefore does not lead to optimal mathematical analyses. However, by slightly expanding the integration support, depending on the case, it is possible to demonstrate optimal convergence and stability results.

2.2.2. Discreet mixed formulation

The equivalent mixed problem is introduced:

\[\]

: label: eq-20

textrm {Find}

{u} ^ {h}in {V} ^ {h}

textrm {and \({\lambda }^{h}\in {M}^{h}\) such as}

a ({u} ^ {h}, {v} ^ {h}) -b ({mathrm {lambda}} ^ {h}, {v} ^ {h}) =l ({v} ^ {h}) =l ({v} ^ {h})text {v} ^ {h}) =l ({v} ^ {h}) =l ({v} ^ {h}) =l ({v} ^ {h}) =l ({v} ^ {h}))text {} ^ {h})text {} ^ {h})text {} ^ {h})text {} ^ {h})

b ({mathrm {mu}}} ^ {h} - {mathrm {lambda}} ^ {h}, {u} ^ {h})ge 0text {}forall {forall {mathrm {mu}} - {mathrm {mu}}} ^ {h}} ^ {h}

with:

\[\]

: label: eq-21

{M} ^ {h} ={{mu} ^ {h} ^ {h}in {X} _ {1} ^ {h}mathrm {:} {mu} ^ {h}le 0text {on} {text {on} {on} {Gamma} _ {C}}

Where:

(2.18)\[ {X} _ {1} ^ {h} =\ {{\ mathrm {\ mu}}} ^ {h}\ in {L} ^ {2} ({\ Gamma} _ {C})\ mathrm {::} {\ mathrm {:} {\ mathrm {\ mu}}} {\ mathrm {\ mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}}} ^ {mu}} ^ {mu}}} ^ {mu}} ^ {mu}}} ^ {mu}} ^ {mu}}} ^ {mu}} ^ {0}}} ^ {0}} {T} ^ {m}),\ forall {T} ^ {m}\ in {T} ^ {M}\}\]

We therefore obtain a « mortar \({P}^{0}\) » method that will be automatically stabilized by the definition of the \({T}^{M}\) macro-mesh.

**Note:**With the contact condition in\({K}^{h}`* fixed, the Lagrange multiplier space will always be a subspace of*:math:`{P}^{0}({T}^{m})`*regardless of the type of finite elements used in the displacement approximation space*:math:`{V}^{h}\)*. It is therefore noted that the order of the contact condition remains fix**for any approximation space considered (linear or quadratic), only the definition of the macro-mesh varies to ensure the good mathematical properties of the method.*

2.3. Math results

Here we recall the main mathematical results of method LAC. The fundamental hypothesis of the method defines how to build the macro-mesh is the hypothesis of the degree of internal freedom:

Hypothesis of degree of internal freedom:

For any macro element \({T}^{m}\in {T}^{M}\), there is a \({x}_{i}\) degree of freedom of \({V}_{1}^{h}\) such as \(\mathit{supp}({\mathrm{\varphi }}_{i})\subset {T}^{m}\), where \({\phi }_{i}\) is the base function associated with \({x}_{i}\).

Note: o*n note that there is a minimum size for macro-meshes that allows both to satisfy the previous hypothesis* and to guarantee the locality of the method. It should also be noted that the choice of the trace mesh used to define the macro-mesh is free; here, mesh 1 was chosen. *

We can now recall the two main convergence results of method LAC.

Theorem 1: Be \(u\) and \({u}^{h}\) be the solutions of continuous and discrete contact problems. We assume that*:math:uin ({H}^{s}({mathrm{Omega }}^{1}){)}^{d}times ({H}^{s}({mathrm{Omega }}^{2}){)}^{d}`*with*:math:`d=2,3`*and*:math:`3/2<sle 2`* (*:math:`3/2le s<5/2`*if*:math:`k=2`*). If the hypothesis*of the degree of internal freedom* is satisfied, then there is a constant*:math:`C>0`* *independent of* :math:`{h}_{1} , \({h}_{2}\) (where \({h}_{1}\) and \({h}_{2}\) are the parameters for discretizing the solid \(1\) and the solid and the solid \(2\) respectively) and \(u\) , such that:

(2.19)\[ {\ Green u- {u} ^ {h}\ Green} _ {1, {\ mathrm {\ Omega}} ^ {1}, {\ mathrm {\ Omega}} ^ {2}}\ le C ({h} _ {1} _ {1}} ^ {1} ^ {s-1}} + {h}} _ {2} ^ {s-1}) {\ green u\ green}} _ {s, {\ mathrm} _ {s, {\ mathrm} {\ Omega}} ^ {1}, {\ mathrm {\ Omega}} ^ {2}}\]

Theorem 2: theory*:math:`(u,mathrm{lambda })`*and*:math:`({u}^{h},{mathrm{lambda }}^{h})`*the solutions of continuous and discrete contact problems. We assume that*:math:`uin ({H}^{s}({mathrm{Omega }}^{1}){)}^{d}times ({H}^{s}({mathrm{Omega }}^{2}){)}^{d}`*with*:math:`d=mathrm{2,}3`*and*:math:`3/2<sle 2`* (:math:`3/2<sle 5/2`*if:math:`k=2`*). If the hypothesis*of internal DDL*is satisfied, then there is a constant*:math:`C>0`*independent of*:math:`{h}_{1}`*, :math:`{h}_{2}`*and:math:`u`*, such that: *

(2.20)\[ {\ green\ mathrm {\ lambda}} - {\ mathrm {\ lambda}}} ^ {h}\ green} _ {1/2,\ ast, {\ mathrm {\ Gamma}} _ {C}}} _ {C}}}}} + {\ green u- {u}}} + {\ green u- {u}}} {u}} ^ {u}}} + {\ green u- {u}} ^ {u}} ^ {u}} ^ {u}} ^ {u}} ^ {1}} + {\ green u- {u}} ^ {u}} ^ {u}} ^ {u}} ^ {1}} + {\ green u- {u}} ^ {u}} ^ {u}} ^ {1}} + {\ green Omega}} ^ {2}}\ the C ({h} _ {1} _ {1} ^ {s-1} + {h} _ {2} ^ {s-1}) {\ green or\ green} _ {s, {\ mathrm {\ Omega}} {\ omega}}} ^ {omega}} _ {2}}\]

where \({\Vert \text{.}\Vert }_{1/\mathrm{2,}\ast ,{\mathrm{\Gamma }}_{C}}\) is the dual standard of \({\Vert \text{.}\Vert }_{1/2,{\mathrm{\Gamma }}_{C}}\) .

Note: d In the rest of this document, the following convention will be adopted: the surface on which the macro-meshes have been defined will be called slave surface, that is to say the area carrying the contact Lagrange multipliers in the case of the mixed formulation. The surface facing each other with the latter will be the master surface.

2.4. Matrix formulation

Consider the following matrices \({C}^{E}\in {M}_{e,k}(\mathrm{ℝ})\), \({C}^{M}\in {M}_{m,k}(\mathrm{ℝ})\), and \({M}^{\mathit{LAC}}\in {M}_{k}(\mathrm{ℝ})\), where \({M}_{i,j}\) corresponds to the space of rectangular matrices of size \(i\times j\) and \({M}_{i}={M}_{i,i}\) corresponds to the space of square matrices of size \(i\), defined by:

(2.21)\[ \begin{align}\begin{aligned} {C} _ {\ mathit {ij}}} ^ {E}} = {\ int} = {\ int} _ {j}} {\ left ({\ mathrm {\ varphi}}} _ {N} ^ {E}\ right)} _ {i} d\ mathrm {\ varphi}} _ {\ varphi}} _ {\ varphi}} _ {N} ^ {E}\ right)} _ {i} d\ mathrm {\ varphi}} _ {\ varphi}} _ {N} ^ {E}\ right}\\ {C} _ {\ mathit {ij}}} ^ {M}} = {\ int} = {\ int} _ {j}} {\ left ({\ mathrm {\ varphi}}} _ {N} ^ {M}\ right)} _ {i} d\ mathrm {\ varphi}} _ {\ varphi}} _ {\ varphi}} _ {N} ^ {M}\ right)} _ {i} d\ mathrm {\ varphi}}} _ {\ varphi}} _ {\ varphi}} _ {N} ^ {M}\ right}\\ {M} _ {\ mathit {ij}}} ^ {\ mathit {LAC}} = {\ mathrm {\ delta}} _ {\ mathit {ij}}\ text {|} {|} {T} {T} {T}} ^ {i}\ text {|}\end{aligned}\end{align} \]

where \({T}^{i}\) are the carriers of the basic functions of \({P}^{0}({T}^{M})\), the \({\mathrm{\varphi }}_{N}^{E}\) are the products of the normal with the basic functions associated with the slave surface and the \({\mathrm{\varphi }}_{N}^{M}\) are the products of the normal with the basic functions associated with the master surface.

Let \(U\) be the displacement field and \(\mathrm{\Lambda }\) the Lagrange multiplier, the matrix formulation of the problem is as follows:

(2.22)\[\begin{split} \ left [\ begin {array} {cccc} {K} _ {\ mathit {NN}} _ {\ mathit {NN}}} & {K} _ {\ mathit {NE}}} & 0\\ {K}} & 0\\ {K}} _ {\ mathit {M}} _ {\ mathit {ME}}} & {C} _ {\ mathit {ME}}} & {C}} ^ {M}\\ {K} _ {\ mathit {EN}}} & {K}} _ {\ mathit {EM}} & {K} _ {\ mathit {EE}} & {C} ^ {E}\ end {array}\ {E}\\ {E}\\}\ {E}\\ {E}\\ {E}\\ {E}\\ {E}\\ {E}\\ {E}\\ {U}\\ right]\ right]\ times\ right]\ times\ left [\ begin {array} {EM}} {M}\\ {U} _ {E}\\ mathrm {\ Lambda}\ end {array}\ right] =F\end{split}\]

With the following conditions:

\[\]

: label: eq-27

{M} ^ {{mathit {LAC}}} ^ {-1}}} ^ {-1}}} ({text {}}} ^ {t} {C} ^ {E} {U} _ {E} + {text {}}}} ^ {t}} {t} {t} {T}} {C}} ^ {M}) ≤0

textrm {in \({\mathrm{ℝ}}^{k}\),}

{M} ^ {mathit {LAC}}}mathrm {Lambda} ≤0

textrm {in \({\mathrm{ℝ}}^{k}\),}

{M} ^ {{mathit {LAC}}} ^ {-1}}} ^ {-1}}} ({text {}}} ^ {t} {C}} _ {E} + {text {}}} ^ {t} {t} {C} {C} {C} {C}} ^ {M}})cdot {M}} ^ {mathit {LAC}}}mathrm {Lambda} =0

textrm {in \(\mathrm{ℝ}\).}

Noticing that \({M}^{\mathit{LAC}}\) is a positive definite diagonal, we have equivalence with the following formulation:

\[\]

: label: eq-28

left [begin {array} {cccc} {K} _ {mathit {NN}} _ {mathit {NN}}} & {K} _ {mathit {NE}}} & 0\ {K}} & 0\ {K}} _ {mathit {M}} _ {mathit {ME}}} & {C} _ {mathit {ME}}} & {C}} ^ {M}\ {K} _ {mathit {EN}}} & {K}} _ {mathit {EM}} & {K} _ {mathit {EE}} & {C} ^ {E}end {array}{E}\ {E}\}{E}\ {E}\ {E}\ {E}\ {E}\ {E}\ {E}\ {U}\ right]right]timesright]timesleft [begin {array} {EM}} {M}\ {U} _ {E}\ mathrm {Lambda}end {array}right] =F

With the following conditions:

(2.23)\[ \begin{align}\begin{aligned} {\ text {}} ^ {t} {C} {C} ^ {E} ^ {E} {E} + {\ text {}} ^ {t} {C} ^ {M} {U} _ {M} ≤0\\ \ textrm {in :math:`{\mathrm{ℝ}}^{k}`,}\\ \ mathrm {\ Lambda} ≤0\\ \ textrm {in :math:`{\mathrm{ℝ}}^{k}`,}\\ {\ text {}} ^ {t} {C} {C} ^ {E} {E} {U} {U} _ {E} + {\ text {}} ^ {t} {C} ^ {M} _ {M} _ {M}\ cdot {M}}\ cdot {M}}\ cdot {M}} {M}\ cdot {M} {M}\ cdot {M} {M}\ cdot {M}} {M}\ cdot {M} {M}\ cdot {M}} {M}\ cdot {M}} {M}\ cdot {M} M} {M} LAC\\ \ textrm {in :math:`\mathrm{ℝ}`.}\end{aligned}\end{align} \]

The problem () - () is solved by using an active set strategy coupled with a Newton algorithm.