7. Theoretical convergence results#
For frictionless problems, [1] shows a convergence demonstration for the active stress method.
For problems with friction, convergence results with uniqueness of the solution for the discretized problem are established in [1] for low values of the Coulomb coefficient of friction. The results are obtained using a fixed point algorithm associated with a Lagrange multiplier method. For each contact problem solved, the associated friction problem is studied. Once this one is resolved, a new contact problem is solved and so on. However, these methods are different from those presented here and it is therefore not possible to present theoretical convergence results for the latter.
7.1. Redistribution of the time step#
Theoretically, the convergence of the active constraints method is ensured in a finite number of iterations. In practice, certain digital artifacts can make this convergence difficult. A strategy has therefore been developed to ensure the robustness of the algorithm.
During contact calculations, especially if the load steps taken are too large, undesirable phenomena may occur:
The contact matrix is singular,
Oscillation of the active stress method: a node is detected alternately « stuck » and then « unstuck ».
To overcome these difficulties, the following strategy has been adopted. If:
The contact matrix is singular,
The number of active constraint iterations is greater than a limit that depends on the number of potential links. This number is set to twice the total number of slave nodes for the active constraints method, and to ITER_CONT_MULT times the total number of slave nodes for the other methods.
So we recut the time step i.e. we go back to the previous load step and instead of trying to reach the next load level in one step as we just did, we do several (For more details on this functionality of the STAT_NON_LINE operator, see the documentation [U4.51.03]).
7.2. Compatibility with Dirichlet boundary conditions#
In the case of methods with Lagrange multipliers, we can observe incompatibilities with the fact of imposing Dirichlet-type boundary conditions. Indeed, the problem must physically make sense. We cannot deal with a contact problem in the direction of the \(z\) axis if all the points have zero displacement along \(z\). As we will see, dealing with such a problem leads to a singularity of matrices of the type \(\left[{A}^{\text{c}}\right]\mathrm{.}{\left[K\right]}^{\mathrm{-}1}\mathrm{.}{\left[{A}^{\text{c}}\right]}^{T}\) with the treatment of Dirichlet boundary conditions by double lagrange of Code_Aster.
7.2.1. Writing boundary conditions#
Inspired by the reference documentation [R5.03.01] from STAT_NON_LINE, the dualization of Dirichlet boundary conditions leads to the following system of equations to be solved:
: label: eq-153
mathrm {{}begin {array} {c} {c}left [Kright]mathrm {.} left{delta uright} + {left [Bright]} ^ {T}mathrm {.} left{deltalambdaright}mathrm {=}mathrm {=}left{{L} _ {i} ^ {text {int}}right}mathrm {-}left{-}left\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ left{{u} _ {i}right}right}mathrm {=}left{{u} _ {i} ^ {d}right}mathrm {-}left {-}left [Bright]mathrm {.} left{{u} _ {imathrm {-} 1}right}end {array}
We then denote \(\left[C\right]\) the stiffness matrix of the system such that:
: label: eq-154
left [Cright]mathrm {=}}left [begin {array} {cc}left [Kright] & {left [Bright]} ^ {T}\left [Bright]\ left [Bright] &left [0right]end {array}right]
This matrix has an inverse of the form:
: label: eq-155
{left [Cright]} ^ {mathrm {-} 1}mathrm {-} 1}mathrm {=}left [Cright]}left [Eright] &left [Fright] &left [Fright]left [Fright]left [Fright]left [Fright]left [Fright]
such as: \(\left[E\right]\mathrm{.}{\left[B\right]}^{T}\mathrm{=}\left[0\right]\). We thus check that for each condition at limits \(l\) we have the property \(\left[E\right]\mathrm{.}{\left[{B}_{l}\right]}^{T}\mathrm{=}\left[0\right]\).
7.2.2. Back to the contact problem#
The matrix \(\left[{A}^{\text{c}}\right]\mathrm{.}{\left[K\right]}^{\mathrm{-}1}\mathrm{.}{\left[{A}^{\text{c}}\right]}^{T}\) can also be written \(\left[{A}^{\text{c}}\right]\mathrm{.}\left[E\right]\mathrm{.}{\left[{A}^{\text{c}}\right]}^{T}\) since the link vectors \(\left[A\right]\) only involve the degrees of freedom of movement.
The result then is that if a bond vector \(J\) in the \(\left[A\right]\) matrix is a linear combination of Dirichlet boundary conditions, it verifies the following property: \(\left[E\right]\mathrm{.}{\left[{A}_{J}\right]}^{T}\mathrm{=}\left[0\right]\). The matrix \(\left[{A}^{\text{c}}\right]\mathrm{.}\left[E\right]\mathrm{.}{\left[{A}^{\text{c}}\right]}^{T}\) is then singular because it has a column of zeros. In practice, without any particular treatment, the code ends up with a stop message of the stop type on a singular touch-friction matrix. The detection of these singular columns has been implemented in the code in order to eliminate such types of relationships from contact and friction and to avoid the interruption described above.
The result then is that if a bond vector \(J\) of the \(\left[A\right]\) matrix contains a linear combination of Dirichlet boundary conditions and is written \(\left[{A}_{J}\right]\mathrm{=}\mathrm{\sum }{\alpha }_{i}\mathrm{.}\left[{B}_{i}\right]+\left[{\stackrel{ˉ}{A}}_{J}\right]\), it verifies the following property: \(\left[E\right]\mathrm{.}{\left[{A}_{J}\right]}^{T}\mathrm{=}\left[E\right]\mathrm{.}{\left[\stackrel{ˉ}{{A}_{J}}\right]}^{T}\). We can then have a singular \(\left[{A}^{\text{c}}\right]\mathrm{.}\left[E\right]\mathrm{.}{\left[{A}^{\text{c}}\right]}^{T}\) matrix because it has two identical rows. This detection is currently not available in the code and we end up in the code with a stop message of the stop type on a singular touch-friction matrix.
Note:
This compatibility problem between contact and friction and boundary conditions does not appear with regularized methods insofar as stiffness is added to the overall stiffness and no elimination is done as in the calculation of the m multipliers of L*agrange.*