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General description of the condition

This conditions represents the main part of the algorithm to solve contact problems. The algorithm is based on a surface to surface augmented lagrange/updated penalty formulation using a master and a slave contact surface. It is formulated to solve frictional contact problems in structural mechanics, the algorithm is formulated in a finite deformation context.

This condition and the underlying algorithm for contact problems has not been tested by means of a broader range of benchmark examples so far. It seem tio work quite fine for a number of different examples, but I do not give a guarantee that it is free of errors in its formulation or its implementation.

Theoretical background of the used formulation

Formulation without friction

Two bodies Ωi staying in contact over a contact surface \Gamma^c_i
Slave Γ1 and master Γ2 contact surface, and gap g=\nu\cdot\left(\boldsymbol{y}_2-\boldsymbol{y}_1\right) between the surfaces
Three dimensional view on the contact surfaces and the gap between the two surfaces

The contact of two (or more) bodies in three dimension is a constrained continuum mechanical problem. The surface \partial\Omega =\Gamma of the contacting continua Ω1 and Ω2 are constructed by the Neumann boundary Γσ, the Dirichlet boundary Γu and the contact boundary Γc. For the treated area Ω the following assumptions hold:

  • \Omega = \Omega_1 \cup \Omega_2 and \Omega_1 \cap  \Omega_2=0,
  • having the boundaries \partial\Omega=\Gamma^{\sigma}_i\cup\Gamma^{u}_i\cup\Gamma^{c}_i and \Gamma^{\sigma}_i\cap \Gamma^{u}_i=0, \Gamma^{\sigma}_i\cap \Gamma^{c}_i=0, \Gamma^{u}_i\cap \Gamma^{c}_i=0

The problem is constrained by the prescription of a penetration of the two bodies, e.g. of the penetration of the master surface Γ2 by the slave surface Γ1. This constrained is expressed by the gap function defined as the scalar projection of the distance vector between the point on the slave surface \mathbf{x}\in \Gamma^1 and its closest point projection onto the slave surface \mathbf{y}(\mathbf{x})=\mbox{arg}\;\mbox{min}_{\mathbf{y}\epsilon\Gamma_2^c}\| \mathbf{x}-\mathbf{y}\left(\mathbf{x}\right)\| and the normal vector ν on the master surface:

  • g\left(\mathbf{x}\right)=-\mathbf{\nu}\left(\mathbf{x}-\mathbf{y}\right)

The constrained that defines the contact problem is now stated as:

  • g(x)\leq 0

Within the finite element formulation the problem is now solved by use of an updated penalty treatment of the contact constrained. To solve the problem the Energie of the system must be minimised by fulfilling the penetration constraint. After having introduced a contact stress

  • tN = < λN + εNg >

that acts on the surfaces if they are in contact the constraint can be rewritten in form of Kuhn-Tucker constraints:

  • g\leq 0\;\;\;\;\;\;t_N\geq 0\;\;\;\;\;\;g t_N=0\;\;\;\forall\;\mathbf{x}\epsilon\Gamma^c

Whereas the contact stress tN consists of an (outer) Lagrange value λN, which is constant during the Newton iteration, and an inner penaltization of the gap εNg. The updated penalty method now penalizes violation of the constraint, leading to the following potential:

  • \Pi^c=\sum_{i=1}^2\int_{\Gamma_c^{i}}\frac{1}{2\epsilon_n}<\lambda_N+\epsilon_Ng>^2-\frac{1}{2\epsilon_n}\lambda_N^2d\Gamma^i

The minimization of the potential is now been done by its variation:

  • \delta W^c=\sum_{i=1}^2\int_{\Gamma_c^{i}}\delta g<\lambda_N+\epsilon_Ng>d\Gamma^i

Together with the minimisation of the system energy

  • \delta W^{int, ext}=\sum_{i=1}^2\delta W_i^{int, ext}

the minimisation of the system energie with fulfillment of the penetration constraint can be written as

  • δW = δWint,ext + δWc = 0

This equation states a initial boundary value problem and can be solved by normal discretization methods in time and space. The algorithm for the solution of the contact problem now consists of an inner iteration (Newton) and an outer iteration (Uzawa):

  • start simulation with λ = 0
  • Uzawa loop
    • Solve \delta W=\delta W^{int, ext}+\delta W^c\left(\lambda^{uz}_N=\mbox{const.}, \epsilon_N\right)=0 by using Newtons loop
    • update the Lagrange Multiplier \lambda_N^{uz+1}=<\lambda_N^{uz}+\epsilon_Ng>
    • next Uzawa iteration uz\rightarrow uz+1

Formulation with friction

If now friction does occur between the contacting surfaces a constrained to for the used friction law has to be considered. In case of Coulomb friction this friction law is stated as:  |\boldsymbol{u}_T|\left\{\begin{array}{c}=0\; if t_T\leq\mu t_N\\\geq 0 if t_T>\mu t_N\end{array}\right\}


  • T.A. Laursen Computational Contact and Impact Mechanics, Springer, 2002
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