The contact constraint in the normal direction to the contacting boundaries can be represented by the nonpenetration condition. On the destination boundary Γdst, the normal contact condition can in a weak sense be written as (the contribution of normal contact to the virtual work)

where Tn is the contact pressure, assumed positive in compression. To complete the contact condition, Equation 3-176 is subjected to the Kuhn–Tucker conditions

where the first inequality represents the nonpenetration condition. Equation 3-176 and Equation 3-177 together represents a unilateral constraint that require special techniques to implement. Several different techniques for doing so are available in COMSOL Multiphysics, which can be divided into two main classes: the penalty method, the augmented Lagrangian method, and the Nitsche method.

An alternative way of expressing the contact condition is to view the contact pressure as a force acting on both source and destination described as a distributed load. Making use of Newton’s second law, an alternative to Equation 3-176 can then be formulated as

In principle, Equation 3-176 and Equation 3-178 describes the same contribution to the virtual work. However, when the system of equations are linearized they will produce different results. The main difference originates from the variation δgn with respect to u that, for the ray-tracing mapping strategy used by COMSOL Multiphysics, includes several terms that can be difficult and expensive to evaluate. These are not present if Equation 3-178 is used, which can increase the robustness of the solution. What formulation that is used can be controlled by the Jacobian contribution setting in the Contact node. When set to Symmetric or Legacy, Equation 3-176 is used. For Symmetric, some terms of δgn are omitted to improve the robustness, whereas Legacy uses the full variation of δgn. When the Nonsymmetric option is selected, Equation 3-178 is used.

For the penalty method, the contact pressure in Equation 3-176 is represented using a penalization of the physical gap so that

where pn is the penalty factor, and p0 is the pressure at zero gap. This leads to an approximative solution of the nonpenetration condition. The penalty method as defined by Equation 3-179 is conceptually based on the insertion of a nonlinear spring between the contacting points. The penalty factor can be interpreted as the stiffness of that spring. A high value of the stiffness enforces the contact conditions more strictly, but if it is too high, it makes the problem ill-conditioned and unstable.

The pressure at zero gap, p0, can be used to reduce the overclosure between the contacting boundaries if an estimate to the contact pressure is known. In the default case, when p0 = 0, there must always be some overclosure in order for a contact pressure to develop.

In addition to Equation 3-179, COMSOL Multiphysics also provides an alternative penalization based on a viscous formulation intended for transient simulations, referred to as the Penalty, dynamic method. This formulation is based on the additional persistency condition

where gn,old is the physical gap distance at the previous increment, and <> is the positive parts operator. With the definition in Equation 3-180, Tn is only nonzero if the physical gap is negative during the entire increment, which is important for energy conservation during the initial contact event. However, there will always be some initial penetration before the gap rate is penalized. Hence the observed overclosure will be dependent on the size of the increment when two points come into contact. Note that this definition of the contact pressure implies that pressure contact is dissipative.

It is possible to use the definitions of Tn in Equation 3-179 and Equation 3-180 simultaneously. In such a case, they are additive and conceptually represent a parallel coupling of a spring and a dashpot.

where Tnp is the penalized (or augmented) contact pressure, and Tn is a Lagrange multiplier that can be identified as a contact pressure. Alternatively, Equation 3-178 can be augmented, giving the following result

The Lagrange multiplier Tn is defined as a dependent variable of the contact problem and is typically discretized using Lagrange shape functions. There are two methods for solving the above coupled system of equations, and this choice affects the definition of Tnp.

The first solution method is referred to as a Segregated method and corresponds to the so-called Uzawa algorithm. Here, the system of equations is decoupled, and the displacement field u and the Lagrange multiplier Tn are solved in a segregated way. This leads to two levels of iterations in the solver, where Tn is held constant when solving for u, and vice versa when updating Tn. For this solution method, the penalized contact pressure Tnp is defined with a smooth transition to stabilize the problem when transitioning between contact states. The definition for Tnp for outer iteration j+1 is defined by

The segregated method can then for increment n+1 schematically be described as

The appropriate solver sequence is automatically set up by COMSOL Multiphysics for the Segregated solution method when adding a new solver, or resetting to the default solver.

The alternative is to use a Fully Coupled solution method, which, as the name implies, amounts to solving the coupled system in Equation 3-181 or Equation 3-182 for the two dependent variables u and Tn (and any other variables included in the multiphysics model). When using this method, the penalized contact pressure is defined by

The Fully Coupled method can be convenient as it enforces the nonpenetration condition exactly over the contacting boundaries, while it puts no restriction on how to set up the solver sequence. This flexibility can be especially advantageous when working with multiphysics problems, compared to the segregated approach. However, one should be aware that the method comes with some of the drawbacks of a pure Lagrange multiplier approach, albeit to a lesser extent. One drawback is that the solution typically is a saddle point, and convergence can be sensitive to how Newton iterations are made when solving the nonlinear problem.

In the limit, the Segregated and the Fully Coupled solution methods converge to the same solution, but they can have different convergence properties.

In addition to the standard augmented Lagrangian method outlined above, an Augmented Lagrangian, dynamic method is available for transient simulations. As discussed for the penalty method, this formulation is based on the addition of the persistency condition. A segregated solution method is recommended for this method, and the main difference compared to the standard method lies in the definition of the penalized contact pressure. Both the nonpenetration and persistency conditions are enforced by penalizing the rate of the physical gap distance so that

where gn,old is the physical gap distance at the previous increment, and <> is the positive parts operator. The same segregated solution strategy as described above is applied also when the dynamic method is used.

The Nitsche’s method for contact is an extension of the general method suggested by J. Nitsche in 1971 to impose Dirichlet conditions in a weak sense without having to add Lagrange multipliers. A general derivation of the method for such applications is presented in Continuity Condition. Here the method is applied to large deformation contact using Equation 3-178 as a starting point, and following similar steps as in Ref. 1 to derive the final equations. The first step is to reformulate the contact pressure from Equation 3-179 as

where P is the first Piola-Kirchoff stress, N is the material normal, n is the spatial normal, and Ta is the nominal traction vector. The positive parts operator is used above to indicate that Tn is strictly positive. The term from Equation 3-178 in brackets including δu is also reformulated as

where θ is a parameter used to control the symmetry of the formulation. Using the above definitions, Equation 3-178 can, after some manipulation and simplification, be rewritten to obtain the weak contribution added by the Nitsche method for normal contact

From the above weak equation several variants of Nitsche’s formulation for contact can be set up using different values for the parameter θ:

Note that since the derivation Equation 3-184 started from Equation 3-178, all of the above choices of θ will cause a nonsymmetric stiffness matrix when setting of the global system of equations. Also, the penalty factor pn for the Nitsche’s method can be considered as a stabilization factor, not directly as a spring stiffness as in the penalty method.

The Penalty, dynamic and Augmented Lagrange, dynamic methods are based on a viscous type formulation, and dissipate energy when the contact is active. The accumulated dissipated energy Wcntv can be computed by solving a distributed ODE on the destination boundary: