3.4 Nonlinear Multiple Point Constraints

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Sometimes there are extra constraints that are not covered by the constitutive equations.

The simplest ones are single point constraints, expressing that a degree of freedom has

GEOMETRIC NONLINEAR EFFECTS 155

to assume a specific value. These are simple boundary conditions. In other cases, a relationship

is established among several degrees of freedom. These are called multiple point

constraints (MPC). They can be linear or nonlinear. Linear multiple point constraints were

encountered in Chapter 2, for instance, in Section 2.10 on cyclic symmetry. Examples of

nonlinear equations are given in the following sections and include rigid body motion,

incompressible behavior and others. In Section 2.6, it was shown that a linear multiple

point constraint can be taken care of right away at the creation time of the stiffness matrix

by expressing the dependent degree of freedom as a function of the independent degrees

of freedom. A nonlinear multiple point constraint can be treated in the same way after

linearization.

The linearization follows exactly the scheme sketched in Section 3.1. Let

U := _ui1, ui2, . . . , uin_ (3.55)

be the degrees of freedom involved in the nonlinear multiple point constraint f (U) = F.

Then, a linearization at U = U0 yields

f (U0)+∇fU (U0) ・ (U − U0) = F (3.56)

or

∇fU (U0) ・ _U = F − f (U0) (3.57)

where

_U := U − U0. (3.58)

This equation is updated as soon as a new solution U0 is obtained. Notice that not only can

the coefficients of a linearized multiple point constraint change from iteration to iteration,

but also the degrees of freedom involved. This can lead to a change of the dependent

degrees of freedom as the calculation proceeds.

Accordingly, a stream chart of a nonlinear solution procedure that includes nonlinear

multiple point constraints looks like the one shown in Figure 3.8. The box “update MPC”

not only stands for the update of the multiple point constraints but also for the update of

any solution dependent boundary conditions such as contact areas or radiation heat flux

rates.

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