A rigid domain, or a rigid body, is an idealization of a body in which the deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of any external forces acting on it. An object can be assumed to be perfectly rigid if its flexibility can be neglected in comparison with other flexibilities in the system, and when there is no need to compute the stress in the object. To model a rigid domain, you use the Rigid Material material model.

The Rigid Material is a material model, which is mutually exclusive to all other material models. The only material property needed is the mass density.

In 2D, this is represented by two in-plane translations and the scalar rotation around the z-axis.

In 3D the situation is more complex. Six degrees of freedom are necessary. They are usually selected as three translations and three parameters for the rotation. For finite rotations, any choice of three rotation parameters is will give a singularity at some specific set of angles. For this reason, a four-parameter quaternion representation is used for the rotations in COMSOL Multiphysics. Thus, each rigid domain in 3D actually has seven degrees of freedom: three for the translation, and four for the rotation. The quaternion parameters are called a, b, c, and d. These four parameters are not independent, so an extra equation stating that

The connection between the quaternion parameters and the rotation matrix R is:

In 2D, the rotational degree of freedom is the angle of rotation about the z-axis ϕ, and its relation with the rotation matrix R is:

where X are the material coordinates of any point in the rigid domain, XM is the center of mass of the rigid domain, u is the translation vector at the center of mass, and I is the identity matrix.

The rigid body displacement at the center of mass (u) are degrees of freedom. Thus, the rigid body translational velocity and acceleration can be evaluated by directly taking the time derivatives of u. In the time domain it can be expressed as:

In the frequency domain, they can be expressed in terms of frequency (ω):

The same is true for the rotation in 2D since the rigid body rotation ϕ is the degree of freedom. The rigid body angular velocity and acceleration can be evaluated by directly taking the time derivatives of ϕ.

The parameter a can be considered as measuring the rotation, while b, c, and d can be interpreted as the orientation of the rotation vector. For small rotations, this relation simplifies to:

Here is the conjugate of q, and denotes quaternion multiplication.

The inertial properties mass (m) and moment of inertia tensor (I) of a rigid domain are computed as:

where E3 and XM are the identity matrix and the center of mass of a rigid domain, respectively. The special case for the Shell interface is described in Rigid Domain for Shells.

where the volume integration has been replaced by an area integration multiplied by the out-of-plane thickness h.

Here, the subscripts I and ext denote inertial and external forces, respectively, and R is the current rotation matrix. The inertial forces are contributions from Mass and Moment of Inertia nodes.

The Rigid Material is a separate material model, and it overrides the default Linear Elastic Material node and the default Initial Values node in the physics interface. The initial values are given in a separate Initial Values subnode for each Rigid Material.

Given the initial values of translation (u), rotation (ϕ), translational velocity () and angular velocity (ω) about a center of rotation (Xc), the rigid body displacement and quaternion degrees of freedom are initialized as:

The variable ur is the translation at the center of mass due to a rotation around the center of rotation, and is thus zero when the two points coincide. In the case that you are entering the data using a separate center of rotation, you must pay special attention to how the initial displacement and velocity are composed if initial rotations and rotational velocities are present.

Sometimes a rigid domain needs the added effect of an associated abstract rigid object, which is physically not modeled and where the inertial properties are known. You can model this using Mass and Moment of Inertia, where the inertial properties of this abstract domain (center of mass, mass, and moment of inertia tensor) can be directly entered.

where Xmc is the vector from the center of mass of the rigid domain (XM) to the center of mass of this contribution (Xm),

The Prescribed Displacement/Rotation node can be used to:

The components of this displacement vector are prescribed individually in the selected coordinate system. Through Equation 3-165, a constraint on a translation will impose a relation between translational and rotational degrees of freedom if the center of rotation differs from the center of mass.

where and ϕ0 are the axis of rotation and angle of rotation, respectively.

The loads available for a flexible domain can also be used for a rigid domain. In addition to these boundary conditions, a rigid domain also has global subnodes for applying forces and moments. If you use Applied Force, a force and its location can be prescribed in a selected coordinate system. A force implicitly also contributes to the moment unless it is applied at the center of mass of a rigid domain. If an Applied Moment node is used, a moment can be prescribed in a selected coordinate system.

where X is a coordinate on the boundary. If rotational degrees of freedom are present, which is the case in the Shell and Beam interfaces, the rotations are set equal to those of the rigid domain.