Spherical to Spherical Formulation
This formulation is used to model contact between two spherical rigid bodies. When both source and destination are rigid spheres, a center-based contact method is used. The gap between source and destination boundaries is computed. If the gap is less than zero, a penalty force is applied to prevent penetration.
The gap distance is computed from the spatial positions of the source and destination centers, and the radii of the source and destination spheres. The gap distance computation depends on the location of the source with respect to the destination. If the source is located outside the destination, the outer boundaries of the destination may come in contact with the source. If the source is located inside the destination, the inside boundaries of the destination may come in contact with the source.
When the source is outside the destination, the gap is defined as
Here, d is the distance between source and destination centers, rs is the radius of the source sphere, and rd is the radius of the destination sphere.
Figure 3-25: Rigid body contact between a spherical source and a spherical destination. The source is located outside the destination.
If the source is inside the destination sphere, the gap is defined as
The center distance between the source and destination spheres (d) is calculated as
Here, Xsrc and Xdst are the undeformed locations of the source and destination centers, and usrc and udst are the corresponding displacements.
A direction vector from the source center to the destination center (ec) is defined as
Figure 3-26: Rigid body contact between a spherical source and a spherical destination showing the gap distance. Here, the source is located inside the destination.
The Penalty Method
This method is based on penalization of the physical gap such that an approximate solution for the nonpenetration condition is obtained. The contact force (Fn) is calculated as
where pn is the penalty factor. If the penalty factor control is automatic, the value of pn is calculated from the penalty factor control multiplier fp as
In this expression, diag is the geometric diagonal of the geometry’s bounding box.
The Penalty, Dynamic Method
This method is based on a viscous formulation for transient simulation. Here, the contact force is calculated based on penalization of the rate of the physical gap. The contact force (Fn) is calculated as
where ∂gap / ∂t is the time derivative of the gap distance and pnv is the viscous penalty factor.
If the viscous penalty factor control is automatic, the value of pnv is calculated from the characteristic time τn as τnpn.
If the penalty factor control is viscous only, the stiffness term of the penalty contact are omitted in the formulation. In this case, the contact force (Fn) is calculated as
Here, if viscous penalty factor control is automatic, the value of pnv is calculated from the characteristic time τn as
In this expression, diag is the geometric diagonal of the bounding box of the geometry.