Analytical Rigid Surface Definition

Using analytical rigid surfaces instead of defining element-based rigid surfaces provides two important advantages in contact modeling.

• Many curved geometries can be modeled exactly with analytical rigid surfaces because of the ability to parameterize the surface with curved line segments. The result is a smoother surface description, which can reduce contact noise and provide a better approximation to the physical contact constraint.

• Using analytical rigid surfaces instead of rigid surfaces formed by element faces may result in decreased computational cost incurred by the contact algorithm.

  The use of curved line segments instead of many linear facets will decrease the time spent in contact tracking operations. Additional computational savings may be realized in three dimensions because of the intrinsic two-dimensional descriptions of the analytical surfaces.

There are also some disadvantages to using analytical rigid surfaces for contact modeling.

• An analytical rigid surface must always act as a main surface in a contact interaction. Therefore, contact cannot be modeled between two analytical rigid surfaces.

• Contact forces and pressures cannot be contoured on an analytical rigid surface. However, contact forces and pressures can be plotted on the secondary surface.

• The use of a very large number (thousands) of segments to define an analytical rigid surface can degrade performance. In most cases it is not necessary to use a large number of segments to define an analytical rigid surface, because curved segment types are allowed. In rare cases in which a very large number of segments would be necessary, the analysis may be more efficient if an element-based rigid surface is used instead (see Element-Based Surface Definition).

• An analytical rigid surface does not contribute to the mass and rotary inertia properties of the rigid body with which it is associated. Therefore, if the mass distribution on an analytical rigid surface needs to be accounted for, equivalent mass and rotary inertia properties must be defined for the rigid body by using MASS and ROTARYI elements, or a finite element discretization of the surface should be used instead of an analytical rigid surface (see Rigid Body Definition).

• In Abaqus/Explicit reaction force output for a rigid body containing an analytical rigid surface is calculated only for constraints that are active at the reference node (e.g., constraints specified as boundary conditions). If the net contact force on the rigid body corresponding to an unconstrained degree of freedom is desired, it must be calculated from the rigid body's acceleration and mass.

The surface profile is the collection of line segments defining the cross-section of the surface. The surface type determines whether the profile is swept (cylindrical surfaces), revolved (surfaces of revolution), or, in the two-dimensional case, used as is (planar surfaces).

You construct a profile by providing the endpoint of each line segment in the profile; the starting point is always the endpoint of the previous segment, or, in the case of the first segment, the point specified as the starting point. The center points of circular arcs must be given. Abaqus can define only arcs that are less than 179.74°; thus, it will use the shorter arc defined by the data provided (use two adjacent arcs to define a longer arc). For parabolic arcs you must give a third point that lies on the parabola and within the arc.

To define a planar rigid surface, specify the line segments forming the rigid surface's profile in the global coordinate system. If the analytical surface is being defined inside a part, specify the line segments in the local part coordinate system.

To define a cylindrical rigid surface in a model that is not defined in terms of an assembly of part instances, specify the points a, b, and c shown in Figure 2 that define the local coordinate system.

Cylindrical rigid surface.

Give the coordinates of these points—($X_a,Y_a,Z_a$), ($X_b,Y_b,Z_b$), and ($X_c,Y_c,Z_c$)—in the default global coordinate system. As shown in Figure 2, point a defines the origin of the local system; point b defines the local x-axis; and point c defines the generator vector, which is the negative local z-axis. If the segment $ac$ is not perpendicular to $ab$, Abaqus will automatically adjust point c within the plane defined by points a, b, and c, such that they become perpendicular. The line segments forming the profile of the rigid surface are defined in the local x–y plane. The three-dimensional surface is formed by sweeping this profile along the generator vector. The resulting surface extends to infinity in both the positive and negative directions of the generator vector.

To define a cylindrical rigid surface within a part, specify the line segments forming the profile of the rigid surface in the part coordinate system. For an analytical surface defined within a part (or part instance), point a is located at the origin of the part coordinate system, point b is located on the part x-axis, and point c is located on the negative part z-axis. If the segment $ac$ is not perpendicular to $ab$, Abaqus will automatically adjust point c within the plane defined by points a, b, and c, such that they become perpendicular. You cannot redefine this analytical surface coordinate system; instead, you can position the surface in the model by giving positioning data when you instance the part (see Assembly Definition).

To define a rigid surface of revolution in a model that is not defined in terms of an assembly of part instances, specify the two points a and b shown in Figure 3 that define the local coordinate system.

Rigid surface of revolution.

Give the coordinates of these points—($X_a,Y_a,Z_a$) and ($X_b,Y_b,Z_b$)—in the default global coordinate system. As shown in Figure 3, point a defines the origin of the local system, and the vector from a to b defines the local z-axis, which is the axis of a cylindrical coordinate system. The line segments forming the profile of the surface of revolution are defined in the local r–z plane, where the local r-axis aligns with the radial axis of the cylindrical coordinate system. The three-dimensional surface is formed by revolving this profile about the axis of the cylindrical system, the local z-axis.

To define a rigid surface of revolution within a part, specify the line segments forming the cross-section of the rigid surface in the local part coordinate system. For an analytical surface defined within a part (or part instance), point a is located at the origin of the part coordinate system, the part x-axis aligns with the radial axis of the cylindrical coordinate system, and point b is located on the part y-axis. You cannot redefine this local axis; instead, you can position the surface in the model by giving positioning data when you instance the part (see Assembly Definition).

The outward surface normal for analytical rigid surfaces is determined by the direction of the line segments forming the profile of the surface. The sequence of line segments defines a vector $\mathbf{s}$ along the rigid surface from the starting point of the first segment to the ending point of the last segment. The outward surface normal is created by taking the cross product of the vector ${\mathbf{e}}_3$, the unit normal to the plane in which the surface is defined, and the vector $\mathbf{s}$, the tangent to the surface: $\mathbf{n}={\mathbf{e}}_3\times \mathbf{s}$. Figure 4 shows the vector $\mathbf{s}$ in the definition plane of an analytical rigid surface.

Orientation of surface normals for a rigid surface.

The unit vector ${\mathbf{e}}_3$ is defined such that ${\mathbf{e}}_1$, ${\mathbf{e}}_2$, and ${\mathbf{e}}_3$ form a right-handed orthonormal coordinate system. In-plane coordinate directions ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ depend on the type of analytical rigid surface being defined. For two-dimensional analytical rigid surfaces they correspond to the global X- and Y-axes in planar models and the r- and z-axes in axisymmetric models. For cylindrical rigid surfaces they correspond to the local x- and y-axes, and for rigid surfaces of revolution they correspond to the local r- and z-axes. The outward normals for a cylindrical rigid surface and rigid surface of revolution are shown in Figure 2 and Figure 3, respectively.

If the line segments are specified in the wrong order, the surface normals of a rigid surface will appear in exactly the opposite direction to what was intended. Such a mistake can be corrected only by specifying the line segments in the opposite sequence.

In many cases it can be beneficial to smooth surfaces to more accurately represent the surface geometry. In particular, it can be very difficult to obtain a converged solution in a finite-sliding Abaqus/Standard simulation if the main surface does not have continuous normal and surface tangent vectors (see Contact Formulations in Abaqus/Standard); therefore, it is important to smooth any sharp corners on the main surface so that discontinuities in these vectors are eliminated.

By default, Abaqus does not smooth main surfaces that are analytical rigid surfaces. Smooth transitions between adjacent line segments can always be created by manually inserting additional curved line segments. Alternatively, smooth surfaces can be generated automatically by Abaqus. You specify the radius of curvature, r, in the units of length used in the model, that Abaqus will use to construct a smooth transition between any discontinuous line segments forming the rigid surface. The default value of zero provides no smoothing of the surface.

The effect of a fillet radius on adjoining line segments and on adjoining line and circular arc segments is illustrated in Figure 5.

Effect of fillet radius on an analytical rigid surface.

The sharp corners have been smoothed using the fillet radius so that the normal and tangent surface vectors are continuous along the entire main surface. Any value r can be used in a model. However, if r is greater than the length of either of the two adjacent segments, no smoothing will occur. Therefore, a practical limit on the size of r is the length of the smallest line segment forming the surface.

Abaqus forms analytical rigid surfaces such that the first surface tangent, ${\mathbf{t}}_1$, is always along the direction of the line segments forming the surface $\mathbf{s}$. The second surface tangent, ${\mathbf{t}}_2$, is defined such that the outward surface normal and the two surface tangents form a right-handed orthonormal system, as shown in Figure 6.

Surface tangent and outward normal definitions for analytical rigid surfaces.

To define a planar rigid surface, define the line segments forming the rigid surface's cross-section in the global coordinate system. You must provide the endpoint of each line segment; the starting point is always the endpoint of the previous segment, or, in the case of the first segment, the point specified as the starting point. The centers of the circular arcs, points c and f in Figure 7, must be given. Abaqus can define only arcs that are less than, but not equal to, 179.74°; thus, it will use the shorter arc defined by the data provided (use two adjacent arcs to define a longer arc). For parabolic arcs you must give a third point that lies on the parabola and within the arc.

Two-dimensional analytical rigid surface contacting a deformable body.

To define a cylindrical rigid surface, specify the points a, b, and c shown in Figure 2 that define the local coordinate system. Give the coordinates of these points—($X_a,Y_a,Z_a$), ($X_b,Y_b,Z_b$), and ($X_c,Y_c,Z_c$)—in the default global coordinate system. As shown in Figure 2, point a defines the origin of the local system; point b defines the local x-axis; and point c defines the generator vector, which is the negative local z-axis. The line segments forming the cross-section of the rigid surface are defined in the local x–y plane. The three-dimensional surface is formed by sweeping this cross-section along the generator vector. The resulting surface extends to infinity in both the positive and negative directions of the generator vector.