Bezier Geometry

Computational geometry methods such as Bezier, B-spline, and NURBS used in CAD systems are based on polynomial representations (Dierckx, 1993; Farin, 1999; Piegl and Tiller, 1997; Rogers, 2001). The coefficients of the polynomials are replaced by coordinates of points, called control points, that form what is called a control polygon. Not all of the control points must lie on the curve or the surface; therefore, they are not necessarily material points as is the case in the FE representation. The use of control points that are not necessarily material points is a fundamental difference between computational geometry methods and FE formulations that use nodal coordinates to represent material points. Computational geometry methods are rooted in Bezier representation. Bezier representation, however, does not allow for domain discretization and for the use of piecewise polynomials; more complex shapes can be represented in Bezier geometry by increasing the polynomial order. Therefore, Bezier description, which does not allow for efficient shape manipulation, can be considered the counterpart of the Raleigh–Ritz analysis method, which uses modes that describe the deformation of the entire structure.

B-Spline Geometry

B-spline, in contrast, allows for domain discretization and the use of piecewise polynomials that define shapes over domain intervals (subdomains). B-spline geometry, therefore, is described by segments connected at points called breakpoints. For this reason, B-spline geometry description, which uses control points, can be considered the counterpart of the FE method. Being able to describe independently the geometry of segments over subdomains allows for efficient local shape manipulation because geometric changes can be made over small regions without affecting the geometry away from those regions. B-spline also uses the concept of knot vector and knot multiplicity, which allows for adjusting the degree of continuity at the breakpoints at which B-spline segments are connected. As the knot multiplicity at a given breakpoint decreases, the degree of continuity at that point increases. For example, a cubic B-spline curve representation with knot multiplicity of four at a point makes the curve discontinuous (C⁻¹) at this point; knot multiplicity of three is associated with coordinate continuity (C⁰); knot multiplicity of two is associated with coordinate and gradient continuity (C¹); and so on. B-spline representation is based on a recurrence structure that allows for automatically adjusting the degree of continuity by changing the knot multiplicity. Such a representation also allows for knot insertion and for adding control points to locally manipulate the geometry.

NURBS Geometry

There are shapes such as circular and conic segments the geometry of which cannot be described exactly using nonrational polynomials as those used in Bezier and B-splines; these shapes can be described exactly by using only rational functions (Dierckx, 1993; Farin, 1999; Piegl and Tiller, 1997; Rogers, 2001). NURBS geometry uses rational polynomials, allowing for exact geometric representation of circular and conic sections. NURBS geometry is based on a recurrence formula similar to that used in B-spline representation and it uses similar knot-vector and knot-multiplicity concepts. In Bezier, B-spline, and NURBS geometries, rigid-body displacement of a shape can be achieved by simply operating on the control points. A rigid-body transformation applied to the control points does not distort the geometry that undergoes the same transformation. For this reason, Bezier, B-spline, and NURBS geometry does not have the problems associated with some FE formulations in which the shape is not preserved under arbitrary rigid-body displacements.

ANCF Element Geometry

In the case of the ANCF Euler–Bernoulli cable element, the global position vector r of an arbitrary point on the FE centerline can be defined using the element shape function matrix and the nodal coordinate vector as follows:

7.1

where S is the element shape function matrix expressed in terms of the element spatial coordinate x; e is the vector of nodal coordinates that consist of absolute nodal position coordinates and position coordinate gradients; and t is time. The element shape function matrix used in Equation 1 for the three-dimensional cable elements can be written as

7.2

where I is the identity matrix and s_i, i = 1, 2, 3, 4, are appropriate shape functions that can be introduced using polynomials or other geometric representations. The vector of the element nodal coordinates can be written as follows:

7.3

where superscripts A and B refer to the first and second nodes of the element, respectively; and the element nodal coordinate vector at node N is defined as follows:

7.4

The displacement field of this element can be defined using polynomial representation (described previously) as

7.5

In this equation, l is the length of the finite element, and ξ = x/l.

Control-Point Representation

Whereas ANCF finite elements use gradients of absolute position vectors as nodal coordinates, a gradient vector always can be expressed in terms of the position vectors of two points divided by a scaling-length factor. Therefore, the ANCF geometric description presented in this section in terms of gradient vectors always can be converted to an equivalent geometric representation expressed solely in terms of position vectors of points that do not have to lie on the beam space curve or the plate and shell midsurface. As the finite element deforms, the location of these points will change with time. To demonstrate this coordinate transformation, the simple cable element defined previously is considered. For the cubic representation of Equation 5, one can define four points whose position vectors are defined by the vectors P₀, P₁, P₂, and P₃. Using these position vectors, the following coordinate transformation can be defined:

7.6

Substituting the coordinate transformation of this equation into Equation 5, one can show that the displacement field of the ANCF finite element can be written in terms of the point coordinates instead of the gradient coordinates as

7.7

The two representations of Equations 5 and 7 are equivalent and each can be used in the description of the geometry as well as the displacement, including deformations and rigid-body motion. Figure 1 is a graphical representation demonstrating the relationship between the gradient coordinates and point coordinates. In computational geometry, the points P₀, P₁, P₂, and P₃ are called control points. These points, which form the control polygon, do not have to represent material points. Equation 7, therefore, demonstrates how the geometric description of the ANCF finite elements can be expressed in terms of control points. The inverse relationship, in which the ANCF position and gradients coordinates are expressed in terms of the control points, can be obtained as explained in the following section. Such a relationship allows for converting CAD models to ANCF analysis meshes without geometry distortion.

Control Points and Degree of Continuity

For a given polynomial order p, one can always define at a breakpoint p + 1 vectors. These vectors represent the position vector and its derivatives up to the pth derivative. For example, in the case of a cubic Bezier curve (p = 3), one can define r, ∂r/∂u, ∂²r/∂u², and ∂³r/∂u³. Higher derivatives will be equal to zero. Therefore, in the case of two cubic Bezier segments i and j, one can have different conditions. If the two segments are not connected, the number of control points totals eight because each Bezier segment is represented using four (p + 1) control points. This is the case of C⁻¹ continuity. In this case, all of the vectors r, ∂r/∂u, ∂²r/∂u², and ∂³r/∂u³ that correspond to the first segment at the breakpoint are not related to the vectors r, ∂r/∂u, ∂²r/∂u², and ∂³r/∂u³ that correspond to the second segment at this breakpoint. This case of C⁻¹ continuity corresponds to the case of knot multiplicity of four (p + 1). If continuity is to be imposed on the position at the breakpoint at which the two segments are connected, one must have at the breakpoint the condition rⁱ = r^j, where i and j are the segment numbers. This algebraic equation, which ensures C⁰ continuity, can be used to write a control point of one segment in terms of the remaining control points. This leads to a reduction of the number of control points by one. This case of C⁰ continuity corresponds to knot multiplicity of three. Similarly, in the case of C¹ continuity, two conditions are imposed at the breakpoint: the positions must be the same, rⁱ = r^j; and the gradients must be the same, (∂r/∂u)ⁱ = (∂r/∂u)^j. These two vector algebraic equations allow for reducing the number of control points by two. This case corresponds to knot multiplicity of two in the case of cubic polynomials. Using a similar procedure, one can show that in the case of cubic polynomials, C² continuity at a breakpoint corresponds to knot multiplicity of one and C³ continuity corresponds to knot multiplicity of zero. In the latter case of C³ continuity, the two segments blend together to form one larger cubic segment; four control points are eliminated, thereby reducing the number of control points for the newly created segment to four.

The concept of the knot multiplicity and the B-spline recurrence formula can be used to achieve automatically a higher degree of continuity by reducing the number of independent control points. Understanding the role of the knot multiplicity in B-spline representation is important when relating geometry methods to FE analysis methods that use the concept of nodes and degrees of freedom. If r + 1 is the number of knots in U, then in the B-spline geometry, one must have r = n + p + 1. This relationship can be verified easily starting with one Bezier segment. It is clear from this relationship that for a given polynomial degree p, reducing the knot multiplicity, to increase the degree of continuity, leads to a similar reduction in the number of control points, and increasing the knot multiplicity leads to a similar increase in the number of control points.

Illustrative Example

Figure 2 shows a cubic B-spline curve that has the six control points P₀, P₁, P₂, P₃, P₄, and P₅ and the knot vector U = {0 0 0 0 1 2 3 3 3 3} (Lan and Shabana, 2010). The number of segments in this example is s = 3 and the distinct breakpoints are ū₀ = 0, ū₁ = 1, ū₂ = 2, and ū₃ = 3. Note that if the knot vector has the form , the B-spline curve reduces to a p-order Bezier curve; therefore, the Bezier curve can be considered a special case of the B-spline curve representation, as previously mentioned. At ū₀ = 0, the knot multiplicity is four, which implies that there is no continuity condition imposed at this breakpoint, and r, ∂r/∂u, ∂²r/∂u², and ∂³r/∂u³ are discontinuous. At the breakpoint ū₁ = 1, the knot multiplicity is one and the curve is C² continuous at this point. In this case, r, ∂r/∂u, and ∂²r/∂u² are continuous, whereas the continuity of ∂³r/∂u³ is not ensured. The B-spline recurrence formula with the knot vector defined previously can be used to demonstrate this fact. The breakpoint ū₂ = 2 has knot multiplicity of one; therefore, this point has continuity conditions similar to the breakpoint ū₁ = 1. The breakpoint ū₃ = 3 has knot multiplicity of four; therefore, this point has continuity conditions similar to ū₀ = 0.

Knot Insertion

By inserting knots into a B-spline knot vector, one can increase the number of control points as well as the knot multiplicity at selected points. For example, the S-shaped cubic B-spline curve shown in Figure 3, which has three segments, can be described by the knot vector U = {0 0 0 0 1 2 3 3 3 3} and the control points P₀, P₁, P₂, P₃, P₄, and P₅. Changing the knot vector to U = {0 0 0 0 1 1 1 2 2 2 3 3 3 3} by increasing the multiplicity at ū₂ = 1 and ū₂ = 2 to three reduces the degree of continuity at these breakpoints. At these points, the B-spline curve has C⁰ continuity, which ensures the continuity of the position vector only. Because the knot multiplicity is increased by two at two breakpoints, the number of control points increases by four. The corresponding control points for the new B-spline curve , and also are shown in Figure 3. This B-spline curve, which ensures only C⁰ continuity at the inner breakpoints, is formed from three cubic Bezier curves that have the control points , and . In fact, one can verify that any p-order C⁰ B-spline curve can be constructed using Bezier curves if the knot vector takes the form . Decreasing the knot multiplicity at the inner breakpoints automatically increases the degree of continuity and eliminates the redundant control points. This process is built automatically in the powerful recurrence formula defined by Equations 16 and 17. Using these equations, one can also show that when the multiplicity at a breakpoint reduces to zero, two B-spline segments blend together to define one knot span with a length equal to the sum of the lengths of the two original segments. In addition to this powerful feature of the B-spline representation that provides the flexibility of easily changing the degree of continuity, the B-spline basis functions have unique desirable features such as the local support property, the nonnegativity, and the partition of unity (Dierckx, 1993; Farin, 1999; Piegl and Tiller, 1997; Rogers, 2001).

Comparison with FE Formulations

Because of the problems associated with the kinematic description of existing FE formulations, computational geometry methods are being adopted as analysis tools. This led to a new research field known as isogeometric analysis, which is based on using CAD systems for both geometric modeling and analysis. As discussed in a subsequent section, computational geometry methods have serious limitations as analysis tools. These methods do not provide the flexibility offered by FE analysis methods and fail to capture certain types of discontinuities that define the geometric shapes of many mechanical and structural components. In the remainder of this section, the focus is on basic differences between the descriptions used by computational geometry and FE methods (Lan and Shabana, 2010).

The concept of nodes that represent material points on a curve or a surface is not used in B-spline geometry and other computational geometry methods; instead, computational geometry methods use control points that do not have to represent material points. The definition of nodes, is fundamental in the FE analysis. Different coordinate types such as positions, angles, gradients, curvatures, and stresses, among others, can be used as the nodal variables. The use of some of these coordinate types can impose restrictions on the geometric shapes that can be assumed by the finite elements. In fact, some of the conventional finite elements do not preserve the shapes of the finite elements under an arbitrary rigid-body transformation, as previously discussed. The kinematic description used for these elements, therefore, is not equivalent to the descriptions used by computational geometry methods such as the B-spline representation. For this reason, the geometry of the FE mesh used in the analysis in the reference configuration is not always identical to the geometry created in CAD systems. B-spline geometry, however, can always be converted to ANCF geometry using simple linear transformation. The converse is not always true, as demonstrated later in this chapter.

Another fundamental difference between the geometric descriptions used in the FE and computational geometry methods is the domain definition. Most FE formulations use the domain in the reference configuration such as the undeformed-length or cross-sectional dimensions of a beam to carry out the integrations required to formulate the FE inertia and stiffness forces. The concept of reference configuration or element domain as used in FE analysis is not used in computational geometry methods. Therefore, it is important to address this issue to successfully convert the B-spline geometry to an FE mesh with the exact geometric properties as that developed in a CAD system (Lan and Shabana, 2010).

B-Spline Surfaces

B-spline surfaces are defined using the product of B-spline base functions, two parameters, and two knot vectors. B-spline surfaces can be defined in the following parametric form (Piegl and Tiller, 1997):

7.22

where u and v are the parameters; N_i,p(u) and N_j,q(v) are B-spline basis functions of degrees p and q, respectively; and P_i,j are a set of bidirectional net of control points. The B-spline basis functions N_i,p(u) are defined as

7.23

where u_i,j = 0, 1, 2,…, n + p + 1 are called the knots and u_i ≤ u_i+1. The vector U = {u₀, u₁…u_n+p+1} is called the knot vector associated with the u parameter. Similar definitions can be introduced for N_j,q(v) with another knot vector V = {v₀ v₁ · · · v_m+q+1}. Note that the orders of the polynomials in the u and v directions can be different; for example, a cubic interpolation can be used along u, whereas a linear interpolation can be used along v. As in the case of B-spline curves, the knots of B-spline surfaces do not have to be distinct; distinct knots are called breakpoint and define surface segments with nonzero dimensions. The number of the nondistinct knots in U and V at a point is referred to as the knot multiplicity associated, respectively, with the parameters u and v at this point. At a given breakpoint, the multiplicity associated with u can be different from the multiplicity associated with v, allowing for different degrees of continuity for the derivatives with respect to u and v. For cubic N_i,p (p = 3), C⁰, C¹, or C² conditions correspond, respectively, to knot multiplicity of three, two, and one; whereas in the case of linear interpolation of N_j,q, the highest continuity degree that can be demanded is continuity of the gradients. When zero multiplicity is used at a breakpoint, the segments blend together at this point.

In B-spline surface representation, there is a relationship among the polynomial degree, the number of knots, and the number of control points. This relationship must be fully understood if B-spline geometry is used as an analysis tool. If r + 1 is the number of knots in U and s + 1 is the number of knots in V, then in B-spline geometry, one must have

7.24

As in the case of B-spline curves, these formulas imply that for a given polynomial order, if the number of knots decreases, the number of control points that defines the number of degrees of freedom used in the analysis must also decrease. This also can be equivalent to increasing the degree of continuity because eliminating a control point can be the result of imposing algebraic equations that relate the derivatives at a certain breakpoint.

From the bidirectional structure used in Equation 22, a surface segment that has cubic interpolation along u(p = 3, n = 3, r + 1 = 8) and a linear interpolation along v(q = 1, m = 1, s + 1 = 4) should have (n + 1) × (m + 1) = 8 control points; this is true regardless of whether the surface is two- or three-dimensional. Manipulation of the B-spline surface defined by Equation 22 shows that these eight control points are the result of using the alternate basis set 1, u, v, uv, (u)², (u)²v, (u)³, (u)³v. That is, B-spline representation and the formulas of Equation 24 do not allow for the use of the basis set 1, u, v, uv, (u)², (u)³, which can be used effectively to develop a shear deformable beam model. If a cubic interpolation is used for both u and v (case of thin plate), the B-spline representation will require 16 control points because the expansion must include all terms (u)^k (v)^l; k, l = 0, 1, 2, 3, regardless of whether the shape of deformation of the plate is simple or complex; one must follow strictly the B-spline rigid structure. This can be a disadvantage in the analysis because such a geometric representation can increase unnecessarily the dimensions of the model and lead to the loss of flexibility offered by the FE method or modal analysis techniques. Another important and interesting issue regarding the use of B-spline as an analysis tool is capturing discontinuities, which is discussed later in this chapter.

ANCF Surfaces

Whereas B-spline geometry always can be converted to ANCF geometry, the converse is not true. ANCF geometry does not impose restrictions on the basis functions that must be included in the interpolating polynomials. This allows for developing finite elements that have a small number of nodal coordinates as compared to those developed using the B-spline representation. Furthermore, ANCF geometry can be used to model both structural and nonstructural discontinuities, whereas the rigid B-spline recurrence representation cannot be used to model structural discontinuities in a straightforward manner, as discussed in the following section.

A basic difference between ANCF and B-spline geometries is demonstrated in this section using a planar-beam example. The displacement field of the shear deformable beam used can be written as r(x₁, x₂) = S(x₁, x₂) e(t), where x₁ and x₂ are the element spatial coordinates, t is time, S is the element shape function matrix, and e is the vector of the element nodal coordinates. The shape function matrix for the element considered in this section is defined as

7.25

where the shape functions s_i, i = 1, 2, …, 6, are defined as (Omar and Shabana, 2001):

7.26

In this equation, ξ = x₁/l, and η = x₂/l. ANCF finite elements use gradient vectors as nodal coordinates. For the element used in this section, the vector of nodal coordinates is defined as

7.27

where superscript k, k = 1, 2, indicates variables evaluated at node k of the element. The element defined by the preceding equations is based on a cubic interpolation for x₁ and a linear interpolation for x₂.

The finite element described in this section is an example of ANCF elements that cannot be converted to B-spline representation. This element is based on a polynomial expansion that does not have the two basis functions (x₁)² x₂ and (x₁)³ x₂. These terms can be systematically included in ANCF geometry by adding nodal coordinates, which allows for converting B-spline representation to ANCF representation. Similar comments apply to ANCF thin-plate elements that do not require including all of the basis functions (x₁)^k (x₂)^l; k, l = 0, 1, 2, 3. This flexibility offered by ANCF geometry allows for developing finite elements that have a smaller number of nodal coordinates compared to those elements developed by B-spline geometry.

B-Spline Model

As previously mentioned, reducing the knot multiplicity by one at a curve breakpoint leads to C⁰ continuity and to the elimination of one control point. This is equivalent to the multibody system (MBS) pin-joint-constraint definition in planar analysis and to the MBS spherical-joint definition in spatial analysis. This type of C⁰ continuity captured by B-spline is of the nonstructural-discontinuity type that leads to a rigid-body mode and to a nonunique state of the strain at the discontinuity node. The B-spline recurrence formula structure leads automatically to this type of discontinuity. The structural discontinuity, although it is also of the C⁰ type, requires additional algebraic equations to define a unique strain state by eliminating the relative rotation at the joint definition point. These algebraic equations can be used to eliminate other control points and such elimination is not embedded in the rigid B-spline geometry representation. Only one type of C⁰ continuity can be described using B-spline formula; reducing the knot multiplicity by one in B-spline representation does not capture structural discontinuity. More fundamental problems are encountered when B-spline surfaces are considered in MBS applications as explained in the literature (Yu and Shabana, 2015).

ANCF Model

One can show that ANCF finite elements describe both structural and nonstructural discontinuities. Nonstructural discontinuities that allow for large rigid-body rotations can be described using a C⁰ model obtained by imposing constraints only on the position coordinates. For example, if two elements i and j are connected by a pin joint at a node, one can apply the algebraic equations rⁱ = r^j at this node. These algebraic equations can be imposed at a preprocessing stage to eliminate the dependent variables and to define an FE mesh that has a constant mass matrix and zero Coriolis and centrifugal forces, despite the finite rotations allowed between the finite elements of the mesh. As previously mentioned, nonstructural discontinuities also can be described using B-spline geometry by reducing the knot multiplicity by one at the joint node. In the case of nonstructural discontinuities, no constraints are imposed on the gradient vectors; therefore, the state of strain is not unique at the joint node. Each of the Lagrangian strains , , and has two values at the joint node: one defined on element i and the other on element j. Here, , .

The concept of degrees of freedom widely used in mechanics is not considered in developing the recurrence relationships on which B-spline and NURBS geometry are based. This represents another serious limitation when these computational geometry methods are used as analysis tools, which is evident by the fact that B-spline geometry cannot describe structural discontinuities. As previously mentioned, this type, although it is of the C⁰ continuity type, requires imposing additional gradients constraints, which cannot be captured by the B-spline recurrence formula because they require the elimination of additional vectors. In the case of B-spline, C⁰ continuity is achieved by reducing the knot multiplicity by one, which eliminates one control point leading to the definition of a pin joint (nonstructural discontinuity). ANCF geometry, conversely, allows for imposing constraints on the gradients using the tensor transformation (∂r/∂x) = (∂r/∂y)A, where and are two sets of coordinate lines and A is the matrix of coordinate-line transformation. Using this tensor-gradient transformation, the structural discontinuities can be modeled systematically using ANCF finite elements, as described in the literature (Shabana and Mikkola, 2003; Shabana, 2011).