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Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56www.elsevier.com/locate/cma

Algebraic distance estimations for enriched isogeometric analysis

K. Upreti, T. Song, A. Tambat, G. Subbarayan∗

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, United States

Received 30 January 2014; received in revised form 10 July 2014; accepted 10 July 2014Available online 19 July 2014

Abstract

In problems with evolving boundaries, interfaces or cracks, blending functions are used to enrich the underlying domain withthe known behavior on the enriching entity. The blending functions are typically dependent on the distance from the propagatingboundaries. For boundaries defined by free form curves or surfaces, the distance fields have to be constructed numerically. This mayrequire either a polytope approximation to the boundary and/or an iterative solution to determine the exact distance to the boundary.In this paper a purely algebraic, and computationally efficient technique is described for constructing distance measures from Non-Uniform Rational B-Splines (NURBS) boundaries that retain the geometric exactness of the boundaries while eliminating the needfor iterative and non-robust distance calculation. The constructed distance measures are level sets of the implicitized constituentBezier patches of the NURBS surfaces that are obtained purely algebraically. Since, in general, the implicitized functions extendbeyond the parametric range of the generating Bezier patch, algorithmic procedures are developed to trim these global implicitfunctions to the boundaries of the Bezier patch. Boolean compositions are then carried out between adjoining Bezier patches toconstruct a composite distance field over the domain. The compositions rely on R-functions that are also algebraic in nature. Thedeveloped technique is demonstrated by constructing algebraic distance field for complex geometries and by solving a variety ofexamples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three-dimensional cracks.c⃝ 2014 Elsevier B.V. All rights reserved.

Keywords: Algebraic geometry; NURBS; Implicitization; R-functions; Distance field; Enrichments

1. Introduction

In general, in moving boundary problems, the motion of complex boundaries need to be tracked within the domain.Examples of such problems arise in many fields, including fluid mechanics, solid mechanics, optimal design, computervision and image processing. Commonly, an Eulerian framework in which the geometry of the boundaries is inferredimplicitly as the zero level set of an evolving field is used to numerically solve these class of problems [1]. Since thelevel sets implicitize both the geometry of the boundary as well as the distance from the boundary, in other computa-tional procedures for moving boundary problems, there is a common need for explicitly calculated distance fields.

In general, in these problems, distance from the boundary or interface serves as a measure of influence of thebehavior on the boundary at a point in the underlying domain. Computational procedures relying on distance fields are

∗ Corresponding author. Tel.: +1 765 494 9770; fax: +1 765 494 9770.E-mail address: ganeshs@purdue.edu (G. Subbarayan).

http://dx.doi.org/10.1016/j.cma.2014.07.0120045-7825/ c⃝ 2014 Elsevier B.V. All rights reserved.

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 29

many. For example, the use of distance fields for image processing is well established [2]. In fluid–structure interactionproblems, signed distance functions have been used to represent fluid boundaries such as the fluid–structure interfaceor the free surface within the computational domain [3,4]. Distance fields are also useful in contact problems [5–7]for defining the gap function for contact detection. Recently, in the Isogeometric Analysis (IGA, [8,9]) literature,an approximate distance field has been used to enrich the base approximations with those on lower-dimensionalgeometrical features, enabling application of boundary conditions as well as simulations of crack propagation [10].Mathematical representation of graded materials have also used distance fields to describe the desired blended materialdistributions [11,12]. Thus, while the use of distance field as a measure of influence of the boundaries on the domainhas enabled the numerical solution to complex problems, inexpensive distance field calculations are essential for theanalysis to be computationally efficient.

Since the notion of distance fields is fundamental to many computational solution procedures, there exists asignificant established literature aimed at improving the techniques for calculating the distance field. Iterative methodssuch as the Newton–Raphson technique are usually required for finding continuously varying distance from spatialpoints to a parametric surface [13–16]. While the iterative schemes make distance computations expensive, thecalculated distance field is unfortunately not sufficiently smooth for many engineering applications. This is sincethe distance function is not differentiable at points that are equidistant to two or more points on the surface (cut locusof the surface).

Iterative distance calculation methods in general become necessary due to the geometrical nonlinearity of thesurface. Hence, piecewise planar approximations of a parametric surface using, for instance, triangular mesh arepopular in computer graphics applications (see for instance [17]). The piecewise linearization of geometry makesthe distance calculation relatively straightforward, albeit by introducing a new combinatorial problem — that ofdetermining the nearest planar surface from among the many possible ones at a spatial point. Therefore, the algorithmsrelating to piecewise planar approximations mainly deal with efficient data structures to determine the triangular faceclosest to the point of interest (see for instance, [18,19]) or algorithmically propagating distances from calculationscloser to the surface [20,21]. The disadvantage of the piecewise planar approximations is that the geometricalexactness of the boundary is not preserved making the calculated distance accurate only in the limit of refinementof the planar approximations. But, more critically, the calculated distance field is non-smooth since a subset of thespatial points will project to an edge or vertex of the triangular face. This in turn implies that one cannot rely ondistance calculations on planar approximations to generate smooth, continuous distance measures.

In general, distance measures may be thought of as extending approximations of fields from the boundaries intothe underlying domain. For constructing such a distance measure, the exact distance to the boundary is not as criticalas a smooth and monotonically increasing function of distance that serves as a measure of influence of the boundaryon the underlying domain. Smoothness of such an approximate distance measure would ensure robustness of anycalculations based on the constructed field. Hence, an approximate distance measure that is smooth is sufficient forthese applications. Towards building such an approximate measure, Biswas and Shapiro [22] described a methodusing piecewise planar geometric approximations of a parametric curve. They constructed an approximate distancefield for each linear segment and combined the distance field of each linear segment using R-functions into a globalapproximate distance field. The advantage of this technique over the previous distance calculation techniques is thatthis enables desired smoothness in the distance field. However, this technique relies on piecewise linearization of thegeometry and hence, compromises the exactness of the boundary.

In this paper, a purely algebraic, and therefore computationally efficient, technique for constructing an approximatedistance function that avoids the iterative, and therefore inefficient, distance computations is developed. The techniquepreserves the geometric exactness of low-degree (two or three) NURBS surfaces. Such purely algebraic distancecomputation procedures that preserve the geometric exactness do not appear to exist in prior literature. The proposedtechnique overcomes the need for the iterative exact distance computations at every quadrature point during analysiswhile providing smoother, more robust distance field relative to the numerically computed distance.

The rest of the paper is organized as follows. In Section 2, a general theoretical formulation for the algebraicdistance field construction technique is presented and illustrated using rational Bezier as well as NURBS curves. InSection 3, a detailed algorithm is proposed for algebraic distance field construction technique for Bezier and NURBScurves. The properties of the algebraic distance field are discussed next in Section 4. In Section 5, the algebraicdistance field algorithm is extended to three dimensional Bezier and NURBS surfaces. In Section 6, examples ofalgebraic distance field are demonstrated. The paper is finally summarized in Section 7.

30 K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56

Fig. 1. Illustration of the different types of boundaries and interfaces encountered during engineering analysis that may be modeled as enrichmentson an underlying field.

2. Theoretical development of algebraic distance field

In this section, the theoretical concepts essential for development of the algebraic distance field are presented.Rational Bezier and NURBS parametrizations are considered as they are most popular in Computer aided geometricdesign (CAGD) and Isogeometric Analysis.

2.1. Distance field: definition and properties

Given a set S ∈ Rn , for any point x ∈ Rn a distance field d(x) of S is defined as the minimum distance from thepoint x to an element p ∈ S given by

dS(x) = infp∈S

∥x − p∥. (1)

The distance function d is continuous at all x ∈ Rn and is differentiable almost everywhere in Rn . The points wherethe field is not differentiable are those where minimum distance corresponds to non-unique points p ∈ S. An importantproperty associated with the gradient of the field d is

∥∇dS(x)∥ = 1. (2)

Thus, by definition, the distance field refers to exact distance to the given geometry. In this paper, the sameterminology is followed, and any approximation to the exact distance field is referred to as approximate distancefield, distance measure, algebraic level sets, or algebraic distance field depending on the context.

2.2. Requirements on distance field for explicit modeling of evolving boundaries

As a motivation for identifying the requirements on distance measures so they may be useful for analysis, varioustypes of boundaries and interfaces that occur in engineering design and analysis problems are summarized in Fig. 1.Ideally, the distance measure would preserve the geometric exactness of the explicit boundary so that the behavioron these complex boundaries is accurately captured by the analysis scheme. Hence, a fundamental requirement thatdrives the current work is that the distance field needs to be exact on and very near the boundaries. At the same time,it is also required that the distance field is sufficiently smooth for modeling the analysis problem. Thus, the followingrequirements on distance field are imposed in this study:

1. Exact on the boundary.2. Monotonic function of exact distance.3. Sufficiently smooth for engineering applications within the domain.4. Efficiently obtained through non-iterative calculations.

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 31

2.3. Implicitization of parametric geometry using elimination theory

An implicit representation of a geometry is an equation of the form f (x1, . . . , xn) = 0 in n-dimensionalspace. Given the implicit representation f (x1, . . . , xn) = 0, a level set of the function f is of the form L( f ) =

(x1, . . . , xn)| f (x1, . . . , xn) = c, that is a set for which the function f takes on some constant value c. Such levelsets of a geometry give monotonic measure of distance from the geometry as shown in Fig. 2. Hence, by obtaining animplicit representation of the parametric geometry, it is possible to construct a utilizable distance measure from thegeometry.

Algebraic techniques based on elimination theory [23,24] enable the conversion of parametric geometry to itsimplicit representation f (x1, . . . , xn) = 0. Elimination theory investigates the conditions under which sets of polyno-mials have common roots. For instance, a linear system of equations Ax = 0 will have a non-trivial solution if and onlyif the determinant of the coefficient matrix vanishes, i.e., |A| = 0. An expression of this form |A| involving the coeffi-cients of the polynomials is a Resultant. The vanishing of this resultant is a necessary and sufficient condition for theset of polynomials to have a common non-trivial root. Implicitization tools are based on construction of such resultants.

2.3.1. Implicitization of parametric curvesSederberg [23] discussed in detail the various resultants that may be used for implicitization. In this work, Bezout’s

resultant is used for implicitization of parametric curves as it is efficient to compute compared to the other resultants.We briefly illustrate Bezout’s resultant with an example but, prior to that, we observe that for a rational parametriccurve C(X (t), Y (t), W (t)) or C(x, y) with x =

X (t)W (t) , y =

Y (t)W (t) , two auxiliary polynomials may be formed as:

q1(x, t) = W (t)x − X (t) = 0q2(y, t) = W (t)y − Y (t) = 0.

(3)

The resultant of the above polynomials (treating x and y as constants) yields the implicit curve corresponding to theparametric entity. Now, specifically, consider the following two polynomials

q1(t) = a3t3+ a2t2

+ a1t + a0

q2(t) = b3t3+ b2t2

+ b1t + b0(4)

where, from the form of Eq. (3), it is clear that ai and b j are linear functions of x and y respectively. By the followingalgebraic manipulations, three equations are obtained in t2, t and 1.

a3q2 − b3q1 = (a3b2)t2+ (a3b1)t + (a3b0) = 0

(a3t + a2)q2 − (b3t + b2)q1 = (a3b1)t2+ [(a3b0) + (a2b1)]t + (a2b0) = 0

(a3t2+ a2t + a1)q2 − (b3t2

+ b2t + b1)q1 = (a3b0)t2+ (a2b0)t + (a1b0) = 0.

(5)

In matrix form, this system of equations can be written as Ax = 0, or(a3b2) (a3b1) (a3b0)

(a3b1) (a3b0) + (a2b1) (a2b0)

(a3b0) (a2b0) (a1b0)

t2

t1

= 0. (6)

Then, Bezout’s resultant of the two polynomials q1(t) and q2(t) is |A| by definition.In general, Bezout’s resultant of two degree r polynomials is determinant of a r × r symmetric matrix:(ar br−1) · · · (ar b0)

......

(ar b0) · · · (a1b0)

tr−1

...

1

= 0 (7)

where (ai b j ) = ai b j − a j bi , a and b are linear functions of x and y respectively. For the values of t for whichq1(t) = 0 and q2(t) = 0, there exists points P(x, y) lying on the parametric curve C(X (t), Y (t), W (t)) such thatf (x, y) = 0. This is the implicit equation of the parametric curve C . Fig. 3 illustrates implicitized curves of differentdegrees.

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Fig. 2. Level curves of an ellipse with implicit equation f (x, y) = x2+ y2

− xy − 8.

Fig. 3. Implicitization examples. (a) Degree 2 Bezier curve with homogeneous control points (−3, −2, 0, 1), (0, 2, 0, 1), (4, 1, 0, 1), (b) degree 3Bezier curve with homogeneous control points (4, −3, 0, 1), (−1, 2, 0, 4), (5, 1, 0, 1.5), (2, 6, 0, 2), (c) degree 4 Bezier curve with homogeneouscontrol points (0, −3, 0, 1), (4, 4, 0, 2), (−3, 7, 0, 4), (5, 4, 0, 1.5), (2, 6, 0, 2).

2.3.2. Implicitization of parametric surfacesImplicitization of parametric surfaces is carried out using Dixon’s resultant [23]. Given a rational parametric surface

S(X (u, v), Y (u, v), Z(u, v), W (u, v)) of degree r1 × r2, three auxiliary polynomials are formed as:

q1(x, u, v) = W (u, v)x − X (u, v) = 0q2(y, u, v) = W (u, v)y − Y (u, v) = 0q3(z, u, v) = W (u, v)z − Z(u, v) = 0.

(8)

As before, holding x , y and z to be constants, if some (u′, v′) simultaneously satisfy q1 = q2 = q3 = 0, then for(u, v) = (u′, v′) and any (α, β), the following determinant always vanishes [24]:

Det (u, v, α, β) =

q1(u, v) q2(u, v) q3(u, v)

q1(α, v) q2(α, v) q3(α, v)

q1(α, β) q2(α, β) q3(α, β)

. (9)

Also, the determinant vanishes for u = α or v = β. Hence, (u − α) and (v − β) are factors of the determinant. Now,these factors may be eliminated to obtain the following determinant:

δ(u, v, α, β) =Det (u, v, α, β)

(u − α)(v − β)=

2r1−1i=0

r2−1j=0

Qi, j (u, v)αiβ j . (10)

The polynomial δ is further expressed as a polynomial in α and β whose coefficients Qi, j are polynomials in u andv. The polynomial δ vanishes for (u, v) = (u′, v′) with any choice of (α, β), which is possible if and only if every

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 33

Fig. 4. Level set as distance measure. Level curves of implicit representation of a degree-3 Bezier curve.

Qi, j (u′, v′) = 0. This generates 2r1r2 polynomials and gives rise to the following system of equations [23]:

α0β0

.

.

.

αi β j

· · ·

α2r1−1βr2−1

A(0, 0, 0, 0) · · · A(0, 0, k, l) · · · A(0, 0, r1 − 1, 2r2 − 1)

.

.

....

.

.

.

A(i, j, 0, 0) · · · A(i, j, k, l) · · · A(i, j, r1 − 1, 2r2 − 1)

.

.

....

.

.

.

A(2r1 − 1, r2 − 1, 0, 0) · · · A(2r1 − 1, r2 − 1, k, l) · · · A(2r1 − 1, r2 − 1, r1 − 1, 2r2 − 1)

u0v0

.

.

.

ukvl

· · ·

ur1−1v2r2−1

= 0 (11)

where terms A(i, j, k, l) are linear functions of x , y and z. The determinant of the above 2r1r2 × 2r1r2 matrix isDixon’s resultant.

2.4. Challenges to implicit representation

The implicit equation obtained using Bezout’s resultant gives distance measure (level sets) from the implicitrepresentation. Fig. 4 shows the level curves of the implicit representation of a parametric cubic curve. These levelcurves form an approximate distance field.

However, the implicit representation of a parametric curve is global, meaning that it extends beyond the parametricrange of the original curve. The distance measure formed by the level sets of the implicit representation essentiallycorresponds to the shortest distance from the implicit curve. But the shortest distance to implicit curve may not alwaysbe the shortest distance to the parametric curve. This is illustrated in Fig. 5. The parametric curve is a segment of thecircle and the implicit curve is the circle (Fig. 5(a)). The implicitized distance function is plotted against exact distancefrom the Bezier curve in Fig. 5(d). Implicit distance function is not monotonic with respect to exact distance from theparametric curve. Hence, the implicitized distance field does not qualify as a valid distance measure for the purposesof analysis.

2.5. Boolean operations using R-functions

In order to obtain a valid distance measure, it is essential to obtain an implicit or function representation of theparametric geometry that is defined only within the parametric range of the geometry. For instance, given a rationalcurve C(X (t), Y (t), W (t)) with the parametric range t ∈ [t0, tn], the implicit representation must be of the formF (X (t), Y (t)) = 0 ∀ t ∈ [t0, tn]. The level curves of such an implicit function (F (X (t), Y (t)) = c) will provide avalid distance measure for the given parametric curve.

In this work, an algebraic distance measure is proposed based on the following trimming procedure [25]. A convextrimming region φ ≥ 0 is defined such that its intersection with the implicit curve f = 0 is the parametric curvesegment C(t) as shown in Fig. 6. The algebraic representation of this trimmed parametric curve segment C(t) isobtained by boolean operations on the fields of trimming region φ and implicit representation f . R-functions [25] arechosen here to enable boolean operations on functional description of domains.

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Fig. 5. Implicitized distance function. (a) An example of a Bezier curve and the corresponding implicit representation, (b) level curves of implicitrepresentation as distance field, (c) axis with markings of exact distance from the Bezier curve, (d) plot of implicitized distance field with respectto exact distance from Bezier curve.

The theory of R-functions [25] provides smooth functional equivalents of boolean operations and is thereforeappropriate in an algebraic procedure. Some basic R-functions and their companion boolean operations are as follows:

1. R-conjunction is the functional equivalent to boolean intersection operation. The R-conjunction function valueessentially corresponds to the minimum of functions g1 and g2 such that the resulting function has desiredsmoothness.

g1 ∧ g2 = g1 + g2 −

g2

1 + g22 . (12)

2. R-disjunction is the functional equivalent to boolean union operation. The R-disjunction function corresponds tothe maximum of functions g1 and g2 with desired smoothness.

g1 ∨ g2 = g1 + g2 +

g2

1 + g22 . (13)

In this work, the R-function for the trimming procedure is adopted from Ref. [22]. This R-function as proposed byRvachev [26] is of the form

g =

f 21 + f 2

2 (14)

where, f1 is a distance measure to the infinite curve and f2 is a distance measure to the trimming region such that f2is identically zero in the interior of the trimming region. Then, the function g defines the distance measure from anypoint to the trimmed curve.

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 35

Fig. 6. Parametric curve C(t) as an intersection of region φ ≥ 0 and implicitized curve f = 0.

Let f (x, y) be the implicit representation for the given parametric boundary and φ be the algebraic representationof the convex trimming region. Then, functions f1 and f2 in Eq. (14) are constructed as follows.

f1 = f

f2 =(|φ| − φ)

2.

Substituting the functions f1 and f2 constructed from the function representation of implicit curve and trimmingregion respectively, a valid distance field g(d) is constructed by the following R-function.

g =

f 2 +

(|φ| − φ)2

4. (15)

This trimming operation enables construction of a valid approximate distance field g(x, y) for a parametric curvewith implicit representation f (x, y) = 0. Inside region φ, the distance field corresponds to the implicit representationsuch that

g = f ∀ x ∈ φ ≥ 0. (16)

For points outside the trimming region φ, distance field is obtained as a composition of the fields φ and f .

g =

f 2 + φ2 ∀ x ∈ φ < 0. (17)

2.6. Normalization and composition of algebraic distance fields

The parametric representation of a spline geometry is a piecewise polynomial representation. Hence, the algebraicdistance measure of the spline is constructed through union of the algebraic distance fields of the constituentparametric geometries. However, boolean composition of algebraic distance fields may result in a non-monotonicdistance measure as illustrated in Fig. 7(a). This is because the gradient of algebraic distance measure may notalways satisfy the property, ∥∇d∥ = 1. Hence, the adjoining distance fields of piecewise polynomial curve maygrow at different rates and exhibit non-monotonic behavior when composed. Also, an algebraic function that givesan approximate measure of distance from a boundary could be locally converted into a proper distance function byscaling. For both of these reasons, normalization is of importance when constructing distance fields. The normalizationprocess is described below.

Let d(x1, . . . , xn) be an exact distance function from a boundary and s(x1, . . . , xn) be an approximate distancefield to the boundary. Then, s is a function of the exact distance d from the boundary. Let n be the variable in thenormal direction at the boundary. Taylor series expansion of s(d) with respect to d near the boundary can be writtenas

s (d) = s (0) +∂s

∂n(0) d +

12!

∂2s

∂n2 (0) d2+

13!

∂3s

∂n3 (0) d3+ · · · . (18)

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Fig. 7. Normalization of distance field. (a) The composition of non-normalized distance fields can lead to a non-monotonic approximate distancefield, (b) first order normalization yields a monotonic distance field.

Ignoring the higher order terms in the expansion, s(d) can be assumed to be a linear approximation locally suchthat scaling the function by ∂s

∂n (0) gives a first order normalized distance function.

s (d) =s (d)∂s∂n (0)

. (19)

Higher order normalization can be obtained by assuming a better approximation of the function s(d) [19]. Ingeneral, the first order normalized distance fields possess the properties that make them suitable for analysis [22].

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 37

Fig. 8. Bounds on the gradient of R-conjunction composition of normalized algebraic distance fields.

Thus, normalization enables the construction of smooth boolean compositions of the algebraic distance fields. Theextent of smooth blending depends on the order of normalization (Fig. 7(b)).

R-function operations such as R-conjunction are used to compose the normalized algebraic distance fields intoa single algebraic distance measure. R-function operations preserve normalization of the constituent fields almosteverywhere. Consider two normalized functions s1 and s2 and let s be obtained as an R-conjunction composition ofs1 and s2. Then,

s = s1 + s2 −

s2

1 + s22

∂ s

∂x=

∂ s1

∂x+

∂ s2

∂x−

s1∂ s1∂x + s2

∂ s2∂x

s21 + s2

2

,∂ s

∂y=

∂ s1

∂y+

∂ s2

∂y−

s1∂ s1∂y + s2

∂ s2∂y

s21 + s2

2

.

It can be seen that when s1 = 0 and s2 = 0, ∂ s∂x =

∂ s1∂x and ∂ s

∂y =∂ s1∂y . Thus,

|∇ s|2 = |∇ s1|2

= 1.

Similarly, |∇ s|2 = 1 when s1 = 0 and s2 = 0. Thus, R-conjunction preserves normalization near the boundary exceptwhen s1 = 0 and s2 = 0. At such points, the gradient is bounded and the bounds are derived in Appendix A as m − 1

√1 + m2

≤ |∇ s| ≤

2 −m + 1

√1 + m2

(20)

where

m = lims1→0

s2

s1.

The maximum value of the upper bound is 3 and the minimum value of lower bound is 0 (Fig. 8).Hence, once the constituent algebraic fields are normalized, the normalization is carried forward in further boolean

compositions except at points where the parametric curves intersect. However, the gradient is bounded at such points.

3. Algebraic distance field construction algorithm for a NURBS curve

Based on the theory described in the previous section, a detailed algorithm is proposed for constructing approximatedistance fields from NURBS and Bezier curves in this section. The flowchart in Fig. 9 illustrates the steps for thealgebraic distance computation from a NURBS curve. These steps are explained in detail below. The algorithmperforms an initial check for repeated control points of the NURBS curve, in which case the NURBS curve reduces toa degenerate point. In such a case the distance is computed to the degenerate point and the execution of the algorithmis stopped. For a non-degenerate case, the algorithm proceeds as described below.

3.1. Bezier decomposition

Since NURBS geometries are piecewise parametric representations, they are not suitable for direct implicitization.Therefore, the NURBS curve is first decomposed into its Bezier segments. Given a NURBS curve of degree p, theBezier segments can be obtained by inserting additional knots at interior knot locations until the multiplicity becomesp. Established subdivision techniques using knot insertion described in [14,27,28] were used. The decompositionprocess is illustrated in Fig. 10 using a degree two NURBS curve.

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Fig. 9. Flowchart describing the algebraic distance field construction for a NURBS curve.

Fig. 10. Decomposition of a NURBS curve into its Bezier segments. (a) Degree two NURBS curve with control polygon, (b) Bezier curves withrespective convex hulls.

3.2. Algebraic distance field construction for Bezier curve

After Bezier decomposition, the algebraic distance field is constructed for each Bezier segment. The flow of controlfor the construction of distance fields from a Bezier curve is shown in Fig. 11. This algorithm also begins with a checkfor the case of repeated control points that reduces the Bezier curve to a degenerate point. In such a case distancebetween the given point and the degenerate point is computed and the algorithm is terminated. For a non-degeneratecase, the convex hull is constructed for the Bezier curve using its control points.

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 39

Fig. 11. Flowchart describing the algebraic distance field construction for a Bezier curve.

3.2.1. Normalized distance field of convex hullThe algebraic distance field is based on the trimming procedure described in Section 2.5. The trimming procedure

requires the following condition to be met — a convex trimming region φ ≥ 0 (Fig. 6) that encloses the parametriccurve segment such that the end points of the curve segment lie on the trimming boundary. In this work, the convexhull of the Bezier segments are proposed as the trimming region since the convex hull satisfies the requirement onconvexity and the condition on curve endpoints. The convex hull is also straightforward to compute from the Beziercontrol points.

Let (xi , yi ) and (xi+1, yi+1) be the end points of the i th edge of the convex hull, then the algebraic representationof that edge is given by its normalized implicit equation

hi (x, y) = ±(x − xi )(yi+1 − yi ) − (y − yi )(xi+1 − xi )

(xi+1 − xi )2 + (yi+1 − yi )2. (21)

The implicit equation hi (x, y) gives signed distance to an infinite line passing through the edge. The sign of theimplicit field is chosen so as to ensure hi (x, y) ≥ 0 inside the convex hull. Then, the distance field of the convexpolygon is obtained using the R-conjunction operation of Eq. (12). Such a process is equivalent to determining theminimum distance of the implicit fields of the individual edges. The resulting field is a signed distance measure to theconvex hull. The field for the convex hull of a Bezier curve is shown in Fig. 12.

In the case of linear or nearly linear Bezier curve, it is not possible to construct a convex hull from the controlpoints. For such a case, a circle with the line segment joining the end points of the Bezier curve as diameter is chosenas the trimming region φ ≥ 0.

3.2.2. Normalized distance field of implicit curveAn implicit representation f of each Bezier curve is obtained by the implicitization process using Bezout resultant.

The implicit function is normalized locally near the curve by scaling it by the gradient value at a point on the curve.

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Fig. 12. Function representation of convex hull. (a) Convex hull as trimming region φ ≥ 0 for the Bezier curve, (b) implicit field of convex hull ofthe Bezier curve.

Fig. 13. Composition of distance field of Bezier curves. (a) Distance field of each Bezier curve, (b) composed distance field of the NURBS curve.

The gradient value at one of the end control points of the Bezier curve was chosen here for scaling the implicitfunction. The derivative of the implicit function for scaling is calculated by exploiting the Bezout resultant matrix Adescribed earlier in Eq. (7). The derivative is obtained using Jacobi’s formula as:

∂|A|

∂xi= tr

ad j (A)

∂ A

∂xi

(22)

where, ∂∂xi

is the partial derivative with respect to the i th coordinate variable, tr is the trace operator, and ad j is theadjugate or the transpose of the cofactor matrix. Since each term in the Bezout matrix A is linear in x and y, thederivative of Bezout resultant matrix in Eq. (22) is a constant. In the case of linear or nearly linear Bezier curve,implicit representation f is the line passing through the end points of the Bezier curve. Then, the normalized distancerepresentation of the line is given by Eq. (21).

3.2.3. Trimming operationAfter the normalized fields are computed for the trimming region as well as the implicit curve, the algebraic

distance field for Bezier curve is calculated by the trimming operation on the normalized fields using Eq. (15). Here,the convex hull of the Bezier curve is used as the trimming region φ for the implicitized representation f of the Beziercurve. An example of the calculated algebraic distance field for a Bezier curve is shown in Fig. 13(a).

3.3. Compositions using R-conjunction

Further compositions are then carried out between adjoining Bezier curve fields using the R-conjunction operationgiven in Eq. (12). The resulting distance field, given below, corresponds to the nearest Bezier segment:

g1 ∧ g2 = g1 + g2 −

g2

1 + g22 (23)

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 41

Fig. 14. Illustration of singular points on a degree three Bezier curve. The homogeneous control points are (−10, −8, 0, 1), (−8, −1.2, 0, 1.2),(5, −2, 0, 2), and (6, −8, 0, 1). (a) The lone singular point lies outside the convex hull of Bezier curve allowing one to construct, (b) the algebraicdistance field of Bezier curve.

where, g1 and g2 are algebraic distance fields of adjoining Bezier curves. The composed distance field for the NURBScurve is shown in Fig. 13(b).

4. Properties of the algebraic distance field

The algebraic distance field proposed in this work is a non-iterative and efficient approximation to the exactdistance function obtained through Newton–Raphson iterations. Further, the algebraic distance field preserves thegeometric exactness of the explicit boundary. Unlike the exact distance function that is often not uniquely definednear points of high curvature, the algebraic distance field is unique and sufficiently smooth for use during analysis.The gradients of the field are bounded on the geometric entity and are derived in Appendix A. The properties of thealgebraic distance are detailed below, and the performance and robustness of the algebraic measure are comparedagainst Newton–Raphson iterations that is used to determine the orthogonal distance.

4.1. Double points and their elimination

The methodology described in this work exploits algebraic geometry concepts that are briefly elaborated here.Given an irreducible plane algebraic curve C p of degree p, by Bezout’s theorem [29], it can be shown that thecurve of algebraic degree p may possess as many as (p−1)(p−2)

2 self-intersections or double points (see Fig. 14).The critical question of concern here is whether the implicitized version of the rational parametric curve will possessself-intersections. This possibility depends on the inverse mapping, given an algebraic curve, to its equivalent rationalparametric form. The existence of the rational parametric form depends on the genus G of the curve defined as

G =(p − 1)(p − 2)

2− D P (24)

where, D P is the number of double points present in the algebraic curve. In order for a curve to have a rationalparametrization, its genus needs to be zero. Thus, an implicitized curve obtained from a rational parametric curveof degree three will possess a single double point when implicitized. However, if the singular point lies outside theconvex hull region (see Fig. 14), then the algebraic distance field is not affected by the singular point as the convexhull is used to trim the distance field. But, if the singular point falls inside the convex hull, then it affects the algebraicdistance construction. In such a case, the parametric curve is subdivided at the singular point to force the singular pointto lie outside the trimming region (Fig. 15). The parameter t corresponding to the singular point is calculated usingmoving line techniques [30–32]. Subdivision is performed by using the knot insertion algorithm at the parameter t onthe curve. Algebraic distance field is then constructed for each sub-curve and composed using R-functions.

The above technique can be extended to higher degree curves with multiple singularities. However, higher thedegree of the curve, the greater the number of singular points per Eq. (24). Therefore, in this work, the methodologyis restricted to low degree NURBS curves (d ≤ 3).

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Fig. 15. Subdivision of curve to remove the singular point from the convex hull.

Fig. 16. Test curves with corresponding control net and algebraic distance field. (a) A quadratic NURBS curve, (b) a cubic curve and (c) a quarticcurve.

4.2. Numerical efficiency relative to Newton–Raphson iterations

The time complexity of the algebraic distance construction algorithm is derived in Appendix B. In general, the costof computing the algebraic distance field from a NURBS curve is O(np3), where n is the number of Bezier segmentsin the NURBS curve and p is the degree of the curve. With the assumption of low algebraic degree, p may be assumedto be fixed. Hence, in practice, the computational cost is dependent on n alone.

Next, the computational efficiency of the algebraic distance field is compared to the Newton–Raphson iterationsthrough the following examples. Three NURBS curves (Fig. 16) of degrees two, three and four respectively arechosen for the test. The distance field is constructed both algebraically and using Newton–Raphson iterations. Thecorresponding computational expense is plotted in Fig. 17. As can be seen, the algebraic distance estimation costsapproximately 12%–14% of the Newton–Raphson iterations.

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 43

Fig. 17. Comparison of algebraic distance method and Newton–Raphson iterative scheme for curves of varying degree in Fig. 16 (tested usingcomputer with single Intel i5 processor and 4G memory).

Fig. 18. Comparison of distance fields. (a) Exact distance field with the marked regions corresponding to points that are equidistant to two differentpoints on the curve, and (b) algebraic distance field.

4.3. Uniqueness of solution and robustness

The exact distance field and the algebraic distance field for an S-shaped curve are compared in Fig. 18. The circularregions marked in the figure indicate points that are equidistant from the curve. At these points, the exact distancecalculations are non-unique leading to non-smooth distance fields overall. The algebraic distance field, on the otherhand, gives a smoother estimate of the distance to the curve while preserving exactness to the parametric curve. Thus,the algebraic procedure provides a more robust estimate of the distance.

5. Extension to NURBS surfaces

The algebraic distance field construction technique for planar NURBS and Bezier curves is extended to three-dimensional NURBS and Bezier surfaces in this section. The basic theory and algorithm remain the same, onlydifference being the underlying technique used for implicitization and the technique used for constructing thetrimming region. Bezier surfaces are implicitized using Dixon’s resultant described in Section 2.3.2.

5.1. Trimming region for Bezier surface

For the case of a three-dimensional parametric surface, the convex hull is constructed from the control point gridof the Bezier patch (Fig. 19). Each face of the convex hull is described by the equation of a plane. Let (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3) be three non-collinear vertices of the i th face of the convex hull. Then, the normalized

44 K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56

Fig. 19. Three-dimensional convex hull as the trimming region for a Bezier patch.

Fig. 20. Trimming region for planar Bezier patch.

field of the face is constructed using the following implicit equation

hi (x, y, z) = ±Ax + By + Cz + D√

A2 + B2 + C2(25)

where,

A =

1 y1 z11 y2 z21 y3 z3

, B =

x1 1 z1x2 1 z2x3 1 z3

, C =

x1 y1 1x2 y2 1x3 y3 1

, D = −

x1 y1 z1x2 y2 z2x3 y3 z3

.The distance measure corresponding to the convex hull is obtained by R-conjunction operation on the normalized

fields of the faces. In the case of planar, or nearly planar Bezier patch, since its control point grid degenerates to aplane, it is not possible to construct a three-dimensional convex hull from the control points. In such a planar case, theimplicit representation of the patch is the plane containing the patch. Hence, a trimming region has to be constructedsuch that its intersection with the plane gives the planar Bezier patch. This implies that the boundary of the patch mustlie on the trimming region. Using this insight, the trimming region is chosen to be the boundary Γ of the planar patchas shown in Fig. 20.

Now, the trimming region φ corresponds to the planar region bounded by Γ and its implicit representation isobtained by using the algebraic distance field computation procedure for planar curves. The R-function in Eq. (14)is used to construct the algebraic distance field of the planar patch. First, the normalized implicit field f1 is obtained

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 45

Fig. 21. Planar patch. (a) Planar NURBS patch, (b) algebraic distance field of the planar patch on a plane slicing the patch in three dimensionalspace.

as the implicit equation of the plane containing the patch. The implicit function f2 is constructed from the implicitrepresentation of trimming region φ such that it is identically zero in the interior of Γ . At any point in the domain, f2is computed at its corresponding projected point on the plane of the Bezier patch.

This process is illustrated in Fig. 20 using two test points P1 and P2. In each case, the test point is projected ontothe plane of the patch and then algebraic distance is calculated to the bounding curve Γ . If the projected point P proj

1lies outside the bounding curve then the algebraic distance field of the planar patch is computed as

g =

f 21 + f 2

2

where, f2 is the distance from the projected point P proj1 to Γ . If the projected point P proj

2 is on or inside the boundingcurve then Eq. (14) reduces to normalized distance from plane, i.e., g = f1. The distance field for a planar patch isillustrated in Fig. 21.

5.2. Normalization of three-dimensional distance fields

In general, the distance fields computed on the three-dimensional convex hull as well as the implicitized Bezierpatch needed greater care in normalization to ensure that the resulting distance fields resembled the exact ones.

5.2.1. Normalized distance field of three-dimensional convex hullThe distance field of the three-dimensional convex hull was constructed by R-conjunction operations on normalized

implicit field of its planar faces. As mentioned in Section 2.6, the R-conjunction operation preserves normalizationat almost all points except at points on the intersection edge between any two faces. Hence, after each R-conjunctionoperation, the resulting distance field was normalized prior to further compositions. The field φk , obtained after eachpairwise R-conjunction operation on the φk−1 normalized field and kth face of the convex hull, was normalized by itsgradient as follows:

φk =φk

|∇φk |. (26)

For normalized fields of the convex hull faces, |∇φk | = 1 near the faces except at their intersection. In Section 2.6,the bounds on the gradient of field obtained using R-conjunction operation near the intersection region was derivedas 0 < |∇φk | ≤ 3. Here, normalization of the R-conjunction fields is carried out by a scaling operation using a

46 K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56

Fig. 22. Normalization of field of trimming region. (a) Field of trimming region before scaling, (b) field of trimming region after scaling.

Fig. 23. Algebraic distance field of a bicubic Bezier patch. (a) Distance field of Bezier patch before scaling of implicit field, (b) distance field ofBezier patch after scaling of implicit field.

representative value of ∇φk such that 1 < |∇φk | ≤ 3. In this work, ∂φk∂xi

is chosen to be 1 such that |∇φk | =√

3. Thealgebraic distance field of the convex hull with and without the proposed scaling are compared in Fig. 22.

5.2.2. Normalized distance field of implicit surfaceThe implicit function f of the Bezier patch obtained using Dixon’s resultant is normalized by scaling it using the

ratio of the implicitized distance s(xr ) and the exact distance d(xr ) at a representative point xr . Here, a control pointthat does not lie on the Bezier patch is chosen as the representative point xr . The scaled implicit function f is thengiven by

f =f

s(xr )d(xr )

. (27)

This strategy was used to avoid computing the gradient |∇ f | since that is computationally more expensive tocarryout. The algebraic distance field for a Bezier patch with non-normalized and normalized implicit function arecompared in Fig. 23.

6. Examples of algebraic distance field

In this section, algebraic distance field is demonstrated on complex geometries first. Next, the algebraic distancefield is applied to analyze the convergence of stress intensity factors of curved cracks in planar solids, and tosimulate crack propagation. Finally, the developed procedure is applied to model heat conduction in solids with three-dimensional cracks.

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 47

Fig. 24. Hip prosthesis. (a) Model of hip implant constructed in this study for demonstration, (b) algebraic distance field over a cross-section ofthe hip implant.

Fig. 25. The Utah Teapot. (a) The teapot modeled using 32 bicubic Bezier patches, (b) algebraic distance field on a plane slicing the teapot.

6.1. Algebraic distance field for complex three-dimensional geometries

In this example, the algebraic distance field is demonstrated on the three-dimensional geometry of a hip prosthesis.A representative model of the hip implant (Fig. 24(a)) was constructed using Bezier patches. The algebraic distancefield over a cross-section of the prosthesis is shown in Fig. 24(b).

A more complex geometry, of that of the Utah teapot [33], is demonstrated next. The teapot model consisted of 32bicubic Bezier patches (Fig. 25(a)). The algebraic distance field for Utah teapot is demonstrated on a plane slicing thegeometry (Fig. 25(b)).

6.2. Modeling cracks through material enrichment

A heat conduction application is considered here in which heat flows from one face of the solid into the oppositeface that is cooled. In the presence of a thermal defect, normal heat flow is blocked and therefore causes a temperaturerise behind the defect.

In this example, the defect is modeled implicitly as a material enrichment [10] of zero thermal conductivity overthe underlying domain. The enriched thermal conductivity of the domain is modeled as:

k = (1 − w)k0 + wke (28)

48 K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56

a b

c d

Fig. 26. Thermal conduction in the presence of infinite strip defect. (a) Infinite strip defect modeled as a two-dimensional domain with a line crack,(b) enriched thermal conductivity over the domain, (c) validation for rise in temperature behind defect, (d) error in rise in temperature as a functionof distance scaling factor.

where, k0 is the thermal conductivity of the underlying material, ke is the thermal conductivity of the defect. Theweight function w smears the loss in thermal conductivity over a finite distance. The weight function is constructedas a monotonically decreasing function of distance from the defect. A Gaussian form of weight field is used in thisexample:

w = e−

s

s0

2

(29)

where s is the algebraic distance field from the defect and s0 is a scaling factor.An infinite strip defect was modeled as an equivalent two-dimensional problem with a line defect. A two-

dimensional square domain of size 2 × 2 units was considered (see Fig. 26(a)). Constant temperature boundaryconditions were enforced on top and bottom edges of the domain as shown in Fig. 26(a). The obtained numericalsolution for rise in temperature behind the defect is plotted against the analytical solution [34] in Fig. 26(c).

The accuracy of the material enrichment model is a function of the distance scaling factor s0. As s0 → 0, thesmeared region approaches a sharp crack configuration. Refinement studies were performed for three scaling factorss0 = 0.2, 0.1, 0.05 (Fig. 26(c)). In the limit, as s0 → 0, the solution approaches the analytical one with refinement ofthe control point spacing h. The convergence in error as a function of the distance scaling factor, s0 for three pointson the crack at distance x from the crack center is shown in Fig. 26(d).

6.3. Analysis of a curved crack

The problem of an infinite plate with a curved center crack under uniaxial tensile stress is modeled here and thenumerical convergence of the stress intensity factors is demonstrated. This problem was adapted from Ref. [35]. The

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 49

a b

c

Fig. 27. Plate with a curved crack. (a) Infinite plate modeled as a two-dimensional finite domain with a center curved crack, (b) geometry, loadingand boundary conditions, (c) error in numerical stress intensity factors as a function of number of control points.

analytical stress intensity factors are [36]:

K I =σ

2

π R sin(β)

1 − sin2 (β/2) cos2 (β/2)

cos (β/2)

1 + sin2 (β/2)+ cos (3β/2)

(30)

K I I =σ

2

π R sin(β)

1 − sin2 (β/2) cos2 (β/2)

sin (β/2)

1 + sin2 (β/2)+ sin (3β/2)

(31)

where σ is the uniaxial tensile stress, R is the radius of the circular arc, and 2β is the angle subtended by the arc atthe center as shown in Fig. 27(a).

The problem is modeled using a finite square domain of size 2 × 2 units and a circular crack of radius R = 1.005units, center angle 2β = 11.42 such that the ratio of the square side length to crack length is reasonably large (≥10)(see Fig. 27(a)). Only one half of the plate with the crack was modeled due to symmetry across the y-axis. The loading

50 K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56

and boundary conditions are described in Fig. 27(b). Elastic material properties with modulus E = 100 and Poisson’sratio ν = 0.3 were chosen.

Previously developed method for enriched isogeometric approximations [10] was adopted to model the crack. Ageneral enriched approximation for modeling fracture is of the form:

f = (1 − we) fΩ + we fe (32)

where, the continuous field ( fΩ ) and discontinuous enriching field ( fe) are approximated independent of each otherand composed to obey partition of unity. The weight field we is defined as an exponentially decaying function ofalgebraic distance. The discontinuity due to the crack is modeled using Heaviside step function H(x) as:

fe(P(x) → t) =

I

NI (t)u I H(x). (33)

Uniform control point grid discretizations of 20 × 20, 25 × 25, 31 × 31 and 34 × 34 were chosen for theconvergence study. The analytical solution for Mode I and Mode II stress intensity factors (for the chosen parameters)are K I = 0.5556 and K I I = 0.0556 respectively. The numerical stress intensity factors were calculated using thedisplacement correlation method. The resulting expression for estimating the stress intensity factors under plane strainconditions (and under plane stress conditions if ν is replaced by ν = ν/(1 + ν)) are:

K I =µ

√2π (vb − va)

√r (2 − 2ν)

(34)

K I I =µ

√2π (ub − ua)

√r (2 − 2ν)

(35)

where, µ is the shear modulus, ν is Poisson’s ratio, r is the distance from the crack tip to the correlation point,(ua, va) and (ub, vb) are the orthogonal displacements in x and y directions at correlation points a and b respectively.The convergence in the stress intensity factors is shown in Fig. 27(c).

6.4. Simulations of crack propagation

Crack propagation in a 45inclined edge-cracked specimen under uniform tension is modeled next. Geometry andloading conditions were as shown in Fig. 28. The elastic modulus was assumed to be 100 units and Poisson’s ratiowas 0.3.

The crack was modeled as a material enrichment [10] with a modulus value of zero. The elastic modulus of theunderlying material was enriched with a cohesive damage description to nucleate and propagate the crack. Thus, theenriched elastic modulus of the domain was of the form:

E = (1 − w)E0 + w(1 − D)E00 ≤ D ≤ 1

(36)

where, E0 is the elastic modulus of the undamaged material, D is the measure of damage with D = 0 being thepristine state and D = 1 being the fully damaged state of the material at which it does not bear any load (fracturedstate). The weight function w smears the loss in modulus over a finite spatial region. The algebraic distance field wasused to determine the weight field of the crack over the rectangular domain.

The crack propagation was based on maximum damage at points on a circular path ahead of the crack tip. Ingeneral, the damage measure D is defined as the ratio:

D =G

Γ(37)

where G is the energy release rate and Γ is the fracture toughness. In this example, a bilinear irreversible cohesivedamage description was used [10]. Snapshots of crack propagation steps are shown in Fig. 29. It can be observed thatthe crack eventually becomes horizontal, which is expected due to the symmetric tension loading on the specimen.

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 51

Fig. 28. Plate with inclined crack under tension. (a) Geometry and boundary conditions, (b) underlying regular control point grid independent ofinclined crack.

Fig. 29. Snapshots of crack propagation steps in order: Step 2 −→ Step 6 −→ Step 10 −→ Step 15.

6.5. Analysis of three-dimensional cracks

The heat conduction problem considered in Section 6.2 is reconsidered here. Specifically, a three-dimensionaldomain with a planar circular defect is analyzed. Geometry and boundary conditions are shown in Fig. 30(a). Theconverged numerical solution for rise in temperature behind the defect is compared to the analytical solution fordistance scaling factors s0 = 0.1, 0.05, 0.03 in Fig. 30(b). As before, as s0 → 0, the numerical solution approachesthe analytical solution [34]. Again, the convergence in error as a function of the distance scaling factor, s0 for threepoints on the crack at distance x from the crack center is shown in Fig. 30(c).

Finally, a complex, three-dimensional freeform NURBS defect geometry was chosen to illustrate the applicationof algebraic distance field for modeling an implicit crack. Geometry and boundary conditions are shown in Fig. 31(a).The algebraic distance field for the complex defect geometry is plotted in Fig. 31(b) over three planes slicing thegeometry in x, y and z directions respectively. The numerical solution for temperature in the domain is plotted over

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Fig. 30. Thermal conduction in the presence of a circular defect. (a) Three-dimensional domain with a circular defect: geometry and boundaryconditions, (b) validation for rise in temperature behind defect, (c) error in rise in temperature as a function of distance scaling factor.

the same three slices in Fig. 31(c). Through this example, it is reinforced that an approximate distance function asdeveloped in this study is sufficient to enrich underlying domain with known discontinuous behavior.

7. Summary

For problems containing enriching boundaries with known behavior, often an approximate distance function issufficient to capture the influence of such boundaries on the underlying domain. In the present study, a purely algebraictechnique for computing distance measures from complex non-linear boundaries was proposed. The procedure wasbased on the fact that the resultant of a parametric entity obtained by algebraic implicitization possesses the propertiesof distance. The theoretical development of the resultant based implicit function to construct distance measures for theparametric geometry is shown using normalization techniques and boolean operations. Algorithms were developed foralgebraic distance field construction for NURBS and Bezier geometries. The proposed technique overcomes the needfor iterative exact distance computations at every quadrature point during isogeometric analysis while enabling greatersmoothness of the field and robustness in the distance estimation. Algebraic distance field construction was illustratedon complex geometries such as the Utah teapot. The application of algebraic distance field was illustrated by modeling

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 53

a

b

c

Fig. 31. Thermal conduction in the presence of complex freeform defect geometry. (a) Geometry and boundary conditions, (b) algebraic distancefield plotted over planes slicing the defect geometry in the three principal directions, (c) temperature solution plotted over the same planes as thealgebraic distance field.

fracture and crack propagation examples. Three-dimensional implementation of the method was demonstrated througharbitrary shaped three-dimensional cracks. The cracks were represented by NURBS surfaces as a surface of zeromaterial property smeared over the underlying domain as a function of algebraic distance from the crack.

Acknowledgments

K. Upreti and T. Song were partially supported under Semiconductor Research Corporation Tasks 1292.075 and1292.090. The authors are thankful for this support.

Appendix A. Derivation of bounds on gradient of field obtained by R-function operation

The following two-dimensional analysis motivated by the analysis in [22] enables the estimation of the bounds forgradient of approximate distance field in the neighborhood of a boundary. Let u be the approximate distance function

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Fig. A.32. Quadratic transformation of φ1–φ2 plane to φ′1–φ′

2 plane.

obtained by R-function operation F on normalized fields φ1 and φ2.

u = F(φ1, φ2).

The partial derivatives and the magnitude of gradient of u are

∂ u

∂x=

∂ F

∂φ1

∂φ1

∂x+

∂ F

∂φ2

∂φ2

∂x,

∂ u

∂y=

∂ F

∂φ1

∂φ1

∂y+

∂ F

∂φ2

∂φ2

∂y

|∇u|2

=

∂ F

∂φ1

∂φ1

∂x+

∂ F

∂φ2

∂φ2

∂x

2

+

∂ F

∂φ1

∂φ1

∂y+

∂ F

∂φ2

∂φ2

∂y

2

.

If φ1 and φ2 are normalized, i.e., |∇φ1| = 1 and |∇φ2| = 1,

|∇u|2

=

∂ F

∂φ1

2

+

∂ F

∂φ2

2

+ 2∂ F

∂φ1

∂ F

∂φ2

∂φ1

∂x

∂φ2

∂x+

∂φ1

∂y

∂φ2

∂y

|∇u|

2=

∂ F

∂φ1

2

+

∂ F

∂φ2

2

+ 2∂ F

∂φ1

∂ F

∂φ2(∇φ1.∇φ2) . (A.1)

Since, φ1 and φ2 are normalized, ∇φ1 · ∇φ2 varies between −1 and +1. Hence, the theoretical bounds on |∇u| are∂ F

∂φ1−

∂ F

∂φ2

2

≤ |∇u|2

≤

∂ F

∂φ1+

∂ F

∂φ2

2

. (A.2)

Using R-conjunction equation (12), F(φ1, φ2) is given by

F = φ1 + φ2 −

φ2

1 + φ22 .

Now, the partial derivatives of F with respect to φ1 and φ2 are

∂ F

∂φ1= 1 −

φ1φ2

1 + φ22

,∂ F

∂φ2= 1 −

φ2φ2

1 + φ22

∂ F

∂φ1−

∂ F

∂φ2=

φ2 − φ1φ2

1 + φ22

,∂ F

∂φ1+

∂ F

∂φ2= 2 −

φ2 + φ1φ2

1 + φ22

. (A.3)

In the φ1–φ2 coordinate plane, the point of interest here is the origin, where φ1 = φ2 = 0. A quadratictransformation [29] of the φ1–φ2 plane to φ′

1–φ′

2 plane is defined as φ′

1 = φ1 and φ′

2 = φ2/φ1 as shown inFig. A.32. Under this transformation the origin is mapped onto the entire φ′

2-axis. At the origin in the φ1–φ2 plane, anindeterminate form φ′

2 =00 exists.

By approaching the origin along the line of slope m (line φ2 = mφ1),

φ′

2 =φ2

φ1=

mφ1

φ1= m, ∀ φ1 = 0

K. Upreti et al. / Comput. Methods Appl. Mech. Engrg. 280 (2014) 28–56 55

hence,

limφ1→0

φ′

2 = m ⇒ limφ1→0

φ2

φ1= m.

Substituting in Eq. (A.3),

∂ F

∂φ1−

∂ F

∂φ2=

m − 1√

1 + m2,

∂ F

∂φ1+

∂ F

∂φ2= 2 −

m + 1√

1 + m2. (A.4)

Thus the bounds for gradient of distance function in the neighborhood of the boundary are given by m − 1√

1 + m2

≤ |∇u| ≤

2 −m + 1

√1 + m2

. (A.5)

Thus, the maximum value of the upper bound is 3 and the minimum value of lower bound is 0 (Fig. 8).

Appendix B. Time complexity of algebraic distance field algorithm

Consider a NURBS curve of degree p. Further, assume that the NURBS curve decomposes into n Bezier curves.Then, the time complexity of this decomposition process is O(n). Each Bezier curve comprises of p+1 control points.Hence, for each step in constructing algebraic distance field of the Bezier curve, the time complexity is a function ofits degree p as follows:

• Time complexity of convex hull construction is O(p log(p)).• Time complexity of computing normalized distance field of convex hull is O(p).• Time complexity of computing the resultant depends on the determinant calculation and is O(p3).• Time complexity of trimming operation is O(1).

Thus, the time complexity of constructing algebraic distance field of Bezier curve is O(p3) and that for the NURBScurve is O(np3).

Given a NURBS surface of degree p × p that decomposes into n Bezier patches, the time complexity can becomputed as follows:

• Time complexity of convex hull construction is O(p2 log(p)).• Time complexity of computing normalized distance field of convex hull is O(p2).• Time complexity of computing the resultant depends on the determinant calculation and is O(p6).• Time complexity of trimming operation is O(1).

Thus, the time complexity of constructing algebraic distance field of Bezier surface is O(p6) and that for the NURBSsurface is O(np6).

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