Applying 2D Tensor Field Topology to Solid Mechanics ... · Applying 2D Tensor Field Topology to Solid Mechanics Simulations 3 semi-positive-deﬁnite tensors, such as the diffusion

Applying 2D Tensor Field Topology to SolidMechanics Simulations

Yue Zhang, Xiaofei Gao, and Eugene Zhang

Abstract There has been much work in the topological analysis of symmetric tensorfields, both in 2D and 3D. However, there has been relatively little work in the phys-ical interpretations of the topological analysis, such as why wedges and trisectorsappear in stress and strain tensors. In this chapter, we explore the physical meaningsof degenerate points and describe some results made during our initial investigation.

1 Introduction

The analysis of symmetric tensor fields has seen much advance in the last twodecades. Topology-based tensor field analysis has found applications in not onlyscientific visualization, but also computer graphics [21] and geometry process-ing [1, 21].

Delmarcelle and Hesselink [6] study the singularities in a 2D symmetric tensor field,which they term degenerate points, i.e., points in the domain where the tensor fieldhas two identical eigenvalues (i.e., degeneracy). Delmarcelle and Hesselink pointout that there are two types of degenerate points, wedges and trisectors, which theyclassify using a descriptor that they introduce.

Yue ZhangSchool of Electrical Engineering and Computer Science, 3117 Kelley Engineering Center, OregonState University, Corvallis, OR 97331, USA. e-mail: [email protected] GaoSchool of Electrical Engineering and Computer Science, 1148 Kelley Engineering Center, OregonState University, Corvallis, OR 97331, USA. e-mail: [email protected] ZhangSchool of Electrical Engineering and Computer Science, 2111 Kelley Engineering Center, OregonState University, Corvallis, OR 97331, USA. e-mail: [email protected]

1

2 Yue Zhang, Xiaofei Gao, and Eugene Zhang

Both wedges and trisectors represent directional discontinuities in the tensor field.However, why are there only two types of fundamental degenerate points in a 2Dtensor field and why they appear in some locations? To the best of our knowledge,these questions are well understood only for the curvature tensor, which describesthe bending of surfaces. For example, wedges tend to appear at the tips of protru-sions in the surface, while trisectors tend to appear at the joints of the surface [2].

While the above interpretation of the degenerate points is well understood in shapemodeling, it is not clear how to adapt this interpretation to the stress tensor and straintensor in solid mechanics. The curvature tensor naturally relates to the distributionof the Gaussian curvature of surfaces considered and of the degenerate points inthe tensor field. In solid and fluid mechanics, the study of tensor field topology isrelatively under-utilized.

In this book chapter, we describe our ongoing effort and some initial results in apply-ing 2D symmetric tensor field topology to the stress tensor fields. Our approach in-cludes the generation of simulation scenarios with controlled shapes, material prop-erties, external forces, and boundary conditions. The occurrence of the degeneratepoints becomes a function of the physical properties of the shapes studied. In addi-tion, we present an enhanced topological description for 2D symmetric tensor fields,based on the concepts of isotropy index and deviator variability index.

Section 2 reviews past research in symmetric tensor field visualization. In Section 3we review relevant mathematical background about tensor fields. In Section 4 wedescribe our approach. We present our enhanced topological descriptors for 2D sym-metric tensor fields in Section 5 before concluding in Section 6.

2 Previous Work

There has been much work on the topic of 2D and 3D tensor fields for medicalimaging, scientific visualization, and geometry modeling. We refer the readers to therecent survey by Kratz et al. [11]. Here we only refer to the research most relevantto this chapter.

Most of the earlier research on symmetric tensor field analysis and visualization fo-cused on the diffusion tensor, a semi-positive-definite tensor extracted from brainimaging. The main focus is two-folds. First, fibers following the eigenvectors ofthe diffusion tensor are computed. Second, appropriate glyphs are designed to helpthe user understand the diffusion tensor. This has led to various measures for theanisotropy in the diffusion tensor, such as the relative anisotropy and the fractionalanisotropy [3]. Unfortunately these measures do not distinguish between the lin-ear and planar types of tensors. Westin et al. [19] overcome this by modeling theanisotropy using three coefficients that measure the linearity, planarity, and spher-icalness of a tensor, respectively. The aforementioned measures are designed for

Applying 2D Tensor Field Topology to Solid Mechanics Simulations 3

semi-positive-definite tensors, such as the diffusion tensor. We refer interested read-ers to the book [5] and the survey by Zhang et al. [23] on research related to thisarea.

At the same time, there have been a number of approaches to visualize 2D and3D symmetric tensor fields. Delmarcelle and Hesselink [7] introduce the notionof hyperstreamlines for the visualization of 2D and 3D symmetric tensor fields.Zheng and Pang [25] visualize hyperstreamlines by adapting the well-known LineIntegral Convolution (LIC) method of Cabral and Leedom [4] to symmetric ten-sor fields which they term HyperLIC [25]. Zheng and Pang also deform an objectto demonstrate the deformation tensor [24]. These visualization techniques havebeen later used for geomechanics data sets [14]. Hotz et al. [10] use an physically-driven approach to produce LIC like visualization. One of the fundamental differ-ences between the diffusion tensor and the other symmetric tensors from mechanics(stress, strain, symmetric part of the velocity gradient tensor) is that the former issemi-positive-definite (no negative eigenvalues) while the latter can have both semi-positive and negative eigenvalues. Schultz and Kindlmann [16] extend ellipsoidalglyphs that are traditionally used for semi-positive-definite tensors to superquadricglyphs which can be used for general symmetric tensors.

Delmarcelle and Hesselink [7, 8] introduce the topology of 2D symmetric tensorfields symmetric tensor fields as well as conduct some preliminary studies on 3Dsymmetric tensors in the context of flow analysis. Hesselink et al. later extend thiswork to 3D symmetric tensor fields [9] and study the degeneracies in such fields.Zheng and Pang [26] point out that triple degeneracy, i.e., a tensor with three equaleigenvalues, cannot be extracted in a numerically stable fashion. They further showthat double degeneracies, i.e., only two equal eigenvalues, form lines in the domain.In this work and subsequent research [28], they provide a number of degeneratecurve extraction methods based on the analysis of the discriminant function of thetensor field. Furthermore, Zheng et al. [27] point out that near degenerate curves thetensor field exhibits 2D degenerate patterns and define separating surfaces whichare extensions of separatrices from 2D symmetric tensor field topology. Tricocheet al. [17] convert the problem of extracting degenerate curves in a 3D tensor fieldto that of finding the ridge and valley lines of an invariant of the tensor field, thusleading to a more robust extraction algorithm. Tricoche and Scheuermann [18] in-troduce a topological simplification operation which removes two degenerate pointswith opposite tensor indexes from the field. Zhang et al. [21] propose an algorithmto perform this pair cancellation operation by converting the tensor field to a vectorfield and reusing similar operations in vector field topological simplification [22].

Finally, a number of researchers have investigated the visualization and analysisof data in solid and fluid mechanics using non-topological approaches, such as thestress classification using Mohr’s diagram [12], stress streamlets [20], and machinelearning [13].


3 Background on Tensors and Tensor Fields

In this section we review the most relevant background on 2D symmetric tensorsand tensor fields.

3.1 Tensors

A K-dimensional (symmetric) tensor T has K real-valued eigenvalues: λ1 ≥ λ2 ≥...≥ λK . When all the eigenvalues are non-negative, the tensor is referred to as semi-positive-definite. The largest and smallest eigenvalues are referred to as the majoreigenvalue and minor eigenvalue, respectively. When K = 3, the middle eigenvalueis referred to as the medium eigenvalue. An eigenvector belonging to the majoreigenvalue is referred to as a major eigenvector. Medium and minor eigenvectorscan be defined similarly. Eigenvectors belonging to different eigenvalues are mutu-ally perpendicular.

The trace of a tensor T=(Ti j) is trace(T)=∑Ki=1 λi. T can be uniquely decomposed

as D+A where D= trace(T)K I (I is the K-dimensional identity matrix) and A=T−D.

The deviator A is a traceless tensor, i.e., trace(A) = 0. Note that T and A havethe same set of eigenvectors. Consequently, the anisotropy in a tensor field can bedefined in terms of its deviator tensor field. Another nice property of the set oftraceless tensors is that it is closed under matrix addition and scalar multiplication,making it a linear subspace of the set of tensors.

The magnitude of a tensor T is ||T||=√

∑1≤i, j≤K T 2i j =

√∑

Ki λ 2

i , while the determi-

nant is |T|= ∏Ki=1 λi. For a traceless tensors T, when K = 2 we have ||T||2 =−2|T|.

This observation infers a simplification in tensor invariant calculations when onlythe traceless portions of the tensors are investigated, and it becomes advantageousto decompose the tensor into the trace and deviator parts.

A tensor is degenerate when there are repeating eigenvalues. In this case, thereexists at least one eigenvalue whose corresponding eigenvectors form a higher-dimensional space than a line. When K = 2 a degenerate tensor must be a mul-tiple of the identity matrix. In 2D, the aforementioned trace-deviator decomposi-tion can turn any tensor into the sum of a degenerate tensor (isotropic) and a non-degenerate tensor (anisotropic). For example, when the tensor is the curvature tensorof two-dimensional manifolds embedded in 3D, the isotropic-deviator decomposi-tion amounts to

K =U ′(

κ1 00 κ2

)U (1)


Fig. 1 Surface classification scheme based on the shape index φ ∈ [π/2,π/2] is color mapped tothe [BLUE,RED] arc in HSV color space: Left top: continuous mapping. Bottom: binned classi-fication. The legend (right) shows surfaces patches which are locally similar to points with givenvalues. This figure is a courtesy of [15], c©2012 IEEE.

where κ1 ≥ κ2 are the principal curvatures and the columns of U are the corre-sponding principal curvature directions. Rewriting the above equation we have

(κ1 00 κ2

)=

√κ2

1 +κ22√

2

[sinφ

(1 00 1

)+ cosφ

(1 00 −1

)](2)

where κ21 +κ2

2 = ||K||2 is the total curvature and φ = arctan(κ1+κ2κ1−κ2

) measures therelative strength between the isotropic and anisotropic components in the curvaturetensor. Figure 1 shows the canonical shapes corresponding to various representativeφ values and how φ can be used as a classification of surface geometry on the bunnysurface.

If all the eigenvalues of a tensor T are positive, the tensor is positive-definite. Ex-amples of semi-positive-definite tensors include the diffusion tensor from medicalimaging and the metric tensor from differential geometry. Tensors that are not semi-positive-definite include the curvature tensor from differential geometry and thestress and strain tensors from solid and fluid mechanics.


3.2 Tensor Fields

We now review tensor fields, which are tensor-valued functions over some domainΩ ⊂ RK . A tensor field can be thought of as K eigenvector fields, correspondingto the K eigenvalues. A hyperstreamline with respect to an eigenvector field ei(p)is a 3D curve that is tangent to ei everywhere along its path. Two hyperstreamlinesbelonging to two different eigenvalues can only intersect at the right angle, sinceeigenvectors belonging to different eigenvalues must be mutually perpendicular.

Hyperstreamlines are usually curves. However, they can occasionally consist of onlyone point, where there are more than one choice of lines that correspond to theeigenvector field. This is precisely where the tensor field is degenerate. A pointp0 ∈ Ω is a degenerate point if T(p0) is degenerate. The topology of a tensor fieldconsists of its degenerate points.

In 2D, the set of degenerate points of a tensor field are isolated points under numer-ically stable configurations, when the topology does not change given sufficientlysmall perturbation in the tensor field. An isolated degenerate point can be measuredby its tensor index [21], defined in terms of the winding number of one of the eigen-vector fields on a loop surrounding the degenerate point. The most fundamentaltypes of degenerate points are wedges and trisectors, with a tensor index of 1

2 and− 1

2 , respectively. Let LTp0(p) be the local linearization of T(p) at a degenerate

point p0 =

(x0y0

), i.e.,

LTp0(p) =(

a11(x− x0)+b11(y− y0) a12(x− x0)+b12(y− y0)a12(x− x0)+b12(y− y0) a22(x− x0)+b22(y− y0)

)(3)

Then δ =

∣∣∣∣( a11−a222 a12

b11−b222 b12

)∣∣∣∣ is invariant under the change of basis. Moreover, p0 is a

wedge when δ > 0 and a trisector when δ < 0. When δ = 0, p0 is a higher-orderdegenerate point. A major separatrix is a hyperstreamline emanating from a de-generate point following the major eigenvector field. A minor separatrix is definedsimilarly. The directions in which a separatrix can occur at a degenerate point p0can be computed as follows.

Let v = (x,y) be a unit vector. Let LTp0(p) be the local linearization at p0. A majorseparatrix can leave p0 in the direction of v if v is parallel to e1(p0,αv) for someα > 0. Here, e1(p0,αv) is a major eigenvector of LTp0(p0 +αv). Note that theLTp0(p) is a linear tensor field. Consequently, LTp0(p0 +αv) = αLTp0(p0 + v),and it is sufficient to choose α = 1. Similarly, a minor separatrix can leave p0 in thedirection of v if v is parallel to e2(p0,αv) where e2(p0,αv) is a minor eigenvector ofLTp0(p0 +αv). Finding either major separatrix or minor separatrix directions leadsto a cubic polynomial with either one or three solutions under stable conditions.It is known that around a trisector there must be three solutions, corresponding to


Fig. 2 A wedge (left) and a trisector (right).

three separatrices that divide the neighborhood of p0 into three sectors, thus thenamesake. Around a wedge there can be either one sector or three sectors.

The total tensor index of a continuously tensor field over a two-dimensional mani-fold is equal to the Euler characteristic of the underlying manifold. Consequently, itis not possible to remove one degenerate point. Instead, a pair of degenerate pointswith opposing tensor indexes (a wedge and trisector pair) must be removed simul-taneously [21]. Figure 2 shows a wedge pattern (left) and a trisector pattern (right),respectively.

4 Our Approach

The classification in Equation 2 is applicable to tensor fields other than the curvaturetensor. In this book chapter, we apply it to the stress and strain tensors from 2D solidmechanics simulations.

Our research is inspired by the following question: in the stress tensor, what tensorproperties decide the location and type of a degenerate point? To explore this ques-tion, we develop a number of simulation scenarios, using a commercial softwaretool for continuum mechanics. Currently, we are focusing on 2D scenarios. Further-more, we implement a visualization system which takes as input a 2D symmetrictensor field and computes a number of derived quantities from the tensor field, suchas the components of the tensors as well as their traces, determinants, magnitudes,eigenvalues, and eigenvectors. The eigenvectors (either major, or minor, or both) aredisplayed using texture-based methods [21] while the scalar quantities are displayedusing colors. For our initial geometry, we choose a relative simple case, i.e., a squareshape with uniform linear elastic material property. After deforming this shape, weobtain nontrivial stress tensors over the domain. In our visualization, the textures


Fig. 3 The visualization of the stress tensor from a simulation in which the left edge is fixed, andthe top and bottom edges are pulled to expand the shape. No boundary condition is applied to theright edge. The colors show the trace based on the rainbow map: red (high), yellow (medium high),green (zero), cyan (medium low), and blue (low). The textures show the major eigenvectors. In theleft, the object is undeformed, while in the right it is deformed.

Fig. 4 Visualization of three deformed shapes from the same initial shape by varying the boundaryconditions. Even though different amplitudes and locations of the boundary conditions lead todifferent topology, a possible pattern of when the trisectors (shown here for this family of scenarios)seems to exist.

and colors can be displayed on the mesh before and after the deformation. Figure 3shows one example of our visualization applied to a simulated scenario before andafter the deformation. Figure 4 shows different tensor field topology for three suchsimulation scenarios with the same initial domain and homogeneous material prop-erties but different boundary conditions and external force configurations.

Our study of these data sets has led to a new tensor classification approach as wellas some observations.


5 Tensor Classification

Equation 2 is applicable to not only the curvature tensor in geometry processing, butalso the stress and strain tensors in solid and fluid mechanics. In this case, degeneratetensors correspond to points where the stress is isostropic, with either a positive trace(pressure) or a negative trace (pressure). Degenerate points in the stress tensor aretherefore where isotropic stress occurs, i.e., only hydrostatic pressure and no shear.

This straightforward application of 2D symmetric tensor field topology to stress ten-sors has two shortcomings. First, pure shear locations, where no hydrostatic pressureexists, are not included as important features in tensor field topology. Second, thetype of the degenerate points (wedge or trisector) is not considered. To address this,we introduce an enhanced version of the tensor field topology.

As mentioned earlier, there is not a curvature tensor quantity in stress tensor fieldsand this hinders developing topology over these fields. To divide the material re-gion into meaningful topological domains, we explore various measurement crite-ria. Here, we consider the following two quantities: we consider the following twoquantities: isotropy index φ , and deviator variation index δ . The isotropy index φ isthe same as the shape index in Equation 2. Next, we describe the deviator variationindex.

Given a symmetric tensor field T (x,y) defined on a domain D⊂ R2, the deviator of

T (x,y) introduces a map η : D→ F where F = (

a bb −a

)‖a,b ∈ R2 is the set of

2D traceless, symmetric tensors. We define the anisotropic variation index at a point(x0,y0) ∈ D as:

δ (x0,y0) = limA(Γ )→0

A(η(Γ ))

A(Γ )(4)

where Γ is a region enclosing (x0,y0), η(Γ ) is the image of Γ under the map η ,and A(K) is the signed area of K, which in this case refers to Γ and η(Γ ), respec-tively. This quantity measures the spatial variability of the deviatoric stress tensor,which also shows spatial variability in the (major or minor) eigenvector fields around(x0,y0). Note that δ (x0,y0) can be negative, indicating Γ and η(Γ ) are oppositelyoriented.

Assuming T (x,y) is sufficiently differentiable, it can be shown that

δ (x0,y0) = |

(12 (

∂T11∂x (x0,y0)− ∂T22

∂x (x0,y0))∂T12∂x (x0,y0)

12 (

∂T11∂y (x0,y0)− ∂T22

∂y (x0,y0))∂T12∂y (x0,y0)

)| (5)

Notice that δ is exactly the same quantity that Delmarcelle and Hesselink [6] usedto classify a degenerate point in a 2D symmetric tensor field. In their analysis, the


Fig. 5 Three simulation scenarios visualized using the deviator variability index. As expected,the deviator variability index is positive for wedges (e.g., (left)) and negative for trisectors (e.g.,(middle)). However, the absolute value of the deviator variability index is a useful tool to measurethe importance of degenerate points (middle: the two trisectors near the right edge have a higherindex value than those of the three trisectors near the left edge, indicating great variability in theeigenvectors around the degenerate points). Moreover, even in the absence of degenerate points(right), the spatial variation of the deviator variability index is also a characterization of the tensorfield itself.

Fig. 6 The tensor magnitude (left), isotropy index (middle), and deviator variability index (right)of a simulated scenario. Note that the three descriptors together provide more insight into the tensorfield.

sign of the deviator variation index is used to classify a degenerate point (positivefor wedges and negative for trisectors). We wish to point out that the absolute valueof the deviator variation index is also important as it measures how spatially vary-ing the eigenvector fields are around the point of interest, which can be either adegenerate point or a regular point.

Figure 5 demonstrates this with three examples. This leads to the following char-acterization of a 2D stress tensor field, based on the magnitude, the isotropy index,and the deviator variability index.

In addition to the enhanced description of tensor fields, we also propose to add theset of zero isotropy index points and the set of zero deviator variability index pointsto 2D tensor field features. For convenience, we refer to a zero isotropy index pointas a pure shear point and a zero deviator variability index point as a transition point.Note that the latter is somewhat a misuse of the term transition point, which refers


Fig. 7 The sign functions of the isotropy index (left) and the deviator variability index (right) ofa simulated scenario. The boundaries between red and blue regions in both figures correspond topure shear points (left) and zero deviator variability points (right), respectively. These two sets ofpoints form curves, which, collectively divide the domain in four types of regions.

to a structurally unstable degenerate point. Not all points that have a zero deviatorvariability index is a degenerate point. However, for convenience we will overloadthe term in the remainder of this book chapter. The sets of pure shear points andtransition points are both curves under structurally stable conditions. Together, theydivide the domain into four types of regions, which we refer to as (1) expansionwedge region, (2) expansion trisector region, (3) compression wedge region, and(4) compression trisector region. Again, note that we use wedge and trisector in thenames of these regions for convenience, instead of terms such as positive deviatorvariability index and negative deviator variability index. The names do not suggestthat there must be any degenerate point in any of these regions.

Figure 7 shows the partitions of the domain by the set of pure shear points (left) andthe transition points (right) of a simulation.

It is interesting that when we set up the boundary conditions for our simulations, theexpected number, location, and type of degenerate points have often deviated fromthe actual outcome. This highlights the need to understand the physical interpreta-tions of degenerate points, such as why they appear in certain locations. In addition,the knowledge can help domain scientists control the number, type, and location ofdegenerate points when setting up the simulation.

6 Conclusion

The notion of tensor field topology originated from fluid and solid mechanics [6].Yet, after years of research in tensor field topology, its uses in engineering appli-cation are still rather limited. One of the main difficulties is the lack of physicalinterpretation of tensor field topology in these applications. Inspired by this, wehave started the exploration of the physical interpretations of tensor field topology


in terms of stress tensors. Our approach includes the generation of a large numberof simulation scenarios with controlled shapes and material properties and externalforce as well as topological analysis of the resulting stress tensor fields. As a firststep of this exploration, we enhanced existing 2D symmetric tensor fields by in-cluding the set of pure shear points and the set of transition points, which divide thedomain into four types of regions. At part of this, we point out the geometric mean-ing of the deviator variability index, which was limited to classifying degeneratepoints.

As future work, we plan to adapt our analysis to 2D asymmetric tensor fields and3D symmetric tensor fields.

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Applying 2D Tensor Field Topology to Solid Mechanics ... · Applying 2D Tensor Field Topology to Solid Mechanics Simulations 3 semi-positive-deﬁnite tensors, such as the diffusion