A SPECT reconstruction method for extending parallel to non-parallel geometries Junhai Wen and Zhengrong Liang

Transcription

1 This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: This content was downloaded on 03/04/2019 at 09:09 Please note that terms and conditions apply. You may also be interested in: Measurement and Instrumentation Principles Alan S Morris Design and instrumentation of a magnetic nerve stimulator D McRobbie Simple boxcar integrator covering a large input frequency range P Williams and H W Lam New books on physics and related sciences Elena V Zakharova New books on physics and related sciences Elena V Zakharova A SPECT reconstruction method for extending parallel to non-parallel geometries Junhai Wen and Zhengrong Liang Voigt wave propagation in biaxial composite materials Tom G Mackay and Akhlesh Lakhtakia New books on physics and related sciences Elena V Zakharova

2 IOP Publishing Orthogonal tensors in computational engineering mechanics R M Brannon Chapter 1 Introduction Each practicing engineer must, at some point, characterize reorientations of a body or find vector and tensor components with respect to a rotated coordinate system. Additionally, rotations and reflections can contribute to overall engineering system response in combination with other predominant physical effects. Consequently, many important theorems (such as the polar decomposition and the time derivative of logarithmic strain) involve rotations as a part of a larger problem. A vast literature is available for the most common rotation-related tasks such as simple orthogonal Cartesian coordinate changes. Most reference books, however, tend to cover one or two methods for handling rotations, which makes it difficult for a researcher to translate between different methods used in different books. Resource information for less common rotation-related tasks is even scarcer. For example, mesoscale material simulations (in which crystals are explicitly modeled in a representative volume element) require methods to establish a uniform distribution of initial crystal orientations, which is a non-trivial task that is rarely explained in textbooks. Obscure rotation-related research applications can be found in disparate journal articles, but self-contained comprehensive review articles that cover both elementary and advanced engineering concepts written in a manner comprehensible to engineers are rare. Morawiec [1] offers an outstanding list of references on the topic of rotation and covers many of the topics reviewed here with greater mathematical rigor. Available rotation review articles and textbooks seem to be aimed more at mathematicians than applications specialists. While it is true that this book will cover several mathematically advanced topics, our target audience is practicing engineers, especially those who lack a guru-level background in abstract algebra, topology, differential geometry, hyper-complex numbers, and group theory, that unfortunately seems to be a prerequisite for understanding existing resource publications. Hence, most of our discussions are limited to ordinary three-dimensional (3D) vectors of the type discussed in first calculus courses. Detailed proofs are provided only when they are non-trivial (and not readily available in the literature) doi: / ch1 1-1 ª IOP Publishing Ltd 2018

3 or when the proof itself clarifies a concept. In this book, the assumed level of the reader s mathematical background varies considerably. Some chapters are quite elementary (first-year graduate student level), while other chapters are aimed at a target audience of mathematically adept researchers who are working on the arcane topics of contemporary interest. This book s topics and examples tend to favor materials mechanics issues. Rather than organizing this book using increasing levels of complexity, the topics are grouped according to subject area. Each chapter (and each section within a chapter) opens with an elementary (more intuitive) approach to the subject at hand, and then it progresses into greater levels of sophistication and rigor. Consequently, if you have only limited background in mathematics, you are encouraged to drill into each chapter only to the depth appropriate to your background. Each chapter is (as much as practical) self-contained, so it is not necessary to read preceding chapters to learn about subjects in later chapters. One aim of this book is to catalog the many different ways to define and quantify rotations and reflections. A rotation may be described mathematically by a special kind of orthogonal tensor. By using the special kind proviso, we are implying that rotation tensors are always orthogonal, but orthogonal tensors are not necessarily rotations. A tensor Q is orthogonal if and only if T Q Q = I, where Ĩ is the identity tensor 1. The tensor Q is orthogonal if and only if its component matrix [Q] is orthogonal. A real matrix [Q] is orthogonal if and only if where [I ] is the identity matrix. In indicial form, T [ Q] [ Q] = [ I], (1.1) QkiQ kj = δij (1.2) where δ ij is the Kronecker delta 2 and (using standard tensor notation conventions summarized in chapter 2) repeated indices are understood to be summed from 1 to 3. An elementary identity in tensor analysis states that a tensor is zero iff3 Z u Z v = 0 vectors u and v (1.3) Accordingly, two tensors, Ã and B, are equal iff u A v = u B v vectors ũ and v. A tensor Q is therefore orthogonal iff u Q T Q v = u I v vectors u and v (1.4) 1 This book uses a self-defining notation in which scalars, vectors and second-order tensors are written with zero, one, and two under-tildes, respectively (as in s, ṽ, and ). The composition of two tensors, denoted symbolically by A T B, has ij components given by AikBkj in which the repeated index is implicitly summed. The notation AB : = AB ij ij is the tensor inner product. The notations A u and v B are vectors with ith components of, respectively, A u ik k and v k B ki. These definitions immediately give the often-used identity A u = u A T. 2 I.e., components of the 3 3 identity matrix. Thus δ ij equals 1 if i = j and 0 if i j. 3 I.e., if and only if. 1-2

4 To interpret this assertion, consider the star vectors in figure 1.1, which are defined to be orthogonal transformations of non-starred fiducial vectors: u* = Q u and v* = Q v. (1.5) Noting that T u Q = Q u, equation (1.4) then reduces to u * v*= u v vectors u and v. (1.6) As a special case, taking the two arbitrary vectors to be identical to each other proves that an orthogonal Q must preserve lengths u * = * u and v = v. (1.7) Denoting these magnitudes by u and v, respectively, recall from elementary vector analysis that a vector dot product satisfies u θ v = uv cos, where θ is the angle between the vectors. Consequently, equation (1.6) shows that an orthogonal transformation Q preserves the magnitude but not necessarily the sign of the angle between two vectors: cos θ*= cos θ θ*=± θ (1.8) In two dimensions, the angle θ between ũ and ṽ can be arbitrarily declared to be positive if the vector ũ must be rotated counter-clockwise by θ to become aligned with ṽ. As indicated in figure 1.1, a 2D orthogonal transformation is called proper if it preserves the angle in both magnitude and sign; it is otherwise an improper transformation that will preserve the magnitude of the angle, but not the sign. An improper-orthogonal transformation is seen in the example of figure 1.1 to not only rotate the vectors but also to reflect them into different relative ordering (from clockwise to counter-clockwise or vice versa). These statements may be generalized to higher dimensions based on properties of the [Q] orthogonal transformation matrix itself. Figure 1.1. Effects of an orthogonal transformation. (a) The word fiducial simply means basis of comparison, which refers to the pre-transformation starting vectors (b) shows a change in orientation without a change in length or angle and (c) shows a change in both orientation and numerical sign of the angle. 1-3

5 It is fairly common for writers to refer to the set of all tensors as lin, and the subset of all orthogonal tensors is called orth [2]. Second-order engineering tensors have nine independent components, so they are said to belong to a 9D space. Moreover, because any linear combination of tensors is itself a tensor, lin is a 9D linear manifold and therefore it is possible to set up a Cartesian coordinate system for all members of lin. To rigorously prove that a second-order tensor on 3D space is, in fact, also a 9D vector, one must simply confirm axioms (additive commutativity, associativity, existence of an additive inverse, etc) that must be met for something to be called a vector. Just as the Cartesian representation for a vector x = x1e + e + e 1 x2 x (in which the ẽi are orthonormal basis vectors) may be represented in array form as x x 2 = x1 0 + x2 1 + x3 0, (1.9) x the Cartesian expansion of a tensor, T = T11e1e1+ T12e1e2+ + T33e e 3 3 (in which the ee i j are basis dyads), may be represented in matrix form as T 11 T12 T T 21 T22 T23 = T T T (1.10) T T T This expansion has nine terms involving nine orthonormal basis tensors. The space of ordinary 3D vectors is Euclidian (and thus has a Cartesian basis), but it can contain lower-dimensional spaces, such as the surface of a sphere, that are not Euclidean which means that a linear combination of points in that curvilinear space might not produce a point in the space. A point on the unit sphere, for example, is a vector of unit magnitude. Adding unit vectors does not generally produce a unit vector, so the set of all unit vectors is non-euclidean. This does not mean they are not vectors. They simply belong to a 2D curvilinear space contained within the larger 3D Euclidean space of general vectors. Similarly, we will find that the set of all orthogonal tensors belongs to a 3D curvilinear space (a hyper-sphere) embedded within the larger 9D Euclidean space of general second-order tensors. Unit vectors have only two independent components (given any two components, the third is constrained to produce a unit vector). Even though the components of a unit vector are not independent, it is nevertheless most intuitive to work with them using all three components. Similarly, even though rotations have only three independent components, it is nevertheless most intuitive to work with them in their full 3 3 form. A reduction to fewer components is useful mostly to conserve computer memory, but typically at the penalty of increased CPU cost. The fact that orth is made of tensors having only three independent components follows because columns (or rows) of an orthogonal matrix must form an orthonormal triad. The first column has two independent components (because it is a unit vector), and the second column has only one independent component because it must 1-4

6 be a unit vector and also have a zero inner product with the first column. Once the first and second columns of a rotation matrix are set, the third is the cross product of the first two, giving no additional independent components (other than sign if reflections are allowed). Because a linear combination of unit vectors is not generally a unit vector, it follows that a linear combination of orth tensors is not generally an orth tensor itself. Consequently, orth is a 3D nonlinear (curvilinear) subset of lin. As a nonlinear space, orth is not a flat higher-dimensional plane; instead, it will be seen to belong to a subset of a hyper-sphere in lin. Before doing any high-math to prove this, you should be able to guess the radius of this sphere: just as the magnitude of a vector v is vv k k, the magnitude of a tensor à is AA ij ij. If the tensor happens to be orthogonal Q, its magnitude (and hence the radius of the hyper-sphere) is T QQ = Q Q = δ = 3, (1.11) ij ij ji ij which applies if [Q] isa3 3 orthogonal matrix. In general, an N N orthogonal matrix has magnitude N. An orthogonal matrix [Q] with a positive determinant (equal to +1) is called a rotation or proper orthogonal matrix [3]. The set of all proper-orthogonal tensors is often referred to as orth +. An orthogonal matrix [Q] with a negative determinant (equal to 1) is said to be an improper-orthogonal tensor and is often misleadingly called reflection but it may actually be a reflection in combination with a rotation (see chapter 6). The set of improper-orthogonal tensors is often referred to as orth. A proper-orthogonal tensor is the most useful means of characterizing a rotation even though it is inefficient to store as nine components when only three are independent. Euler angles are also popular, but of limited value. These and many other lesser-known representations will be reviewed. Orthogonal tensors emerge as the superior quantification of a rotation or reflection 4, so each definition of an alternative representation of a rotation will be accompanied by instructions for converting from that representation to an orthogonal tensor and vice versa. Algorithms and source code are provided (and may be downloaded from In addition to cataloging the many ways to quantify rotation, this book covers several rotation-related applications. The topics are arranged as summarized here. For a smaller list that highlights the most significant contributions of this text, see the closing remarks in chapter 28. ij 4 Having nine components, an orthogonal tensor is not superior to other methods if minimizing storage is the metric. Instead, it is the structure and ease of computation that tends to make orthogonal tensors convenient. In the same way, a unit vector ñ is still most conveniently described in terms of three components ( n 1, n 2, n 3 ) satisfying one constraint, n1 2 + n2 2 + n3 2 = 1, even though they may be described in terms of two spherical angles as n1 = cos φ sin ϑ, n2 = sin φ sin ϑ, n3 = cos ϑ. This representation, though efficient in storage, has an ugly imbalance as well as numerical inefficiency in rebuilding the components. Similar assertions apply to rotations. The most compelling advantage of building a full three-component unit vector is that the set of unit vectors forms a 2D curvilinear subset (surface of a sphere) contained within a larger 3D Euclidean space. Euclidean representations are needed to compute Jacobians (needed to quantify cluster density of unit normals on a sphere or, in a higher-dimensional analog, texture of crystallographic orientations in materials). 1-5

7 Chapter 2 outlines notation and key mathematical prerequisites, as well as explaining the linear fractional transformation notation commonly used in robotics/controls. Chapter 3 reviews how orthogonal direction-cosine matrices are used to transform vector and tensor components from one orthonormal basis to another, and how these transformations are used to define engineering vectors or tensors as closure tests in the abstract mathematician s definitions that make no reference to length or direction. The distinction between a position vector s components and coordinates is made clear. Chapter 4 points out the distinction between rotation and coordinate transformation. Analyzing rotation demands awareness of your desired perspective. You can rotate an object, while you stay still, or you can keep the object fixed while you rotate yourself. It is important to be aware of which of these perspectives applies for your problem of interest. The distinction between these fundamentally different transformations goes beyond one being the same as the other with an opposite rotation angle. Chapter 5 describes nonlinear rotations (using torsion and vortex as examples), how to convert an axis and angle of rotation into a rotation tensor and vice versa, properties of skew tensors associated with rotation axes, and spinor vector representations of rotations. Chapter 6 reiterates that a rotation is characterized by a proper-orthogonal tensor i.e., one having a determinant equal to +1. An orthogonal tensor with a determinant equal to 1 should be regarded (in general) as a reflection, possibly in combination with a rotation. Chapter 7 presents the representation of a rotation in terms of a unit quaternion. Quaternions, which (being generalizations of complex numbers) are predecessors to modern vectors, and they are enjoying a resurgent vogue in rotationrelated applications. Even though a rotation tensor can be constructed uniquely from a unit quaternion (having one real component and three hyper-imaginary parts), it is explained that this does not imply that the smallest Euclidean space containing all possible rotation tensors is itself 4D. Chapter 8 describes how any linear vector-to-vector operator (in 3D) can be expressed as a sum of three dyads, and this result is specialized to rotations. This chapter provides yet another way to describe rotations that is based on a colorful nautical analogy between embedded triads in an object and embedded directions in a sailboat (mast, starboard, bow). Any orientation of these embedded directions can be specified through the boat s latitude, longitude and heading. Applications of dyadic representations of tensors include user interfaces for controlling rotation of a 3D object on a computer image, finding basis orientations that most simplify a tensor s component matrix, and constructing a rotation that will transform a specified vector into another specified vector. Chapter 9 shows that any general rotation may be expressed as a (noncommuting) sequence of three rotation operations. When defining a rotation in this way, it is essential to know whether subsequent rotations are defined with respect to fixed axes or follower axes. Rotations applied sequentially 1-6

8 about the fixed laboratory basis are different from rotations applied sequentially about axes of a triad that moves with the body as quantified by Euler angles, but the two descriptions are intimately related. Chapter 10 provides the series expansion representation of a rotation, which turns out to be a tensor analog to the series expansion of an exponential. This chapter also describes the Cayley transformation that constructs a rotation tensor from a skew tensor, as well as the Rayleigh transformation (which is used to apply a superimposed rotation to a tensor of any order). Chapter 11 derives the eigenvalues and eigenvectors of any rotation tensor. Chapter 12 presents the polar-decomposition theorem, which shows how a general deformation can be broken into two distinct (non-commuting) steps, one being a rigid rotation, and the other embodying pure material distortion and/or volume changes with no net material rotation. Chapter 13 describes how the polar decomposition can be used to strip away overall material rotation from material deformation, quantified by various strain measures commonly used in engineering applications. The spatial stretch which generates spatial strain measures is altered in the presence of rotation. The reference stretch which generates reference strain measures is unaffected by rotation. Chapter 14 discusses candidate strategies for advecting rotations in Eulerian physics codes or in Lagrangian codes that remap the mesh. The issues are similar to the challenge of mixing or interpolating unit vectors. Chapter 15 shows how the rate of a rotation tensor is related to the conventional angular rotation vector, vorticity tensor, and vorticity vector. This chapter includes derivatives of a rotation with respect to the rotation angle and with respect to the rotation axis as well as numerical algorithms for accurately integrating a rate of rotation to update material orientation in a physics code. The famous Dienes algorithm for computing polar spin from vorticity is included, along with a debunking of the common (but erroneous) assertion that eigenvectors of the symmetric part of the velocity gradient rotate according to the vorticity tensor defined by the skew part of the velocity gradient. Chapter 16 shows how to compute the time rate (or other derivative) of a principal function of a symmetric tensor, with emphasis on finding the time rate of logarithmic strain. A principal function (such as the Hencky logarithmic strain used in materials modeling) requires casting the argument (a symmetric tensor) in its principal basis to apply the function to the eigenvalues, after which the result is cast back to the physical basis. Finding the rate of such functions is crucial for linearizing numerical solvers, yet is extraordinarily complicated by the fact that principal directions not only evolve but are not even unique when there are repeated eigenvalues. An algorithm for taking rates of principal functions is presented and shown to be valid even when eigenvalues are repeated. The results are furthermore used to express the rate of reference Hencky strain as a linear transform of un-rotated rate of deformation (defined by a Rayleigh un-rotation of the symmetric 1-7

9 part of the velocity gradient and proved in this chapter to not be a pathindependent rate of any measure of deformation). Chapter 17 shows how to generate a uniformly random rotation tensor, which is useful for generating grain orientations for microscale simulations. We also discuss how to find the average of a tensor over all possible (uniform) rotations of that tensor. 5 This chapter includes extensive tutorials on elementary statistical sampling methods, as well as clear explanations about how to construct a non-square Jacobian matrix for transforming random numbers to random realizations and to then use it to prove whether or not an algorithm produces a uniform sampling. A case study, for example, proves that uniform sampling of Euler angles will not produce a uniformly random rotation, as is needed (for example) in mesoscale polycrystalline constitutive modeling. Chapter 18 defines isotropic tensors, which are very special tensors whose components do not change upon an orthogonal change in basis. A distinction is made between proper isotropy and strict isotropy, both of which are useful concepts. The set of all isotropic tensors of a given class 6 is a linear manifold, which means that it must possess a finite number of isotropic tensors (called primitive isotropic tensors) that form a basis for all isotropic tensors of that class. In three dimensions, there is only one primitive isotropic second-order tensor the identity tensor. There are two primitive proper-isotropic tensors in two dimensions. Fourth-order tensors in three dimensions have three primitive isotropic tensors. Chapter 19 provides an elementary introduction to the principle of materialframe indifference (PMFI), which requires that material constitutive models must be invariant under rigid rotation. An important point is that there do exist scalar invariants (which, by definition, will not change under a basis rotation) that will change upon a change in frame (i.e., a superimposed rotation). Likewise, a vector or tensor never changes under a basis rotation (the components change, but the sum of components times basis vectors or basis dyads remains invariant); whether or not a vector or tensor will change under a superimposed rotation depends on its physical definition. The discussion of PMFI includes a review of objective rates of spatial tensors, which are often employed in materials modeling (though we argue against this approach in favor of genuine rates within un-distorted or un-rotated material reference frames). Chapter 20 explains the difference between tensor symmetry and material symmetry, and then proceeds to prove that isotropic tensors are not always limited to mere multiples of the identity. Transverse symmetry (for which a tensor s components are unchanged upon rotation about a fixed axis) is covered in detail, including a rather brute force (hence accessible to a broader audience) explanation of why double-symmetric fourth-order transverse 5 Spoiler alert: it is proved to be the isotropic part of that tensor. 6 Where class refers to a tensor order (also called rank ) and the dimension of the underlying physical space. 1-8

10 stiffness tensors have five independent components, but only four distinct eigenvalues. The corresponding eigenprojectors are derived and a closed-form solution is derived for necessary and sufficient conditions for two such tensors to have the same eigenspaces (useful, for example, to force elastic and plastic stiffness tensors to have the same spectral symmetries). Application to jointed rock is also discussed. Chapter 21 defines scalar invariants of tensors, which are scalar-valued properties of tensors whose values do not change upon a change of basis. Not all real numbers are scalars. Chapter 22 returns to the topic of frame indifference as it applies to incremental constitutive models, first explaining the effect of rotation rate using an easily understood spring model, then moving towards various co-rotational rates in the literature, finishing with the obligatory derivation of oscillatory stress using certain rates (and explaining why it probably is not the rate that is the core issue) along with some guidance about how to run frame indifference testing in computational mechanics. Chapter 23 specializes the general continuum laws of mechanics to rigid-body mechanics. By applying the full power of tensor analysis, this presentation distinguishes itself from the derivations normally found in elementary dynamics textbooks. For example, with direct-notation tensor analysis, Euler s equations and the parallel-axis theorem can be presented in a single line. Chapter 24 returns to the problem of deformable media, but considers the case that bulk rigid motion is the primary motion, with relatively small deformations with respect to an appropriately defined rigidly moving observer frame (e.g., turbine blade, human bodies in a braking automobile, etc). Phantom body forces are derived to allow solving the problem in a non-inertial material frame. Chapter 25 covers some basic equations for 3D computer graphics visualization, in particular providing equations governing mapping from a virtual 3D body and providing recommendations for 3D visualization that would not require a person to wear 3D goggles. Chapter 26 defines conventional Voigt components for arranging the six independent components of symmetric tensors into a 6 1 array and demonstrates that these are in fact components of a 6D vector whose associated basis is not normalized (which is why manipulations using Voigt components always contain unruly factors of 2 that are actually the metrics associated with the non-normalized Voigt basis). Normalization of the Voigt basis produces the Mandel basis with associated Mandel components that are not plagued by factors of 2 and which therefore obey standard linear algebra, permitting more efficient use of computer library functions such as eigenvalue solvers for fourth-order tensors. Using Mandel components, the 6 6 component matrix associated with a tensor rotation operation in physical space is derived. 1-9

11 Chapter 27 completes the main text by providing a brief discussion of rotations in higher dimensions, which is applicable to constitutive material modeling. Ordinary physical tensor rotation operations are shown to correspond to double-plane rotations in stress space, where part of the tensor rotates by an amount equal to the rotation angle, another part rotates by twice this angle (as in Mohr diagrams), and a third part of the tensor does not rotate at all in tensor space. Appendix A provides FORTRAN and Python source listings that perform most of the computations presented in this book. Additional source code in these and other programming languages are available. References [1] Moraweic A 2004 Orientations and Rotations: Computations in Crystallographic Textures (Berlin: Springer) [2] Bigoni D 2012 Nonlinear Solid Mechanics (New York: Cambridge University Press) [3] Gurtin M E 1981 An Introduction to Continuum Mechanics (New York: Academic) 1-10