When a Fluid is Said to Deform Continuously Under the Action of a Shearing Stress This Means That

Continuous Deformation

Volume 2

Diogo S. Pinto , ... Joaquim M.S. Cabral , in Encyclopedia of Tissue Engineering and Regenerative Medicine, 2019

Shear stress

Fluid is defined as a substance that undergoes continuous deformation when subjected to a shearing force. Such shearing forces act tangentially to the surfaces over which they are applied. In MSC dynamic cultures, those surfaces can be either planar surfaces, such as the plate surface of parallel plate bioreactors, or 3D surfaces, including scaffolds, 3D spheroids or microcarriers. Thus, MSC growing adherent to a surface are exposed to shear forces from the moving fluid. Shear stress is known to be one of the main culture factors affecting MSC expansion (i.e., proliferation) and differentiation. Shear stress strongly depends on the viscosity of the fluid and, in the case of MSC culture medium, the fluids are considered as laminar Newtonian fluids. Under these conditions, the flow regimen in which a bioreactor can operate can be either laminar (viscous forces dominate inertial ones and Reynolds number, Re [conceptually the ratio of these two forces]   <   2   ×   10⁴) or turbulent (inertial forces dominate viscous ones, Re  >   2   ×   10⁴). Examples of bioreactor configurations operating in a laminar flow regimen are packed/fixed-bed bioreactors, parallel plate bioreactors, and hollow fiber bioreactors. Conversely, types of bioreactors operating under turbulent or transiently turbulent regimes for MSC expansion include STR, wave-mixed bioreactors, rotating bed bioreactors, and the Vertical-Wheel™ bioreactors.

Parallel plate bioreactors have been well-investigated regarding shear stress, which is low in their case. In 1996, researchers studied the influence of the bioreactor geometry on fluid flow and the resulting growth and differentiation of BM stem cells. The results indicated a higher cell density and uniformity in the radial-flow bioreactor due to a more uniform environment caused by the hyperbolic velocity and tube-like shear stress contribution, as well as the absence of walls parallel to the flow paths creating slow flowing regions. Flow pattern and shear stress levels were also studied in the Integrity™ Xpansion™ Multiplate Bioreactor, using a computational fluid dynamics (CFD) approach. The analysis revealed the occurrence of gentle laminar flow and a maximum wall shear stress lower than 10   mPa, which was 1000 times lower than in stirred bioreactors. The influence of the superficial velocity of the culture medium on MSC expansion in a fixed-bed bioreactor was also studied. The fact that MSC are exposed to shear stress caused by the medium flow in this type of bioreactors demands the determination of an upper limit for the superficial velocity. Results showed a decrease of more than 50% of the mean growth rate when the superficial velocity was increased from 2.65   ×   10^{−   4}  m/s to 1.59   ×   10^{−   3}  m   s^{−   1}, demonstrating the importance of controlling the fluid velocity on laminar flow-operating bioreactors.

Local shear rates in stirred bioreactors vary within the vessel. It is therefore more difficult to associate cellular effects (like cell differentiation or damage) with the magnitude of the prevailing shear rate or the associated shear stress. Several concepts have been proposed to describe the effect of shear stress on cells growing on microcarrier cultures in bioreactor systems. Researchers estimated the theoretical values of maximum shear stress under stirred conditions, τ _max, as result of flow through Kolmogorov eddies, in a 1.3   L Bioflo® bioreactor system. A maximum shear stress of 1.5   dyn/cm² (0.15   N   m^{−   2}) was reported for a 0.8   L working volume of BM MSC culture, which was lower than the one determined for a spinner flask culture. Using the same approach, fluid flow and suspension were characterized inside a 125   mL spinner flask. Researchers found that AT MSC cultured in this system tolerate mean and maximum shear stresses in the order of 0.004 to 0.2   N   m^{−   2}, respectively. Shear stress levels were also calculated for a BM MSC culture in a Vertical Wheel™ 3   L bioreactor system. Under the culture conditions studied, the shear stress rate was 0.021   N   m^{−   2}, which is roughly an order-of-magnitude lower than the levels reported as detrimental for cell growth. Other authors proposed different approaches to estimate the shear stress in stirred bioreactors, including the concept of an "Integrated Shear Factor" (ISF), a measure of strength of the shear field between the impeller and the walls. In wave-mixed bioreactor systems, shear stress is highest at the lowest filling level together with the highest rocking rate and rocking angle. Furthermore, shear stress pattern was more homogeneous in wave-mixed bioreactors with 1D motion than in stirred bioreactors. For this reason, wave-mixed bioreactors are good candidates for the manufacturing of shear sensitive cells.

To summarize, shear effects need to be minimized to successfully expand MSC and prevent undesirable differentiation or cell damage. Controversial results regarding shear stress tolerance of MSC might be due to protective effects of medium components, such as serum, or differences in the type of surface or cell density. For example, cells growing on macroporous microcarriers are better protected against shear than cells growing on nonporous carriers. To minimize shear stress levels in a bioreactor culture, it is important to understand which parameters affect the mixing efficiency.

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Unimodal B-Spline Registration

James Shackleford , ... Gregory Sharp , in High Performance Deformable Image Registration Algorithms for Manycore Processors, 2013

2.1 Introduction

B-spline registration is a method of deformable registration that uses B-spline curves to define a continuous deformation field that maps each and every voxel in a moving image to a corresponding voxel within a fixed or reference image ( Rueckert et al., 1999). An optimal deformation field accurately describes how the voxels in the moving image have been displaced with respect to their original positions in the fixed image. Naturally, this assumes that the two images are of the same scene taken at different times using similar or different imaging modalities. This chapter deals with unimodal registration which is the process of matching images obtained via the same imaging modality. Figure 2.1 shows an example of deformable registration of two 3D CT images using B-splines, where registration is performed between an inhaled lung image and an exhaled image taken at two different times. Prior to registration, the image difference shown is quite large, highlighting the motion of the diaphragm and pulmonary vessels during breathing. Registration is performed to generate the vector or displacement field. After registration, the image difference is much smaller, demonstrating that the registration has successfully matched tissues of similar density.

Figure 2.1. Deformable registration of two 3D CT volumes. Images of an inhaled lung and an exhaled lung taken at different times from the same patient serve as the fixed and moving images, respectively. The registration algorithm iteratively deforms the moving image in an attempt to minimize the intensity difference between the images. The final result is a vector field describing how voxels in the moving image should be shifted in order to make it match the fixed image. The difference between the fixed and moving images with and without registration is also shown.

In the case of B-spline registration, the dense deformation field can be parameterized by a sparse set of control points which are uniformly distributed throughout the moving image's voxel grid. This results in the formation of two grids that are aligned with one another: a dense voxel grid and a sparse control point grid. Individual voxel movement between the two images is parameterized in terms of the coefficient values provided by these control points, and the displacement vectors are obtained via interpolation of these control point coefficients using piecewise continuous B-spline basis functions. Registration of images can then be posed as a numerical optimization problem wherein the spline coefficients are refined iteratively until the warped moving image closely matches the fixed image. Gradient descent optimization is often used, meaning either analytic or numeric gradient estimates must be available to the optimizer after each iteration. This requires that we evaluate (i) a cost function corresponding to a given set of spline coefficients that quantifies the similarity between the fixed and moving images and (ii) the change in this cost function with respect to the coefficient values at each individual control point which we will refer to as the cost function gradient. The registration process then becomes one of iteratively defining coefficients, performing B-spline interpolation, evaluating the cost function, calculating the cost function gradient for each control point, and performing gradient descent optimization to generate the next set of coefficients.

The above-described process has two time-consuming steps: B-spline interpolation, wherein a coarse array of B-spline coefficients is taken as the input and a fine array of displacement values is computed as the output defining the vector field from the moving image to the reference image, and the cost function gradient computation that requires evaluating the partial derivatives of the cost function with respect to each spline-coefficient value. Recent work has focused on accelerating these steps within the overall registration process using multicore processors. For example, the authors Rohlfing et al. (2003), Rohrer et al. (2008), Zheng et al. (2009), and Saxena et al. (2010) have developed parallel deformable registration algorithms using mutual information between the images as the similarity measure. Results reported by Zheng et al. (2009) for B-splines show a speedup of n/2 for n processors compared to a sequential implementation; two $512\times 512\times 459$ images are registered in 12   min using a cluster of 10 computers, each with a 3.4-GHz CPU, compared to 50   min for a sequential program. Rohfling et al. (2003) present a parallel design and implementation of a B-spline registration algorithm based on mutual information for shared-memory multiprocessor machines. Rohrer et al. (2008) describe a design for the Cell processor and a GPU-based design is discussed in Saxena et al. (2010).

This chapter describes how to develop GPU-based designs to accelerate both steps in the B-spline registration process, and its main contribution with respect to the state of the art lies in the design of the second step: the cost function gradient computation. We show how to optimize the GPU-based designs to achieve coalesced accesses to GPU global memory, a high compute to memory access ratio (number of floating point calculations performed for each memory access), and efficient use of shared memory. The resulting design, therefore, computes and aggregates the large amount of intermediate values needed to obtain the gradient very efficiently and can process large data sets.

We follow a systematic approach to accelerating B-spline registration algorithms. First, we develop a fast reference (sequential) implementation by developing a grid-alignment technique and accompanying data structure that greatly reduces redundant computation in the registration algorithm. We then show how to identify the data parallelism of the grid-aligned algorithm and how to restructure it to fit the single instruction, multiple data (SIMD) model, necessary to effectively utilize the large number of processing cores available in modern GPUs. The SIMD model can exploit the fine-grain parallelism present in registration algorithms, wherein operations can be performed on individual voxels in parallel. For complex spline-based algorithms, however, there are many ways of structuring the same algorithm within the SIMD model, making the problem quite challenging. A number of SIMD versions must therefore be developed and their performance analyzed to discover the optimal implementation. We introduce a carefully optimized implementation that avoids redundant computations while exhibiting regular memory access patterns that are highly conducive to the GPU's memory architecture. We also evaluate other design options with speedup implications such as using a lookup table (LUT) on the GPU to store precomputed spline parameterization data versus computing this information on the fly.

Finally, single-core CPU, multicore CPU, and many-core GPU-based implementations are benchmarked for performance as well as registration quality. The NVidia Tesla C1060 and 680 GTX GPU platforms are used for the GPU versions. Though speedup varies by image size, in the best case, the 680 GTX achieves a speedup of 39 times over the reference implementation and the multicore CPU algorithm achieves a speedup of 8 times over the reference when executed on eight CPU cores. Furthermore, the registration quality achieved by the GPU is nearly identical to that of the CPU in terms of the RMS differences between the vector fields.

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Introduction

Maurice Herlihy , ... Sergio Rajsbaum , in Distributed Computing Through Combinatorial Topology, 2014

1.1.1 Distributed computing and topology

In the past decade, exciting new techniques have emerged for analyzing distributed algorithms. These techniques are based on notions adapted from topology, a field of mathematics concerned with properties of objects that are innate , in the sense of being preserved by continuous deformations such as stretching or twisting, although not by discontinuous operations such as tearing or gluing. For a topologist, a cup and a torus are the same object; Figure 1.1 shows how one can be continuously deformed into the other. In particular, we use ideas adapted from combinatorial topology, a branch of topology that focuses on discrete constructions. For example, a sphere can be approximated by a figure made out of flat triangles, as illustrated in Figure 1.2.

Figure 1.1. Topologically identical objects.

Figure 1.2. Starting with a shape constructed from two pyramids, we successively subdivide each triangle into smaller triangles. The finer the degree of triangulation, the closer this structure approximates a sphere.

Although computer science itself is based on discrete mathematics, combinatorial topology and its applications may still be unfamiliar to many computer scientists. For this reason, we provide a self-contained, elementary introduction to the combinatorial topology concepts needed to analyze distributed computing. Conversely, although the systems and models used here are standard in computer science, they may be unfamiliar to readers with a background in applied mathematics. For this reason, we also provide a self-contained, elementary description of standard notions of distributed computing.

Distributed computing encompasses a wide range of systems and models. At one extreme, there are tiny GPUs and specialized devices, in which large arrays of simple processors work in lock-step. In the middle, desktops and servers contain many multithreaded, multicore processors, which use shared memory communication to work on common tasks. At the other extreme, "cloud" computing and peer-to-peer systems may encompass thousands of machines that span every continent. These systems appear to have little in common besides the common concern with complexity, failures, and timing. Yet the aim of this book is to reveal the astonishing fact that they do have much in common, more specifically, that computing in a distributed system is essentially a form of stretching one geometric object to make it fit into another in a way determined by the task. Indeed, topology provides the common framework that explains essential properties of these models.

We proceed to give a very informal overview of our approach. Later, we will give precise definitions for terms like shape and hole, but for now, we appeal to the reader's intuition.

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Persistent homology

Leigh Metcalf , William Casey , in Cybersecurity and Applied Mathematics, 2016

9.3 Holes

We are going to digress a bit and discuss the nature of holes in data. We start by defining a region or a subset of ${\mathbb{R}}^2$ . For example, a square, a circle, several circles, a ring, a square with several rings inside of it, a triangle, or even a hexagon with three triangles in it are all regions.

A continuous deformation of a region is where we can shrink or twist the region but without tearing. Gluing is also not allowed. This allows us to transform one region into another. In other words, it is terraforming, or earth shaping, with restrictions. We can expand our land, shrink our land, or even change the borders completely. We also can expand a water feature or shrink it, but we cannot remove it entirely. Removing it is essentially gluing a patch on the region, which is not allowed. We cannot split the land into two pieces or take two pieces of land and put them together to form one.

For example, suppose we have a circle in ${\mathbb{R}}^2$ . We can shrink it or expand it or even push it until it looks like a square. We cannot rip it open and create a line out of it though. Think of it as if the region was enclosed with elastic strings. We can shrink the string or expand it, but we cannot tear it. So if we had a ring with two strings, we could push the inner string out so that it matches the outer string, but we cannot remove the strings. Similarly, suppose we have two separate circles. There is no way to join them into one circle without tearing them open.

Let us look at another example. Consider the letter A. We can push the legs up until they go away, which leaves us with a triangle. We can then deform the triangle so that it looks like the letter O. Conversely, we can deform the letter O so that it looks like a small triangle then pull on the triangle to form legs. On the other hand, we cannot continuously deform an A to form a H. We would have to tear the triangle apart to create the H.

We can also look at this in terms of graphs. The holes in a graph are the fundamental cycles. If an edge is not part of a fundamental cycle, then it is not part of any cycle in the graph. This is because any cycle in the graph can be created as a linear combination of the fundamental cycles in the graph. So we can push the edge that is not part of any cycle in the graph back into the graph by using a continuous deformation. In a similar fashion, two components of a graph cannot be joined together using a continuous deformation to create a new graph.

In summary, a continuous deformation cannot remove a hole in a region, nor can it put two pieces of a region back together. This means that we can classify a region by the number of holes that it has. A region with two holes can be deformed into a similar region with two holes, but cannot be deformed into a region with three holes. Similarly, if a region has two distinct parts, we cannot combine them to form one nor rip one apart to form three.

We have two kinds of holes possible in a region of ${\mathbb{R}}^2$ . The first is the number of distinct subregions in the region. These are separate in that we cannot glue them together to form a new region. For example, the letters o o are two distinct regions that cannot be continuously deformed to form a new region. For example, we cannot put them together to form a letter B. The second type of hole is the region that cannot be removed without tearing. For example, the letter o cannot be continuously deformed to remove the hole. These holes are the features of the region, also known as the topological features.

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Topological Knot Theory and Macroscopic Physics

L. Boi , in Encyclopedia of Mathematical Physics, 2006

Influence of Geometry and Topology on Fluid Flows

Ideal topological fluid mechanics deals essentially with the study of fluid structures that are continuously deformed from one configuration to another by ambient isotopies. Since the fluid flow map ϕ is both continuous and invertible, then ${\phi }_{t_1}\left(K\right)$ and ${\phi }_{t_2}\left(K\right)$ generate isotopies of a fluid structure K (e.g., a vortex filament) for any $\left\{t_1,t_2\right\}\in I$ . Isotopic flows generate equivalence classes of (linked and knotted) fluid structures. In the case of (vortex or magnetic) fluid flux tubes, fluid actions induce continuous deformations in D. One of the simplest deformations is local stretching of the tube. From a mathematical viewpoint, this deformation corresponds to a time-dependent, continuous reparametrization of the tube centerline. This reparametrization (via homotopy classes) generates ambient isotopies of the flux tube, with a continuous deformation of the integral curves.

Moreover, in the context of the Euler equations, the Reidemeister moves (or isotopic plane deformations), whose changes conserves the knot topology, are performed quite naturally by the action of local flows on flux tube strands. If the fluid in (D − K) is irrotational, then these fluid flows (with velocity u ) must satisfy the Dirichlet problem for the Laplacian of the stream function ϕ, that is,

[7] $\mathit{u}=\nabla \phi \text{in}(D-K){\nabla }^2\phi =0$

with normal component of the velocity to the tube boundary ${\mathit{u}}_{\perp }$ given. Equations [7] admit a unique solution in terms of local flows, and these flows are interpretable in terms of Reidemeister's moves performed on the tube strands. Note that boundary conditions prescribe only ${\mathit{u}}_{\perp }$ , whereas no condition is imposed on the tangential component of the velocity. This is consistent with the fact that tangential effects do not alter the topology of the physical knot (or link). The three type of Reidemeister's moves are therefore performed by local fluid flows, which are solutions to [7], up to arbitrary tangential actions.

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DIGITAL PROCESSING

Weyrich Tim , ... Leif Kobbelt , in Point-Based Graphics, 2007

5.3.1 OVERVIEW

Modeling the shape of 3D objects is one of the central techniques in geometry processing. This section discusses two fundamental modeling approaches for point-sampled geometry: Boolean operations and free-form deformation. While the former are concerned with building complex objects by combining simpler shapes [Hof89 ], the latter defines a continuous deformation field in space to smoothly deform a given surface [ SP86] (see Figure 5.29).

Figure 5.29. Boolean operations ( left ) and free-form deformation ( right ).

Boolean operations are most easily defined on implicit surface representations, since the required inside-outside classification can be directly evaluated on the underlying scalar field. On the other hand, free-form deformation is a very intuitive modeling paradigm for explicit surface representations. For example, mesh vertices or NURBS control points can be directly displaced according to the deformation field. For point-based representations, the hybrid structure of the surface model defined in Chapter 4 can be exploited to integrate these two modeling approaches into a unified shape-modeling framework. Boolean operations can utilize the approximate signed distance function defined by the MLS projection (see Section 4.2) for inside-outside classification, while free-form deformations operate directly on the point samples.

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Abelian Higgs Vortices

J.M. Speight , in Encyclopedia of Mathematical Physics, 2006

Introduction

For the purpose of this article, vortices are topological solitons arising in field theories in (2+1)-dimensional spacetime when a complex-valued field ϕ is allowed to acquire winding at infinity, meaning that the phase of ϕ(t, x ), as x traverses a large circle in the spatial plane, changes by 2πn, where n is a nonzero integer. Such winding cannot be removed by any continuous deformation of ϕ (hence "topological") and traps a considerable amount of energy which tends to coalesce into smooth, stable lumps with highly particle-like characteristics (hence "solitons"). Clearly, the universe is (3+1) dimensional. Nonetheless, planar field theories are of physical interest for two main reasons. First, the theory may arise by dimensional reduction of a (3+1)-dimensional model under the assumption of translation invariance in one direction. Vortices are then transverse slices through straight tube-like objects variously interpreted as magnetic flux tubes in a superconductor or cosmic strings. Second, a crucial ingredient of the standard model of particle physics is spontaneous breaking of gauge symmetry by a Higgs field. As well as endowing the fundamental gauge bosons and chiral fermions with mass, this mechanism can potentially generate various types of topological solitons (monopoles, strings, and domain walls) whose structure and interactions one would like to understand. Vortices in (2+1) dimensions are interesting in this regard because they arise in the simplest field theory exhibiting the Higgs mechanism, the abelian Higgs model (AHM). They are thus a useful theoretical laboratory in which to test ideas which may ultimately find application in more realistic theories. This article describes the properties of abelian Higgs vortices and explains how, using a mixture of numerical and analytical techniques, a good understanding of their dynamical interactions has been obtained.

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Liquid Crystals

O.D. Lavrentovich , in Encyclopedia of Mathematical Physics, 2006

Experimental Observations

When a thick UN sample (say, 100μm thick) with no special aligning layers is viewed under the microscope, one usually observes a number of mobile flexible lines, the so-called disclinations. The disclinations are seen as thin and thick threads (see Figure 5 ). Thin threads strongly scatter light and show up as sharp lines. These are truly topologically stable defect lines, along which the nematic symmetry of rotation is broken. The disclinations are topologically stable in the sense that no continuous deformation can transform them into a uniform state, n ( r ) = const. Thin disclinations are singular in the sense that the director is not defined along the core of the defect line. Thick threads are line defects only in appearance; they are not singular disclinations. The director is smoothly curved and well defined everywhere, except, perhaps, at a number of point defects, the so-called hedgehogs (see Figure 5 ).

Figure 5. (a) Thin singular disclinations and thick nonsingular threads in the nematic (n-pentylcyanobiphenyle (5CB)) bulk. Crossed polarizers; (b, c) typical director configurations associated with thin and thick lines; thick lines are often associated with point defects in the nematic bulk – hedgehogs.

In thin UN samples (1–50 μm) with the director tangential to the bounding plates, the disclinations are often perpendicular to the plates. Under a microscope with two crossed polarizers, one can see the ends of the disclinations as centers with emanating pairs of dark brushes (see Figure 6 ) giving rise to the so-called "Schlieren texture." The dark brushes display the areas where n is either in the plane of polarization of light or in the perpendicular plane. The director rotates by an angle ±π when one goes around the end of the disclination at the surface. Centers with four emanating brushes are also observed; they correspond to point defects located at the surface, the so-called boojums, (see Figure 6 ). The director undergoes a ±2π rotation around these four-brush centers. The principal difference between the centers with two brushes (ends of singular lines) and centers with four brushes (surface point defects) can be seen after a gentle shift of one of the bounding plates with respect to the other. Upon shear-induced separation in the plane of observation, the centers with two brushes are clearly seen as connected by a singular trace – disclination, while the centers with four brushes separate without a visible singularity between them.

Figure 6. Schlieren texture of a thin (13μm) slab of 5CB. Centers with two and four brushes are the ends of singular disclinations and point defects – boojums, respectively. Tangential director orientation. Crossed polarizers.

The intensity of linearly polarized light coming through a uniform UN slab depends on the angle β between the polarization direction and the projection of the director n onto the slab's plane:

[10] $I=I_0{sin}^{2}2\beta {sin}^2\left[\frac{\pi h}{\lambda }(n_{\text{e},\text{eff}}-n_{\text{o}})\right]$

where I ₀ is the intensity of incident light, λ is the wavelength of the light, n _{e, eff} is the effective refractive index that depends on the ordinary index n _o, extraordinary index n _e, and the director orientation. Equation [10] allows one to relate the number |k| of director rotations by ±2π around the defect core, to the number B of brushes:

[11] $\left|k\right|=B/4$

Taken with a sign that specifies the direction of rotation, k is called the "strength of disclination," and is related to a more general concept of a topological charge (but does not coincide with it). Note that I = 0 when n is perpendicular to the plates (so-called homeotropic state), as n _{e, eff} = n _o. The homeotropic state is used as one of the ground states in modern flat-panel TV sets. By applying the electric field, one tilts the director so that n _{e, eff} ≠ n _o and the cell (or the corresponding pixel in the liquid crystal panel) becomes transparent.

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Visualization Using Virtual Reality

R. BOWEN LOFTIN , ... LARRY ROSENBLUM , in Visualization Handbook, 2005

24.2.2 Examples

Well over 100 extant publications address the application of virtual reality in visualization. Below are brief descriptions of specific projects that demonstrate the breadth of applicability of virtual reality to visualization. The examples below are not meant to be exhaustive or even to be a uniform sampling of the available literature. Inclusion or exclusion of a specific application does not imply a value judgment on the part of the authors of this chapter.

24.2.2.1 Archeology

A number of groups have used virtual reality to visualize archeological data. One group [1] used a CAVE to visualize the locations of lamps and coins discovered in the ruins of the Petra Great Temple site in Jordan.

24.2.2.2 Architectural Design

The Electronic Visualization Laboratory (EVL) at the University of Illinois in Chicago [23] has utilized virtual reality in architectural design and collaborative visualization to exploit virtual reality's capability for multiple perspectives on the part of users. These perspectives, including multiple mental models and multiple visual viewpoints, allow virtual reality to be applied in the early phases of the design process rather than during a walkthrough of the final design.

24.2.2.3 Battlespace Visualization

Work done at Virginia Tech and the Naval Research Laboratory [12, 18] resulted in virtual-reality-based Battlespace visualization applications using both a CAVE and a projection workbench. The modern Battlespace extends from the bottom of the ocean into low earth orbit. Thus, 3D visualizations that support powerful direct interaction techniques offer significant value to military planners, trainers, and operators.

24.2.2.4 Cosmology

Song and Norman [36] demonstrated, as early as 1993, the utility of virtual reality as a tool for visualizing numerical and observational cosmology data. They have implemented an application that supports multiscale visualization of large, multilevel time-dependent datasets using both an immersive display and a gesture interface that facilitates direct interaction with the data.

24.2.2.5 Genome Visualization

Kano et al. [19] have used virtual reality to develop an application for pair-wise comparison between cluster sets generated from different gene expression datasets. Their approach displays the distribution of overlaps between two hierarchical cluster sets, based on hepatocellular carcinomas and hepatoblastomas.

24.2.2.6 Meteorology

Meteorologists typically use 2D plots or text to display their data. Such an approach makes it difficult to visualize the 3D atmosphere. Ziegler et al. [40] have tackled the problem of comparing and correlating multiple layers by using an immersive virtual environment for true 3D display of the data.

24.2.2.7 Oceanography

A multidisciplinary group of computer scientists and oceanographers [14] has developed a tool for visualizing ocean currents. The c-thru system uses virtual reality to give researchers the ability to interactively alter ocean parameters and communicate those changes to an ocean model calculating the solution.

24.2.2.8 Protein Structures

Protein structures are large and complex. Large-format virtual-reality systems support not only the visualization of such data, but the collaboration of small teams that analyze the data. One group [2 ] has visualized four geometric protein models: space-filling spheres, the solvent accessible surface, the molecular surface, and the alpha complex. Relationships between the different models are represented via continuous deformations.

24.2.2.9 Software Systems

Many computer programs now exceed one million lines of code. The ability to truly understand programs of such magnitude is rare. Visualizations of such systems offer a means of both comprehending the system and collaboratively extending or modifying it. An example of such a visualization is the work of

Amari et al. [3]. In this case, a visualization of static structural data and execution trace data of a large software application's functional units was developed. Further, the visualization approach supported the direct manipulation of graphical representations of code elements in a virtual-reality setting.

24.2.2.10 Statistical Data

A group at Iowa State University [4] has developed a virtual-reality-based application for the analysis of high-dimensional statistical data. Moreover, the virtual-reality approach proved superior to a desktop approach in terms of structural-detection tasks.

24.2.2.11 Vector Fields

Real-time visualization of particle traces using virtual environments can aid in the exploration and analysis of complex 3D vector fields. Kuester et al. [21] have demonstrated a scalable method for the interactive visualization of large time-varying vector fields.

24.2.2.12 Vehicle Design

The use of increasingly complex finite element (FE) simulations of vehicles during crashes has led to the use of virtual-reality techniques to visualize the results of the computations [22]. A program called VtCrash was designed to enable intuitive and interactive analyses of large amounts of crash-simulation data. The application receives geometry and physical-properties data as input and provides the means for the user to enter a virtual crash and to interact with any part of the vehicle to better understand the implications of the simulation.

24.2.2.13 Virtual Wind Tunnel

One of the earliest successful demonstrations of virtual reality as a visualization tool was the development of the Virtual Wind Tunnel at the NASA Ames Research Center [5, 6, 8]. Steve Bryson precomputed complex fluid flows around various aerodynamic surfaces. To view these flow fields, the user employed a tracked, head-mounted display (Fakespace's BOOM) that used relatively high-resolution color displays, one for each eye. These displays were attached to the head but were supported by a counterweighted boom to relieve the user of bearing the weight of the system. Optical encoders in the boom joints provided real-time, precise tracking data on the user's head position. Tools were developed to enable the user to explore the flow field using a tracked glove (a DataGlove) on one hand. For example, the user could use the gloved hand to identify the source point for streamlines that would allow visualization of the flow field in specific regions. A great deal of work went into developing both the precomputed data and the software that supported the visualization system. The software framework for the virtual wind tunnel was extensible and had interactive (i.e., real-time) performance. Fig. 24.1 shows Bryson examining the flow fields around an experimental vehicle.

Figure 24.1. Steve Bryson interacting with the Virtual Wind Tunnel developed at the NASA Ames Research Center. (See Fig. 21.2 in color insert.)

Others, for example Severance [33], have extended the work of Bryson's group by fusing the data from several wind-tunnel experiments into a single, coherent visualization. Given the high cost of maintaining and operating wind tunnels and the limited regimes (of both wind speed and aerodynamic surface size), the virtual wind tunnel offers a significant potential to reduce the cost and expand the availability of wind-tunnel experiments.

24.2.2.14 Hydrocarbon Exploration and Production

In recent years, virtual reality has had a growing impact on the exploration and production of hydrocarbons, specifically oil and gas. In 1997, there were only two large-scale visualization centers in the oil and gas industry, but by 2000, the number had grown to more than 20. In spite of this growth, the use of virtual reality technology was largely limited to 3D displays—interaction was still typically done via the keyboard and mouse. Lin et al. [24, 25] created an application that supported more direct interaction between the user and the data in a CAVE. Not only could three to four users share an immersive 3D visualization, one of them could also interact directly with the data via natural gestures. Fig. 24.2 shows a user in a CAVE working with objects representing geophysical surfaces within a salt dome in the Gulf of Mexico. The user holds a tracked pointing/interaction device in the dominant hand (the right hand, in the figure) while a virtual menu is attached to the nondominant hand. Studies were done to determine the effectiveness of this approach when compared with typical desktop applications.

Figure 24.2. A user in a CAVE interacting with geophysical data describing a salt dome in the Gulf of Mexico.

Additional work was done by Loftin et al. [26] to implement powerful interaction techniques on a projection workbench. In Fig. 24.3, the user is again using both hands for navigation, menu interaction, and detailed manipulation of the data. The success of these efforts and similar work by other groups has led the oil and gas industries to make major investments in virtual-reality-based visualization systems. More importantly, these systems have had a demonstrable return on investment in terms of improved speed and success of decision making for both exploration and production activities.

Figure 24.3. A user interacting with geophysical data on a projection workbench.

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Complex Variable Theory

Frank E. Harris , in Mathematics for Physical Science and Engineering, 2014

17.4 Contour Integrals

Much of the power of complex analysis stems from a set of integral theorems, largely due to Cauchy. These theorems deal with line integrals of analytic functions $f(z)$ over closed curves denoted by symbols such as $C$ , generically written in notations like

$\underset{C}{\oint }f(z)\mathit{dz}\text{.}$

Here the curve $C$ is often called a contour, and the line integral is referred to as a contour integral. As in earlier sections of this book, the circle on the integral sign serves as a reminder that the curve $C$ is a closed loop. Unless it is specifically indicated otherwise, it is understood that the contour is to be traversed in the counterclockwise (mathematically positive) direction. A reversal of the direction of traverse changes the sign of the integral.

Cauchy's Theorem

Cauchy's theorem states that if $C$ is any simple closed curve ¹ in the complex plane and $f(z)$ is analytic on $C$ and everywhere in the area enclosed by $C$ , then

(17.7) $\underset{C}{\oint }f(z)\mathit{dz}=0\text{.}$

To prove Cauchy's theorem we use Green's theorem in the plane, as given in Eq. (6.46). Restating that theorem,

(17.8) ${\int }_{\partial A}\left[P(x\text{,}y)\mathit{dy}+Q(x\text{,}y)\mathit{dx}\right]={\int }_A\left[\frac{\partial P}{\partial x}-\frac{\partial Q}{\partial y}\right]\mathit{dA}\text{.}$

As before, the notation $\partial A$ denotes the closed curve that is the boundary of the area $A$ ; the line integral is to be evaluated along that curve.

Writing $f(z)=u(x\text{,}y)+\mathit{iv}(x\text{,}y)$ and $\mathit{dz}=\mathit{dx}+\mathit{idy}$ , the line integral of Eq. (17.7) takes the form

(17.9) $\underset{C}{\oint }f(z)\mathit{dz}=\underset{C}{\oint }(u+iv)(\mathit{dx}+i\mathit{dy})=\underset{C}{\oint }\left[u\mathit{dx}-v\mathit{dy}\right]+i\underset{C}{\oint }\left[u\mathit{dy}+v\mathit{dx}\right]\text{.}$

We have separated the real and the imaginary contributions to Eq. (17.9).

Our next step is to use Eq. (17.8) to rewrite the line integrals in the right-hand member of Eq. (17.9) as integrals over the area $A$ enclosed by $C$ :

(17.10) ${\displaystyle \underset{C}{\oint }}f(z)\mathit{dz}={\int }_A\left[-\frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}\right]\mathit{dA}+i{\int }_A\left[\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right]\mathit{dA}\text{.}$

A rigorous proof of Cauchy's theorem includes a demonstration that the functions $u$ and $v$ must be differentiable if $f(z)$ is analytic; assuming that to be the case, we accept Eq. (17.10) and continue by noting that each integrand in Eq. (17.10) is an expression that vanishes by virtue of the Cauchy-Riemann equations, Eqs. (17.2). Note that these integrands vanish at all points within $A$ because it has been assumed that $f(z)$ is analytic throughout $A$ . If that analyticity is absent anywhere within $A$ , even if only at a single point, the theorem does not apply.

Implications of Cauchy's Theorem

Line integral independent of path–Consider two different paths (denoted $C_1$ and $C_2$ ), both starting at a point $A$ and ending at a point $B$ , where $A\text{,}B$ , both paths, and the region between them are all in the same region of analyticity of a function $f(z)$ . See Fig. 17.6. Then

Figure 17.6. Two paths from $A$ to $B$ .

(17.11) ${\int }_{C_1}f(z)\mathit{dz}-{\int }_{C_2}f(z)\mathit{dz}=\oint f(z)\mathit{dz}=0\text{.}$

We have used the fact that taking minus the integral over $C_2$ is equivalent to traversing $C_2$ starting at $B$ and ending at $A$ . We can rearrange Eq. (17.11) to

(17.12) ${\int }_{C_1}f(z)\mathit{dz}={\int }_{C_2}f(z)\mathit{dz}\text{.}$

Equation (17.12) states that the line integral of $f(z)\mathit{dz}$ between $A$ and $B$ is invariant with respect to deformation of the path $C$ , so long as the deformation is entirely within a region in which $f(z)$ is analytic.

Deformed path enclosing singularities–Consider now the situation illustrated in the left panel of Fig. 17.7, in which we are integrating $f(z)\mathit{dz}$ on either of the closed paths $C_1$ or $C_2$ in a region of analyticity that encloses one or more singularities. These singularities can be of any type (poles, branch points, or essential singularities), but if there are branch cuts, they must lie entirely within the "island" containing the singularities. Cauchy's theorem does not let us conclude that the value of this integral is zero, but we are nevertheless able to show that its value is unchanged by deformations that can be reached without leaving the region of analyticity of $f(z)$ , and that therefore the integral over $C_2$ has the same value as that over $C_1$ . Note that even if the singularity is only at a single point, the deformations under discussion here do not include those that move the contour through the singularity.

Figure 17.7. Left: paths enclosing singularities. Right: modified path for use of cauchy's theorem.

To prove that the integrals on paths $C_1$ and $C_2$ of Fig. 17.7 have the same value (if traversed in the same direction), we modify those paths as shown in the right panel of the figure, by cutting them open at the points marked $A$ and reconnecting them by the lines marked $B$ and $B^{\prime }$ . We then integrate over $C_1\text{,}B\text{,}{C}_2$ , and $B^{\prime }$ in the directions shown by the arrows, thereby forming a closed loop about a region in which $f(z)$ is entirely analytic, and to which Cauchy's theorem applies. Because the segments $B$ and $B^{\prime }$ lie in a region where $f(z)$ is analytic and are close together and traversed in opposite directions, they make no net contribution to our contour integral, while the contributions of $C_1$ and $C_2$ (which are traversed in opposite directions) add to zero, as required by Cauchy's theorem. If both $C_1$ and $C_2$ are traversed in the same direction, their contributions are therefore shown to be equal.

Summarizing, we have shown that a contour integral enclosing singularities can be subjected to an arbitrary continuous deformation within its region of analyticity without changing its value. For example, a contour surrounding a pole or an essential singularity can be contracted to a circle of arbitrarily small radius without changing the value of the integral.

Cauchy's Integral Formula

We now examine a specific contour integral enclosing a singularity. Cauchy's integral formula states that

(17.13) $f(z_0)=\frac{1}{2\pi i}{\oint }_C\frac{f(z)}{z-z_0}\mathit{dz}\text{,}$

where $f(z)$ is a function that is analytic on and within $C$ , and $z_0$ is a complex number that is in the region enclosed by $C$ . Let's prove this formula.

The integrand in Eq. (17.13) is analytic everywhere on and within $C$ except for a simple pole at $z=z_0$ , so our first step in analyzing the integral in that equation will be to contract $C$ to a circle of some small radius $r$ about $z_0$ and to make a change of variables to $z-z_0={\mathit{re}}^{i\theta }$ , where the integration is over $0\le \theta \le 2\pi$ . We have

(17.14) $\frac{1}{z-z_0}=\frac{e^{-i\theta }}{r}\text{and}\mathit{dz}=\frac{\partial \left(z_0+{\mathit{re}}^{i\theta }\right)}{\partial \theta }d\theta ={\mathit{ire}}^{i\theta }d\theta \text{,}$

and therefore

$\begin{array}{cc}\hfill {\displaystyle {\oint }_C}\frac{f(z)}{z-z_0}\mathit{dz}=& {\int }_{\theta =0}^{2\pi }f(z_0+{\mathit{re}}^{i\theta })\frac{e^{-i\theta }}{r}{\mathit{ire}}^{i\theta }d\theta =i{\int }_0^{2\pi }f(z_0+{\mathit{re}}^{i\theta })d\theta \hfill \\ \hfill =& \mathit{if}(z_0){\int }_0^{2\pi }d\theta =2\pi if(z_0)\text{.}\hfill \end{array}$

The second line of this equation is obtained by letting $r$ approach zero; $f(z_0+{\mathit{re}}^{i\theta })$ can then be replaced by $f(z_0)$ because $f(z)$ is analytic at $z_0$ . The final result is equivalent to Eq. (17.13).

A trivial generalization of Eq. (17.13) is that its contour integral evaluates to zero if $z_0$ is not enclosed by $C$ . In that case we simply have an instance of Cauchy's theorem.

An interesting aspect to Cauchy's integral formula is that it shows that the value of $f(z)$ at any point $z_0$ within the curve $C$ is entirely determined by the values of $f(z)$ on $C$ . There is no corresponding result in real-variable theory; real functions are not determined over an entire interval by their values at the ends of the interval.

Another Integral Formula

The technique used to derive Cauchy's integral formula can be used to evaluate integrals of the generic type

(17.15) $I_n=\frac{1}{2\pi i}{\oint }_C{(z-z_0)}^n\mathit{dz}\text{,}$

where $C$ encloses $z_0$ and $n$ is an integer (of either sign) or zero. If $n\ge 0$ , the integrand is analytic and, by Cauchy's theorem, $I_n=0$ . If $n$ is a negative integer, we make the substitution $z=z_0+{\mathit{re}}^{i\theta }$ , use the expression for $\mathit{dz}$ from Eq. (17.14), and integrate over a circle of radius $r$ . Equation (17.15) becomes

(17.16) $I_n=\frac{1}{2\pi i}{\int }_0^{2\pi }\left(r^n{e}^{\mathit{in}\theta }\right){\mathit{ire}}^{i\theta }d\theta =\frac{r^{n+1}}{2\pi }{\int }_0^{2\pi }{e}^{i(n+1)\theta }d\theta \text{.}$

If $n$ is any integer other than $-1$ , the integral over $\theta$ evaluates to zero (irrespective of the value of $r$ ); we have

${\int }_0^{2\pi }{e}^{i(n+1)\theta }d\theta ={\left.\frac{e^{i(n+1)\theta }}{i(n+1)}\right|}_0^{2\pi }=0\text{,}d(n\ne -1)\text{,}$

while for $n=-1$ we have a result corresponding to Cauchy's integral formula for $f(z)=1$ , namely $I_{-1}=1$ . Summarizing this important result,

(17.17) $I_n=\frac{1}{2\pi i}\underset{C}{\oint }{(z-z_0)}^n\mathit{dz}={\delta }_{n\text{,}-1}\text{.}$

We can now use Eq. (17.17) to derive a formula for the coefficients in a Laurent (or Taylor) series. If the contour $C$ is chosen to lie entirely within the region where a Laurent (or Taylor) series converges, and we write the series in the form

(17.18) $f(z)={\displaystyle \sum _n}{a}_n{(z-z_0)}^n\text{,}$

where the range of $n$ is whatever is needed to represent $f(z)$ , we can then obtain the coefficient $a_m$ (for any integer $m$ , positive, negative, or zero) by evaluating the following contour integral:

(17.19) $a_m=\frac{1}{2\pi i}{\oint }_C{(z-z_0)}^{-m-1}f(z)\mathit{dz}\text{.}$

Remember that in using Eq. (17.19), $C$ must be in the region of analyticity for the series whose coefficients are to be calculated.

The proof of Eq. (17.19) is direct; substitution of the series for $f(z)$ , Eq. (17.18), produces a series of integrals all of which vanish according to Eq. (17.17) except that with $n=m$ , for which the integral (including its prefactor) evaluates to $a_m$ .

As observed previously, direct evaluation of the integral in Eq. (17.19) will often not be the easiest way to establish a Laurent or Taylor series; it is of course legitimate to obtain the series in other ways.

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