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Project Assignment: Python Code for Visualizing Vector Fields using VTK Tool

Related Work

This is a Project Assignment, in this assignment you should want to do in python code to visulize. the code must be compiled and executed in Spyder programing tool version 5.0.5 ,that tool should be installed by using Anaconda navigator . and in this assignment we will use VTK TOOL. after install Anaconda navigator then open Anaconda navigator command prompt and setup the installation of VTK. in project assignment you want to buld the program to focous on Application to Engine Simulation Data Set in project assignment.pdf. you should want to implement program in that area. Specific requirements:

1) i want python code.

2) output screen shot in word file.

3) and the data file that you want are implement to buld the program

4) ppt of the project (detail implementation of the project and how the vector flow)

5) the implementation page(like paper publish).

Vector fields arise as models in almost all scientific and engineering endeavors which involve systems that change continuously with time. In the case of two-dimensional systems that can be modelled by vector fields defined on surfaces, visualization can play an important role in understanding the essential features in the system. This is also true for twodimensional vector fields that are linked to potentially noisy data, such as a velocity field extracted from experiments or numerical simulations of fluids. In both cases, there are occasions in which one wishes to simplify the dynamic structure in a coherent admissible manner [1]. This latter step requires the ability to edit the underlying vector field. Furthermore, there are problems where the construction and modification of a vector field represents a preliminary step towards a larger goal such as texture synthesis [2], [3], [4] and fluid simulation for special effects [5].

There is substantial literature on the subject of vector field topology extraction and simplification, with considerable focus on the identification and manipulation of fixed points (see [6] and references therein). On the other hand, periodic orbits are essential structures of non-gradient vector fields, such as those in electromagnetism, chemical reactions, fluid dynamics, locomotion control, population modelling, and economics. There is a fundamental need to be able to incorporate them into the subject of vector field visualization and design. For example, Figure 1 shows the swirl motion of fluid in a combustion chamber using simulation [7]. Periodic orbits appear in some planar slices along the main axis of the chamber (middle) as well as the boundary geometry (Figure 13). The existence and locations of the periodic orbits provide clues to the swirl motion inside the chamber. Efficient periodic orbit detection and vector field visualization can help design engineers better understand how the shape of the chamber and the initial speed of the fluid through the intake ports impact engine efficiency. Many of the above mentioned applications involve systems of nonlinear ordinary differential equations, for which explicit analytic solutions do not exist.

## Vector Field Design

The lack of analytic expressions led to the development of the subject of dynamical systems where the focus is on the qualitative structure of solutions. In the case of two-dimensional vector fields, the classical theoretical description of the dynamics is based on identifying fundamental topological and geometric structures such as fixed points, periodic orbits, separatrices, and their relationships [8], [9]. However, in practice there are at least two essential difficulties with this approach. First, unambiguously identifying all the topological structures for an arbitrary system is impossible. For example, Hilbert’s 16th problem, that of bounding the number of isolated periodic orbits for a polynomial vector field on the plane, remains essentially unsolved [10]. Second, the existence of noise reduces the importance of objects such as fixed points and periodic orbits.

Fig. 1. Visualizing the simulation of flow in a diesel engine: the combustion chamber (leftmost) and four planar slices of the flow inside the chamber for which the plane normals are along the main axis of the chamber. From left to right are slices cut at 10%, 25%, 50%, and 75% of the length of the cylinder from the top where the intake ports meet the chamber. The vector fields are defined as zeros on the boundary of the geometry (no-slip condition). The automatic extraction and visualization of flow topology allows the engineer to gain insight into where the ideal pattern of swirl motion is realized inside the combustion chamber. In fact, the behavior of the flow and its associated topology, including periodic orbits, is much more complicated than the ideal. Figure 13 provides complementary visualization of the flow on the boundary of the diesel engine.

Vector field visualization, analysis, simplification, and design have received much attention from the Visualization community over the past twenty years. Much excellent work exists, and to review it all is beyond the scope of this paper. Here, we only refer to the most relevant work. Interested readers can find a complete survey in [14], [15], [16].

There has been some work in creating vector fields on the plane and surfaces, most of which is for graphics applications such as texture synthesis [2], [3], [4] and fluid simulation [5]. These methods do not address vector field topology, such as fixed points. There are a few vector field design systems that make use of topological information. For instance, Rockwood and Bunderwala [17] use ideas from geometric algebra to create vector fields with desired fixed points. Van Wijk [18] develops a vector field design system to demonstrate his image-based flow visualization technique (IBFV). The basic idea of this system is the use of basis vector fields that correspond to various types of fixed points. This system is later extended to surfaces [19], [20].

Vector Field Topology and Analysis Helman and Hesselink [23] introduce vector field topology for the visualization of vector fields. They also propose efficient algorithms to extract vector field topology. Following their footsteps, much research has been conducted in topological analysis of vector fields. For example, Scheuermann et al. [24] use clifford algebra to study the non-linear fixed points of a vector field and propose an efficient algorithm to merge nearby first-order fixed points. Tricoche et al. [1] and Polthier and Preuβ [26] give efficient methods to locate fixed points in a vector field. Wischgoll and Scheuermann [27] develop a method to extract closed streamlines in a 2D vector field defined on a triangle mesh. Note that closed streamlines are in fact attracting and repelling periodic orbits. Theisel et al. [28] propose a mesh-independent periodic orbit detection method for planar domains. In contrast to these approaches, our automatic detection algorithm is extended to surfaces. Furthermore, this is the first time periodic orbit extraction and visualization has found utility in a real application. C. Vector Field Simplification Vector field simplification refers to reducing the complexity of a vector field.

There are two classes of simplification techniques: topology-based (TB), and non-topologybased (NTB) [6]. Existing NTB techniques are usually based on performing Laplacian smoothing on the potential of a vector field inside the specified region. One example of these work is by Tong et al. [29], who decompose a vector field using Hodge-decomposition and then smooth each-component independently before summing them. TB techniques simplify the topology of a vector field explicitly. Tricoche et al. [1] simplify a planar vector field by performing a sequence of cancelling operations on fixed point pairs that are connected by a separatrix. They refer to this operation as pair annihilation. A similar operation, named pair cancellation, has been used to remove a wedge and trisector pair in a tensor field [30]. We will follow this convention and refer to such an operation as fixed point pair cancellation. Zhang et al. [6] provide a fixed point pair cancellation method based on Conley theory.