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Geometric ModelingGeometric modeling is a crucial aspect of computer graphics and design, allowing for the creation of realistic and visually appealing 3D models. It involves the representation of geometric shapes and structures in a digital format, which can then be manipulated and rendered on a computer screen. Geometric modeling is used in a wide range of applications, including video games, animation, virtual reality, and industrial design. One of the key benefits of geometric modeling is its ability to accurately represent complex shapes and structures that would be difficult or impossible to create by hand. By using mathematicalequations and algorithms, designers can create detailed and realistic 3D modelsthat can be viewed from any angle and manipulated in real-time. This level of precision and flexibility is essential for many modern design projects, where accuracy and realism are paramount. In addition to its practical applications, geometric modeling also plays a crucial role in the artistic and creative process. Designers and artists can use geometric modeling tools to explore new ideas, experiment with different shapes and forms, and push the boundaries of traditional design aesthetics. By combining mathematical precision with artistic vision, geometric modeling allows for the creation of stunning and innovative visual experiences that captivate and inspire audiences. Furthermore, geometric modeling is an essential tool for collaboration and communication in design and engineering fields. By creating digital models that can be easily shared and manipulated, designers and engineers can work together more effectively, exchanging ideas and feedback in real-time. This collaborative approach not only streamlines the design process but also ensures that all stakeholders are on the same page, reducing the risk of miscommunication and errors. Moreover, geometric modeling has revolutionized the manufacturing industry, allowing for the rapid prototyping and production of complex parts and components. By creating digital models of products and machines, engineers can simulate their behavior under various conditions, identify potential issues, and make necessary adjustments before moving to production. This level of virtual testing and validation not only saves time and money but also improves the overall quality and reliability of the final product. Overall, geometric modeling is a versatile and powerful tool that has transformedthe way we design, create, and communicate. From its practical applications in computer graphics and engineering to its artistic and creative potential, geometric modeling continues to push the boundaries of what is possible in the world of design. By embracing this technology and harnessing its capabilities, designers and engineers can unlock new opportunities for innovation and discovery, shaping the future of design and manufacturing for years to come.。
a fushion modeling method -回复[A fusion modeling method]In recent years, the field of modeling has seen significant advancements in various techniques and methods. Among these, one particular approach that has gained popularity is fusion modeling. Fusion modeling is a process that combines different modeling techniques to enhance the accuracy and effectiveness of models. In this article, we will explore the step-by-step process of fusion modeling and discuss its benefits and applications.Step 1: Selection of Modeling TechniquesThe first step in fusion modeling is to identify and select the most appropriate modeling techniques to be utilized. This requires a careful analysis of the problem at hand and an understanding of the strengths and weaknesses of different modeling approaches. Common modeling techniques include statistical modeling, machine learning, neural networks, and mathematical modeling. The selection of techniques should be based on the specific requirements of the problem and the data available.Step 2: Preprocessing and Data IntegrationOnce the modeling techniques have been chosen, it is essential to preprocess and integrate the data before feeding it into the models. This step involves cleaning the data, eliminating missing values, and removing outliers. Additionally, if the data originates from multiple sources, it needs to be integrated into a unified structure. Integration can involve techniques such as data harmonization, feature extraction, and data transformation.Step 3: Model Training and EvaluationThe next step is to train the selected models on the preprocessed and integrated data. This involves dividing the data into training and testing sets and using the training set to build the models. The models are then evaluated using the testing set to assess their accuracy and performance. Evaluation metrics can vary depending on the specific problem, but commonly used ones include accuracy, precision, recall, and F1 score.Step 4: Model FusionAfter the individual models have been trained and evaluated, the fusion process begins. Fusion can be performed in several ways, including model stacking, model averaging, or ensemble methods. Model stacking involves training a meta-model that takes thepredictions of the individual models as input and generates the final prediction. Model averaging combines the predictions of the models by taking the average or weighted average. Ensemble methods use techniques such as bagging or boosting to combine the models' outputs.Step 5: Optimization and ValidationOnce the fusion models have been created, optimization techniques can be applied to fine-tune their performance. This involves adjusting the parameters and hyperparameters of the models to improve accuracy and reduce any bias or overfitting. Additionally, validation techniques such as cross-validation can be used to validate the fusion models' performance on different subsets of the data.Step 6: Deployment and ApplicationThe final step in the fusion modeling process is the deployment and application of the models. Depending on the problem, the models can be used in various ways, such as predicting outcomes, making decisions, or generating insights. Fusion models have found applications in various fields, including finance, healthcare, marketing, and image recognition. Their ability to combine thestrengths of different modeling techniques makes them particularly valuable in complex and dynamic environments.In conclusion, fusion modeling is a powerful approach that combines different modeling techniques to enhance model accuracy and performance. By carefully selecting modeling techniques, preprocessing and integrating data, training and evaluating models, performing model fusion, optimizing and validating models, and deploying them in real-world applications, fusion modeling allows for more accurate predictions and better decision-making. As the field of modeling continues to evolve, fusion modeling is likely to become an increasingly important tool for researchers and practitioners alike.。
AI model may speed up document analysis for the banking, financial and insurance industriesMarch 7 2022, by Peter WarzynskiThe new Xceptor system extracts customised key information from invoices. Credit: Loughborough UniversityResearchers have developed an AI-based solution that can automatically analyze and extract large amounts of information from computer documents.The team from Loughborough University and Xceptor—which describes itself as a "no-code data automation platform"—has created a deep learning model for natural language processing (NLP) that can analyze the content and structure of invoices, tax forms and other digitaldocuments and sort the information into categories.The improved system will streamline processes, such as setting up bank accounts, approving mortgages, responding to customer queries and processing insurance claims by speeding up fraud checking and extracting details from identity documents.Lead developer Dr. Chao Zhang, of Loughborough's department of Computer Science, said the technology was faster and cheaper than current systems, which perform the same task and would benefit similar tasks in the banking, financial service and insurance sectors.He said: "Compared with the traditional rule-based or pattern matching approaches, the developed NLP can identify terms, learn language structures, extract contextual correlation and classify texts into semantic groups and clauses, such as invoice numbers, payee addresses, counterparty names as well as distinguishing due date [from] invoice date."The AI model was trained to deal with complex freeform contents and robustly extract information linking with context rather than relying on pre-defined templates in texts and is built on state-of-the-art deep learning technology.The concept of graph modeling was introduced in the learning process to improve the model performance on complex documents, which may include tables and blocked texts with spatial alignment information. Such documents are more difficult to process than plain texts in paragraphs. The academic lead Professor Baihua Li, from Loughborough's School of Science, said: "Extracting required information from a large number of documents is currently a very time-consuming manual process. Developing AI solutions to learn contextual meaning and correlationpresented in complexly structured documents is extremely challenging."We are pleased that Loughborough University's specialists in NLP and machine learning are working with Xceptor on this game-changing innovation, and can successfully integrate the AI automation function into the company's smart document analysis platform for improved speed and accuracy."Dr. Rob Lowe, Chief Architect at Xceptor, said: "The power of this AI-based technology is that it can adapt to work with a wide range of documents."For the complex and rapidly changing environments that are inherent in banking, financial services and insurance, this technology makes it simpler and faster to automate processes and keep them working efficiently over time."Ultimately our customers become more responsive and agile, and their experts can concentrate on higher-value tasks."Former Loughborough academic Professor Eran Edirisinghe, now at Keele University, added: "We hope such collaborative projects will help the industry improve their competitiveness and develop new services through the better use of knowledge, technology and skills that transfer from research."Provided by Loughborough UniversityCitation: AI model may speed up document analysis for the banking, financial and insurance industries (2022, March 7) retrieved 23 December 2023 fromhttps:///news/2022-03-ai-document-analysis-banking-financial.htmlThis document is subject to copyright. Apart from any fair dealing for the purpose of private study or research, no part may be reproduced without the written permission. The content is provided for information purposes only.。
A Facial Aging Simulation Method Using flaccidity deformation criteriaAlexandre Cruz Berg Lutheran University of Brazil.Dept Computer ScienceRua Miguel Tostes, 101. 92420-280 Canoas, RS, Brazil berg@ulbra.tche.br Francisco José Perales LopezUniversitat les Illes Balears.Dept Mathmatics InformaticsCtra Valldemossa, km 7,5E-07071 Palma MallorcaSpainpaco.perales@uib.esManuel GonzálezUniversitat les Illes Balears.Dept Mathmatics InformaticsCtra Valldemossa, km 7,5E-07071 Palma MallorcaSpainmanuel.gonzales@uib.esAbstractDue to the fact that the aging human face encompasses skull bones, facial muscles, and tissues, we render it using the effects of flaccidity through the observation of family groups categorized by sex, race and age. Considering that patterns of aging are consistent, facial ptosis becomes manifest toward the end of the fourth decade. In order to simulate facial aging according to these patterns, we used surfaces with control points so that it was possible to represent the effect of aging through flaccidity. The main use of these surfaces is to simulate flaccidity and aging consequently.1.IntroductionThe synthesis of realistic virtual views remains one of the central research topics in computer graphics. The range of applications encompasses many fields, including: visual interfaces for communications, integrated environments of virtual reality, as well as visual effects commonly used in film production.The ultimate goal of the research on realistic rendering is to display a scene on a screen so that it appears as if the object exists behind the screen. This description, however, is somewhat ambiguous and doesn't provide a quality measure for synthesized images. Certain areas, such as plastic surgery, need this quality evaluation on synthesized faces to make sure how the patient look like and more often how the patient will look like in the future. Instead, in computer graphics and computer vision communities, considerable effort has been put forthto synthesize the virtual view of real or imaginary scenes so that they look like the real scenes.Much work that plastic surgeons put in this fieldis to retard aging process but aging is an inevitable process. Age changes cause major variations in the appearance of human faces [1]. Some aspects of aging are uncontrollable and are based on hereditary factors; others are somewhat controllable, resulting from many social factors including lifestyle, among others [2].1.1.Related WorkMany works about aging human faces have been done. We can list some related work in the simulation of facial skin deformation [3].One approach is based on geometric models, physically based models and biomechanical models using either a particle system or a continuous system.Many geometrical models have been developed, such as parametric model [4] and geometric operators [5]. The finite element method is also employed for more accurate calculation of skin deformation, especially for potential medical applications such as plastic surgery [6]. Overall, those works simulate wrinkles but none of them have used flaccidity as causing creases and aging consequently.In this work is presented this effort in aging virtual human faces, by addressing the synthesis of new facial images of subjects for a given target age.We present a scheme that uses aging function to perform this synthesis thru flaccidity. This scheme enforces perceptually realistic images by preserving the identity of the subject. The main difference between our model and the previous ones is that we simulate increase of fat and muscular mass diminish causing flaccidity as one responsible element for the sprouting of lines and aging human face.In the next section will plan to present the methodology. Also in section 3, we introduce the measurements procedure, defining structural alterations of the face. In section 4, we present a visual facial model. We describe age simulation thrua deformation approach in section 5. In the last section we conclude the main results and future work.2.MethodologyA methodology to model the aging of human face allows us to recover the face aging process. This methodology consists of: 1) defining the variations of certain face regions, where the aging process is perceptible; 2) measuring the variations of those regions for a period of time in a group of people and finally 3) making up a model through the measurements based on personal features.That could be used as a standard to a whole group in order to design aging curves to the facial regions defined.¦njjjpVM2.1Mathematical Background and AnalysisHuman society values beauty and youth. It is well known that the aging process is influenced by several parameters such: feeding, weight, stress level, race, religious factors, genetics, etc. Finding a standard set of characteristics that could possibly emulate and represent the aging process is a difficult proposition.This standard set was obtained through a mathematical analysis of some face measurements in a specific group of people, whose photographs in different ages were available [7]. To each person in the group, there were, at least, four digitized photographs. The oldest of them was taken as a standard to the most recent one. Hence, some face alterations were attained through the passing of time for the same person.The diversity of the generated data has led to the designing of a mathematical model, which enabled the acquiring of a behavior pattern to all persons of the same group, as the form of a curve defined over the domain [0,1] in general, in order to define over any interval [0,Į] for an individual face. The unknown points Įi are found using the blossoming principle [8] to form the control polygon of that face.The first step consisted in the selection of the group to be studied. Proposing the assessment of the face aging characteristics it will be necessary to have a photographic follow-up along time for a group of people, in which their face alterations were measurable.The database used in this work consisted of files of patients who were submitted to plastic surgery at Medical Center Praia do Guaíba, located in Porto Alegre, Brazil.3.MeasurementsAccording to anatomic principles [9] the vectors of aging can be described aswhich alter the position and appearance of key anatomic structures of the face as can be shown in figure 1 which compares a Caucasian mother age 66 (left side) with her Caucasian daughters, ages 37 (right above) and 33 (right below) respectively.Figure 1 - Observation of family groupsTherefore, basic anatomic and surgical principles must be applied when planning rejuvenative facial surgery and treating specific problems concomitantwith the aging process.4.Visual Facial ModelThe fact that human face has an especially irregular format and interior components (bones, muscles and fabrics) to possess a complex structure and deformations of different face characteristics of person to person, becomes the modeling of the face a difficult task. The modeling carried through in the present work was based on the model, where the mesh of polygons corresponds to an elastic mesh, simulating the dermis of the face. The deformations in this mesh, necessary to simulate the aging curves, are obtained through the displacement of the vertexes, considering x(t) as a planar curve, which is located within the (u,v ) unit square. So, we can cover the square with a regular grid of points b i,j =[i/m,j/n]T ; i=0,...,m; j=0,...,n. leading to every point (u,v ) asfrom the linear precision property of Bernstein polynomials. Using comparisons with parents we can distort the grid of b i,j into a grid b'i,j , the point (u,v )will be mapped to a point (u',v') asIn order to construct our 3D mesh we introduce the patch byAs the displacements of the vertexes conform to the certain measures gotten through curves of aging and no type of movement in the face is carried through, the parameters of this modeling had been based on the conformation parameter.4.1Textures mappingIn most cases the result gotten in the modeling of the face becomes a little artificial. Using textures mapping can solve this problem. This technique allows an extraordinary increase in the realism of the shaped images and consists of applying on the shaped object, existing textures of the real images of the object.In this case, to do the mapping of an extracted texture of a real image, it is necessary that the textureaccurately correspond to the model 3D of that is made use [9].The detected feature points are used for automatic texture mapping. The main idea of texture mapping is that we get an image by combining two orthogonal pictures in a proper way and then give correct texture coordinates of every point on a head.To give a proper coordinate on a combined image for every point on a head, we first project an individualized 3D head onto three planes, the front (x, y), the left (y, z) and the right (y, z) planes. With the information of feature lines, which are used for image merging, we decide on which plane a 3D-head point on is projected.The projected points on one of three planes arethen transferred to one of feature points spaces suchas the front and the side in 2D. Then they are transferred to the image space and finally to the combined image space.The result of the texture mapping (figure 2) is excellent when it is desired to simulate some alteration of the face that does not involve a type of expression, as neutral. The picture pose must be the same that the 3D scanned data.¦¦¦ mi nj lk n j m i lk k j i w B v B u B b w v u 000,,)()()(')',','(¦¦ m i nj n jmij i v B u B b v u 00,)()(),(¦¦ m i nj n j m i j i v B u B b v u 00,)()(')','(¦¦¦ mi nj lk n j m i lk k j i w B v B u B b w v u 000,,)()()(')',','(Figure 2 - Image shaped with texturemapping5.Age SimulationThis method involves the deformation of a face starting with control segments that define the edges of the faces, as¦¦¦ mi nj lk n j m i lk k j i w B v B u B b w v u 000,,)()()(')',','(Those segments are defined in the original face and their positions are changed to a target face. From those new positions the new position of each vertex in the face is determined.The definition of edges in the face is a fundamental step, since in that phase the applied aging curves are selected. Hence, the face is divided in influencing regions according to their principal edges and characteristics.Considering the face morphology and the modeling of the face aging developed [10], the face was divided in six basic regions (figure 3).The frontal region (1) is limited by the eyelids and the forehead control lines. The distance between these limits enlarges with forward aging.The orbitary region (2) is one of the most important aging parameters because a great number of wrinkles appears and the palpebral pouch increases [11]. In nasal region (3) is observed an enlargement of its contour.The orolabial region (4) is defined by 2 horizontal control segments bounding the upper and lower lips and other 2 segments that define the nasogenian fold. Figure 3 - Regions considering the agingparametersThe lips become thinner and the nasogenian fold deeper and larger. The mental region (5) have 8 control segments that define the low limit of the face and descend with aging. In ear curve (6) is observed an enlargement of its size. The choice of feature lines was based in the characteristic age points in figure 6.The target face is obtained from the aging curves applied to the source face, i.e., with the new control segment position, each vertex of the new image has its position defined by the corresponding vertex in the target face. This final face corresponds to the face in the new age, which was obtained through the application of the numerical modeling of the frontal face aging.The definition of the straight-line segment will control the aging process, leading to a series of tests until the visual result was adequate to the results obtained from the aging curves. The extremes of the segments are interpolated according to the previously defined curves, obtained by piecewise bilinear interpolation [12].Horizontal and vertical orienting auxiliary lines were defined to characterize the extreme points of the control segments (figure 4). Some points, that delimit the control segments, are marked from the intersection of the auxiliary lines with the contour of the face, eyebrow, superior part of the head and the eyes. Others are directly defined without the use of auxiliary lines, such as: eyelid hollow, eyebrow edges, subnasion, mouth, nasolabial wrinkle andnose sides.Figure 4 - Points of the control segmentsOnce the control segments characterize the target image, the following step of the aging process can be undertaken, corresponding to the transformations of the original points to the new positions in the target image. The transformations applied to the segments are given by the aging curves, presented in section 4.In the present work the target segments are calculated by polynomial interpolations, based on parametric curves [12].5.1Deformation approachThe common goal of deformation models is to regulate deformations of a geometric model by providing smoothness constraints. In our age simulation approach, a mesh-independent deformation model is proposed. First, connected piece-wise 3D parametric volumes are generated automatically from a given face mesh according to facial feature points.These volumes cover most regions of a face that can be deformed. Then, by moving the control pointsof each volume, face mesh is deformed. By using non-parallel volumes [13], irregular 3D manifolds are formed. As a result, smaller number of deformvolumes are necessary and the number of freedom incontrol points are reduced. Moreover, based on facialfeature points, this model is mesh independent,which means that it can be easily adopted to deformany face model.After this mesh is constructed, for each vertex on the mesh, it needs to be determined which particularparametric volume it belongs to and what valueparameters are. Then, moving control points ofparametric volumes in 3D will cause smooth facialdeformations, generating facial aging throughflaccidity, automatically through the use of the agingparameters. This deformation is written in matricesas , where V is the nodal displacements offace mesh, B is the mapping matrix composed ofBernstein polynomials, and E is the displacementvector of parametric volume control nodes.BE V Given a quadrilateral mesh of points m i,j ,, we define acontinuous aged surface via a parametricinterpolation of the discretely sampled similaritiespoints. The aged position is defined via abicubic polynomial interpolation of the form with d m,n chosen to satisfy the known normal and continuity conditions at the sample points x i,j .>@>M N j i ,...,1,...,1),(u @@>@>1,,1,),,( j j v i i u v u x ¦3,,),(n m n m n m v u d v u x An interactive tool is programmed to manipulate control points E to achieve aged expressions making possible to simulate aging through age ranges. Basic aged expression units are orbicularis oculi, cheek, eyebrow, eyelid, region of chin, and neck [14]. In general, for each segment, there is an associated transformation, whose behavior can be observed by curves. The only segments that do not suffer any transformation are the contour of the eyes and the superior side of the head.5.2Deformation approachThe developed program also performs shape transformations according to the created aging curves, not including any quantification over the alterations made in texture and skin and hair color. Firstly, in the input model the subjects are required to perform different ages, as previouslymentioned, the first frame needs to be approximately frontal view and with no expression.Secondly, in the facial model initialization, from the first frame, facial features points are extracted manually. The 3D fitting algorithm [15] is then applied to warp the generic model for the person whose face is used. The warping process and from facial feature points and their norms, parametric volumes are automatically generated.Finally, aging field works to relieve the drifting problem in template matching algorithm, templates from the previous frame and templates from the initial frame are applied in order to combine the aging sequence. Our experiments show that this approach is very effective. Despite interest has been put in presenting a friendly user interface, we have to keep in mind that the software system is research oriented. In this kind of applications an important point is the flexibility to add and remove test facilities. 6.Results The presented results in the following figuresrefer to the emulations made on the frontalphotographs, principal focus of this paper, with theobjective to apply the developed program to otherpersons outside the analyzed group. The comparisonswith other photographs of the tested persons dependon their quality and on the position in which theywere taken. An assessment was made of the new positions, of the control segments. It consisted in: after aging a face, from the first age to the second one, through the use of polynomial interpolation of the control segments in the models in the young age, the new positions are then compared with the ones in the model of a relative of older age (figure 5). The processed faces were qualitatively compared with theperson’s photograph at the same age. Figure 5 - Synthetic young age model,region-marked model and aged modelAlso the eyelid hollow, very subtle falling of the eyebrow, thinning of the lips with the enlarging of the nasion and the superior part of the lip, enlargingof the front and changing in the nasolabial wrinkle.7.ConclusionsModelling biological phenomena is a great deal of work, especially when the biggest part of the information about the subject involves only qualitative data. Thus, this research developed had has a challenge in the designing of a model to represent the face aging from qualitative data.Due to its multi-disciplinary character, the developed methodology to model and emulate the face aging involved the study of several other related fields, such as medicine, computing, statistics and mathematics.The possibilities opened by the presented method and some further research on this field can lead to new proposals of enhancing the current techniques of plastic face surgery. It is possible to suggest the ideal age to perform face lifting. Once the most affected aging regions are known and how this process occurs over time. Also missing persons can be recognized based on old photographs using this technique. AcknowledgementsThe project TIN2004-07926 of Spanish Government have subsidized this work.8. References[1] Burt, D. M. et al., Perc. age in adult Caucasianmale faces, in Proc. R. Soc., 259, pp 137-143,1995.[2] Berg, A C. Aging of Orbicularis Muscle inVirtual Human Faces. IEEE 7th InternationalConference on Information Visualization, London, UK, 2003a.[3] Beier , T., S. Neely, Feature-based imagemetamorphosis, In Computer Graphics (Proc.SIGGRAPH), pp. 35-42, 1992.[4] Parke, F. I. P arametrized Models for FacialAnimation, IEEE Computer & Graphics Applications, Nov. 1982.[5] Waters, K.; A Muscle Model for Animating ThreeDimensional Facial Expression. Proc SIGGRAPH'87,Computer Graphics, Vol. 21, Nº4, United States, 1987. [6] Koch, R.M. et alia.. Simulation Facial SurgeryUsing Finite Element Models, Proceedings of SIGGRAPH'96, Computer Graphics, 1996.[7] Kurihara, Tsuneya; Kiyoshi Arai, ATransformation Method for Modeling and Animation of the Human Face from Photographs, Computer Animatio n, Springer-Verlag Tokyo, pp.45-58, 1991.[8] Kent, J., W. Carlson , R. Parent, ShapeTransformation for Polygon Objects, In Computer Graphics (Proc. SIGGRAPH), pp. 47-54, 1992. [9] Sorensen, P., Morphing Magic, in ComputerGraphics World, January 1992.[10]Pitanguy, I., Quintaes, G. de A., Cavalcanti, M.A., Leite, L. A. de S., Anatomia doEnvelhecimento da Face, in Revista Brasileira deCirurgia, Vol 67, 1977.[11]Pitanguy, I., F. R. Leta, D. Pamplona, H. I.Weber, Defining and measuring ageing parameters, in Applied Mathematics and Computation , 1996.[12]Fisher, J.; Lowther, J.; Ching-Kuang S. Curveand Surface Interpolation and Approximation: Knowledge Unit and Software Tool. ITiCSE’04,Leeds, UK June 28–30, 2004.[13]Lerios, A. et al., Feature-Based VolumeMetamorphosis, in SIGGRAPH 95 - Proceedings,pp 449-456, ACM Press, N.Y, 1995.[14]Berg, A C. Facial Aging in a VirtualEnvironment. Memória de Investigación, UIB, Spain, 2003b.[15]Hall, V., Morphing in 2-D and 3-D, in Dr.Dobb's Journal, July 1993.。
Geometric ModelingGeometric modeling is a fundamental aspect of computer-aided design (CAD) and computer graphics, playing a crucial role in the creation of virtual 3D models and simulations. This technology has revolutionized various industries, including architecture, engineering, animation, and gaming, by enabling designers and developers to visualize and manipulate complex geometric shapes with precision and efficiency. In this response, we will explore the historical background, different perspectives, case studies, and critical evaluation of geometric modeling, as well as its future implications and recommendations. The development of geometric modeling can be traced back to the early 1960s when Ivan Sutherland created Sketchpad, a revolutionary computer program that allowed users to draw and manipulate basic geometric shapes on a screen. This marked the beginning of computer-aided design (CAD) and laid the foundation for the development of geometric modeling as we know it today. Over the years, geometric modeling has evolved significantly, with the introduction of advanced algorithms, rendering techniques, and modeling tools that have expanded its applications across various industries. From a historical perspective, the evolution of geometric modeling has been driven by the increasing demand for more sophisticated and realistic 3D models in fields such as architecture, automotive design, industrial engineering, and entertainment. As technology has advanced, so too has the complexity and realism of geometric models, leading to a greater emphasis on precision, detail, and interactivity in the design and visualization process. From a technological perspective, geometric modeling has undergone a paradigm shift with the advent of parametric and non-parametric modeling techniques. Parametric modeling allows designers to create models based on a set of parameters, enabling them to make changes and updates to the design easily. On the other hand, non-parametric modeling focuses on creating freeform shapes and surfaces, providing greater flexibility and creativity in the design process. These different perspectives have led to debates within the industry about which approach is more effective for specific applications and design requirements. One example of the impact of geometric modeling can be seen in the field of architecture. With the use of CAD software and advanced geometric modeling tools, architects can create highlydetailed and realistic 3D models of buildings and structures, allowing them to visualize the final product and make necessary adjustments before construction begins. This not only improves the design process but also helps in conveying the design intent to clients and stakeholders, leading to better communication and decision-making. In the automotive industry, geometric modeling hasrevolutionized the design and manufacturing process, enabling engineers and designers to create complex 3D models of vehicles and their components with precision and accuracy. This has led to significant improvements in vehicle safety, performance, and aesthetics, as well as streamlined production processes and reduced time-to-market for new vehicle models. While geometric modeling has brought about numerous benefits, it also poses certain challenges and drawbacks. One of the main drawbacks is the steep learning curve associated with advanced modeling tools and techniques, which can be a barrier for newcomers to the field. Additionally, the complexity of geometric models can lead to performance issuesand computational challenges, especially when dealing with large-scale models or simulations. Looking ahead, the future implications of geometric modeling arevast and promising. As technology continues to advance, we can expect to see even more realistic and interactive 3D models that push the boundaries of visualfidelity and immersion. Furthermore, the integration of geometric modeling with other emerging technologies such as virtual reality (VR) and augmented reality (AR) holds great potential for creating new and innovative applications in fields such as education, training, and entertainment. In conclusion, geometric modeling has played a pivotal role in shaping the way we design, visualize, and interact with3D models across various industries. While it has brought about significant advancements and benefits, there are also challenges and considerations that need to be addressed. By understanding the historical background, different perspectives, and case studies related to geometric modeling, we can better appreciate its impact and potential for the future. As technology continues to evolve, it is essential to stay informed and adaptable to the latest trends and developments in geometric modeling, in order to harness its full potential and drive innovation in design and visualization.。
地理学博士论文英语作文This dissertation focuses on the in-depth study of geographical phenomena and their underlying mechanisms. Through extensive literature review and fieldwork, I have attempted to uncover the multi-faceted aspects of our planet's physical and human geographies.The first part of this thesis examines the spatial patterns and temporal dynamics of natural processes such as climate change, landform evolution, and ecosystem functioning. By employing advanced mapping and modeling techniques, I aim to provide a better understanding of these processes and their implications for sustainable development.The second section delves into the complex interplay between human activities and the environment. This includes analyzing the impact of urbanization, economic development, and population growth on land use, resource availability, and environmental quality.Furthermore, this work also explores the role of geography in shaping social and cultural dynamics. It considers how geographical factors influence migration patterns, cultural diversity, and political boundaries.To conclude, this dissertation provides a comprehensive analysis of the diverse and interrelated aspects of geography. It offers insights that can contribute to informed decision-making in various fields, ranging from environmental management to urban planning and beyond.。
Fluid-Structure Interaction Fluid-structure interaction (FSI) is a complex and interdisciplinary fieldthat involves the interaction between a fluid flow and a solid structure. This interaction can occur in various engineering applications such as aerospace, civil engineering, biomechanics, and biomedical engineering. The study of FSI is crucial for understanding the behavior of structures under fluid loading, and for designing more efficient and reliable engineering systems. However, it also presents significant challenges due to the nonlinear and coupled nature of the problem. One of the key challenges in FSI is the accurate modeling and simulation of the interaction between the fluid and the structure. This requires the development of advanced numerical methods that can capture the complex behavior of both the fluid and the structure, as well as their interaction. The development of such methods is crucial for improving the accuracy and reliability of FSI simulations, and for enabling the design of more efficient and safe engineering systems. Another challenge in FSI is the experimental validation of numerical simulations. While numerical simulations can provide valuable insights into the behavior of FSI systems, experimental validation is essential for ensuring the accuracy and reliability of the simulations. However, conducting experiments for FSI systems can be challenging due to the complex nature of the interaction, and the need for advanced experimental techniques and equipment. Overcoming these challenges requires close collaboration between researchers and engineers from different disciplines, as well as the development of innovative experimental techniques. In addition to the technical challenges, FSI also presentssignificant practical and economic challenges. The design and analysis of FSI systems often require significant computational resources and expertise, which can be costly and time-consuming. Furthermore, the development of FSI simulations and experimental techniques requires close collaboration between researchers and engineers from different disciplines, which can be challenging due to thedifferent perspectives and approaches of each discipline. Overcoming these challenges requires a multidisciplinary approach and a strong collaboration between researchers, engineers, and industry partners. Despite the challenges, the study of FSI offers numerous opportunities for advancing engineering knowledgeand technology. By understanding and modeling the complex interaction betweenfluids and structures, researchers and engineers can develop more efficient and reliable engineering systems, and improve the safety and performance of existing systems. Furthermore, the study of FSI can also lead to the development of innovative technologies and solutions for a wide range of engineering applications, from aerospace and civil engineering to biomechanics and biomedical engineering. In conclusion, fluid-structure interaction is a complex and interdisciplinaryfield that presents significant challenges, but also offers numerous opportunities for advancing engineering knowledge and technology. By addressing the technical, practical, and economic challenges of FSI, researchers and engineers can develop more accurate and reliable simulations, and improve the design and performance of engineering systems. However, overcoming these challenges requires a multidisciplinary approach, close collaboration between researchers and engineers, and the development of innovative technologies and solutions.。
空间数据模型英文文献空间数据模型英文文献Title: A Comprehensive and Insightful Analysis of Spatial Data ModelsAbstract:Spatial data models play a crucial role in organizing and analyzing geographical information. This paper presents a comprehensive review of various spatial data models to shed light on their capabilities and applications. It highlights the importance of spatial data modeling in addressing the complexities of spatial relationships and provides valuable insights for researchers and practitioners in the field.Introduction:With the increasing availability and volume of spatial data, effective modeling techniques are essential for capturing, representing, and analyzing geographical information. Spatial data models provide a structured framework for organizing and managing diverse spatial datasets, enabling efficient spatial analysis and decision-making processes. This paper examines several prominentspatial data models and discusses their strengths, weaknesses, and applications.1. Vector Model:The vector model represents spatial features using points, lines, and polygons. It is widely used for analyzing discrete objects and is suitable for representing maps, transportation networks, and administrative boundaries. The vector model offers precise geometric accuracy but may struggle with representing continuous phenomena and large-scale datasets.2. Raster Model:The raster model divides geographic space into a regular grid of cells and assigns values to each cell. This model is suitable for continuous data such as elevation, temperature, and satellite imagery. Raster models enable efficientanalysis operations but may face challenges in handlingvector-oriented queries and spatial relationships.3. Object-Based Model:The object-based model represents spatial entities as objects with attributes and behaviors, allowing for the modeling of complex spatial relationships. This model iswell-suited for modeling urban environments, naturallandscapes, and ecological systems. The object-based model offers high-level concept representation but may require extensive data preprocessing and expert knowledge.4. Field Model:The field model focuses on representing continuousspatial phenomena as continuous functions over a spatial domain. It is particularly useful for environmental modeling, geographic analysis, and simulation. Field models enable efficient interpolation and analysis of continuous data but may lack precision in representing discrete objects.Conclusion:Spatial data modeling is a fundamental component of geographic information systems and spatial analysis. This paper reviewed four prominent spatial data models, namely the vector, raster, object-based, and field models. Each model has its strengths and weaknesses, making them suitable for specific applications and data types. By understanding the capabilities and limitations of these models, researchers and practitioners can make informed decisions when selecting the most appropriate model for their spatial analysis tasks. Additionally, this article highlights the need for further research and development in spatial data modeling to addressemerging challenges and enhance the effectiveness of spatial analysis techniques.。
IntroductionSpatial models of land use change are important tools to analyse the possible trajectories of land use change in the near future. The results of land use models are important to evaluate policy options and assess the impact of land use change on natural and socio-economic conditions.The Conversion of Land Use and its Effects modeling framework (CLUE) (Veldkamp and Fresco, 1996; Verburg et al., 1999) was developed to simulate land use change using empirically quantified relations between land use and its driving factors in combination with dynamic modeling of competition between land use types. The model was developed for the national and continental level and applications for Central America, Ecuador, China and Java, Indonesia are available. For study areas with such a large extent the spatial resolution for analysis was coarse and, as a result, each land use is represented by assigning the relative cover of each land use type to the pixels.Land use data for study areas with a relatively small spatial extent is often based on land use maps or remote sensing images that denote land use types respectively by homogeneous polygons or classified pixels. This results in only one dominant land use type occupying one unit of analysis. Because of the differences in data representation and other features that are typical for regional applications, the CLUE model can not directly be applied at the regional scale. Therefore the modelling approach has been modified and is now called CLUE-S (the Conversion of Land Use and its Effects at Small regional extent). CLUE-S is specifically developed for the spatially explicit simulation of land use change based on an empirical analysis of location suitability combined with the dynamic simulation of competition and interactions between the spatial and temporal dynamics of land use systems.More information on the development of the CLUE-S model can be found in Verburg et al. (2002) and Verburg and Veldkamp (2003).。
目录摘要 (2)第1章引言 (6)1.1. 我国机器人研究现状 (8)1.2. 工业机器人概述： (9)1.3. 本论文研究的主要内容 (10)第2章机器人方案的设计 (15)2.1. 机器人机械设计的特点 (15)2.2. 与机器人有关的概念 (15)2.3. 工业机器人的组成及各部分关系概述 (16)2.4. 工业机器人的设计分析 (17)2.5. 方案设案 (18)2.6. 自由度分析 (18)2.7. 机械传动装置的选择 (20)2.7.1. 滚珠丝杠的选择 (20)第3章零部件设计与建模 (22)3.1. Croe软件介绍 (22)3.2. 关键零部件建模 (22)3.3. 各部分的装配关系 (36)第4章仿真分析 (39)第5章致谢 (43)参考文献 (44)摘要工业技术水平是工业用机器人现代化水平的重要指标，从研究和研究领域发展的结论，提高现代产业的要求，提高产业控制和控制任务的复杂性，提出了很高的要求。
理论上，我国末期输送能力和定位精确度高、小误差、惯性误差、反应速度快、工业工作并行、快速准确、现有工业工程预计会进一步增加，本文将研究并行研究、实用化并行以企业工学实用化为目标。
从摩擦接口、外乱和不确定性来看，如果没有连锁和动力学模型化的负担，传统的控制战略将难以得到基于控制有效性模型的预期。
通常，与一系列平行于更复杂的运动模型相比，动态测试和控制机制将更加复杂。
因此，有必要研究并联机构的动力学建模及其控制问题。
这是一个新的机器人，机器人的刚性。
承载能力高。
高精度。
小负荷的重量。
具有良好的性能和广泛的应用，是robotów.spokojnie系列的补充。
有一个固定的一部分，在特点和实验室条件下的动力学加速度（重力加速度），.终端控制机制，原来的三角洲是最有效的机制平行安装“电子项目机器人是机器人的控制和规划动力学研究的基础上，发挥着重要的作用，在“.badania kinematykę反向动力学和由简单到przodu.odwrotnie相对平行前进，kinematykę相对skomplikowane.na结构分析的基础上，建立了三角洲机器人模型，机器人的机器人。
Modeling Complex Spatial Dynamics of Two-Population Interaction in Urbanization ProcessYanguang Chen, Feng XuCollege of Urban and Environmental Sciences, Peking University, Beijing 100871, PRC. Email:chenyg@Abstract: This paper is mainly devoted to lay an empirical foundation for further researching into complex spatial dynamics of two-population interaction. Based on the US population census data, a rural and urban population interaction model is built. Subsequently a logistic equation on percentage urban is derived from the urbanization model so that spatial interaction can be connected with logistic growth mathematically. The numerical simulation by using the discretized urban-rural population interaction model of urbanization shows a period-doubling bifurcation and chaotic behavior, which is identical in patterns to those from the simple mathematical models of logistic growth in ecology. This suggests that the complicated dynamics of logistic growth may come from the nonlinear interaction unbeknown to us.Key words: complex dynamics; bifurcation; chaos; two-population interaction; logistic growth; urbanization1. IntroductionThe study of the logistic equation as viewed from ecology indicates that a simple deterministic system can present periodic oscillation and chaotic behavior along with the model parameter change (May, 1976). However, why the simple model contains complex dynamics still remains ambiguous. Urban study can provide us with facilities for exploring the springhead of complicated dynamics of logistic process. In the urbanization process, the level of urbanization (percentage urban) follows the logistic curve and can be described with the logistic function (Karmeshu, 1988; United Nations, 1980; United Nations, 1993). Moreover, the urban system and ecological system show comparability in several aspects (Dendrinos, 1992; Dendrinos and Mullally, 1985), whichimplies that the process of urbanization might have period-doubling bifurcation or chaotic dynamics. In theory, urban system and the process of urbanization can generate complex behaviors such as chaos (e.g. Dendrinos, 1996; Dendrinos and El Naschie, 1994; Nijkamp, 1990; Nijkamp and Reggiani, 1998; Van der Leeuw and McGlade, 1997; Wong and Fotheringham, 1990). Many studies of chaotic cities are relative to spatial interaction and logistic growth.On the other hand, a great number of simulation analyses and empirical researches show that urban system bears the fractal structure (e.g. Batty and Longley, 1994; Chen and Zhou, 2003; Frankhauser, 1994; White and Engelen, 1994). Fractal structure and chaotic behavior coexist in lots of systems. Fractal property of urban systems suggests complex dynamics of urban evolution. What we concern is not only the bifurcation and chaos in the sheer numerical simulation experiments but also the ones that can be captured from the observation data. One of the viewpoints is that fractal actually appears at the edge of chaos and the coexistence phenomenon of fractal and chaos does not imply the certain correlation between them (Bak, 1996). Perhaps this is true, but we still intend to investigate it from the standpoint of urban systems and urbanization dynamics in order to reveal the relation between the chaotic behavior and fractal structure of nonlinear systems.Now chaotic cities and fractal cities have become important branch ranges of self-organized cities (Portugali, 2000). The studies of fractal cities and systems of cities are supported by a great number of observations (e.g. Batty and Longley, 1994; Chen and Zhou, 2004; Chen and Zhou, 2006; Frankhauser, 1994; White et al, 1997). However, most applications of chaos theory in the social sciences lack empirical content (Nijkamp and Reggiani, 1992). This thing has changed little for more than ten years. In fact, cities and networks of cities are typical complex systems suitable for exploring complicated dynamics (Allen, 1997; Wilson, 2000). The key lies in how to associate theory with practice and reality. The principal aim of this paper is at two aspects. One is to lay an empirical foundation for researching chaotic cities, and the other is to make preparations for revealing the essence of complicated behaviors of simple models and the relation between chaotic cities and fractal cities.The following parts of this paper are structured as follows. In section 2, we build a nonlinear dynamics model about the urban-rural interaction based on the population census data of the United States of America (USA), and then derive the logistic equation of urbanization level fromthe model. In section 3, with the aid of the US census data, we demonstrate the feasibility and rationality of the model based on statistical analysis, logistic analysis and numerical simulation analysis. This part is used to consolidate the empirical foundation of the model. In section 4, we implement numerical simulation experiment with the model of urbanization dynamics, testifying whether or not such a model presents all the behavior characters of the logistic equation, including periodic oscillation and chaotic behavior. Finally, in Section 5, the discussion is concluded by making some remarks on the significance of the complicated dynamics research from the aspect of the two-population interaction in urban geography.2 Choice and Transform of Mathematical Models2.1 Urban-rural interaction modelsA variety of mathematical models have been made to describe the spatial dynamics of the urban-rural population migration. Among these models, two are attention-attracting. One is the Keyfitz-Rogers linear model (Keyfitz, 1980; Rogers, 1968), and the other, the United Nations nonlinear model (United Nations, 1980; Karmeshu, 1988). The United Nations adopted a pair of nonlinear equations to characterize the urbanization dynamics. That is⎪⎪⎩⎪⎪⎨⎧+++=+−+=)()()()()()(d )(d )()()()()()(d )(d t u t r t u t r d t r t cu t t u t u t r t u t r b t u t ar t t r ψϕ, (1) where r (t ) and u (t ) denotes the rural and urban population in time t respectively, a , b , c , d , φ and ψ are parameters. If parameters φ=ψ=0, we can derive the logistic model of urbanization level from the UN model. For many years, the United Nations has been using the logistic curve to forecast the level of urbanization of each country in the world (United Nations, 1993; United Nations, 2004).However, empirical study and statistical analysis show that the urbanization dynamics of many countries such as America, China, and India can neither be effectively described by the Keyfitz-Rogers model nor by the United Nations model. In short, rural population couldn’t migrate into urban regions and vice versa without spatial interaction between urban and rural population. In other words, population migration and exchange between urban and rural regions depends only onurban-rural population interaction. Consequently, two items of the United Nations model are actually excrescent and equation (1) should be simplified to such a form⎪⎪⎩⎪⎪⎨⎧++=+−=)()()()()(d )(d )()()()()(d )(d t u t r t u t r d t cu tt u t u t r t u t r b t ar t t r . (2) According to equation (2), the rural population can not spontaneously flow into the cities and vice versa . The exchange of urban and rural population relies mainly on the urban-rural interaction. For a close region, in theory, it is expected b =d . As will be shown later, the US model of urbanization dynamics might be simpler than equation (2). That is c =0 in reality.2.2 Derivation of the logistic modelIn order to research into the above model, we need to examine it from two ways: one is the logical analysis, and the other empirical analysis. The logical analysis involves at least two aspects. First, whether or not the level of urbanization derived from the above model is close to the logistic increase, and whether or not the total population in a region is limited. Second, whether or not the result of the numerical simulation is coincident with that of the mathematical deduction.First of all, we derive the well-known logistic model on the level of urbanization, i.e. percentage urban . The level of urbanization is defined as the proportion or share of urban population in relation to the total population in a region (United Nations, 2004). Thus we have)(1)()()()()()()(t V t V t u t r t u t P t u t L +=+==, (3) where L (t ) refers to the level of urbanization, P (t )=r (t )+u (t ) to the total population, and V (t )=u (t )/r (t ) to the urban-rural ratio of population. Differentiating, we get⎦⎤⎢⎣⎡++−+=t t u t t r t u t r t u t u t r t t u t t L d )(d d )(d )]()([)()()(d /)(d d )(d 2. (4) Substituting equation (2) into equation (4) yields⎥⎦⎤⎢⎣⎡+−+++−+++=)()()()()()()()]()([)()]()([)()()()()(d )(d 22t u t r t u t r b d t cu t ar t u t r t u t u t r t u t dr t u t r t cu t t L . (5) For simplicity, taking a region as a close system, then we have b =d . In terms of the definition of urbanization level, equation (5) can be transformed into the following form[]2)()()()()(1)(d )(d t L t u t r a d t L t cL t t L −+−=. (6) According to equation (3), we have an urban-rural ratio of population)(1)()()()(t L t L t r t u t V −==. (7) That implies 1/V (t )=r (t )/u (t )=1/L (t )-1. Therefore, equation (6) can be transformed into logistic equation[][])(1)()()()(1)()(1)(d )(d 2t L t L a d c t L t V a d t L t cL t t L −−+=−+−=. (8) Thus, we have constructed the mathematical relation between models for two interacting population and the logistic equation. Let k =c +d -a =c +b -a represent the intrinsic /original rate of growth . Then equation (7) can be simplified as the usual form[])(1)(d )(d t L t kL tt L −=. (9) Solving equation (8) yields the well-known expression of the logistic curvekte L t L −−+=)1/1(11)(0, (10) where L 0 represents the initial value of L (t ) theoretically. That is, when t =0, we have L (t )= L 0.A key criterion to judge the urbanization model is the rationality of the increase curve of the total population. Taking derivative of population P (t ) with respect to time t givestt u t t r t t P d )(d d )(d d )(d +=. (11) Substituting equation (2) into equation (11) yields)()(d )(d t cu t ar tt P +=. (12) Obviously, from equation (12) we can get two inconsistent equations as follows)()()(d )(d t r c a t cP tt P −+=, (13) )()()(d )(d t u a c t aP tt P −+=. (14) According to equation (13), when a >c , the total population grows more quickly; while according to equation (14), when a >c, the total regional population grows slower. These two equations collide with each other. The inconsistency can be eliminated by two conditions: a =c or c =0. If a =cas given, then the total population will grow infinitely in the exponential way predicted by Malthus (1798/1996); On the other, if c=0, the total population will stop growing when it increases to certain extent. Under the latter circumstance, according to equation (13), since the rural population r(t)→0, the growth rate of the total population P(t) will gradually decrease to 0; According to equation (14), because the whole population will be completely urbanized, i.e. u(t)→P(t), the growth rate of the total population will tend toward to 0 ultimately. In the real world, we do have c=0, as will be illustrated in the following empirical analysis.It is easy to see that b or d is a very significant parameter in equation (2). On the one hand, it controls the developing trend and quantity of the total population; on the other hand, it affects the parameter k value of the logistic equation on level of urbanization. As we know, parameter k dominates the behavior characters of the dynamical system. When k>2.57, the logistic map coming from the discretization of equation (9) will present very complicated behaviors (May, 1976). So what is the case in reality? In the next section, we will validate the above models in virtue of the US observation data. Then we perform numerical simulation experiment to unfold some intrinsic regularity of the urbanization dynamics.3 Empirical Foundation of Two-population Interaction Model3.1 Data and methodThe main purpose of this study, as indicated above, is to lay an empirical foundation for further research into complex spatial dynamics of urban-rural interaction. So it is necessary to make relevant statistical analysis of the dynamical equations. There are two central variables in the study of spatial dynamics of urban evolvement: population and wealth (Dendrinos, 1992). According to our theme, we only choose the first variable, population, to test the models. Generally speaking, the population measure falls roughly into four categories: rural population r(t), urban population u(t), total population P(t)= r(t)+u(t), and level of urbanization or percentage urban, L(t)= u(t)/ P(t). The American data comes from the population censuses whose interval is about 10 years. Although the website of American population census offers 22 times of census data from 1790 to 2000, we only use the data from 1790 to 1960 (table 1). The reason is that the US changed the definition of cities in 1950, and the new definition became effective in 1970. From then on, theAmerican urban population was measured with the new standard. As a result, the statistic caliber of the population data from 1970 to 2000 might be different from those before 1970 although they approximately join with each other (figure 1).Table 1 The US rural and urban population and the related data (1790-1960)Time (year) [t] Interval(month)[∆t]Ruralpopulation[r(t)]Urbanpopulation[u(t)])()()()(tut rtut r+Rural rateof growth[∆r(t)]Urban rateof growth[∆u(t)]1790 10 3727559 201655191305.67125855.30 12071.60 1800 10 4986112 322371302794.21172831.00 20308.80 1810 10 6714422 525459487322.03223077.60 16779.60 1820 9.8125 8945198 693255643391.97284153.58 44228.48 1830 10 11733455 11272471028443.23348484.30 71780.80 1840 10 15218298 18450551645549.78439908.20 172944.10 1850 10 19617380 35744963023569.39560942.30 264202.20 1860 10 25226803 62165184987478.10342920.70 368584.30 1870 10 28656010 99023617359287.97740346.40 422737.40 1880 10 360594741412973510151800.00481402.70 797653.00 1890 10 408735012210626514346837.12512383.50 810856.70 1900 9.7917 45997336 3021483218235956.49425582.20 1210127.90 1910 9.7917 50164495 4206400122879255.97163788.26 1244862.74 1920 10.25 51768255 5425328226490822.68221831.22 1454372.39 1930 10 540420256916059930336844.29341720.60 554473.90 1940 10 574592317470533832478532.68373837.30 1542285.60 1950 10 611976049012819436448706.03506197.80 2293539.90 1960 10 6625958211306359341776788.81Source: /population.The data displayed in table 1 are fitted to the discretization expressions of the United Nations model and the Lotka-V olterra-type model respectively (r.e. Dendrinos and Mullally, 1985; Lotka, 1956; V olterra, 1931). Since the Keyfitz-Rogers model and the American urbanization model are both special cases of the United Nations model, there is no need to try Keyfitz-Rogers model particularly. The parameters of models are made by the least squares computation, which can make the key parameter, slope, lies in the most reasonable range.After estimating the model parameters, we should make tests in two ways. One is the well-known statistical test, and the other is the logical test, which is often ignored in practice. If the model fails to pass the statistical test, it has problems such as incomplete or redundant variables,or inaccurate parameter values; if the model cannot pass the logical test, it has structure problem so that it cannot explain the phenomena at present and predict the developing trend in future. Statistical tests can be made in definite procedure, while the logical test needs to be done with the help of mathematical transformation and numerical simulation experiment.6000000012000000018000000024000000030000000017751800182518501875190019251950197520002025YearP o p u l a t i o nFigure 1. The changing trend of the US urban, rural and total population (1790-2000) (Notes : The solid points are data from 1790 to 1960; the hollow points are data from 1970 to 2000. The definitionof the city after 1960 is different from before, but the two calibers generally fit with each other.)3.2 Parameters estimation and model selectionIn order to make statistical analysis, we must discretize the United Nations model so that it transform from differential equations into difference expressions, i.e., a 2-dimension map. Then the analysis of continuous dynamics changes to that of discrete dynamics. If ∆t =10 as taken, then d x /d t ∝∆x /∆t . Let r (t ), u (t ) and r (t )*u (t )/[r (t )+u (t )] be independent variables, and ∆u (t )/∆t and ∆r (t )/∆t be dependent variables. A multivariate stepwise regression analysis based on least squares computation gives the following model⎪⎪⎩⎪⎪⎨⎧+=ΔΔ+−=ΔΔ)()()()(05044.0)()()()()(03615.0)(02584.0)(t u t r t u t r tt u t u t r t u t r t r t t r . (15) This is a pair of difference equations of which all kinds of statistics including F statistic , P value (or t statistic), variance inflation factor (VIF) value and Durbin-Watson (DW) value, etc, can pass the tests under the significance of α=0.01 (Appendix 1). In this model, c =0. Although in theory weshould have b =d , they are not equal in the empirical results. There might be two reasons for this. One is that the US is not a truly closed system because of mass foreign migration; the other is that the natural growth of the urban population is dependent on the urban-rural interaction. The second reason might be more important. But as a whole, the equations as a special case of the United Nations model can better describe the American urban and rural population migration process in the recent 200 years.In light of equation (10), the level of urbanization should follow the logistic curve. It is easily to calculate the percentage of urban population using the data in table 1. A least squares computation involving the percentage urban data gives the following resultst et L 02238.041573.2011)(−+=. (16) The goodness of fit is R 2=0.9839. For convenience, we set t =year-1790 (figure 2). Thus we have k =0.02238 as the estimated value of the intrinsic growth rate. On the other hand, we could estimate the original rate of growth k value by equation (15): one is k 1=b -a =0.03615-0.02584=0.01031, and the other is k 2=d -a =0.05044-0.02584=0.02460. The intrinsic growth rate should come into between k 1=0.01031 and k 2=0.03615 and indeed it does. The parameter values estimated from the dynamical system model, equation (15), are similar to that from the logistic model, equation (16). There are some differences between different estimated results due mainly to three factors. The first is non-closed region, the second imprecise data, and the third the computation error resulting from transformation from continuous equation to discrete expression. For comparison and selection, we also fit the American rural and urban data to the discretization of the predator-prey interaction model. Let r (t ), u (t ) and r (t )*u (t ) be independent variables and ∆u (t )/∆t or ∆r (t )/∆t dependent variables. The multivariable stepwise regression based on least squares computation gives an abnormal result, which cannot be accepted. If we loosen the requirements, then the American urbanization process could be expressed with the Keyfitz model. However, this mathematical expression has two vital shortcomings, which defies us to accept the Keyfitz model for the US urbanization. In short, neither the linear Keyfitz-Rogers model nor the usual non-linear Lotka-V olterra model is as good as the United Nations model in terms of logic sense and statistic effect (Appendix 2).0.00.20.40.60.81.017751800182518501875190019251950197520002025YearP e r c e n t a g e u r b a nFigure 2 Logistic process of the US level of urbanization (1790-2000) (Note : The solid points are the data from 1790 to 1960, and the hollow points are the data from 1970 to 2000.)We can generate the data of American urban, rural and total population and the urbanization level by using discrete dynamics model, and then draw a comparison between the simulation value and observed data. Figures 3 and 4 respectively show the simulation results based on equations(15). It is easy to see that the change of the urban and total population approximately follow the path of the S-curve, while the rural population first increases, then decreases, and finally turns itself into the urban population completely (figure 3). Moreover, the level of urbanization increases in the logistic way. The changing trend of the numerical simulation results displayed in figures 3 and 4 is roughly coincident with the actual observation data (figures 1 and 2). Although it is unpractical that the saturation value of the urbanization level is 100%, the characters of evolvement of the urban and rural population reflected by the discrete dynamical model, i.e., equation (15), comply with logic rules well. The total population converges, and the change of the percentage urban conforms to the logistic curve.To sum up, the American model of urban-rural population interaction can be expressed by equation (2) except parameter c =0. This is the experimental foundation of theoretical analysis of discrete urbanization dynamics. So far, we have finished the building work of the model of urbanization based on the population observation in the real world. In the following section, we will discuss the complicated behaviors of the above model of urbanization dynamics in the possible world in theory.Figure 3 Numerical simulation curve of rural, urban, and total population in the Americanurbanization process(Notes: The numerical simulation results are based on the discrete dynamical equations of urbanization, equation(15), the unit of population is taken as 10,000 persons.)Figure 4 Numerical simulation curve of American urbanization level (1790-2400) (Notes: The numerical simulation based on equation (15). The saturation value is 1. The curve is identical inshape to that of logistic growth indicated by equation (16).)4 Complex Behaviors of Urbanization Dynamics ModelOne of the purposes of this work is to make preparations for revealing the essence of complicated behaviors of simple models. The discrete model of two-population interaction between urban and rural systems can exhibit all the complex dynamics arising from the logistic map, includingperiod-doubling bifurcation and chaos. Moreover, the discrete urban-rural interaction model can show richer details of complicated behaviors than what logistic map does, and especially, it can offer a new way of looking at complex dynamics of simple mathematical models.According to equation (15), the parameter c =0, thus equation (8) can be reduced to[][])(1)()(1)()(d )(d t L t kL t L t L a b tt L −=−−=, (17) where the intrinsic rate of growth is that k =b -a . The discretization of equation (17) is a finite-difference equation21)1(t t t kL L k L −+=+, (18)Defining a new variable x t =kL t /(1+k ), we can turn equation (18) into the familiar parabola, i.e., a 1-dimension map x t +1=(1+k )x t (1-x t ).As we know, according to May (1976), the quadratic map can present periodic oscillation and even more complicated chaotic behaviors under certain conditions. Since equation (17) is derived from equation (2), the behavior characters of equation (18) should be able to be produced by the discretization of equation (2). For testing this hypothesis, we can perform some numerical simulation experiment by using equation (2), which can be discretized as a 2-dimension map⎪⎪⎩⎪⎪⎨⎧+++=++−+=+)()()()()()1()1()()()()()()1()1(t u t r t u t r d t u c t u t u t r t u t r b t r a t r . (19) The conversion between differential equation and difference will lead to some subtle change of parameter values. But for simplicity, we don’t modify the parameter symbols after converting equation (2) into equation (19). The numerical solutions of equation (19) shows that when the difference between b and a increases (please notice k =b -a ), the growth curve of urbanization level L t indeed changes from simplicity to complexity, from S shape to periodic oscillation and even to chaos. In short, all the behaviors of logistic map revealed by May (1976) can be exhibited by the discrete two-population interaction model (Figure 5).0.00.20.40.60.81.01.20102030405060t L (t)0.00.20.40.60.81.01.21.40102030405060t L (t )a Logistic growth (b =c =0.25) b Two-period oscillation (b =c=2.25)0.00.20.40.60.81.01.21.40102030405060t L (t)0.00.20.40.60.81.01.21.40102030405060t L (t )c Four-period oscillation (b =c =2.55) d Chaotic state (b =c =2.85)Figure 5 Four types of changes of urbanization level by urban-rural interaction model: fromfixed state to chaos(Notes : The parameter values are taken as a =0.025, c =0. In order to correspond to the logistic model, we make b =d .The original urban and rural population values are based on the US census in 1790, i.e., r (0)=3,727,559,u (0)=201,655. It is easy to see that the numerical simulation results from the two-population interaction model areidentical in curves to those from the logistic model by May in 1976.)As a matter of conciseness, we may as well set b =d based on the theoretical hypothesis. According to the estimated results of the US urbanization model, let a =0.025 and c =0. In addition, the US census data in 1790 are taken as the original urban and rural population values. Then, we increase the value of b and d continually. The numerical simulation result shows that when b =d <1.31, the urbanization level presents the S-shaped curve growth, i.e. a fixed state curve; when b =d >2.025 (k =b -a >2), the dynamical system comes into 2-period oscillation state; when b =d >2.475 (k =b -a >2.45), the system takes on 4-period oscillation state; then the system will fall into 8, 16, and 2n -period state as the values of b and d increase (n is a positive integer); when b =d >2.6 (k =b -a >2.575), the system will perform random period or chaotic state. The growinglimit of parameters is b=d=3.03. Compared with the work of May (1976), the period-doubling bifurcation route to chaos of the discrete urban-rural interaction model is identical in patterns to that of the logistic map. Of course, there might be subtle difference sometimes. Why the 2-dimension map exhibits the same complex dynamics with that arising from the 1-dimension map? Maybe the two-population interaction model poses a new question about the essence of chaos (Figure 6).Figure 6 A 1-dimension map and a 2-dimension map reach the same goal by different routesFurther, if we ignore the connection between the urban-rural interaction model and the logistic equation by permitting b≠d, then the behavior features of the dynamical system will become much richer. When we fix a, c, and d, the system will exhibit periodic oscillation or even chaos; however, when we fix b, the behavior characters of the system do not change along with the changes of the other parameters. It is obvious that the key parameter that determines the system behavior is b, or strictly speaking, is the difference between a and b. In detail, for instance, let’s consider a=0.025, c=0, and d=0.05 according to the aforementioned empirical analysis. The curve of the percentage urban changes along with b is in the same way with the result based on b=d, but the critical values of the period-doubling bifurcation route to chaos increases.Under such circumstance, when the system comes into the chaotic state, it still presents periodic oscillation. However, the period is not only a multiple of 2 any more, but a random integer. For example, when b=3.2, system will enter into period 5 state (figure 7). More experimental results。
