Rozprawy doktorskie na temat „Non Uniform Rational B-Splines”

With the continuous growth in computing power, numerical optimisation is increasingly applied in shape optimisation using Computational Fluid Dynamics (CFD). Since CFD computations are expensive, gradient-based optimisation is preferable when the number of design variables is large. In particular the recent progress with adjoint solvers is important, as these solvers allow to compute the gradients at constant computational cost regardless of the number of design variables, and as a consequence enable the use of automatically derived and rich design spaces. One of the crucial steps in shape optimisation is the parametrisation of the geometry, which directly determines the design space and thus the nal results. This thesis focuses on CAD-based parametrisations with the CAD model continuously updated in the design loop. An existing approach that automatically derives a parametrisation from the control points of a net of B-Spline patches is extended to include NURBS. Continuity constraints for water-tightness, tangency and curvature across patch interfaces are evaluated numerically and a basis for the resulting design space is computed using Singular Value Decomposition (SVD). A CAD-based shape optimisation framework is developed, coupling a flow solver, an adjoint solver, the in-house CAD kernel and a gradient-based optimiser. The flow sensitivities provided by the adjoint solver and the geometric sensitivities computed through automatic differentiation (AD) are assembled and provided to the optimiser. An extension to maintain the design space and hence enables use of a quasi-Newton method such as the BFGS algorithm is also presented and the convergence improvements are demonstrated. The framework is applied to three shape optimisation cases to show its effectiveness. The performance is assessed and analysed. The effect of parameters that can be chosen by the user are analysed over a range of cases and best practice choices are identified.

Malgré des décennies d’incontestables progrès dans le domaine des sciences informatiques, un certain nombre de problèmes restent difficiles à traiter en raison, soit de leur complexité numérique (problème d’optimisation, …), soit de contraintes spécifiques telle que la nécessité de traitement en temps réel (réalité virtuelle, augmentée, …). Dans ce contexte, il existe des méthodes de réduction de modèle qui permettent de réduire les temps de calcul de simulations multi-champs et/ou multi-échelles complexes. Le processus de réduction de modèle consiste à paramétrer un métamodèle qui requiert moins de ressources pour être évalué que le modèle complexe duquel il a été obtenu, tout en garantissant une certaine précision. Les méthodes actuelles nécessitent, en général, soit une expertise de l’utilisateur, soit un grand nombre de choix arbitraires de sa part. De plus, elles sont bien souvent adaptées à une application spécifique mais difficilement transposable à d’autres domaines. L’objectif de notre approche est donc d’obtenir, s'il n'est pas le meilleur, un bon métamodèle quel que soit le problème considéré. La stratégie développée s’appuie sur l’utilisation des hypersurfaces NURBS et se démarque des approches existantes par l’absence d’hypothèses simplificatrices sur les paramètres de celles-ci. Pour ce faire, une méta heuristique (de type algorithme génétique), capable de traiter des problèmes d’optimisation dont le nombre de variables n’est pas constant, permet de déterminer automatiquement l’ensemble des paramètres de l’hypersurface sans transférer la complexité des choix à l’utilisateur
Despite undeniable progress achieved in computer sciences over the last decades, some problems remain intractable either by their numerical complexity (optimisation problems, …) or because they are subject to specific constraints such as real-time processing (virtual and augmented reality, …). In this context, metamodeling techniques can minimise the computational effort to realize complex multi-field and/or multi-scale simulations. The metamodeling process consists of setting up a metamodel that needs less resources to be evaluated than the complex one that is extracted from by guaranteeing, meanwhile, a minimal accuracy. Current methods generally require either the user’s expertise or arbitrary choices. Moreover, they are often tailored for a specific application, but they can be hardly transposed to other fields. Thus, even if it is not the best, our approach aims at obtaining a metamodel that remains a good one for whatever problem at hand. The developed strategy relies on NURBS hypersurfaces and stands out from existing ones by avoiding the use of empiric criteria to set its parameters. To do so, a metaheuristic (a genetic algorithm) able to deal with optimisation problems defined over a variable number of optimisation variables sets automatically all the hypersurface parameters so that the complexity is not transferred to the user

For many years, incompatible computer-aided design (CAD) packages that are based on Non-uniform Rational B-Spline (NURBS) technology carried out the exchange of models and data through either neutral file formats (IGES or STEP) or proprietary formats that have been accepted as quasi industry standards. Although it is the only available solution at the current time, the exchange process most often produces unsatisfactory results. Models that are impeccable in the original modeling system usually end up with gaps or intersections between surfaces on another incompatible system. Issues such as loss of information, change of data accuracy, inconsistent tolerance, and misinterpretation of the original design intent are a few examples of problems associated with migrating models between different CAD systems. While these issues and drawbacks are well known and cost the industry billions of dollars every year, a solution to eradicate problems from their sources has not been developed. Meanwhile, researchers along with the industries concerned with these issues have been trying to resolve such problems by finding means to repair the migrated models either manually or by using specialized software. Designing in recent years is becoming more knowledge intensive and it is essential for NURBS to take its share of the ever increasing use of knowledge. NURBS are very powerful modeling tools and have become the de facto standard in modeling. If we stretch their strength and make them knowledge driven, benefits beyond current expectations can be achieved easily. This dissertation introduces knowledge guided NURBS with theoretical and practical foundations for supporting design intent capturing, retrieval, and exchange among dissimilar CAD systems. It shows that if NURBS entities are tagged with some knowledge, we can achieve seamless data exchange, increase robustness, and have more reliable computations, all of which are ultimate objectives many researchers in the field of CAD have been trying to accomplish for decades. Establishing relationships between a NURBS entity and its origin and destinations can aid with seamless CAD model migration. The type of the NURBS entity and the awareness of any irregularities can lead to more intelligent decisions on how to proceed with many computations to increase robustness and achieve a high level of reliability. As a result, instead of having models that are hardly modifiable because of migrating raw numerical data in isolation, the knowledge driven migration process will produce models that are editable and preserve design intent. We have addressed the issues not only theoretically but also by developing a prototype system that can serve as a test bed. The developed system shows that a click of a button can regenerate a migrated model instead of repairing it, avoiding delay and corrective processes that only limit the effective use of such models.

En esta tesis se presenta un método para analizar antenas montadas sobre estructuras arbitrarias. La Optica Geométrica (GO) y la Teoría Uniforme de la difraccion (UTD), han sidoempleadas para analizar los efectos que la estructura produce sobre el diagrama de radiación de la antena emisora. Para la descripción geométrica de la estructura, han sido utilizados parches NURBS (Non Uniform Rational B-Spline), por lo que el método presentado, es compatible con la mayoría de los programas gráficos disponibles en el mercado.EL tratamiento de geometrías arbitrarias requiere un código eficiente en el análisis de tres dimensiones.Por otro lado, para obtener resultados satisfactorios, la descripción de la superficie de la estructura, debe ser muy próxima al modelo real, sin embargo, esto complica el tratamiento computacional. Aquí la estructura es modelada mediante un conjunto de parches NURBS, que unidos entre sí, describen el modelo completo. Esta descripción permite manipular superficies arbitrarias con un bajo numero de parches, lo que significa un volumen de información reducido.La descripción inicial por NURBS del modelo, es acompañada con información complemetaria como por ejemplo: la tipología de las superficies, las curvas frontera, el tipo de material, etc. Esto es imprescindible para la aplicación de criterios de selección dedicados a la aceleración del proceso.El método tras leer la descripción del modelo, descompone los parches NURBS en superficies racionales de Bezier. Un parche de Bezier es también una superficie paramétrica definida en términos de una combinación lineal de los polinomios de Bernstein.Las antenas son modeladas usando modelos numéricos simples basados en agrupaciones de dipolos infinitesimales eléctricos y magnéticos. Esta caracterización de la antena es muyventajosa ya que con un numero reducido de datos de entrada, la fuente queda definida para cualquier dirección del espacio y el valor del campo radiado puede ser calculado fácilmente.El análisis electromagnético de los efectos que contribuyen al campo dispersado por la geometría comienza con una selección rigurosa de la geometría iluminada desde la fuente.Unicamente los parches de Bezier iluminados serán almacenados por el ordenador durante el análisis. La filosofía de este proceso es descartar aquella parte de la geometría que no contribuye a los efectos de dispersión.El campo total calculado es la superposición de los siguientes efectos pertenecientes a la GO y a la UTD: campo directo procedente de la fuente, campo reflejado por los parches de Bezier, campo difractado por las aristas del modelo definidas como curvas de Bezier, ondas de superficie, dobles reflexiones, reflexione-difraccion y difraccion-reflexión. El método ha sido diseñado para analizar campo cercano y lejano. El mayor gasto computacional se debe a la búsqueda de los puntos de dispersión, por lo que antes de emplear los algoritmos de intersección es necesario aplicar un conjunto de criterios rápidos dependientes de la dirección de observación.El principio de Fermat en combinación con el Gradiente Conjugado (CGM) es usado para obtener de manera eficiente los puntos de dispersión sobre la estructura. Para cada efecto, laposible ocultación de la trayectoria completa del rayo es examinada, por ello, si el rayo corta alguno de los parches de Bezier su contribución será descartada. Los dobles efectos son tratados como una generalización de los simples efectos.El método desarrollado es eficiente ya que precisa de un numero reducido de superficies para modelar objetos complejos lo que se traduce en bajos requerimientos de memoria y reducidos tiempos de calculo.
In this thesis a method to analyze antennas on board of complex bodies is presented. The Geometrical Optics (GO) and Uniform Theory of Diffraction (UTD) have been used to analyze the effect of the structure in the radiation pattern of the antennas. The bodies are geometrically modelled by using NURBS (Non Uniform Rational B-Spline) surfaces. In addition to be accurate and efficient, the method is compatible with most of the modern CAGD (Computer Aided Geometric Design) available programs.The treatment of arbitrary geometries requires a code which can carry out an efficient 3D analysis. To obtain accurate results the description of the surface must be close to the real model, however this complicates the computational procedure. Here the structure is modeled by a collection of individual N.U.R.B.S. surface patches joined to form a complete description of the surface model. The NURBS description is able to manipulate free form surfaces with a low number of patches, and therefore, with a low amount of information. The initial description of the model by NURBS surfaces is accompanied with other complementary data for example : the topology of the surfaces, the boundary curves, the types of material and other inputs. It is very interesting to apply criteria to make the complete analysis faster.The method reads the NURBS description of the model and transforms the NURBS into the rational BEZIER surfaces. A rational BEZIER patch is also a parametric surface defined in terms of a linear combination of Bernstein polynomials.The antennas are modelled using simple numerical models based on arrays of electric and magnetic infinitesimal dipoles. This antenna modelization is very advantageous because with a little input data, the source is defined in any direction and the field value is readily accessible.The electromagnetic analysis of the contributive effects to the scattering field by the geometry, starts with the rigorous selection of the geometry illuminated from the source. Only the Bezier patches illuminated will be in memory of the computer during the analysis. The philosophy of this previous process is to discard in the process the part of the geometry which does not contribute to the scattering effects.The total field is the superposition of the following GO and UTD field components: direct field from the source, reflected fields from the Bezier patches of the model, diffracted fields from the arbitrary edges defined as a Bezier curves, creeping waves, double reflected field and diffracted-reflected and reflected-diffracted fields. The search of specular and diffraction points are the most CPU time consuming, thus before using the intersection algorithms it is necessary to apply a set of fast selection criteria which depend on the observation direction.The Fermat principle in conjunction with the Conjugate Gradient Method (CGM) is used for obtaining efficiently the reflection points and diffraction points on the structure. For each effect the complete ray path is examined to see whether or not it is interrupted by any Bezier patch of the model, in this case the field component is not computed. The double effects are treated using a generalization of the single effects algorithms. The method has been developed to analyze the near and far field cases for different frequencies.The developed method is quite efficient because it makes use of a small number of surfaces to model complex bodies, so it requires few memory and low computing time.

Aquesta tesi proposa una millora del clàssic mètode dels elements finits (finite element method, FEM) per a un tractament eficient de dominis amb contorns corbs: el denominat NURBS-enhanced finite element method (NEFEM). Aquesta millora permet descriure de manera exacta la geometría mitjançant la seva representació del contorn CAD amb non-uniform rational B-splines (NURBS), mentre que la solució s'aproxima amb la interpolació polinòmica estàndard. Per tant, en la major part del domini, la interpolació i la integració numèrica són estàndard, retenint les propietats de convergència clàssiques del FEM i facilitant l'acoblament amb els elements interiors. Només es requereixen estratègies específiques per realitzar la interpolació i la integració numèrica en elements afectats per la descripció del contorn mitjançant NURBS.

La implementació i aplicació de NEFEM a problemes que requereixen una descripció acurada del contorn són, també, objectius prioritaris d'aquesta tesi. Per exemple, la solució numèrica de les equacions de Maxwell és molt sensible a la descripció geomètrica. Es presenta l'aplicació de NEFEM a problemes d'scattering d'ones electromagnètiques amb una formulació de Galerkin discontinu. S'investiga l'habilitat de NEFEM per obtenir solucions precises amb malles grolleres i aproximacions d'alt ordre, i s'exploren les possibilitats de les anomenades malles NEFEM, amb elements que contenen singularitats dintre d'una cara o aresta d'un element. Utilitzant NEFEM, la mida de la malla no està controlada per la complexitat de la geometria. Això implica una dràstica diferència en la mida dels elements i, per tant, suposa un gran estalvi tant des del punt de vista de requeriments de memòria com de cost computacional. Per tant, NEFEM és una eina poderosa per la simulació de problemes tridimensionals a gran escala amb geometries complexes. D'altra banda, la simulació de problemes d'scattering d'ones electromagnètiques requereix mecanismes per aconseguir una absorció eficient de les ones scattered. En aquesta tesi es discuteixen, optimitzen i comparen dues tècniques en el context de mètodes de Galerkin discontinu amb aproximacions d'alt ordre.

La resolució numèrica de les equacions d'Euler de la dinàmica de gasos és també molt sensible a la representació geomètrica. Quan es considera una formulació de Galerkin discontinu i elements isoparamètrics lineals, una producció espúria d'entropia pot evitar la convergència cap a la solució correcta. Amb NEFEM, l'acurada imposició de la condició de contorn en contorns impenetrables proporciona resultats precisos inclús amb una aproximació lineal de la solució. A més, la representació exacta del contorn permet una imposició adequada de les condicions de contorn amb malles grolleres i graus d'interpolació alts. Una propietat atractiva de la implementació proposada és que moltes de les rutines usuals en un codi d'elements finits poden ser aprofitades, per exemple rutines per realitzar el càlcul de les matrius elementals, assemblatge, etc. Només és necessari implementar noves rutines per calcular les quadratures numèriques en elements corbs i emmagatzemar el valor de les funciones de forma en els punts d'integració. S'han proposat vàries tècniques d'elements finits corbs a la literatura. En aquesta tesi, es compara NEFEM amb altres tècniques populars d'elements finits corbs (isoparamètics, cartesians i p-FEM), des de tres punts de vista diferents: aspectes teòrics, implementació i eficiència numèrica. En els exemples numèrics, NEFEM és, com a mínim, un ordre de magnitud més precís comparat amb altres tècniques. A més, per una precisió desitjada NEFEM és també més eficient: necessita un 50% dels graus de llibertat que fan servir els elements isoparamètrics o p-FEM per aconseguir la mateixa precisió. Per tant, l'ús de NEFEM és altament recomanable en presència de contorns corbs i/o quan el contorn té detalls geomètrics complexes.
This thesis proposes an improvement of the classical finite element method (FEM) for an efficient treatment of curved boundaries: the NURBSenhanced FEM (NEFEM). It is able to exactly represent the geometry by means of the usual CAD boundary representation with non-uniform rational Bsplines (NURBS), while the solution is approximated with a standard piecewise polynomial interpolation. Therefore, in the vast majority of the domain, interpolation and numerical integration are standard, preserving the classical finite element (FE) convergence properties, and allowing a seamless coupling with standard FEs on the domain interior. Specifically designed polynomial interpolation and numerical integration are designed only for those elements affected by the NURBS boundary representation.

The implementation and application of NEFEM to problems demanding an accurate boundary representation are also primary goals of this thesis. For instance, the numerical solution of Maxwell's equations is highly sensitive to geometry description. The application of NEFEM to electromagnetic scattering problems using a discontinuous Galerkin formulation is presented. The ability of NEFEM to compute an accurate solution with coarse meshes and high-order approximations is investigated, and the possibilities of NEFEM meshes, with elements containing edge or corner singularities, are explored. With NEFEM, the mesh size is no longer subsidiary to geometry complexity, and depends only on the accuracy requirements on the solution, whereas standard FEs require mesh refinement to properly capture the geometry. This implies a drastic difference in mesh size that results in drastic memory savings, and also important savings in computational cost. Thus, NEFEM is a powerful tool for large-scale scattering simulations with complex geometries in three dimensions. Another key issue in the numerical solution of electromagnetic scattering problems is using a mechanism to perform the absorption of outgoing waves. Two perfectly matched layers are discussed, optimized and compared in a high-order discontinuous Galerkin framework.

The numerical solution of Euler equations of gas dynamics is also very sensitive to geometry description. Using a discontinuous Galerkin formulation and linear isoparametric elements, a spurious entropy production may prevent convergence to the correct solution. With NEFEM, the exact imposition of the solid wall boundary condition provides accurate results even with a linear approximation of the solution. Furthermore, the exact boundary representation allows using coarse meshes, but ensuring the proper implementation of the solid wall boundary condition. An attractive feature of the proposed implementation is that the usual routines of a standard FE code can be directly used, namely routines for the computation of elemental matrices and vectors, assembly, etc. It is only necessary to implement new routines for the computation of numerical quadratures in curved elements and to store the value of shape functions at integration points.

Several curved FE techniques have been proposed in the literature. In this thesis, NEFEM is compared with some popular curved FE techniques (namely isoparametric FEs, cartesian FEs and p-FEM), from three different perspectives: theoretical aspects, implementation and performance. In every example shown, NEFEM is at least one order of magnitude more accurate compared to other techniques. Moreover, for a desired accuracy NEFEM is also computationally more efficient. In some examples, NEFEM needs only 50% of the number of degrees of freedom required by isoparametric FEs or p-FEM. Thus, the use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details.

Mestrado em Engenharia Mecânica
A few years ago drawings were made in the drawing boards and using pencils on vellum. There were no computers helping the designers in the parts modeling. After designing the object, the design was passed to the analysts. The designers and analysts were in constant communication. Nowadays, the designers used Computer Aided Design (CAD) tools in the parts modeling. For application the analysis at the geometries, initially a mesh to approximate the geometries is generated. After this, on the mesh the Finite Element Method (FEM) was applied. In complex engineering design, the generation and manipulation of meshes in FEA was estimated to take over 80% of the overall analysis time. The form to break down the barriers between engineering design and the analysis is with reconstruction the entire process, but at the same time maintaining compatibility with existing practices. Create only one geometric model is the focus of reconstruction process. This geometric model is used in the representation of the geometry, as well as in the analysis, and this concept is designated by Isogeometric Analysis (IGA). In this present work the development of the tools for generate the CAD and calculate the basis function for representation the object are proposed. Initially, the mathematical formulations for Bézier, B-Spline and NURBS, for curves and surfaces are presented. The algorithms developed to generate the curves and surfaces are demonstrated. The IGA and FEM formulation for tridimensional and bidimensional spaces are introduced. In this work, a development of a tools for application this method are proposed. The convergence of the results for FEM and IGA programs are studied and compared to the theoretical values and Abaqus comercial program. The results obtained with IGA formulation converge to the reference values.
Há alguns anos atrás, os objectos eram feitos pelos designers e a criação do desenho era feita com lápis e papel vegetal. Não existiam computadores nos gabinetes de desenho para ajudar na modelação dos objectos. Após o desenho estar concluído este era entregue aos analistas para calcularem a resistência do mesmos quando solicitados por cargas externas. Assim, o gabinete de design e o gabinete de análise estavam em constante comunicação. Nos tempos de hoje os designers utilizam as ferramentas de Computer-Aided Design (CAD) para gerar os objectos, representando assim a geometria original. Por outro lado, os analistas fazem a análise baseada no Método dos Elementos Finitos (MEF). Neste método, inicialmente, gera-se uma malha para fazer a aproximação do objecto e utiliza-se esta malha gerada na análise. A forma de combater esta barreira é a construção de um novo processo de análise, mas ao mesmo tempo manter a compatibilidade com a análise do Método de Elementos Finitos. Este novo método foca-se na geração de um modelo geométrico, sendo este modelo utilizado tanto para a representação da geometria como para a análise. A principal sustentação deste novo método é a utilização das funções de base da criação e representação dos objectos, posteriormente, utilizadas na análise dos mesmos. Este novo conceito é designado por Análise Isogeométrica. Neste trabalho é exposto o desenvolvimento de ferramentas para gerar curvas e superfícies utilizando as formulações de Bézier, B-spline e NURBS. Assim, desenvolveram-se sub-rotinas para calcular as funções de base. Inicialmente apresentaram-se as formulações matemáticas e posteriormente os algoritmos desenvolvidos para a representação das curvas e superfícies. O desenvolvimento de ferramentas de análise para problemas no espaço bidimensional e tridimensional utilizando o Método de Elementos Finitos e a Análise Isogeométrica também é abordado neste trabalho. Para ser mais fácil a sua aplicação, foi desenvolvida um interface. Por fim utilizaram-se problemas e estudaram-se as curvas de convergência dos resultados e compararando-os com as referência analíticas e com o programa Abaqus. Em termos de conclusão, os resultados obtidos com a Análise Isogeométrica convergem mais rapidamente para os valores de referência do que o Abaqus e o programa desenvolvido com base no método de elementos finitos.

Mestrado em Engenharia Mecânica
In the present day most product development industries uses the Finite Element Method (FEM) for structural analysis. Designers model the product geometries using Computer-Aided Design (CAD) software, the geometries are then fitted for analysis, by the analysts, with a mesh approximation that inevitably results in loss of accuracy. Achieving the best geometry description for complex components can be a complex task and it can take a lot of time. Considering this drawback, a new method was developed which takes advantages of curve representation tools and uses them as bases for analysis. Aiming for no loss of geometrical precision, this new method has been called "Isogeometric Analysis" (IGA). The smoothness of Spline representations used in Isogeometric Analysis can be useful for a particular branch of structural analysis which is the analysis of plates and shells. The classic thin plate theory developed by Gustav Kirchhoff requires a geometry description with C1 continuity between elements which is normally defined by high order polynomial functions, which typically represents a problem with the piecewise Lagrangian shape functions used in conventional FEM. The present work explores parametric descriptions used as basis for Isogeometric Analysis, such as Bézier curves, B-splines and NURBS, taking advantage of its smoothness to develop formulations for thin plate elements. The 4-node rectangular derived by Melosh, O. Zienkiewicz and Y. Chung called MCZ thin plate element based on Kirchhoff assumptions, was the starting point to build up to a NURBS-based thin plate element. MCZ thin plate elements, NURBS-based thin plate elements (with different order geometries) and Abaqus commercial software shell elements are evaluated by means of classical plate benchmarks comparing the elements convergences and overall performance. It can be shown that the proposed NURBS-based formulation is reliable for the analysis of thin structures.
Nos dias de hoje a maioria da indústria de desenvolvimento do produto utiliza o Método dos Elementos Finitos (MEF) na análise estrutural. Os desenhistas modelam o produto através de ferramentas de Computer-Aided Design (CAD). As geometrias são depois ajustadas para a análise pelos analistas que constroem uma aproximação através de uma malha de elementos finitos, o que inevitavelmente resulta numa perda de precisão geométrica. Para conseguir a melhor aproximação à geometria original para componentes complexos o processo pode ser complicado e pode consumir muito tempo. Considerando esta desvantagem foi desenvolvido um novo método que tira partido da descrição geométrica das ferramentas de desenho e utiliza as funções base das curvas para analise, com o objectivo de não haver perda de precisão geométrica, este novo método tem o nome de “Análise Isogeométrica” (IGA). A suavidade das geometrias Splines usadas na análise isogeometrica pode ser muito útil num ramo particular da análise estrutural, no estudo das placas e cascas. A teoria clássica de análise de placas finas de Kirchhoff requer uma descrição geométrica que tenha continuidade C1 entre elementos, que é normalmente definida por polinómios de ordem elevada, que são tipicamente um problema para as funções de forma Lagrangeanas usadas em MEF. O presente trabalho explora as descrições geométricas utilizadas como funções de forma para a análise isogeométrica como as curvas de Bézier, as B-splines e as NURBS, tirando vantagem da facilidade de estas conseguirem a requerida continuidade entre elementos para criar elementos de placas finas com as funções de base NURBS como funções de forma. É utilizado o elemento de placa fina MCZ desenvolvido por Melosh, O. Zienkiewicz e Y. Chung com base nas premissas de Kirchhoff como ponto de partida para desenvolver o elemento com base em NURBS. No fim os elementos de placas finas MCZ, os elementos com funções de base NURBS (com geometrias de diferentes ordens) e elementos do tipo casca do software comercial Abaqus são avaliados através de uma série de diferentes problemas clássicos de placas, comparando a convergência e o desempenho global. É possivel ver que a formulação proposta é fidedigna na análise de estruturas de placa fina.

博士
國立成功大學
電機工程學系碩博士班
99
An in-depth study on the Non-Uniform B-splines (NURBS) neural networks and their applications is conducted in this dissertation. Firstly, based on the concept of the NURBS curve, the NURBS Curve Neural Network (NURBSCNN) is proposed. Since the characteristic curve that describes the relationship between the input voltage and the output speed for the traveling wave ultrasonic motor (TWUSM) is highly complex and nonlinear, the proposed NURBSCNN is applied to implement the feedforward compensator and speed controller for the TWUSM. Secondly, exploiting the idea of NURBS surface, the NURBS Surface Neural Network (NURBSSNN) is proposed. Since a digital image can be represented by a NURBS surface, the proposed NURBSSNN is employed to cope with the image compression and image restoration problems in this dissertation. Both the proposed NURBSCNN and NURBSSNN belong to the category of feedforward neural networks. Compared with other commonly used neural networks, the most significant difference is that the activation functions of the first hidden layers in the proposed neural networks are blending functions rather than the commonly used sigmoid functions. The back-propagation algorithm is exploited to learn appropriate values of the control points and weights in the proposed NURBSCNN and NURBSSNN. Moreover, the selection methods for the values of the corresponding parameter and knot vector in the NURBSCNN and NURBSSNN, as well as their application flowcharts, are elaborated upon and discussed in detail. The feasibility and effectiveness of the proposed approaches are demonstrated by several illustrative examples in this dissertation.

碩士
國立交通大學
機械工程系所
105
In this thesis, we developed a selective laser sintering (SLS) system with NURBS curve interpolator, and sintering the silver nanoparticles ink on a flexible substrate. Take advantage of laser for reducing thermal damage of substrate, which allow us to use lower laser power to sinter the silver nanoparticles ink. To ensure that sintered silver nanoparticles can be used for conductive line, the experiment are designed for laser power in the range of 100 ~250 mW and the scanning speed in the range of 2~5 mm/s. The best electrical conductivity of the experiment result is used in the following experiment. In order to show if NURBS interpolator can effectively improve the continuity of laser scanning speed, by using a butterfly-shaped of NURBS curve as example to analyze the path of the motion stage. After sintering, traditional method (G01) with purposed method (G06) for the change rate of linewidth in the corner of the path are compared. At last, a planar spiral inductance for application is demonstrated.

碩士
國立臺灣科技大學
電子工程系
88
B-Splines and Non-Uniform Rational B-Splines (NURBS) have become the essential modeling primitives in computer graphics and geometric modeling applications. In this thesis, we propose a modified NURBS algorithm incorporated with two useful properties, sum up to one and dynamic denominator. This novel algorithm provides less order and fewer division operations than the traditional algorithm reported in the literature. Based on this algorithm, a unified architecture for the computation of various types of B-Spline curves and surfaces is presented. The resultant chip, consisting of approximately 752 K transistors, occupies 3.1 mm by 3.1 mm area in the 0.35-μm SPQM CMOS technology. It operates at 100 MHz with two 16-bit data outputs and consumes only 920mW at a supply voltage of 3.3V. The output data rate is two 16-bit words per cycle, which corresponds to a pair of the coordinate values of a point and its normal on a curve/surface.

The thesis describes a new method for obtaining minimizers for optimal control problems whose minima serve as control policies for guiding nonlinear dynamical systems to achieve prescribed goals under imposed trajectory and actuator constraints. One of the major contributions of the present work resides in the approximation of such minimizers by piecewise polynomial functions expressed in terms of a linear combination of non-uniform rational B-spline (NURBS) basis functions and the judicious exploitation of the properties of the resulting NURBS curves to improve the computational effort often associated with solving optimal control problems for constrained dynamical systems.

In particular, by exploiting the two structures combined in a NURBS curve, NURBS basis functions and an associated union of overlapping polytopes constructed from the coefficients of the linear combination, we are able to separate an optimal control problem into two subproblems | guidance and obstacle avoidance, making the original problem tractable. This is accomplished by laying out the union of overlapping polytopes in such a way that they delineate a section of space that avoids all obstacles and then manipulating the NURBS basis functions to obtain trajectories that are guaranteed to remain bounded by this section of space without explicitly including the conjunction of disjunctions naturally induced from obstacles into the guidance problem.

In addition, we show how one can construct systematically a feasible region that corresponds to a NURBS parameterization starting from an ordered union of pairwise adjacently overlapping nonempty compact convex sets. Specifically, we show how to setup a nonlinear programming problem to solve for the feasible region in terms of an ordered union of pairwise adjacently overlapping polytopes with nonempty interiors by maximizing the sum of their volumes and starting from a feasible region described by an ordered union of pairwise adjacently overlapping nonempty convex compact simi-algebraic sets. Finally, we show how this strategy can be implemented practically for an autonomous system traversing an urban environment.

Finally, this work culminated in the filing of patent US20070179685 on behalf of Northrop Grumman for the Space Technology sector and in the development of the NURBS-based OTG software package. This C++ package contains the theoretical results of this thesis in the form of an object-oriented implementation optimized for real-time trajectory generation.

Isogeometric analysis (IGA) is a family of numerical methods to solve boundary value problems. The founding principle of isogeometric methods is aimed at integrating computer aided design (CAD) with nite element method (FEM). In general, CAD platforms employ NURBS (Non Uniform Rational B-Splines) functions to model geometry and the same functions are adopted by isogeometric methods to approximate the unknown eld variables. Apart from their ability to model complex geometries, NURBS functions possess better approximation properties and high inter-element continuity properties. In its earliest days, isogeometric analysis comprised of Galerkin-isogeometric method alone. Galerkin-isogeometric method in essence is NURBS-based isoparametric nite element method. Research e orts in IGA further led to the development of new numerical methods like isogeometric collocation method. Isogeometric collocation o ers the geometric exibility of an isogeometric method and the computational advantage of a collocation scheme. The present thesis focuses on development of new computational approaches based on isogeometric methods for the bending analysis of laminated structural elements, namely, laminated composite plates and beams. A standard primal approach, shear locking free approach and a mixed approach based on isogeometric collocation are developed for the bending analysis of laminated composite plates governed by rst order shear deformation laminated plate (FSDT) theory. Variational asymptotic method (VAM) within the framework of isogeometric analysis is presented for the bending analysis of laminated composite beams

Micro Aerial Vehicles (MAVs) are battery operated, remote controlled miniature flying vehicles. MAVs are required in military missions, traffic management, hostage situation surveillance, sensing, spying, scientific, rescue, police and mapping applications. The essential characteristics required for MAVs are: light weight, maneuverability, ease of launch in variety of conditions, ability to operate in very hostile environments, stealth capabilities and small size. There are three main classes of MAVs : fixed, rotary and flapping wing MAV’s. There are some MAVs which are combinations of these main classes. Each class has its own advantage and disadvantage. Different scenarios may call for different types of MAV. Amongst the various classes, flapping wing class of MAVs offer the required potential for miniaturisation and maneuverability, necessitating the need to understand flapping wing flight. In the case of flapping winged flight, the thrust required for the vehicle flight is obtained due to the flapping of the wing. Hence for efficient flapping flight, optimising the flap motion is necessary. In this thesis work, an algorithm for motion optimisation of plunging airfoils is developed in a parallel framework. An evolutionary optimisation algorithm, PSO (Particle Swarm Optimisation), is coupled with an unsteady flow solver to develop a generic motion optimisation tool for plunging airfoils. All the unsteady flow computations in this work are done with the HIFUN1 code, developed in–house in the Computational Aerodynamics Laboratory, IISc. This code is a cell centered finite volume compressible flow solver. The motion optimisation algorithm involves starting with a population of motion curves from which an optimal curve is evolved. Parametric representation of curves using NURBS is used for efficient handling of the motion paths. In the present case, the motion paths of a plunging NACA 0012 airfoil is optimised to give maximum flight efficiency for both inviscid and laminar cases. Also, the present analysis considers all practically achievable plunge paths, si- nusoidal and non–sinusoidal, with varying plunge amplitudes and slopes. The results show promise, and indicate that the algorithm can be extended to more realistic three dimension motion optimisation studies.

Micro Aerial Vehicles (MAVs) are battery operated, remote controlled miniature flying vehicles. MAVs are required in military missions, traffic management, hostage situation surveillance, sensing, spying, scientific, rescue, police and mapping applications. The essential characteristics required for MAVs are: light weight, maneuverability, ease of launch in variety of conditions, ability to operate in very hostile environments, stealth capabilities and small size. There are three main classes of MAVs : fixed, rotary and flapping wing MAV’s. There are some MAVs which are combinations of these main classes. Each class has its own advantage and disadvantage. Different scenarios may call for different types of MAV. Amongst the various classes, flapping wing class of MAVs offer the required potential for miniaturisation and maneuverability, necessitating the need to understand flapping wing flight. In the case of flapping winged flight, the thrust required for the vehicle flight is obtained due to the flapping of the wing. Hence for efficient flapping flight, optimising the flap motion is necessary. In this thesis work, an algorithm for motion optimisation of plunging airfoils is developed in a parallel framework. An evolutionary optimisation algorithm, PSO (Particle Swarm Optimisation), is coupled with an unsteady flow solver to develop a generic motion optimisation tool for plunging airfoils. All the unsteady flow computations in this work are done with the HIFUN1 code, developed in–house in the Computational Aerodynamics Laboratory, IISc. This code is a cell centered finite volume compressible flow solver. The motion optimisation algorithm involves starting with a population of motion curves from which an optimal curve is evolved. Parametric representation of curves using NURBS is used for efficient handling of the motion paths. In the present case, the motion paths of a plunging NACA 0012 airfoil is optimised to give maximum flight efficiency for both inviscid and laminar cases. Also, the present analysis considers all practically achievable plunge paths, si- nusoidal and non–sinusoidal, with varying plunge amplitudes and slopes. The results show promise, and indicate that the algorithm can be extended to more realistic three dimension motion optimisation studies.