Добірка наукової літератури з теми "Discontinuous spline"

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

The paper examines methods for constructing mathematical models of two variables discontinuous functions using various information about them: one-sided values at points and one-sided traces along a given system of lines. The case is considered when the domain of the required function is triangulated by right-angled triangles. If interpolation or approximation methods are used, then for their construction the values of the function at given points must be given; if we use interlination methods, then traces of the desired function along a given system of lines. In this work, we construct a discontinuous interpolation and approximation splines for approximating a discontinuous function of two variables with given one-sided values in a given system of points (in our case, at the vertices of right-angled triangles), and prove theorems on the estimation of the approximation error by constructed discontinuous structures. In the paper a discontinuous interlination spline, which uses completely different information about the discontinuous function, namely one-sided traces along a given system of lines (in our case, along the sides of right-angled triangles) is also built. Interlination of functions can find wide application in the aircraft and automobile body design automation; when receiving and processing the results of sonar and radar, when solving problems of computed tomography, in digital signal processing and in many other areas. In the paper theorems on the integral form and an estimate of the approximation error by the constructed discontinuous interlination operator are also proved. Computational experiments that compare the results of the approximation of a discontinuous function of two variables by different information operators using triangular elements are presented. In the future, it is planned to apply the constructed operators of discontinuous approximation and interlination to solve a two-dimensional problem of computed tomography with a significant use of the inhomogeneity of the internal structure of the body, which must be reconstructed.

L’Analyse isogéométrique (AIG) est une méthode innovante de résolution numérique des équations différentielles, proposée à l’origine par Thomas Hughes, Austin Cottrell et Yuri Bazilevs en 2005. Cette technique de discrétisation est une généralisation de l’analyse par éléments finis classiques (AEF), conçue pour intégrer la conception assistée par ordinateur (CAO), afin de combler l’écart entre la description géométrique et l’analyse des problèmes d’ingénierie. Ceci est réalisé en utilisant des B-splines ou des B-splines rationnelles non uniformes (NURBS), pour la description des géométries ainsi que pour la représentation de champs de solutions inconnus.L’objet de cette thèse est d’étudier la méthode isogéométrique dans le contexte des problèmes hyperboliques en utilisant les fonctions B-splines comme fonctions de base. Nous proposons également une méthode combinant l’AIG avec la méthode de Galerkin discontinue (GD) pour résoudre les problèmes hyperboliques. Plus précisément, la méthodologie de GD est adoptée à travers les interfaces de patches, tandis que l’AIG traditionnelle est utilisée dans chaque patch. Notre méthode tire parti de la méthode de l’AIG et la méthode de GD.Les résultats numériques sont présentés jusqu’à l’ordre polynomial p= 4 à la fois pour une méthode deGalerkin continue et discontinue. Ces résultats numériques sont comparés pour un ensemble de problèmes de complexité croissante en 1D et 2D
Isogeometric Analysis (IGA) is a modern strategy for numerical solution of partial differential equations, originally proposed by Thomas Hughes, Austin Cottrell and Yuri Bazilevs in 2005. This discretization technique is a generalization of classical finite element analysis (FEA), designed to integrate Computer Aided Design (CAD) and FEA, to close the gap between the geometrical description and the analysis of engineering problems. This is achieved by using B-splines or non-uniform rational B-splines (NURBS), for the description of geometries as well as for the representation of unknown solution fields.The purpose of this thesis is to study isogeometric methods in the context of hyperbolic problems usingB-splines as basis functions. We also propose a method that combines IGA with the discontinuous Galerkin(DG)method for solving hyperbolic problems. More precisely, DG methodology is adopted across the patchinterfaces, while the traditional IGA is employed within each patch. The proposed method takes advantageof both IGA and the DG method.Numerical results are presented up to polynomial order p= 4 both for a continuous and discontinuousGalerkin method. These numerical results are compared for a range of problems of increasing complexity,in 1D and 2D

Un corps solide soumis à des contraintes, subit une déformation décrite par la connaissance du champ de déplacements en chacun de ses points. Si l'on considère un corps parfaitement plastique, le champ de déplacements recherché est solution d'un problème non linéaire (problème de Hencky) et présente, a priori, une perte de régularité l'existence de telles discontinuités est d'ailleurs confirmée par de nombreux résultats expérimentaux (observation de surfaces de glissement, plis, angles. . . ). Les solutions recherchées étant supposées régulières de part et d'autre d'une courbe dont la position reste à déterminer, nous sommes ainsi ramenés à un problème d'optimisation de domaines, que nous résolvons par des méthodes numériques d'analyse convexe, à l'aide d'une représentation discrète des espaces fonctionnels (éléments finis) et de la ligne de glissement (B-Splines)
The strain of a solid body submitted to stresses may be described by the displacements field in each point. If we consider field, which is and,a priori, discontinuities a perfectly plastic material, we search a displacements a solution of a non-linear problem (the Hencky's problem) exhibits a loss of regularity: the existence of such is otherwise confirmed by many experimental results (slip surfaces, angles,). If we suppose that the solutions are smooth on both sides of a curve whose exact location remains to be determined, we are led to an Optimal Design problem which we can solve by convex analysis and numerical methods after a well-suited discretization of the functional spaces (by finite elements) and the slip line (by B-Splines. )

Thesis (MSc)--Stellenbosch University, 2013.
ENGLISH ABSTRACT: Gully erosion is a major environmental problem not only having direct influences on site but also indirect influences felt further down the catchment. Combating gully erosion has proven to be elusive due to the difficulty in finding the causal factors and developing mechanisms involved. Soil is the medium in which gully erosion occurs yet few research have investigated it as a driving factor behind gully development and those that have has done it in a very elementary way. The first aim of this project was to physically and chemically characterise and classify the discontinuous gully system at Malansdam to establish the relationship between landscape hydrology and geomorphologic gully development with a focus on control factors. This was done by field observations, physical measurements and spatial and hydrological analyses with a Geographical Information System (GIS). The Malansdam gully system was the first ever recorded Strahler stream order (SSO) 5 classical gully system with the most active region being in the upper reaches where a steeper slope is experienced. Although piping was observed the V-shape channels and SAR data from traditional wet analysis indicated runoff to be the dominant formation process. A duo of factors, consisting of one anthropogenic and one natural factor respectively, was found that the major control factors behind the gully formation. A unique anthropogenic factor that has never been published beforehand was found to be the anthropogenic driving factor namely the ploughed contour cultivation technique employed by the farmers in the Sandspruit catchment. The ploughed contours act as channels firstly collecting and secondly moving water that would have drained naturally downwards in the valley to one exit point in the gully system. This allows increased erosive energy because of the larger volumes of water entering one single point in the upper reaches of the gully system where a steeper slope is experienced. The driving factor in the natural group was determined to be weak soil structure due to an abundant amount of exchangeable Mg2+ cations occupying the exchange sites on the clay fraction. This would cause soil to disperse in the presence of water even with a low amount of exchangeable Na+. Combat methods would accordingly exist in the form of rectifying the soil structure and finding an alternative to the ploughed contour system currently employed, but also planting vegetation especially grass or wheat in the gully channels. The second aim of this project was to determine the capability of Near Infrared (NIR) spectrometry, with wavenumbers 12 500 – 4 000 cm-1, to predict indicators used in soil science to establish the dispersive nature of a soil. These indices included the Exchangeable Sodium Percentage (ESP), Sodium Absorption Ratio (SAR), Magnesium Saturation Percentage (MS%), Electrical Conductivity (EC), Potential Hydrogen (pH) as well as the four main exchangeable cations namely calcium (Ca2+), potassium (K+), sodium (Na+) and magnesium (Mg2+). Surface and subsurface soil samples were collected from active gully heads. These samples were minimally pre-processed thus only dried, milled and sieved. Thereafter it was subject to NIR analysis making use of the Bruker multi-purpose FT-NIR Analyser (MPA; Bruker Optik GmbH, Germany) with a spectral range of 12 500cm-1 to 4000cm-1 which is. Partial Least Square Regression (PLSR) models were built for each index and the exchangeable cations making use of QUANT 2 utility of OPUS 6.5 (MPA; Bruker Optik GmbH, Germany) software. Five different regrssion statistics namely the coefficient of determination (r2), Root Mean Square Error of Cross Validation (RMSECV), Ratio of Performance to Deviation (RPD), Bias and the Ratio of Performance of Quartiles (RPIQ) were used to assess the legitimacy of each PLSR model. Upon validation all the PLSR models performed in line with previously published work and in certain cases better. The only exception was MS% which would require further investigation. NIR thus possess the capability to predict a soil’s dispersive nature in a fast, reliable, inexpensive and non- destructive way, thus implying whether or not it contributes to gully erosion at a significant level or only minimally.
AFRIKAANSE OPSOMMING: Donga erosie is 'n groot omgewingsprobleem. Dit het nie net ‘n direkte invloed op die area waar dit geleë is nie, maar het ook ‘n indirekte invloed wat elders in die opvangsgebied ervaar word. Bekamping van donga erosie is moeisaam aangesien die faktore wat aanleiding gee tot die vorming en dryf daarvan moeilik is om te bepaal. Grond is die medium waarin erosie plaasvind, maar daar is nogtans steeds min navorsing wat grond ondersoek het as ‘n moontlike faktor aanleiding gee tot donga erosie. Die wat dit al wel ingesluit het, het dit slegs op n baie elemntêre manier ondersoek. Die eerste doel van hierdie projek was om die diskontinue donga stelsel fisies en chemise te karakteriseer en klassifiseer om soedoende die verhouding tussen die landskap hidrologie en geomorfologiese donga ontwikkeling te bepaal met n fokus op die faktore wat dit dryf. Dit was gedoen deur middel van observasies gedoen terwyl veldwerk uitgerig was, fisiese metings asook ruimtelike en hidrologiese analises deur gebruik te maak van n Geografiese Inligting Stelsel (GIS). Die klassieke Malansdam donga stelsel is ‘n Strahler stroomorde (SSO) van 5 toegeken en is die eerste een ooit wat dit behaal het. Die mees aktiefste area was in die bolope waar die steilste helling ervaar was. Alhoewel ondergrondse pyp formasie waargeneem was het die V-vormige donga kanale en SAR data van die tradisionele nat analise aangedui dat afloop die dominante vorming proses was. Daar was gevind dat 'n duo van faktore, wat bestaan uit een menslike en een natuurlike faktor onderskeidelik, die faktore was wat donga ontwikkeling in die area dryf. 'n Unieke menslike faktor wat nog nie vantevore gepubliseer is, was bevind as die menslike faktor wat aanleiding gee tot donga erosie. Hierdie faktor is die bewerkiingsmetode wat in die Sandspruit opvangsgebied gebruik word naamlik geploegde kontoerbewerking. Die geploegde kontoere tree op as kanale om eerstens water te versamel en tweedens om die vloeirigting daarvan te wysig. Water wat onder natuurlike toestande afwaarts sou dreineer tot in die vallei word vasgevang deur die kontoere en gekanaliseer na een invloei punt in die donga. Hierdie proses verhoog die erosiekrag van die water aangesien groter volumes by 'n enkele punt in die steiler bolope van die donga stelsel invloei. Die dryf faktor in die natuurlike groep was swak grond struktuur. Die oorsaak hiervan was die besetting van ‘n grootmaat uitruilbare Mg2+ katione op die uitruil plekke van die kleifraksie. Dit sou veroorsaak dat grond in die teenwoordigheid van water maklik sou dispergeer, selfs in die teenwoordigheid van 'n lae hoeveelheid uitruilbare Na+ katione. Metodes om donga erosie te bekamp sal dienooreenkomstig bestaan uit die herstel van die grondstruktuur en die toepassing van 'n alternatiewe gondbewerkings stelsel. Die aanplanting van plantegroei, veral gras en koring binne die donga kanale sal verder help met die veg tot bekamping Die tweede doel van hierdie projek was om te bepaal indien naby infrarooi (NIR) spektrometrie (met golfnommer van 12 500 – 4 000cm-1) oor die vermoë beskik om aanwysers wat traditioneel in grondkunde gebruik word om die dispergering van grond te meet te voorspel. Hierdie aanwysers sluit vyf indekse in naamlik die Veranderlike Natrium Persentasie (ESP), Natrium Absorpsie Verhouding (SAR), Magnesium Versadiging Persentasie (MS%), Elektriese Geleidingsvermoë (EC) en die Potensiële Waterstof (pH) sowel as die vier hoof uitruilbare katione naamlik kalsium (Ca2+) , kalium (K+), natrium (Na+) en magnesium (Mg2+). Oppervlak en ondergrondse grondmonsters is ingesamel by die punt van oorsprong by aktiewe dongas. Hierdie monsters is minimaal voorberei, dus slegs gedroog, gemaal en gesif. Daarna was dit onderworpe aan die NIR analise. Die Bruker meerdoelige FT-NIR Analiseerder (MPA; Bruker Optik GmbH, Duitsland) met 'n spektrale omvang van 12 500cm 1 4000cm-1 is hiervoor gebruik. Parsiële kleinste kwadraat regressie (PLSR) modelle is gebou vir elke indeks asook die uitruilbare katione deur gebruik te maak van die nutsprogram Quant 2 van die OPUS 6.5 (MPA; Bruker Optik GmbH, Duitsland) sagteware. Vyf verskillende regressie statistieke naamlik die bepalingskoëffisiënt (r2), vierkantswortel fout tydens kruis validasie (RMSECV), verhouding van prestasie teenoor voorspellingsafwyking (RPD), sydigheid en die verhouding van prestasie van kwartiele (RPIQ) was gebruik om die geldighied van elke PLSR model te asseseer. Alle PLSR modelle het goed presteer, behalwe vir MS% wat verdere navorsing vereis. NIR beskik dus oor die vermoë om die aard van dispergering van grond te bepaal op 'n vinnige, betroubare, goedkoop en nie afbrekende manier. Dit kan dus effektief aangewend word as ‘n substitusie vir die traditionele metodes om te bepaal as grond a beduidende faktor is of nie.

Objective: We consider the following problem from credit risk modeling: Our sample (Xi; Yi), 1 < i < n, consists of pairs of variables. The first variable Xi measures the creditworthiness of individual i. The second variable Yi is the default indicator of individual i. It has two states: Yi = 1 indicates a default, Yi = 0 a non-default. A default occurs, if individual i cannot meet its contractual credit obligations, i. e. it cannot pay back its outstandings regularly. In afirst step, our objective is to estimate the threshold between good and bad creditworthiness in the sense of dividing the range of Xi into two rating classes: One class with good creditworthiness and a low probability of default and another class with bad creditworthiness and a high probability of default. Methods: Given observations of individual creditworthiness Xi and defaults Yi, the field of change point analysis provides a natural way to estimate the breakpoint between the rating classes. In order to account for dependency between the observations, the literature proposes a combination of three model classes: These are a breakpoint model, a linear one-factor model for the creditworthiness Xi, and a Bernoulli mixture model for the defaults Yi. We generalize the dependency structure further and use a generalized link between systematic factor and idiosyncratic factor of creditworthiness. So the systematic factor cannot only change the location, but also the form of the distribution of creditworthiness. Results: For the case of two rating classes, we propose several estimators for the breakpoint and for the default probabilities within the rating classes. We prove the strong consistency of these estimators in the given non-i.i.d. framework. The theoretical results are illustrated by a simulation study. Finally, we give an overview of research opportunities.

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.

La détermination de la forme d'un obstacle élastique immergé dans un milieu fluide à partir de mesures du champ d'onde diffracté est un problème d'un vif intérêt dans de nombreux domaines tels que le sonar, l'exploration géophysique et l'imagerie médicale. A cause de son caractère non-linéaire et mal posé, ce problème inverse de l'obstacle (IOP) est très difficile à résoudre, particulièrement d'un point de vue numérique. De plus, son étude requiert la compréhension de la théorie du problème de diffraction direct (DP) associé, et la maîtrise des méthodes de résolution correspondantes. Le travail accompli ici se rapporte à l'analyse mathématique et numérique du DP élasto-acoustique et de l'IOP. En particulier, nous avons développé un code de simulation numérique performant pour la propagation des ondes associée à ce type de milieux, basé sur une méthode de type DG qui emploie des éléments finis d'ordre supérieur et des éléments courbes à l'interface afin de mieux représenter l'interaction fluide-structure, et nous l'appliquons à la reconstruction d'objets par la mise en oeuvre d'une méthode de Newton régularisée.

碩士
國立高雄應用科技大學
模具工程系碩士班
96
This study is to construct the discontinuous B-Spline surfaces of three dimensions by using the reverse software we had developed. In some curves or surfaces of industrial products, we need to alter the B-Spline curves by handed manner in the reverse software in order to create the geometric feature line. That feature line with discontinuity (Positional continuity) is represented on some specific surfaces or regions, and the other unmodified surfaces remain the continuity (Slope continuity). However, it takes a lot of time to construct the feature line by hand. Therefore, we use the proposal algorithm to alter the B-Spline curves or surfaces. In this research, the 3D metrology system is used to get the 3D cloud data from the reflector surfaces of the automotive headlamp in order to construct the B-Spline curves in the fitting process. The vertices of the curve were coincided the control points of the curve vertices position, which automatically searched by the developed reverse software programs and turned the knots into the open uniform. It makes the curve’s smooth areas produce the discontinuity with above statements to create IGES files. Finally, The discontinuity B-Spline curves of three dimensions has shown in the reverse software systems from the IGES files. In this method, it can construct the curves into the free-form surfaces as well as check its quality. The producers could manufacture the headlamp’s reflector surfaces by CAM /RP systems after analysing by CAE. The results show that the feature lines of the reflector surfaces were established rapidly in the modified areas that we expected. The results of this study are also applied to the other alteration of the modeling surfaces to save the research and development (R&D) time for the engineers.

A method for calculating Fourier coefficients of functions of three variables using tomograms is proposed and investigated. Earlier Lytvyn O.M. proposed and introduced the method of calculating Fourier coefficients of the function of two variables using integrals of this function along a given system of direct. It is believed that these integrals - projections or projection data arrive from a computer tomograph.

A triangular B-spline (DMS-spline)-based finite element method (TBS-FEM) is proposed along with possible enrichment through discontinuous Galerkin, continuous-discontinuous Galerkin finite element (CDGFE) and stabilization techniques. The developed schemes are also numerically explored, to a limited extent, for weak discretizations of a few second order partial differential equations (PDEs) of interest in solid mechanics. The presently employed functional approximation has both affine invariance and convex hull properties. In contrast to the Lagrangian basis functions used with the conventional finite element method, basis functions derived through n-th order triangular B-splines possess (n ≥ 1) global continuity. This is usually not possible with standard finite element formulations. Thus, though constructed within a mesh-based framework, the basis functions are globally smooth (even across the element boundaries). Since these globally smooth basis functions are used in modeling response, one can expect a reduction in the number of elements in the discretization which in turn reduces number of degrees of freedom and consequently the computational cost. In the present work that aims at laying out the basic foundation of the method, we consider only linear triangular B-splines. The resulting formulation thus provides only a continuous approximation functions for the targeted variables. This leads to a straightforward implementation without a digression into the issue of knot selection, whose resolution is required for implementing the method with higher order triangular B-splines. Since we consider only n = 1, the formulation also makes use of the discontinuous Galerkin method that weakly enforces the continuity of first derivatives through stabilizing terms on the interior boundaries. Stabilization enhances the numerical stability without sacrificing accuracy by suitably changing the weak formulation. Weighted residual terms are added to the variational equation, which involve a mesh-dependent stabilization parameter. The advantage of the resulting scheme over a more traditional mixed approach and least square finite element is that the introduction of additional unknowns and related difficulties can be avoided. For assessing the numerical performance of the method, we consider Navier’s equations of elasticity, especially the case of nearly-incompressible elasticity (i.e. as the limit of volumetric locking approaches). Limited comparisons with results via finite element techniques based on constant-strain triangles help bring out the advantages of the proposed scheme to an extent.

Chapter 2 investigates a specific configuration (dubbed “remnant-internal relativization”) in which the head noun and some modifier/complement related to it appear discontinuously (as in the so-called split or discontinuous noun phrases). It is argued that the analysis of remnant-internal relativization is of particular interest from the theoretical and diachronic point of view. Theoretically it can illuminate the long-standing debate between the right adjunction and the head raising analyses of RRCs, providing evidence in favor of the latter. From a diachronic perspective, it is argued that the loss of remnant-internal relativization with the modifier/complement in the left periphery of the Portuguese relative clauses might be due to a restriction on movement that emerges inside the DP, which blocks the extraction of the modifier/complement to a left peripheral position.

This chapter investigates syntactic change regarding the availability of split noun phrases in relative clauses in the diachrony of Portuguese. In earlier stages of the language an element that is thematically dependent on the head noun (either as a complement or as a modifier) may not appear adjacent to it but in a relative clause internal position. In Contemporary European Portuguese, noun phrase discontinuity also arises in relative clauses, but only with the modifier/complement in the rightmost position. The word order with the modifier/complement at the left periphery of the relative clause is not allowed. The change is explained as being due to the loss of a left-peripheral position for contrastive focus within relative clauses (and possibly other types of subordinate clauses). Hence, the contraction of clause structure and the concomitant loss of movement are taken to constrain the possibilities of phrasal discontinuity found in earlier periods.

The vast majority of metamodeling demonstrations focuses on problems composed of continuous variables. However, important engineering design problems often include one or more discontinuous variables that require special attention. Previous work demonstrated the ability of Non-Uniform Rational B-spline HyPerModels to represent highly nonlinear functions composed of continuous variables. With minor modifications those capabilities can be extended to include functions defined by combinations of discontinuous input and output variables of different types, including discrete integer variables, feasibility variables and membership functions. Examples are used to demonstrate these modeling capabilities including applications developed from real engineering design problems such as the optimal positioning of a construction site crane and the optimal lay-up of a composite material I-beam.

The spline joint is a kind of widely used joint structures in the rotor system. Its discontinuous mechanical characteristics results from the contact surface. The surface will slide and deform when the spool deforms. As a consequence, the joint stiffness is always smaller than that of the integral configuration. This affects the rotor stiffness distribution and the rotor dynamics further. The objective of this study is to investigate the stiffness mechanical characteristics of the spline joint and their affecting factors. Based on the characteristics of structure and mechanical state, a spline joint stiffness mechanical model is built to explain the stiffness loss and its affecting factors. The cylindrical coordinate contact model is used to describe the contact of the cylindrical centering surface. And then a spline joint simulation model, taking the characteristics of the contact into account, is built by the nonlinear finite element method. At last, in the static stiffness experiment, the displacement of the spline joints is measured in different assembling and loading conditions. The results show that, the linear and angular stiffness loss of spline joint is significant and has a close relationship with the load and assembling condition. The study shows the effectiveness in controlling the mechanical properties of the rotor with spline joints by carefully adjusting structural design, load and assembling parameters.

This paper proposes a new model for cogging forces of linear motor systems. Sinusoidal functions of positions are used to capture the largely periodic nature of cogging forces with respect to position effectively while B-spline functions are employed to account for the additional aperiodic part of cogging forces. This model is experimentally demonstrated to be able to capture both the periodic and non-periodic characteristics of cogging force while having a linear parametrization form which makes effective on-line adaptive compensation of cogging forces possible. A discontinuous projection based desired compensation adaptive robust controller (DCARC) is then constructed for linear motors, which makes full use of the proposed cogging force model for an improved cogging force compensation. Comparative experimental results are obtained on both axes of a linear motor driven Anorad industrial gantry having a linear encoder resolution of 0.5 μm. Experiments are done with each axis running separately to compare the three algorithms: DCARC without cogging force compensation, DCARC with sinusoidal cogging force model compensation, and DCARC with the proposed cogging force model compensation. The results show that DCARC with proposed model compensation achieves the best tracking performance among the three algorithms tested, validating the proposed cogging force model. The excellent tracking performances obtained in experiments also verify the effectiveness of the proposed ARC control algorithms in practical applications.

The typical goals for defining a control net for a Non-Uniform Rations B-spline (NURBs) based metamodel from a given set of data the desired result is the smallest set of control points in the least possible time while minimizing local and/or global error. Current metamodel fitting algorithms iteratively find and eliminate the largest sources of local error, thus creating a very accurate control net with sub-optimal size. Since, control net size is directly related to the speed the control net can be searched for global optima, the size must be reduced as much as possible without increasing local or global error. Current algorithms can model discontinuous portions of data by clustering numerous control points close together. This is both inefficient to search and may cause the search algorithm to become numerically unstable and crash because of control points placed too closely together. Furthermore, many current algorithms do not take advantage of the weight property of each control point. In this paper, a Genetic Algorithm (GA) is used to optimize existing control nets as well as create new control nets from data sets. In order to offer a comparable creation time for the control net, parallel programming techniques are used incorporating the CUDA GPU Architecture. CUDA was chosen because it is low cost, highly parallel architecture available on millions of computers. The code is intended for a single desktop computer, running a maximum of 4 CUDA devices, not a CUDA cluster. This approach may have fitting applications to both metamodels and geometric fitting of NURBs objects.

In order to overcome the limits due to the fact that homogeneous layers of soft material placed over robotic limbs behave differently with respect to biological models, this paper suggests the adoption of soft covers (pads) with differentiated structure. In particular, it is proposed to divide the allowable pad thickness into two layers: a continuous external layer (skin) and a discontinuous internal layer, so that the overall stiffness can be adjusted by properly shaping the discontinuous layer. The methodology adopted for designing the internal layer is composed of two steps. Firstly, the cover surface is conceptually split into finite elementary triangular sub-regions. Secondly, the internal layer of each triangular element is designed in order to replicate the shape of the non-linear compression law which is typical of endoskeletal structures covered by pulpy tissues. A series of symmetrically-disposed inclined micro-beams is used for the purpose. Once the compression law of each triangular element is known, the overall pad compliance can be modulated by correctly choosing the number and size of the elements composing the pad. Equipment and results of a combined experimental and numerical analysis (FEM) are presented. The results confirm that the proposed concept can be an effective solution when designing soft covers whose behavior need to match the compliance of the biological counterpart. As an example, artificial pads which mimic the human finger behavior are presented.