On multi-degree splines

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to B\'ezier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines

Matrix representations for multi-degree B-splines

The paper is concerned with computing the B-spline basis of a multi-degree spline space, namely a space of piecewise functions comprised of polynomial segments of different degrees. To this aim, we provide a general method to work out a matrix representation relating the sought basis with another one easier to compute. This will allow us, for example, to calculate a multi degree B-spline basis starting from local Bernstein bases of different degrees or from the B-spline basis of a spline space where all sections have the same degree. This change of basis can be translated into a conceptually simple and computationally efficient algorithm for the evaluation of multi-degree B-splines

Modellazione Geometrica con MD-Spline

Le curve B-spline comunemente usate in modellazione geometrica sono costituite da segmenti polinomiali aventi tutti lo stesso grado. Nella pratica, è spesso necessario modellare curve composte da segmenti polinomiali di grado differente. In questi casi l’utilizzo delle curve B-spline classiche complica inutilmente la modellazione ed necessita di informazioni superflue alla descrizione matematica della curva. Le curve MD-Spline (Multi Degree Spline) sono una generalizzazione delle curve B-spline costituite da segmenti polinomiali di gradi differenti. Con le MD-Spline un progettista CAD ha la possibilità di gestire il grado di ogni segmento in maniera indipendente dagli altri e può rappresentare una determinata forma specificando solo le informazioni strettamente necessarie a definirla. La conseguenza è che modellare con curve MD-Spline si rivela più semplice e rapido rispetto a quanto possibile con curve B-spline. La tesi tratta le MD-Spline descrivendone gli aspetti teorici alla base ed i principali algoritmi di modellazione. In particolare il lavoro di tesi ha prodotto il primo software di modellazione con curve MD-Spline

Progettazione e Sviluppo di una Web App per curve MD-spline

La tesi si occupa della progettazione e sviluppo di una Web App scritta in HTML5 e canvas e JavaScript che ha l'obiettivo di permettere di sperimentare le potenzialità di una nuova classe di curve 2D utili per il disegno denominata C1 MD-spline

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms

Evolutionary structural pptimisation based on boundary element representation of b-spline geometry

Evolutionary Structural Optimisation (ESO) has become a well-established technique for determining the optimum shape and topology of a structure given a set of loads and constraints. The basic ESO concept that the optimum topology design evolves by slow removal and addition of material has matured over the last ten years. Nevertheless, the development of the method has almost exclusively considered finite elements (FE) as the approach for providing stress solutions. This thesis presents an ESO approach based on the boundary element method. Non-uniform rational B-splines (NURBS) are used to define the geometry of the component and, since the shape of these splines is governed by a set of control points, use can be made of the locations of these control points as design variables. The developed algorithm creates internal cavities to accomplish topology changes. Cavities are also described by NURBS and so they have similar behaviour to the outside boundary. Therefore, both outside and inside are optimised at the same time. The optimum topologies evolve allowing cavities to merge between each other and to their closest outer boundary. Two-dimensional structural optimisation is investigated in detail exploring multi-load case and multi-criteria optimisation. The algorithm is also extended to three-dimensional optimisation, in which promising preliminary results are obtained. It is shown that this approach overcomes some of the drawbacks inherent in traditional FE-based approaches, and naturally provides accurate stress solutions on smooth boundary representations at each iteration

Computer simulation of a motorcycle and dummy rider in impact

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis is concerned with the simulation model of an OPAT dummy rider on a Norton motorcycle in different configured impacts with a rigid barrier. The mathematical equations used in describing the mass-spring-damper-based impacts have been given. The software used in designing the mathematical model have also been outlined. The simulation model was then calibrated against full scale crash tests by means of film analysis and the processed digitised measurements. This led to the investigations into numerical processing of differentiation and integration. A parametric study was also conducted to examine injury to the dummy rider based on some varying parameters. The simulation model was further verified by different configurations and also an introduction of an airbag. Finally, the model was extended to a HYBRID3 dummy rider on the same motorcycle in different configured impacts with a motor car. It is hoped that after the validations and verifications have been performed to examine the robustness of the simulation model, it can assist in the analyses of motorcycle impacts with the less frequent need of conducting a full scale crash test, so that safety design of a motorcycle can be established

Vertical accuracy of data on the topographic maps and their application in mining

The paper describes some experiences and the methodology used for evaluation the vertical accuracy of data on the official (state) topographic maps. The vertical accuracy is an important element of quality 3D representation on the topographic maps in digital, as well as in analogous forms. A comparative way of testing is usually applied for determining the vertical accuracy, namely the comparison of the measured values of topographic maps with 'true' or 'conditionally true' values (numeric data of the geodetic and height points). This method gives a direct accuracy assessment of the printed maps and 3D terrain models, especially when used in industries such as mining. To a large extent, the vertical accuracy of data and geomorphology (geographic) fidelity relief on the topographic maps is not always sufficient quality. The type of survey and how the matter affects the quality, or the geodetic control points are drawn precisely, where the accuracy of the contour lines is lower. However, the application of a new technology is changes the 3D visualization of terrain, thus increasing the vertical accuracy of data

Aylık yağışın konumsal dağılımının modellenmesinde farklı enterpolasyon yöntemlerinin karşılaştırmalı analizi

For many water resources planning and management studies such as water budget and hydrological modeling, it is very important to estimate areal precipitation from point observation stations. There are many deterministic and geostatistical methods for determining the spatial distribution of precipitation. In this study, the most widely used methods, inverse distance weighting (IDW), Simple Kriging (SK) and Co-Kriging (CK) are applied. It is the main objective of the study that Geographic Information Systems (GIS) techniques are used to compare widely preferred interpolation methods and to model the spatial distribution of monthly precipitation values for prediction in ungauged areas in Akarcay Sinanpasa and Suhut sub-basins, Turkey. At the same time, the effects of number of stations, basin area, characteristics and secondary data usage such as elevation on model performance are investigated. The IDW, a deterministic method and the SK-CK, geostatistical methods are compared with each other by cross validation technique and the applicability of the interpolation techniques for the study areas is analyzed. According to the cross validation test results of IDW, SK and CK methods, the mean RMSE (root mean square error) values of Sinanpasa sub-basin are respectively 13,76 mm, 9,32 mm and 8,72 mm while these values are 9,43 mm, 7,82 mm and 7,90 mm for Suhut sub-basin. Then, uncertainty analysis by means of PSE (prediction standard error) is applied to SK-CK methods with clear advantages over the IDW method and with the close RMSE values. In consideration of the results of the uncertainty analysis, the SK method with the mean PSE values 10,30 mm and 8,54 mm has a little superiority to the CK method whose average PSE values are 11,03 mm and 9,02 mm for both Sinanpasa and Suhut sub-basins, respectively. When the findings are evaluated, it can be seen that all three methods can be used for the study areas. The determination of the spatial distribution of precipitation in this way is considered to be beneficial for many water resources engineering studies in areas of ungauged/sparsely gauged.Su bütçesi ve hidrolojik modelleme gibi birçok su kaynakları planlama ve yönetim çalışmaları için noktasal yağış gözlemlerinden alansal yağışın tahmin edilmesi çok önemlidir. Yağışın konumsal dağılımının belirlenmesi için deterministik ve jeoistatistik birçok yöntem bulunmaktadır. Bu çalışmada en yaygın kullanılan uzaklığın tersi ile ağırlıklandırma (IDW), Simple Kriging (SK) ve CoKriging (CK) yöntemleri uygulanmıştır. Akarçay Sinanpaşa ve Şuhut alt havzalarında, Coğrafi Bilgi Sistemleri (CBS) teknikleri ile yaygın olarak tercih edilen enterpolasyon yöntemlerinin karşılaştırılması ve aylık yağış değerlerinin konumsal dağılımının ölçüm yapılmayan alanlarda tahmin yapılması için modellenmesi çalışmanın ana amacını oluşturmaktadır. Aynı zamanda istasyon sayısı, havza alanı, karakteristikleri ve yükseklik gibi ikincil veri kullanımının model performansları üzerindeki etkileri araştırılmıştır. Deterministik bir yöntem olan IDW ve jeoistatistik yöntemler olan SK-CK yöntemlerinin çapraz doğrulama tekniği ile performansları test edilerek karşılaştırılmış ve çalışma alanları için enterpolasyon tekniklerinin kullanılabilirliği incelenmiştir. IDW, SK ve CK yöntemlerinin çapraz doğrulama test sonuçlarına göre Sinanpaşa alt havzası için sırasıyla RMSE (karesel ortalama hata) değerleri 13,76 mm, 9,32 mm ve 8,72 mm iken; Şuhut alt havzası için 9,43 mm, 7,82 mm ve 7,90 mm'dir. IDW yöntemine kıyasla açık üstünlükleri olan ve yakın RMSE değerlerine sahip SK-CK yöntemlerine, ek olarak PSE (tahmin standart hatası) ile belirsizlik analizi uygulanmıştır. Belirsizlik analizi sonuçlarına göre hem Sinanpaşa hem de Şuhut alt havzaları için SK yöntemi sırasıyla 10,30 mm ve 8,54 mm PSE değerleriyle, 11,03 mm ve 9,02 mm PSE değerlerine sahip CK yöntemine az da olsa üstünlük sağlamıştır. Elde edilen bulgulara göre her üç yönteminde çalışma alanları için kullanılabilir olduğu görülmektedir. Bu şekilde yağışın konumsal dağılımının belirlenmesinin ölçüm yapılmayan veya kıt ölçüm yapılan alanlarda birçok su kaynakları mühendisliği çalışmaları için faydalı olacağı düşünülmektedir