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Grundlagen der geometrischen Datenverarbeitung
... Here n and m define the polynomial degree of the Bernstein polynomials along the parameters t and s. This results in OEn + 1 m + 1 Bézier points bij, which define a bidirectional net of control points [9][10][11]. Fig. 3 shows the The Bernstein polynomials Bin(t) form a basis system of linearly independent functions, which are calculated according to Eq. 4. ...
... Due to the nature of the Bernstein polynomials, which have a function value greater than 0 on the range t, s in (0,1), the location of a single Bézier point influences the shape of the entire surface. In addition, the number of Bézier points is directly coupled to the polynomial degree [9][10][11]. ...
... There are various mathematical methods for solving this minimization problem. The most common method is the least square fit [10], which is used in BECAL for the surface approximation with Bézier surfaces [1,5]. ...
More and more simulation tools are being used in the development of gears in order to save development time and costs while improving the gears. BECAL is a comprehensive software tool for the tooth contact analysis (TCA) of bevel, hypoid, beveloid and spur gears. The gear geometry is provided by a manufacturing simulation or a geometry import. To determine the exact contact conditions in the TCA, the discrete flank points are converted into a continuous and differentiable surface representation. At present, it is an approximation by means of Bézier tensor product surfaces. With this surface representation, significant deviations to the target points can occur depending on the tooth geometry. In particular tip, root and end relief, strongly curved tooth root geometries or discontinuous topological measurement data due to e.g. micro-pitting can only be considered insufficiently.
Hence, a new method for surface approximation with non-uniform rational b‑spline surfaces (NURBS) is presented. Its application can significantly improve the surface representation compared to the target geometry, leading to more realistic results regarding contact stress, tooth root stress and transmission error. To illustrate the advantages, NURBS-based surfaces are compared with the Bézier tensor product surfaces. Finally, the potential of the new approach regarding the prediction of lifetime and acoustics is demonstrated by application to different gear geometries.
... Furthermore, a global procedure of strain calculation based on smoothing DIC displacement data using the FEM is given in [2,30]. Another way is strain calculation based on global functions obtained by approximation using, e.g., polynomials [19,27] or B-splines (approximation method-see [15], strain calculation on the basis of this method-see [22][23][24]). ...
... In particular, the non-linear stress-strain behavior, the Mullins effect, hysteresis, and permanent set can be captured by this model. The Kirchhoff stress tensor τ of the MORPH model is composed out of three contributions according to Eq. (15). Therein, the first term represents a neo-Hookean like contribution, τ A is an additional stress defined by the ordinary differential equation (16), and the last term is a volumetric contribution according to [12] with the bulk modulus K M : ...
... Furthermore, for comparison purposes, the evaluation method can be applied to FE data, as well. The coordinates measured by DIC are approximated using B-spline surfaces (for basic principle, see [15]). Using this procedure, functions for the 3D coordinates are obtained, which enable strain calculation and smoothing of noise given by the measurement data. ...
In this paper, experimental and numerical investigations on cord–elastomer composites are presented. A finite-element model is introduced, which was developed within the framework of an industrial project. The model is able to simulate an elastomer matrix with inserted cords as load bearing elements and to predict the strains and stresses in cord and elastomer sections. The inelastic material behavior of the elastomer matrix and the yarns is described by corresponding material models suitable for large deformation processes. With the help of a specially developed demonstrator bellows, which is similar to an air spring, the simulation results are compared with experiments. For this purpose, the digital image correlation method is used to determine the deformations on the outer surface of the demonstrator bellows and to calculate the strains on and between the cords. The comparison of the results shows that the employed simulation method is very well suited to predict the strains in these cord–elastomer composites.
... Die Beschreibung einer Werkstückoberfläche in Form einer Punktmenge ist dabei am umfassendsten, da diese Be schreibungsform sowohl manuell einfach erstellt, als auch von Scannereinrichtungen und CAD-Systemen generiert werden kann. Zur Bestimmung von Werten zwischen Stützpunkten, sind in der Literatur mehrere Verfahren bekannt [64], Die Interpolation von Oberflächenprofilen mittels einer einzigen Funktion ist hier ungeeignet, da sie aufgrund des hohen Polynomgrades stark zum Oszillieren neigt. Als Alternative bietet sich an, die gegebene Punktmenge in kleinere Teilintervalle zu zerlegen. ...
... Als Alternative bietet sich an, die gegebene Punktmenge in kleinere Teilintervalle zu zerlegen. Die so entstandenen Flächensegmente können durch Funktionen mit niedrigerem Polynomgrad interpoliert werden [64], Wird als Funktion zwischen vier in einem Rechteck angeordneten Stützwerten ein Polynom dritten Grades gewählt, so kann eine Oberfläche nachgebildet werden, die über alle Flächen segmente sowohl stetig als auch stetig differenzierbar ist. Die Gesamtheit aller ermittelter Po lynome wird als bikubische Splinefunktion bezeichnet [64], Gesucht wird eine die Stützpunkte (x;,yj) interpolierende Splinefläche f(x,y) mit den Teil funktionen [64] fij(x,y) = Z;Eaijki(x-xi)k(y-yj) 1 Gl. 3 ...
... Die so entstandenen Flächensegmente können durch Funktionen mit niedrigerem Polynomgrad interpoliert werden [64], Wird als Funktion zwischen vier in einem Rechteck angeordneten Stützwerten ein Polynom dritten Grades gewählt, so kann eine Oberfläche nachgebildet werden, die über alle Flächen segmente sowohl stetig als auch stetig differenzierbar ist. Die Gesamtheit aller ermittelter Po lynome wird als bikubische Splinefunktion bezeichnet [64], Gesucht wird eine die Stützpunkte (x;,yj) interpolierende Splinefläche f(x,y) mit den Teil funktionen [64] fij(x,y) = Z;Eaijki(x-xi)k(y-yj) 1 Gl. 3 ...
Eine oftmals hohe Variantenvielfalt, geringe Produktlebenszeiten und eine hohe Produktqualität kennzeichnen gegenwärtig die Situation der produzierenden Unternehmen in der Bundesrepublik Deutschland. Vor dem Hintergrund der hohen Lohnkosten bestehen keinerlei Alternativen zu einer weiteren, flexiblen Automatisierung, um Qualitäts- und Produktivitätssteigerungen zu erzielen. Gerade Industrieroboter, die eine flexible Anpassung an wechselnde Fertigungsbedingungen ermöglichen und außerdem ein hohes Automatisierungspotential besitzen, können dazu einen wichtigen Beitrag leisten. Letztendlich entscheiden aber nicht alleine die Fähigkeiten des Industrieroboters über den Erfolg oder Mißerfolg von Automatisierungsbestrebungen, sondern die Roboterperipherie und Programmierwerkzeuge tragen wesentlich dazu bei. Insbesondere bei der Integration von externen Sensoren an Roboteranlagen bestehen weitreichende Defizite, die den Einsatz von Industrierobotern in einigen Bereichen einschränken, wenn nicht ganz verhindern. So existieren bisher nur eingeschränkte Möglichkeiten zur flexiblen und schnellen Integration externer Sensordaten, um die Bewegungsbahn von Industrierobotern an die aktuellen Form- und Lagetoleranzen des Werkstück anzupassen. Im Rahmen der vorliegenden Arbeit wird deshalb ein Konzept entwickelt und realisiert, daß eine konsequente Unterstützung sowohl bei der Auswahl von qualifizierten Sensoren als auch deren schnelle und konfigurierbare Integration in eine Robotersteuerung beinhaltet. Hauptziel ist der Entwurf eines integrierten Gesamtsystems, das den Anwender insbesondere beim Entwurf und der Durchführung einer schnellen Sensorregelung eines Industrieroboters weitreichend und effizient unterstützt. Verschiedene Sensorregelungs- und Integrationskonzepte werden hinsichtlich ihres dynamischen Systemverhaltens untersucht, um Aussagen zur Leistungsfähigkeit von Sensorregelungen bei Bearbeitungsverfahren zu erlangen. Zunächst wird die Situation der Sensorintegration an heutigen, robotergestützten Fertigungsanlagen analysiert und die Notwendigkeit und deren technologisch-ökonomischer Nutzen beleuchtet. Die bekannten Strategien zur Sensorintegration und Sensorregelung lassen sich systematisch strukturieren und bewerten. Eine Diskussion der Defizite bestehender Lösungen runden die Darstellung ab. In einem 2. Schritt werden die Anforderungen an ein konfigurierbares, integriertes Gesamtsystem abgeleitet. Es gilt dabei insbesondere Fragen zu technologischen Anforderungen zu analysieren und zu strukturieren. Die technologischen Zieldefinitionen lassen sich auf die entsprechenden Komponenten eines Gesamtsystems projizieren und bilden die Basis zur Definition des Anforderungsprofils. Diese Überlegungen münden in das Konzept des integrierten Gesamtsystems ISRS, das den Ausfuhrenden bei der Programmerstellung, dem Entwurf von Sensorregelungen und dem Test von Sensorintegrationslösungen weitgehend unterstützt. Der Kern des Gesamtsystems ISRS stellt die flexibel parametrierbare Sensorsteuerung SC dar, die maßgeblich die Sensor Wertverarbeitung, die Übertragung von Sensordaten zur Robotersteuerung und die Kommunikationssteuerung ausfuhrt, sowie weitgehend von der Robotersteuerung aus bedienbar ist. Ein speziell auf die Belange der schnellen Sensorregelung optimiertes Transputersystem dient zur optionalen Erweiterung der Sensorsteuerung SC für den Fall der schnellen Sensorintegration im Lageregeltakt der Robotersteuerung. Der Robotersimulator RC-SIM, eine weitere Komponente des integrierten Gesamtsystems ISRS, erlaubt zudem eine Emulation der Robotersteuerung und deren Kommunikationsfunktionalität, um Sensorprogramme offline ohne die Gefahr einer Beschädigung der Anlage zu testen. Auf der Basis des integrierten Gesamtsystems werden Untersuchungen, die eine Adaption der Bewegungsbahn des Roboters mit Hilfe unterschiedlicher Sensorregelungs- und Integrationskonzepte zum Ziel haben, durchgeführt. Verschiedene Sensorregler können dabei das dynamische Systemverhalten der Sensorregelung im Interpolationstakt verbessern, jedoch wird das dynamische Verhalten von einer totzeitdominanten Regelstrecke geprägt. Im Gegensatz dazu läßt sich mit einfachen Reglern bei der Sensorregelung im Lageregeltakt ein sehr gutes dynamisches Verhalten erzielen, das nur mehr durch die verbleibende Dynamik des Roboterantriebs begrenzt ist. Die resultierenden Schleppfehler bei unterschiedlichen Regelungs- und Integrationsstrategien werden analytisch wie praktisch ermittelt und ein Bezug zu den entsprechenden Anforderungen der Bearbeitungstechnologien hergestellt. Das Gesamtsystem wird zudem um die Fähigkeit zur Integration und Fusion multisensorieller Daten erweitert. Eine Analyse der Defizite bestehender Systeme mündet in ein Konzept, daß eine flexible und je nach den bestehenden Erfordernissen frei konfigurierbare Integration von multisensoriellen Daten innerhalb des integrierten Gesamtsystems erlaubt. An Hand einer exemplarischen Anwendung kann der Nachweis erbracht werden, daß sich das Konzept auch für sensorgeregelte Applikationen eignet. Vor der eigentlichen Integration von Sensoren in eine Roboteranlage müssen zunächst in einem mühsamen und zeitaufwendigen Prozeß qualifizierte Sensoren ausgewählt werden. Eine Analyse des Auswahlvorgangs und der bestehenden Defizite münden in ein Konzept für eine strukturierte Auswahlsystematik. Auf der Basis der Auswahlsystematik wird ein rechnergestütztes Planungssystem entworfen, das den Ausführenden bei Auswahl eines geeigneten Sensors unterstützt.
... Kirbach et al. (2015), Pierron et al. (2007)) or B-splines can be used for this purpose. Basics of B-spline approximation can be found in (Hoschek & Lasser, 1992) and a selection of application examples are given in (Lehmann et al., 2018;Peretzki et al., 2022;Kanzenbach et al., 2022). Using an approximation-based method, strains are easily determined by derivatives and smoothing is achieved by the continuous mathematical description due to the functions (in most cases the displacement). ...
... The polynomial degree can be set higher than necessary without directly noticing the disadvantages. Especially for numerical methods, lower polynomial degrees are often easier and more stable to handle [17]. In general, the lowest possible polynomial degree should always be chosen with regard to the necessary accuracy. ...
An essential aspect in the whole engine modelling of modern aircraft engines is the preliminary design of the meridional gas path definition. Especially the annulus dimensioning and topological characterisation of engine core subsystems are usually time-consuming. This paper presents a holistic and parametric approach to pre-design the geometric 2D annulus contours of compressor and turbine subsystems without complex thermody-namic calculations. It is based on distribution functions of the axial and radial dimensions, as well as the variations of compressor and turbine stages and realises a time-efficient re-dimensioning and topological variation of the complex annulus geometry. Parametric methods for an optimised arrangement of the blade stages in a turbomachinery subsystem configuration are shown and integrated into a mathematical model.
... From a geometric point of view, the most natural choice for Γ is an offset to Γ . Offset curves are defined as the locus of points that are at a constant distance d along the normal vector from the so-called progenitor curve [42,43], and there are several techniques in the literature to generate them; see e.g. [44,45]. ...
We present a novel isogeometric method, namely the Immersed Boundary-Conformal Method (IBCM), that features a layer of discretization conformal to the boundary while employing a simple background mesh for the remaining domain. In this manner, we leverage the geometric flexibility of the immersed boundary method with the advantages of a conformal discretization, such as intuitive control of mesh resolution around the boundary, higher accuracy per degree of freedom, automatic satisfaction of interface kinematic conditions, and the ability to strongly impose Dirichlet boundary conditions. In the proposed method, starting with a boundary representation of a geometric model, we extrude it to obtain a corresponding conformal layer. Next, a given background B-spline mesh is cut with the conformal layer, leading to two disconnected regions: an exterior region and an interior region. Depending on the problem of interest, one of the two regions is selected to be coupled with the conformal layer through Nitsche’s method. Such a construction involves Boolean operations such as difference and union, which therefore require proper stabilization to deal with arbitrarily cut elements. In this regard, we follow our precedent work called the minimal stabilization method (Antolin et al in SIAM J Sci Comput 43(1):A330–A354, 2021). In the end, we solve several 2D benchmark problems to demonstrate improved accuracy and expected convergence with IBCM. Two applications that involve complex geometries are also studied to show the potential of IBCM, including a spanner model and a fiber-reinforced composite model. Moreover, we demonstrate the effectiveness of IBCM in an application that exhibits boundary-layer phenomena.
... Franke (1982) provides a brief review and classification of 32 algorithms. Hoschek and Lasser (1992) give a comprehensive discussion of theories in geometrical data processing and extensive references in the area of data interpolation and Akin and Siemes (1988) and Davis (1973) provide necessary mathematical background skills on the statistics and data analysis in geology. ...
... From a geometric point of view, the most natural choice forΓ is an offset to Γ . Offset curves are defined as the locus of points that are at a constant distance d along the normal vector from the so-called progenitor curve [30,31], and there are several techniques in the literature to generate them, see e.g. [32,33]. ...
We present a novel isogeometric method, namely the Immersed Boundary-Conformal Method, that features a layer of discretization conformal to the boundary while employing a simple background mesh for the remaining domain. In this manner, we leverage the geometric flexibility of the immersed boundary method with the advantages of a conformal discretization, such as intuitive control of mesh resolution around the boundary, higher accuracy per degree of freedom, automatic satisfaction of interface kinematic conditions, and the ability to strongly impose Dirichlet boundary conditions. In the proposed method, starting with a boundary representation of a geometric model, we extrude it to obtain a corresponding conformal layer. Next, a given background B-spline mesh is cut with the conformal layer, leading to two disconnected regions: an exterior region and an interior region. Depending on the problem of interest, one of the two regions is selected to be coupled with the conformal layer through Nitsche's method. Such a construction involves Boolean operations such as difference and union, which therefore require proper stabilization to deal with arbitrarily cut elements. In this regard, we follow our precedent work called the minimal stabilization method [1]. In the end, we solve several 2D benchmark problems to demonstrate improved accuracy and expected convergence with the proposed method. Moreover, we present two applications that involve complex geometries to show the potential of the proposed method, including a spanner model and a fiber-reinforced composite model.
... The idea of geometric planar spline fitting, i.e., fitting splines based on clothoids as building blocks can be traced through the literature for more than 50 years [BdB65,Sto82,Meh74,MW92,HL92,Coo93]. Further recent contributions are for instance [BF18] presenting an alternative to [Sto82] for the computation of a C 2 interpolating clothoid spline. ...
We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To define clothoidal averaging, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present results produced by the proposed schemes and discuss their features. In particular, we demonstrate that the proposed schemes yield visually convincing curves.
... As a result, the multiple continuous segments of a vehicle path must have the same tangents in the connecting points. This requirement ensures a G 1 curve, as the resulting curve will be continuously differentiable at least once [11]. ...
When operating automated guided vehicles (AGVs) in a logistics or production environment, they are likely to face unknown obstacles, e.g. logistics containers or human personnel. In order to ensure a high productivity of the AGVs, instead of stopping, they are supposed to circumvent such obstacles if possible. At the same time, the jerk in the circumventing movement must be kept small to reduce wear on the drive system and to ensure that the AGV’s load does not become disordered or damaged. Circumventing obstacles is done using local path planning. Local path planning aims to find a short-term path planning solution in an unknown environment.
However, common algorithms for local path planning, like vector field histogram or dynamic window, generate a steering command for the next interval without respect to smoothness of the path. If the generated paths are smoothed, this happens by statistically reducing jitter in the results. Such a smoothing method does not take the jerk into account.
This paper shows that paths of G² differentiability are needed to reduce jerk on a path planning level. A path planning concept for the vector field histogram and for dynamic window and an evaluation method for this concept are shown.
... Franke and Nielson 1980;Franke 1982) and textbooks (e.g. Hoschek and Lasser 1992). Besides the wellknown polynomial and spline interpolation, also radial basis function-based methods, such as kriging and linear prediction interpolation, are used to solve different geodetic interpolations problems. ...
The thin plate spline (TPS) is an interpolation approach that has been developed to investigate a frequently occurring problem in geosciences: the modelling of scattered data. In this paper, we carry over the concept of the thin plate spline from the plane to the sphere. To develop the spherical TPS, we utilize the idea of an elastic shell that is attributed with the bending energy and the external energy. The bending energy describes the shape of the membrane, while the external energy reflects deviations between the shell and the data to be modelled. Minimizing both energy terms leads to the variational problem with the solution in the form of the Euler–Lagrange equations. We provide the solution of the variational problem for two cases: (1) total energy minimization over the whole sphere and (2) total energy minimization over a closed region of the sphere. In case (1) we found a closed analytical solution in the form of collocation in a reproducing kernel Hilbert space. The local case (2) solution is based on a discretization of the corresponding Euler–Lagrange equation using the spherical Laplace operator. The performance of the introduced spherical TPS is demonstrated on two real world data sets. It is shown quantitative that the thin plate approach is significantly more effective than Gaussian filter in terms of the GRACE data de-striping. We also show that the TPS can be used effectively for the modelling of the vertical total electron content. It allows the reduction of the computational effort in comparison with well-established planar TPS approximation. Moreover, the harmonicity property of the TPS can be utilized to solve various issues related to Earth gravity modelling.
... B-Spline volumes are defined just like B-Spline surfaces [HL92,Far90], except that the dimension of the domain is the same as that of the image. In our case, since we are looking for an interpolating function f : R 3 → R 3 , we define a tensor product B-spline volume as ...
Chapter 2 presents an object-oriented framework for describing (authoring) virtual environments (VEs). After reviewing briefly some related approaches, a new paradigm is developed. This paradigm allows to describe VEs without programming and it is intuitive enough to be used by non-programmers. In ad- dition, it facilitates specialized high-level graphical user-interfaces for building applications in special application domains. Furthermore, it has been designed with multi-user VR in mind. This framework has proven to be suitable, pow- erful, and flexible; all VR applications for manufacturing customers have been built on top of this framework.
Chapter 3 presents several algorithms for detecting collisions of polygo- nal objects. This is the major bottleneck of many simulations, in particular physically-based simulations. In order to achieve real-time behavior of objects, this problem must be solved. This work presents several algorithms for sev- eral classes of objects, which are among the fastest algorithms known today.6 Algorithms for finding quickly pairs of objects possibly intersecting are pre- sented. Several ways of parallelizing collision detection algorithms have been implemented and are evaluated. Finally, a collision detection pipeline is devel- oped comprising all the algorithms presented earlier. In addition, issues such as robustness, concurrency and other implementation issues are discussed.
Chapter 4 deals with various issues relating to user interaction. A frame- work for integration of input devices is presented and lessons learnt using and working with it are discussed. One of the problems of electro-magnetic track- ing (a common tracking technique) is noise. In order to reduce this I present a filtering pipeline which have been implemented in our VR system. Another problem with electro-magnetic tracking is distortion which leads to warped im- ages and can cause interaction difficulties in precision tasks. A simple and fast method to correct these distortions is provided. The precision of this method is evaluated both with real data and with mathematical experiments. In addition, the amount of distortion of different tracking systems has been evaluated. In the second part of this chapter, a framework for navigation is presented (which has been implemented in the VR system), as well as a framework for the user’s head. The discussion is focused in particular on practical issues of interaction paradigms.
Finally, Chapter 5 describes several applications, which have been built on top of the frameworks and algorithms presented earlier. They prove the use- fulness, flexibility, and power of the algorithms and frameworks developed in this thesis, helping to make VR more practicable than it used to be.
This paper deals with a method for strain determination around boreholes in plastic specimens in the context of residual stress analyses. In contrast to the frequently used hole drilling method with strain gauges, the strains are determined using digital image correlation (DIC). In the investigations presented, load stresses occur instead of residual stresses, which is caused by defined loading conditions. Furthermore, an adaption of the hole drilling method is developed, called staggered machining, which includes the successive drilling of a larger and a smaller borehole to obtain deformation data in larger depths compared to the conventional approach. For strain evaluation, a special method is used with parametrisation of an annulus around the borehole. In this region, B‐spline approximation of the DIC displacements is carried out using an optimisation procedure of the smoothing parameter. The tests of the entire method show very plausible and precise results. Furthermore, a comparison with simulation demonstrates good correlation of the strain distributions.
Interpolating unstructured data using barycentric coordinates becomes infeasible at high dimensions due to the prohibitive memory requirements of building a Delaunay triangulation. We present a new algorithm to construct ad-hoc simplices that are empirically guaranteed to contain the target coordinates, based on a nearest neighbor heuristic and an iterative dimensionality reduction through projection. We use these simplices to interpolate the astrophysical cooling function Λ and show that this new approach produces good results with just a fraction of the previously required memory.
Bézier curves and surfaces are important to computer-aided design applications. This paper presents algorithms for checking the injectivity of 2D and 3D Bézier curves. An injective Bézier curve or surface is one that has no self-intersections. The proposed algorithms use recently proposed sufficient and necessary conditions under which Bézier curves are guaranteed to be non-self-intersecting. As well as a rigorous derivation of the proposed algorithms, we present a series of examples and derive the complexity and computation times of the proposed algorithms. We find that the complexity our algorithms is approximately O(m) , representing an improvement over previous injectivity-checking algorithms.
In an NC controller, the processing of the geometrical data plays a key role as this processing converts the traversing information, which is specified in the form of NC programs, to machining motions. As well as generating a path according to the programmed data, this task also involves monitoring functions and taking corrections into account. Corrections and monitoring functions must be performed online. It is not possible to carry out a precalculation offline.
We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.
Der Führungsgrößenerzeugung bzw. Interpolation von NC-Steuerungen kommt mit der Verarbeitung von geometrischen Daten zur Realisierung eines Verfahrwegs bei der Fertigung eines Produkts eine zentrale Schlüsselrolle zu. Diese Schlüsselrolle definiert sich im Kern über die Generierung von vorgegebenen Lage-, Geschwindigkeits- und Beschleunigungswerten zur Bearbeitung von Werkstücken mit numerisch gesteuerten Werkzeugmaschinen bei Einhaltung vorgegebener Randbedingungen wie tolerierte Bahnabweichungen, maschinenkinematische Transformationen sowie Bahnkorrekturen. Des Weiteren werden Korrektur- und Überwachungsfunktionen in den heutigen Funktionsumfang mit eingeschlossen, wodurch eine echtzeitfähige Datenverarbeitung erforderlich ist.
In motion control, the generation of motion profiles for non-linear kinematics is usually computationally complex. In order to minimize the workload on the machine’s control system, the approach pursued is outsourcing complex calculation tasks to the offline area. In this offline motion preparation, predefined criteria have to be taken into account to guarantee process stability on the real machine. During the motion preparation, a high performance is desired, characterized by less data generated and at the same time little computing effort. The evaluation will use the example of a motion specification, which is characterized by a large amount of data compared to conventional motion specifications. Thus, the demands on performance become even higher. This paper examines the performance of different motion preparation approaches known from literature. On the one hand, selected spline-based algorithms are discussed and compared. A recursive algorithm based on monomial splines is recommended for use in the example. On the other hand, a very simple approach based on the linearization of the non-linear workspace of the mechanism is presented and applied on the algorithms. With this, the performance increased significantly again.
In diesem Kapitel wird die Implementierung einer CAD/CAM-Prozesskette beschrieben. Dies beinhaltet einen Überblick über CAx-Systeme, welche beim rechnerunterstützten Konstruieren und Fertigen zur Anwendung kommen. Ausgangspunkt für die CAD/CAM-Kopplung bildet ein rechnerinternes (3D-)CAD-Modell, dessen Möglichkeiten der Modellierung bzw. rechnerinternen Abspeicherung genauer betrachtet werden.
The procedure of positioning of pipes, for example for the automobile exhaust construction, by the use of 3D CAD systems is difficult and time consuming because of obstacles in an engine compartment. There are also further strong technical requirements and restrictions. An automatic generation procedure based on the level set method and the numerical solution of the Eikonal equation is proposed. The positions of pipes which satisfy the technical requirements are obtained using spline functions.
Wichtige Aufgaben im Zuge des Produktentstehungsprozesses bestehen darin, durch Modelle bestimmte Fragen, insbesondere zur Absicherung der Eigenschaften neuer Produkte, zu beantworten. Dazu steht heute eine breite Palette an CAx-Systemen und –Anwendungen zur Verfügung. Zu ihnen gehören unter anderen Systeme und Anwendungen für CAD, CAE, CAM, CAP, CAQ wie auch zur Verformungs- und Spannungsanalyse auf Basis der Methode der Finiten Elemente (FEM, Kap. 6), zur Modellierung des dynamischen Verhaltens von Maschinen als Mehrkörpersysteme (MKS, Kap. 7) oder weitere Modellierungstechniken (Kap. 8), wie beispielsweise zur Untersuchung von Strömungs- und Wärmeübertragungsprozessen durch CFD (Computational Fluid Dynamics, Abschn. 8.2) und zur Bauteiloptimierung (Kap. 9).
Kurzfassung
Die Simulation von Zerspanprozessen erfordert Modelle für das Werkzeug und Werkstück, die den jeweiligen Genauigkeitsanforderungen an die Simulationsergebnisse gerecht werden. Für die Fräs- und Schleifbearbeitung von Freiformflächen haben sich zwei Modellierungsansätze, das Multidexelvolumenmodell und das Versatzvektorfeld, als geeignet erwiesen. Zur Schließung der simulativen Prozesskette können diese Modelle ineinander überführt werden.
Harmonic maps are a certain kind of an optimal map projection which has been developed for map projections of the sphere. Here we generalize it to the “ellipsoid of revolution”. The subject of an optimization of a map projection is not new for a cartographer. For instance, in Sect. 5-25, we compute the minimum distortion energy for mapping the “sphere-to-plane”.
Conventionally, conformal coordinates, also called conformal charts, representing the surface of the Earth or any other Planet as an ellipsoid-of-revolution, also called the Geodetic Reference Figure, are generated by a two-step procedure. First, conformal coordinates (isometric coordinates, isothermal coordinates) of type UMP (Universal Mercator Projection, compare with Example 15.1) or of type UPS (Universal Polar Stereographic Projection, compare with Example 15.2) are derived from geodetic coordinates such as surface normal ellipsoidal longitude/ellipsoidal latitude. UMP is classified as a conformal mapping on a circular cylinder, while UPS refers to a conformal mapping onto a polar tangential plane with respect to an ellipsoid-of-revolution, an azimuthal mapping.
In this chapter, we present a collection of most widely used map projections in the polar aspect in which meridians are shown as a set of equidistant parallel straight lines and parallel circles (parallels) by a system of parallel straight lines orthogonally crossing the images of the meridians. As a specialty, the poles are not displayed as points but straight lines as long as the equator. First, we derive the general mapping equations for both cases of (i) a tangent cylinder and (ii) a secant cylinder and describe the construction principle.
At the beginning of this chapter, let us briefly refer to Chap. 8, where the data of the best fitting “ellipsoid-of-revolution to Earth” are derived in form of a table. Here, we specialize on the mapping equations and the distortion measures for mapping an ellipsoid-of-revolution to a cylinder, equidistant on the equator. Section 14-1 concentrates on the structure of the mapping equations, while Sect. 14-2 gives special cylindric mappings of the ellipsoid-of-revolution, equidistant on the equator. At the end, we shortly review in Sect. 14-3 the general mapping equations of a rotationally symmetric figure different from an ellipsoid-of-revolution, namely the torus.
A special mapping, which was invented by Gauss (1822, 1844), is the double projection of the ellipsoid-of-revolution to the sphere and from the sphere to the plane. These are conformal mappings. A very efficient compiler version of the Gauss double projection was presented by Rosenmund (1903) (ROM mapping equations) and applied for mapping Switzerland and the Netherlands, for example. An alternative mapping, called “authalic”, is equal area, first ellipsoid-of-revolution to sphere, and second sphere to plane.
In the world of conformal mappings of the Earth or other celestial bodies, the Mercator projection plays a central role. The Mercator projection of the sphere or of the ellipsoid-of-revolution beside conformality is characterized by the equidistant mapping of the equator. In contrast, the transverse Mercator projection is conformal and maps the transverse meta-equator, the meridian of reference, equidistantly. Accordingly, the Mercator projection is very well suited for regions which extend East–West around the equator, while the transverse Mercator projection fits well to those regions which have a South–North extension. Obviously, several geographical regions are centered along lines which are neither equatorial, parallel circles, or meridians, but may be taken as central intersection of a plane and the reference figure of the Earth or other celestial bodies, the ellipsoid-of-revolution (spheroid).
Up to now, we treated various mappings of the ellipsoid and the sphere, for instance of type conformal, equidistant, or equal areal or perspective and geodetic.
Among cylindrical projections, mappings in the transverse aspect play the most important role. Although many worldwide adopted legal map projections use the ellipsoid-of-revolution as the reference figure for the Earth, the spherical variant forms the basis for the Universal Transverse Mercator (UTM) grid and projection. In the subsequent chapter, we first introduce the general concept of a cylindrical projection in the transverse aspect. Following this, three special map projections are presented: (i) the equidistant mapping (transverse Plate Carrée projection), (ii) the conformal mapping (transverse Mercator projection), and (iii) the equal area mapping (transverse Lambert projection). The transverse Mercator projection is especially appropriate for regions with a predominant North-South extent. As in previous chapters, the two possible cases of a tangent and a secant cylinder are treated simultaneously by introducing the meta-latitude B = ±B1 of a meta-parallel circle which is mapped equidistantly. For a first impression, have a look at Fig. 11.1.
In Chap. 21, we already transformed from a global three- dimensional geodetic network into a regional or local geodetic network. We aimed at the analysis of datum parameters, namely seven parameters of type translation, rotation and scale, as elements of the global conformal group C7(3).
Cylindrical projections in the oblique aspect are mainly used to display regions which have a predominant extent in the oblique direction, neither East-West nor North-South. In addition, they form the most general cylindrical projections because mapping equations for projections in the polar and the transverse aspect can easily be derived from it. This is done by setting the corresponding latitude of the meta-North Pole Φ
0 to a specific value: Φ
0 = 90∘ generates cylindrical projections in the polar aspect, Φ
0 = 0∘ result in cylindrical projections in the transverse aspect. As an introductory part, we present the equations for general cylindrical mappings together with the equations for the principal stretches, before derivations for specific cylindrical map projections of the sphere (oblique equidistant projection, oblique conformal projection and oblique equal area projection) are given. For a first impression, have a look at Fig. 12.1.
Mapping the ellipsoid-of-revolution to a tangential plane. Azimuthal projections in the normal aspect (polar aspect): equidistant, conformal, equiareal, and perspective mapping.
Mapping the sphere to a tangential plane: meta-azimuthal projections in the oblique aspect. Equidistant, conformal (oblique UPS), and equal area (oblique Lambert) mappings.
One of the key challenges faced by engineers is finding, concretizing, and optimizing solutions for a specific technical problem in the context of requirements and constraints (Pahl et al. 2007). Depending on the technical problem’s nature, specifically designed products and processes can be its solution with product and processes depending on each other. Although products are usually modeled within the context of their function, consideration of the product’s life cycle processes is also essential for design. Processes of the product’s life cycle concern realization of the product (e.g., manufacturing processes), processes that are realized with the help of the product itself (e.g., use processes) and processes at the end of the product’s life cycle (recycling or disposal). Yet, not just product requirements have to be considered during product development, as requirements regarding product life cycle processes need to be taken into account, too. Provision for manufacturing process requirements plays an important role in realizing the product’s manufacturability, quality, costs, and availability (Chap. 3). Further life cycle demands, such as reliability, durability, robustness, and safety, result in additional product and life cycle process requirements. Consequently, the engineer’s task of finding optimal product and process solutions to solve a technical problem or to fulfill a customer need is characterized by high complexity, which has to be handled appropriately (Chaps. 5 and 6).
Anfangs wird die Stellung der Freiformflächen in der Prozesskette Karosseriebau kurz beleuchtet. Um die CAD-Werkzeuge für diese Flächen zweckentsprechend einsetzen zu können, wird das notwendige Minimum an theoretischem Grundwissen zu Kurven und Flächen (basierend auf [14] und [15]) vermittelt. Die Vor- und Nachteile der mathematischen Ansätze (Bézier und B-Spline) werden gegenübergestellt, damit der CAD-Nutzer je nach Anwendungsgebiet selbst den richtigen Ansatz auswählen kann. Für tiefere Studien wird auf die entsprechende mathematische Literatur verwiesen. Weitere Unterabschnitte behandeln die Oberflächenklassifizierung, – gebräuchlicherweise mit Classes A, B und C englisch bezeichnet – und bewährte Regeln und Empfehlungen zum so genannten Straken im Falle von Class A-Flächen. An einem Beispiel wird aufzeigt, wie man importierte B-Spline-Geometrien erforderlichenfalls (zwecks Qualitätsverbesserung) durch stückweise Konvertierung in zusammengesetzte Béziergeometrien verwandeln kann.
POMOS — POint based Modelling System — developed in the Research Centre Karlsruhe is designed to close the gap between measuring devices and following CAX-applications. The capabilities of POMOS are focused on point operations, surface generation, surface analysis and interfaces from and to neutral file formats.
POMOS is not intended to replace a CAD-system but to be an add-on. Therefore typical CAD-functions such as fillets, intersections, projections etc. are not realised in the system.
POMOS is able to handle a large amount of digitised point data. This point data can be manipulated, modified, analysed and sorted. By defining three or more borders, the system is able to approximate free form surfaces. These generated surfaces can then be fitted together with either an edge or a smooth joint.
Before exporting the generated surfaces, the quality of these surfaces can be analysed using point-to-surface deviation, cutting lines, isophotes, reflection lines and curvature analysis.
Methods for surface design based on certain desired properties will be presented. Working in this way enable CAD-users to entirely concentrate on prescribing properties, leaving the task to fulfil them to the computer. In this way, the operator may almost instantaneously attain design solutions than would takes days or weeks with traditional means — if reached at all.
A user may, however, prescribe properties for which there does not exist any physically realisable surface having them, as will be illustrated. From a mathematical point, this is to be expected, since most styling properties gives rise to nonlinear partial differential equations that in general, does not have smooth solutions. It is shown that for several visual- and curvature properties, there are unique perturbation solutions admitting a user, adhering to a few basic principles, to work rather freely.
SYRKO is the CAD/CAM system developed and used by Mercedes-Benz for car body design and construction. The name is derived from “System für rechnerunterstützte Konstruktion und Fertigung”, which simply means “CAD/CAM system”. Important stages in the growth of the system were
1967
Batch programs for curve and surface design
1972
CAD representation and NC milling of a test car model
1976
Mainframe version on vector screen with interactive user interface
1980
SYRKO used in the design-, preproduction- and tool design departments in Sindelfingen plant
1985
Colour raster screen
1986
SYRKO used in Wörth and Bremen plants 1990 SYRKO on UNIX workstations
Nowadays the body of a new car is almost completely described as a SYRKO surface model. The system is used in 5 plants of Mercedes-Benz as well as by a few supplying companies.
Mathematik ist die Wissenschaft vom Formalisieren; sie widmet sich der Untersuchung und Charakterisierung der hierdurch definierbaren Kalküle, Räume und Strukturen, entwickelt Beweis- und Rechenmethoden und leitet gültige Eigenschaften her. Hierbei haben sich prinzipielle Grenzen herausgestellt: Die Widerspruchsfreiheit läßt sich i. allg. nicht innerhalb von Kalkülen beweisen, viele Probleme sind mit Rechenmethoden unlösbar oder nicht effizient lösbar, die meisten nichtlinearen Systeme besitzen keine Lösung in knapper geschlossener mathematischer Darstellung usw. Solche Grenzen gelten auch für die Informatik, die sich mit maschinell bearbeitbaren Lösungsverfahren (also mit Algorithmen), ihren Darstellungen, ihren Eigenschaften, ihrer Realisierung und ihren Anwendungen befaßt. Bis in die 60er Jahre wurde die Informatik noch als ein Bereich aufgefaßt, der sich in die Mathematik eingliedern läßt. In der Tat entwickelt die Informatik (wie die Mathematik) grundlegende, meist formale Methoden und Techniken, die in anderen Wissenschaften benötigt und in immer stärkerem Maße dort eingesetzt werden. Doch ebenso, wie die Ingenieurwissenschaften nicht mehr als Teil der Naturwissenschaften angesehen werden, haben die Probleme bei der Realisierung und beim Einsatz und die in großen Mengen entwickelten Systeme imd Werkzeuge aus der Informatik eine eigenständige, seit 1970 vorwiegend ingenieurwissenschaftlich arbeitende Disziplin gemacht, die sich aber nicht mit den Grundstoffen „Materie“ und „Energie“, sondern mit der „Information“ beschäftigt. Da auf der Information persönliche und gesellschaftliche Entscheidungs- und Lernprozesse beruhen, wirkt die Informatik nachhaltig in fast alle Bereiche des menschlichen Lebens und Zusammenlebens hinein.
Mathematik ist die Wissenschaft vom Formalisieren; sie widmet sich der Untersuchung und Charakterisierung der hierdurch definierbaren Kalküle, Räume und Strukturen, entwickelt Beweis- und Rechenmethoden und leitet gültige Eigenschaften her. Hierbei haben sich prinzipielle Grenzen herausgestellt: Die Widerspruchsfreiheit läßt sich i.allg. nicht innerhalb von Kalkülen beweisen, viele Probleme sind mit Rechenmethoden unlösbar oder nicht effizient lösbar, die meisten nichtlinearen Systeme besitzen keine Lösung in knapper geschlossener mathematischer Darstellung usw. Solche Grenzen gelten auch für die Informatik, die sich mit maschinell bearbeitbaren Lösungsverfahren (also mit Algorithmen), ihren Darstellungen, ihren Eigenschaften, ihrer Realisierung und ihren Anwendungen befaßt. Bis in die 60er Jahre wurde die Informatik noch als ein Bereich aufgefaßt, der sich in die Mathematik eingliedern läßt. In der Tat entwickelt die Informatik (wie die Mathematik) grundlegende, meist formale Methoden und Techniken, die in anderen Wissenschaften benötigt und in immer stärkerem Maße dort eingesetzt werden. Doch ebenso, wie die Ingenieurwissenschaften nicht mehr als Teil der Naturwissenschaften angesehen werden, haben die Probleme bei der Realisierung und beim Einsatz und die in großen Mengen entwickelten Systeme und Werkzeuge aus der Informatik eine eigenständige, seit 1970 vorwiegend ingenieurwissenschaftlich arbeitende Disziplin gemacht, die sich aber nicht mit den Grundstoffen „Materie“ und „Energie“, sondern mit der „Information“ beschäftigt. Da auf der Information persönliche und gesellschaftliche Entscheidungs- und Lernprozesse beruhen, wirkt die Informatik nachhaltig in fast alle Bereiche des menschlichen Lebens und Zusammenlebens hinein.
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