Reverse engineering of CAD models via clustering and approximate implicitization

In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse engineering. In this work we develop a novel method to determine these primitive shapes by combining clustering analysis with approximate implicitization. The proposed method is automatic and can recover algebraic hypersurfaces of any degree in any dimension. In exact arithmetic, the algorithm returns exact results. All the required parameters, such as the implicit degree of the patches and the number of clusters of the model, are inferred using numerical approaches in order to obtain an algorithm that requires as little manual input as possible. The effectiveness, efficiency and robustness of the method are shown both in a theoretical analysis and in numerical examples implemented in Python

Changing representation of curves and surfaces: exact and approximate methods

Το κύριο αντικείμενο μελέτης στην παρούσα διατριβή είναι η αλλαγή αναπαράστασης γεωμετρικών αντικειμένων από παραμετρική σε αλγεβρική (ή πεπλεγμένη) μορφή. Υπολογίζουμε την αλγεβρική εξίσωση παρεμβάλλοντας τους άγνωστους συντελεστές του πολυωνύμου δεδομένου ενός υπερσυνόλου των μονωνύμων του. Το τελευταίο υπολογίζεται απο το Newton πολύτοπο της αλγεβρικής εξίσωσης που υπολογίζεται από μια πρόσφατη μέθοδο πρόβλεψης του συνόλου στήριξης της εξίσωσης. H μέθοδος πρόβλεψης του συνόλου στήριξης βασίζεται στην αραιή (ή τορική) απαλοιφή: το πολύτοπο υπολογίζεται από το Newton πολύτοπο της αραιής απαλοίφουσας αν θεωρίσουμε την παραμετροποίηση ως πολυωνυμικό σύστημα. Στα μονώνυμα που αντιστοιχούν στα ακέραια σημεία του Newton πολυτόπου δίνονται τιμές ώστε να σχηματίσουν έναν αριθμητικό πίνακα. Ο πυρήνα του πίνακα αυτού, διάστασης 1 σε ιδανική περίπτωση, περιέχει τους συντελεστές των μονωνύμων στην αλγεβρική εξίσωση. Υπολογίζουμε τον πυρήνα του πίνακα είτε συμβολικά είτε αριθμητικά εφαρμόζοντας την μέθοδο του singular value decomposition (SVD). Προτείνουμε τεχνικές για να διαχειριστούμε την περίπτωση ενός πολυδιάστατου πυρήνα το οποίο εμφανίζεται όταν το προβλεπόμενο σύνολο στήριξης είναι ένα υπερσύνολο του πραγματικού. Αυτό δίνει έναν αποτελεσματικό ευαίσθητο-εξόδου αλγόριθμο υπολογισμού της αλγεβρικής εξίσωσης. Συγκρίνουμε διαφορετικές προσεγγίσεις κατασκευής του πίνακα μέσω των λογισμικών Maple και SAGE. Στα πειράματά μας χρησιμοποιήθηκαν ρητές καμπύλες και επιφάνειες καθώς και NURBS. Η μέθοδός μας μπορεί να εφαρμοστεί σε πολυώνυμα ή ρητές παραμετροποιήσεις επίπεδων καμπυλών ή (υπερ)επιφανειών οποιασδήποτε διάστασης συμπεριλαμβανομένων και των περιπτώσεων με παραμετροποίηση σεσημεία βάσης που εγείρουν σημαντικά ζητήματα για άλλες μεθόδους αλγεβρικοποίησης. Η μέθοδος έχει τον εξής περιορισμό: τα γεωμετρικά αντικείμενα πρέπει να αναπαριστώνται από βάσεις μονωνύμων που στην περίπτωση τριγωνομετρικών παραμετροποιήσεων θα πρέπει να μπορούν να μετασχηματιστούν σε ρητές συναρτήσεις. Επιπλέον η τεχνική που προτείνουμε μπορεί να εφαρμοστεί σε μη γεωμετρικά προβλήματα όπως ο υπολογισμόςτης διακρίνουσας ενός πολυωνύμου με πολλές μεταβλητές ή της απαλοίφουσας ενός συστήματος πολυωνύμων με πολλές μεταβλητές.The main object of study in our dissertation is the representation change of the geometric objects from the parametric form to implicit. We compute the implicit equation interpolating the unknown coefficients of the implicit polynomial given a superset of its monomials. The latter is derived from the Newton polytope of the implicit equation obtained by the recently developed method for support prediction. The support prediction method we use relies on sparse (or toric) elimination: the implicit polytope is obtained from the Newton polytope of the sparse resultant of the system in parametrization, represented as polynomials. The monomials that correspond to the lattice points of the Newton polytope are suitably evaluated to build a numeric matrix, ideally of corank 1. Its kernel contains their coefficients in the implicit equation. We compute kernel of the matrix either symbolically, or numerically, applying singular value decomposition (SVD). We propose techniques for handling the case of the multidimensional kernel space, caused by the predicted support being a superset of the actual. This yields an efficient, output-sensitive algorithm for computing the implicit equation. We compare different approaches for constructing the matrix in Maple and SAGE software. In our experiments we have used classical algebraic curves and surfaces as well as NURBS. Our method can be applied to polynomial or rational parametrizations of planar curves or (hyper)surfaces of any dimension including cases of parameterizations with base points which raise important issues for other implicitization methods. The method has its limits: geometric objects have to be presented using monomial basis; in the case of trigonometric parametrizations they have to be convertible to rational functions. Moreover, the proposed technique can be applied for nongeometric problems such as the computation of the discriminant of a multivariate polynomial or the resultant of a system of multivariate polynomials

Parameterization adaption for 3D shape optimization in aerodynamics

When solving a PDE problem numerically, a certain mesh-refinement process is always implicit, and very classically, mesh adaptivity is a very effective means to accelerate grid convergence. Similarly, when optimizing a shape by means of an explicit geometrical representation, it is natural to seek for an analogous concept of parameterization adaptivity. We propose here an adaptive parameterization for three-dimensional optimum design in aerodynamics by using the so-called "Free-Form Deformation" approach based on 3D tensorial B\'ezier parameterization. The proposed procedure leads to efficient numerical simulations with highly reduced computational costs

Polynomial cubic splines with tension properties

In this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems

Efficient and High-Quality Rendering of Higher-Order Geometric Data Representations

Computer-Aided Design (CAD) bezeichnet den Entwurf industrieller Produkte mit Hilfe von virtuellen 3D Modellen. Ein CAD-Modell besteht aus parametrischen Kurven und Flächen, in den meisten Fällen non-uniform rational B-Splines (NURBS). Diese mathematische Beschreibung wird ebenfalls zur Analyse, Optimierung und Präsentation des Modells verwendet. In jeder dieser Entwicklungsphasen wird eine unterschiedliche visuelle Darstellung benötigt, um den entsprechenden Nutzern ein geeignetes Feedback zu geben. Designer bevorzugen beispielsweise illustrative oder realistische Darstellungen, Ingenieure benötigen eine verständliche Visualisierung der Simulationsergebnisse, während eine immersive 3D Darstellung bei einer Benutzbarkeitsanalyse oder der Designauswahl hilfreich sein kann. Die interaktive Darstellung von NURBS-Modellen und -Simulationsdaten ist jedoch aufgrund des hohen Rechenaufwandes und der eingeschränkten Hardwareunterstützung eine große Herausforderung. Diese Arbeit stellt vier neuartige Verfahren vor, welche sich mit der interaktiven Darstellung von NURBS-Modellen und Simulationensdaten befassen. Die vorgestellten Algorithmen nutzen neue Fähigkeiten aktueller Grafikkarten aus, um den Stand der Technik bezüglich Qualität, Effizienz und Darstellungsgeschwindigkeit zu verbessern. Zwei dieser Verfahren befassen sich mit der direkten Darstellung der parametrischen Beschreibung ohne Approximationen oder zeitaufwändige Vorberechnungen. Die dabei vorgestellten Datenstrukturen und Algorithmen ermöglichen die effiziente Unterteilung, Klassifizierung, Tessellierung und Darstellung getrimmter NURBS-Flächen und einen interaktiven Ray-Casting-Algorithmus für die Isoflächenvisualisierung von NURBSbasierten isogeometrischen Analysen. Die weiteren zwei Verfahren beschreiben zum einen das vielseitige Konzept der programmierbaren Transparenz für illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle und zum anderen eine neue hybride Methode zur Reprojektion halbtransparenter und undurchsichtiger Bildinformation für die Beschleunigung der Erzeugung von stereoskopischen Bildpaaren. Die beiden letztgenannten Ansätze basieren auf rasterisierter Geometrie und sind somit ebenfalls für normale Dreiecksmodelle anwendbar, wodurch die Arbeiten auch einen wichtigen Beitrag in den Bereichen der Computergrafik und der virtuellen Realität darstellen. Die Auswertung der Arbeit wurde mit großen, realen NURBS-Datensätzen durchgeführt. Die Resultate zeigen, dass die direkte Darstellung auf Grundlage der parametrischen Beschreibung mit interaktiven Bildwiederholraten und in subpixelgenauer Qualität möglich ist. Die Einführung programmierbarer Transparenz ermöglicht zudem die Umsetzung kollaborativer 3D Interaktionstechniken für die Exploration der Modelle in virtuellenUmgebungen sowie illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle. Die Erzeugung stereoskopischer Bildpaare für die interaktive Visualisierung auf 3D Displays konnte beschleunigt werden. Diese messbare Verbesserung wurde zudem im Rahmen einer Nutzerstudie als wahrnehmbar und vorteilhaft befunden.In computer-aided design (CAD), industrial products are designed using a virtual 3D model. A CAD model typically consists of curves and surfaces in a parametric representation, in most cases, non-uniform rational B-splines (NURBS). The same representation is also used for the analysis, optimization and presentation of the model. In each phase of this process, different visualizations are required to provide an appropriate user feedback. Designers work with illustrative and realistic renderings, engineers need a comprehensible visualization of the simulation results, and usability studies or product presentations benefit from using a 3D display. However, the interactive visualization of NURBS models and corresponding physical simulations is a challenging task because of the computational complexity and the limited graphics hardware support. This thesis proposes four novel rendering approaches that improve the interactive visualization of CAD models and their analysis. The presented algorithms exploit latest graphics hardware capabilities to advance the state-of-the-art in terms of quality, efficiency and performance. In particular, two approaches describe the direct rendering of the parametric representation without precomputed approximations and timeconsuming pre-processing steps. New data structures and algorithms are presented for the efficient partition, classification, tessellation, and rendering of trimmed NURBS surfaces as well as the first direct isosurface ray-casting approach for NURBS-based isogeometric analysis. The other two approaches introduce the versatile concept of programmable order-independent semi-transparency for the illustrative and comprehensible visualization of depth-complex CAD models, and a novel method for the hybrid reprojection of opaque and semi-transparent image information to accelerate stereoscopic rendering. Both approaches are also applicable to standard polygonal geometry which contributes to the computer graphics and virtual reality research communities. The evaluation is based on real-world NURBS-based models and simulation data. The results show that rendering can be performed directly on the underlying parametric representation with interactive frame rates and subpixel-precise image results. The computational costs of additional visualization effects, such as semi-transparency and stereoscopic rendering, are reduced to maintain interactive frame rates. The benefit of this performance gain was confirmed by quantitative measurements and a pilot user study

A Note on Ribbon-based Biharmonic Surface Patches

In this short note we describe a simple adaptation of biharmonic surfaces to interpolate boundary cross-derivatives given in ribbon form, and compare with the recently proposed Generalized B-spline patches

Isogeometric Design, Analysis and Optimisation of Lattice-Skin Structures

The advancements in additive manufacturing techniques enable novel designs using lattice structures in mechanical parts, lightweight materials, biomaterials and so forth. Lattice-skin structures are a class of structures that couple thin-shells with lattices, which potentially combine the advantages of the thin-shell and the lattice structure. A new and systematic isogeometric analysis approach that integrates the geometric design, structural analysis and optimisation of lattice-skin structures is proposed in the dissertation. In the geometric design of lattice-skin structures, a novel shape interrogation scheme for splines, specifically subdivision surfaces, is proposed, which is able to compute the line/surface intersection efficiently and robustly without resorting to successive refinements or iterations as in Newton-Raphson method. The line/surface intersection algorithm involves two steps: intersection detection and intersection computation. In the intersection detection process, a bounding volume tree of k-dops (discrete oriented polytopes) for the subdivision surface is first created in order to accelerate the intersection detection between the line and the surface. The spline patches which are detected to be possibly intersected by the line are converted to Bézier representations. For the intersection computation, a matrix-based algorithm is applied, which converts the nonlinear intersection computation into solving a sequence of linear algebra problems using the singular value decomposition (SVD). Finally, the lattice-skin geometry is generated by projecting selected lattice nodes to the nearest intersection points intersected by the lattice edges. The Stanford bunny example demonstrates the efficiency and accuracy of the developed algorithm. The structural analysis of lattice-skin structures follows the isogeometric approach, in which the thin-shell is discretised with spline basis functions and the lattice structure is modelled with pin-jointed truss elements. In order to consider the lattice-skin coupling, a Lagrange multiplier approach is implemented to enforce the displacement compatibility between the coupled lattice nodes and the thin-shell. More importantly, the parametric coordinates of the coupled lattice nodes on the thin-shell surface are obtained directly from the lattice-skin geometry generation, which integrates the design and analysis process of lattice-skin structures. A sandwich plate example is analysed to verify the implementation and the accuracy of the lattice-skin coupling computation. In addition, a SIMP-like lattice topology optimisation method is proposed. The topology optimisation results of lattice structures are analysed and compared with several examples adapted from the benchmark examples commonly used in continuum topology optimisation. The SIMP-like lattice topology optimisation proposed is further applied to optimise the lattice in lattice-skin structures. The lattice-skin topology optimisation is fully integrated with the lattice-skin geometry design since the sensitivity analysis in the proposed method is based on lattice unit cells which are inherited from the geometry design stage. Finally, shape optimisation of lattice-skin structures using the free-form deformation (FFD) technique is studied. The corresponding shape sensitivity of lattice-skin structures is derived. The geometry update of the lattice-skin structure is determined by the deformation of the FFD control volume, and in this process the coupling between lattice nodes and the thin-shell is guaranteed by keeping the parametric coordinates of coupled lattice nodes which are obtained in the lattice-skin geometry design stage. A pentagon roof example is used to explore the combination of lattice topology optimisation and shape optimisation of lattice-skin structures

On Triangular Splines:CAD and Quadrature

The standard representation of CAD (computer aided design) models is based on the boundary representation (B-reps) with trimmed and (topologically) stitched tensor-product NURBS patches. Due to trimming, this leads to gaps and overlaps in the models. While these can be made arbitrarily small for visualisation and manufacturing purposes, they still pose problems in downstream applications such as (isogeometric) analysis and 3D printing. It is therefore worthwhile to investigate conversion methods which (necessarily approximately) convert these models into water-tight or even smooth representations. After briefly surveying existing conversion methods, we will focus on techniques that convert CAD models into triangular spline surfaces of various levels of continuity. In the second part, we will investigate efficient quadrature rules for triangular spline space