Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial

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

Algebraic level sets for CAD/CAE integration and moving boundary problems

Boundary representation (B-rep) of CAD models obtained from solid modeling kernels are commonly used in design, and analysis applications outside the CAD systems. Boolean operations between interacting B-rep CAD models as well as analysis of such multi-body systems are fundamental operations on B-rep geometries in CAD/CAE applications. However, the boundary representation of B-rep solids is, in general, not a suitable representation for analysis operations which lead to CAD/CAE integration challenges due to the need for conversion from B-rep to volumetric approximations. The major challenges include intermediate mesh generation step, capturing CAD features and associated behavior exactly and recurring point containment queries for point classification as inside/outside the solid. Thus, an ideal analysis technique for CAD/CAE integration that can enable direct analysis operations on B-rep CAD models while overcoming the associated challenges is desirable. ^ Further, numerical surface intersection operations are typically necessary for boolean operations on B-rep geometries during the CAD and CAE phases. However, for non-linear geometries, surface intersection operations are non-trivial and face the challenge of simultaneously satisfying the three goals of accuracy, efficiency and robustness. In the class of problems involving multi-body interactions, often an implicit knowledge of the boolean operation is sufficient and explicit intersection computation may not be needed. Such implicit boolean operations can be performed by point containment queries on B-rep CAD models. However, for complex non-linear B-rep geometries, the point containment queries may involve numerical iterative point projection operations which are expensive. Thus, there is a need for inexpensive, non-iterative techniques to enable such implicit boolean operations on B-rep geometries. ^ Moreover, in analysis problems with evolving boundaries (ormoving boundary problems), interfaces or cracks, blending functions are used to enrich the underlying domain with the known behavior on the enriching entity. The blending functions are typically dependent on the distance from the evolving boundaries. For boundaries defined by free form curves or surfaces, the distance fields have to be constructed numerically. This may require either a polytope approximation to the boundary and/or an iterative solution to determine the exact distance to the boundary. ^ In this work a purely algebraic, and computationally efficient technique is described for constructing signed distance measures from Non-Uniform Rational B-Splines (NURBS) boundaries that retain the geometric exactness of the boundaries while eliminating the need for iterative and non-robust distance calculation. The proposed technique exploits the NURBS geometry and algebraic tools of implicitization. Such a signed distance measure, also referred to as the Algebraic Level Sets, gives a volumetric representation of the B-rep geometry constructed by purely non-iterative algebraic operations on the geometry. This in turn enables both the implicit boolean operations and analysis operations on B-rep geometries in CAD/CAE applications. Algebraic level sets ensure exactness of geometry while eliminating iterative numerical computations. Further, a geometry-based analysis technique that relies on hierarchical partition of unity field compositions (HPFC) theory and its extension to enriched field modeling is presented. The proposed technique enables direct analysis of complex physical problems without meshing, thus, integrating CAD and CAE. The developed techniques are demonstrated by constructing algebraic level sets for complex geometries, geometry-based analysis of B-rep CAD models and a variety of fracture examples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three-dimensional cracks. ^ The proposed techniques are lastly applied to investigate the risk of fracture in the ultra low-k (ULK) dies due to copper (Cu) wirebonding process. Maximum damage induced in the interlayer dielectric (ILD) stack during the process steps is proposed as an indicator of the reliability risk. Numerical techniques based on enriched isogeometric approximations are adopted to model damage in the ULK stacks using a cohesive damage description. A damage analysis procedure is proposed to conduct damage accumulation studies during Cu wirebonding process. Analysis is carried out to identify weak interfaces and potential sites for crack nucleation as well as damage nucleation patterns. Further, the critical process condition is identified by analyzing the damage induced during the impact and ultrasonic excitation stages. Also, representative ILD stack designs with varying Cu percentage are compared for risk of fracture

A Line/Trimmed NURBS Surface Intersection Algorithm Using Matrix Representations

International audienceWe contribute a reliable line/surface intersection method for trimmed NURBS surfaces, based on a novel matrix-based implicit representation and numerical methods in linear algebra such as singular value decomposition and the computation of generalized eigenvalues and eigenvectors. A careful treatment of degenerate cases makes our approach robust to intersection points with multiple pre-images. We then apply our intersection algorithm to mesh NURBS surfaces through Delaunay refinement. We demonstrate the added value of our approach in terms of accuracy and treatment of degenerate cases, by providing comparisons with other intersection approaches as well as a variety of meshing experiments

Numerical proper reparametrization of parametric plane curves

We present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. More precisely, given a tolerance ϵ>0 and a rational parametrization P of a plane curve C with perturbed float coefficients, we present an algorithm that computes a parametrization Q of a new plane curve D such that Q is an ϵ –proper reparametrization of D. In addition, the error bound is carefully discussed and we present a formula that measures the “closeness” between the input curve C and the output curve D

Kommutative Algebra

The workshop brought together researchers on various areas of Commutative Algebra. New results in combinatorial commutative algebra, homological methods and invariants, characteristic p-methods, and in general commutative algebra and algebraic geometry were presented in the lectures of the workshop