MANTZAFLARIS , Angelos

Geometric computation in computer aided geometric design and solid modeling calls for solving non-linear polynomial systems in an approximate-yetcertified manner. We introduce new subdivision algorithms that tackle this fundamental problem. In particular, we generalize the univariate so-called continued fraction solver to general dimension....

G+Smo (pronounced gismo or gizmo) is a C++ library for isogeometric analysis (IGA). Geometry plus simulation modules aims at the seamless integration of Computer-aided Design (CAD) and Finite Element Analysis (FEA).

We give an overview of the open-source library "G+Smo". G+Smo is a C++ library that brings together mathematical tools for geometric design and numerical simulation. It implements the relatively new paradigm of isogeometric analysis, which suggests the use of a unified framework in the design and analysis pipeline. G+Smo is an object-oriented,...

Singular zeros of systems of polynomial equations constitute a bottleneck when it comes to computing, since several methods relying on the regularity of the Jacobian matrix of the system do not apply when the latter has a non-trivial kernel. Therefore they require special treatment. The algebraic information regarding an isolated singularity...

Semi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them. In this work we present an algorithm to efficiently and in a certified way compute the connected components of semi-algebraic sets given by intersection or union of conjunctions of bi-variate equalities and inequalities. For any given...

In this work, we develop a specialized quadrature rule for trimmed domains , where the trimming curve is given implicitly by a real-valued function on the whole domain. We follow an error correction approach: In a first step, we obtain an adaptive subdivision of the domain in such a way that each cell falls in a pre-defined base case. We then...

We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems is presented, which avoids redundant computation and reduces the size of the intermediate linear systems...

We consider the adaptive refinement of bivariate quartic C 2-smooth box spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quar-tic polynomials, which will be called the space of special quartics. Given a bounded domain Ω ⊂ R 2 and finite sequence (G...

In this work we analyze the space of geometrically smooth biquintic Bézier polynomials defined on a quadrilateral mesh M generated by the G1 Approximate Catmull-Clark scheme (G1 ACC) in [27]. With the use of quadratic gluing data functions, an explicit construction for an efficient set of basis functions generating the G1 ACC space is provided...

In this work, we develop a specialized quadrature rule for trimmed domains , where the trimming curve is given implicitly by a real-valued function on the whole domain. We follow an error correction approach: In a first step, we obtain an adaptive subdivision of the domain in such a way that each cell falls in a pre-defined base case. We then...

We elaborate on a correspondence between the coeffcients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only integer arithmetic (in contrast to Bernstein basis) and are feasible over unbounded regions. Then, we study an...

We exploit structure in polynomial system solving by considering polyno-mials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We...

In this paper a construction of a globally G1 family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of Bézier patches, which are bicubic C2 around regular vertices and biquintic G1 around extraordinary vertices (and C1 on...

Hierarchical B-splines that allow local refinement have become a promising tool for developing adaptive isogeometric methods. Unfortunately, similar to tensor-product B-splines, the computational cost required for assembling the system matrices in isogeometric analysis with hierarchical B-splines is also high, particularly if the spline degree...

The construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since...

Bases and dimensions of trivariate spline functions possessing first order geometric continuity on two-patch domains were studied in [4]. It was shown that the properties of the spline space depend strongly on the type of the gluing data that is used to specify the relation between the partial derivatives along the interface between the...

We present a completeness characterization of box splines on three-directional triangulations, also called Type-I box spline spaces, based on edge-contact smoothness properties. For any given Type-I box spline, of specific maximum degree and order of global smoothness, our results allow to identify the local linear subspace of polynomials...

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f_1, \ldots, f_N)\in C[x_1, \ldots, x_n]^N$, we present a Newton iteration on an extended deflated system that locally converges, under regularity...

System matrix assembly for isogeometric (i.e., spline-based) discretizations of partial differential equations is more challenging than for classical finite elements, due to the increased polynomial degrees and the larger (and hence more overlapping) supports of the basis functions. The global tensor-product structure of the discrete spaces...

This paper is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the efficient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline...

Isolated singularities typically occur at self-intersection points of planar algebraic curves, curve offsets, intersections between spatial curves and surfaces, and so on. The information characterizing the singularity can be captured in a local dual basis, expressing combinations of vanishing derivatives at the singular point. Macaulay's...

In this paper we present a space-time isogeometric analysis scheme for the discretization of parabolic evolution equations with diffusion coefficients depending on both time and space variables. The problem is considered in a space-time cylinder in R +1 , with = 2, 3 and is discretized using higher-order and highly-smooth spline spaces. This...

We analyze the spaces of trivariate C 1-smooth isogeometric functions on two-patch domains. Our aim is to generalize the corresponding results from the bivariate [25] to the trivariate case. In the first part of the paper, we introduce the notion of gluing data and use it to define glued spline functions on two-patch domains. Applying the...

We bound the Boolean complexity of computing isolating hyperboxes for all complex roots of systems of bilinear polynomials. The resultant of such systems admits a family of determinantal Sylvester-type formulas, which we make explicit by means of homological complexes. The computation of the determinant of the resultant matrix is a bottleneck...