Angelos Mantzaflaris

Multipatch Discontinuous Galerkin Isogeometric Analysis

Isogeometric Analysis (IgA) uses the same class of basis functions for both representing the geometry of the computational domain and approximating the solution of the boundary value problem under consideration. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This multi-patch representation corresponds to a decomposition of the computational domain into non-overlapping subdomains also called patches in the geometrical framework. We will present discontinuous Galerkin (dG) methods that allow for discontinuities across the subdomain (patch) boundaries. The required interface conditions are weakly imposed by the dG terms associated with the boundary of the subdomains. The construction and the corresponding discretization error analysis of such dG multi-patch IgA schemes is given for heterogeneous diffusion model problems in volumetric 2d and 3d domains as well as on open and closed surfaces. The theoretical results are confirmed by numerous numerical experiments which have been performed in G+SMO. The concept and the main features of the IgA library G +SMO are also described.

Deflation and certified isolation of singular zeros of polynomial systems

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 to solve. We derive a one-step deflation technique, from the description of the multiplicity structure in terms of differentials. The deflated system can be used in Newton-based iterative schemes with quadratic convergence. Starting from a polynomial system and a sufficiently small neighborhood, we obtain a criterion for the existence and uniqueness of a singular root of a given multiplicity structure, applying a well-chosen symbolic perturbation. Standard verification methods, based e.g. on interval arithmetic and a fixed point theorem, are employed to certify that there exists a unique perturbed system with a singular root in the domain. Applications to topological degree computation and to the analysis of real branches of an implicit curve illustrate the method.

Adaptively refined multi-patch B-splines with enhanced smoothness

A spline space suitable for Isogeometric Analysis (IgA) on multi-patch domains is presented. Our construction is motivated by emerging requirements in isogeometric simulations. In particular, IgA spaces should allow for adaptive mesh refinement and they should guarantee the optimal smoothness of the discretized solution, even across interfaces of adjacent patches.Given a domain manifold M consisting of individual patches (isomorphic to the unit square or cube) that are glued together along interfaces, we present a construction of multi-patch B-splines defined on them. Their smoothness is enhanced by locally modifying or merging basis functions around the boundary of each patch. The resulting multi-patch B-splines with enhanced smoothness (MPBES) possess the property of local linear independence and form a non-negative partition of unity. Moreover, their span can be characterized as the linear space of all piecewise polynomial functions on the domain manifold that possess certain smoothness properties.Subsequently, adaptively refined MPBES are obtained by generalizing the construction of truncated hierarchical (TH) B-splines. More precisely, a nested sequence of spaces spanned by MPBES is considered, corresponding to steps of local enrichment. In addition, an inversely nested sequence of subdomains (which are submanifolds of M ) is used to specify the local refinement level of functions in these spaces. Finally, truncated hierarchical MPBES are obtained by means of the selection and truncation mechanism of THB-splines. The desired properties of linear independence and convex partition of unity are maintained.The paper presents several numerical examples which demonstrate potential applications of the new basis in isogeometric analysis.

On continued fraction expansion of real roots of polynomial systems, complexity and condition numbers

We elaborate on a correspondence between the coefficients 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 the Bernstein basis) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and obtain a subdivision scheme for the domain of multivariate polynomial functions. This implies a new algorithm for real root isolation, MCF, that generalizes the Continued Fraction (CF) algorithm of univariate polynomials. A partial extension of Vincents Theorem for multivariate polynomials is presented, which allows us to prove the termination of the algorithm. Bounding functions, projection and preconditioning are employed to speed up the scheme. The resulting isolation boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. Finally, we present new complexity bounds for a simplified version of the algorithm in the bit complexity model, and also bounds in the real RAM model for a family of subdivision algorithms in terms of the real condition number of the system. Examples computed with our C++ implementation illustrate the practical aspects of our method.

Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation

It has been observed that the task of matrix assembly in Isogeometric Analysis (IGA) is more challenging than in the case of traditional finite element methods. The additional difficulties associated with IGA are caused by the increased degree and the larger supports of the functions that occur in the integrals defining the matrix elements. Recently we introduced an interpolation-based approach that approximately transforms the integrands into piecewise polynomials and uses look-up tables to evaluate their integrals [17, 18]. The present paper relies on this earlier work and proposes to use tensor methods to accelerate the assembly process further. More precisely, we show how to represent the matrices that occur in IGA as sums of a small number of Kronecker products of auxiliary matrices that are defined by univariate integrals. This representation, which is based on a low-rank tensor approximation of certain parts of the integrands, makes it possible to achieve a significant speedup of the assembly process without compromising the overall accuracy of the simulation.

Exploring Matrix Generation Strategies in Isogeometric Analysis

An important step in simulation via isogeometric analysis (IGA) is the assembly step, where the coefficients of the final linear system are generated. Typically, these coefficients are integrals of products of shape functions and their derivatives. Similarly to the finite element analysis (FEA), the standard choice for integral evaluation in IGA is Gaussian quadrature. Recent developments propose different quadrature rules, that reduce the number of quadrature points and weights used. We experiment with the existing methods for matrix generation. Furthermore we propose a new, quadrature-free approach, based on interpolation of the geometry factor and fast look-up operations for values of B-spline integrals. Our method builds upon the observation that exact integration is not required to achieve the optimal convergence rate of the solution. In particular, it suffices to generate the linear system within the order of accuracy matching the approximation order of the discretization space. We demonstrate that the best strategy is one that follows the above principle, resulting in expected accuracy and improved computational time.

Continued fraction expansion of real roots of polynomial systems

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a fully analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditionning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincents theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Examples computed with a preliminary C++ implementation illustrate the approach.

Multihomogeneous resultant formulae for systems with scaled support

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bezout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.

Mathemagix: the quest of modularity and efficiency for symbolic and certified numeric computation?

We describe the goals and architecture of the software project Mathemagix, the main list of packages it contains, the main characteristics of its programming language, and connections with existing frontends.

Mesh grading in isogeometric analysis

This paper is concerned with the construction of graded meshes for approximating so-called singular solutions of elliptic boundary value problems by means of multipatch discontinuous Galerkin Isogeometric Analysis schemes. Such solutions appear, for instance, in domains with re-entrant corners on the boundary of the computational domain, in problems with changing boundary conditions, in interface problems, or in problems with singular source terms. Making use of the analytic behavior of the solution, we construct the graded meshes in the neighborhoods of such singular points following a multipatch approach. We prove that appropriately graded meshes lead to the same convergence rates as in the case of smooth solutions with approximately the same number of degrees of freedom. Representative numerical examples are studied in order to confirm the theoretical convergence rates and to demonstrate the efficiency of the mesh grading technology in Isogeometric Analysis.