Academic literature on the topic 'Enriched polynomial space'

Elastodynamic problems are investigated in this work by employing the enriched finite element method (EFEM) with various enrichment functions. By performing the dispersion analysis, it is confirmed that for elastodynamic analysis, the amount of numerical dispersion, which is closely related to the numerical error from the space domain discretization, can be suppressed to a very low level when quadric polynomial bases are employed to construct the local enrichment functions, while the amount of numerical dispersion from the EFEM with other types of enrichment functions (linear polynomial bases or first order of trigonometric functions) is relatively large. Consequently, the present EFEM with a quadric polynomial enrichment function shows more powerful capacities in elastodynamic analysis than the other considered numerical techniques. More importantly, the attractive monotonic convergence property can be broadly realized by the present approach with the typical two-step Bathe temporal discretization technique. Three representative numerical experiments are conducted in this work to verify the abilities of the present approach in elastodynamic analysis.

The traditional finite element method (FEM) could only provide acceptable numerical solutions for the Helmholtz equation in the relatively small wave number range due to numerical dispersion errors. For the relatively large wave numbers, the corresponding FE solutions are never adequately reliable. With the aim to enhance the numerical performance of the FEM in tackling the Helmholtz equation, in this work an extrinsic enriched FEM (EFEM) is proposed to reduce the inherent numerical dispersion errors in the standard FEM solutions. In this extrinsic EFEM, the standard linear approximation space in the linear FEM is enriched extrinsically by using the polynomial and trigonometric functions. The construction of this enriched approximation space is realized based on the partition of unity concept and the highly oscillating features of the Helmholtz equation in relatively large wave numbers can be effectively captured by the employed specially-designed enrichment functions. A number of typical numerical examples are considered to examine the ability of this extrinsic EFEM to control the dispersion error for solving Helmholtz problems. From the obtained numerical results, it is found that this extrinsic EFEM behaves much better than the standard FEM in suppressing the numerical dispersion effects and could provide much more accurate numerical results. In addition, this extrinsic EFEM also possesses higher convergence rate than the conventional FEM. More importantly, the formulation of this extrinsic EFEM can be formulated quite easily without adding the extra nodes. Therefore, the present extrinsic EFEM can be regarded as a competitive alternative to the traditional finite element approach in dealing with the Helmholtz equation in relatively high frequency ranges.

PsdVoigt2 function was used to fit SGW function and a series of bar beam elements based on the theory of SGW and wavelet finite element were constructed. Traditional finite element polynomial interpolation was replaced by the SGW scaling function and transformation matrix was utilized to transform wavelet interpolation coefficients to physical space. Thereby the shape function and element were constructed. The precision of a series of bar beam elements constructed with the SGW scale function as the interpolation function were verified by the calculation cases. The calculation results showed that the precision of the SGW-based element constructed was higher in calculating the deformation and strain. And the SGW-based bar beam element constructed enriched the element library of wavelet-finite element method.

This paper presents a novel and efficient solution procedure to improve 3D solid finite element analysis with an enrichment scheme. To this end, we employ finite elements enriched by polynomial cover functions, which can expand their solution space without requiring mesh refinement or additional nodes. To facilitate this solution procedure, an error estimation method and cover function selection scheme for 3D solid finite element analysis are developed. This enables the identification of nodes with suboptimal solution accuracy, allowing for the adaptive application of cover functions in a systematic and efficient manner. Furthermore, a significant advantage of this procedure is its consistency, achieved by excluding arbitrary coefficients from the formulations employed. The effectiveness of the proposed procedure is demonstrated through several numerical examples. In the majority of the examples, it is observed that the stress prediction error is reduced by more than half after applying the proposed procedure.

An improved finite volume scheme based on variational reconstruction and function enrichment has been proposed in this paper. By incorporating the law-of-the-wall into the variational reconstruction, the proposed method can resolve turbulent flow accurately on grids much coarser than those needed by traditional methods. The usual reconstruction in a finite volume scheme assumes that the solution is belonging to a polynomial function space, which is inaccurate to resolve the velocity profile within the turbulent boundary layer unless the grid in wall-normal direction is fine enough. In the present paper, this function space is “enriched” by adding a basis function that is derived from the logarithmic law of the turbulent boundary layer. Then variational reconstruction procedure is applied to find the “best” solution belonging to the expanded function space. The advantage of the present method over the traditional wall function model is that the turbulent flow within the boundary layer is resolved rather than modeled. The algorithms and the implementations are discussed in detail. The proposed method is applied to the turbulent flow over a flat plate at a Reynolds number of [Formula: see text] and the turbulent flow over a plate with a bump at a Reynolds number of [Formula: see text]. The results of second- and third-order schemes are presented for the turbulent velocity profile and the skin friction coefficients. The numerical results suggest that this approach not only resolves the near wall turbulent accurately on very coarse grids but also reduces the computational time significantly.

Standard Boundary Element Methods (BEM) for time-harmonic acoustics, using piecewise polynomial finite-element type approximation spaces, have a computational cost that grows rapidly with frequency, to ensure at least a fixed number of degrees of freedom per wavelength. Hybrid Numerical-Asymptotic (HNA) BEMs, based on enriched approximation spaces consisting of the products of piecewise polynomials with carefully chosen oscillatory functions, have a computational cost that is almost frequency-independent for some problem classes (e.g. Chandler-Wilde, Graham, Langdon, Spence, Acta Numerica 2012), but the technology is largely undeveloped for problems where multiple scattering is important. In this paper we present computational experiments, supported by mathematical analysis, which suggest that multiple scattering configurations may be within reach. Specifically, we solve, by a HNA BEM, scattering by a pair of screens in an arbitrary configuration, which we anticipate may serve as a building block towards algorithms for general multiple scattering problems with computational cost independent of frequency. The specific configuration considered, as we discuss, is relevant to the efficient simulation of multiple outdoor noise barriers.

Purpose – The purpose of this paper is to present an application of the Generalized Finite Element Method (GFEM) for modal analysis of 2D wave equation. Design/methodology/approach – The GFEM can be viewed as an extension of the standard Finite Element Method (FEM) that allows non-polynomial enrichment of the approximation space. In this paper the authors enrich the approximation space with sine e cosine functions, since these functions frequently appear in the analytical solution of the problem under study. The results are compared with the ones obtained with the polynomial FEM using higher order elements. Findings – The results indicate that the proposed approach is able to obtain more accurate results for higher vibration modes than standard polynomial FEM. Originality/value – The examples studied in this paper indicate a strong potential of the GFEM for the approximation of higher vibration modes of structures, analysis of structures subject to high frequency excitations and other problems that concern high frequency oscillatory phenomena.

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text], and by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text]. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise [Formula: see text] polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element [Formula: see text] plus [Formula: see text] in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

The article delves into the intricate relationship between symmetry and mathematical imaging, spanning various mathematical disciplines. Symmetry, a concept deeply ingrained in mathematics, manifests in art, nature, and physics, providing a powerful tool for understanding complex structures. The paper explores three types of symmetriesreflection, rotational, and translationalexemplified through concrete mathematical expressions. Evariste Galoiss Group Theory emerges as a pivotal tool, providing a formal framework to understand and classify symmetric operations, particularly in the roots of polynomial equations. Galois theory, a cornerstone of modern algebra, connects symmetries, permutations, and solvability of equations. Group theory finds practical applications in cryptography, physics, and coding theory. Sophus Lie extends group theory to continuous spaces with Lie Group Theory, offering a powerful framework for studying continuous symmetries. Lie groups find applications in robotics and control theory, streamlining the representation of transformations. Benoit Mandelbrots fractal geometry, introduced in the late 20th century, provides a mathematical framework for understanding complex, self-similar shapes. The applications of fractal geometry range from computer graphics to financial modeling. Symmetrys practical applications extend to data visualization and cryptography. The article concludes by emphasizing symmetrys foundational role in physics, chemistry, computer graphics, and beyond. A deeper understanding of symmetry not only enriches perspectives across scientific disciplines but also fosters interdisciplinary collaborations, unveiling hidden order and structure in the natural and designed world. The exploration of symmetry promises ongoing discoveries at the intersection of mathematics and diverse fields of study.

AbstractIn this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one- and two-dimensional elliptic PDEs with random coefficients.

Un problème très courant en science computationnelle est la détermination d'une approximation, dans un intervalle fixe, d'une fonction dont les évaluations ne sont connues que sur un ensemble fini de points. Une approche courante pour résoudre ce problème repose sur l'interpolation polynomiale. Un cas d'un grand intérêt pratique est celui où ces points suivent une distribution équidistante dans l'intervalle considéré. Dans ces hypothèses, un problème lié à l'interpolation polynomiale est le phénomène de Runge, caractérisé par une augmentation de l'erreur d'interpolation près des extrémités de l'intervalle. En 2009, J. Boyd et F. Xu ont démontré que le phénomène de Runge pouvait être éliminé en interpolant la fonction que sur un sous-ensemble approprié formé par les noeuds les plus proches des noeuds de Chebyshev-Lobatto, communément appelés noeuds de mock-Chebyshev.Cependant, cette stratégie implique de ne pas utiliser presque toutes les données disponibles. Afin d'améliorer la précision de la méthode proposée par Boyd et Xu, tout en tirant pleinement parti des données disponibles, S. De Marchi, F. Dell'Accio et M. Mazza ont introduit une nouvelle technique appelée constrained mock-Chebyshev least squares approximation. Dans cette méthode, le rôle du polynôme nodal, est crucial. Son extension au cas bivarié nécessite cependant des approches alternatives. La procédure développée par F. Dell'Accio, F. Di Tommaso et F. Nudo, utilisant la méthode des multiplicateurs de Lagrange, permet également de définir l'approximation des moindres carrés de mock-Chebyshev sur une grille uniforme de points. Cette technique innovante, équivalente à la méthode univariée précédemment introduite en termes analytiques, se révèle également plus précise en termes numériques. La première partie de la thèse est consacrée à l'étude de cette nouvelle technique et à son application à des problèmes de quadrature et de différenciation numérique.Dans la deuxième partie de cette thèse, nous nous concentrons sur le développement d'un cadre unifié et général pour l'enrichissement de l'élément fini linéaire triangulaire standard en deux dimensions et de l'élément fini linéaire simplicial standard en dimensions supérieures. La méthode des éléments finis est une approche largement adoptée pour résoudre numériquement les équations aux dérivées partielles qui se posent en ingénierie et en modélisation mathématique [55]. Sa popularité est attribuable en partie à sa polyvalence pour traiter diverses formes géométriques. Cependant, les approximations produites par cette méthode s'avèrent souvent inefficaces pour résoudre des problèmes présentant des singularités. Pour surmonter ce problème, diverses approches ont été proposées, l'une des plus célèbres reposant sur l'enrichissement de l'espace d'approximation des éléments finis en ajoutant des fonctions d'enrichissement appropriées. Un des éléments finis le plus simple est l'élément fini triangulaire linéaire standard, largement utilisé dans les applications. Dans cette thèse, nous introduisons un enrichissement polynomial de l'élément fini triangulaire linéaire standard et utilisons ce nouvel élément fini pour introduire une amélioration de l'opérateur triangulaire de Shepard. Ensuite, nous introduisons une nouvelle classe d'éléments finis en enrichissant l'élément triangulaire linéaire standard avec des fonctions d'enrichissement qui ne sont pas nécessairement polynomiales, mais qui satisfont la condition d'annulation aux sommets du triangle.Nous généralisons les résultats présentés dans le cas bidimensionnel au cas de l'élément fini simplicial linéaire standard, en utilisant également des fonctions d'enrichissement qui ne satisfont pas la condition d'annulation aux sommets du simplexe.Enfin, nous appliquons ces nouvelles stratégies d'enrichissement pour définir une version plus généralede l'enrichissement de l'élément fini linéaire vectoriel simplicial développé par Bernardi et Raugel
A very common problem in computational science is the determination of an approximation, in a fixed interval, of a function whose evaluations are known only on a finite set of points. A common approach to solving this problem relies on polynomial interpolation, which consists of determining a polynomial that coincides with the function at the given points. A case of great practical interest is the case where these points follow an equispaced distribution within the considered interval. In these hypotheses, a problem related to polynomial interpolation is the Runge phenomenon, which consists in increasing the magnitude of the interpolation error close to the ends of the interval. In 2009, J. Boyd and F. Xu demonstrated that the Runge phenomenon could be eliminated by interpolating the function only on a proper subset formed by nodes closest to the Chebyshev-Lobatto nodes, the so called mock-Chebyshev nodes.However, this strategy involves not using almost all available data. In order to improve the accuracy of the method proposed by Boyd and Xu, while making full use of the available data, S. De Marchi, F. Dell'Accio, and M. Mazza introduced a new technique known as the constrained mock-Chebyshev least squares approximation. In this method, the role of the nodal polynomial, essential for ensuring interpolation at mock-Chebyshev nodes, is crucial. Its extension to the bivariate case, however, requires alternative approaches. The recently developed procedure by F. Dell'Accio, F. Di Tommaso, and F. Nudo, employing the Lagrange multipliers method, also enables the definition of the constrained mock-Chebyshev least squares approximation on a uniform grid of points. This innovative technique, equivalent to the previously introduced univariate method in analytical terms, also proves to be more accurate in numerical terms. The first part of the thesis is dedicated to the study of this new technique and its application to numerical quadrature and differentiation problems.In the second part of this thesis, we focus on the development of a unified and general framework for the enrichment of the standard triangular linear finite element in two dimensions and the standard simplicial linear finite element in higher dimensions. The finite element method is a widely adopted approach for numerically solving partial differential equations arising in engineering and mathematical modeling [55]. Its popularity is partly attributed to its versatility in handling various geometric shapes. However, the approximations produced by this method often prove ineffective in solving problems with singularities. To overcome this issue, various approaches have been proposed, with one of the most famous relying on the enrichment of the finite element approximation space by adding suitable enrichment functions. One of the simplest finite elements is the standard linear triangular element, widely used in applications. In this thesis, we introduce a polynomial enrichment of the standard triangular linear finite element and use this new finite element to introduce an improvement of the triangular Shepard operator. Subsequently, we introduce a new class of finite elements by enriching the standard triangular linear finite element with enrichment functions that are not necessarily polynomials, which satisfy the vanishing condition at the vertices of the triangle.Later on, we generalize the results presented in the two-dimensional case to the case of the standard simplicial linear finite element, also using enrichment functions that do not satisfy the vanishing condition at the vertices of the simplex.Finally, we apply these new enrichment strategies to extend the enrichment of the simplicial vector linear finite element developed by Bernardi and Raugel

In this thesis we aim for a systematic way of extending Stone-Halmos duality theorems to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval [0, 1] and present duality theory for ordered compact spaces and (suitably defined) finitely cocomplete categories enriched in [0, 1]. In the second part, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one — intuitively, the topological analogue of a Kripke polynomial functor.
Nesta tese pretendemos estender de forma sistemática dualidades de StoneHalmos para categorias que incluem todos os espaços de Hausdorff compactos. Para atingir este objectivo combinamos teoria de dualidades e teoria de categorias enriquecidas em quantais. A nossa ideia principal é que ao passar do espaço discreto com dois elementos para um cogerador da categoria de espaços de Hausdorff compactos, todas as restantes estruturas envolvidas devem ser substituídas por versões enriquecidas correspondentes. Desta forma, consideramos o intervalo unitário [0, 1] e desenvolvemos teoria de dualidades para espaços ordenados compactos e categorias enriquecidas em [0, 1] finitamente cocompletas (apropriadamente definidas). Na segunda parte da tese estudamos limites em categorias de coalgebras cujo functor subjacente é um functor de Vietoris polinomial — intuitivamente, uma versão topológica de um functor polinomial de Kripke.
Programa Doutoral em Matemática

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

Abstract We present an overview of stabilized finite element methods and of the standard Galerkin method enriched with residual-free bubble functions. The inadequacy of the standard Galerkin method using piecewise polynomials is discussed for different applications; the treatment using stabilized methods in their different versions is reviewed; and the connection to the standard Galerkin method with richer subspaces follows using the subgrid method or the residual-free-bubbles viewpoint. We close with a discussion on how to approximate the exact problem suggested by residual-free bubbles. The standard Galerkin method can be roughly described as being an approximation of the variational formulation of a PDE (or system of PDE’s) in a space of functions that is spanned by piecewise polynomials. This simple idea presents several advantages: first, the discrete system of equations that arise from such an approximation is going to be “banded” since the piecewise polynomials can be constructed to have a “small” support, and therefore the matrices involved are sparse.

For a system with the parameters modeled as uncertain, polynomial approximations such as polynomial chaos expansion provide an effective way to estimate the statistical behavior of the eigenvalues and eigenvectors, provided the eigenvalues are widely spaced. For a system with a set of clustered eigenvalues, the corresponding eigenvalues and eigenvectors are very sensitive to perturbation of the system parameters. An enrichment scheme to the polynomial chaos expansion is proposed here in order to capture the behavior of such eigenvalues and eigenvectors. It is observed that for judiciously chosen enrichment functions, the enriched expansion provides better estimate of the statistical behavior of the eigenvalues and eigenvectors.

The Robust Stable Matching (RSM) problem involves finding a stable matching that allows getting another stable matching within a minimum number of changes when a pair becomes forbidden. It has been shown that such a problem is NP-Hard. In this paper, we enrich the mathematical model for the RSM problem based on new theoretical properties. We derive from these properties new polynomial time pre-solving algorithms which both reduce the search space and speed up the exploration. We also extend our results to the instances of the Many-to-Many problem and give its corresponding constraint programming (CP) model. We show how the use of our algorithms improve the state-of-the-art results and make it possible to obtain proofs of optimality on large instances via the CP model.

As widely argued [HG97; Sat96], transitive roles play an important role in the adequate representation of aggregated objects: they allow these objects to be described by referring to their parts without specifying a level of decomposition. In [HG97], the Description Logic (DL) ALCHR+ is presented, which extends ALC with transitive roles and a role hierarchy. It is argued in [Sat98] that ALCHR+ is well-suited to the representation of aggregated objects in applications that require various part-whole relations to be distinguished, some of which are transitive. However, ALCHR+ allows neither the description of parts by means of the whole to which they belong, or vice versa. To overcome this limitation, we present the DL SHI which allows the use of, for example, has part as well as is part of. To achieve this, ALCHR+ was extended with inverse roles. It could be argued that, instead of defining yet another DL, one could make use of the results presented in [DL96] and use ALC extended with role expressions which include transitive closure and inverse operators. The reason for not proceeding like this is the fact that transitive roles can be implemented more efficiently than the transitive closure of roles (see [HG97]), although they lead to the same complexity class (ExpTime-hard) when added, together with role hierarchies, to ALC. Furthermore, it is still an open question whether the transitive closure of roles together with inverse roles necessitates the use of the cut rule [DM98], and this rule leads to an algorithm with very bad behaviour. We will present an algorithm for SHI without such a rule. Furthermore, we enrich the language with functional restrictions and, finally, with qualifying number restrictions. We give sound and complete decision proceduresfor the resulting logics that are derived from the initial algorithm for SHI. The structure of this report is as follows: In Section 2, we introduce the DL SI and present a tableaux algorithm for satisfiability (and subsumption) of SI-concepts—in another report [HST98] we prove that this algorithm can be refined to run in polynomial space. In Section 3 we add role hierarchies to SI and show how the algorithm can be modified to handle this extension appropriately. Please note that this logic, namely SHI, allows for the internalisation of general concept inclusion axioms, one of the most general form of terminological axioms. In Section 4 we augment SHI with functional restrictions and, using the so-called pairwise-blocking technique, the algorithm can be adapted to this extension as well. Finally, in Section 5, we show that standard techniques for handling qualifying number restrictions [HB91;BBH96] together with the techniques described in previous sections can be used to decide satisfiability and subsumption for SHIQ, namely ALC extended with transitive and inverse roles, role hierarchies, and qualifying number restrictions. Although Section 5 heavily depends on the previous sections, we have made it self-contained, i.e. it contains all necessary definitions and proofs from scratch, for a better readability. Building on the previous sections, Section 6 presents an algorithm that decides the satisfiability of SHIQ-ABoxes.