The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Douglas N. Arnold - One of the best experts on this subject based on the ideXlab platform.
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The Serendipity Family of Finite Elements
Foundations of Computational Mathematics, 2011Co-Authors: Douglas N. Arnold, Gerard AwanouAbstract:We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s − r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r . The degrees of freedom are given by moments of degree at most r −2 d on each face of dimension d . We establish unisolvence and a Geometric Decomposition of the space.
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The Serendipity Family of Finite Elements
Foundations of Computational Mathematics, 2011Co-Authors: Douglas N. Arnold, Gerard AwanouAbstract:We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a Geometric Decomposition of the space.Comment: 7 pages, 1 figure, 1 table. To appear in Foundations in Computational Mathematics, 201
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Geometric Decompositions and local bases for spaces of finite element differential forms
Computer Methods in Applied Mechanics and Engineering, 2009Co-Authors: Douglas N. Arnold, Richard S. Falk, Ragnar WintherAbstract:Abstract We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart–Thomas, Brezzi–Douglas–Marini, and Nedelec spaces. In the present paper, we derive Geometric Decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a Geometric Decomposition of a finite element space leading to a local basis.
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Geometric Decompositions and local bases for spaces of finite element differential forms
Computer Methods in Applied Mechanics and Engineering, 2009Co-Authors: Douglas N. Arnold, Richard S. Falk, Ragnar WintherAbstract:We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nedelec spaces. In the present paper, we derive Geometric Decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a Geometric Decomposition of a finite element space leading to a local basis.Comment: 25 pages, 2 figures, submitted to Computer Methods in Applied Mechanics and Engineerin
Gerard Awanou - One of the best experts on this subject based on the ideXlab platform.
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The Serendipity Family of Finite Elements
Foundations of Computational Mathematics, 2011Co-Authors: Douglas N. Arnold, Gerard AwanouAbstract:We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s − r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r . The degrees of freedom are given by moments of degree at most r −2 d on each face of dimension d . We establish unisolvence and a Geometric Decomposition of the space.
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The Serendipity Family of Finite Elements
Foundations of Computational Mathematics, 2011Co-Authors: Douglas N. Arnold, Gerard AwanouAbstract:We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a Geometric Decomposition of the space.Comment: 7 pages, 1 figure, 1 table. To appear in Foundations in Computational Mathematics, 201
Francesco Maggi - One of the best experts on this subject based on the ideXlab platform.
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a new approach to counterexamples to l 1 estimates korn s inequality Geometric rigidity and regularity for gradients of separately convex functions
Archive for Rational Mechanics and Analysis, 2005Co-Authors: Sergio Conti, Daniel Faraco, Francesco MaggiAbstract:The derivation of counterexamples to L1 estimates can be reduced to a Geometric Decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korn’s inequality, and of the corresponding Geometrically nonlinear rigidity result, in L1. Secondly, we construct a function f:ℝ2→ℝ which is separately convex but whose gradient is not in BVloc, in the sense that the mixed derivative ∂2f/∂x1∂x2 is not a bounded measure.
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A New Approach to Counterexamples to L^1 Estimates: Korn’s Inequality, Geometric Rigidity, and Regularity for Gradients of Separately Convex Functions
Archive for Rational Mechanics and Analysis, 2005Co-Authors: Sergio Conti, Daniel Faraco, Francesco MaggiAbstract:The derivation of counterexamples to L ^1 estimates can be reduced to a Geometric Decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korn’s inequality, and of the corresponding Geometrically nonlinear rigidity result, in L ^1. Secondly, we construct a function f :ℝ^2→ℝ which is separately convex but whose gradient is not in BV _loc, in the sense that the mixed derivative ∂^2 f /∂ x _1∂ x _2 is not a bounded measure.
Slobodan Žumer - One of the best experts on this subject based on the ideXlab platform.
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Topological and Geometric Decomposition of nematic textures.
Physical review. E Statistical nonlinear and soft matter physics, 2012Co-Authors: Simon Copar, Slobodan ŽumerAbstract:Directional media, such as nematic liquid crystals and ferromagnets, are characterized by their topologically stabilized defects in directional order. In nematics, boundary conditions and surface-treated inclusions often create complex structures, which are difficult to classify. Topological charge of point defects in nematics has ambiguously defined sign, and its additivity cannot be ensured when defects are observed separately. We demonstrate how the topological charge of complex defect structures can be determined by identifying and counting parts of the texture that satisfy simple Geometric rules. We introduce a parameter called the defect rank and show that it corresponds to what is intuitively perceived as a point charge based on the properties of the director field. Finally, we discuss the role of free-energy constraints in the validity of the classification with the defect rank.
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Topological and Geometric Decomposition of nematic textures
Physical review. E Statistical nonlinear and soft matter physics, 2012Co-Authors: Slobodan Žumer, Simon ČoparAbstract:Directional media, such as nematic liquid crystals and ferromagnets, are characterized by their topologically stabilized defects in directional order. In nematics, boundary conditions and surface-treated inclusions often create complex structures, which are difficult to classify. Topological charge of point defects in nematics has ambiguously defined sign, and its additivity cannot be ensured when defects are observed separately. We demonstrate how the topological charge of complex defect structures can be determined by identifying and counting parts of the texture that satisfy simple Geometric rules. We introduce a parameter called the defect rank and show that it corresponds to what is intuitively perceived as a point charge based on the properties of the director field. Finally, we discuss the role of free-energy constraints in the validity of the classification with the defect rank. © 2012 American Physical Society.
Nicolas Hudon - One of the best experts on this subject based on the ideXlab platform.
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Geometric Decomposition, potential-based representation and integrability of non-linear systems
IMA Journal of Mathematical Control and Information, 2020Co-Authors: Martin Guay, Nicolas Hudon, Kai HöffnerAbstract:Abstract This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a Decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The Decomposition method, based on the Hodge Decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed Decomposition.
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CDC - Stabilization of nonlinear systems via potential-based realization
IEEE Transactions on Automatic Control, 2016Co-Authors: Martin Guay, Nicolas HudonAbstract:This technical note considers the problem of representing a sufficiently smooth control affine system as a structured potential-driven system and to exploit the obtained representation for stability analysis and state feedback controller design. These problems have been studied in recent years for particular classes of potential-driven systems. To recover the advantages of those representations for the stabilization of general nonlinear systems, the present note proposes a Geometric Decomposition technique, based on the Hodge Decomposition theorem, to re-express a given vector field into a potential-driven form. Using the proposed Decomposition technique, stability conditions are developed based on the convexity of a computed potential. Finally, stabilization is studied in the context of the proposed Decomposition by reshaping the Hessian matrix of the obtained potential using damping feedback.
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CDC - ℒ 2 -stability for a class of nonlinear systems via potential-based realizations
53rd IEEE Conference on Decision and Control, 2014Co-Authors: Martin Guay, Nicolas HudonAbstract:This paper considers the problem of representing a sufficiently smooth control affine system as a structured potential-driven system and to exploit the obtained representation to study ℒ 2 -stability and stabilization. The representation problem has been studied extensively in recent years for particular classes of potential-driven systems, however exploiting these structures, for example generalized Hamiltonian systems, to study input-output stability was not fully investigated in the literature. The present note proposes a Geometric Decomposition technique, based on the Hodge Decomposition theorem, to reexpress a given vector field into a potential-driven form. Using the proposed Decomposition technique, finite gain stability conditions are developed, in the form of Hamilton-Jacobi inequalities, based on the convexity of a computed potential.
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Geometric Decomposition and potential based representation of nonlinear systems
American Control Conference, 2013Co-Authors: Martin Guay, Nicolas Hudon, Kai HöffnerAbstract:This paper considers the problem of representing a sufficiently smooth nonlinear dynamical as a structured potential-driven system. The proposed approach is based on a Decomposition of a differential one-form that encodes the divergence of the given vector fields into its exact and anti-exact components, and into its co-exact and anti-coexact components. The Decomposition method, based on the Hodge Decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems, and generalized Hamiltonian systems are given to illustrate the proposed approach.
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dissipative based dynamic state feedback design using a Geometric Decomposition
American Control Conference, 2013Co-Authors: Nicolas Hudon, Jie BaoAbstract:This paper presents a constructive approach to dynamic state feedback controller design for control affine systems such that the closed-loop system exhibits some desired dissipative properties. The proposed approach relies on the construction of a storage function for a given control affine system together with a dynamic controller structure. Using this approach, it is then possible to shape the Hessian of a potential function associated to the system in the extended space. A damping feedback controller with an injection term is used to ensure desired dissipative properties of the closed-loop system. An example, studied previously in the context of dynamic power-shaping passivity-based control and stabilization, is presented to illustrate the design methodology.