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Dejan Slepčev - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit
Archive for Rational Mechanics and Analysis, 2021Co-Authors: Antonio Esposito, Francesco S. Patacchini, André Schlichting, Dejan SlepčevAbstract:We consider dynamics driven by Interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-Interaction Equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity Equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-Interaction energy the nonlocal nonlocal-Interaction Equation (NL $$^2$$ 2 IE). We develop the existence theory for the solutions of the NL $$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL $$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.
Antonio Esposito - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit
Archive for Rational Mechanics and Analysis, 2021Co-Authors: Antonio Esposito, Francesco S. Patacchini, André Schlichting, Dejan SlepčevAbstract:We consider dynamics driven by Interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-Interaction Equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity Equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-Interaction energy the nonlocal nonlocal-Interaction Equation (NL $$^2$$ 2 IE). We develop the existence theory for the solutions of the NL $$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL $$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.
André Schlichting - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit
Archive for Rational Mechanics and Analysis, 2021Co-Authors: Antonio Esposito, Francesco S. Patacchini, André Schlichting, Dejan SlepčevAbstract:We consider dynamics driven by Interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-Interaction Equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity Equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-Interaction energy the nonlocal nonlocal-Interaction Equation (NL $$^2$$ 2 IE). We develop the existence theory for the solutions of the NL $$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL $$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.
Francesco S. Patacchini - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit
Archive for Rational Mechanics and Analysis, 2021Co-Authors: Antonio Esposito, Francesco S. Patacchini, André Schlichting, Dejan SlepčevAbstract:We consider dynamics driven by Interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-Interaction Equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity Equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-Interaction energy the nonlocal nonlocal-Interaction Equation (NL $$^2$$ 2 IE). We develop the existence theory for the solutions of the NL $$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL $$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.
Talebian Nima - One of the best experts on this subject based on the ideXlab platform.
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Upright Frame Shear Stiffness and Upright Biaxial Bending in the Design of Cold-Formed Steel Storage Rack-Supported Buildings
Griffith University, 2018Co-Authors: Talebian NimaAbstract:Steel storage racks are commonly used worldwide to store goods on pallets and represent freestanding structures to design. Recently, a new type of storage systems has gained popularity in which the rack system supports both the building enclosure and the stored goods. These new rack structures are referred to as “rack-supported buildings” or “clad racks”. Due to combined actions of wind loading and stored pallets, uprights undergo a combination of biaxial bending and compression. Existing design rules may not be adequate for this type of combined loading. Furthermore in clad racks, as the outer rack frames must withstand cross-aisle horizontal actions due to wind loading, accurately determining the transverse shear stiffness of the upright frames is essential. Indeed, this stiffness is needed in calculating the elastic buckling load, performing earthquake design and serviceability checks. This thesis is motivated by the two aforementioned aspects relative to clad racks and investigates first the factor affecting the transverse shear stiffness of steel storage rack upright frames and second the biaxial bending behaviour of the uprights. International racking design specifications recommend different approaches to evaluate the shear stiffness. The Rack Manufacturers Institute (RMI) specification conservatively uses an analytical solution based on Timoshenko and Gere's theory while the European (EN15512) and Australian (AS4084) specifications recommend experimental testing to be conducted. Discrepancy between Finite Element Analyses (FEA) and experimental test results is likely attributed to the local deformations occurring at the bolted joints. In the first part of this thesis, an advanced FEA model to accurately capture the transverse shear stiffness of upright frames is developed and verified against published experimental test results. Based on the FE model, the factors contributing to the transverse shear deformation of the frames with Cee-bracing members are quantified and discussed for lip-to-lip and back-to-back bracing patterns. In cold-formed steel structures international specifications, a linear Interaction Equation is typically used to account for members subject to biaxial bending and may be inaccurate. In the second part of this thesis, the biaxial bending capacity of the uprights is experimentally investigated and the actual interactive relationship between bending of the uprights about the major and minor axes, for local and distortional buckling is determined. Two types of regularly perforated and non-perforated storage rack uprights are investigated. An advanced finite element model to determine the biaxial bending capacity of cold-formed steel storage rack upright sections is validated against the experimental tests and parametric studies are performed to analyse the biaxial response of slender, semi-compact and compact unperforated storage rack upright cross-sections in local and distortional buckling failure modes only. The results from the parametric studies are used to verify the accuracy of different forms of published direct strength method (DSM) Equations.Thesis (PhD Doctorate)Doctor of Philosophy (PhD)School of Eng & Built EnvScience, Environment, Engineering and TechnologyFull Tex
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Local and Distortional Biaxial Bending Capacities of Cold-Formed Steel Storage Rack Uprights
ASCE-AMER SOC CIVIL ENGINEERS, 2018Co-Authors: Talebian Nima, Gilbert Benoit, Cao Hung, Chariere Romain, Karampour HassanAbstract:Cold-formed steel storage rack-supported buildings, also referred to as clad racks, support both the building enclosure and the stored goods. Because of combined actions of wind loading and stored pallets, uprights undergo a combination of biaxial bending and compression. The focus of attention of this paper is only on pure biaxial bending capacity of the uprights. In cold-formed steel structures international specifications, a linear Interaction Equation is typically used to account for members subject to biaxial bending and may be inaccurate. In order to produce safe and economical design guidelines, this paper experimentally investigates the actual interactive relationship between bending of the uprights about the major and minor axes, for local and distortional buckling. Two types of regularly perforated and nonperforated storage rack uprights are investigated. Results show that a nonlinear interactive relationship governs the biaxial bending of the studied uprights, and the linear Interaction Equation in design specifications underestimates the biaxial bending capacity by up to 44 and 68% for local and distortional buckling, respectively. Also, the accuracy of the direct strength method (DSM) to directly predict the local and distortional buckling strengths of the uprights under biaxial bending is investigated. Results show that DSM Equations provide better predictions but still underestimate the biaxial bending capacity by up to 27 and 36% for local and distortional buckling, respectively.Full Tex