31 research outputs found

    Adaptive techniques for solving chaotic system of parabolic-type

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    Time-dependent partial differential equations of parabolic type are known to have a lot of applications in biology, mechanics, epidemiology and control processes. Despite the usefulness of this class of differential equations, the numerical approach to its solution, especially in high dimensions, is still poorly understood. Since the nature of reaction-diffusion problems permit the use of different methods in space and time, two important approximation schemes which are based on the spectral and barycentric interpolation collocation techniques are adopted in conjunction with the third-order exponential time-differencing Runge-Kutta method to advance in time. The accuracy of the method is tested by considering a number of time-dependent reaction-diffusion problems that are still of current and recurring interests in one and high dimensions.© 2022 The Authors. Published by Elsevier B.V. on behalf of African Institute of Mathematical Sciences / Next Einstein Initiative. This is an open access article under the CC BY license.http://www.elsevier.com/locate/sciafhj2023Mathematics and Applied Mathematic

    International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

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    The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

    New Optimal Periodic Control Policy for the Optimal Periodic Performance of a Chemostat Using a Fourier-Gegenbauer-Based Predictor-Corrector Method

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    In its simplest form, the chemostat consists of microorganisms or cells which grow continually in a specific phase of growth while competing for a single limiting nutrient. Under certain conditions on the cells' growth rate, substrate concentration, and dilution rate, the theory predicts and numerical experiments confirm that a periodically operated chemostat exhibits an "over-yielding" state in which the performance becomes higher than that at the steady-state operation. In this paper we show that an optimal control policy for maximizing the chemostat performance can be accurately and efficiently derived numerically using a novel class of integral-pseudospectral methods and adaptive h-integral-pseudospectral methods composed through a predictor-corrector algorithm. Some new formulas for the construction of Fourier pseudospectral integration matrices and barycentric shifted Gegenbauer quadratures are derived. A rigorous study of the errors and convergence rates of shifted Gegenbauer quadratures as well as the truncated Fourier series, interpolation operators, and integration operators for nonsmooth and generally T-periodic functions is presented. We introduce also a novel adaptive scheme for detecting jump discontinuities and reconstructing a discontinuous function from the pseudospectral data. An extensive set of numerical simulations is presented to support the derived theoretical foundations.Comment: 35 pages, 20 figure

    Structure-Preserving Model Reduction of Physical Network Systems

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    This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

    Multiscale aeroelastic modelling in porous composite structures

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    Driven by economic, environmental and ergonomic concerns, porous composites are increasingly being adopted by the aeronautical and structural engineering communities for their improved physical and mechanical properties. Such materials often possess highly heterogeneous material descriptions and tessellated/complex geometries. Deploying commercially viable porous composite structures necessitates numerical methods that are capable of accurately and efficiently handling these complexities within the prescribed design iterations. Classical numerical methods, such as the Finite Element Method (FEM), while extremely versatile, incur large computational costs when accounting for heterogeneous inclusions and high frequency waves. This often renders the problem prohibitively expensive, even with the advent of modern high performance computing facilities. Multiscale Finite Element Methods (MsFEM) is an order reduction strategy specifically developed to address such issues. This is done by introducing meshes at different scales. All underlying physics and material descriptions are explicitly resolved at the fine scale. This information is then mapped onto the coarse scale through a set of numerically evaluated multiscale basis functions. The problems are then solved at the coarse scale at a significantly reduced cost and mapped back to the fine scale using the same multiscale shape functions. To this point, the MsFEM has been developed exclusively with quadrilateral/hexahedral coarse and fine elements. This proves highly inefficient when encountering complex coarse scale geometries and fine scale inclusions. A more flexible meshing scheme at all scales is essential for ensuring optimal simulation runtimes. The Virtual Element Method (VEM) is a relatively recent development within the computational mechanics community aimed at handling arbitrary polygonal (potentially non-convex) elements. In this thesis, novel VEM formulations for poromechanical problems (consolidation and vibroacoustics) are developed. This is then integrated at the fine scale into the multiscale procedure to enable versatile meshing possibilities. Further, this enhanced capability is also extended to the coarse scale to allow for efficient macroscale discretizations of complex structures. The resulting Multiscale Virtual Element Method (MsVEM) is originally applied to problems in elastostatics, consolidation and vibroacoustics in porous media to successfully drive down computational run times without significantly affecting accuracy. Following this, a parametric Model Order Reduction scheme for coupled problems is introduced for the first time at the fine scale to obtain a Reduced Basis Multiscale Virtual Element Method. This is used to augment the rate of multiscale basis function evaluation in spectral acoustics problems. The accuracy of all the above novel contributions are investigated in relation to standard numerical methods, i.e., the FEM and MsFEM, analytical solutions and experimental data. The associated efficiency is quantified in terms of computational run-times, complexity analyses and speed-up metrics. Several extended applications of the VEM and the MsVEM are briefly visited, e.g., VEM phase field Methods for brittle fracture, structural and acoustical topology optimization, random vibrations and stochastic dynamics, and structural vibroacoustics

    Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

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    This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

    Realizability-preserving discretization strategies for hyperbolic and kinetic equations with uncertainty

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    Applications

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    Analysis and applications of dynamic density functional theory

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    Classical fluid mechanics and, in particular, the general compressible Navier-Stokes-Fourier equations, have long been of great use in the prediction and understanding of the flow of fluids in various scenarios. While the classical theory is well established in increasingly rigorous mathematical frameworks, the atomistic properties and microscopic processes of fluids must be considered by other means. A central problem in fluid mechanics concerns capturing microscopic effects in meso/macroscopic continuum models. With more attention given to the non-Newtonian properties of many naturally occurring fluid flows, resolving the gaps between the atomistic viewpoint and the continuum approach of Navier-Stokes-Fourier is a rich and open field. This thesis centres on the modelling, analysis and computation of one continuum method designed to resolve the highly multiscale nature of non-equilibrium fluid flow on the particle scale: Dynamic Density Functional Theory (DDFT). A generalised version of DDFT is derived from first principles to include: driven flow, inertia and hydrodynamic interactions (HI) and it is observed that the equations reproduce known dynamics in heuristic overdamped and inviscid limits. Also included are rigorous, analytical derivations of the short-range lubrication forces on particles at low Reynolds number, with accompanying asymptotic theory, uniformly valid in the entire regime of particle distances and size ranges, which were previously unknown. As well as demonstrating an improvement on known classical results, these calculations were determined necessary to comply with the continuous nature of the integro-differential equations for DDFT. The numerical implementation of the driven, inertial equations with short range HI for a range of colloidal systems in confining geometries is also included by developing the pseudo-spectral collocation scheme 2DChebClass [67]. A further area of interest for non-equilibrium fluids is mathematical well–posedness. This thesis provides, for the first time, the existence and uniqueness of weak solutions to an overdamped DDFT with HI, as well as a rigorous investigation of its equilibrium behaviour
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