11 research outputs found

    MAGNETOHİDRODİNAMİK KANAL AKIŞLARININ KARŞILIKLI SINIR ELEMANLARI METODU İLE ÇÖZÜMÜ

    Get PDF
    In the thesis, four different MHD duct flow problems are solved by using the Dual Reciprocity Boundary Element Method (DRBEM) with the suitable boundary conditions according to the physics of the problem. The two-dimensional, steady or unsteady, fully-developed MHD flow of a viscous, incompressible and electrically conducting fluid is considered in a long pipe of rectangular cross-section (duct) under the effect of an externally applied magnetic field which is either uniform or time-dependent or axially changing. The inductionless MHD flow with temperature dependent viscosity and heat transfer is the first considered problem. In this problem, the induced magnetic field is neglected due to the small magnetic Reynolds number assumption. Secondly, the MHD duct flow under a time-varied external magnetic field is studied. Then, we turn our concern to MHD flow problems under an axial-dependent magnetic field varying in the streamwise direction (pipe-axis direction) in the third and the fourth problems. Specifically, the inductionless MHD flow with electric potential is considered under the effect of the axially-changing magnetic field as the third problem. Adding the induced magnetic field to the velocity and electric potential equations as a triple is the last MHD flow problem considered in the thesis. The parametrix BEM implementation is also presented for the solution of the variable coefficient convection-diffusion type equations. The influence of the magnetic fields on the MHD flows is investigated and simulated in terms of the velocity, temperature, induced magnetic field and electric potential contours for several values of physical parameters.Bu tezde, dört farklı Magnetohidrodinamik (MHD) kanal akış problemi, problemin fiziğine göre uygun sınır koşulları ile birlikte karşılıklı sınır elemanları metodu (DRBEM) kullanılarak çözülmüştür. Viskoz, sıkıştırılamaz ve elektrik ileten sıvının dikdörtgen kesitli bir kanal içerisindeki iki boyutlu, zamana bağlı veya zamandan bağımsız tam gelişmiş akışı dışarıdan uygulanan bir manyetik alan etkisinde incelenmiştir. Akışı etkileyen manyetik alan ya tek düzedir ya zamana bağlıdır ya da eksenel olarak değişmektedir. Ele alınan ilk problem, sıcaklığa bağlı viskoziteye ve ısı transferine sahip indüksiyonsuz MHD akışıdır. Bu problemde, indüklenen manyetik alan küçük manyetik Reynolds sayısı varsayımından dolayı ihmal edilmiştir. İkinci problem olarak, dışarıdan uygulanan ve zamana bağlı manyetik alan etkisindeki MHD akış çalışılmıştır. Daha sonra ise, üçüncü ve dördüncü problem olarak akım yönündeki eksen boyunca değişen bir manyetik alan etkisindeki MHD akış problemleri çözülmüştür. Üçüncü problemdeki MHD akışı elektrik potansiyeline sahip fakat indüksiyonsuz bir akıştır. Dördüncü problemde ise üçüncü problemdeki MHD akışa indüklenen manyetik alan eklenerek problem denklemleri hız, elektrik potansiyel ve indüklenen manyetik alan olarak üçlü çözülmüştür. Değişken katsayılı konveksiyon-difüzyon tipi denklemlerin çözümü için parametre sınır elemanı metodu (parametrix BEM) da kullanılmıştır. Uygulanan manyetik alanların MHD akışlarına etkisi, çeşitli fiziksel problem parametre değerleri için hız, sıcaklık, indüklenen manyetik alan ve elektrik potansiyeli açısından incelenmiş ve simülasyonları yapılmıştır.Ph.D. - Doctoral Progra

    FastSVD-ML-ROM\textit{FastSVD-ML-ROM}: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applications

    Full text link
    Digital twins have emerged as a key technology for optimizing the performance of engineering products and systems. High-fidelity numerical simulations constitute the backbone of engineering design, providing an accurate insight into the performance of complex systems. However, large-scale, dynamic, non-linear models require significant computational resources and are prohibitive for real-time digital twin applications. To this end, reduced order models (ROMs) are employed, to approximate the high-fidelity solutions while accurately capturing the dominant aspects of the physical behavior. The present work proposes a new machine learning (ML) platform for the development of ROMs, to handle large-scale numerical problems dealing with transient nonlinear partial differential equations. Our framework, mentioned as FastSVD-ML-ROM\textit{FastSVD-ML-ROM}, utilizes (i)\textit{(i)} a singular value decomposition (SVD) update methodology, to compute a linear subspace of the multi-fidelity solutions during the simulation process, (ii)\textit{(ii)} convolutional autoencoders for nonlinear dimensionality reduction, (iii)\textit{(iii)} feed-forward neural networks to map the input parameters to the latent spaces, and (iv)\textit{(iv)} long short-term memory networks to predict and forecast the dynamics of parametric solutions. The efficiency of the FastSVD-ML-ROM\textit{FastSVD-ML-ROM} framework is demonstrated for a 2D linear convection-diffusion equation, the problem of fluid around a cylinder, and the 3D blood flow inside an arterial segment. The accuracy of the reconstructed results demonstrates the robustness and assesses the efficiency of the proposed approach.Comment: 35 pages, 22 figure

    Computational and numerical analysis of differential equations using spectral based collocation method.

    Get PDF
    Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally efficient spectral collocation-based methods, both modified and new, and apply them to solve differential equations. Spectral collocation-based methods are the most commonly used methods for approximating smooth solutions of differential equations defined over simple geometries. Procedurally, these methods entail transforming the gov erning differential equation(s) into a system of linear algebraic equations that can be solved directly. Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported in the literature, researchers often transform their models to reduce the number of variables or narrow them down to problems with fewer dimensions. Such a process is accomplished by making a series of assumptions that limit the scope of the study. To address this deficiency, the present study explores the development of numerical algorithms for solving ordinary and partial differential equations defined over simple geometries. The solutions of the differential equations considered are approximated using interpolating polynomials that satisfy the given differential equation at se lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the computational domain is particularly emphasized as it plays a key role in determining the number of grid points that are used; a feature that dictates the accuracy and the computational expense of the spectral method. To solve differential equations defined on large computational domains much effort is devoted to the development and application of new multidomain approaches, based on decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con firms the superiority of these multiple domain techniques in terms of accuracy and computational efficiency over the single domain approach when applied to problems defined over large domains. The structure of the thesis indicates a smooth sequence of constructing spectral collocation method algorithms for problems across different dimensions. The process of switching between dimensions is explained by presenting the work in chronological order from a simple one-dimensional problem to more complex higher-dimensional problems. The preliminary chapter explores solutions of or dinary differential equations. Subsequent chapters then build on solutions to partial differential i equations in order of increasing computational complexity. The transition between intermediate dimensions is demonstrated and reinforced while highlighting the computational complexities in volved. Discussions of the numerical methods terminate with development and application of a new method namely; the trivariate spectral collocation method for solving two-dimensional initial boundary value problems. Finally, the new error bound theorems on polynomial interpolation are presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical algorithms. The numerical results of the study confirm that incorporating domain decomposition techniques in spectral collocation methods work effectively for all dimensions, as we report highly accurate results obtained in a computationally efficient manner for problems defined on large do mains. The findings of this study thus lay a solid foundation to overcome major challenges that numerical analysts might encounter

    Нестаціонарні теплові процеси в анізотропних твердих тілах

    Get PDF
    Дисертацію присвячено дослідженню теплових процесів в анізотропних твердих тілах за допомогою безсіткового методу розв’язання тривимірних задач нестаціонарної теплопровідності. Серед великого розмаїття задач математичної фізики, які нині успішно вирішуються, особливу роль займають задачі теплопровідності в анізотропних матеріалах. Насамперед це пов’язано з активним використанням анізотропних матеріалів при виготовленні великої кількості сучасних приладів та пристроїв, деталей конструкцій та машин – наприклад, трансформаторів із сердечниками з текстурованої сталі (в електротехніці), лопаток газотурбінних двигунів із жароміцних нікелевих сплавів з монокристалічною структурою (в авіації), п’єзоперетворювачів, електрооптичних модуляторів та рідкокристалічних індикаторів (в електронному приладобудуванні). Сучасні анізотропні матеріали зі складною структурою (наприклад, композитні матеріали, багатошарові матеріали, покриття, нанесені на підкладки) все частіше використовуються в новітніх інженерних розробках, а також в якості конструкційних матеріалів.У різних технологічних процесах і пристроях дані матеріали піддаються тепловому впливу, внаслідок чого в них відбуваються фізико-хімічні явища, зокрема зміна геометричних розмірів. Неконтрольоване теплове розширення конструкційних матеріалів може призвести до погіршення експлуатаційних характеристик пристрою, а також до аварійних ситуацій. Тому при створенні та використанні таких матеріалів необхідно враховувати анізотропію їх теплофізичних властивостей, а також досліджувати теплові процеси, які в них протікають. The dissertation deals with the study of thermal processes in anisotropic solids by meshless method for solving three-dimensional non-stationary heat conduction problems. Heat conduction problems in anisotropic solids play a significant role among the wide variety of problems of mathematical physics which are currently being successfully solved. First of all, it is associated with the active use of anisotropic materials in the manufacture of a large number of modern instruments and devices, structural parts and machines. For example, transformers with textured steel cores (in electrical engineering), gas-turbine engine blades of heat-resistant nickel alloys with a single-crystal structure (in aviation), piezoelectric transducers, electro-optic modulators and liquid crystal indicators (in electronic instrument engineering). Modern anisotropic materials with a complex structure (composite materials, multilayered materials, coatings on substrates, etc.) are increasingly used in advanced engineering designs and as structural materials. In various technological processes and devices, these materials are exposed to thermal effects, resulting in physical and chemical phenomena, including changes in geometric parameters. Uncontrolled thermal expansion of structural materials may lead to device performance degradation, as well as to emergency situations. Therefore, when creating and using such materials, it is necessary to take into account the anisotropy of their thermophysical properties, as well as to study the thermal processes occurring in them

    Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

    Get PDF
    We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

    Solidification and Gravity VII

    Get PDF
    International audienc

    MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

    Get PDF
    Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

    Generalized averaged Gaussian quadrature and applications

    Get PDF
    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

    An explicit meshless point collocation method for electrically driven magnetohydrodynamics (MHD) flow

    No full text
    In this paper, we develop a meshless collocation scheme for the numerical solution of magnetohydrodynamics (MHD) flow equations. We consider the transient laminar flow of an incompressible, viscous and electrically conducting fluid in a rectangular duct. The flow is driven by the current produced by electrodes placed on the walls of the duct. The method combines a meshless collocation scheme with the newly developed Discretization Corrected Particle Strength Exchange (DC PSE) interpolation method. To highlight the applicability of the method, we discretize the spatial domain by using uniformly (Cartesian) and irregularly distributed nodes. The proposed solution method can handle high Hartmann (Ha) numbers and captures the boundary layers formed in such cases, without the presence of unwanted oscillations, by employing a local mesh refinement procedure close to the boundaries. The use of local refinement reduces the computational cost. We apply an explicit time integration scheme and we compute the critical time step that ensures stability through the Gershgorin theorem. Finally, we present numerical results obtained using different orientation of the applied magnetic field
    corecore