Academic literature on the topic 'Uniformly accurate numerical methods'
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Journal articles on the topic "Uniformly accurate numerical methods"
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Chartier, Philippe, Loïc Le Treust, and Florian Méhats. "Uniformly accurate time-splitting methods for the semiclassical linear Schrödinger equation." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 2 (2019): 443–73. http://dx.doi.org/10.1051/m2an/2018060.
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This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schrödinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.
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Shishkin, G. I. "ROBUST NOVEL HIGH-ORDER ACCURATE NUMERICAL METHODS FOR SINGULARLY PERTURBED CONVECTION‐DIFFUSION PROBLEMS." Mathematical Modelling and Analysis 10, no. 4 (2005): 393–412. http://dx.doi.org/10.3846/13926292.2005.9637296.
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For singularly perturbed boundary value problems, numerical methods convergent ϵ‐uniformly have the low accuracy. So, for parabolic convection‐diffusion problem the order of convergence does not exceed one even if the problem data are sufficiently smooth. However, already for piecewise smooth initial data this order is not higher than 1/2. For problems of such type, using newly developed methods such as the method based on the asymptotic expansion technique and the method of the additive splitting of singularities, we construct ϵ‐uniformly convergent schemes with improved order of accuracy. Straipsnyje nagrinejami nedidelio tikslumo ϵ‐tolygiai konvertuojantys skaitmeniniai metodai, singuliariai sutrikdytiems kraštiniams uždaviniams. Paraboliniam konvekcijos‐difuzijos uždaviniui konvergavimo eile neviršija vienos antrosios, jeigu uždavinio duomenys yra pakankamai glodūs. Tačiau trūkiems pradiniams duomenims eile yra ne aukštesne už 2−1. Šio tipo uždaviniams, naudojant naujai išvestus metodus, darbe sukonstruotos ϵ‐tolygiai konvertuojančios schemos aukštesniu tikslumu.
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Su, Chunmei, and Xiaofei Zhao. "On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (2020): 1491–508. http://dx.doi.org/10.1051/m2an/2020006.
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In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.
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DEBELA, HABTAMU GAROMA, and GEMECHIS FILE DURESSA. "Fitted Operator Finite Difference Method for Singularly Perturbed Differential Equations with Integral Boundary Condition." Kragujevac Journal of Mathematics 47, no. 4 (2003): 637–51. http://dx.doi.org/10.46793/kgjmat2304.637d.
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This study presents a fitted operator numerical method for solving singularly perturbed boundary value problems with integral boundary condition. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, a model problem is considered for numerical experimentation and solved for different values of the perturbation parameter, ε and mesh size, h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result and it also improves the results of the methods existing in the literature.
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Debela, Habtamu Garoma, and Gemechis File Duressa. "Uniformly Convergent Nonpolynomial Spline Method for Singularly Perturbed Robin-Type Boundary Value Problems with Discontinuous Source Term." Abstract and Applied Analysis 2021 (October 22, 2021): 1–12. http://dx.doi.org/10.1155/2021/7569209.
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In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε -uniformly convergent for h ≥ ε where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.
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Cai, Yongyong, and Yan Wang. "A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 2 (2018): 543–66. http://dx.doi.org/10.1051/m2an/2018015.
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A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (ε2) and O (1) in time and space, respectively. In the nonrelativistic regime,i.e., 0 < ε ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds inε ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as hm0+τ2/ε2andhm0 + τ2 + ε2, where his the mesh size, τis the time step and m0depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O (τ2) in the regimes when either ε = O (1) or 0 < ε ≲ τ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.
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Yoon, Daegeun, and Donghyun You. "An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative." Fractional Calculus and Applied Analysis 24, no. 5 (2021): 1356–79. http://dx.doi.org/10.1515/fca-2021-0058.
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Abstract A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the accuracy of the fractional derivative, a novel numerical method for the Caputo fractional derivative is proposed. The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating convolution weights to function values at past time points which can be non-uniformly distributed in time. The superior accuracy of the present method to the accuracy of the previously reported methods is identified by deriving numerical errors analytically. The sub-diffusion process of a time-fractional diffusion equation is simulated to demonstrate the accuracy as well as the computational efficiency of the present method.
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A.B., Kerimov,. "Accuracy comparison of signal recognition methods on the example of a family of successively horizontally displaced curves." Informatics and Control Problems, no. 2(6) (November 18, 2022): 80–91. http://dx.doi.org/10.54381/icp.2022.2.10.
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In some cases, to compare recognition methods the criterion of the total percentage ratio of the proximity of recognized signals to the reference ones is applied. This study proposes a slightly different approach, involving a numerical evaluation when comparing two or more signal recognition methods on the example of an artificially created family of successively and uniformly horizontally displaced curves.
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Xu, Jian-Zhong, and Wen-Sheng Yu. "On the Slightly Reduced Navier-Stokes Equations." Journal of Fluids Engineering 119, no. 1 (1997): 90–95. http://dx.doi.org/10.1115/1.2819124.
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In this paper the so-called slightly reduced Navier-Stokes (SRNS) equations with most streamwise viscous diffusion and heat conduction terms are investigated in detail. It is proved that the SRNS equations are hyperbolic-parabolic in mathematics, which is the same as the current RNS or PNS equations. The numerical methods for solving the RNS equations are, therefore, applicable to the present SRNS equations. It is further proved that the SRNS equations have a uniformly convergent solution with accuracy of 0 (ε2) or 0 (Re−1) which is higher than that of the RNS equations, and for a laminar flow past a flat plate the SRNS solution is regular at the point of separation and is a precise approximation to that of the complete Navier-Stokes equations. The numerical results demonstrate that the SRNS equations may give accurate picture of the flow and are an effective tool in analyzing complex flows.
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Han, Houde, Min Tang, and Wenjun Ying. "Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with Boundary and Interface Layers." Communications in Computational Physics 15, no. 3 (2014): 797–826. http://dx.doi.org/10.4208/cicp.130413.010813a.
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AbstractThis paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime, which is valid up to the boundary and interface layers. A five-point node-centered and a four-point cell-centered tailored finite point schemes (TFPS) are introduced. The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system. Numerically, both methods can not only capture the diffusion limit, but also exhibit uniform convergence in the diffusive regime, even with boundary layers. Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy, uniformly with respect to the mean free path. Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.
Dissertations / Theses on the topic "Uniformly accurate numerical methods"
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Bouchereau, Maxime. "Modélisation de phénomènes hautement oscillants par réseaux de neurones." Electronic Thesis or Diss., Université de Rennes (2023-....), 2024. http://www.theses.fr/2024URENS034.
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Cette thèse porte sur l’application du Machine Learning à l’étude d’équations différentielles fortement oscillantes. Plus précisément, on s’intéresse à une manière d’approcher de manière précise et avec le moins de calculs possible la solution d’une équation différentielle en s’aidant de réseaux de neurones. Tout d’abord, le cas autonome est étudié, où les propriétés de l’analyse rétrograde et des réseaux de neurones sont utilisés afin d’améliorer des méthodes numériques existantes, puis une généralisation au cas fortement oscillant est proposée afin d’améliorer un schéma numérique d’ordre un spécifique à ce cas de figure. Ensuite, les réseaux de neurones sont utilisés afin de remplacer les calculs préalables nécessaires à l’implémentation de méthodes numériques uniformément précises permettant d’approcher les solutions d’équations fortement oscillantes, que ce soit en partant des travaux mis en œuvre pour le cas autonome, ou bien en utilisant une structure de réseau de neurone intégrant directement la structure de l’équation<br>This thesis focuses on the application of Machine Learning to the study of highly oscillatory differential equations. More precisely, we are interested in an approach to accurately approximate the solution of a differential equation with the least amount of computations, using neural networks. First, the autonomous case is studied, where the proper- ties of backward analysis and neural networks are used to enhance existing numerical methods. Then, a generalization to the strongly oscillating case is proposed to improve a specific first-order numerical scheme tailored to this scenario. Subsequently, neural networks are employed to replace the necessary pre- computations for implementing uniformly ac- curate numerical methods to approximate so- lutions of strongly oscillating equations. This can be done either by building upon the work done for the autonomous case or by using a neural network structure that directly incorporates the equation’s structure
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Baumstark, Simon [Verfasser]. "Uniformly Accurate Methods for Klein-Gordon type Equations / Simon Baumstark." Karlsruhe : KIT-Bibliothek, 2018. http://d-nb.info/1171315880/34.
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Stewart, Dawn L. "Numerical Methods for Accurate Computation of Design Sensitivities." Diss., Virginia Tech, 1998. http://hdl.handle.net/10919/30561.
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This work is concerned with the development of computational methods for
approximating sensitivities of solutions to boundary value problems.
We focus on the continuous sensitivity equation method and
investigate the application of adaptive meshing and smoothing
projection techniques to enhance the basic scheme. The fundamental
ideas are first developed for a one dimensional problem and then
extended to 2-D flow problems governed by the incompressible
Navier-Stokes equations. Numerical experiments are conducted to test the
algorithms and to investigate the benefits of adaptivity and smoothing.<br>Ph. D.
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Pasdunkorale, Arachchige Jayantha. "Accurate finite volume methods for the numerical simulation of transport in highly anistropic media." Thesis, Queensland University of Technology, 2003.
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Hübner, Thomas [Verfasser]. "A monolithic, off-lattice approach to the discrete Boltzmann equation with fast and accurate numerical methods / Thomas Hübner." Dortmund : Universitätsbibliothek Technische Universität Dortmund, 2011. http://d-nb.info/1011570777/34.
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Zhao, Wei [Verfasser], Martin [Akademischer Betreuer] Stoll, Martin [Gutachter] Stoll, and Benny Y. C. [Akademischer Betreuer] Hon. "Accurate and efficient numerical methods for nonlocal problems / Wei Zhao ; Gutachter: Martin Stoll ; Martin Stoll, Benny Y.C. Hon." Chemnitz : Technische Universität Chemnitz, 2019. http://d-nb.info/1215909780/34.
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Sharify, Meisam. "Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis : Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues." Palaiseau, Ecole polytechnique, 2011. http://pastel.archives-ouvertes.fr/docs/00/64/38/36/PDF/thesis.pdf.
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L'Algèbre tropicale peut être considérée comme un domaine relativement nouveau en mathématiques. Elle apparait dans plusieurs domaines telles que l'optimisation, la synchronisation de la production et du transport, les systèmes à événements discrets, le contrôle optimal, la recherche opérationnelle, etc. La première partie de ce manuscrit est consacrée a l'étude des applications de l'algèbre tropicale à l'analyse numérique matricielle. Nous considérons tout d'abord le problème classique de l'estimation des racines d'un polynôme univarié. Nous prouvons plusieurs nouvelles bornes pour la valeur absolue des racines d'un polynôme en exploitant les méthodes tropicales. Ces résultats sont particulièrement utiles lorsque l'on considère des polynômes dont les coefficients ont des ordres de grandeur différents. Nous examinons ensuite le problème du calcul des valeurs propres d'une matrice polynomiale. Ici, nous introduisons une technique de mise à l'échelle générale, basée sur l'algèbre tropicale, qui s'applique en particulier à la forme compagnon. Cette mise à l'échelle est basée sur la construction d'une fonction polynomiale tropicale auxiliaire, ne dépendant que de la norme des matrices. Les raciness (les points de non-différentiabilité) de ce polynôme tropical fournissent une pré-estimation de la valeur absolue des valeurs propres. Ceci se justifie en particulier par un nouveau résultat montrant que sous certaines hypothèses faites sur le conditionnement, il existe un groupe de valeurs propres bornées en norme. L'ordre de grandeur de ces bornes est fourni par la plus grande racine du polynôme tropical auxiliaire. Un résultat similaire est valable pour un groupe de petites valeurs propres. Nous montrons expérimentalement que cette mise à l'échelle améliore la stabilité numérique, en particulier dans des situations où les données ont des ordres de grandeur différents. Nous étudions également le problème du calcul des valeurs propres tropicales (les points de non-différentiabilité du polynôme caractéristique) d'une matrice polynômiale tropicale. Du point de vue combinatoire, ce problème est équivalent à trouver une fonction de couplage: la valeur d'un couplage de poids maximum dans un graphe biparti dont les arcs sont valués par des fonctions convexes et linéaires par morceaux. Nous avons développé un algorithme qui calcule ces valeurs propres tropicales en temps polynomial. Dans la deuxième partie de cette thèse, nous nous intéressons à la résolution de problèmes d'affectation optimale de très grande taille, pour lesquels les algorithms séquentiels classiques ne sont pas efficaces. Nous proposons une nouvelle approche qui exploite le lien entre le problème d'affectation optimale et le problème de maximisation d'entropie. Cette approche conduit à un algorithme de prétraitement pour le problème d'affectation optimale qui est basé sur une méthode itérative qui élimine les entrées n'appartenant pas à une affectation optimale. Nous considérons deux variantes itératives de l'algorithme de prétraitement, l'une utilise la méthode Sinkhorn et l'autre utilise la méthode de Newton. Cet algorithme de prétraitement ramène le problème initial à un problème beaucoup plus petit en termes de besoins en mémoire. Nous introduisons également une nouvelle méthode itérative basée sur une modification de l'algorithme Sinkhorn, dans lequel un paramètre de déformation est lentement augmenté. Nous prouvons que cette méthode itérative(itération de Sinkhorn déformée) converge vers une matrice dont les entrées non nulles sont exactement celles qui appartiennent aux permutations optimales. Une estimation du taux de convergence est également présentée<br>Tropical algebra, which can be considered as a relatively new field in Mathematics, emerged in several branches of science such as optimization, synchronization of production and transportation, discrete event systems, optimal control, operations research, etc. The first part of this manuscript is devoted to the study of the numerical applications of tropical algebra. We start by considering the classical problem of estimating the roots of a univariate complex polynomial. We prove several new bounds for the modulus of the roots of a polynomial exploiting tropical methods. These results are specially useful when considering polynomials whose coefficients have different orders of magnitude. We next consider the problem of computing the eigenvalues of a matrix polynomial. Here, we introduce a general scaling technique, based on tropical algebra, which applies in particular to the companion form. This scaling is based on the construction of an auxiliary tropical polynomial function, depending only on the norms of the matrices. The roots (non-differentiability points) of this tropical polynomial provide a priori estimates of the modulus of the eigenvalues. This is justified in particular by a new location result, showing that under assumption involving condition numbers, there is one group of large eigenvalues, which have a maximal order of magnitude, given by the largest root of the auxiliary tropical polynomial. A similar result holds for a group of small eigenvalues. We show experimentally that this scaling improves the backward stability of the computations, particularly in situations when the data have various orders of magnitude. We also study the problem of computing the tropical eigenvalues (non-differentiability points of the characteristic polynomial) of a tropical matrix polynomial. From the combinatorial perspective, this problem can be interpreted as finding the maximum weighted matching function in a bipartite graph whose arcs are valued by convex piecewise linear functions. We developed an algorithm which computes the tropical eigenvalues in polynomial time. In the second part of this thesis, we consider the problem of solving very large instances of the optimal assignment problems (so that standard sequential algorithms cannot be used). We propose a new approach exploiting the connection between the optimal assignment problem and the entropy maximization problem. This approach leads to a preprocessing algorithm for the optimal assignment problem which is based on an iterative method that eliminates the entries not belonging to an optimal assignment. We consider two variants of the preprocessing algorithm, one by using the Sinkhorn iteration and the other by using Newton iteration. This preprocessing algorithm can reduce the initial problem to a much smaller problem in terms of memory requirements. We also introduce a new iterative method based on a modification of the Sinkhorn scaling algorithm, in which a deformation parameter is slowly increased We prove that this iterative method, referred to as the deformed-Sinkhorn iteration, converges to a matrix whose nonzero entries are exactly those belonging to the optimal permutations. An estimation of the rate of convergence is also presented
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Zhao, Wei. "Accurate and efficient numerical methods for nonlocal problems." 2018. https://monarch.qucosa.de/id/qucosa%3A33818.
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In this thesis, we study several nonlocal models to obtain their numerical solutions accurately and efficiently. In contrast to the classical (local) partial
differential equation models, these nonlocal models are integro-differential equations that do not contain spatial derivatives. As a result, these nonlocal
models allow their solutions to have discontinuities. Hence, they can be widely used for fracture problems and anisotropic problems.
This thesis mainly includes two parts. The first part focuses on presenting accurate and efficient numerical methods. In this part, we first
introduce three meshless methods including two global schemes, namely the radial basis functions collocation method (RBFCM) and the radial ba-
sis functions-based pseudo-spectral method (RBF-PSM) and a localized scheme, namely the localized radial basis functions-based pseudo-spectral
method (LRBF-PSM), which also gives the development process of the RBF methods from global to local. The comparison of these methods
shows that LRBF-PSM not only avoids the Runge phenomenon but also has similar accuracy to the global scheme. Since the LRBF-PSM uses only
a small subset of points, the calculation consumes less CPU time. Afterwards, we improve this scheme by adding enrichment functions so that it
can be effectively applied to discontinuity problems. This thesis abbreviates this enriched method as LERBF-PSM (Localized enriched radial basis
functions-based pseudo-spectral method).
In the second part, we focus on applying the derived methods from the first part to nonlocal topics of current research, including nonlocal
diffusion models, linear peridynamic models, parabolic/hyperbolic nonlocal phase field models, and nonlocal nonlinear Schrödinger equations
arising in quantum mechanics. The first point worth noting is that in order to verify the meshless nature of LRBF-PSM, we apply this method to
solve a two-dimensional steady-state continuous peridynamic model in regular, irregular (L-shaped and Y-shaped) domains with uniform and non-uniform discretizations and even extend this method to three dimensions. It is also worth noting that before solving nonlinear nonlocal Schrödinger equations, according to the property of the convolution, these partial integro-differential equations are transformed into equivalent or approximate partial differential equations (PDEs) in the whole space and then the LRBF-PSM is used for the spatial discretization in a finite domain with suitable boundary conditions. Therefore, the solutions can be quickly approximated.
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(8718126), Duo Cao. "Efficient and accurate numerical methods for two classes of PDEs with applications to quasicrystals." Thesis, 2020.
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This dissertation is a summary of the graduate study in the past few years. In first part, we develop efficient spectral methods for the spectral fractional Laplacian equation and parabolic PDEs with spectral fractional Laplacian on rectangular domains. The key idea is to construct eigenfunctions of discrete Laplacian (also referred to Fourier-like basis) by using the Fourierization method. Under this basis, the nonlocal fractional Laplacian operator can be trivially evaluated, leading to very efficient algorithms for PDEs involving spectral fractional Laplacian. We provide a rigorous error analysis for the proposed methods, as well as ample numerical results to show their effectiveness.<div><br>In second part, we propose a method suitable for the computation of quasiperiodic interface, and apply it to simulate the interface between ordered phases in Lifschitz-Petrich model, which can be quasiperiodic. The function space, initial and boundary conditions are carefully chosen such that it fix the relative orientation and displacement, and we follow a gradient flow to let the interface and its optimal structure. The gradient flow is discretized by the scalar auxiliary variable (SAV) approach in time, and spectral method in space using quasiperiodic Fourier series and generalized Jacobi<br>polynomials. We use the method to study interface between striped, hexagonal and dodecagonal phases, especially when the interface is quasiperiodic. The numerical examples show that our method is efficient and accurate to successfully capture the interfacial structure.</div>
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Jaisankar, S. "Accurate Computational Algorithms For Hyperbolic Conservation Laws." Thesis, 2008. https://etd.iisc.ac.in/handle/2005/905.
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The numerics of hyperbolic conservation laws, e.g., the Euler equations of gas dynamics, shallow water equations and MHD equations, is non-trivial due to the convective terms being highly non-linear and equations being coupled. Many numerical methods have been developed to solve these equations, out of which central schemes and upwind schemes (such as Flux Vector Splitting methods, Riemann solvers, Kinetic Theory based Schemes, Relaxation Schemes etc.) are well known. The majority of the above mentioned schemes give rise to very dissipative solutions. In this thesis, we propose novel low dissipative numerical algorithms for some hyperbolic conservation laws representing fluid flows. Four different and independent numerical methods which give low diffusive solutions are developed and demonstrated.
The first idea is to regulate the numerical diffusion in the existing dissipative schemes so that the smearing of solution is reduced. A diffusion regulator model is developed and used along with the existing methods, resulting in crisper shock solutions at almost no added computational cost. The diffusion regulator is a function of jump in Mach number across the interface of the finite volume and the average Mach number across the surface. The introduction of the diffusion regulator makes the diffusive parent schemes to be very accurate and the steady contact discontinuities are captured exactly. The model is demonstrated in improving the diffusive Local Lax-Friedrichs (LLF) (or Rusanov) method and a Kinetic Scheme. Even when employed together with accurate methods of Roe and Osher, improvement in solutions is demonstrated for multidimensional problems.
The second method, a Central Upwind-Biased Scheme (CUBS), attempts to reorganize a central scheme such that information from irrelevant directions is largely reduced and the upwind biased information is retained. The diffusion co-efficient follows a new format unlike the use of maximum characteristic speed in the Local Lax-Friedrichs method and the scheme results in improved solutions of the flow features. The grid-aligned steady contacts are captured exactly with the reorganized format of diffusion co-efficient. The stability and positivity of the scheme are discussed and the procedure is demonstrated for its ability to capture all the features of solution for different flow problems.
Another method proposed in this thesis, a Central Rankine-Hugoniot Solver, attempts to integrate more physics into the discretization procedure by enforcing a simplified Rankine-Hugoniot condition which describes the jumps and hence resolves steady discontinuities very accurately. Three different variants of the scheme, termed as the Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), based on a single wave (MOVERS-1), multiple waves (MOVERS-n) and limiter based diffusion (MOVERS-L) are presented. The scheme is demonstrated for scalar Burgers equation and systems of conservation laws like Euler equations, ideal Magneto-hydrodynamics equations and shallow water equations. The new scheme uniformly improves the solutions of the Local Lax-Friedrichs scheme on which it is based and captures steady discontinuities either exactly or very accurately.
A Grid-Free Central Solver, which does not require a grid structure but operates on any random distribution of points, is presented. The grid-free scheme is generic in discretization of spatial derivatives with the location of the mid-point between a point and its neighbor being used to define a relevant coefficient of numerical dissipation. A new central scheme based on convective-pressure splitting to solve for mid-point flux is proposed and many test problems are solved effectively. The Rankine-Hugoniot Solver, which is developed in this thesis, is also implemented in the grid-free framework and its utility is demonstrated.
The numerical methods presented are solved in a finite volume framework, except for the Grid-Free Central Solver which is a generalized finite difference method. The algorithms developed are tested on problems represented by different systems of equations and for a wide variety of flow features. The methods presented in this thesis do not need any eigen-structure and complicated flux splittings, but can still capture discontinuities very accurately (sometimes exactly, when aligned with the grid lines), yielding low dissipative solutions.
The thesis ends with a highlight on the importance of developing genuinely multidimensional schemes to obtain accurate solutions for multidimensional flows. The requirement of simpler discretization framework for such schemes is emphasized in order to match the efficacy of the popular dimensional splitting schemes.
Books on the topic "Uniformly accurate numerical methods"
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Li, Wanai. Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids. Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-43432-1.
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Li, Wanai. Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids. Springer Berlin / Heidelberg, 2014.
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Li, Wanai. Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids. Springer, 2016.
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Li, Wanai. Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids. Springer London, Limited, 2014.
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Fontanarosa, Phil B., and Stacy Christiansen. Units of Measure. Oxford University Press, 2009. http://dx.doi.org/10.1093/jama/9780195176339.003.0018.
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The presentation of quantitative scientific information is an integral component of biomedical publication. Accurate communication of scientific knowledge and presentation of numerical data require a scientifically informative system for reporting units of measure. The International System of Units (Le Système International d'Unités or SI) represents a modified version of the metric system that has been established by international agreement and currently is the official measurement system of most nations of the world.1 The SI promotes uniformity of quantities and units, minimizes the number of units and multiples used in other measurement systems, and can express virtually any measurement in science, medicine, industry, and commerce. In 1977, the World Health Organization recommended the adoption of the SI by the international scientific community. Since then, many biomedical publications throughout the world have adopted SI units as their preferred and primary method for reporting scientific measurements...
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Coolen, A. C. C., A. Annibale, and E. S. Roberts. Graphs with hard constraints: further applications and extensions. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198709893.003.0007.
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This chapter looks at further topics pertaining to the effective use of Markov Chain Monte Carlo to sample from hard- and soft-constrained exponential random graph models. The chapter considers the question of how moves can be sampled efficiently without introducing unintended bias. It is shown mathematically and numerically that apparently very similar methods of picking out moves can give rise to significant differences in the average topology of the networks generated by the MCMC process. The general discussion in complemented with pseudocode in the relevant section of the Algorithms chapter, which explicitly sets out some accurate and practical move sampling approaches. The chapter also describes how the MCMC equilibrium probabilities can be purposely deformed to, for example, target desired correlations between degrees of connected nodes. The mathematical exposition is complemented with graphs showing the results of numerical simulations.
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Коллектив, авторов. Труды Физико-технологического института. T. 29: Квантовые компьютеры, микро- и наноэлектроника: физика, технология, диагностика и моделирование. ФГУП «Издательство «Наука», 2020. http://dx.doi.org/10.7868/9785020408081.
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Настоящий том посвящен актуальным проблемам квантовых технологий и микроэлектроники. Рассмотрены различные численные и аналитические подходы к моделированию и контролю элементной базы квантовых компьютеров и симуляторов с учетом декогерентизации и квантовых шумов. Представлены современные методы, направленные на инжиниринг различных квантовых состояний, а также их адекватный, полный и точный контроль. Представлены разработки, обеспечивающие существенное улучшение процедур томографии квантовых состояний и операций с учетом несовершенства технологий и измерений. Кроме того, рассмотрены некоторые вопросы, связанные с разработкой и моделированием приборов микроэлектроники и наноэлектроники. Для широкого круга специалистов в области квантовых информационных технологий, микро- и наноэлектроники, а также студентов и аспирантов, обучающихся по соответствующим специальностям. This volume is devoted to topical problems of quantum technologies and microelectronics. Various numerical and analytical approaches to modeling and control of the element base of quantum computers and simulators, taking into account decoherence and quantum noise, are considered. The modern methods aimed at engineering various quantum states, as well as their adequate, complete and accurate control are presented. Developments are presented that provide a significant improvement in the procedures for tomography of quantum states and operations, taking into account the imperfection of technologies and measurements. In addition, some issues related to the development and modeling of microelectronic and nanoelectronic devices are considered. Intended for a wide range of specialists in the field of quantum information technologies, as well as in the field of micro- and nanoelectronics; it can also be recommended to undergraduate and graduate students of relevant specialties.вЃ
Book chapters on the topic "Uniformly accurate numerical methods"
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Brayanov, Iliya, and Ivanka Dimitrova. "Uniformly Convergent High-Order Schemes for a 2D Elliptic Reaction-Diffusion Problem with Anisotropic Coefficients." In Numerical Methods and Applications. Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-36487-0_44.
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Hafeez, Muhammad Ali, Tetsunori Inoue, Hiroki Matsumoto, Tomoyuki Sato, and Yoshitaka Matsuzaki. "Application of Building Cube Method to Reproduce High-Resolution Hydrodynamics of a Dredged Borrow Pit in Osaka Bay, Japan." In Lecture Notes in Civil Engineering. Springer Nature Singapore, 2024. http://dx.doi.org/10.1007/978-981-99-7409-2_26.
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AbstractOsaka Bay is a semi-enclosed water body suffering from water quality deterioration caused by numerous borrow pits being excavated during the era of the industrial revolution. To mitigate such conditions, backfilling is under consideration, and performance assessment of backfilling requires accurate numerical simulations of hydrodynamics. However, it is challenging to reproduce the hydrodynamics in a borrow pit due to its smaller size, and a coarse-resolution simulation is not enough to accurately capture bathymetry as well as the flow characteristics on a spatial scale. In this study, a non-hydrostatic 3-D hydrodynamic model is successfully used to accurately reproduce the hydrodynamics on both coarse and high-resolution mesh configurations. The advantage of this modeling framework is the inclusion of the building cube method that can locally modify the uniform structural grid to a part of the full simulation domain. This capability made this model quite handy and less time-consuming when it comes to high-resolution simulations focusing on small-scale barrow pits.
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Filbet, Francis, and Giovanni Russo. "Accurate numerical methods for the Boltzmann equation." In Modeling and Computational Methods for Kinetic Equations. Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-0-8176-8200-2_4.
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Roos, H. G., D. Adam, and A. Felgenhauer. "A Nonconforming Uniformly Convergent Finite Element Method in Two Dimensions." In Numerical methods for the Navier-Stokes equations. Vieweg+Teubner Verlag, 1994. http://dx.doi.org/10.1007/978-3-663-14007-8_22.
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Bradji, Abdallah. "A Second Order Time Accurate SUSHI Method for the Time-Fractional Diffusion Equation." In Numerical Methods and Applications. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-10692-8_22.
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van Buuren, R., J. G. M. Kuerten, B. J. Geurts, and P. J. Zandbergen. "Time accurate simulations of supersonic unsteady flow." In Sixteenth International Conference on Numerical Methods in Fluid Dynamics. Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0106603.
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Koren, B., and H. T. M. van der Maarel. "Monotone, higher-order accurate, multi-dimensional upwinding." In Thirteenth International Conference on Numerical Methods in Fluid Dynamics. Springer Berlin Heidelberg, 1993. http://dx.doi.org/10.1007/3-540-56394-6_198.
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Osher, Stanley, and Chi-Wang Shu. "High Order Accurate Modern Numerical Methods Applicable to Stellar Pulsations." In The Numerical Modelling of Nonlinear Stellar Pulsations. Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-009-0519-1_15.
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Schöll, E., and H. H. Frühauf. "An Accurate and Efficient Implicit Upwind Solver for the Navier-Stokes Equations." In Numerical methods for the Navier-Stokes equations. Vieweg+Teubner Verlag, 1994. http://dx.doi.org/10.1007/978-3-663-14007-8_26.
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Catalano, L. A., P. De Palma, G. Pascazio, and M. Napolitano. "Matrix fluctuation splitting schemes for accurate solutions to transonic flows." In Fifteenth International Conference on Numerical Methods in Fluid Dynamics. Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/bfb0107123.
Full textConference papers on the topic "Uniformly accurate numerical methods"
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Xu, X. Y., T. Ma, M. Zeng, and Q. W. Wang. "Numerical Study of the Effects of Different Buoyancy Models on Supercritical Flow and Heat Transfer." In ASME 2013 Heat Transfer Summer Conference collocated with the ASME 2013 7th International Conference on Energy Sustainability and the ASME 2013 11th International Conference on Fuel Cell Science, Engineering and Technology. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/ht2013-17295.
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Due to the dramatic changes in physical properties, the flow and heat transfer in supercritical fluid are significantly affected by buoyancy effects, especially when the ratio of inlet mass flux and wall heat flux is relatively small. In this study, the heat transfer of supercritical water in uniformly heated vertical tube is numerically investigated with different buoyancy models which are based on different calculation methods of the turbulent heat flux. The applicabilities of these buoyancy models are analyzed both in heat transfer enhancement and deterioration conditions. The simulation results show that these buoyancy models make few differences and give good wall temperature prediction in heat transfer enhancement condition when the ratio of inlet mass flux and wall heat flux is very small. With the increase of wall heat flux, the accuracy of wall temperature prediction reduces, and the differences between these buoyancy models become larger. No buoyancy model can currently make accurate wall temperature prediction in deterioration condition in this study.
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Koo, P. C., F. H. Schlereth, R. L. Barbour, and H. L. Graber. "Efficient Numerical Method for Quantifying Photon Distributions in the Interior of Thick Scattering Media." In Advances in Optical Imaging and Photon Migration. Optica Publishing Group, 2022. http://dx.doi.org/10.1364/aoipm.1994.ncpdir.187.
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In this report we present preliminary results towards developing an efficient numerical scheme for computing photon migration in complex scattering media such as body tissues. Our motivation for these studies is the appreciation that development of practical imaging schemes will require that solutions to the forward problem be computationally tractable, physically accurate and have a format that maps well to the inverse problem. The object here is to optimize the tradeoffs and still obtain an acceptable answer. It is known, for example, that Monte Carlo methods can provide accurate estimates of photon distributions, even for complex media, but the computing times can be unacceptable long. Numerical solutions to the diffusion are computationally much more efficient, but of course, the physically accuracy of these solutions are subject to the constraints of the diffusion approximation (i.e., away from boundaries and source and the absence of strong discontinuities). We further recognize that for iterative inversion methods, solution of the forward problem on a uniform grid is desirable, as it permits use of global instead of local interpolation functions. Solutions to the forward problem that employ nonuniform grids certainly can be considered, but add to computing times for the inverse problem.
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Yang, R. J., L. Gu, L. Liaw, et al. "Approximations for Safety Optimization of Large Systems." In ASME 2000 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2000. http://dx.doi.org/10.1115/detc2000/dac-14245.
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Abstract This paper presents four approximation methods for the construction of safety related functions. These methods are: Enhanced Multivariate Adaptive Regression Splines, Stepwise Regression, Artificial Neural Network, and the Moving Least Square. The optimal Latin Hypercube Sampling method is used to distribute the sampling points uniformly over the entire design space. Four benchmark problems used in crash and occupant simulation are employed to investigate the accuracy of the approximate or surrogate models. An occupant safety optimization problem is solved using these four response surfaces. Based on numerical results, a best, applicable approximation strategy for safety optimization is proposed in the end.
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Kalis, Harijs, Ilmars Kangro, and Aivars Aboltins. "Numerical analysis for system of parabolic equations with periodic functions." In 22nd International Scientific Conference Engineering for Rural Development. Latvia University of Life Sciences and Technologies, Faculty of Engineering, 2023. http://dx.doi.org/10.22616/erdev.2023.22.tf157.
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Solving of parabolic partial differential equations (PDE) is closely connected to many practical studies of mathematical physics, environmental science, chemistry, etc. – modelling of heavy metal distribution in peat layer’s block; solving heat transfer problems in multilayer environments. Despite the current great capabilities of software, the development of accurate and effective numerical technique algorithms is still ongoing, particularly in areas 2-D and 3-D involving periodic boundary conditions (PBC). The solutions of some linear and nonlinear problems for parabolic type equations and systems with (PBCs) are obtained using the method of lines (MOL) to approach the partial differential equations (PDS) in the time and discretization in space applying the finite difference scheme (FDS) and the finite difference scheme with the exact spectrum (FDSES). As an application of the described mathematical models the 3-D diffusion problem of peat block is solved. The FDS method in the uniform grid is used to approximate the differential operator of the second and the first order derivatives in the space, using multi-point stencil. The solution in the time is obtained analytically with continuous and discrete Fourier methods and numerically, using MATLAB.
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Thompson, Lonny L., and Yuhuan Tong. "Hybrid Least Squares Finite Element Methods for Reissner-Mindlin Plates." In ASME 1999 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/imece1999-0185.
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Abstract An assumed-stress hybrid 4-node plate element is developed based on the Hellinger-Reissner variational principle modified with a generalized least-squares operator for accurate vibration and wave propagation response of Reissner-Mindlin plates. The least-squares operator is proportional to a weighted integral of a differential operator acting on the residual of the steady-state equations of motion for Reissner-Mindlin plates. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent framework for enhancing the accuracy of mixed Reissner-Mindlin plate elements that have no shear locking or spurious modes. Improved methods are designed such that the complex wave-number finite element dispersion relations closely match the analytical relations for all wave angle directions. For uniform meshes, optimal methods are designed to achieve zero dispersion error along given wave directions. Comparisons of finite element dispersion relations demonstrate the superiority of the new hybrid least-squares plate element over the underlying hybrid element, and standard Galerkin elements based on selectively reduced integration. Numerical experiments validate these conclusions.
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Li, Like, Renwei Mei, and James F. Klausner. "Heat Transfer in Thermal Lattice Boltzmann Equation Method." In ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-87990.
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The evaluation of the boundary heat flux and total heat transfer in the lattice Boltzmann equation (LBE) simulations is investigated. The boundary heat fluxes in the discrete velocity directions of the thermal LBE (TLBE) model are obtained directly from the temperature distribution functions at the lattice nodes. With the rectangular lattice uniformly spaced the effective surface area for the discrete heat flux is the unit spacing distance, thus the heat flux integration becomes simply a summation of all the discrete heat fluxes with constant surface areas. The present method for the evaluation of total heat transfer is very efficient and robust for curved boundaries because it does not require the determination of the normal heat flux on the boundary and the surface area. To validate its applicability and accuracy, several numerical tests with analytical solutions are conducted, including 2-dimensional (2D) steady thermal flow in a channel, 1-D transient heat conduction in an inclined semi-infinite solid, 2-D transient conduction inside a circle, and 3-D steady thermal flow in a circular pipe. For straight boundaries perpendicular to one of the discrete velocity vectors, the total heat transfer is second-order accurate. For curved boundaries only first-order accuracy is obtained for the total heat transfer due to the irregularly distributed lattice fractions cut by the curved boundary.
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Zhang, J. "A coupled thermo-mechanical and neutron diffusion numerical model for irradiated concrete." In AIMETA 2022. Materials Research Forum LLC, 2023. http://dx.doi.org/10.21741/9781644902431-4.
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Abstract: Neutron irradiation plays an important role in nuclear-induced degradation for concrete shielding materials, specifically in determining the radiation induced volume expansion (RIVE) phenomenon driving its failure. When analyzing at the structural level the effects of nuclear radiation on concrete, a non-uniformed distribution of neutron radiation must be considered. This can be done via particle transport calculations preventive to the thermo-mechanic study, or by solving numerically the coupled set of governing equations of the problem. In this work the second approach is pursued in the theoretical framework of the Finite Element Method (FEM). The proposed formulation not only considers an accurate neutron transport model based on the two-group theory, but also it includes the effects induced by thermal neutrons to the temperature field. The formulation lends itself to include RIVE and the other relevant radiation induced effects on the mechanical field. The governing equations are presented and discussed, and some results obtained by using the general 3D numerical formulation proposed herein are compared to results from literature obtained via analytical methods addressing simplified 1D problems.
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Chen, P. L., S. F. Chang, T. Y. Wu, and Y. H. Hung. "A Thermal Network Approach for Predicting Thermal Characteristics of Three-Dimensional Electronic Packages." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-13755.
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In the present study, a numerical approach for characterizing three-dimensional (3-D) electronic packages is presented, based on the steady-state solution of the thermal network method for generalized rectangular geometries, where boundary conditions are uniformly specified over specific regions of the package. As we know, the thermal-network method is very powerful on thermal analysis of electronic packaging because of its feasibility and flexibility. Accordingly, the numerical approach with thermal-network method to simulate heat transfer characteristics for 3-D package geometries becomes important in the modem microelectronic applications. The thermal analyses are presented with a general overview of the thermal network method, boundary conditions and solution procedures. Furthermore, the application of boundary conditions at the fluid-solid, package-board and layer-layer interfaces provides a means for obtaining a unique numerical result for 3-D complex electronic packages. The complex geometries found in most 3-D electronic package configurations are modeled using numerical method through the careful use of simplifying assumptions. Comparisons of the present numerical results with the existing experimental data for 3-D electronic package of pin grid arrays and multi-chip modules are made with a satisfactory agreement. Thus, the present study demonstrates that the numerical thermal-network approach can offer an accurate and efficient solution procedure for evaluating the thermal characterization of 3-D electronic packages.
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Kim, C. M., and R. V. Ramaswamy. "Nonuniform Finite-Difference Method for Modeling of Quasi-TM Smal1-Mode-Size Ti:LiNbO3 Channel Waveguides." In Numerical Simulation and Analysis in Guided-Wave Optics and Opto-Electronics. Optica Publishing Group, 1989. http://dx.doi.org/10.1364/gwoe.1989.sc4.
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Among the various computional methods, the finite-difference method (FDM) has been frequently applied to several kinds of structures due to its easy application and simple manupulation. Recently, semi-vectorial FDM was introduced for the rib waveguide, characterized by piecewise constant refractive index distribution with uniform grid size.[1] In this paper, we present a versatile FDM for quasi-TM modes, which can handle the boundary conditions across abrupt index-changes. This method is flexible in that it uses the nonuniform discretization (Fig.1), which allows us to slice the grid finely in the region of the waveguide and sparcely in the region outside. It also allows us to extend the boundary of the metal box to arbitrary points yet using a limited number of grid lines. We can place grid lines at the positions of index discontinuity flexibly. By setting up grid lines and corresponding cell structure in a judicious way, we can reduce the redundant computer calculations and obtain the desirable accuracy with less memory size as little as 1/4 of that needed otherwise and, therefore, calculation amount is by far reduced. Increasing the number of grid lines ,of course, leads to more accurate solutions. Using this nonuniform FDM, we model the low-loss, minimum-mode-size Ti:LiNbO3 diffused channel waveguides fabricated in our laboratory. [2] The waveguides were fabricated on the z_ surface of optical grade LiNbO3 wafers and characterized at 1.3 μm. The smallest quasi-TM mode size (1/e intensity full width of 3.9 μm and full depth of 2.8 μm) was obtained for waveguides fabricated at 1025°C for 6 hours with Ti thickness of 800° A and Ti strip width of 4 μm.
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Andersen, Pål Østebø. "Intercept Method for Accurately Estimating Critical Fluid Saturation and Approximate Transient Solutions with Production Time Scales in Centrifuge Core Plug Experiments." In SPE EuropEC - Europe Energy Conference featured at the 84th EAGE Annual Conference & Exhibition. SPE, 2023. http://dx.doi.org/10.2118/214402-ms.
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Abstract The centrifuge experiment is used to measure capillary pressure in core plugs by forced displacement (imbibition or drainage): strong gravitational forces (imposed by rotation) displace fluid held in place by capillary forces. This setup is also used to measure and establish critical saturation, the saturation where a fluid loses connectivity and can no longer flow. Obtaining this saturation is challenging as the capillary end effect causing outlet fluid accumulation theoretically only vanishes at infinite rotation speed. Practical speed limitations include maintaining core integrity and avoiding unrepresentative capillary desaturation. In tight or strongly wetted media the capillary forces are strong and more challenging to overcome. Firstly, we demonstrate an ‘intercept method’ to estimate critical saturation. It states that average saturation is proportional to inverse squared rotation speed (at high speeds) allowing to determine critical saturation by linear extrapolation of a few measurements to the intercept where inverse squared speed is zero. The linear trend is valid once the core saturation profile contains the critical saturation. The result follows as the saturation profile near the outlet is invariant and only compressed while the other saturations equal the critical saturation. Although it was assumed the gravitational acceleration is uniform (reasonable for short cores and long centrifuge arm), the result was highly accurate even for extremely non-uniform gravity along the core: the data are linear and the correct critical saturation value is estimated. This was justified by that the end effect profile is uniformly compressed even under those conditions since most of it is located in a narrow part of the core. Secondly, an analytical solution is derived for transient production after the rotation speed is increased starting from an arbitrary initial state towards equilibrium. For this result we assume the outlet profile compresses also during the transient stage. The two regions have fixed mobilities, while the regions occupy different lengths with time. Time as function of production has a linear term and logarithmic term (dominating late time behavior). An analytical time scale is derived which scales all production curves to end (99.5 % production) at same scaled time. We validate the intercept method for high rotation speed data with synthetical and experimental data. For the synthetical data, the input critical saturation is reproduced both for uniform and highly non-uniform gravity along the core. Given the same input as a reservoir simulator, including saturation functions, the analytical transient solution is able to predict similar time scales and trends in time scale (with e.g. rotation speed and viscosity) as numerical simulations. The numerical simulations however indicate that the saturations travel with highly different speeds rather than as a uniformly compressed profile. Especially saturations near the critical saturation are very slow and caused production to span 5 log units of time (the analytical solution predicted 2-3) when the critical saturation was in the core. The correlation better matched low speed data where the critical saturation had not entered the core.
Reports on the topic "Uniformly accurate numerical methods"
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Jenkins, Eleanor W. Air/Water Flow in Porous Media: A Comparison of Accurate and Efficient Numerical Methods. Defense Technical Information Center, 2009. http://dx.doi.org/10.21236/ada518697.
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Cobb, J. W. Third-order-accurate numerical methods for efficient, large time-step solutions of mixed linear and nonlinear problems. Office of Scientific and Technical Information (OSTI), 1995. http://dx.doi.org/10.2172/29360.
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Nobile, F., Q. Ayoul-Guilmard, S. Ganesh, et al. D6.5 Report on stochastic optimisation for wind engineering. Scipedia, 2022. http://dx.doi.org/10.23967/exaqute.2022.3.04.
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This report presents the latest methods of optimisation under uncertainties investigated in the ExaQUte project, and their applications to problems related to civil and wind engineering. The measure of risk throughout the report is the conditional value at risk. First, the reference method is presented: the derivation of sensitivities of the risk measure; their accurate computation; and lastly, a practical optimisation algorithm with adaptive statistical estimation. Second, this method is directly applied to a nonlinear relaxation oscillator (FitzHugh–Nagumo model) with numerical experiments to demonstrate its performance. Third, the optimisation method is adapted to the shape optimisation of an airfoil and illustrated by a large-scale experiment on a computing cluster. Finally, the benchmark of the shape optimisation of a tall building under a turbulent flow is presented, followed by an adaptation of the optimisation method. All numerical experiments showcase the open-source software stack of the ExaQUte project for large-scale computing in a distributed environment.
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Li, Honghai, Mitchell Brown, Lihwa Lin, et al. Coastal Modeling System user's manual. Engineer Research and Development Center (U.S.), 2024. http://dx.doi.org/10.21079/11681/48392.
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The Coastal Modeling System (CMS) is a suite of coupled 2D numerical models for simulating nearshore waves, currents, water levels, sediment transport, morphology change, and salinity and temperature. Developed by the Coastal Inlets Research Program of the US Army Corps of Engineers, the CMS provides coastal engineers and scientists a PC-based, easy-to-use, accurate, and efficient tool for understanding of coastal processes and for designing and managing of coastal inlets research, navigation projects, and sediment exchange between inlets and adjacent beaches. The present technical report acts as a user guide for the CMS, which contains comprehensive information on model theory, model setup, and model features. The detailed descriptions include creation of a new project, configuration of model grid, various types of boundary conditions, representation of coastal structures, numerical methods, and coupled simulations of waves, hydrodynamics, and sediment transport. Pre- and post-model data processing and CMS modeling procedures are also described through operation within a graphic user interface—the Surface- water Modeling System.
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Hart, Carl R., D. Keith Wilson, Chris L. Pettit, and Edward T. Nykaza. Machine-Learning of Long-Range Sound Propagation Through Simulated Atmospheric Turbulence. U.S. Army Engineer Research and Development Center, 2021. http://dx.doi.org/10.21079/11681/41182.
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Conventional numerical methods can capture the inherent variability of long-range outdoor sound propagation. However, computational memory and time requirements are high. In contrast, machine-learning models provide very fast predictions. This comes by learning from experimental observations or surrogate data. Yet, it is unknown what type of surrogate data is most suitable for machine-learning. This study used a Crank-Nicholson parabolic equation (CNPE) for generating the surrogate data. The CNPE input data were sampled by the Latin hypercube technique. Two separate datasets comprised 5000 samples of model input. The first dataset consisted of transmission loss (TL) fields for single realizations of turbulence. The second dataset consisted of average TL fields for 64 realizations of turbulence. Three machine-learning algorithms were applied to each dataset, namely, ensemble decision trees, neural networks, and cluster-weighted models. Observational data come from a long-range (out to 8 km) sound propagation experiment. In comparison to the experimental observations, regression predictions have 5–7 dB in median absolute error. Surrogate data quality depends on an accurate characterization of refractive and scattering conditions. Predictions obtained through a single realization of turbulence agree better with the experimental observations.
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Engel, Bernard, Yael Edan, James Simon, Hanoch Pasternak, and Shimon Edelman. Neural Networks for Quality Sorting of Agricultural Produce. United States Department of Agriculture, 1996. http://dx.doi.org/10.32747/1996.7613033.bard.
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The objectives of this project were to develop procedures and models, based on neural networks, for quality sorting of agricultural produce. Two research teams, one in Purdue University and the other in Israel, coordinated their research efforts on different aspects of each objective utilizing both melons and tomatoes as case studies. At Purdue: An expert system was developed to measure variances in human grading. Data were acquired from eight sensors: vision, two firmness sensors (destructive and nondestructive), chlorophyll from fluorescence, color sensor, electronic sniffer for odor detection, refractometer and a scale (mass). Data were analyzed and provided input for five classification models. Chlorophyll from fluorescence was found to give the best estimation for ripeness stage while the combination of machine vision and firmness from impact performed best for quality sorting. A new algorithm was developed to estimate and minimize training size for supervised classification. A new criteria was established to choose a training set such that a recurrent auto-associative memory neural network is stabilized. Moreover, this method provides for rapid and accurate updating of the classifier over growing seasons, production environments and cultivars. Different classification approaches (parametric and non-parametric) for grading were examined. Statistical methods were found to be as accurate as neural networks in grading. Classification models by voting did not enhance the classification significantly. A hybrid model that incorporated heuristic rules and either a numerical classifier or neural network was found to be superior in classification accuracy with half the required processing of solely the numerical classifier or neural network. In Israel: A multi-sensing approach utilizing non-destructive sensors was developed. Shape, color, stem identification, surface defects and bruises were measured using a color image processing system. Flavor parameters (sugar, acidity, volatiles) and ripeness were measured using a near-infrared system and an electronic sniffer. Mechanical properties were measured using three sensors: drop impact, resonance frequency and cyclic deformation. Classification algorithms for quality sorting of fruit based on multi-sensory data were developed and implemented. The algorithms included a dynamic artificial neural network, a back propagation neural network and multiple linear regression. Results indicated that classification based on multiple sensors may be applied in real-time sorting and can improve overall classification. Advanced image processing algorithms were developed for shape determination, bruise and stem identification and general color and color homogeneity. An unsupervised method was developed to extract necessary vision features. The primary advantage of the algorithms developed is their ability to learn to determine the visual quality of almost any fruit or vegetable with no need for specific modification and no a-priori knowledge. Moreover, since there is no assumption as to the type of blemish to be characterized, the algorithm is capable of distinguishing between stems and bruises. This enables sorting of fruit without knowing the fruits' orientation. A new algorithm for on-line clustering of data was developed. The algorithm's adaptability is designed to overcome some of the difficulties encountered when incrementally clustering sparse data and preserves information even with memory constraints. Large quantities of data (many images) of high dimensionality (due to multiple sensors) and new information arriving incrementally (a function of the temporal dynamics of any natural process) can now be processed. Furhermore, since the learning is done on-line, it can be implemented in real-time. The methodology developed was tested to determine external quality of tomatoes based on visual information. An improved model for color sorting which is stable and does not require recalibration for each season was developed for color determination. Excellent classification results were obtained for both color and firmness classification. Results indicted that maturity classification can be obtained using a drop-impact and a vision sensor in order to predict the storability and marketing of harvested fruits. In conclusion: We have been able to define quantitatively the critical parameters in the quality sorting and grading of both fresh market cantaloupes and tomatoes. We have been able to accomplish this using nondestructive measurements and in a manner consistent with expert human grading and in accordance with market acceptance. This research constructed and used large databases of both commodities, for comparative evaluation and optimization of expert system, statistical and/or neural network models. The models developed in this research were successfully tested, and should be applicable to a wide range of other fruits and vegetables. These findings are valuable for the development of on-line grading and sorting of agricultural produce through the incorporation of multiple measurement inputs that rapidly define quality in an automated manner, and in a manner consistent with the human graders and inspectors.
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Russo, David, Daniel M. Tartakovsky, and Shlomo P. Neuman. Development of Predictive Tools for Contaminant Transport through Variably-Saturated Heterogeneous Composite Porous Formations. United States Department of Agriculture, 2012. http://dx.doi.org/10.32747/2012.7592658.bard.
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The vadose (unsaturated) zone forms a major hydrologic link between the ground surface and underlying aquifers. To understand properly its role in protecting groundwater from near surface sources of contamination, one must be able to analyze quantitatively water flow and contaminant transport in variably saturated subsurface environments that are highly heterogeneous, often consisting of multiple geologic units and/or high and/or low permeability inclusions. The specific objectives of this research were: (i) to develop efficient and accurate tools for probabilistic delineation of dominant geologic features comprising the vadose zone; (ii) to develop a complementary set of data analysis tools for discerning the fractal properties of hydraulic and transport parameters of highly heterogeneous vadose zone; (iii) to develop and test the associated computational methods for probabilistic analysis of flow and transport in highly heterogeneous subsurface environments; and (iv) to apply the computational framework to design an “optimal” observation network for monitoring and forecasting the fate and migration of contaminant plumes originating from agricultural activities. During the course of the project, we modified the third objective to include additional computational method, based on the notion that the heterogeneous formation can be considered as a mixture of populations of differing spatial structures. Regarding uncertainly analysis, going beyond approaches based on mean and variance of system states, we succeeded to develop probability density function (PDF) solutions enabling one to evaluate probabilities of rare events, required for probabilistic risk assessment. In addition, we developed reduced complexity models for the probabilistic forecasting of infiltration rates in heterogeneous soils during surface runoff and/or flooding events Regarding flow and transport in variably saturated, spatially heterogeneous formations associated with fine- and coarse-textured embedded soils (FTES- and CTES-formations, respectively).We succeeded to develop first-order and numerical frameworks for flow and transport in three-dimensional (3-D), variably saturated, bimodal, heterogeneous formations, with single and dual porosity, respectively. Regarding the sampling problem defined as, how many sampling points are needed, and where to locate them spatially in the horizontal x₂x₃ plane of the field. Based on our computational framework, we succeeded to develop and demonstrate a methdology that might improve considerably our ability to describe quntitaively the response of complicated 3-D flow systems. The results of the project are of theoretical and practical importance; they provided a rigorous framework to modeling water flow and solute transport in a realistic, highly heterogeneous, composite flow system with uncertain properties under-specified by data. Specifically, they: (i) enhanced fundamental understanding of the basic mechanisms of field-scale flow and transport in near-surface geological formations under realistic flow scenarios, (ii) provided a means to assess the ability of existing flow and transport models to handle realistic flow conditions, and (iii) provided a means to assess quantitatively the threats posed to groundwater by contamination from agricultural sources.
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Zhang, Renduo, and David Russo. Scale-dependency and spatial variability of soil hydraulic properties. United States Department of Agriculture, 2004. http://dx.doi.org/10.32747/2004.7587220.bard.
Full textAbstract:
Water resources assessment and protection requires quantitative descriptions of field-scale water flow and contaminant transport through the subsurface, which, in turn, require reliable information about soil hydraulic properties. However, much is still unknown concerning hydraulic properties and flow behavior in heterogeneous soils. Especially, relationships of hydraulic properties changing with measured scales are poorly understood. Soil hydraulic properties are usually measured at a small scale and used for quantifying flow and transport in large scales, which causes misleading results. Therefore, determination of scale-dependent and spatial variability of soil hydraulic properties provides the essential information for quantifying water flow and chemical transport through the subsurface, which are the key processes for detection of potential agricultural/industrial contaminants, reduction of agricultural chemical movement, improvement of soil and water quality, and increase of agricultural productivity. The original research objectives of this project were: 1. to measure soil hydraulic properties at different locations and different scales at large fields; 2. to develop scale-dependent relationships of soil hydraulic properties; and 3. to determine spatial variability and heterogeneity of soil hydraulic properties as a function of measurement scales. The US investigators conducted field and lab experiments to measure soil hydraulic properties at different locations and different scales. Based on the field and lab experiments, a well-structured database of soil physical and hydraulic properties was developed. The database was used to study scale-dependency, spatial variability, and heterogeneity of soil hydraulic properties. An improved method was developed for calculating hydraulic properties based on infiltration data from the disc infiltrometer. Compared with the other methods, the proposed method provided more accurate and stable estimations of the hydraulic conductivity and macroscopic capillary length, using infiltration data collected atshort experiment periods. We also developed scale-dependent relationships of soil hydraulic properties using the fractal and geostatistical characterization. The research effort of the Israeli research team concentrates on tasks along the second objective. The main accomplishment of this effort is that we succeed to derive first-order, upscaled (block effective) conductivity tensor, K'ᵢⱼ, and time-dependent dispersion tensor, D'ᵢⱼ, i,j=1,2,3, for steady-state flow in three-dimensional, partially saturated, heterogeneous formations, for length-scales comparable with those of the formation heterogeneity. Numerical simulations designed to test the applicability of the upscaling methodology to more general situations involving complex, transient flow regimes originating from periodic rain/irrigation events and water uptake by plant roots suggested that even in this complicated case, the upscaling methodology essentially compensated for the loss of sub-grid-scale variations of the velocity field caused by coarse discretization of the flow domain. These results have significant implications with respect to the development of field-scale solute transport models capable of simulating complex real-world scenarios in the subsurface, and, in turn, are essential for the assessment of the threat posed by contamination from agricultural and/or industrial sources.
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SECOND-ORDER ANALYSIS OF BEAM-COLUMNS BY MACHINE LEARNING-BASED STRUCTURAL ANALYSIS THROUGH PHYSICS-INFORMED NEURAL NETWORKS. The Hong Kong Institute of Steel Construction, 2023. http://dx.doi.org/10.18057/ijasc.2023.19.4.10.
Full textAbstract:
The second-order analysis of slender steel members could be challenging, especially when large deflection is involved. This paper proposes a novel machine learning-based structural analysis (MLSA) method for second-order analysis of beam-columns, which could be a promising alternative to the prevailing solutions using over-simplified analytical equations or traditional finite-element-based methods. The effectiveness of the conventional machine learning method heavily depends on both the qualitative and the quantitative of the provided data. However, such data are typically scarce and expensive to obtain in structural engineering practices. To address this problem, a new and explainable machine learning-based method, named Physics-informed Neural Networks (PINN), is employed, where the physical information will be utilized to orientate the learning process to create a self-supervised learning procedure, making it possible to train the neural network with few or even no predefined datasets to achieve an accurate approximation. This research extends the PINN method to the problems of second-order analysis of steel beam-columns. Detailed derivations of the governing equations, as well as the essential physical information for the training process, are given. The PINN framework and the training procedure are provided, where an adaptive loss weight control algorithm and the transfer learning technic are adopted to improve numerical efficiency. The practicability and accuracy of which are validated by four sets of verification examples.