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作者： Motamarri, Phani University of Michigan

学位级别：Ph.D.

Past few decades have seen an increasingly important role played by quantum mechanical calculations based on Kohn-Sham density functional theory (DFT) in the investigation of wide variety of materials properties. However, large-scale DFT calculations are computationally very demanding and hence have been primarily associated with either plane-wave basis or atomic-orbital basis sets, imposing severe restrictions on the permissible boundary conditions and the type of materials systems simulated. On the other hand, finite-element (FE) discretization of DFT, among the real-space techniques is versatile and is amenable for unstructured coarse-graining, allows for consideration of complex geometries and boundary conditions, and further is scalable on parallel computing platforms. However, the inherent shortcomings in the use of finite-element discretization for DFT have made it less attractive for large-scale simulations restricting the materials system sizes to few hundreds of electrons. This thesis addresses the inherent shortcomings and presents the development of new computationally efficient and robust parallel algorithms to enable large-scale DFT simulations using finite-element basis (DFT-FE). The proposed DFT-FE enabled for the first time, simulation of electronic structure of materials systems as large as 7000 atoms (~14000 valence electrons) using finite-element basis. The key ideas in the development of DFT-FE include (i) an adaptive higher-order spectral finite-element based self-consistent framework which can handle all-electron and pseudopotential calculations with complex boundary conditions on a single footing, (ii) a subspace projection method to reduce the computational complexity of DFT calculations while treating metallic and insulating systems in a single finite-element framework and (iii) a configurational force approach to efficiently compute forces on atoms to find the geometry of a given materials system in the most stable state. The numerical inv

关键词： Density functional theory finite-element discretization Computational efficiency Subspace projection Thesis

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Stability and numerical results for a suspension bridge of Timoshenko type with second sound

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第4期43卷 1-32页

作者： Aouragh, My Driss El Baz, Mustapha Segaoui, M'hamed Moulay Ismail Univ Meknes Dept Math AM2CSI Grp MSISI LabFST Errachidia Meknes Morocco Moulay Ismail Univ Meknes MSISI Lab AM2CSI Grp FST Errachidia Meknes Morocco

In this paper, we discuss the asymptotic behavior of a linear problem that describes the vibrations of a coupled suspension bridge. The single-span road-bed is modeled as an extensible thermoelastic damped beam, which is simply supported at the ends. The heat conduction is governed by Cattaneo's law. The main cable is modeled as a damped string and is connected to the road-bed by a distributed system of one-sided elastic springs. First, utilizing the theory of semigroups, we prove the existence and uniqueness of the solution. Second, by constructing an appropriate Lyapunov functional, we establish exponential stability using the energy method. Numerically, we introduce fully discrete approximations based on the finite-element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Then, we show that the discrete energy decays, and a priori error estimates are established. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.

关键词： Suspension bridge Timoshenko beam Cattaneo's law Exponential stability finite-element discretization

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Perspectives on the formation of peakons in the stochastic Camassa-Holm equation

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PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 2021年第2250期477卷

作者： Bendall, Thomas M. Cotter, Colin J. Holm, Darryl D. Dynam Res Met Off Exeter Devon England Imperial Coll Math Dept London England

A famous feature of the Camassa-Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

关键词： stochastic partial differential equation nonlinear waves finite-element discretization

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A fast Fourier transform based direct solver for the Helmholtz problem

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2020年第3期27卷 e2283-e2283页

作者： Toivanen, Jari Wolfmayr, Monika Univ Jyvaskyla Fac Informat Technol POB 35 Agora FI-40014 Jyvaskyla Finland

This article is devoted to the efficient numerical solution of the Helmholtz equation in a two- or three-dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT-based direct solver is O(NlogN) operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed.

关键词： absorbing boundary conditions fast direct solver finite-element discretization Fourier transform Helmholtz equation

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finite element analysis of an inflatable torus considering air mass structural element

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ADVANCES IN SPACE RESEARCH 2014年第1期53卷 163-173页

作者： Gajbhiye, S. C. Upadhyay, S. H. Harsha, S. P. Indian Inst Technol Dept Mech & Ind Engn Vibrat & Noise Control Lab Roorkee 247667 Uttarakhand India

Inflatable structures, also known as gossamer structures, are at high boom in the current space technology due to their low mass and compact size comparing to the traditional spacecraft designing. Internal pressure becomes the major source of strength and rigidity, essentially stiffen the structure. However, inflatable space based membrane structure are at high risk to the vibration disturbance due to their low structural stiffness and material damping. Hence, the vibration modes of the structure should be known to a high degree of accuracy in order to provide better control authority. In the past, most of the studies conducted on the vibration analysis of gossamer structures used inaccurate or approximate theories in modeling the internal pressure. The toroidal shaped structure is one of the important key element in space application, helps to support the reflector in space application. This paper discusses the finite-element analysis of an inflated torus. The eigen-frequencies are obtained via three-dimensional small-strain elasticity theory, based on extremum energy principle. The two finite-element model (model-1 and model-2) have cases have been generated using a commercial finite-element package. The structure model-1 with shell element and model-2 with the combination of the mass of enclosed fluid (air) added to the shell elements have been taken for the study. The model-1 is computed with present analytical approach to understand the convergence rate and the accuracy. The convergence study is made available for the symmetric modes and anti-symmetric modes about the centroidal-axis plane, meeting the eigen-frequencies of an inflatable torus with the circular cross section. The structural model-2 is introduced with air mass element and analyzed its eigen-frequency with different aspect ratio and mode shape response using in-plane and out-plane loading condition are studied. (C) 2013 COSPAR. Published by Elsevier Ltd. All rights reserved.

关键词： Modeling and simulation of torus Structural element Aspect ratio Inflation pressure finite-element discretization

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The stabilized extrapolated trapezoidal finite-element method for the Navier-Stokes equations

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2009年第9-12期198卷 958-974页

作者： Labovsky, Alexandr Layton, William J. Manica, Carolina C. Neda, Monika Rebholz, Leo G. Clemson Univ Dept Math Sci Clemson SC 29631 USA Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA Univ Fed Rio Grande do Sul Dept Math Porto Alegre RS Brazil Univ Nevada Dept Math Las Vegas NV USA

We consider a fully discrete stabilized finite-element method for the 2 Navier-Stokes equations which is unconditionally stable and has second order temporal accuracy of O(k(2) + hk+ spatial error). The method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti-diffusion of its effects at the old time level, lowering the cell Reynolds number of the Oseen problem to O(1). The method requires only the solution of one linear system (arising from an Oseen problem) per time step. The effectiveness of the method is illustrated in several numerical experiments. (C) 2008 Elsevier B.V. All rights reserved

关键词： Navier-Stokes equations finite-element discretization Artificial viscosity stabilization

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CONVERGENCE OF A SEMIDISCRETE TWO-GRID ALGORITHM FOR THE CONTROLLABILITY OF THE 1-d WAVE EQUATION

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2008年第6期46卷 3233-3263页

作者： Negreanu, Mihaela Univ Complutense Madrid Fac Math Dept Appl Math E-28040 Madrid Spain

The problem of exact controllability of elastic strings has been extensively studied during the last years. We consider the problem of computing numerically the boundary control for a finite-dimensional system obtained by discretizing in space the 1-d wave equation. More precisely, we analyze whether the controls of numerical approximation schemes converge to the control of the continuous wave equation as the mesh size tends to zero. It is by now well known that, due to high-frequency spurious oscillations, numerical instabilities occur and may lead to the failure of convergence of some apparently natural numerical algorithms. In other words, the classical convergence property of numerical schemes does not guarantee a stable and convergent approximation of controls. Several remedies have been proposed in the literature to compensate for this fact: Tychonoff regularization, Fourier filtering, and mixed finite elements. In this paper we prove that the two-grid method proposed by Glowinski in [J. Comput. Phys., 103 ( 1992), pp. 189-221] to numerically approximate the control of the wave equation converges in 1-d. We prove this result in the context of the finite-element space semidiscretization. Our method of proof relies essentially on the particular properties of the Fourier representation of the initial data of the coarse mesh when projected into the. ne one. The explicit representation formula of the solutions shows that the high-frequency components are modulated by some weights that diminish the effect of these spurious components. This fact, combined with discrete multipliers techniques, allows us to prove uniform observability inequalities. Classical arguments then allow proving the uniform boundedness of the controls and passing to the limit as the mesh size tends to zero. In this way we prove the convergence of the controls of the finite-element semidiscrete approximation of the 1-d wave equation with a boundary control on one extreme. It is important to unde

关键词： wave equation boundary controllability uniform observability two-grid algorithm finite-element discretization

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Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2003年第14期57卷 2007-2025页

作者： Heikkola, E Rossi, T Toivanen, J Univ Jyvaskyla Dept Math Informat Technol FIN-40014 Jyvaskyla Finland

We consider the efficient numerical solution of the Helmholtz equation in a rectangular domain with a perfectly matched layer (PML) or an absorbing boundary condition (ABC). Standard bilinear (trilinear) finite-element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction-type fast direct solver. We present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a PML, and we optimize certain parameters of the layer to minimize the reflection. Copyright (C) 2003 John Wiley Sons, Ltd.

关键词： Helmholtz equation perfectly matched layer absorbing boundary conditions finite-element discretization fast direct solver

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A domain embedding method for scattering problems with an absorbing boundary or a perfectly matched layer

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JOURNAL OF COMPUTATIONAL ACOUSTICS 2003年第2期11卷 159-174页

作者： Heikkola, E Rossi, T Toivanen, J Univ Jyvaskyla Dept Math Informat Technol Jyvaskyla 40014 Finland

The efficient numerical solution of the exterior Helmholtz equation modeling the acoustic scattering by an obstacle is considered. The exterior problem is approximated by truncating the domain with a rectangular boundary and using either a perfectly matched layer or a second-order absorbing boundary condition to reduce reflections from the artificial boundary. The scattering problem is solved iteratively by using a domain embedding method with an efficient separable preconditioner. With this approach, it is possible to realize the GMRES iterations in a low-dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner. The accuracy and the efficiency of the proposed solution technique are studied with numerical experiments.

关键词： Helmholtz equation perfectly matched layer absorbing boundary conditions finite-element discretization domain embedding method

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A boundary functional for the least-squares finite-element solution of neutron transport problems

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2000年第2期37卷 556-586页

作者： Manteuffel, TA Ressel, KJ Starke, G Univ Colorado Program Appl Math Boulder CO 80309 USA German Aerosp Res Estab German Remote Sensing Data Ctr D-82234 Oberpfaffenhofen Germany Univ Essen Gesamthsch Fachbereich Math & Informat 6 D-45117 Essen Germany

The least-squares finite-element framework for the neutron transport equation is based on the minimization of a least-squares functional applied to the properly scaled neutron transport equation. This approach is extended by incorporating the boundary conditions into the least-squares functional. The proof of the V-ellipticity and continuity of the new functional leads to bounds of the discretization error for different regimes. For a P-1 approximation of the angular dependence the resulting system of partial differential equations for the moments is explicitly derived. In the diffusion limit this system is essentially a Poisson equation for the zeroth moment and has a divergence structure for the set of moments of order 1. One of the key features of the least-squares approach is that it produces a posteriori error bounds. The use of these bounds is demonstrated in numerical examples for a spatial discretization using trilinear finite elements on a uniform tessellation into cubes.

关键词： Boltzmann equation neutron transport equation least-squares variational formulation finite-element discretization P-N approximation boundary functional

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