1,721,036 research outputs found
On the fundamental solutions-based inversion of Laplace matrices
No full textThe discretisation of the Laplacian results into the well-known Laplace matrix. In the case of a one dimensional problem, an explicit formula for its inverse is derived on the basis of fundamental solutions (Green's functions) for general boundary conditions. For a linear reaction-diffusion equation, approximations of the inverse are given. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Core-annular flow through a horizontal pipe: hydrodynamic counterbalancing of buoyancy force on core
No full text
Fast Iterative Methods for The Incompressible Navier-Stokes Equations
No full textApplied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Two-level preconditioned conjugate gradient methods with applications to bubbly flow problems
No full textThe Preconditioned Conjugate Gradient (PCG) method is one of the most popular iterative methods for solving large linear systems with a symmetric and positive semi-definite coefficient matrix. However, if the preconditioned coefficient matrix is ill-conditioned, the convergence of the PCG method typically deteriorates. Instead, a two-level PCG method can be used. The corresponding two-level preconditioner usually treats unfavorable eigenvalues of the coefficient matrix effectively, so that the two-level PCG method is expected to converge faster than the original PCG method. Many two-level preconditioners are known in the fields of deflation, multigrid and domain decomposition methods. Several of them are discussed in this thesis, where the main focus is on the deflation method. We show some theoretical properties of the deflation method, which give insights into the effectiveness of this method. A crucial component of the deflation preconditioner is the choice of projection vectors. Several choices are discussed and examined. We advocate that subdomain projection vectors, which are based on disjoint and piecewise-constant vectors, are among the best choices for a class of problems. Subsequently, we examine the application of the deflation method to linear systems with singular coefficient matrices. Several mathematically equivalent variants of the original deflation method are proposed to deal with the possible singularity of this coefficient matrix. In addition, two approaches are discussed in order to handle coarse linear systems with a Galerkin matrix, which are involved in each iteration of the deflation method. After the discussion of the implementation and efficiency issues of the deflation method, it is demonstrated that this method is usually faster than the original PCG method. Moreover, we present a comparison between the deflation method and other well-known two-level PCG methods, among them the balancing-Neumann-Neumann, additive coarse-grid correction, and multigrid methods based on symmetric and nonsymmetric V-cycles. As the parameters of the corresponding two-level preconditioners are abstract, we show that these methods are strongly connected to each other. The comparison is also done where the different two-level PCG methods adopt their typical and optimized set of parameters. Numerical experiments show that some multigrid methods are attractive in addition to the deflation method. The major application of this thesis is the Poisson equation with a discontinuous coefficient, which is derived from 2-D and 3-D bubbly flow problems. Most of the performed numerical experiments in this thesis are based on this equation. Both stationary and time-dependent experiments are carried out to emphasize the theoretical results. We show that two-level PCG methods are significantly faster than the original PCG method in almost all experiments. Hence, computations involved in bubbly flows can be performed very efficiently using these PCG methods.Electrical Engineering, Mathematics and Computer Scienc
Smoothness-Increasing Accuracy-Conserving Filters for Discontinuous Galerkin Methods: Challenging the Assumptions of Symmetry and Uniformity
No full textIn this dissertation, we focus on exploiting superconvergence for discontinuous Galerkin methods and constructing a superconvergence extraction technique, in particular, Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. The SIAC filtering technique is based on the superconvergence property of discontinuous Galerkin methods and aims to achieve a solution with higher accuracy order, reduced errors and improved smoothness. The main contributions described in this dissertation are: 1) an efficient one-sided SIAC filter for both uniform and nonuniform meshes; 2) one-sided derivative SIAC filters for nonuniform meshes; 3) the theoretical and computational foundation for using SIAC filters for nonuniform meshes; and 4) the application of SIAC filters for streamline integration. One-sided SIAC filtering is a technique that enhances the accuracy and smoothness of the DG solution near boundary regions. Previously introduced one-sided filters are not directly useful for most applications since they are limited to uniform meshes, linear equations, and the use of multi-precision packages in the computation. Also, the theoretical proofs relied on a periodic boundary assumption. We aim to overcome these deficiencies and develop a new fast one-sided filter for both uniform and nonuniform meshes. By studying B-splines and the negative order norm analysis, we generalized the structure of SIAC filters from a combination of central B-splines to using more general B-splines. Then, a "boundary shape" B-spline (using multiple knots at the boundary) was used to construct a new one-sided filter. We also presented the first theoretical proof of convergence for SIAC filtering over nonuniform meshes (smoothly-varying meshes). One purpose of SIAC filtering is to improve the smoothness of DG solutions. Because of the increased smoothness, we can obtain a better approximation for the derivatives of DG solutions. Derivative filtering over the interior region of uniform meshes was previously studied. However, nonuniform meshes and boundary regions remain a significant challenge. We extended the one-sided filter to a one-sided derivative filter. To deal with nonuniform meshes, we investigated the negative order norm over arbitrary meshes and proposed to scale the one-sided derivative filter with scaling hµ. For arbitrary nonuniform rectangular meshes, we proved that the one-sided derivative filter can enhance the order of convergence for the ?th derivative of the DG solution from k + 1 - ? to µ(2k + 2), where µ ? 2/3. The most challenging part of this project is recovering the superconvergence of the DG solution over nonuniform meshes through SIAC filtering. Typically, most theoretical proofs for SIAC filters are limited to uniform meshes (or translation invariant meshes). The only theoretical investigations for nonuniform meshes were included in our one-sided and derivative filtering studies. Although our earlier research for nonuniform meshes provides good engineering accuracy, we want to do better mathematically. This is not an easy task since unstructured meshes give DG solutions irregular performance under the negative order norm. In our work, we introduced a parameter to measure the unstructuredness of a given nonuniform mesh. Then, by adjusting the scaling of the SIAC filter based on this unstructuredness parameter, we can obtain the optimal filtered approximation (best accuracy) over a given nonuniform mesh. SIAC filtering for streamline integration is an attempt to use SIAC filters in a realistic engineering application. By using the one-sided filter and one-sided derivative filter, we designed an efficient algorithm: filtering the velocity field along the streamline and then use a backward differentiation formula for integration. Compared to the traditional method of filtering the entire field (multi-dimensional algorithm), the computational cost drops dramatically since its complexity corresponds to a one-dimensional algorithm. We finally note that most of the work presented originates from published and submitted papers for the past four years of this PhD research.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Modeling bone regeneration around endosseous implants
No full textApplied mathematicsElectrical Engineering, Mathematics and Computer Scienc
The Deflated Preconditioned Conjugate Gradient Method Applied to Composite Materials
No full textSimulations with composite materials often involve large jumps in the coefficients of the underlying stiffness matrix. These jumps can introduce unfavorable eigenvalues in the spectrum of the stiffness matrix. We show that the rigid body modes; the translations and rotations, of the disjunct rigid bodies in the composite material correspond to these unfavorable eigenvalues. The stiffness matrix is symmetric positive definite and therefore the preconditioned conjugate gradient (PCG) method is the method of choice for solving the linear systems involved. The unfavorable eigenvalues that correspond to the rigid body modes, slow down the convergence of PCG. By using rigid body mode deflation, the unfavorable eigenvalues are removed from the spectrum of the stiffness matrix. We construct a robust and fast iterative solver, the deflated preconditioned conjugate gradient (DPCG) method, to efficiently solve the linear systems involved. The DPCG method is well suited for parallel computations. Moreover, we show that the DPCG method is a very competitive method compared to other state-of-the-art (parallel) linear solvers such as smoothed aggregation algebraic multigrid and sparse direct solvers.Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
The Density-Enthalpy Method Applied to Model Two–phase Darcy Flow
No full textIn this thesis, we use a more recent method to numerically solve two-phase fluid flow problems. The method is developed at TNO and it is presented by Arendsen et al. in [1] for spatially homogeneous systems. We will refer to this method as the densityenthalpy method (DEM) because the density-enthalpy phase diagrams play an important role in this approach. Multiphase flow occurs in numerous natural and industrial processes. These processes (or flow systems) are typically modeled by one or more sets of PDEs. In the literature, a huge variety of mathematical models for flow and transport in porous media are presented and used to simulate these processes. Many authors classify them into moving grid/free boundary methods and fixed grid methods. The method we use falls in the latter category. Although the DEM is developed for multiphase flow problems but this thesis is limited only to two–phase fluid flow of one substance (Propane). As the name indicates, density and enthalpy are our primary variables. The mathematical model for our approach consists of a mass balance, an energy balance, Darcy’s law and other thermodynamic relations. We solve the mass and energy balances for two state variables, the density (p) and the enthalpy (h). Other solution variables (such as pressure, temperature, and gas mass fraction, etc) are obtained from given p-h phase diagrams. These diagrams are obtained from thermodynamic properties of a substance (in our case the substance is Propane). For the spatial discretization, we use the finite-element method. An Euler Backward method is used for the time integration. The finite-element method is used for the spatial discretization of the system over 1D and 2D grids. This method is selected because of its ability to handle complex domain geometries. In particular, SUPG (Streamline Upwind Petrove-Galerkin) is used in the initial chapters. The use of SUPG is related to the numerical wiggles as discussed in the coming chapters. Later on, a standard Galerkin algorithm is applied where no spurious oscillations are observed. We use piecewise (bi-)linear basis functions to approximate solution variables and test functions appearing in the weak forms of the PDEs. The backward Euler method is used for the time integration of the PDEs. The use of piecewise (bi-)linear basis functions and Euler backward time integration scheme implies a numerical error of order O(?x2 + ?t), where ?x and ?t are the spatial and time steps, respectively. This is verified numerically. The 0D model works fine as shown by Arendsen [1, 2, 3]. The first attempt to solve a multidimensional flow system by using density-enthalpy phase diagrams was made by Abouhafc¸ in [12]. He used the IMEX (IMplicit-EXplicit) scheme for system linearization. For his solver, it was necessary to use very small time-steps. This also means that a large computation time is required for a relatively small process time. At the start of our thesis, the real cause behind a small time-step necessity was unknown. We started with IMEX to verify the results obtained by Abouhafc¸. Next, we applied Euler backward with the Picard iteration method but the challenge of a small timestep remained there. Later on, we show that our criterion for choosing the initial guess, as required at the start of each Picard iteration, makes it equivalent to IMEX linearization scheme, which imposes an upper bound on the time-step for convergence. This motivated us to use the Newton-Raphson method for the system linearization. In general, the Newton method is more sensitive to the initial guess. In our case, we use the system variables from the previous time-step as initial guess at the start of a fresh Newton-Raphson iteration. We have shown that this scheme successfully allows a reasonably large time-step (independent of the spatial grid size). Another challenge at the start of our thesis was to deal with solution variables exhibiting nonphysical steep gradients and sudden variations near the boundary. This problem also appeared in [12] but it was left untreated. We have shown that the use of nonhomogeneous boundary conditions makes the problem ill-posed. This is responsible for nonphysical behavior of the solution variables. We propose some guidelines to use nonzero boundary fluxes to keep the problem well-posed. From the simulation results of this fluid system, we conclude that the new method can successfully be applied to numerically solve multi-phase fluid systems. However we need to consider certain aspects regarding this approach. One issue is that density-enthalpy phase diagrams are not widely available for many multi-phase systems. There are also certain issues with non-homogenous boundary conditions (such as well-posedness and selection criterion of certain parameters). These issues are treated partially but are also included in the recommendation for future work.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner And A Multigrid Deflation Technique
No full textThe Helmholtz equation is the simplest possible model for the wave propagation. Perhaps this is the reason, despite denying traditional iterative methods like Krylov sub-space methods, Multigrids, etcetera, numerical solution of the Helmholtz equation has been an interesting and abundant problem to researchers since years. The work in this dissertation is also classified as an attempt to develop fast and robust iterative methods for the solution of the Helmholtz equation. This works is specified for applications in seismic imaging-Geophysics, where usually high frequency are used. Thus we will be targeting large wavenumber Helmholtz problems. The finite difference discretization of the Helmholtz equation with typically given Absorbing (Sommerfeld) boundary conditions gives rise to symmetric, non-Hermitian, indefinite linear systems. Resolution of large wavenumber requires larger number of grid points, thus large linear systems. Many (sparse) direct solvers and hybrid (direct and iterative) solvers have been proposed, but it is quite obvious for very large problems that (sparse) direct solvers have been too much depending upon memory, which makes them less acceptable. Quite a lot of work has been invested in researching iterative solution methods for the Helmholtz equation since many decades. The indefiniteness, which increases with respect to an increase in the wavenumber, poses extra problems for iterative solvers and robust solution of indefinite (large) linear system forms an important research activity. Many iterative techniques like domain decomposition methods, multigrid methods and preconditioners for Krylov subspace methods have been proposed but non of them has been quite robust. For multigrid methods, indefiniteness arises difficulties in having both good smoothing property and constructing appropriate coarse-grid approximations of the problem, which are responsible for further reduction of low frequency errors. Many attempts have been spent in algebraic variants of multigrid methods. Some of them works well with limitation of homogeneity. Most of them fails to show satisfactory convergence. The same holds for Krylov subspace methods. One of the difficulties for Krylov methods is to find a cheap and performing preconditioner for the indefinite Helmholtz equation. An overview of preconditioners, ranging from classical to matrix based, for indefinite Helmholtz linear system has been give in this thesis. A matrix-based complex shifted Laplace preconditioner (CSLP) has been seen as best in the available ones. However, with increasing wavenumbers CSLP shows a slow convergence behavior. We address this issue continuing using CSLP while taking care of its requirement of specific complex shifts. The projection-type preconditioners have been widely investigated by researchers in numerical analysis community. We propose the projection-type deflation preconditioner to tackle the near-singular nodes, which are the cause of the decay the convergence of, this otherwise well performing, CSLP. Like multigrid, this deflation pre-conditioner, named as ADEF1, requires to solve coarse problems at different coarser levels. An optimized algorithm has been tested and proposed suggesting iterative solution of coarse problems at different levels. This finalizes as a multilevel preconditioner. The re-discretization coarsening strategy that we propose and investigate in this thesis is aimed at reducing the memory size and maintaining stencil size. The multilevel Krylov method (MLKM) has also been investigated and compared with its counterpart ADEF1. The rigorous Fourier analysis (RFA) to investigate the convergence of iterative methods forms a separate research theme, which is included in the thesis. We analyse the proposed multilevel preconditioners ADEF1 and MLKM for two-levels. Analysis shows spectral behavior of the preconditioner, which can be taken as favorable for Krylov methods. RFA points out near-singular modes and highlights their contribution in prevailing stagnation. Further the convergence can be enhanced by adapting coarse grid operator at different levels. The proposed preconditioners have been tested on academic as well as the bench mark Marmousi problem. A huge reduction in number of iterations can be noticed. A comparison in the amount of iterations and solve time, specially for three-dimensional problem, shows that the invested work has paid-off. Proposed preconditioners has been uniformly performing for one- to three-dimensions as well as for heterogeneous medium problems.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
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