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| 2. |
- Fryklund, Fredrik, 1988-
(författare)
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Integral equations and function extension techniques for numerical solution of PDEs
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Today many phenomena from science and engineering can be simulated accurately thanks to computational methods. Still, many challenges remain, one of them being close interface interactions when simulating e.g. the dynamics of a substance concentration in multiphase flows at the micro level. The challenge is to maintain high accuracy and efficiency as drops, vesicles, etc. are very close to each other, which many numerical methods struggle with. Also, the drops' geometries undergo changes over time. Thus far there is no standardized method for solving the equation modeling a concentration on time-dependent geometries efficiently and accurately. Boundary integral methods are powerful in handling moving and complex geometries, and maintaining high accuracy throughout the domain, even for close interactions. However, they are only efficient for a limited class of problems, and thus do not apply to our problem at hand.The focus of this dissertation is to expand the class of problems boundary integral methods are applicable to, without sacrificing their most attractive properties, plus presenting how the resulting equations time-dependent geometries can be solved on time-dependent geometries. This is achieved by the development of the algorithm partition of unity extension (PUX). It smoothly extends data from its domain of definition, with compact support, which allows for application of established fast methods.With our PUX method and state-of-the-art computational algorithms new problems could be studied, and to new levels of accuracy. In the process new underlying dynamics that were previously obscured by large errors appeared. These findings spurred a new set of questions, leading to the design of accurate algorithms for a class of problems that appear when applying boundary integral methods in conjunction with discretizing the governing equations first in time.In sum, we are now closer to a complete solver for the evolution of the substance concentration on time-dependent geometries. In this endeavor, methods have been development and studied that have applications outside our scope, and have already been applied successfully in other researchers' work. The work with PUX has thus been fruitful and will be developed and investigated further in the future, with adaptivity as the goal.
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| 3. |
- af Klinteberg, Ludvig, et al.
(författare)
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A fast integral equation method for solid particles in viscous flow using quadrature by expansion
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.
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| 4. |
- af Klinteberg, Ludvig, et al.
(författare)
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Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions
- 2018
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 40:3, s. A1225-A1249
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Tidskriftsartikel (refereegranskat)abstract
- When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in two dimensions (2D) denoted AQBX—adaptive quadrature by expansion—which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss–Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard.
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| 5. |
- af Klinteberg, Ludvig, et al.
(författare)
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Estimation of quadrature errors in layer potential evaluation using quadrature by expansion
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.
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| 6. |
- af Klinteberg, Ludvig
(författare)
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Fast and accurate integral equation methods with applications in microfluidics
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.The first, and smaller, part of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an interface decomposition resulting in local, Eulerian grid representations. This provides a natural setting for solving the PDE governing the surfactant concentration on the interface.The second, and larger, part of the thesis is concerned with a framework for simulating large systems of rigid particles in three-dimensional, periodic viscous flow using a boundary integral formulation. This framework can solve the underlying flow equations to high accuracy, due to the accurate nature of surface quadrature. It is also fast, due to the natural coupling between boundary integral methods and fast summation methods.The development of the boundary integral framework spans several different fields of numerical analysis. For fast computations of large systems, a fast Ewald summation method known as Spectral Ewald is adapted to work with the Stokes double layer potential. For accurate numerical integration, a method known as Quadrature by Expansion is developed for this same potential, and also accelerated through a scheme based on geometrical symmetries. To better understand the errors accompanying this quadrature method, an error analysis based on contour integration and calculus of residues is carried out, resulting in highly accurate error estimates.
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| 7. |
- af Klinteberg, Ludvig, et al.
(författare)
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Fast Ewald summation for free-space Stokes potentials
- 2017
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Ingår i: Research in the Mathematical Sciences. - : Springer. - 2197-9847. ; 4:1
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Tidskriftsartikel (refereegranskat)abstract
- We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems (Lindbo and Tornberg in J Comput Phys 229(23):8994–9010, 2010. doi: 10.1016/j.jcp.2010.08.026 ), with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid (Vico et al. in J Comput Phys 323:191–203, 2016. doi: 10.1016/j.jcp.2016.07.028 ). Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of $$O(N \log N)$$ O ( N log N ) for problems with N sources and targets. Comparison is made with a fast multipole method to show that the performance of the new method is competitive.
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| 8. |
- af Klinteberg, Ludvig, et al.
(författare)
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Fast Ewald summation for Stokesian particle suspensions
- 2014
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Ingår i: International Journal for Numerical Methods in Fluids. - : John Wiley & Sons. - 0271-2091 .- 1097-0363. ; 76:10, s. 669-698
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Tidskriftsartikel (refereegranskat)abstract
- We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to O(N log N), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions.
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| 9. |
- Lindbo, Dag, et al.
(författare)
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Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems
- 2012
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 136:16, s. 164111-1-164111-16
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Tidskriftsartikel (refereegranskat)abstract
- A new method for Ewald summation in planar/slablike geometry, i.e., systems where periodicity applies in two dimensions and the last dimension is "free" (2P), is presented. We employ a spectral representation in terms of both Fourier series and integrals. This allows us to concisely derive both the 2P Ewald sum and a fast particle mesh Ewald (PME)-type method suitable for large-scale computations. The primary results are: (i) close and illuminating connections between the 2P problem and the standard Ewald sum and associated fast methods for full periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method for the 2P k-space Ewald sum that uses vastly less memory than traditional PME methods; (iii) errors that decouple, such that parameter selection is simplified. We give analytical and numerical results to support this.
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| 10. |
- Saffar Shamshirgar, Davood, et al.
(författare)
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A comparison of the Spectral Ewald and Smooth Particle Mesh Ewald methods in GROMACS
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The smooth particle mesh Ewald (SPME) method is an FFT based methodfor the fast evaluation of electrostatic interactions under periodic boundaryconditions. A highly optimized implementation of this method is availablein GROMACS, a widely used software for molecular dynamics simulations.In this article, we compare a more recent method from the same family ofmethods, the spectral Ewald (SE) method, to the SPME method in termsof performance and efficiency. We consider serial and parallel implementa-tions of both methods for single and multiple core computations on a desktopmachine as well as the Beskow supercomputer at KTH Royal Institute ofTechnology. The implementation of the SE method has been well optimized,however not yet comparable to the level of the SPME implementation thathas been improved upon for many years. We show that the SE method isvery efficient whenever used to achieve high accuracy and that it already atthis level of optimization can be competitive for low accuracy demands.
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