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• 151.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Dynamic subgrid modeling for  scalar convection-diffusion-reaction equations with fractal coefficients2000Konferensbidrag (Refereegranskat)

In this paper we propose and study a subgrid model for linear convection-diffusion-reaction problems with fractal rough coefficients. The subgrid model is based on extrapolation model of a modeling residual from coarser scales using a computed solution without subgrid model on a finest scale as reference. We present a priori and a posteriori error estimates, and we show in experiments that a solution with subgrid model on a scale h corresponds to a solution without subgrid model on a scale less than h/4

• 152.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Dynamic subgrid modeling for time dependent convection-diffusion-reaction equations with fractal solutions2001Konferensbidrag (Refereegranskat)

A dynamic scale similarity model is proposed. The subgrid model is tested for model problems related to time dependent non-linear convection-diffusion-reaction systems with fractal solutions. The error of an approximate solution with subgrid model on a scale h is typically smaller than that of a solution without subgrid model on the scale h/2. We also consider the problem of a posteriori error estimation for fractal solutions, splitting the total computational error into a numerical error, related to the discretization of the continuous equations, and a modelling error, taking into account the quality of the subgrid model.

• 153.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin G22009Ingår i: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 59, nr 11, s. 1241-1258Artikel i tidskrift (Refereegranskat)

General Galerkin (G2) is a new computational method for turbulent flow. where a stabilized Galerkin finite element method is used to compute approximate weak solutions to the Navier-Stokes equations directly, without any filtering of the equations as in a standard approach to turbulence simulation. such as large eddy simulation, and thus no Reynolds stresses are introduced, which need modelling. In this paper, G2 is used to compute the drag coefficient c(D) for the flow Past a circular cylinder at Reynolds number Re=3900, for which the flow is turbulent. It is found that it is possible to approximate c(D) to an accuracy of a few percent, corresponding to the accuracy in experimental results for this problem, using less than 10(5) mesh points, which makes the simulations possible using a standard PC. The mesh adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, which in this paper is c(D). Both the stopping criterion and the mesh-refinement strategy are based on a posteriori error estimates, in the form of a space-time integral of residuals times derivatives of the solution of it dual problem, linearized at the approximate solution, and with data coupling to the output of interest.

• 154.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Simulating drag crisis for a sphere using skin friction boundary conditions2006Ingår i: Proceedings ECCOMAS CFD 2006 / [ed] P. Wesseling, E. Oñate, J. Périaux, TU Delft , 2006Konferensbidrag (Refereegranskat)

In this paper we use a General Galerkin (G2) method to simulate drag crisis for a sphere, where the unresolved turbulent boundary layer is modeled as a decreasing skin friction.

• 155.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Simulation of turbulent flow past bluff bodies on coarse meshes using General Galerkin methods: drag crisis and turbulent Euler solutions2006Ingår i: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 38, nr 05-apr, s. 390-402Artikel i tidskrift (Refereegranskat)

In recent years adaptive stabilized finite element methods, here referred to as General Galerkin (G2) methods, have been developed as a general methodology for the computation of mean value output in turbulent flow. In earlier work, in the setting of bluff body flow, the use of no slip boundary conditions has been shown to accurately capture the separation from a laminar boundary layer, in a number of benchmark problems. In this paper we extend the G2 method to problems with turbulent boundary layers, by including a simple wall-model in the form of a friction boundary condition, to account for the skin friction of the unresolved turbulent boundary layer. In particular, we use G2 to simulate drag crisis for a circular cylinder, by adjusting the friction parameter to match experimental results. By letting the Reynolds number go to infinity and the skin friction go to zero, we get a G2 method for the Euler equations with slip boundary conditions, which we here refer to as an EG2 method. The only parameter in the EG2 method is the discretization parameter, and we present computational results indicating that EG2 may be used to model very high Reynolds numbers flow, such as geophysical flow.

• 156.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Subgrid Modeling for Convection-Diffusion-Reaction in Two Space Dimensions Using a Haar Multiresolution Analysis2003Ingår i: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 13, nr 10, s. 1515-1536Artikel i tidskrift (Refereegranskat)

In this paper we study a subgrid model based on extrapolation of a modeling residual, in the case of a linear convection-diffusion-reaction problem Lu=f in two dimensions. The solution u to the exact problem satisfies an equation Lhu=[f]h+Fh(u), where Lh is the operator used in the computation on the finest computational scale h, [f]h is the approximation of f on the scale h, and Fh(u) is a modeling residual, which needs to be modeled. The subgrid modeling problem is to compute approximations of Fh(u) without using finer scales than h. In this study we model Fh(u) by extrapolation from coarser scales than h, where Fh(u) is directly computed with the finest scale h as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a mesh of size less than h/4.

• 157.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Weak uniqueness of the navier-stokes equations and adaptive turbulence simulation2005Konferensbidrag (Refereegranskat)

We consider the problem of computational simulation of turbulence, where we study turbulent solutions to the incompressible Navier- Stokes equations. We construct approximate weak solutions using a stabilized Galerkin finite element method, here referred to as General Galerkin G2, for which we investigate uniqueness in output (or weak uniqueness ) by solving an associated dual problem computationally, with data coupling to the particular output we are interested in. For simulation of turbulent flow we refer to the adaptive version of G2 as Adaptive DNS/LES, with part of the flow being resolved in a Direct Numerical Simulation DNS, and part of the flow being left unresolved in a Large Eddy Simulation LES , with the stabilization in G2 acting as a dissipative subgrid model. We present computational results using Adaptive DNS/LES, where we find that for the problem of simulating the turbulent flow past various bluff bodies we are able to compute mean value output, such as drag, using  10-100 times less degrees of freedom than in typical LES computations using ad hoc mesh refinement. We further use Adaptive DNS/LES to simulate the turbulent flow past a cylinder rolling along ground.

• 158.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation2011Ingår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, nr 37-40, s. 2758-2767Artikel i tidskrift (Refereegranskat)

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier-Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.

• 159.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Unicorn: a unified continuum mechanics solver; in automated solution pf differential equations by the finite element method2011Ingår i: Automated Solution of Differential Equations by the Finite Element Method, Springer Berlin/Heidelberg, 2011Kapitel i bok, del av antologi (Refereegranskat)
• 160.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Turbulent flow and fluid-structure interaction; in automated solution of differental equations by the finite element method2011Ingår i: Automated Solution of Differential Equations by the Finite Element Method / [ed] Anders Logg Kent-Andre Mardal, Garth Wells, Springer Berlin/Heidelberg, 2011Kapitel i bok, del av antologi (Refereegranskat)
• 161.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
A General Galerkin Finite Element Method for the Compressible Euler Equations2008Ingår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197Artikel i tidskrift (Refereegranskat)

In this paper we present a General Galerkin (G2) method for the compressible Euler equations, including turbulent ow. The G2 method presented in this paper is a nite element method with linear approximation in space and time, with componentwise stabilization in the form  of streamline diusion and shock-capturing modi cations. The method conserves mass, momentum  and energy, and we prove an a posteriori version of the 2nd Law of thermodynamics for the method.  We illustrate the method for a laminar shock tube problem for which there exists an exact analytical  solution, and also for a turbulent flow problem

• 162.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Unicorn: Parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry2011Rapport (Övrigt vetenskapligt)

We present a framework for adaptive finite element computation of turbulent flow and fluid-structure interaction, with focus on general algorithms that allow for complex geometry and deforming domains. We give basic models and finite element discretization methods, adaptive algorithms and strategies for e cient parallel implementation. To illustrate the capabilities of the computational framework, we show a number of application examples from aerodynamics, aero-acoustics, biomedicine and geophysics. The computational tools are free to download open source as Unicorn, and as a high performance branch of the finite element problem solving environment DOLFIN, both part of the FEniCS project

• 163.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
A computational study of turbulent flow separation for a circular cylinder using skin friction boundary conditions2011Ingår i: Quality And Reliability Of Large-Eddy Simulations II, Springer Netherlands, 2011, Vol. 16, nr 1, s. 57-68Konferensbidrag (Refereegranskat)

In this paper we present a computational study of turbulent flow separation for a circular cylinder at high Reynolds numbers. We use a stabilized finite element method together with skin friction boundary conditions, where we study flow separation with respect to the decrease of a friction parameter. In particular, we consider the case of zero friction corresponding to pure slip boundary conditions, for which we observe an inviscid separation mechanism of large scale streamwise vortices, identified in our earlier work. We compare our computational results to experiments for very high Reynolds numbers. In particular, we connect the pattern of streamwise vorticity in our computations to experimental findings of spanwise 3d cell structures reported in the literature.

• 164.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Adaptive finite element methods for incompressible fluid flow2001Konferensbidrag (Refereegranskat)

We present recent work on the following issues of CFD:(i) discretization of the non-stationary incompressible Navier-Stokes equations, (ii) solution of the discrete system at each time step, (iii) hydrodynamic stability, (iv) adaptive error control and a posteriori error estimates, (v) transition of turbulence, (vi) turbulence modeling

• 165.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Adaptive finite element methods for incompressible fluid flow2002Ingår i: Lecture notes in computational Science and EngineeringArtikel i tidskrift (Refereegranskat)

We present recent work on the following issues of CFD:(i) discretization of the non-stationary incompressible Navier-Stokes equations, (ii) solution of the discrete system at each time step, (iii) hydrodynamic stability, (iv) adaptive error control and a posteriori error estimates, (v) transition of turbulence, (vi) turbulence modeling

• 166.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Mathematics and computation2005Konferensbidrag (Refereegranskat)
• 167.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Topics in adaptive computational methods for differential equations2001Konferensbidrag (Refereegranskat)

We discuss two topics of adaptive computational methods for differential equations: (i) individual time-stepping (ii) subgrid modeling, and we present some applications including the computability and predictability of the Solar System and aspects of subgrid modeling in convection-diffusion-reaction systems.

• 168.
Courant Institute, New York University.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
A new approach to computational turbulence modeling2006Ingår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, nr 23-24, s. 2865-2880Artikel i tidskrift (Refereegranskat)

We present a new approach to computational fluid dynamics (CFD) using adaptive stabilized Galerkin finite element methods with duality based a posteriori error control for chosen output quantities of interest. We address the basic question of computability in CFD: For a given flow, what quantity is computable to what tolerance to what cost? We focus on incompressible Newtonian flow with medium to large Reynolds numbers involving both laminar and turbulent flow features. We estimate a posteriori the output of the computed solution with the output based on the exact solution to the Navier–Stokes equations, thus circumventing introducing and modeling Reynolds stresses in averaged Navier–Stokes equations. Our basic tool is a representation formula for the error in the quantity of interest in terms of a space–time integral of the residual of a computed solution multiplied by weights related to derivatives of the solution of an associated dual problem with data connected to the output. We use the error representation formula to derive an a posteriori error estimate combining residuals with computed dual weights, which is used for mesh adaptivity in space–time with the objective of satisfying a given error tolerance with minimal computational effort. We show in a concrete example that outputs such as a mean value in time of drag of a turbulent flow around a bluff body are computable on a PC with a tolerance of a few percent using a few hundred thousand mesh points in space. We refer to our methodology as adaptive DNS/LES, where automatically by adaptivity certain features of the flow are resolved in a direct numerical simulation (DNS), while certain other small scale turbulent features are left unresolved in a large eddy simulation (LES). The stabilization of the Galerkin method giving a weighted least square control of the residual acts as the subgrid model in the LES. The a posteriori error estimate takes into account both the error from discretization and the error from the subgrid model. We pay particular attention to the stability of the dual solution from (i) perturbations replacing the exact convection velocity by a computed velocity, and (ii) computational solution of the dual problem, which are the crucial aspects entering by avoiding using averaged Navier–Stokes equations including Reynolds stresses. A crucial observation is that the contribution from subgrid modeling in the a posteriori error estimation is small, making it possible to simulate aspects of turbulent flow without accurate modeling of Reynolds stresses.

• 169.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Adaptive multiscale computational modeling of complex incompressible ﬂuid ﬂow2002Ingår i: Conference proceedings of WCCM Fifth World Congress on Computational Mechanics, 2002Konferensbidrag (Refereegranskat)
• 170.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Analysis of separation in turbulent on incompressible flow2011Manuskript (preprint) (Övrigt vetenskapligt)

We present a scenario of stable 3d rotational separation in high Reynolds number slightly viscous incompressible turbulent flow around a solid body such as an airplane, car or boat, supported by computation, mathematical analysis and experimental observation. Our scenario is fundamentally different from the scenario of unstable 2d irrotational separation in viscous flow advocated by Prandtl in 1904 signifying the birth of modern fluid mechanics of viscous flow as a development of classical fluid mechanics of inviscid potential flow presented by Euler and d’Alembert. Our scenario allows for a description of high Reynolds number slightly viscous flow past a solid body as inviscid potential flow before separation followed by 3d rotational separation, thus providing a solution to the basic problem of theoretical fluid mechanics of describing the main characteristics of slightly viscous flow in mathematical terms. We also show that the pressure distribution of3d rotational separation is a determining factor for both drag and lift of a wing

• 171.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Blow up of incompressible Euler solutions2008Ingår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 48, nr 2, s. 285-307Artikel i tidskrift (Refereegranskat)

We present analytical and computational evidence of blowup of initially smooth solutions of the incompressible Euler equations into non-smooth turbulent solutions. We detect blowup by observing increasing L-2-residuals of computed solutions under decreasing mesh size.

• 172.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Computability and adaptivity in CFD2007Ingår i: Encyclopedia of computational mechanics / [ed] E. Stein, R. de Borst,T.J.R. Hughes, Wiley , 2007Kapitel i bok, del av antologi (Refereegranskat)

We briefly recall research on adaptive computational methods for laminar compressible and incompressible flow, and then move on to present recent work on computability and adaptivity for turbulent incompressible flow, based on adaptive stabilized Galerkin finite element methods with duality-based a posteriori error control for chosen output quantities of interest, here referred to asgeneral Galerkin G2 methods.

We show in concrete examples that outputs such as mean values in time of drag and lift of turbulent flow around a bluff body are computable on a PC with a tolerance of a few percent using a few hundred thousand mesh points in space.

• 173.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Computational turbulent incompressible flow: Applied mathematics: Body and soul 42007Bok (Refereegranskat)

This is Volume 4 of the book series of the Body & Soul mathematics education reform program, and presents a unified new approach to computational simulation of turbulent flow starting from the general basis of calculus and linear algebra of Vol 1-3. The book puts the Body & Soul computational finite element methodology in the form of General Galerkin (G2), up against the challenge of computing turbulent solutions of the inviscid Euler equations and the Navier-Stokes equations with small viscosity. The book shows that direct application of G2 without any turbulence or wall modeling, allows reliable computation on a PC of mean value quantities of turbulent flow such as drag and lift. The power of G2 is demonstrated by resolving several classical scientific paradoxes of fluid flow and by uncovering secrets of flying, sailing, racing and ball sports. The book presents new aspects on both mathematics and computation of turbulent flow, and challenges established approaches. The book is directed to a wide audience of computational mathematicians fluid dynamicists and scientists. The G2 solver is available as part the free software project FEniCS at www.fenics.org. The book has a dedicated dynamic web page, including movies from a wide variety of simulations, at www.bodysoulmath.org. The book is focussed on incompressible flow, but opens to compressible flow continued in Vol 5 on thermodynamics. The authors are experts on computational mathematics and technology.

• 174.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Irreversibility in reversible systems2006Ingår i: HERMIS The international journal of computer mathematics and its applications, ISSN 1108-7609, Vol. 6, s. 12-33Artikel i tidskrift (Refereegranskat)
• 175.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Resolution of d'Alembert's Paradox2010Ingår i: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 12, nr 3, s. 321-334Artikel i tidskrift (Refereegranskat)

We propose a resolution of d'Alembert's Paradox comparing observation of substantial drag/lift in fluids with very small viscosity such as air and water, with the mathematical prediction of zero drag/lift of stationary irrotational solutions of the incompressible inviscid Euler equations, referred to as potential flow. We present analytical and computational evidence that (i) potential flow cannot be observed because it is illposed or unstable to perturbations, (ii) computed viscosity solutions of the Euler equations with slip boundary conditions initiated as potential flow, develop into turbulent solutions which are wellposed with respect to drag/lift and which show substantial drag/lift, in accordance with observations.

• 176.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Stability of the dual Navier-Stokes equations and efficient computation of mean output in turbulent flow using adaptive DNS/LES2006Ingår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, nr 13-16, s. 1709-1721Artikel i tidskrift (Refereegranskat)

We discuss aspects of adaptive DNS/LES, where adaptive finite element methods are used to accurately compute chosen output from a turbulent flow with the computational power of a PC. The key to this break-through is: (i) application of the general approach to adaptive error control in Galerkin methods based on duality.. coupled with (ii) crucial properties of turbulent flow allowing accurate computation of mean value quantities such as drag and lift without full resolution of all scales.

• 177.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
The mathematical secret of flight2009Ingår i: Normat, ISSN 0801-3500, Vol. 57, nr 4, s. 145-169Artikel i tidskrift (Övrigt vetenskapligt)
• 178.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
The mathematical theory of flight2009Ingår i: Normat, ISSN 0801-3500, Vol. 57, s. 145-169Artikel i tidskrift (Refereegranskat)

We show by computational solution of the incompressible Navier-Stokes equations with friction force boundary conditions, that the classical inviscid circulation theory by Kutta-Zhukovsky for lift of a wing and laminar viscous boundary layer theory by Prandtl for drag, which have dominated 20th century flight mechanics, do not correctly describe the real turbulent airflow around a wing. We show that lift and drag essentially originate from a turbulent wake of counter-rotating rolls of low-pressure streamwise vorticity generated by a certain instability mechanism of potential flow at rear separation. The new theory opens the possibility of computational prediction of flight characteristics of an airplane using millions of meshpoints without resolving thin boundary layers, instead of the imposssible quadrillions required according to state-of-the-art for boundary layer resolution.

• 179.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Algorithms and Codes for Wave Propagation Problems2011Rapport (Övrigt vetenskapligt)

This technical report is a summary of selected numerical methods formultiscale wave propagation problems. The main topic is the discussionof nite dierence schemes, kernels for computing the mean value of oscil-latory functions and how to compute coecients in an eective equationfor long time wave propagation.

• 180.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Multi-scale methods for wave propagation in heterogeneous media2009Licentiatavhandling, monografi (Övrigt vetenskapligt)

Multi-scale wave propagation problems are computationally costly to    solve by traditional techniques because the smallest scales must be    represented over a domain determined by the largest scales of the    problem.  We have developed new numerical methods for multi-scale wave    propagation in the framework of heterogeneous multi-scale methods.  The    numerical methods couples simulations on macro and micro scales with    data exchange between models of different scales.  With the new method    we are able to consider a general class of problems including some    problems where a homogenized equation is unknown.  We show that the    complexity of the new method is significantly lower than that of    traditional techniques.  Numerical results are presented from problems    in one, two and three dimensional and for finite and long time.  We also    analyze the method, in one and several dimensions and for finite time,    using Fourier analysis.

• 181.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Multiscale Methods for Wave Propagation Problems2011Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

Simulering av högfrekventa vågor i heterogena material är viktigt i många tillämpningar, till exempel seismologi, elektromagnetism, akustik och  strömningsmekanik. Dessa tillämpningar är exempel på klassiska multiskalproblem och har typiskt en för hög beräkningskostnad, i form av datortid och minne, för en direkt numerisk simulering. De minsta skalorna i problemet måste vara upplösta över ett område som representeras av dom största skalorna och detta innebär en hög beräkningskostnad. Vi har utvecklat och analyserat numeriska metoder för vågekvationer med snabbt oscillerande lösningar $u^{\varepsilon}$ där $\varepsilon$ representerar storleken på den minsta skalan. Metoderna är baserade på ramverket \emph{heterogena multiskalmetoden} (HMM). I dessa metoder approximeras den hastigt oscillerande mikroskalan med små lokala mikroproblem av storleksordning $\varepsilon$ i tids- och rumsriktning. Lösningen till mikroproblemen är kopplade till en global modell på makroskalan i divergensform $u_{tt} = \nabla \cdot F$, där flödet $F$ ges av mikroproblemen. De hastiga oscillationerna kan härröras från snabba variationer i hastighetsfältet, begynnelsevillkor eller randvillkor. Vi har utvecklat algoritmer som kopplar mikro- och makroskalor i bägge fallen. Valet av makroskalvariabler inspireras av de analytiska metoderna homogenisering och geometrisk optik. I det första fallet används lokala medelvärden $u \approx u^{\varepsilon}$ på makroskalnivån. I det andra fallet är fas $\phi$ och energi bra val av makroskalvariabler. Det finns två huvudmål med vår forskning. Ett mål är att utveckla och analysera algoritmer för simulering av vågproblem med multipla skalor med låg beräkningskostnad (om möjligt, oberoende av $\varepsilon$) för problem över begränsad tid. Vi visar numeriska resultat från multiskalproblem i en, två och tre dimensioner. Det andra målet är att att använda vågutbredning som en modell för att bättre förstå HMM ramverket. Ett exempel på detta är simulering med oscillerande hastighetsfält över lång tid. Efter lång tid så uppträder dispersion. Vi har demonstrerat att vår HMM-metod, som ursprungligen var formulerad för begränsad tid, även kan appliceras på detta fall. För att få den rätta dispersionen krävs högre noggrannhetsordning, men metoden ändrar inte form. Detta visar på metodens robusthet.

• 182.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method2012Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length.  The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort.

We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme.  We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions.

The stability properties of coefficient modifications are essential for practical usability.  We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated.

Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves.

To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale.

The implementations and analysis of the developed methods are validated through systematic numerical tests.

• 183.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
University of Texas.
Consistent modeling of boundaries in acoustic finite-difference time-domain simulations2012Ingår i: Journal of the Acoustical Society of America, ISSN 0001-4966, E-ISSN 1520-8524, Vol. 132, nr 3, s. 1303-1310Artikel i tidskrift (Refereegranskat)

The finite-difference time-domain method is one of the most popular for wave propagation in the time domain. One of its advantages is the use of a structured staggered grid, which makes it simple and efficient on modern computer architectures. A drawback however is the difficulty in approximating oblique boundaries, having to resort to staircase approximations.  In many scattering problems this means that the grid resolution required to obtain an accurate solution is much higher than what is dictated by propagation in a homogeneous material.  In this paper zero boundary data is considered, first for the velocity and then the pressure. These two forms of boundary conditions model perfectly rigid and pressure-release boundaries, respectively.  A simple and efficient method to consistently model curved rigid boundaries in two dimensions was developed in [A.-K. Tornberg and B. Engquist, J. Comput. Phys. 227, 6922--6943 (2008)].  Here this treatment is generalized to three dimensions.  Based on the approach of this method, a technique to model pressure-release surfaces with second order accuracy and without additional restriction on the timestep is also introduced.  The structure of the standard method is preserved, making it easy to use in existing solvers.  The effectiveness is demonstrated in several numerical tests.

• 184.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Centra, SeRC - Swedish e-Science Research Centre.
Accuracy of staircase approximations in finite-difference methods for wave propagation2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, nr 4, s. 741-771Artikel i tidskrift (Refereegranskat)

While a number of increasingly sophisticated numerical methods have been developed for time-dependent problems in electromagnetics, the Yee scheme is still widely used in the applied fields, mainly due to its simplicity and computational efficiency. A fundamental drawback of the method is the use of staircase boundary approximations, giving inconsistent results. Usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result. In this paper, we derive exact discrete solutions to the Yee scheme close to the staircase approximated boundary, enabling a detailed theoretical study of the amplitude, phase and frequency errors created. Furthermore, we show how evanescent waves of amplitude occur along the boundary. These characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why, in practice, the Yee scheme works as well as it does. The analysis is supported by detailed proofs and numerical examples.

• 185.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Simulation of 3D unsteady incompressible flow past a NACA 0012 wing section2012Rapport (Övrigt vetenskapligt)

We present computational simulations of three-dimensional unsteady high Reynolds number incompressible flow past a NACA 0012 wing profile, for a range of angles of attack, from low lift through stall. A stabilized finite element method is used, referred to as General Galerkin (G2), with adaptive mesh refinement with respect to the error in target output, such as aerodynamic forces. Computational predictions of aerodynamic forces are validated against experimental data.

• 186.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Simulation of 3d unsteady incompressible flow past a NACA 0012 wing sectionManuskript (preprint) (Övrigt vetenskapligt)
• 187.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC). KTH, Skolan för datavetenskap och kommunikation (CSC). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). MIT, USA. KTH, Skolan för datavetenskap och kommunikation (CSC), Människa-datorinteraktion, MDI (stängd 20111231). KTH, Skolan för datavetenskap och kommunikation (CSC), Människa-datorinteraktion, MDI (stängd 20111231).
Gestural 3D Interaction with a Beating Heart: Simulation Visualization and Interaction2011Ingår i: Proceedings of SIGRAD 2011: Evaluations of Graphics and Visualization— Efficiency, Usefulness, Accessibility, Usability / [ed] Thomas Larsson, Lars Kjelldahl & Kai-Mikael Jää-Aro, Linköping University Electronic Press, 2011Konferensbidrag (Refereegranskat)

The KTH School of Computer Science and Communication (CSC) established a strategic platform in Simulation-Visualization-Interaction (SimVisInt) in 2009, focused on the high potential in bringing together CSC core com-petences in simulation technology, visualization and interaction. The main part of the platform takes the form aset of new trans-disciplinary projects across established CSC research groups, within the theme of ComputationalHuman Modeling and Visualization: (i) interactive virtual biomedicine (HEART), (ii) simulation of human mo-tion (MOTION), and (iii) virtual prototyping of human hand prostheses (HAND). In this paper, we present recentresults from the HEART project that focused on gestural and haptic interaction with a heart simulation.

• 188.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Algorithms and Data Structures for Multi-Adaptive Time-Stepping2008Ingår i: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 35, nr 3Artikel i tidskrift (Refereegranskat)

Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the multi-adaptive methods allow individual and adaptive time steps to be used for different components or in different regions of space. We present algorithms for efficient multi-adaptive time-stepping, including the recursive construction of time slabs and adaptive time step selection. We also present data structures for efficient storage and interpolation of the multi-adaptive solution. The efficiency of the proposed algorithms and data structures is demonstrated for a series of benchmark problems.

• 189.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
High performance adaptive finite element methods for turbulent fluid flow2011Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)

Understanding the mechanics of turbulent fluid flow is of key importance for industry and society as for example in aerodynamics and aero-acoustics. The massive computational cost for resolving all turbulent scales in a realistic problem makes direct numerical simulation of the underlying Navier-Stokes equations impossible. Recent advances in adaptive finite element methods offer a new powerful tool in Computational Fluid Dynamics (CFD). The computational cost for simulating turbulent flow can be minimized where the mesh is adaptively resolved, based on a posteriori error control. These adaptive methods have been implemented for efficient serial computations, but the extension to an efficient parallel solver is a challenging task.

This work concerns the development of an adaptive finite element method for modern parallel computer architectures. We present efficient data structures and data decomposition methods for distributed unstructured tetrahedral meshes. Our work also concerns an efficient parallellization of local mesh refinement methods such as recursive longest edge bisection.

We also address the load balance problem with the development of an a priori predictive dynamic load balancing method. Current results are encouraging with almost linear strong scaling to thousands of cores on several modern architectures.

• 190.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
A Hybrid MPI/PGAS Finite Element Solver2012Rapport (Övrigt vetenskapligt)

We present our work on developing a hybrid parallel programming model for a general finite element solver. The main focus of our work is to demonstrate that legacy codes with high latency, two-sided communication in the form of message passing can be improved using lightweight one-sided communication. We introduce a new hybrid MPI/PGAS implementation of the open source finite element framework FEniCS, replacing the linear algebra backend (PETSc) with a new library written in UPC.  A detailed description of the linear algebra backend implementation and the hybrid interface to FEniCS is given. We also present a detailed analysis of the performance of this hybrid solver on the Cray XE6 Lindgren at PDC/KTH including a comparison with the MPI only implementation, where we find that the hybrid implementation results in improvements of up to 33% in communication intensive parts of the solver.

• 191.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Computer simulation of incompressible flow past a circular cylinder at a very high Reynolds numbers2011Manuskript (preprint) (Övrigt vetenskapligt)
• 192.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Framework For Massively Parallel Adaptive Finite Element Computational Fluid Dynamics On Tetrahedral Meshes2012Ingår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 34, nr 1, s. C24-C42Artikel i tidskrift (Refereegranskat)

In this paper we describe a general adaptive finite element framework for unstructured tetrahedral meshes without hanging nodes suitable for large scale parallel computations. Our framework is designed to scale linearly to several thousands of processors, using fully distributed and efficient algorithms. The key components of our implementation, local mesh refinement and load balancing algorithms, are described in detail. Finally, we present a theoretical and experimental performance study of our framework, used in a large scale computational fluid dynamics computation, and we compare scaling and complexity of different algorithms on different massively parallel architectures.

• 193.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
Adaptive simulation of turbulent flow past a full car model2011Ingår i: State of the Practice Reports, SC'11, 2011Konferensbidrag (Refereegranskat)

The massive computational cost for resolving all turbulent scales makes a direct numerical simulation of the underlying Navier-Stokes equations impossible in most engineering applications. We present recent advances in parallel adaptive finite element methodology that enable us to efficiently compute time resolved approximations for complex geometries with error control. In this paper we present a LES simulation of turbulent flow past a full car model, where we adaptively refine the unstructured mesh to minimize the error in drag prediction. The simulation was partly carried out on the new Cray XE6 at PDC/KTH where the solver shows near optimal strong and weak scaling for the entire adaptive process.

• 194.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Adaptive finite element computational fluid dynamics for large scale massiverly parallel computing2012Ingår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197Artikel i tidskrift (Refereegranskat)
• 195.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Convergence factors of Newton methods for nonlinear eigenvalue problems2012Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, nr 10, s. 3943-3953Artikel i tidskrift (Refereegranskat)

Consider a complex sequence convergent to λC with order pN. The convergence factor is typically defined as the fraction ck:=(λk+1-λ)/(λk-λ)p in the limit k. In this paper, we prove formulas characterizing ck in the limit k for two different Newton-type methods for nonlinear eigenvalue problems. The formulas are expressed in terms of the left and right eigenvectors.

The two treated methods are called the method of successive linear problems (MSLP) and augmented Newton and are widely used in the literature. We prove several explicit formulas for ck for both methods. Formulas for both methods are found for simple as well as double eigenvalues. In some cases, we observe in examples that the limit ck as k does not exist. For cases where this limit does not appear to exist, we prove other limiting expressions such that a characterization of ck in the limit is still possible.

• 196.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
The Lambert W function and the spectrum of some multidimensional time-delay systems2007Ingår i: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 43, nr 12, s. 2124-2128Artikel i tidskrift (Refereegranskat)

In this note we find an explicit expression for the eigenvalues of a retarded time-delay system with one delay, for the special case that the system matrices are simultaneously triangularizable, which includes the case where they commute. Using matrix function definitions we define a matrix version of the Lambert W function, from which we form the expression. We prove by counter-example that some expressions in other publications on Lambert W for time-delay systems do not always hold. (c) 2007 Elsevier Ltd. All rights reserved.

• 197.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
An Arnoldi method with structured starting vectors for the delay eigenvalue problem2010Ingår i: Proceedings of the 9th IFAC Workshop on Time Delay Systems, 2010Konferensbidrag (Refereegranskat)

The method called Arnoldi is currently a very popular method to solve largescale eigenvalue problems. The general purpose of this paper is to generalize Arnoldi to the characteristic equation of a time-delay system, here called adelay eigenvalue problem . The presented generalization is mathematically equivalent to Arnoldi applied to the problem corresponding to a Taylor approximation of the exponential. Even though the derivation of the result is with a Taylor approximation, the constructed method can be implemented in such a way that it is independent of the Taylor truncation paramater N . This is achieved by exploiting properties of vectors with a special structure, the vectorization of a rank one matrix plus the vectorization of a matrix which right-most columns are zero. It turns out that this set of vectors is closed under the matrix vector product as well as orthogonalization. Moreover, both operations can be efficiently computed. Since Arnoldi only consists of these operations, if Arnoldi is started with the special vector structure, the method can be efficiently executed. The presented numerical experiments indicate that the method is very efficient in comparison to methods in the literature.

• 198.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
A linear eigenvalue algorithm for the nonlinear eigenvalue problem2012Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 122, nr 1, s. 169-195Artikel i tidskrift (Refereegranskat)

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted . We consider the Arnoldi method for the operator and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.

• 199. Kanevsky, Alex
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA. KTH, Skolan för teknikvetenskap (SCI), Centra, Linné Flow Center, FLOW.
Modeling simple locomotors in Stokes flow2010Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, nr 4, s. 958-977Artikel i tidskrift (Refereegranskat)

Motivated by the locomotion of flagellated micro-organisms and by recent experiments of chemically driven nanomachines, we study the dynamics of bodies of simple geometric shape that are propelled by specified tangential surface stresses. We develop a mathematical description of the body dynamics based on a mixed-type boundary integral formulation. We also derive analytic axisymmetric solutions for the case of a single locomoting sphere and ellipsoid based on spherical and ellipsoidal harmonics, and compare our numerical results to these. The hydrodynamic interactions between two spherical and ellipsoidal swimmers in an infinite fluid are then simulated using second-order accurate spatial and temporal discretizations. We find that the near-field interactions result in complex and interesting changes in the locomotors' orientations and trajectories. Stable as well as unstable pairwise swimming motions are observed, similar to the recent findings of Pooley et al. [C.M. Pooley, G.P. Alexander, J.M. Yeomans, Hydrodynamic interaction between two swimmers at low Reynolds number, Phys. Rev. Lett. 99 (2007) 228103].

• 200. Khatri, Shilpa
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
A numerical method for two phase flows with insoluble surfactants2011Ingår i: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 49, nr 1, s. 150-165Artikel i tidskrift (Refereegranskat)

In many practical multiphase flow problems, i.e. treatment of gas emboli and various microfluidic applications, the effect of interfacial surfactants, or surface reacting agents, on the surface tension between the fluids is important. The surfactant concentration on an interface separating the fluids can be modeled with a time dependent differential equation defined on the moving and deforming interface. The equations for the location of the interface and the surfactant concentration on the interface are coupled with the Navier-Stokes equations. These equations include the singular surface tension forces from the interface on the fluid, which depend on the interfacial surfactant concentration. A new accurate and inexpensive numerical method for simulating the evolution of insoluble surfactants is presented in this paper. It is based on an explicit yet Eulerian discretization of the interface, which for two dimensional flows allows for the use of uniform one dimensional grids to discretize the equation for the interfacial surfactant concentration. A finite difference method is used to solve the Navier-Stokes equations on a regular grid with the forces from the interface spread to this grid using a regularized delta function. The timestepping is based on a Strang splitting approach. Drop deformation in shear flows in two dimensions is considered. Specifically, the effect of surfactant concentration on the deformation of the drops is studied for different sets of flow parameters.

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