| 1. |
- Bonito, Andrea, et al.
(författare)
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Numerical Simulations of Surface Quasi-Geostrophic Flows on Periodic Domains
- 2021
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 43:2, s. B405-B430
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Tidskriftsartikel (refereegranskat)abstract
- We propose a novel algorithm for the approximation of surface quasi-geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit strong-stability-preserving three-stage Runge-Kutta method while a flux-corrected-transport (FCT) method coupled with Dunford-Taylor representations of fractional operators is advocated for the space discretization. Standard continuous piecewise linear finite elements are employed, and the algorithm does not have restrictions on the mesh structure or on the computational domain. In the inviscid case, we show that the resulting scheme satisfies a discrete maximum principle property under a standard CFL condition and observe, in practice, its second order accuracy in space. The algorithm successfully approximates several benchmarks with sharp transitions and fine structures typical of SQG flows. In addition, theoretical Kolmogorov energy decay rates are observed on a freely decaying atmospheric turbulence simulation.
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| 2. |
- Bouderlique, Thibault, et al.
(författare)
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Surface flow for colonial integration in reef-building corals
- 2022
-
Ingår i: Current Biology. - : Elsevier. - 0960-9822 .- 1879-0445. ; 32:12, s. 2596-2609
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Tidskriftsartikel (refereegranskat)abstract
- Reef-building corals are endangered animals with a complex colonial organization. Physiological mechanisms connecting multiple polyps and integrating them into a coral colony are still enigmatic. Using live imaging, particle tracking, and mathematical modeling, we reveal how corals connect individual polyps and form integrated polyp groups via species-specific, complex, and stable networks of currents at their surface. These currents involve surface mucus of different concentrations, which regulate joint feeding of the colony. Inside the coral, within the gastrovascular system, we expose the complexity of bidirectional branching streams that connect individual polyps. This system of canals extends the surface area by 4-fold and might improve communication, nutrient supply, and symbiont transfer. Thus, individual polyps integrate via complex liquid dynamics on the surface and inside the colony.
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| 3. |
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| 4. |
- Dao, Tuan Anh, et al.
(författare)
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A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations
- 2022
-
Ingår i: Journal of Scientific Computing. - : Springer Nature. - 0885-7474 .- 1573-7691. ; 92:3
-
Tidskriftsartikel (refereegranskat)abstract
- We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.
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| 5. |
- Dao, Tuan Anh, et al.
(författare)
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A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics
- 2024
-
Ingår i: Journal of Computational Physics. - 0021-9991 .- 1090-2716. ; 512, s. 113146-113146
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Tidskriftsartikel (refereegranskat)abstract
- We present a novel high-order nodal artificial viscosity approach designed for solving Magnetohydrodynamics (MHD) equations. Unlike conventional methods, our approach eliminates the need for ad hoc parameters. The viscosity is mesh-dependent, yet explicit definition of the mesh size is unnecessary. Our method employs a multimesh strategy: the viscosity coefficient is constructed from a linear polynomial space constructed on the fine mesh, corresponding to the nodal values of the finite element approximation space. The residual of MHD is utilized to introduce high-order viscosity in a localized fashion near shocks and discontinuities. This approach is designed to precisely capture and resolve shocks. Then, high-order Runge-Kutta methods are employed to discretize the temporal domain. Through a comprehensive set of challenging test problems, we validate the robustness and high-order accuracy of our proposed approach for solving MHD equations.
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| 6. |
- Dao, Tuan Anh, et al.
(författare)
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A structure preserving numerical method for the ideal compressible MHD system
- 2024
-
Ingår i: Journal of Computational Physics. - 0021-9991 .- 1090-2716. ; 508
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.
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| 7. |
- Dao, Tuan Anh, et al.
(författare)
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Energy stable and accurate coupling of finite element methods and finite difference methods
- 2022
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 449
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are present. The proposed coupling technique requires minimal changes in the existing schemes while maintaining strict stability, accuracy, and energy conservation. Results are demonstrated on linear and nonlinear scalar conservation laws in two spatial dimensions.
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| 8. |
- Dao, Tuan Anh, 1994-
(författare)
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Invariant domain preserving schemes for magnetohydrodynamics
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Magnetohydrodynamics (MHD) studies the behaviors of ionized gases, such as plasmas, in the presence of a magnetic field. MHD is used in many applications, such as geophysics, space physics, and nuclear fusion.Despite intensive research in recent decades, many physical and numerical aspects of MHD are not well understood. The challenges inherent in solving MHD stem from the obstacles encountered in ordinary hydrodynamics, such as those described by the compressible Euler/Navier-Stokes equations, along with the intricacies arising from electromagnetism. A characteristic of compressible flows is their tendency to develop shocks/discontinuities over time. This often leads to unphysical traits in numerical approximations if the capturing scheme is not constructed properly. By physical laws, the magnetic field is solenoidal. However, in practice, numerical schemes seldom ensure this property precisely, which may lead to instability and convergence to wrong solutions. In numerical simulation of many applications, positive physical quantities such as density and pressure can easily become negative. On the whole, preserving the physical relevance of the numerical solutions poses a significant challenge in MHD.This thesis presents several numerical schemes based on Galerkin approximations to solve MHD. The schemes rely on viscous regularization, a technique to remove mathematical singularities by adding a vanishing viscosity term to the MHD equations. At the continuous level, we propose several choices of viscous regularization and rigorously show that they are consistent with thermodynamics. Based on these choices, we construct numerical schemes of which robustness is confirmed through many challenging benchmarks. Finally, we propose a nonconventional algorithm that simultaneously preserves many desirable physical properties, including positivity of density and internal energy, conservation of total energy, minimum entropy principle, and zero magnetic divergence.
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| 9. |
- Dao, Tuan Anh, et al.
(författare)
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Monolithic parabolic regularization of the MHD equations and entropy principles
- 2022
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 398
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Tidskriftsartikel (refereegranskat)abstract
- We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.
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| 10. |
- Dao, Tuan Anh, et al.
(författare)
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Viscous Regularization of the MHD Equations
- 2024
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Ingår i: SIAM Journal on Applied Mathematics. - : Society for Industrial and Applied Mathematics. - 0036-1399 .- 1095-712X. ; 84:4, s. 1439-1459
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Tidskriftsartikel (refereegranskat)abstract
- Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations that holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments, including contact waves and magnetic reconnection.
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| 11. |
- Guermond, Jean-Luc, et al.
(författare)
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Finite element-based invariant-domain preserving approximation of hyperbolic systems : Beyond second-order accuracy in space
- 2024
-
Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 418
-
Tidskriftsartikel (refereegranskat)abstract
- This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a highorder finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting.
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| 12. |
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| 13. |
- Guermond, Jean-Luc, et al.
(författare)
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Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting
- 2018
-
Ingår i: SIAM Journal on Scientific Computing. - : SIAM PUBLICATIONS. - 1064-8275 .- 1095-7197. ; 40:5, s. A3211-A3239
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Tidskriftsartikel (refereegranskat)abstract
- A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy, and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, guaranteed maximum speed method using a graph viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method, which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where the 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.
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| 14. |
- Hoffman, Johan, et al.
(författare)
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A General Galerkin Finite Element Method for the Compressible Euler Equations
- 2008
-
Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197.
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Tidskriftsartikel (refereegranskat)abstract
- 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
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| 15. |
- Hoffman, Johan, et al.
(författare)
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Unicorn : Parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry
- 2011
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- 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
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| 16. |
- Hoffman, Johan, et al.
(författare)
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Unicorn : Parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry
- 2013
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Ingår i: Computers & Fluids. - : Elsevier BV. - 0045-7930 .- 1879-0747. ; 80:SI, s. 310-319
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Tidskriftsartikel (refereegranskat)abstract
- 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 efficient 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.
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| 17. |
- Hoffman, Johan, 1974-, et al.
(författare)
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Unicorn : A unified continuum mechanics solver
- 2012
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Ingår i: Lecture Notes in Computational Science and Engineering. - : Springer Science and Business Media Deutschland GmbH. ; , s. 339-361
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Bokkapitel (refereegranskat)abstract
- This chapter provides a description of the technology of Unicorn focusing on simple, efficient and general algorithms and software for the Unified Continuum (UC) concept and the adaptive General Galerkin (G2) discretization as a unified approach to continuum mechanics. We describe how Unicorn fits into the FEniCS framework, how it interfaces to other FEniCS components, what interfaces and functionality Unicorn provides itself and how the implementation is designed. We also present some examples in fluid–structure interaction and adaptivity computed with Unicorn.
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| 18. |
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| 19. |
- Jansson, Niclas, et al.
(författare)
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Adaptive simulation of turbulent flow past a full car model
- 2011
-
Ingår i: State of the Practice Reports, SC'11. - New York, NY, USA : ACM. - 9781450311397
-
Konferensbidrag (refereegranskat)abstract
- 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.
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| 20. |
- Leitenmaier, Lena, et al.
(författare)
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A finite element based Heterogeneous Multiscale Method for the Landau-Lifshitz equation
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We present a Heterogeneous Multiscale Method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, a simple model for a ferromagnetic composite. A finite element macro scheme is combined with a finite difference micro model to approximate the effective equation corresponding to the original problem. This makes it possible to obtain effective solutions to problems with rapid material variations on a small scale, described by ε << 1, which would be too expensive to resolve in a conventional simulation.
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| 21. |
- Leitenmaier, Lena, et al.
(författare)
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A finite element based Heterogeneous Multiscale Method for the Landau-Lifshitz equation
- 2023
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 486
-
Tidskriftsartikel (refereegranskat)abstract
- We present a Heterogeneous Multiscale Method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, a simple model for a ferromagnetic composite. A finite element macro scheme is combined with a finite difference micro model to approximate the effective equation corresponding to the original problem. This makes it possible to obtain effective solutions to problems with rapid material variations on a small scale, described by ε << 1, which would be too expensive to resolve in a conventional simulation.
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| 22. |
- Lu, Li, et al.
(författare)
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Nonlinear artificial viscosity for spectral element methods
- 2019
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Ingår i: Comptes rendus. Mathematique. - : Elsevier BV. - 1631-073X .- 1778-3569. ; 357:7, s. 646-654
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Tidskriftsartikel (refereegranskat)abstract
- We present a filter-based approach to computing artificial viscosities for spectral element methods. A number of applications for this approach are presented.
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| 23. |
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| 24. |
- Lundgren, Lukas, 1994-, et al.
(författare)
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A high-order artificial compressibility method based on Taylor series time-stepping for variable density flow
- 2023
-
Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 421
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis functions in space. Taylor series time-stepping relies on time derivative correction terms to achieve high-order accuracy. We provide detailed algorithms to approximate the time derivatives of the variable density Navier-Stokes equations. Numerical validations confirm a fourth-order accuracy for smooth problems. We also numerically illustrate that the Taylor series method is unsuitable for problems where regularity is lost by solving the 2D Rayleigh-Taylor instability problem.
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| 25. |
- Lundgren, Lukas, 1994-, et al.
(författare)
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A high-order residual-based viscosity finite element method for incompressible variable density flow
- 2024
-
Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 497
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce a high-order accurate finite element method for incompressible variable density flow. The method uses high-order Taylor-Hood velocity-pressure elements in space and backward differentiation formula (BDF) time stepping in time. This way of discretization leads to two main issues: (i) a saddle point system that needs to be solved at each time step; a stability issue when the viscosity of the flow goes to zero or if the density profile has a discontinuity. We address the first issue by using Schur complement preconditioning and artificial compressibility approaches. We observed similar performance between these two approaches. To address the second issue, we introduce a modified artificial Guermond-Popov viscous flux where the viscosity coefficients are constructed using a newly developed residual-based shock-capturing method. Numerical validations confirm high-order accuracy for smooth problems and accurately resolved discontinuities for problems in 2D and 3D with varying density ratios.
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| 26. |
- Lundgren, Lukas, 1994-
(författare)
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High-order finite element methods for incompressible variable density flow
- 2023
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution.Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness.Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. In this thesis, we develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.
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| 27. |
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| 28. |
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| 29. |
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| 30. |
- Nazarov, Murtazo, 1980-
(författare)
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Adaptive Algorithms and High Order Stabilization for Finite Element Computation of Turbulent Compressible Flow
- 2011
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This work develops finite element methods with high order stabilization, and robust and efficient adaptive algorithms for Large Eddy Simulation of turbulent compressible flows. The equations are approximated by continuous piecewise linear functions in space, and the time discretization is done in implicit/explicit fashion: the second order Crank-Nicholson method and third/fourth order explicit Runge-Kutta methods. The full residual of the system and the entropy residual, are used in the construction of the stabilization terms. These methods are consistent for the exact solution, conserves all the quantities, such as mass, momentum and energy, is accurate and very simple to implement. We prove convergence of the method for scalar conservation laws in the case of an implicit scheme. The convergence analysis is based on showing that the approximation is uniformly bounded, weakly consistent with all entropy inequalities, and strongly consistent with the initial data. The convergence of the explicit schemes is tested in numerical examples in 1D, 2D and 3D. To resolve the small scales of the flow, such as turbulence fluctuations, shocks, discontinuities and acoustic waves, the simulation needs very fine meshes. In this thesis, a robust adjoint based adaptive algorithm is developed for the time-dependent compressible Euler/Navier-Stokes equations. The adaptation is driven by the minimization of the error in quantities of interest such as stresses, drag and lift forces, or the mean value of some quantity. The implementation and analysis are validated in computational tests, both with respect to the stabilization and the duality based adaptation.
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| 31. |
- Nazarov, Murtazo, et al.
(författare)
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An adaptive finite element method for inviscid compressible flow
- 2010
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Ingår i: International Journal for Numerical Methods in Fluids. - : Wiley. - 0271-2091 .- 1097-0363. ; 64:10-12, s. 1102-1128
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Tidskriftsartikel (refereegranskat)abstract
- We present an adaptive finite element method for the compressible Euler equations, based on a posteriori error estimation of a quantity of interest in terms of a dual problem for the linearized equations. Continuous piecewise linear approximation is used in space and time, with componentwise weighted least-squares stabilization of convection terms and residual-based shock-capturing. The adaptive algorithm is demonstrated numerically for the quantity of interest being the drag force on a body.
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| 32. |
- Nazarov, Murtazo
(författare)
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An adaptive finite element method for the compressible Euler equations
- 2009
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This work develops a stabilized finite element method for the compressible Euler equations and proves an a posteriori error estimate for the approximated solution. The equations are approximated by the cG(1)cG(1) finite element method with continuous piecewise linear functions in space and time. cG(1)cG(1) gives a second order accuracy in space, and corresponds to a Crank-Nicholson type of discretization in time, resulting in second order accuracy in space, without a stabilization term. The method is stabilized by componentwise weighted least squares stabilization of the convection terms, and residual based shock capturing. This choice of stabilization gives a symmetric stabilization matrix in the discrete system. The method is successfully implemented for a number of benchmark problems in 1D, 2D and 3D. We observe that cG(1)cG(1) with the above choice of stabilization is robust and converges to an accurate solution with residual based adaptive mesh refinement. We then extend the General Galerkin framework from incompressible to compressible flow, with duality based a posteriori error estimation of some quantity of interest. The quantities of interest can be stresses, strains, drag and lift forces, surface forces or a mean value of some quantity. In this work we prove a duality based a posteriori error estimate for the compressible equations, as an extension of the earlier work for incompressible flow [25]. The implementation and analysis are validated in computational tests both with respect to the stabilization and the duality based adaptation
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| 33. |
- Nazarov, Murtazo, et al.
(författare)
-
An adaptive finite element method for the compressible Euler equations
- 2010
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Ingår i: INT J NUMER METHOD FLUID. ; , s. 1102-1128
-
Konferensbidrag (refereegranskat)abstract
- We present an adaptive finite element method for the compressible Euler equations, based on a posteriori error estimation of a quantity of interest in terms of a dual problem for the linearized equations. Continuous piecewise linear approximation is used in space and time, with componentwise weighted least-squares stabilization of convection terms and residual-based shock-capturing. The adaptive algorithm is demonstrated numerically for the quantity of interest being the drag force on a body.
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| 34. |
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| 35. |
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| 36. |
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| 37. |
- Nazarov, Murtazo, et al.
(författare)
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On the stability of the dual problem for high Reynolds number flow past a circular cylinder in two dimensions
- 2012
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 34:4, s. A1905-A1924
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present a computational study of the stability of time dependent dual problems for compressible flow at high Reynolds numbers in two dimensions. The dual problem measures the sensitivity of an output functional with respect to numerical errors and is a key part of goal oriented a posteriori error estimation. Our investigation shows that the dual problem associated with the computation of the drag force for the compressible Euler/Navier-Stokes equations, which are approximated numerically using different temporal discretization and stabilization techniques, is unstable and exhibits blow-up for several Mach regimes considered in this paper.
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| 38. |
- Nazarov, Murtazo, et al.
(författare)
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Regularity of almost periodic solutions of Poisson equation
- 2020
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Ingår i: Ufa Mathematical Journal. - : Institute of Mathematics, USC RAS. - 2074-1863 .- 2074-1871. ; 12:2, s. 97-107
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Tidskriftsartikel (refereegranskat)abstract
- This paper discusses some regularity of almost periodic solutions of the Poisson equation −Δu=f in Rn, where f is an almost periodic function. It was proved by Sibuya [Almost periodic solutions of Poisson's equation. Proc. Amer. Math. Soc., 28:195–198, 1971.] that if u is a bounded continuous function and solves the Poisson equation in the distribution sense, then u is an almost periodic function. In this work, we weaken the assumption of the usual boundedness to boundedness in the sense of distribution, which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, assuming that u is a bounded generalized function and solves the Poisson equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution ∂u/∂xi, i=1,…,n, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using the Green function for the Poisson equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used in studying other elliptic problems.
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| 39. |
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| 40. |
- Stiernström, Vidar, et al.
(författare)
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A residual-based artificial viscosity finite difference method for scalar conservation laws
- 2021
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 430
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.
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| 41. |
- Tominec, Igor, 1991-, et al.
(författare)
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Residual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws
- 2023
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 94
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we solve nonlinear conservation laws using the radial basis function generated finite difference (RBF-FD) method. Nonlinear conservation laws have solutions that entail strong discontinuities and shocks, which give rise to numerical instabilities when the solution is approximated by a numerical method. We introduce a residual-based artificial viscosity (RV) stabilization framework adjusted to the RBF-FD method, where the residual of the conservation law adaptively locates discontinuities and shocks. The RV stabilization framework is applied to the collocation RBF-FD method and the oversampled RBF-FD method. Computational tests confirm that the stabilized methods are reliable and accurate in solving scalar conservation laws and conservation law systems such as compressible Euler equations.
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| 42. |
- Tominec, Igor, 1991-, et al.
(författare)
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Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete ℓ2-norm intrinsic to each of the three methods.The results show that Kansa's method and RBF-PUM can be ℓ2-stable in time under a sufficiently large oversampling of the discretized system of equations. On the other hand, the RBF-FD method is not ℓ2-stable by construction, no matter how large the oversampling is. We show that this is due to the jumps (discontinuities) in the RBF-FD cardinal basis functions. We also provide a stabilization of the RBF-FD method that penalizes the spurious jumps. Numerical experiments show an agreement with our theoretical observations.
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| 43. |
- Weber, Ivy, 1995-, et al.
(författare)
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Stability analysis of high order methods for the wave equation
- 2022
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 404
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we investigate the stability of a numerical method for solving the wave equation. The method uses explicit leap-frog in time and high order continuous and discontinuous (DG) finite elements using the standard Lagrange and Hermite basis functions in space. Matrix eigenvalue analysis is used to calculate time-step restrictions. We show that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, such as equidistributed Lagrange nodes and Gauss–Lobatto nodes. We show that the time-step restriction for the symmetric interior penalty DG schemes with the usual penalty terms is tighter than for continuous Lagrange finite elements. Finally, we conclude that the best time-step restriction is obtained for continuous Hermite finite elements up to polynomial degrees.
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| 44. |
- Ålund, Oskar, 1987-
(författare)
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Applications of summation-by-parts operators
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Numerical solvers of initial boundary value problems will exhibit instabilities and loss of accuracy unless carefully designed. The key property that leads to convergence is stability, which this thesis primarily deals with. By employing discrete differential operators satisfying a so called summation-by-parts property, it is possible to prove stability in a systematic manner by mimicking the continuous analysis if the energy has a bound. The articles included in the thesis all aim to solve the problem of ensuring stability of a numerical scheme in some context. This includes a domain decomposition procedure, a non-conforming grid coupling procedure, an application in high energy physics, and two methods at the intersection of machine learning and summation-by-parts theory.
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