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- Fehn, Niklas, et al.
(författare)
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Numerical evidence of anomalous energy dissipation in incompressible Euler flows : towards grid-converged results for the inviscid Taylor-Green problem
- 2021
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Ingår i: Journal of Fluid Mechanics. - : Cambridge University Press. - 0022-1120 .- 1469-7645. ; 932
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Tidskriftsartikel (refereegranskat)abstract
- The well-known energy dissipation anomaly in the inviscid limit, related to velocity singularities according to Onsager, still needs to be demonstrated by numerical experiments. The present work contributes to this topic through high-resolution numerical simulations of the inviscid three-dimensional Taylor-Green vortex problem using a novel high-order discontinuous Galerkin discretisation approach for the incompressible Euler equations. The main methodological ingredient is the use of a discretisation scheme with inbuilt dissipation mechanisms, as opposed to discretely energy-conserving schemes, which - by construction - rule out the occurrence of anomalous dissipation. We investigate effective spatial resolution up to 8192(3) (defined based on the 2 pi-periodic box) and make the interesting phenomenological observation that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but that the sequence of discrete solutions seemingly converges to a solution with non-zero kinetic energy dissipation rate. Taking the fine-resolution simulation as a reference, we measure grid-convergence with a relative L-2-error of 0.27% for the temporal evolution of the kinetic energy and 3.52% for the kinetic energy dissipation rate against the dissipative fine-resolution simulation. The present work raises the question of whether such results can be seen as a numerical confirmation of the famous energy dissipation anomaly. Due to the relation between anomalous energy dissipation and the occurrence of singularities for the incompressible Euler equations according to Onsager's conjecture, we elaborate on an indirect approach for the identification of finite-time singularities that relies on energy arguments.
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| 5. |
- Kronbichler, Martin, et al.
(författare)
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A next-generation discontinuous Galerkin fluid dynamics solver with application to high-resolution lung airflow simulations
- 2021
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Ingår i: SC21. - St. Louis, MO, USA : Association for Computing Machinery (ACM). - 9781450384421 ; , s. 1-15
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Konferensbidrag (refereegranskat)abstract
- We present a novel, highly scalable and optimized solver for turbulent flows based on high-order discontinuous Galerkin discretizations of the incompressible Navier-Stokes equations aimed to minimize time-to-solution. The solver uses explicit-implicit time integration with variable step size. The central algorithmic component is the matrix-free evaluation of discretized finite element operators. The node-level performance is optimized by sum-factorization kernels for tensor-product elements with unique algorithmic choices that reduce the number of arithmetic operations, improve cache usage, and vectorize the arithmetic work across elements and faces. These ingredients are integrated into a framework scalable to the massive parallelism of supercomputers by the use of optimal-complexity linear solvers, such as mixed-precision, hybrid geometric-polynomial-algebraic multigrid solvers for the pressure Poisson problem. The application problem under consideration are fluid dynamical simulations of the human respiratory system under mechanical ventilation conditions, using unstructured/structured adaptively refined meshes for geometrically complex domains typical of biomedical engineering.
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| 6. |
- Sudhakar, Yogaraj, et al.
(författare)
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On the use of compressed polyhedral quadrature formulas in embedded interface methods
- 2017
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 39:3, s. B571-B587
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Tidskriftsartikel (refereegranskat)abstract
- The main idea of this paper is to apply a recent quadrature compression technique to algebraic quadrature formulas on complex polyhedra. The quadrature compression substantially reduces the number of integration points but preserves the accuracy of integration. The compression is easy to achieve since it is entirely based on the fundamental methods of numerical linear algebra. The resulting compressed formulas are applied in an embedded interface method to integrate the weak form of the Navier-Stokes equations. Simulations of flow past stationary and moving interface problems demonstrate that the compressed quadratures preserve accuracy and rate of convergence and improve the efficiency of performing the weak form integration, while preserving accuracy and order of convergence.
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