<b>Adaptive Multiresolution Methods: Practical issues on Data Structures, Implementation and Parallelization</b>

Brix, Kolja; Melian, Silvia Sorana; Müller, Siegfried; Bachmann, Mathieu

doi:10.1051/proc/201134003

Cited by 18 publications

(26 citation statements)

References 61 publications

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“…To account for a changing load due to grid adaptation, we perform rebalancing after each adaptation. For more details on the parallelization concept and its realization as well as performance studies, we refer to [13,14].…”

Section: Parallelizationmentioning

confidence: 99%

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Investigation of Hypersonic Intakes Using Reynolds Stress Modeling and Wavelet-Based Adaptation

et al. 2014

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The simulation of hypersonic flows is computationally demanding due to the large gradients of the flow variables at hand, caused both by strong shock waves and thick boundary or shear layers. The resolution of those gradients imposes the use of extremely small cells in the respective regions. Taking turbulence into account intensifies the variation in scales even more. Furthermore, hypersonic flows have been shown to be extremely grid sensitive. For the simulation of fully three-dimensional configurations of engineering applications, this results in a huge amount of cells and, as a consequence, prohibitive computational time. Therefore, modern adaptive techniques can provide a gain with respect to both computational costs and accuracy, allowing the generation of locally highly resolved flow regions where they are needed and retaining an otherwise smooth distribution. In this paper, an h-adaptive technique based on wavelets is employed for the solution of hypersonic flows. The compressible Reynolds-averaged Navier-Stokes equations are solved using a differential Reynolds stress turbulence model, well suited to predict shock-wave/ boundary-layer interactions in high-enthalpy flows. Two test cases are considered: a compression corner at 15 deg and a scramjet intake. The compression corner is a classical test case in hypersonic flow investigations because it poses a shock-wave/turbulent-boundary-layer interaction problem. The adaptive procedure is applied to a twodimensional configuration as validation. The scramjet intake is first computed in two dimensions. Subsequently, a three-dimensional geometry is considered. Both test cases are validated with experimental data and compared to nonadaptive computations. The results show that the use of an adaptive technique for hypersonic turbulent flows at high-enthalpy conditions can strongly improve the performance in terms of memory and CPU time while at the same time maintaining the required accuracy of the results. Nomenclature b ij= anisotropy tensor c p = specific heat at constant pressure, pressure coefficient D ij = diffusion tensor for the Reynolds stresses, m 2 ∕s 3 δ ij = Kronecker delta d = spatial dimension E = specific total energy, m 2 ∕s 2 H = total specific enthalpy, m 2 ∕s 2 II = second invariant of the anisotropy tensor k = turbulent kinetic energy, m 2 ∕s 2 L = maximum refinement level l = local refinement level M = Mach number M ij = turbulent mass flux tensor for the Reynolds stresses, m 2 ∕s 3 μ = molecular viscosity, kg∕m · s P ij = production tensor for the Reynolds stresses, m 2 ∕s 3 p = pressure, Pa p t = total pressure, Pa q i = component of heat flux vector, W∕m 2 q t k = turbulent heat flux, W∕m 2 R ij = Reynolds stress tensor, m 2 ∕s 2 Re = Reynolds number, 1∕m Res drop = averaged density residual at which the adaptations are performed ρ = density, kg∕m 3 S ij = strain-rate tensor, 1∕s St = Stanton number T = temperature, K T w = wall temperature, K T 0 = total temperature, K t = time, s U = local velocity, m∕s u i = velocity component, m∕s W ij...

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Section: Parallelizationmentioning

confidence: 99%

“…On the contrary, they have been designed as one program package to efficiently solve aerodynamic problems with a wide variation of scales. Recently, this solver has been parallelized on distributed memory architectures using Message Passing Interface [13,14]. For load-balancing, we use the concept of space-filling curves [15].…”

Section: Introductionmentioning

confidence: 99%

Investigation of Hypersonic Intakes Using Reynolds Stress Modeling and Wavelet-Based Adaptation

et al. 2014

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“…The encoding of tree structures usually relies either on hash-tables of keys which refers directly to the positions of the nodes [11] either on real tree structures built with pointers. In Fortran, these pointers can be emulated due to tables of indices [60].…”

Section: Six-dimensional Adaptive Hierarchical Basis For Vlasov Equatmentioning

confidence: 99%

“…Such refinement is often associated with finite volume methods [40,52,60]. We can distinguish two different approaches among AMR methods: in patch-based AMR the data structure is a collection of grids of varying sizes and accuracies, which are embedded within each other [42,56], whereas in the fully-threaded tree AMR in which the points or cells, or groups thereof, are stored in the nodes or leaves of a large tree structure [11,36,52,53,60]. We use the latter method.…”

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Six-Dimensional Adaptive Simulation of the Vlasov Equations Using a Hierarchical Basis

Deriaz¹,

Peirani²

2018

Multiscale Model. Simul.

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Abstract. We present an original adaptive scheme using a dynamically refined grid for the simulation of the six-dimensional Vlasov-Poisson equations. The distribution function is represented in a hierarchical basis that retains only the most significant coefficients. This allows considerable savings in terms of computational time and memory usage. The proposed scheme involves the mathematical formalism of Multiresolution Analysis and computer implementation of Adaptive Mesh Refinement. We apply a finite difference method to approximate the Vlasov-Poisson equations although other numerical methods could be considered. Numerical experiments are presented for the d-dimensional Vlasov-Poisson equations in the full 2d-dimensional phase space for d = 1, 2 or 3. The six-dimensional case is compared to a Gadget N-body simulation.keywords: adaptive mesh refinement, hierarchical basis, wavelet, multiresolution analysis, finite differences, phase-space simulations, Vlasov-Poisson equations 1. Introduction. The general idea of Adaptivity in Numerical Analysis lies in the representation of a complex system by a reduced number of elements. It aims to decrease the computational time and memory resources needed to simulate the system. Often, the situation boils down to a trade-off between a numerical volume (computer memory, number of elementary operations etc.) and an increase in the implementation complexity (data structures or algorithms).The most common adaptive technique used in Mechanics [24,40,45,52,53], Physics [7,42] and Astrophysics [1,59,60] is called the Adaptive Mesh Refinement (AMR). In AMR, a Cartesian grid refines locally into finer Cartesian grids. Such refinement is often associated with finite volume methods [40,52,60]. We can distinguish two different approaches among AMR methods: in patch-based AMR the data structure is a collection of grids of varying sizes and accuracies, which are embedded within each other [42,56], whereas in the fully-threaded tree AMR in which the points or cells, or groups thereof, are stored in the nodes or leaves of a large tree structure [11,36,52,53,60]. We use the latter method.In the present study, we are particularly interested in using adaptive methods to tackle the simulation of large systems. Most of these rely on reduced functional basis discretization. This implicates finite elements where the underlying mesh does not have to be Cartesian [41,55]; wavelets associated with a vast non-linear approximation theory [17]; spectral methods that show a good numerical accuracy [31,51]. The hierarchical bases [7] are special kinds of wavelet bases for which the primal scaling function and the primal wavelet are identical, although their duals remain different. A sparse grid method itself [7,50] is a sub-class of hierarchical bases. It is similar to the hyperbolic -anisotropic-hierarchical base. It allows approximation of highdimensional problems for which there is no medium or high frequency oscillation [10,37]. Tensorization consists of expressing a multi-dimensional function as t...

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An Open and Parallel Multiresolution Framework Using Block-Based Adaptive Grids

Sroka¹,

Engels²,

Krah³

et al. 2018

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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A numerical approach for solving evolutionary partial differential equations in two and three space dimensions on block-based adaptive grids is presented. The numerical discretization is based on high-order, central finite-differences and explicit time integration. Grid refinement and coarsening are triggered by multiresolution analysis, i.e. thresholding of wavelet coefficients, which allow controlling the precision of the adaptive approximation of the solution with respect to uniform grid computations. The implementation of the scheme is fully parallel using MPI with a hybrid data structure. Load balancing relies on space filling curves techniques. Validation tests for 2D advection equations allow to assess the precision and performance of the developed code. Computations of the compressible Navier-Stokes equations for a temporally developing 2D mixing layer illustrate the properties of the code for nonlinear multi-scale problems. The code is open source.

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Adaptive Multiresolution Methods: Practical issues on Data Structures, Implementation and Parallelization

Cited by 18 publications

References 61 publications

Investigation of Hypersonic Intakes Using Reynolds Stress Modeling and Wavelet-Based Adaptation

Investigation of Hypersonic Intakes Using Reynolds Stress Modeling and Wavelet-Based Adaptation

Six-Dimensional Adaptive Simulation of the Vlasov Equations Using a Hierarchical Basis

An Open and Parallel Multiresolution Framework Using Block-Based Adaptive Grids

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