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

Abstract

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.

Wavelet adaptive proper orthogonal decomposition for large-scale flow data

“…In the following, we describe the numerical methods used in the wPOD algorithm and give detailed insight into its implementation, when handling multiple blockbased adaptive grids. The basic wavelet adaptation technique used for our algorithm has been already discussed in [19,48]. We hence limit the presentation here to changes specific to our algorithm.…”

“…Adaptive methods benefit from the sparse representation of the data already in the stage of production. For example, when generating the data numerically using finite element solvers like FEniCS [4,5] or wavelet adaptive solvers such as WABBIT [19,48,49]. In contrast to randomized methods, the representation of the data is seen from an infinite-dimensional perspective, where each snapshot corresponds to a function over an infinite dimensional Hilbert space.…”

“…The approach presented here is integrated as a post-processing routine into an open source software called WABBIT ((W)avelet (A)daptive (B)lock (B)ased Solver for (I)nteractions with (T)urbulence), which is freely available at [49]. The basic adaptation technique grounds on the idea of [14] and the recent works [19,48], in which a blockstructured grid is refined or coarsened using biorthogonal wavelets. The framework is implemented using the MPI Library to exploit parallel computing architectures.…”

Wing Design in Flies: Properties and Aerodynamic Function

“…For this review, we additionally calculated the aerodynamic quantities of revolving ( Figure 3 ) and flapping ( Figure 4 ) wings of a blowfly, as well as simple rectangular and ideal-shaped wings in order to compare their performance. The ideal wing shape was calculated according to the estimation by Prandtl–Betz in Figure 2 e. The numerical simulations were performed using a previously published numerical model [ 83 , 125 ] combined with a wavelet-adaptive solver [ 126 ], and efficiency was calculated as Rankine–Froude efficiency [ 127 ]. Table 1 shows that revolving rectangular and fly wings perform similarly, producing approximately the same amount of lift.…”

A modular massively parallel computing environment for three-dimensional multiresolution simulations of compressible flows

“…Regarding vectorization, block-based schemes have proven their potential in the context of AMR methods [30] as well as for collocation methods [31]. For MR algorithms, blocking strategies have proven efficient for 2D cases using cell-centered FVM with a Threading Building Blocks (TBB) shared-memory parallelization [32] and for point-based finite difference schemes via MPI parallelization [33].…”