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Register a new userEquation-free patch scheme for efficient computational homogenisation via self-adjoint coupling
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Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale system at a desired macroscale system level. In this scheme, the given microscale model is computed in small patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate simulations, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve translational invariance, rotational invariance, and self-adjoint symmetry, thus guaranteeing that conservation laws associated with these symmetries are preserved in the macroscale simulation. Spectral and algebraic analyses of the proposed scheme in both one and two dimensions reveal mechanisms for further improving the accuracy of the simulations. Consistency of the patch schemes macroscale dynamics with the original microscale model is proved. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation in a wide range of multiscale scenarios of interest to scientists and engineers.
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Massive parallelisation has lead to a dramatic increase in available computational power. However, data transfer speeds have failed to keep pace and are the major limiting factor in the development of exascale computing. New algorithms must be developed which minimise the transfer of data. Patch dynamics is a computational macroscale modelling scheme which provides a coarse macroscale solution of a problem defined on a fine microscale by dividing the domain into many nonoverlapping, coupled patches. Patch dynamics is readily adaptable to massive parallelisation as each processor can evaluate the dynamics on one, or a few, patches. However, patch coupling conditions interpolate across the unevaluated parts of the domain between patches, and are typically reevaluated at every microscale time step, thus requiring almost continuous data transfer. We propose a modified patch dynamics scheme which minimises data transfer by only reevaluating the patch coupling conditions at `mesoscale time scales which are significantly larger than the microscale time of the microscale problem. We analyse the error arising from patch dynamics with mesoscale temporal coupling as a function of the mesoscale time interval, patch size, and ratio between the microscale and macroscale.
A new adaptive mesh refinement (AMR) version of the ZEUS-3D astrophysical magnetohydrodynamical (MHD) fluid code, AZEuS, is described. The AMR module in AZEuS has been completely adapted to the staggered mesh that characterises the ZEUS family of codes, on which scalar quantities are zone-centred and vector components are face-centred. In addition, for applications using static grids, it is necessary to use higher-order interpolations for prolongation to minimise the errors caused by waves crossing from a grid of one resolution to another. Finally, solutions to test problems in 1-, 2-, and 3-dimensions in both Cartesian and spherical coordinates are presented.
The Equation-Free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction -- the patch dynamics scheme. A recent novel patch scheme empowered the Equation-Free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the Equation-Free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving macro-scale mesh of patches is typically not stiff given the context of the micro-scale time-steps required for the sub-patch dynamics.
Let $Omega_-$ and $Omega_+$ be two bounded smooth domains in $mathbb{R}^n$, $nge 2$, separated by a hypersurface $Sigma$. For $mu>0$, consider the function $h_mu=1_{Omega_-}-mu 1_{Omega_+}$. We discuss self-adjoint realizations of the operator $L_{mu}=- ablacdot h_mu abla$ in $L^2(Omega_-cupOmega_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_mu$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Sigma$) and study some properties of its unique self-adjoint extension $mathcal{L}_mu:=overline{L_mu}$. If $mu e 1$, then $mathcal{L}_mu$ simply coincides with $L_mu$ and has compact resolvent. If $n=2$, then $mathcal{L}_1$ has a non-empty essential spectrum, $sigma_mathrm{ess}(mathcal{L}_{1})={0}$. If $nge 3$, the spectral properties of $mathcal{L}_1$ depend on the geometry of $Sigma$. In particular, it has compact resolvent if $Sigma$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Sigma$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Sigma$. We discuss some links between the resulting self-adjoint operator $mathcal{L}_mu$ and some effects observed in negative-index materials.
Computational fluid dynamics is a direct modeling of physical laws in a discretized space. The basic physical laws include the mass, momentum and energy conservations, physically consistent transport process, and similar domain of dependence and influence between the physical reality and the numerical representation. Therefore, a physically soundable numerical scheme must be a compact one which involves the closest neighboring cells within the domain of dependence for the solution update under a CFL number $(sim 1 )$. In the construction of explicit high-order compact scheme, subcell flow distributions or the equivalent degree of freedoms beyond the cell averaged flow variables must be evolved and updated, such as the gradients of the flow variables inside each control volume. The direct modeling of flow evolution under generalized initial condition will be developed in this paper. The direct modeling will provide the updates of flow variables differently on both sides of a cell interface and limit high-order time derivatives of the flux function nonlinearly in case of discontinuity in time, such as a shock wave moving across a cell interface within a time step. The direct modeling unifies the nonlinear limiters in both space for the data reconstruction and time for the time-dependent flux transport. Under the direct modeling framework, as an example, the high-order compact gas-kinetic scheme (GKS) will be constructed. The scheme shows significant improvement in terms of robustness, accuracy, and efficiency in comparison with the previous high-order compact GKS.
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