Although some preconditioners are available for solving dense linear systems, there are still many matrices for which preconditioners are lacking, in particular in cases where the size of the matrix N becomes very large. Examples of preconditioners include ILU preconditioners that sparsify the matrix based on some threshold, algebraic multigrid, and specialized preconditioners, e.g., Calderón and other analytical approximation methods when available. Despite these methods, there remains a great need to develop general purpose preconditioners whose cost scales well with the matrix size N . In this paper, we propose a preconditioner, with broad applicability and with cost O(N ), for dense matrices, when the matrix is given by a smooth kernel. Extending the method using the same framework to general H 2 -matrices (i.e., algebraic instead of defined in terms of an analytical kernel) is relatively straightforward, but this won't be discussed here. These preconditioners have a controlled accuracy (e.g., machine accuracy can be achieved if needed) and scale linearly with N . They are based on an approximate direct solve of the system. The linear scaling of the algorithm is achieved by means of two key ideas. First, the H 2 -structure of the dense matrix is exploited to obtain an extended sparse system of equations. Second, fill-ins arising when performing the elimination of the latter are compressed as low-rank matrices if they correspond to well-separated interactions. This ensures that the sparsity pattern of the extended sparse matrix is preserved throughout the elimination, hence resulting in a very efficient algorithm with O(N log(1/ε) 2 ) computational cost and O(N log 1/ε) memory requirement, for an error tolerance 0 < ε < 1. The solver is inexact, although the error can be controlled and made as small as needed. These solvers are related to ILU in the sense that the fill-in is controlled. However, in ILU, most of the fill-in (i.e., below a certain tolerance) is simply discarded, whereas here it is approximated using low-rank blocks, with a prescribed tolerance. Numerical examples are discussed to demonstrate the linear scaling of the method and to illustrate its effectiveness as a preconditioner. ). 1 2. The inverse fast multipole method: key ideas and algorithm.2.1. Extended sparsification. It is the aim to obtain a fast solver for the dense linear system Ax = b, where A is an H 2 -matrix, i.e., a matrix constructed based on strong admissibility criteria (well-separated clusters are assumed to be low-rank only), and with a nested basis for the low-rank representations.The first step in obtaining such a solver is to exploit the H 2 -structure of the matrix in order to convert the original dense system into an extended sparse sys-2 The term fill-in is used here to denote the non-zero matrix entries. 3 The term elimination refers to removing variables from a system of equations and updating the coefficients of the remaining variables appropriately.

Abstract. Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability, but need to be used with an appropriate preconditioner (e.g., ILU, AMG, Gauss-Seidel, etc.) for proper convergence. The choice of an effective preconditioner is highly problem dependent. We propose a novel fully algebraic sparse matrix solve algorithm, which has linear complexity with the problem size. Our scheme is based on the Gauss elimination. For a given matrix, we approximate the LU factorization with a tunable accuracy determined a priori. This method can be used as a stand-alone direct solver with linear complexity and tunable accuracy, or it can be used as a black-box preconditioner in conjunction with iterative methods such as GMRES. The proposed solver is based on the low-rank approximation of fill-ins generated during the elimination. Similar to H-matrices, fill-ins corresponding to blocks that are well-separated in the adjacency graph are represented via a hierarchical structure. The linear complexity of the algorithm is guaranteed if the blocks corresponding to well-separated clusters of variables are numerically low-rank.

The working principle of particle-based solar receivers is to utilize the absorptivity of a dispersed particle phase in an otherwise optically transparent carrier fluid. In comparison to their traditional counterparts, which use a solid surface for radiation absorption, particle-based receivers offer a number of opportunities for improved efficiency and heat transfer uniformity. The physical phenomena at the core of such receivers involve coupling between particle transport, fluid turbulence, and radiative heat transfer. Previous analyses of particle-based solar receivers ignored delicate aspects associated with this three-way coupling. Namely, these investigations considered the flow fields only in the mean sense and ignored turbulent fluctuations and the consequent particle preferential concentration. In the present work, we have performed three-dimensional direct numerical simulations of turbulent flows coupled with radiative heating and particle transport over a range of particle Stokes numbers. Our study demonstrates that the particle preferential concentration has strong implications on the heat transfer statistics. We demonstrate that “for a typical setting” the preferential concentration of particles reduces the effective heat transfer between particles and the gas by as much as 25%. Therefore, we conclude that a regime with Stokes number of order unity is the least preferred for heat transfer to the carrier fluid. We also provide a 1D model to capture the effect of particle spatial distribution in heat transfer.

We study the case of inertial particles heated by thermal radiation while settling by gravity through a turbulent transparent gas. We consider dilute and optically thin regimes in which each particle receives the same heat flux. Numerical simulations of forced homogeneous turbulence are performed taking into account the two-way coupling of both momentum and temperature between the dispersed and continuous phases. Particles much smaller than the smallest flow scales are considered and the point-particle approximation is adopted. The particle Stokes number (based on the Kolmogorov time scale) is of order unity, while the nominal settling velocity is up to an order of magnitude larger than the Kolmogorov velocity, marking a critical difference with previous two-way coupled simulations. It is found that non-heated particles enhance turbulence when their settling velocity is sufficiently high compared to the Kolmogorov velocity. Energy spectra show that the non-heated particle settling impacts both the very small and very large flow scales, while the intermediate scales are weakly affected. When heated, particles shed plumes of buoyant gas, further modifying the turbulence structure. At the considered radiation intensities, clustering is strong but the classic mechanism of preferential concentration is modified, while preferential sweeping is eliminated or even reversed. Particle heating also causes a significant reduction of the mean settling velocity, which is caused by rising buoyant plumes in the vicinity of particle clusters. The turbulent kinetic energy is affected non-monotonically as the radiation intensity is increased due to the competing effects of the downward gravitational force and the upward buoyancy force. The thermal radiation influences all scales of the turbulence. The effects of settling and buoyancy on the turbulence anisotropy are also discussed.

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We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it exploits the low-rank structure of fill-in blocks. Depending on the accuracy of low-rank approximations, the hierarchical solver can be used either as a direct solver or as a preconditioner. The parallel algorithm is based on data decomposition and requires only local communication for updating boundary data on every processor. Moreover, the computation-tocommunication ratio of the parallel algorithm is approximately the volumeto-surface-area ratio of the subdomain owned by every processor. We present various numerical results to demonstrate the versatility and scalability of the parallel algorithm.

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In this study we consider particle-laden turbulent flows with significant heat transfer between the two phases due to sustained heating of the particle phase. The sustained heat source can be due to particle heating via an external radiation source as in the particle-based solar receivers or an exothermic reaction in the particles. Our objective is to investigate the effects of fluid heating by a dispersed phase on the turbulence evolution. An important feature in such settings is the preferential clustering phenomenon which is responsible for non-uniform distribution of particles in the fluid medium. Particularly, when the ratio of particle inertial relaxation time to the turbulence time scale, namely the Stokes number, is of order unity, particle clustering is maximized, leading to thin regions of heat source similar to the flames in turbulent combustion. However, unlike turbulent combustion, a particle-laden system involves a wide range of clustering scales that is mainly controlled by particle Stokes number. To study these effects, we considered a decaying homogeneous isotropic turbulence laden with heated particles over a wide range of Stokes numbers. Using a low-Mach-number formulation for the fluid energy equation and a Lagrangian framework for particle tracking, we performed numerical simulations of this coupled system. We then applied a high-fidelity framework to perform spectral analysis of kinetic energy in a variable-density fluid. Our results indicate that particle heating can considerably influence the turbulence cascade. We show that the pressure-dilatation term introduces turbulent kinetic energy at a range of scales consistent with the scales observed in particle clusters. This energy is then transferred to high wavenumbers via the energy transfer term. For low and moderate levels of particle heating intensity, quantified by a parameter $\unicode[STIX]{x1D6FC}$ defined as the ratio of eddy time to mean temperature increase time, turbulence modification occurs primarily in the dilatational modes of the velocity field. However, as the heating intensity is increased, the energy transfer term converts energy from dilatational modes to divergence-free modes. Interestingly, as the heating intensity is increased, the net modification of turbulence by heating is observed dominantly in divergence-free modes; the portion of turbulence modification in dilatational modes diminishes with higher heating. Moreover, we show that modification of divergence-free modes is more pronounced at intermediate Stokes numbers corresponding to the maximum particle clustering. We also present the influence of heating intensity on the energy transfer term itself. This term crosses over from negative to positive values beyond a threshold wavenumber, showing the cascade of energy from large scales to small scales. The threshold is shown to shift to higher wavenumbers with increasing heating, indicating a growth in the total energy transfer from large scales to small scales. The fundamental energy transfer analysis presented in this paper provides insightful guidelines for subgrid-scale modelling and large-eddy simulation of heated particle-laden turbulence.