arXiv subjects

Maciej Matyka

Publications and source records attributed to Maciej Matyka.

At least 19 recordsLinked to original sources

arXiv2026

Benchmark of no-slip boundary conditions for meshless Lattice Boltzmann Method in Stokes flow

The introduction of the approximated streaming to the standard Lattice Boltzmann Method (LBM) decouples the space and velocity discretizations. Although many no-slip implementations knwon from LBM can be directly adopted to the models with approximated streaming, their exact behavior in the off-lattice setting remains unexplored. In this work, we investigate the meshfree off-lattice D2Q9 model equipped with several no-slip implementations -- non-equilibrium extrapolation, simple and interpolated bounceback, and moment-based boundaries -- applied to a Stokes flow around a cylinder. We compare the pressure, velocity and velocity gradient fields obtained with various no-slips, on polar and hybrid polar-scattered discretizations. We find that non-equilibrium extrapolation performs the best of all the studied no-slip realizations, giving smooth stresses and zero velocity on the solid walls. Other no-slips suffer from oscillations and/or discontinuities in the hydrodynamic fields. Hybrid discretizations are found to give higher errors than the polar grid, but improve the stability of the solution. Finally, we simulate the deformation of an elastic ring under the stresses obtained with the studied no-slips to highlight the importance of the sensible choice of the no-slip realization in more complex problems.

physics.flu-dyn↗

arXiv2026

Physics-informed convolutional neural networks for fluid flow through porous media

Accurate simulation of fluid flow in porous media is challenging due to complex pore-space geometries and the computational cost of solving the Navier-Stokes equations. This difficulty is particularly important when repeated simulations are required, as standard numerical solvers may converge slowly in intricate porous domains. We present a neural-network-based framework for predicting pore-scale velocity fields directly from sample geometry. The method uses a convolutional encoder-decoder architecture with skip connections to preserve spatial detail while extracting multi-scale features. Physical consistency is encouraged through a custom loss function combining velocity reconstruction with incompressibility, no-flow conditions inside solids, periodicity constraints, and agreement with the global tortuosity index. We analyze the influence of the corresponding loss weights and quantify the contribution of individual loss components to prediction accuracy. Several CNN backbones are evaluated to identify architectures providing accurate and robust predictions. The generalization ability of the trained model is tested on samples outside the training distribution, including changes in obstacle geometry, boundary conditions, porosity, and realistic porous structures. Finally, we demonstrate a practical use of the predicted velocity fields as initial conditions for Lattice-Boltzmann simulations. This warm-start strategy accelerates solver convergence, reducing the number of iterations in over 90% of tested cases.

cs.LG↗

arXiv2025

The role of porosity in the transition to inertial regime in porous media flows

In this work, we investigate the fundamental physical mechanism of the transition from Darcy to inertial (Darcy-Forchheimer) regime in steady-state flows through porous media, with the focus on vortex formation. We investigate their influence on the tortuosity--Reynolds number relation during this transition for systems of various porosities. We do so by numerically solving the Navier-Stokes equations within the pore-scale of simple cubic systems and relating the observations made therein to stochastic systems of more complex geometry. We observe that the tortuosity defined by integrals over the whole fluid volume behaves similarly in both types of systems. At the same time, in simple cubic systems, the tortuosity based on averaging of the length of the streamlines diverges from the volume-integrated one when the inertia onset takes place. We show that the discrepancy between those two tortuosities at increasing Reynolds number carries information about the dynamics of the vortex growth in the system. We stipulate that those dynamics are directly governed by the porosity. Our results highlight the utility of various definitions of tortuosity as measures of inertia in porous media flows and explain the reasons for the differences between those definitions. This can lead to a more sensible choice of inertia indicators in more application-oriented problems.

physics.flu-dyn↗

arXiv2025

Cross-validation of meshless Navier-Stokes solvers in porous media flows

In this paper, two mesh-free CFD solvers for pore-scale fluid flow through porous media are considered, namely the Lattice Boltzmann Method with the two relaxation time collision term and the direct Navier-Stokes solver under the artificial compressibility limit. The porous media is built with a regular arrangement of spherical grains with variable radii, which allows control of the porosity. Both solvers use the same $h$-refined meshless spatial discretization to adequately capture the underlying geometry and the same Radial Basis Function (RBF) method to approximate the involved fields and partial differential operators. First, the results are compared with the data from the literature in terms of drag coefficient and permeability at different porosities achieving excellent agreement with the reported results. Next, the simulations are extended beyond the porosity range reported in the literature using proposed $h$-refined CFD solvers. The results are supported by convergence and timing analyses and discussions on meshless parameters such as stencil size and refinement settings.

physics.flu-dyn↗

arXiv2025

Inertial effects on fluid flow through low-porosity media

We investigated the nonlinear effects of gravity-driven fluid flow through a two-dimensional, low-porosity, packed bed of stubby stone grains. We focused on preferential channel formation, tortuosity, spatial distribution of kinetic energy, and vortex formation. We show that nonlinear effects dominate at relatively high Reynolds numbers, even though the deviation from Darcy's law is not visible in friction factor measurements. We further notice an increased flow asymmetry of the flow field revealed by vorticity analysis and surprising correlation between tortuosity and apparent permeability in the inertial flow regime.

physics.flu-dyn↗

arXiv2024

Cauliflower shapes of bacterial clusters in the off-lattice Eden model for bacterial growth in a Petri dish with an agar layer

We developed the off-lattice Eden model to simulate the growth of bacterial colonies in the three-dimensional geometry of a Petri dish. In contrast to its two-dimensional counterpart, our model takes a three-dimensional set of possible growth directions and employs additional constraints on growth, which are limited by access to the nutrient layer. We rigorously tested the basic off-lattice Eden implementation against literature data for a planar cluster. We then extended it to three-dimensional growth. Our model successfully demonstrated the non-trivial dependency of the cluster morphology, non-monotonous dependency of the cluster density, and power law of the thickness of the boundary layer of clusters as a function of the nutrient layer height. Moreover, we revealed the fractal nature of all the clusters by investigating their fractal dimensions. Our density results allowed us to estimate the basic transport properties, namely the permeability and tortuosity of the bacterial colonies.

physics.comp-ph↗

arXiv2024

Modeling inertial flows with meshless Lattice Boltzmann Method

One of the limitations of the Lattice Boltzmann Method in simulating inertial flows is the coupling of the discretization of space to the velocity discretization. It requires an increase of the size of computational lattices in order to increase the Reynolds number at a fixed velocity and viscosity. In this work, we adopt the recently proposed meshless formulation of Lattice Boltzmann Method to the problem of inertial flows. In contrast to the standard algorithm, it allows to decouple space and velocity discretizations. Thus, one can change the conversion factors from lattice to physical units for length, velocity, and body force by scaling the streaming distance. In turn, one increases the Reynolds number without increasing the size of the discretization. We measure the accuracy and efficiency of this approach in the Kármán vortex street behind a circular obstacle and the flow through a porous sample in Darcy and inertial regime. Additionally, we apply the meshless streaming step to the recently proposed fixed relaxation time $τ=1$ LBM to extend its applicability to model inertial flows.

physics.flu-dyn↗

arXiv2024

Inertia onset in disordered porous media flow

We investigate the very onset of the inertial regime in the fluid flow at the pore level in a three-dimensional, disordered, highly porous media. We analyze the flow structure in a wide range of Reynolds numbers starting from 0.01 up to 100. We focus on qualitative and quantitative changes that appear with increasing Reynolds number. To do that, we investigate the weakening of the channeling effect, defined as the existence of preferred flow paths in a system. We compute tortuosity, spatial kinetic energy localization, and the pore-space volume fraction containing negative streamwise velocity to assess accompanying changes quantitatively. Our results of tortuosity and participation number derivatives show that the very onset of inertia is apparent for Reynolds number Re $\sim 0.1$, an order of magnitude lower than indicated by analyzing relations of friction factor with the Reynolds number. Moreover, we show that the vortex structures appear at Reynolds number two orders of magnitude higher than the onset of inertia.

physics.flu-dyn↗

arXiv2023

The lattice Boltzmann method for fluid flows at relaxation time equal one: performance study

Running large-scale computer codes for huge fluid flow problems requires not only large supercomputers but also efficient and well-optimized computer codes that save the resources of those supercomputers. This paper evaluates the high-performance implementation of the recently proposed Lattice Boltzmann Method (LBM) algorithm with a fixed viscosity and relaxation time $τ=1$ called Tau1. We show that the performance of the Tau1 algorithm is almost $4\times$ higher than other state-of-the-art standard LBM implementations. We support this finding by detailed complexity analysis and performance study based on the code with several optimizations, including multithreading, vector processing, significantly decreased number of divisions, and dedicated memory layout. We studied its performance in porous media flow in the three-dimensional model of porosity-varying porous medium to make it sound in the physical context. We find performance drops with porosity, which we link to the fact that memory access patterns change dramatically with the increased complexity of the pore space. In contrast to standard LBM implementations, where the performance drops with the number of lattice links, the processing speed of the new algorithm measured in fluid node updates per second is almost constant regardless of lattice arrangements.

physics.comp-ph↗

arXiv2023

Study of the convergence of the Meshless Lattice Boltzmann Method in Taylor-Green and annular channel flows

The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and, at the same time, decouples space and velocity discretizations. In this study, we investigate the numerical convergence of MLBM in two benchmark tests: the Taylor-Green vortex and annular (bent) channel flow. We compare our MLBM results to LBM and to the analytical solution of the Navier-Stokes equation. We investigate the method's convergence in terms of the discretization parameter, the interpolation order, and the LBM streaming distance refinement. We observe that MLBM outperforms LBM in terms of the error value for the same number of nodes discretizing the domain. We find that LBM errors at a given streaming distance $δx$ and timestep length $δt$ are the asymptotic lower bounds of MLBM errors with the same streaming distance and timestep length. Finally, we suggest an expression for the MLBM error that consists of the LBM error and other terms related to the semi-Lagrangian nature of the discussed method itself.

physics.flu-dyn↗

arXiv2023

Deep learning for diffusion in porous media

We adopt convolutional neural networks (CNN) to predict the basic properties of the porous media. Two different media types are considered: one mimics the sand packings, and the other mimics the systems derived from the extracellular space of biological tissues. The Lattice Boltzmann Method is used to obtain the labeled data necessary for performing supervised learning. We distinguish two tasks. In the first, networks based on the analysis of the system's geometry predict porosity and effective diffusion coefficient. In the second, networks reconstruct the concentration map. In the first task, we propose two types of CNN models: the C-Net and the encoder part of the U-Net. Both networks are modified by adding a self-normalization module [Graczyk \textit{et al.}, Sci Rep 12, 10583 (2022)]. The models predict with reasonable accuracy but only within the data type, they are trained on. For instance, the model trained on sand packings-like samples overshoots or undershoots for biological-like samples. In the second task, we propose the usage of the U-Net architecture. It accurately reconstructs the concentration fields. In contrast to the first task, the network trained on one data type works well for the other. For instance, the model trained on sand packings-like samples works perfectly on biological-like samples. Eventually, for both types of the data, we fit exponents in the Archie's law to find tortuosity that is used to describe the dependence of the effective diffusion on porosity.

physics.comp-ph↗

arXiv2021

Memory-efficient Lattice Boltzmann Method for low Reynolds number flows

The Lattice Boltzmann Method algorithm is simplified by assuming constant numerical viscosity (the relaxation time is fixed at $τ=1$). This leads to the removal of the distribution function from the computer memory. To test the solver the Poiseuille and Driven Cavity flows are simulated and analyzed. The error of the solution decreases with the grid size L as $L^{-2}$. Compared to the standard algorithm, the presented formulation is simpler and shorter in implementation. It is less error-prone and needs significantly less working memory in low Reynolds number flows. Our tests showed that the algorithm is less efficient in multiphase flows. To overcome this problem, further extension and the moments-only formulation was derived, inspired by the Multi-Relaxation Time (MRT) approach for single component multiphase flows.

physics.comp-ph↗

arXiv2020

Predicting Porosity, Permeability, and Tortuosity of Porous Media from Images by Deep Learning

Convolutional neural networks (CNN) are utilized to encode the relation between initial configurations of obstacles and three fundamental quantities in porous media: porosity ($φ$), permeability $k$, and tortuosity ($T$). The two-dimensional systems with obstacles are considered. The fluid flow through a porous medium is simulated with the lattice Boltzmann method. It is demonstrated that the CNNs are able to predict the porosity, permeability, and tortuosity with good accuracy. With the usage of the CNN models, the relation between $T$ and $φ$ has been reproduced and compared with the empirical estimate. The analysis has been performed for the systems with $φ\in (0.37,0.99)$ which covers five orders of magnitude span for permeability $k \in (0.78, 2.1\times 10^5)$ and tortuosity $T \in (1.03,2.74)$.

physics.comp-ph↗

arXiv2020

Pushing a soft body droplet through porous medium

We investigate the problem of transport of a single fluid droplet through a non-wettable superhydrophobic porous medium. A mechanical soft body model is developed and used to simulate the process of pushing fluid droplets through pore space of a random porous medium. Path lines of the center of mass of each droplet are used to calculate tortuosity of the media. Our results show that droplet based tortuosity increases with decreasing porosity of the porous samples. Although qualitatively this agrees with the behaviour observed for tortuosity derived from the fluid flow, the form of this relation is different.

physics.comp-ph↗

arXiv2016

Multiparticle collision dynamics in porous media

Multiparticle collision dynamics (MPCD) is a relatively new algorithm of fluid flow simulations that has been applied mostly to flows around simple objects. One might ask how it behaves in more complex flows. Therefore, we extend MPCD to simulate transport in porous media. For this, a particle-level drag force is introduced into the original algorithm. The force hinders the flow, which results in global resistance to flow and decrease of permeability. The extended algorithm is validated in the flow through a porous channel and compared with an analytical solution. Some basic properties of the solver are investigated.

physics.comp-ph↗

arXiv2015

Power exponential velocity distributions in disordered porous media

Velocity distribution functions link the micro- and macro-level theories of fluid flow through porous media. Here we study them for the fluid absolute velocity and its longitudinal and lateral components relative to the macroscopic flow direction in a model of a random porous medium. We claim that all distributions follow the power exponential law controlled by an exponent $γ$ and a shift parameter $u_0$ and examine how these parameters depend on the porosity. We find that $γ$ has a universal value $1/2$ at the percolation threshold and grows with the porosity, but never exceeds 2.

physics.flu-dyn↗

arXiv2014

Compressed Multi-Row Storage Format for Sparse Matrices on Graphics Processing Units

A new format for storing sparse matrices is proposed for efficient sparse matrix-vector (SpMV) product calculation on modern graphics processing units (GPUs). This format extends the standard compressed row storage (CRS) format and can be quickly converted to and from it. Computational performance of two SpMV kernels for the new format is determined for over 130 sparse matrices on Fermi-class and Kepler-class GPUs and compared with that of five existing generic algorithms and industrial implementations, including Nvidia cuSparse CSR and HYB kernels. We found the speedup of up to $\approx 60%$ over the best of the five alternative kernels.

physics.comp-ph↗

arXiv2013

Anisotropy of flow in stochastically generated porous media

Models of porous media are often applied to relatively small systems, which leads not only to system-size-dependent results, but also to phenomena that would be absent in larger systems. Here we investigate one such finite-size effect: anisotropy of the permeability tensor. We show that a non-zero angle between the external body force and macroscopic flux vector exists in three-dimensional periodic models of sizes commonly used in computer simulations and propose a criterion, based on the system size to the grain size ratio, for this phenomenon to be relevant or negligible. The finite-size anisotropy of the porous matrix induces a pressure gradient perpendicular to the axis of a porous duct and we analyze how this effect scales with the system and grain sizes.

physics.flu-dyn↗