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1. Dirichlet Boundary Conditions for a Meshless Solver of the Discrete Boltzmann Equation

   https://doi.org/10.1115/FEDSM2025-158550  

   Maidenberg, Amandine R; Chen, Leitao (September 2025, American Society of Mechanical Engineers)

   Abstract The discrete Boltzmann equation (DBE), which is rooted in the kinetic theory, has shown its potential as researchers explore alternate methods of solving transport phenomena. Most applications for DBE are solved using the Lattice Boltzmann Method (LBM) that assumes a coupled and uniform time-space discretization, which renders a Lagrangian streaming process that is constrained to a fixed lattice structure. This adhesion to a fixed and uniform grid structure can become cumbersome in handling irregular domains such as porous media. Alternatively, meshless methods with the radial basis function finite difference (RBF-FD) have shown potential to solve partial differential equations without traditional grid underpinnings. The challenge in using an unstructured discretization to solve the DBE is the translation of macroscale properties on the boundaries, such as density and velocity, into the particle density function (PDF) tracked by the DBE. In this paper, the DBE is solved with a RBF-FD meshless method to simulate fluid flows. A review of the Taylor Green Vortex flow is presented to demonstrate the RBF-FD meshless method without actual boundaries, which is used to explore the artificial viscosity created by RBF-FD technique without the presence of boundaries. Then a scheme to resolve the unknown components of the PDF on Dirichlet boundary conditions is proposed. In this scheme, the PDF is separated into equilibrium and non-equilibrium components, and a nearest-neighbor approach is used to interpolate unknown components. This technique is demonstrated on both flat and curved surfaces and shows good agreement with literature values. 

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2. A flux bounce-back scheme for the filtered spectral element lattice Boltzmann method

   https://doi.org/10.1016/j.compfluid.2026.106987  

   Zhao, Chunheng; Patel, Saumil S; Lin, Hai Lu; Min, Misun; Lee, Taehun (March 2026, Computers & Fluids)

   We develop a spectral element lattice Boltzmann method with the flux bounce-back (FBB) scheme, to enable accurate simulations of single-phase fluid dynamics in unstructured mesh. We adopt an Eulerian description of the streaming process in place of the perfect shift in the standard LBM. The spectral element method is used to spatially discretize the convective term, while the strong stability-preserving Runge-Kutta (SSPRK) method is used for time integration. To increase stability, we investigate the use of an explicit filter, particularly in the context of the sensitive double shear layer problem. The results indicate that by using the high-order polynomial, we can effectively eliminate the small vortices around the neck region. We introduce the flux bounce-back scheme to enable the current scheme to handle complex boundaries. The proposed scheme and flux boundary method are validated through benchmark simulations, including the unsteady Couette flow and the planar Poiseuille flow. Further validation is provided through the Taylor-Green vortex problem, demonstrating the accuracy and convergence of the scheme for isotropic turbulence. Finally, we consider a fully developed turbulent flow within a cylindrical pipe and correctly predict the turbulent boundary layer profile. 

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3. Implementation of a ternary lattice Boltzmann model in LAMMPS

   https://doi.org/10.1016/j.cpc.2023.108898  

   Arumugam Kumar, Gokul Raman; Andrews, James P.; Schiller, Ulf D. (September 2023, Computer Physics Communications)

   The properties of multicomponent fluids are governed by the interplay of phase behavior, fluid dynamics, and interfacial thermodynamics. A mixture formulation that leverages this interplay is an important aspect in many fabrication processes based on emulsion templating. The lattice Boltzmann method (LBM) has become a popular approach for simulating hydrodynamic effects in complex fluids and soft matter. Here we present an implementation of a ternary lattice Boltzmann model that allows to simulate a mixture of three immiscible fluids. We build on the LATBOLTZ extension of the open-source package LAMMPS and implement a ternary free energy model recently introduced by Semprebon et al. [Phys. Rev. E 93, 033305 (2016)]. We validate the static and dynamic properties by simulating liquid lenses, double emulsions, and ternary mixtures. From the simulations, we obtain the complete morphology diagram of the ternary mixture in composition space. We further discuss an application of the method to phase segregation of ternary films. The implementation of the ternary LBM in LAMMPS opens vast opportunities for mesoscale simulations of interfacial phenomena and non-equilibrium transport processes in multicomponent fluid mixtures. 

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4. A numerical study of fish adaption behaviors in complex environments with a deep reinforcement learning and immersed boundary–lattice Boltzmann method

   https://doi.org/10.1038/s41598-021-81124-8  

   Zhu, Yi; Tian, Fang-Bao; Young, John; Liao, James C.; Lai, Joseph C. (December 2021, Scientific Reports)

   null (Ed.)

   Abstract Fish adaption behaviors in complex environments are of great importance in improving the performance of underwater vehicles. This work presents a numerical study of the adaption behaviors of self-propelled fish in complex environments by developing a numerical framework of deep learning and immersed boundary–lattice Boltzmann method (IB–LBM). In this framework, the fish swimming in a viscous incompressible flow is simulated with an IB–LBM which is validated by conducting two benchmark problems including a uniform flow over a stationary cylinder and a self-propelled anguilliform swimming in a quiescent flow. Furthermore, a deep recurrent Q-network (DRQN) is incorporated with the IB–LBM to train the fish model to adapt its motion to optimally achieve a specific task, such as prey capture, rheotaxis and Kármán gaiting. Compared to existing learning models for fish, this work incorporates the fish position, velocity and acceleration into the state space in the DRQN; and it considers the amplitude and frequency action spaces as well as the historical effects. This framework makes use of the high computational efficiency of the IB–LBM which is of crucial importance for the effective coupling with learning algorithms. Applications of the proposed numerical framework in point-to-point swimming in quiescent flow and position holding both in a uniform stream and a Kármán vortex street demonstrate the strategies used to adapt to different situations. 

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5. Modeling of Rock Scour using Coupled 3-D Discrete Element and Lattice Boltzmann Methods

   Gardner, Michael H.; Sitar, Nicholas (June 2018, 52nd US Rock Mechanics/Geomechancs Symposium)

   Scour of rock in unlined rock spillway channels is a critical issue facing many of the world’s dams. From a modeling point of view this poses a challenging and interesting problem that combines rock mechanics and hydraulics of turbulent flow. We analyze this interaction between the blocky rock mass and water by directly modeling the solid and fluid phases—the individual polyhedral blocks are modeled using the Discrete Element Method (DEM) while the water is modeled using the Lattice Boltzmann Method (LBM). The LBM mesh is entirely independent of the DEM discretization, making it possible to refine the LBM mesh such that transient and varied fluid pressures acting of the rock surface are directly modeled. This provides the capability to investigate the effect of water pressure inside the fractured rock mass, along potential sliding planes, and can be extended to rock falls and slides into standing bodies of water such as lakes and reservoirs. Herein we present preliminary results to demonstrate the capabilities of the methodology. 

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