Journal of Computational Physics: X最新文献

We have developed a concurrent heterogeneous multiscale method (HMM) framework with a microscale molecular dynamics (MD) model and a macroscale kinetic Vlasov-BGK model. The kinetic model is formulated such that BGK collision times are the closure data obtained from MD. Using the H-theorem, we develop the mathematical link between the MD and the kinetic model. We examine three relaxation processes, energy, momentum, and bump-on-tail, using full microscale MD simulations as a reference solution. We find that solutions computed with the HMM framework offer a significant computational reduction ($14\times -100\times$) compared with computing a full MD solution, with significant improvements in accuracy compared with a kinetic model using analytical collision times.

The properties of liquid crystals can be modelled using an order parameter which describes the variability of the local orientation of rod-like molecules. Defects in the director field can arise due to external factors such as applied electric or magnetic fields, or the constraining geometry of the cell containing the liquid crystal material. Understanding the formation and dynamics of defects is important in the design and control of liquid crystal devices, and poses significant challenges for numerical modelling. In this paper we consider the numerical solution of a Q-tensor model of a nematic liquid crystal, where defects arise through rapid changes in the Q-tensor over a very small physical region in relation to the dimensions of the liquid crystal device. The efficient solution of the resulting six coupled partial differential equations is achieved using a finite element based adaptive moving mesh approach, where an unstructured triangular mesh is adapted towards high activity regions, including those around defects. Spatial convergence studies are presented using a stationary defect as a model test case, and the adaptive method is shown to be optimally convergent using quadratic triangular finite elements. The full effectiveness of the method is then demonstrated using a challenging two-dimensional dynamic Pi-cell problem involving the creation, movement, and annihilation of defects.

A multiphase Darcy-Brinkman approach is proposed to simulate two-phase flow in hybrid systems containing both solid-free regions and porous matrices. This micro-continuum model is rooted in elementary physics and volume averaging principles, where a unique set of partial differential equations is used to represent flow in both regions and scales. The crux of the proposed model is that it tends asymptotically towards the Navier-Stokes volume-of-fluid approach in solid-free regions and towards the multiphase Darcy equations in porous regions. Unlike existing multiscale multiphase solvers, it can match analytical predictions of capillary, relative permeability, and gravitational effects at both the pore and Darcy scales. Through its open-source implementation, hybridPorousInterFoam, the proposed approach marks the extension of computational fluid dynamics (CFD) simulation packages into porous multiscale, multiphase systems. The versatility of the solver is illustrated using applications to two-phase flow in a fractured porous matrix and wave interaction with a porous coastal barrier.

We present a Smoothed Particle Hydrodynamics (SPH) scheme suitable to model spatially resolved flow of arbitrarily shaped rigid bodies within highly viscous fluids. Coupling to other methods is avoided by representing both fluid and solid phase by SPH particles. The scheme consists of two elements, an implicit viscosity solver and a rigid body solver, both of which are adapted from existing literature. We present how both methods can be coupled with ease and little modification. The scheme presented in this paper can be used for simulations of a representative volume element in which the motion of rigid bodies can be studied in defined velocity gradients composed of elongation and/or shear conditions. The scheme only requires stabilization by particle shifting. However, this causes the loss of exact momentum and energy conservation at the boundary between fluid and rigid bodies. Results are shown for both 2-dimensional and 3-dimensional simulations including academic cases with existing analytical solutions and industrially relevant cases of semi-dilute suspensions of rigid bodies of various shapes.

This paper addresses the two-way coupled Euler-Lagrange modelling of dilute particle-laden flows, with arbitrary particle-size to mesh-spacing ratio. Two-way coupled Euler-Lagrange methods classically require particles to be much smaller than the computational mesh cells for them to be accurately tracked. Particles that do not satisfy this requirement can be considered by introducing a source term regularisation operator that typically consists in convoluting the point-wise particle momentum sources with a smooth kernel. Particles that are larger than the mesh cells, however, generate a significant local flow disturbance, which, in turn, results in poor estimates of the fluid forces acting on them.

To circumvent this issue, this paper proposes a new framework to recover the local undisturbed velocity at the location of a given particle, that is the local flow velocity from which the disturbance due to the presence of the particle is subtracted. It relies upon the solution of the Stokes flow through a regularised momentum source and is extended to finite Reynolds numbers based on the Oseen flow solution. Owing to the polynomial nature of the regularisation kernel considered in this paper, a correction for the averaged local flow disturbance can be analytically derived, allowing to filter out scales of the flow motion that are smaller than the particle, which should not be taken into account to compute the interaction/drag forces acting on the particle. The proposed correction scheme is applied to the simulation of a particle settling under the influence of gravity, for varying particle-size to mesh-spacing ratios and varying Reynolds numbers. The method is shown to nearly eliminate any impact of the underlying mesh resolution on the modelling of a particle's trajectory. Finally, optimal values for the scale of the regularisation kernel are provided and their impact on the flow is discussed.

A method for the calculation of two-dimensional particle trajectories is proposed in this work. It makes use of the cylindrical symmetry and the simplification of the static electric field, so that there should be no systematic error for the centered large-orbit rotations nor for the acceleration or deceleration in a uniform electric field. The method also shows a lower error level than the standard Boris method in many cases. Typical applications of this method are for example, electron microscopes, electron guns and collectors of gyro-devices as well as of other vacuum tubes, which can be described in axisymmetric cylindrical coordinates. Besides, the proposed method enforces the conservation of canonical angular momentum by construction, which is expected to show its advantages in the simulation of cusp electron guns and other components relying on non-adiabatic transitions in the externally applied static magnetic field.

In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singularities near the corners which poses a challenge for many existing methods. If the boundary data is smooth on each edge of the polygon, then in the vicinity of each corner the solution to the corresponding boundary integral equation has an expansion in terms of certain (analytically available) singular powers. Using the known behavior of the solution, universal discretizations have been constructed for the solution of the Dirichlet problem. However, the leading order behavior of solutions to the Neumann problem is $O\left(t^{\mu }\right)$ for $\mu \in (-1/2,1/2)$ depending on the angle at the corner (compared to $O(C+t^{\mu })$ with $\mu >1/2$ for the Dirichlet problem); this presents a significant challenge in the design of universal discretizations. Our approach is based on using the discretization for the Dirichlet problem in order to compute a solution in the “weak sense” by solving an adjoint linear system; namely, it can be used to compute inner products with smooth functions accurately, but it cannot be interpolated. Furthermore we present a procedure to obtain accurate solutions arbitrarily close to the corner, by solving a sequence of small local subproblems in the vicinity of that corner. The results are illustrated with several numerical examples.

In this short note, we highlight the sensitivity of the HLL-type Riemann solver with respect to the choice of signal speed estimates and demonstrate a major deficiency of the arithmetic-average estimate. The investigation of two essential Riemann problems and a classical bow shock simulation reveals that inherent inconsistencies of the arithmetic-average estimate may lead to unexpected behavior and erroneous results.

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation.