The adaptation of Crouzeix-Raviart finite element in the context of multi-scale finite element method (MsFEM) is studied and implemented on diffusion and advection-diffusion problems in perforated media. It is known that the approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix-Raviart function space across element edges leads to a natural boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to complex patterns of perforations. Another ingredient to our method is the application of bubble functions which is shown to be instrumental in maintaining high accuracy amid dense perforations. Additionally, the application of penalization method makes it possible to avoid complex unstructured domain and allows extensive use of simpler Cartesian meshes.

The accurate simulation of disperse two-phase flows, where a discrete particulate condensed phase is transported by a carrier gas, is crucial for many applications; Eulerian approaches are well suited for high performance computations of such flows. However when the particles from the disperse phase have a significant inertia compared to the time scales of the flow, particle trajectory crossing (PTC) occurs i.e. the particle velocity distribution at a given location can become multi-valued. To properly account for such a phenomenon many Eulerian moment methods have been recently proposed in the literature. The resulting models hardly comply with a full set of desired criteria involving: 1- ability to reproduce the physics of PTC, at least for a given range of particle inertia, 2- well-posedness of the resulting set of PDEs on the chosen moments as well as guaranteed realizability, 3- capability of the model to be associated with a high order realizable numerical scheme for the accurate resolution of particle segregation in turbulent flows. The purpose of the present contribution is to introduce a multi-variate Anisotropic Gaussian closure for such particulate flows, in the spirit of the closure that has been suggested for out-of-equilibrium gas dynamics and which satisfies the three criteria. The novelty of the contribution is three-fold. First we derive the related moment system of conservation laws with source terms, and justify the use of such a model in the context of high Knudsen numbers, where collision operators play no role. We exhibit the main features and advantages in terms of mathematical structure and realizability. Then a second order accurate and realizable MUSCL/HLL scheme is proposed and validated. Finally the behavior of the method for the description of PTC is thoroughly investigated and its ability to account accurately for inertial particulate flow dynamics in typical configurations is assessed.

Our recently developed lattice Boltzmann model is used to simulate droplet dynamical behaviour governed by thermocapillary force in microchannels. One key research challenge for developing droplet-based microfluidic systems is control of droplet motion and its dynamic behaviour. We numerically demonstrate that the thermocapillary force can be exploited for microdroplet manipulations including synchronisation, sorting, and splitting. This work indicates that the lattice Boltzmann method provides a promising design simulation tool for developing complex droplet-based microfluidic devices.

High-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.

In this paper, we extend the high order finite-difference method with subcell resolution (SR) in [34] for two-species stiff one-reaction models to multispecies and multireaction inviscid chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.

The design optimization and analysis of charged particle beam systems employing intense beams requires a robust and accurate Poisson solver. This paper presents a new type of Poisson solver which allows the effects of space charge to be elegantly included into the system dynamics. This is done by casting the charge distribution function into a series of basis functions, which are then integrated with an appropriate Green's function to find a Taylor series of the potential at a given point within the desired distribution region. In order to avoid singularities, a Duffy transformation is applied, which allows singularity-free integration and maximized convergence region when performed with the help of Differential Algebraic methods. The method is shown to perform well on the examples studied. Practical implementation choices and some of their limitations are also explored.

We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous-approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

A mass-conserving lattice Boltzmann model based on the Bhatnagar-Gross-Krook (BGK) model is proposed for non-Newtonian fluid flows. The equilibrium distribution function includes the local shear rate related with the viscosity and a variable parameter changing with the shear rate. With the additional parameter, the relaxation time in the collision can be fixed invariable to the viscosity. Through the Chapman-Enskog analysis, the macroscopic equations can be recovered from the present mass-conserving model. Two flow problems are simulated to validate the present model with a local computing scheme for the shear rate, and good agreement with analytical solutions and/or other published results are obtained. The results also indicate that the present modified model is more applicable to practical non-Newtonian fluid flows owing to its better accuracy and more robustness than previous methods.

Accurate simulations of high-speed rarefied flows present many physical and computational challenges. Toward this end, the present work extends the Unified Gas Kinetic Scheme (UGKS) to a wider range of Mach and Knudsen numbers by implementing WENO (Weighted Essentially Non-Oscillatory) interpolation. Then the UGKS is employed to simulate the canonical problem of lid-driven cavity flow at high speeds. Direct Simulation Monte Carlo (DSMC) computations are also performed when appropriate for comparison. The effect of aspect ratio, Knudsen number and Mach number on cavity flow physics is examined leading to important insight.

The lattice Boltzmann equation (LBE) methods (both LBGK and MRT) and the discrete unified gas-kinetic scheme (DUGKS) are both derived from the Boltzmann equation, but with different consideration in their algorithm construction. With the same numerical discretization in the particle velocity space, the distinctive modeling of these methods in the update of gas distribution function may introduce differences in the computational results. In order to quantitatively evaluate the performance of these methods in terms of accuracy, stability, and efficiency, in this paper we test LBGK, MRT, and DUGKS in two-dimensional cavity flow and the flow over a square cylinder, respectively. The results for both cases are validated against benchmark solutions. The numerical comparison shows that, with sufficient mesh resolution, the LBE and DUGKS methods yield qualitatively similar results in both test cases. With identical mesh resolutions in both physical and particle velocity space, the LBE methods are more efficient than the DUGKS due to the additional particle collision modeling in DUGKS. But, the DUGKS is more robust and accurate than the LBE methods in most test conditions. Particularly, for the unsteady flow over a square cylinder at Reynolds number 300, with the same mesh resolution it is surprisingly observed that the DUGKS can capture the physical multi-frequency vortex shedding phenomena while the LBGK and MRT fail to get that. Furthermore, the DUGKS is a finite volume method and its computational efficiency can be much improved when a non-uniform mesh in the physical space is adopted. The comparison in this paper clearly demonstrates the progressive improvement of the lattice Boltzmann methods from LBGK, to MRT, up to the current DUGKS, along with the inclusion of more reliable physical process in their algorithm development. Besides presenting the Navier-Stokes solution, the DUGKS can capture the rarefied flow phenomena as well with the increasing of Knudsen number.

In the present contribution we evaluate the heat flux prediction capabilities of second-order accurate Residual Distribution (RD) methods in the context of atmospheric (re-)entry problems around blunt bodies. Our departing point is the computation of subsonic air flows (with air modeled either as an inert ideal gas or as chemically reacting and possibly out of thermal equilibrium gas mixture) around probe-like geometries, as those typically employed into high enthalpy wind tunnels. We confirm the agreement between the solutions obtained with the RD method and the solutions computed with other Finite Volume (FV) based codes.

However, a straightforward application of the same numerical technique to hypersonic cases involving strong shocks exhibits severe deficiencies even on a geometry as simple as a 2D cylinder. In an attempt to mitigate this problem, we derive new variants of RD schemes. A comparison of these alternative strategies against established ones allows us to derive a diagnose for the shortcomings observed in the traditional RD schemes.

The formation of NaH molecules in the lowest rovibrational level of the ground electronic state is investigated using a pump-dump photoassociation (PA) scheme. In short-range region, two colliding atoms Na and H are efficiently associated into the NaH molecule in the rovibrational |0,0〉 state of the ground electronic state via the intermediately rovibrational |10,1〉 state of the excited electronic state. The changes of populations with the electric field amplitudes, frequency detunings, dump pulse duration and delay time between two laser pulses are calculated and discussed. The PA probability reaches 0.623 with a high state-selectivity.

Amid the recent interest in the role of membrane viscosity in the deformation of a fluid-filled capsule, we consider the role of various capsule properties (shear elasticity, membrane bending stiffness and viscosity) in determining the response and recovery times of a spherical capsule in shear flow. These times are determined by fitting exponential functions to results for the Taylor deformation parameter Dxy. We focus on the relationship between the membrane and fluid viscosity ratios, as suggested by Diaz et al, and whether adjustments to the fluid viscosity ratio may be used to approximate the effects of membrane viscosity. Based on its ability to reproduce response and recovery times, our results suggest that such an approach holds promise.

Homogeneous equilibrium model (HEM) has been widely used in cavitating flow simulations. The major feature of this model is that a single equation of state (EOS) is proposed to describe the thermal behavior of bubbly liquid, where both kinematic and thermal equilibrium is assumed between two phases. In this paper, the HEM was coupled with multi-relaxation-time lattice Boltzmann model (MRT-LBM) and the acoustic behavior was simulated. Two approaches were applied alternatively: adjusting speed of sound (Buick, J. Phys. A, 2006, 39:13807-13815) and setting real gas EOS. Both approaches result in high accuracy in acoustic speed predictions for different void (gas) volume of fractions. It is demonstrated that LBM could be successfully applied as a Navier-Stokes equation solver for industrial applications. However, further dissipation and dispersion analysis shows that Shan-Chen type approaches of LBM are deficient, especially in large wave-number region.

Moment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.

Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which suffer not only from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. Amore recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.

Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

In this paper, the fluid flow and heat transfer around a circular cylinder are studied under various conditions (Reynolds number 10 < Re < 200; Prandtl number, 0.1 ≤ Pr ≤ 2). To solve the governing equations, we use the simplified thermal lattice Boltzmann model based on double-distribution function approach, and present a corresponding boundary treatment for both velocity and temperature fields. Extensive numerical results have been obtained to the flow and heat transfer behaviors. The vortices and temperature evolution processes indicate that the flow and temperature fields change synchronously, and the vortex shedding plays a determinant role in the heat transfer. Furthermore, the effects of Reynolds and Prandtl number on the flow and isothermal patterns and local and averaged Nusselt numbers are discussed in detail. Our simulations show that the local and averaged Nusselt numbers increase with the Reynolds and Prandtl numbers, irrespective of the flow regime. However, the minimum value of the local Nusselt number can shift from the rear point at the back of the cylinder with higher Prandtl number even in the steady flow regime, and the distribution of the local Nusselt number is almost monotonous from front stagnation point to rear stagnation point with lower Prandtl number in the unsteady flow regime.

Realistic and accurate computer simulations of the particle-hemodynamics in arterial systems can be a valuable tool for numerous biomedical applications. Examples include optimal by-pass grafting and optimal drug-delivery, as well as best medical management concerning the cardio-vascular system. However, such numerical analyses require large computer resources which may become prohibitive for extended sets of arterial bifurcations. A remedy is to develop a hybrid model where the first few generations of the bifurcating arteries of interest are simulated in full 3-D, while a 1-D model is then coupled for subsequent bifurcations. Alternatively, a 1-D computer model can be directly employed to simulate fluid-particle transport in complex bifurcating networks.

Relying on a representative axial velocity profile, a physiological 1-D model has been developed and validated, which is capable of predicting with reasonable accuracy arterial flow, pressure field and elastic wall interaction as well as particle transport. The usefulness of the novel 1-D simulation approach is demonstrated via a comparison to 3-D blood flow and microsphere transport in a hepatic artery system, featuring as outlets one major branch and four small daughter vessels. Compared to the 3-D simulation, the 1-D analysis requires only about 1% of computational time. The hybrid modeling approach would be also applicable to the human respiratory tract to evaluate the fate of inhaled aerosols.

A simple and cost-effective way to simulate particle-hemodynamics is using a 1-D model for simulating arterial pressures and flow rates as well as microsphere transport, based on assumptions involving the use of a simple algebraic pressure-area relation, an exponential elasticity model for the vessels, and considering only unidirectional flow with a representative skewed velocity profile. In summary, the novel contributions are:

• Particle tracking in arteries via 1-D fluid modeling and selection of an averaged, skewed velocity profile based on 3-D simulation results to provide more realistic friction and inertia term values for modeling a flow system with bifurcations.

• The 1-D model can be coupled to a 3-D model so that simulations can be run for larger regions of vascular or lung-airway systems.

We present a general theoretical framework for the formulation of the nonlinear electromechanics of polymeric and biological active media. The approach developed here is based on the additive decomposition of the Helmholtz free energy in elastic and inelastic parts and on the multiplicative decomposition of the deformation gradient in passive and active parts. We describe a thermodynamically sound scenario that accounts for geometric and material nonlinearities. In view of numerical applications, we specialize the general approach to a particular material model accounting for the behavior of fiber reinforced tissues. Specifically, we use the model to solve via finite elements a uniaxial electromechanical problem dynamically activated by an electrophysiological stimulus. Implications for nonlinear solid mechanics and computational electrophysiology are finally discussed.

We show that a single particle distribution for the “energy-conserving” D2Q13 lattice Boltzmann scheme can simulate coupled effects involving advection and diffusion of velocity and temperature. We consider various test cases: non-linear waves with periodic boundary conditions, a test case with buoyancy, propagation of transverse waves, Couette and Poiseuille flows. We test various boundary conditions and propose to mix bounce-back and anti-bounce-back numerical boundary conditions to take into account velocity and temperature Dirichlet conditions. We present also first results for the de Vahl Davis heated cavity. Our results are compared with the coupled D2Q9-D2Q5 lattice Boltzmann approach for the Boussinesq system and with an elementary finite differences solver for the compressible Navier-Stokes equations. Our main experimental result is the loss of symmetry in the de Vahl Davis cavity computed with the single D2Q13 lattice Boltzmann model without the Boussinesq hypothesis. This result is confirmed by a direct Navier Stokes simulation with finite differences.

An algebraic multilevel method is proposed for efficiently simulating linear wave propagation using higher-order numerical schemes. This method is used in conjunction with the Finite Volume Time Domain (FVTD) technique for the numerical solution of the time-domain Maxwell’s equations in electromagnetic scattering problems. In the multilevel method the solution is cycled through spatial operators of varying orders of accuracy, while maintaining highest-order accuracy at coarser approximation levels through the use of the relative truncation error as a forcing function. Higher-order spatial accuracy can be enforced using the multilevel method at a fraction of the computational cost incurred in a conventional higher-order implementation. The multilevel method is targeted towards electromagnetic scattering problems at large electrical sizes which usually require long simulation times due to the use of very fine meshes dictated by point-per-wavelength requirements to accurately model wave propagation over long distances.