The immersed boundary method (IBM) has been popular in simulating fluid structure interaction (FSI) problems involving flexible structures, and the recent introduction of the lattice Boltzmann method (LBM) into the IBM makes the method more versatile. In order to test the coupling characteristics of the IBM with the multiple-relaxation-time LBM (MRT-LBM), the three-dimensional (3D) balloon dynamics, including inflation, release and breach processes, are simulated. In this paper, some key issues in the coupling scheme, including the discretization of 3D boundary surfaces, the calculation of boundary force density, and the introduction of external force into the LBM, are described. The good volume conservation and pressure retention properties are verified by two 3D cases. Finally, the three FSI processes of a 3D balloon dynamics are simulated. The large boundary deformation and oscillation, obvious elastic wave propagation, sudden stress release at free edge, and recoil phenomena are all observed. It is evident that the coupling scheme of the IBM and MRT-LBM can handle complicated 3D FSI problems involving large deformation and large pressure gradients with very good accuracy and stability.

A novel method is developed for solving the inverse problem of reconstructing the shape of an interior cavity. The boundary of the cavity is assumed to be a small and smooth perturbation of a circle. The incident field is generated by a point source inside the cavity. The scattering data is taken on a circle centered at the source. The method requires only a single incident wave at one frequency. Using a transformed field expansion, the original boundary value problem is reduced to a successive sequence of two-point boundary value problems and is solved in a closed form. By dropping higher order terms in the power series expansion, the inverse problem is linearized and an explicit relation is established between the Fourier coefficients of the cavity surface function and the total field. A nonlinear correction algorithm is devised to improve the accuracy of the reconstruction. Numerical results are presented to show the effectiveness of the method and its ability to obtain subwavelength resolution.

The effective thermal conductivity of composite materials with thermal contact resistance at interfaces is studied by lattice Boltzmann modeling in this work. We modified the non-dimensional partial bounce-back scheme, proposed by Han et al. [Int. J. Thermal Sci., 2008. 47: 1276-1283], to introduce a real thermal contact resistance at interfaces into the thermal lattice Boltzmann framework by re-deriving the redistribution function of heat at the phase interfaces for a corrected dimensional formulation. The modified scheme was validated in several cases with good agreement between the simulation results and the corresponding theoretical solutions. Furthermore, we predicted the effective thermal conductivities of composite materials using this method where the contact thermal resistance was not negligible, and revealed the effects of particle volume fraction, thermal contact resistance and particle size. The results in this study may provide a useful support for materials design and structure optimization.

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

The coordinate transformation offers a remarkable way to design cloaks that can steer electromagnetic fields so as to prevent waves from penetrating into the cloaked region (denoted by Ω0, where the objects inside are invisible to observers outside). The ideal circular and elliptic cylindrical cloaked regions are blown up from a point and a line segment, respectively so the transformed material parameters and the corresponding coefficients of the resulted equations are highly singular at the cloaking boundary ∂Ω0. The electric field or magnetic field is not continuous across ∂Ω0. The imposition of appropriate cloaking boundary conditions (CBCs) to achieve perfect concealment is a crucial but challenging issue.

Based upon the principle that a well-behaved electromagnetic field in the original space must be well-behaved in the transformed space as well, we obtain CBCs that intrinsically relate to the essential “pole” conditions of a singular transformation. We also find that for the elliptic cylindrical cloak, the CBCs should be imposed differently for the cosine-elliptic and sine-elliptic components of the decomposed fields. With these at our disposal, we can rigorously show that the governing equation in Ω0 can be decoupled from the exterior region , and the total fields in the cloaked region vanish under mild conditions. We emphasize that our proposal of CBCs is different from any existing ones.

Using the exact circular (resp., elliptic) Dirichlet-to-Neumann (DtN) non-reflecting boundary conditions to reduce the unbounded domain to a bounded domain, we introduce an accurate and efficient Fourier-Legendre spectral-element method (FLSEM) (resp., Mathieu-Legendre spectral-element method (MLSEM)) to simulate the circular cylindrical cloak (resp., elliptic cylindrical cloak). We provide ample numerical results to demonstrate that the perfect concealment of waves can be achieved for the ideal circular/elliptic cylindrical cloaks under our proposed CBCs and accurate numerical solvers.

Newtonian and non-Newtonian dispersed phase droplet formation in non-Newtonian continuous phase in T-junction and cross junction microchannels are investigated by the immiscible lattice BGK model. The effects of the non-Newtonian fluid power-law exponent, viscosity and interfacial tension on the generation of the droplet are studied. The final droplet size, droplet generation frequency, and detachment point of the droplet change with the power-law exponent. The results reveal that it is necessary to take into account the non-Newtonian rheology instead of simple Newtonian fluid assumption in numerical simulations. The present analysis also demonstrates that the lattice Boltzmann method is of potential to investigate the non-Newtonian droplet generation in multiphase flow.

The split-operator technique for wave packet propagation in quantum systems is expanded here to the case of propagatingwave functions describing Schrödinger particles, namely, charge carriers in semiconductor nanostructures within the effective mass approximation, in the presence of Zeeman effect, as well as of Rashba and Dresselhaus spin-orbit interactions. We also demonstrate that simple modifications to the expanded technique allow us to calculate the time evolution of wave packets describing Dirac particles, which are relevant for the study of transport properties in graphene.

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in [26], and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.

Currently, there exists a lack of confidence in the computational simulation of multiple body high-speed air delivered systems. Of particular interest is the ability to accurately predict the dispersion pattern of these systems under various deployment configurations. Classical engineering-level methods may not be able to predict these patterns with adequate confidence due primarily to accuracy errors attributable to reduced order modeling. In the current work, a new collision modeling capability has been developed to enable multiple-body proximate-flight simulation in the Loci/CHEM framework. This approach maintains high-fidelity aerodynamics and incorporates six degrees of freedom modeling with collision response, and is well-suited for simulation of a large number of projectiles. The proposed simulation system is intended to capture the strong interaction phase early in the projectile deployment, with subsequent transfer of projectile positions and flight states to the more economical engineering-level methods. Collisions between rigid bodies are modeled using an impulse-based approach with either an iterative propagation method or a simultaneous method. The latter is shown to be more accurate and robust for cases involving multiple simultaneous collisions as it eliminates the need to sort and resolve the collisions sequentially. The implementation of both the collision detection methodology and impact mechanics are described in detail with validation studies to demonstrate the efficiency and accuracy of the developed technologies. The studies chronologically detail the findings for simulating simple impacts and collisions between multiple bodies with aerodynamic interference effects.

This paper presents a numerical and experimental study on hydrodynamic behavior of thin liquid films in rectangular domains. Three-dimensional computer simulations were performed using the lattice Boltzmann equation method (LBM). The liquid films laying on solid and liquid substrates are considered. The rupture of liquid films in computations is initiated via the thermocapillary (Marangoni) effect by applying an initial spatially localized temperature perturbation. The rupture scenario is found to depend on the shape of the temperature distribution and on the wettability of the solid substrate. For a wettable solid substrate, complete rupture does not occur: a residual thin liquid film remains at the substrate in the region of pseudo-rupture. For a non-wettable solid substrate, a sharp-peaked axisymmetric temperature distribution induces the rupture at the center of symmetry where the temperature is maximal. Axisymmetric temperature distribution with a flat-peaked temperature profile initiates rupture of the liquid film along a circle at some distance from the center of symmetry. The outer boundary of the rupture expands, while the inner liquid disk transforms into a toroidal figure and ultimately into an oscillating droplet.

We also apply the LBM to simulations of an evolution of one or two holes in liquid films for two-layer systems of immiscible fluids in a rectangular cell. The computed patterns are successfully compared against the results of experimental visualizations. Both the experiments and the simulations demonstrate that the initially circular holes evolved in the rectangular cell undergoing drastic changes of their shape under the effects of the surface tension and gravity. In the case of two interacting holes, the disruption of the liquid bridge separating two holes is experimentally observed and numerically simulated.

An entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper to approximate hyperbolic systems of conservation laws. This numerical scheme involves the solution of a very large nonlinear system of algebraic equations, by a Newton-Krylov method, at every time step. In this paper, we design efficient preconditioners for the large, non-symmetric linear system, that needs to be solved at every Newton step. Two sets of preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are designed. Fourier analysis of the preconditioners reveals their robustness and a large number of numerical experiments are presented to illustrate the gain in efficiency that results from preconditioning. The resulting method is employed to compute approximate solutions of the compressible Euler equations, even for very high CFL numbers.

The method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit method was developed to solve two-dimensional Euler and Navier-Stokes equations by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed to solve the nonlinear system, while the linear system was solved with LU-SGS iteration. The effect of several parameters in the implicit scheme, such as the CFL number, the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated to evaluate the performance of convergence history. Several typical test cases were simulated, and compared with the traditional explicit Runge-Kutta (RK) scheme. Firstly the Couette flow was tested to validate the order of accuracy of the present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel was simulated and the effect of parameters was investigated also. Finally, the implicit algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder and the viscous flow in a square cavity. The numerical results demonstrated that the present implicit scheme can accelerate the convergence history efficiently. Choosing proper parameters would improve the efficiency of the implicit scheme. Moreover, in the same framework, the DG/FV hybrid schemes are more efficient than the same order DG schemes.

Despite its demonstrated importance in the deformation and dynamics of red blood cells, membrane viscosity has not received the same attention in computational models as elasticity and bending stiffness. Recent experiments on red blood cells indicated a power law response due to membrane viscosity. This is potentially much different from the solid viscoelastic models, such as Kelvin-Voigt and standard linear solid (SLS), currently used in computation to describe this aspect of the membrane. Within the context of a framework based on lattice Boltzmann and immersed boundary methods, we introduce SLS and power law models for membrane viscosity. We compare how the Kelvin-Voigt (as approximated by SLS) and power law models alter the deformation and dynamics of a spherical capsule in shear flows.

A biphase mixture continuum mechanics model is derived for neutral heat-shrinkable thermo-sensitive hydrogels in this paper. The mixing free energy of the special mixture is recalculated based on the partition function of Bose system, and it evaluates the contribution of the hydrophilic, hydrophobic interaction and hydrogen bonding to the volume phase transition behaviors. The ideas of the Flory lattice theory and the UNIFAC group contribution method are employed to get the expression of the mixing free energy. Then we deduce a particular model by combining this mixing free energy with the conservation laws equations and constitutive relations of both phases to predict the volume transition behaviors of these special hydrogels.

This paper concerns the analytic structure of the self-consistent field theory (SCFT) energy functional of multicomponent block copolymer systems which contain more than two chemically distinct blocks. The SCFT has enjoyed considered success and wide usage in investigation of the complex phase behavior of block copolymers. It is well-known that the physical solutions of the SCFT equations are saddle points, however, the analytic structure of the SCFT energy functional has received little attention over the years. A recent work by Fredrickson and collaborators [see the monograph by Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, (2006), pp. 203–209] has analysed the mathematical structure of the field energy functional for polymeric systems, and clarified the index-1 saddle point nature of the problem caused by the incompressible constraint. In this paper, our goals are to draw further attention to multicomponent block copolymers utilizing the Hubbard-Stratonovich transformation used by Fredrickson and co-workers. We firstly show that the saddle point character of the SCFT energy functional of multicomponent block copolymer systems may be high index, not only produced by the incompressible constraint, but also by the Flory-Huggins interaction parameters. Our analysis will be beneficial to many theoretical studies, such as the nucleation theory of ordered phases, the mesoscopic dynamics. As an application, we utilize the discovery to develop the gradient-based iterative schemes to solve the SCFT equations, and illustrate its performance through several numerical experiments taking ABC star triblock copolymers as an example.

In this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d’Humières. They extend also the Geier’s cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy.