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.

Coupling the immersed boundary (IB) method and the lattice Boltzmann (LB) method might be a promising approach to simulate fluid-structure interaction (FSI) problems with flexible structures and complex boundaries, because the former is a general simulation method for FSIs in biological systems, the latter is an efficient scheme for fluid flow simulations, and both of them work on regular Cartesian grids. In this paper an IB-LB coupling scheme is proposed and its feasibility is verified. The scheme is suitable for FSI problems concerning rapid flexible boundary motion and a large pressure gradient across the boundary. We first analyze the respective concepts, formulae and advantages of the IB and LB methods, and then explain the coupling strategy and detailed implementation procedures. To verify the effectiveness and accuracy, FSI problems arising from the relaxation of a distorted balloon immersed in a viscous fluid, an unsteady wake flow caused by an impulsively started circular cylinder at Reynolds number 9500, and an unsteady vortex shedding flow past a suddenly started rotating circular cylinder at Reynolds number 1000 are simulated. The first example is a benchmark case for flexible boundary FSI with a large pressure gradient across the boundary, the second is a fixed complex boundary problem, and the third is a typical moving boundary example. The results are in good agreement with the analytical and existing numerical data. It is shown that the proposed scheme is capable of modeling flexible boundary and complex boundary problems at a second-order spatial convergence; the volume leakage defect of the conventional IB method has been remedied by using a new method of introducing the unsteady and non-uniform external force; and the LB method makes the IB method simulation simpler and more efficient.

Orbital-free density functional theory (OFDFT) is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions. We examine some popular algorithms for optimizing the electron density distribution in OFDFT, explaining their suitability, benchmarking their performance, and suggesting some improvements. We start by describing the constrained optimization problem that encompasses electron density optimization. Next, we discuss the line search (including Wolfe conditions) and the nonlinear conjugate gradient and truncated Newton algorithms, as implemented in our open source OFDFT code. We finally focus on preconditioners derived from OFDFT energy functionals. Newly-derived preconditioners are successful for simulation cells of all sizes without regions of low electron-density and for small simulation cells with such regions.

Extreme events play a crucial role in fluid turbulence. Inspired by methods from field theory, these extreme events, their evolution and probability can be computed with help of the instanton formalism as minimizers of a suitable action functional. Due to the high number of degrees of freedom in multi-dimensional fluid flows, traditional global minimization techniques quickly become prohibitive in their memory requirements. We outline a novel method for finding the minimizing trajectory in a wide class of problems that typically occurs in turbulence setups, where the underlying dynamical system is a non-gradient, non-linear partial differential equation, and the forcing is restricted to a limited length scale. We demonstrate the efficiency of the algorithm in terms of performance and memory by computing high resolution instanton field configurations corresponding to viscous shocks for 1D and 2D compressible flows.

A method for gyrokinetic simulation of low frequency (lower than the cyclotron frequency) magnetic compressional modes in general geometry is presented. The gyrokinetic-Maxwell system of equations is expressed fully in terms of the compressional component of the magnetic perturbation, δB∥, with finite Larmor radius effects. This introduces a “gyro-surface” averaging of δB∥ in the gyrocenter equations of motion, and similarly in the perpendicular Ampere’s law, which takes the form of the perpendicular force balance equation. The resulting system can be numerically implemented by representing the gyro-surface averaging by a discrete sum in the configuration space. For the typical wavelength of interest (on the order of the gyroradius), the gyro-surface averaging can be reduced to averaging along an effective gyro-orbit. The phase space integration in the force balance equation can be approximated by summing over carefully chosen samples in the magnetic moment coordinate, allowing for an efficient numerical implementation.

The dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of an adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments.

The unsteady flow separation of airfoil with a local flexible structure (LFS) is studied numerically in Lagrangian frames in detail, in order to investigate the nature of its high aerodynamic performance. For such aeroelastic system, the characteristic-based split (CBS) scheme combined with arbitrary Lagrangian-Eulerian (ALE) framework is developed firstly for the numerical analysis of unsteady flow, and Galerkin method is used to approach the flexible structure. The local flexible skin of airfoil, which can lead to self-induced oscillations, is considered as unsteady perturbation to the flow. Then, the ensuing high aerodynamic performances and complex unsteady flow separation at low Reynolds number are studied by Lagrangian coherent structures (LCSs). The results show that the LFS has a significant influence on the unsteady flow separation, which is the key point for the lift enhancement. Specifically, the oscillations of the LFS can induce the generations of moving separation and vortex, which can enhance the kinetic energy transport from main flow to the boundary layer. The results could give a deep understand of the dynamics in unsteady flow separation and flow control for the flow over airfoil.

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 work deals with the numerical resolution of the M1-Maxwell system in the quasi-neutral regime. In this regime the stiffness of the stability constraints of classical schemes causes huge calculation times. That is why we introduce a new stable numerical scheme consistent with the transitional and limit models. Such schemes are called Asymptotic-Preserving (AP) schemes in literature. This new scheme is able to handle the quasi-neutrality limit regime without any restrictions on time and space steps. This approach can be easily applied to angular moment models by using a moments extraction. Finally, two physically relevant numerical test cases are presented for the Asymptotic-Preserving scheme in different regimes. The first one corresponds to a regime where electromagnetic effects are predominant. The second one on the contrary shows the efficiency of the Asymptotic-Preserving scheme in the quasi-neutral regime. In the latter case the illustrative simulations are compared with kinetic and hydrodynamic numerical results.

This paper presents the implementation, validation and application of TCE (total collision energy) model for simulating hypersonic reactive flows in a parallel direct simulation Monte Carlo code, named PDSC++, using an unstructured grid. A series of benchmarking test cases, which include reproduction of theoretical rate constants in a single cell, 2D hypersonic flow past a cylinder and 2D-axisymmetric hypersonic flow past a sphere, were performed to validate the implementation. Finally, detailed aerothermodynamics of the flown reentry Apollo 6 Command Module at 105 km is simulated to demonstrate the powerful capability of the PDSC++ in treating realistic hypersonic reactive flow at high altitude.

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions, where the wave propagation is governed by the Helmholtz equation. The scattering problem is modeled as a boundary value problem over a bounded domain. Based on the Dirichlet-to-Neumann (DtN) operator, a transparent boundary condition is introduced on an artificial circular boundary enclosing the obstacle. An adaptive finite element based on a posterior error estimate is presented to solve the boundary value problem with a nonlocal DtN boundary condition. Numerical experiments are included to compare with the perfectly matched layer (PML) method to illustrate the competitive behavior of the proposed adaptive method.

The unified lattice Boltzmann model is extended to the quadtree grids for simulation of fluid flow through porous media. The unified lattice Boltzmann model is capable of simulating flow in porous media at various scales or in systems where multiple length scales coexist. The quadtree grid is able to provide a high-resolution approximation to complex geometries, with great flexibility to control local grid density. The combination of the unified lattice Boltzmann model and the quadtree grids results in an efficient numerical model for calculating permeability of multi-scale porous media. The model is used for permeability calculation for three systems, including a fractured system used in a previous study, a Voronoi tessellation system, and a computationally-generated pore structure of fractured shale. The results are compared with those obtained using the conventional lattice Boltzmann model or the unified lattice Boltzmann model on rectangular or uniform square grid. It is shown that the proposed model is an accurate and efficient tool for flow simulation in multi-scale porous media. In addition, for the fractured shale, the contribution of flow in matrix and fractures to the overall permeability of the fractured shale is studied systematically.

A consistent dual-mesh hybrid LES/RANS framework for turbulence modeling has been proposed recently (H. Xiao, P. Jenny, A consistent dual-mesh framework for hybrid LES/RANS modeling, J. Comput. Phys. 231 (4) (2012)). To better enforce componentwise Reynolds stress consistency between the LES and the RANS simulations, in the present work the original hybrid framework is modified to better exploit the advantage of more advanced RANS turbulence models. In the new formulation, the turbulent stresses in the filtered equations in the under-resolved regions are directly corrected based on the Reynolds stresses provided by the RANS simulation. More precisely, the new strategy leads to implicit LES/RANS consistency, where the velocity consistency is achieved indirectly via imposing consistency on the Reynolds stresses. This is in contrast to the explicit consistency enforcement in the original formulation, where forcing terms are added to the filtered momentum equations to achieve directly the desired average velocity and velocity fluctuations. The new formulation keeps the averaging procedure for the filtered quantities and at the same time preserves the ability of the original formulation to conform with the physical differences between LES and RANS quantities. The modified formulation is presented, analyzed, and then evaluated for plane channel flow and flow over periodic hills. Improved predictions are obtained compared with the results obtained using the original formulation.

Numerical aspects of a pore scale model are investigated for the simulation of catalyst layers of polymer electrolyte membrane fuel cells. Coupled heat, mass and charged species transport together with reaction kinetics are taken into account using parallelized finite volume simulations for a range of nanostructured, computationally reconstructed catalyst layer samples. The effectiveness of implementing deflation as a second stage preconditioner generally improves convergence and results in better convergence behavior than more sophisticated first stage pre-conditioners. This behavior is attributed to the fact that the two stage preconditioner updates the preconditioning matrix at every GMRES restart, reducing the stalling effects that are commonly observed in restarted GMRES when a single stage preconditioner is used. In addition, the effectiveness of the deflation preconditioner is independent of the number of processors, whereas the localized block ILU preconditioner deteriorates in quality as the number of processors is increased. The total number of GMRES search directions required for convergence varies considerably depending on the preconditioner, but also depends on the catalyst layer microstructure, with low porosity microstructures requiring a smaller number of iterations. The improved model and numerical solution strategy should allow simulations for larger computational domains and improve the reliability of the predicted transport parameters. The preconditioning strategies presented in the paper are scalable and should prove effective for massively parallel simulations of other problems involving nonlinear equations.

This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime, which is valid up to the boundary and interface layers. A five-point node-centered and a four-point cell-centered tailored finite point schemes (TFPS) are introduced. The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system. Numerically, both methods can not only capture the diffusion limit, but also exhibit uniform convergence in the diffusive regime, even with boundary layers. Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy, uniformly with respect to the mean free path. Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.

Transient diffusion equations arise in many branches of engineering and applied sciences (e.g., heat transfer and mass transfer), and are parabolic partial differential equations. It is well-known that these equations satisfy important mathematical properties like maximum principles and the non-negative constraint, which have implications in mathematical modeling. However, existing numerical formulations for these types of equations do not, in general, satisfy maximum principles and the non-negative constraint. In this paper, we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation. The proposed methodology is based on the method of horizontal lines in which the time is discretized first. This results in solving steady anisotropic diffusion equation with decay equation at every discrete time-level. We also present other plausible temporal discretizations, and illustrate their shortcomings in meeting maximum principles and the non-negative constraint. The proposed methodology can handle general computational grids with no additional restrictions on the time-step. We illustrate the performance and accuracy of the proposed methodology using representative numerical examples. We also perform a numerical convergence analysis of the proposed methodology. For comparison, we also present the results from the standard single-field semi-discrete formulation and the results from a popular software package, which all will violate maximum principles and the non-negative constraint.

The previously developed LB-DF/FD method derived from the lattice Boltzmann method and Direct Forcing/Fictitious Domain method is extended to deal with 3D particle’s Brownian motion. In the model the thermal fluctuations are introduced as random forces and torques acting on the Brownian particle. The hydrodynamic interaction is introduced by directly resolving the fluid motions. A sphere fluctuating in a cubic box with the periodic boundary is considered to validate the present model. By examining the velocity autocorrelation function (VCF) and rotational velocity autocorrelation function (RVCF), it has been found that in addition to the two relaxation times, the mass density ratio should be taken into consideration to check the accuracy and effectiveness of the present model. Furthermore, the fluctuation-dissipation theorem and equipartition theorem have been investigated for a single spherical particle. Finally, a Brownian particle trapped in a harmonic potential has been simulated to further demonstrate the ability of the LB-DF/FD model.

A finite difference scheme with ghost cell technique is used to study viscous fluid sloshing in 2D and 3D tanks with internal structures. The Navier-Stokes equations in a moving coordinate system are derived and they are mapped onto a time-independent and stretched domain. The staggered grid is used and the revised SIMPLEC iteration algorithm is performed. The developed numerical model is rigorously validated by extensive comparisons with reported analytical, numerical and experimental results. The present numerical results were also validated through an experiment setup with a tank excited by an inclined horizontal excitation or a tank mounted by a vertical baffle. The method is then applied to a number of problems including sloshing fluid in a 2D tank with a bottom-mounted baffle and in a 3D tank with a vertical plate. The phenomena of diagonal sloshing waves affected by a vertical plate are investigated in detail in this work. The effects of internal structures on the resonant frequency of a tank with liquid are discussed and the present developed numerical method can successfully analyze the sloshing phenomenon in 2D or 3D tanks with internal structures.

A direct-forcing immersed boundary method (DFIB) with both virtual force and heat source is developed here to solve Navier-Stokes and the associated energy transport equations to study some thermal flow problems caused by a moving rigid solid object within. The key point of this novel numerical method is that the solid object, stationary or moving, is first treated as fluid governed by Navier-Stokes equations for velocity and pressure, and by energy transport equation for temperature in every time step. An additional virtual force term is then introduced on the right hand side of momentum equations in the solid object region to make it act exactly as if it were a solid rigid body immersed in the fluid. Likewise, an additional virtual heat source term is applied to the right hand side of energy equation at the solid object region to maintain the solid object at the prescribed temperature all the time. The current method was validated by some benchmark forced and natural convection problems such as a uniform flow past a heated circular cylinder, and a heated circular cylinder inside a square enclosure. We further demonstrated this method by studying a mixed convection problem involving a heated circular cylinder moving inside a square enclosure. Our current method avoids the otherwise requested dynamic grid generation in traditional method and shows great efficiency in the computation of thermal and flow fields caused by fluid-structure interaction.

In this paper, a new sharp-interface approach to simulate compressible multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative discontinuous Galerkin finite element scheme to evolve an indicator function that tracks the material interface. At the interface our method applies ghost cells to compute the numerical flux, as the ghost fluid method. However, unlike the original ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived from an approximate-state Riemann solver, similar to the approach proposed in [25], but based on a much simpler formulation. Our formulation leads only to one single scalar nonlinear algebraic equation that has to be solved at the interface, instead of the system used in [25]. Away from the interface, we use the new general Osher-type flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann solver, applicable to general hyperbolic conservation laws. The time integration is performed using a fully-discrete one-step scheme, based on the approaches recently proposed in [5,7]. This allows us to evolve the system also with time-accurate local time stepping. Due to the sub-cell resolution and the subsequent more restrictive time-step constraint of the DG scheme, a local evolution for the indicator function is applied, which is matched with the finite volume scheme for the solution of the Euler equations that runs with a larger time step. The use of a locally optimal time step avoids the introduction of excessive numerical diffusion in the finite volume scheme. Two different fluids have been used, namely an ideal gas and a weakly compressible fluid modeled by the Tait equation. Several tests have been computed to assess the accuracy and the performance of the new high order scheme. A verification of our algorithm has been carefully carried out using exact solutions as well as a comparison with other numerical reference solutions. The material interface is resolved sharply and accurately without spurious oscillations in the pressure field.