Flow around a cavity is characterized by a self-sustained mechanism in which the shear layer impinges on the downstream edge of the cavity resulting in a feedback mechanism. Direct Numerical Simulations of the flow at low Reynolds number has been carried out to get pressure and velocity fluctuations, for the case of un-actuated and multi frequency actuation. A Reduced Order Model for the isentropic compressible equations based on the method of Proper Orthogonal Decomposition has been constructed. The model has been extended to include the effect of control. The Reduced Order dynamical system shows a divergence in time integration. A method of calibration based on the minimization of a linear functional of error, to the sensitivity of the modes, is proposed. The calibrated low order model is used to design a feedback control of cavity flows based on an observer design. For the experimental implementation of the controller, a state estimate based on the observed pressure measurements is obtained through a linear stochastic estimation. Finally the obtained control is introduced into the Direct Numerical Simulation to obtain a decrease in spectra of the cavity acoustic mode.

In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.

In this paper, a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species GK model in [A. D. Kotelnikov and D. C. Montgomery, A Kinetic Method for Computing Inhomogeneous Fluid Behavior, J. Comput. Phys. 134 (1997) 364-388]. Different from the conventional BGK model, the collisions between different species are taken into consideration. Based on the Chapman-Enskog expansion, the corresponding macroscopic equations are derived from this two-species model. Because of the relaxation terms in the governing equations, the method of operator splitting is applied. In the hyperbolic part, the integral solutions of the BGK equations are used to construct the numerical fluxes at the cell interface in the framework of finite volume method. Numerical tests are presented in this paper to validate the current approach for the compressible multicomponent flows. The theoretical analysis on the spurious oscillations at the interface is also presented.

In this paper the pressure distribution of the gaseous flow in a microchannel is studied via a lattice Boltzmann equation (LBE) method. With effective relaxation times and a generalized second order slip boundary condition, the LBE can be used to simulate rarefied gas flows from slip to transition regimes. The Knudsen minimum phenomena of mass flow rate in the pressure driven flow is also investigated. The effects of Knudsen number (rarefaction effect), pressure ratio and aspect ratio (compression effect) on the pressure distribution are analyzed. It is found the rarefaction effect tends to the curvature of the nonlinear pressure distribution, while the compression effect tends to enhance its nonlinearity. The combined effects lead to a local minimum of the pressure deviation. Furthermore, it is also found that the relationship between the pressure deviation and the aspect ratio follows a pow-law.

In this paper, we study a class of contact smoothed particle hydrodynamics (SPH) by introducing Riemann solvers and using high-order limiters. In particular, a promising concept of WENO interpolation as limiter is presented in the reconstruction process. The physical values relating interactional particles used as the initial values of the Riemann problem can be reconstructed by the Taylor series expansion. The contact solvers of the Riemann problem at contact points are incorporated in SPH approximations. In order to keep the fluid density at the wall rows to be consistent with that of the inner fluid wall boundaries, several lines of dummy particles are placed outside of the solid walls, which are assigned according to the initial configuration. At last, the method is applied to compressible flows with sharp discontinuities such as the collision of two strong shocks and the interaction of two blast waves and so on. The numerical results indicate that the method is capable of handling sharp discontinuity and efficiently reducing unphysical oscillations.

The ordered patterns formed by microphase-separated block copolymer systems demonstrate periodic symmetry, and all periodic structures belong to one of 230 space groups. Based on this fact, a strategy of estimating the initial values of self-consistent field theory to discover ordered patterns of block copolymers is developed. In particular, the initial period of the computational box is estimated by the Landau-Brazovskii model as well. By planting the strategy into the whole-space discrete method, several new metastable patterns are discovered in diblock copolymers.

Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present a boundary integral method in two dimensions which is specially designed to maintain second order accuracy even if boundaries are arbitrarily close. The method uses a regularization of the integral kernel which admits analytically determined corrections to maintain accuracy. For boundaries with many components we use the fast multipole method for efficient summation. We compute electric potentials on a domain with cells whose conductivity differs from that of the surrounding medium. We first solve an integral equation for a source term on the cell interfaces and then find values of the potential near the interfaces via integrals. Finally we use a Poisson solver to extend the potential to a regular grid covering the entire region. A number of examples are presented. We demonstrate that increased refinement is not needed to maintain accuracy as interfaces become very close.

The local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.

Schwarz waveform relaxation (SWR) algorithm has been investigated deeply and widely for regular time dependent problems. But for time delay problems, complete analysis of the algorithm is rare. In this paper, by using the reaction diffusion equations with a constant discrete delay as the underlying model problem, we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition. The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem, which is more complex than the one arising for regular problems without delay. We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter, which is shown efficient to accelerate the convergence of the SWR algorithm. Numerical results are provided to validate the theoretical conclusions.

Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover the H(div) proposed spaces are completing the De Rham diagram with optimal elements previously constructed for H1 and H(curl) approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed.

We present an accelerated method for stochastically simulating the dynamics of heterogeneous cell populations. The algorithm combines a Monte Carlo approach for simulating the biochemical kinetics in single cells with a constant-number Monte Carlo method for simulating the reproductive fitness and the statistical characteristics of growing cell populations. To benchmark accuracy and performance, we compare simulation results with those generated from a previously validated population dynamics algorithm. The comparison demonstrates that the accelerated method accurately simulates population dynamics with significant reductions in runtime under commonly invoked steady-state and symmetric cell division assumptions. Considering the increasing complexity of cell population models, the method is an important addition to the arsenal of existing algorithms for simulating cellular and population dynamics that enables efficient, coarse-grained exploration of parameter space.

The “analogue gravity formalism”, an interdisciplinary theoretical scheme developed in the past for studying several non relativistic classical and quantum systems through effective relativistic curved space-times, is here applied to largely de-formable elastic bodies described by the nonlinear theory of solid mechanics. Assuming the simplest nonlinear constitutive relation for the elastic material given by a Kirchhoff-St Venant strain-energy density function, it is possible to write for the perturbations an effective space-time metric if the deformation is purely longitudinal and depends on one spatial coordinate only. Theoretical and numerical studies of the corresponding dynamics are performed in selected cases and physical implications of the results obtained are finally discussed.

The Manley-Rowe constants of motion (MRC) are conservation laws written out for a dynamical system describing the time evolution of the amplitudes in resonant triad. In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants (gMRC) and give a constructive method how to compute them. We also give details of a Mathematica implementation of this method. While MRC provide integrability of the underlying dynamical system, gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.

This study mainly focused on the key technologies, the photon dose calculation based on the Monte Carlo Finite-Size Pencil Beam (MCFSPB) model in the Accurate Radiotherapy System (ARTS). In the MCFSPB model, the acquisition of pencil beam kernel is one of the most important technologies. In this study, by analyzing the demerits of the clinical pencil beam dose calculation methods, a new pencil beam kernel model was developed based on the Monte Carlo (MC) simulation and the technology of medical accelerator energy spectrum reconstruction. which greatly improved the accuracy of calculated result. According to the axial symmetry principle, only part of simulation results was used for the data of pencil beam kernel, which greatly reduced the data quantity of the pencil beam and reduced calculated time. Based on the above studies, the MCFSPB method was designed and implemented by the Visual C++ development tool. With several tests including the comparisons among the American Association of Physicists in Medicine (AAPM) No. 55 Report sample and the ion chamber measurement of lung-simulating inhomogeneous phantom in clinical treatment plan, the results showed that the maximum error of most calculated point was less than 0.5% in the homogeneous phantom and less than 3% in the heterogeneous phantom. This method met the clinical criteria, and would be expected to be used as a fast and accurate dose engine for clinic TPS.

An accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of the discretized velocity distribution function in physical space and time. The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Computational examples in one- and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved. Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method. The recovery of quantum statistics to the classical limit is also tested for small fugacity values.

Factorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

In this paper, we present a new model developed in order to analyze phenomena which arise in the solidification of binary mixtures using phase-field method, which incorporates the convection effects and the action of magnetic field. The model consists of flow, concentration, phase field and energy systems which are nonlinear evolutive and coupled systems. It represents the non-isothermal anisotropic solidification process of a binary mixture together with the motion in a melt with the applied magnetic field. To illustrate our model, numerical simulations of the influence of magnetic-field on the evolution of dendrites during the solidification of the binary mixture of Nickel-Copper (Ni-Cu) are developed. The results demonstrate that the dendritic growth under the action of magnetic-field can be simulated by using our model.

In this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE) with a dimensionless parameter ε > 0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.

We study the simulation of specular reflection in a level set method implementation for wavefront propagation in high frequency acoustics using WENO spatial operators. To implement WENO efficiently and maintain convergence rate, a rectangular grid is used over the physical space. When the physical domain does not conform to the rectangular grid, appropriate boundary conditions to represent reflection must be derived to apply at grid locations that are not coincident with the reflecting boundary. A related problem is the extraction of the normal vectors to the boundary, which are required to formulate the reflection condition. A separate level set method is applied to pre-compute the boundary normals which are then stored for use in the wavefront method. Two approaches to handling the reflection boundary condition are proposed and studied: one uses an approximation to the boundary location, and the other uses a local reflection principle. The second method is shown to produce superior results.

Linear/nonlinear and Stokes based-stabilizations for the filter equations for damping out primitive variable (PV) solutions corrupted by uniformly distributed random noises are numerically studied through the natural convection (NC) as well as the mixed convection (MC) environment. The most recognizable filter-scheme is based on a combination of the negative Laplace equation multiplied with the selection of the spatial scale and a linear function in order to preserve the uniqueness of the filtered solution. A more complicated filter-scheme, based on a Stokes problem which couples a filtered velocity and a filtered (artificial) pressure (or Lagrange multiplier) in order to enforce the incompressibility constraint, is also studied. Linear and Stokes based-filters via nested iterative (NI) filters and the consistent splitting scheme (CSS) are proposed for the NC/MC problems. Inspired by the total-variation (TV) model of image diffusion, well preserved feature flow patterns from the corrupted NC/MC environment are obtained by TV-Stokes based-filters together with the CSS. Our experimental results show that our proposed algorithms are effective and efficient in eliminating the unwanted spurious oscillations and preserving the accuracy of thermal convective fluid flows.