Due to the rapid advances in micro-electro-mechanical systems (MEMS), the study of microflows becomes increasingly important. Currently, the molecular-based simulation techniques are the most reliable methods for rarefied flow computation, even though these methods face statistical scattering problem in the low speed limit. With discretized particle velocity space, a unified gas-kinetic scheme (UGKS) for entire Knudsen number flow has been constructed recently for flow computation. Contrary to the particle-based direct simulation Monte Carlo (DSMC) method, the unified scheme is a partial differential equation-based modeling method, where the statistical noise is totally removed. But the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space. Due to the multiscale modeling in the unified method, i.e., the update of both macroscopic flow variables and microscopic gas distribution function, the conventional constraint of time step being less than the particle collision time in many direct Boltzmann solvers is released here. The numerical tests show that the unified scheme is more efficient than the particle-based methods in the low speed rarefied flow computation. The main purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena. In the continuum and free molecular limits, the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions. In the transition regime, the DSMC solution will be used for the validation of UGKS results. This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows. It not only recovers the DSMC results whenever available, but also provides high resolution results in cases where the DSMC can hardly afford the computational cost. In thermal creep flow simulation, surprising solution, such as the gas flowing from hot to cold regions along the wall surface, is observed for the first time by the unified scheme, which is confirmed later through intensive DSMC computation.

In this article we present a new class of high order accurate ArbitraryEulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in [25]. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.

A straightforward method is presented for computing three-dimensional Stokes flow, due to forces on a surface, with high accuracy at points near the surface. The flow quantities are written as boundary integrals using the free-space Green's function. To evaluate the integrals near the boundary, the singular kernels are regularized and a simple quadrature is applied in coordinate charts. High order accuracy is obtained by adding special corrections for the regularization and discretization errors, derived here using local asymptotic analysis. Numerical tests demonstrate the uniform convergence rates of the method.

We use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the "Impulse method" which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency. This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.

Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form  where  are any consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence, a novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.

Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe time step limits, but also take advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

A novel method for boundary constrained tetrahedral mesh generation is proposed based on Advancing Front Technique (AFT) and conforming Delaunay triangulation. Given a triangulated surface mesh, AFT is firstly applied to mesh several layers of elements adjacent to the boundary. The rest of the domain is then meshed by the conforming Delaunay triangulation. The non-conformal interface between two parts of meshes is adjusted. Mesh refinement and mesh optimization are then preformed to obtain a more reasonable-sized mesh with better quality. Robustness and quality of the proposed method is shown. Convergence proof of each stage as well as the whole algorithm is provided. Various numerical examples are included as well as the quality of the meshes.

Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist at each point of space and their relative abundance is described by their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction of one phase goes to zero in part of the computational domain. A small and constant network volume fraction is temporarily added throughout the domain in setting up the discrete linear equations and the same set of equation is solved everywhere. These equations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficient and produces an accurate solution of these equations for the full range of relevant regularization parameter values.

In this paper, a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species BGK 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.

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 $H^1$ 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.

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 were 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.