Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal;that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.

A Full Eulerian Fluid-Membrane Coupling Method with a Smoothed Volume-of-Fluid Approach

A novel full Eulerian fluid-elastic membrane coupling method on the fixed Cartesian coordinate mesh is proposed within the framework of the volume-of-fluid approach.The present method is based on a full Eulerian fluid-(bulk)structure coupling solver(Sugiyama et al.,J.Comput.Phys.,230(2011)596–627),with the bulk structure replaced by elastic membranes.In this study,a closed membrane is considered,and it is described by a volume-of-fluid or volume-fraction information generally called VOF function.A smoothed indicator(or characteristic)function is introduced as a phase indicator which results in a smoothed VOF function.This smoothed VOF function uses a smoothed delta function,and it enables a membrane singular force to be incorporated into a mixture momentum equation.In order to deal with a membrane deformation on the Eulerian mesh,a deformation tensor is introduced and updated within a compactly supported region near the interface.Both the neo-Hookean and the Skalak models are employed in the numerical simulations.A smoothed(and less dissipative)interface capturing method is employed for the advection of the VOF function and the quantities defined on the membrane.The stability restriction due to membrane stiffness is relaxed by using a quasi-implicit approach.The present method is validated by using the spherical membrane deformation problems,and is applied to a pressure-driven flow with the biconcave membrane capsules(red blood cells).

A Hybrid Scheme of Level Set and Diffuse Interface Methods for Simulating Multi-Phase Compressible Flows

We propose a hybrid scheme combing the level set method and the multicomponent diffuse interface method to simulate complex multi-phase flows.The overall numerical scheme is based on a sharp interface framework where the level set method is adopted to capture the material interface,the Euler equation is used to describe a single-phase flow on one side of the interface and the six-equation diffuse interface model is applied to model the multi-phase mixture or gas-liquid cavitation on the other side.An exact Riemann solver,between the Euler equation and the six-equation model with highly nonlinear Mie-Gr¨uneisen equations of state,is developed to predict the interfacial states and compute the phase interface flux.Several numerical examples,including shock tube problems,cavitation problems,air blast and underwater explosion applications are presented to validate the numerical scheme and the Riemann solver.

A General Moving Mesh Framework in 3D and its Application for Simulating the Mixture of Multi-Phase Flows

In this paper, we present an adaptive moving mesh algorithm for meshesof unstructured polyhedra in three space dimensions. The algorithm automaticallyadjusts the size of the elements with time and position in the physical domain to resolvethe relevant scales in multiscale physical systems while minimizing computationalcosts. The algorithm is a generalization of the moving mesh methods basedon harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To make 3D moving mesh simulations possible,the key is to develop an efficient mesh redistribution procedure so that this part willcost as little as possible comparing with the solution evolution part. Since the meshredistribution procedure normally requires to solve large size matrix equations, wewill describe a procedure to decouple the matrix equation to a much simpler blocktridiagonaltype which can be efficiently solved by a particularly designed multi-gridmethod. To demonstrate the performance of the proposed 3D moving mesh strategy,the algorithm is implemented in finite element simulations of fluid-fluid interface interactionsin multiphase flows. To demonstrate the main ideas, we consider the formationof drops by using an energetic variational phase field model which describesthe motion of mixtures of two incompressible fluids. Numerical results on two- andthree-dimensional simulations will be presented.

Logcf: An Efficient Tool for Real Root Isolation

Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem.The authors give a new algorithm for computing an upper bound of positive roots in this paper.The complexity of the algorithm is O(n log(uH-l))additions and multiplications where u is the optimal upper bound satisfying Theorem 3.1 of this paper and n is the degree of the polynomial.The method together w辻h some tricks have been implemented as a software package logcf using C language.Experiments on many benchmarks show that logcf is competitive with Root Intervals of Mathematica and the function realroot of Maple averagely and it is much faster than existing open source real root solvers in many test cases.

Generating Semi-Algebraic Invariants for Non-Autonomous Polynomial Hybrid Systems

Hybrid systems are dynamical systems with interacting discrete computation and continuous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid systems. In the previous work, the fourth author and his coauthors gave a necessary and sufficient condition for a semi-algebraic set being an invariant of a polynomial autonomous dynamical system, which gave a confirmative answer to the open problem. In addition, based on which a complete algorithm for generating all semi-algebraic invariants of a given polynomial autonomous hybrid system with the given shape was proposed. This paper considers how to extend their work to non-autonomous dynamical and hybrid systems. Non-autonomous dynamical and hybrid systems are with inputs, which are very common in practice; in contrast, autonomous ones are without inputs. Furthermore, the authors present a sound and complete algorithm to verify semi-algebraic invariants for non-autonomous polynomial hybrid systems. Based on which, the authors propose a sound and complete algorithm to generate all invariants with a pre-defined template.

Dimension-Reduced Hyperbolic Moment Method for the Boltzmann Equation with BGK-Type Collision

We develop the dimension-reduced hyperbolic moment method for the Boltzmann equation,to improve solution efficiency using a numerical regularized moment method for problems with low-dimensional macroscopic variables and highdimensional microscopic variables.In the present work,we deduce the globally hyperbolic moment equations for the dimension-reduced Boltzmann equation based on the Hermite expansion and a globally hyperbolic regularization.The numbers of Maxwell boundary condition required for well-posedness are studied.The numerical scheme is then developed and an improved projection algorithm between two different Hermite expansion spaces is developed.By solving several benchmark problems,we validate the method developed and demonstrate the significant efficiency improvement by dimension-reduction.

A Multi-Mesh Adaptive Finite Element Approximation to Phase Field Models

In this work,we propose an efficient multi-mesh adaptive finite element method for simulating the dendritic growth in two-and three-dimensions.The governing equations used are the phase field model,where the regularity behaviors of the relevant dependent variables,namely the thermal field function and the phase field function,can be very different.To enhance the computational efficiency,we approximate these variables on different h-adaptive meshes.The coupled terms in the system are calculated based on the implementation of the multi-mesh h-adaptive algorithm proposed by Li(J.Sci.Comput.,pp.321-341,24(2005)).It is illustrated numerically that the multi-mesh technique is useful in solving phase field models and can save storage and the CPU time significantly.

Numerical Regularized Moment Method for High Mach Number Flow

This paper is a continuation of our earlier work[SIAM J.Sci.Comput.,32(2010),pp.2875-2907]in which a numericalmoment method with arbitrary order of moments was presented.However,the computation may break down during the calculation of the structure of a shock wave with Mach number M_(0)≥3.In this paper,we concentrate on the regularization of the moment systems.First,we apply the Maxwell iteration to the infinite moment system and determine the magnitude of each moment with respect to the Knudsen number.After that,we obtain the approximation of high order moments and close the moment systems by dropping some high-order terms.Linearization is then performed to obtain a very simple regularization term,thus it is very convenient for numerical implementation.To validate the new regularization,the shock structures of low order systems are computed with different shock Mach numbers.

A Hybrid Fluid-Solid Interaction Scheme Combining the Multi-Component Diffuse Interface Method and the Material Point Method

We propose a hybrid scheme combing the diffuse interface method and the material point method to simulate the complex interactions between the multiphase compressible flow and elastoplastic solid.The multiphase flow is modelled by the multi-component model and solved using a generalized Godunov method in the Eulerian grids,while the elastoplastic solid is solved by the classical material point method in a combination of Lagrangian particles and Eulerian background grids.In order to facilitate the simulation of fluid-solid interactions,the solid variables are further interpolated to the cell center and coexist with the fluid in the same cell.An instantaneous relaxation procedure of velocity and pressure is adopted to simulate the momentum and energy transfers between various materials,and to keep the system within a tightly coupled interaction.Several numerical examples,including shock tube problem,gasbubble problem,air blast,underwater explosion and high speed impact applications are presented to validate the numerical scheme.

An Edge-Based Anisotropic Mesh Refinement Algorithm and its Application to Interface Problems

Based on an error estimate in terms of element edge vectors on arbitrary unstructured simplex meshes,we propose a new edge-based anisotropic mesh refinement algorithm.As the mesh adaptation indicator,the error estimate involves only the gradient of error rather than higher order derivatives.The preferred refinement edge is chosen to reduce the maximal term in the error estimate.The algorithm is implemented in both two-and three-dimensional cases,and applied to the singular function interpolation and the elliptic interface problem.The numerical results demonstrate that the convergence order obtained by using the proposed anisotropic mesh refinement algorithm can be higher than that given by the isotropic one.

Robust a Simulation for Shallow Flows with Friction on Rough Topography

In this paper,we propose a robust finite volume scheme to numerically solve the shallow water equations on complex rough topography.The major difficulty of this problem is introduced by the stiff friction force term and the wet/dry interface tracking.An analytical integration method is presented for the friction force term to remove the stiffness.In the vicinity of wet/dry interface,the numerical stability can be attained by introducing an empirical parameter,the water depth tolerance,as extensively adopted in literatures.We propose a problem independent formulation for this parameter,which provides a stable scheme and preserves the overall truncation error of δ(Δx^(3)).The method is applied to solve problems with complex rough topography,coupled with h-adaptive mesh techniques to demonstrate its robustness and efficiency.

The Eulerian-Lagrangian Method with Accurate Numerical Integration

This paper is devoted to the study of the Eulerian-Lagrangian method(ELM)for convection-diffusion equations on unstructured grids with or without accurate numerical integration.We first propose an efficient and accurate algorithm to calculate the integrals in the Eulerian-Lagrangian method.Our approach is based on an algorithm for finding the intersection of two non-matching grids.It has optimal algorithmic complexity and runs fast enough to make time-dependent velocity fields feasible.The evaluation of the integrals leads to increased precision and the unconditional stability.We demonstrate by numerical examples that the ELM with our proposed algorithm for accurate numerical integration has the following two features:first it is much more accurate and more stable than the ones with traditional numerical integration techniques and secondly the overall cost of the proposed method is comparable with the traditional ones.

A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids

A recent work of Li et al.[Numer.Math.Theor.Meth.Appl.,1(2008),pp.92-112]proposed a finite volume solver to solve 2D steady Euler equations.Although the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the shock region,the overshoot or undershoot phenomenon can still be observed.Moreover,the numerical accuracy is degraded by using Venkatakrishnan limiter.To fix the problems,in this paper the WENO type reconstruction is employed to gain both the accurate approximations in smooth region and non-oscillatory sharp profiles near the shock discontinuity.The numerical experiments will demonstrate the efficiency and robustness of the proposed numerical strategy.

ARobust Riemann Solver for Multiple Hydro-Elastoplastic Solid Mediums

We propose a robust approximate solver for the hydro-elastoplastic solid material,a general constitutive law extensively applied in explosion and high speed impact dynamics,and provide a natural transformation between the fluid and solid in the case of phase transitions.The hydrostatic components of the solid is described by a family of general Mie-Gruneisen equation of state(EOS),while the deviatoric component includes the elastic phase,linearly hardened plastic phase and fluid phase.The approximate solver provides the interface stress and normal velocity by an iterative method.The well-posedness and convergence of our solver are proved with mild assumptions on the equations of state.The proposed solver is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian girds.Several numerical examples,including Riemann problems,shock-bubble interactions,implosions and high speed impact applications,are presented to validate the approximate solver.

Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis:1-D Lattice Model System

In the first of a series of papers,wewill study a discontinuous Galerkin(DG)framework for many electron quantum systems.The salient feature of this framework is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed piecewise polynomial basis in representing the Hamiltonian of the many body system.Such a flexibility is made possible by using the discontinuous Galerkin method to approximate the Hamiltonian matrix elements with proper constructions of numerical DG fluxes at the finite element interfaces.In this paper,we will apply the DG method to the density matrix minimization formulation,a popular approach in the density functional theory of many body Schrodinger equations.The density matrix minimization is to find the minima of the total energy,expressed as a functional of the density matrixρ(r,r′),approximated by the proposed enriched basis,together with two constraints of idempotency and electric neutrality.The idempotency will be handled with theMcWeeny’s purification while the neutrality is enforced by imposing the number of electrons with a penalty method.A conjugate gradient method(a Polak-Ribiere variant)is used to solve the minimization problem.Finally,the linear-scaling algorithm and the advantage of using the local orbital enriched finite element basis in the DG approximations are verified by studying examples of one dimensional lattice model systems.