A Girsanov particle filter in nonlinear engineering dynamics

In this Letter, we propose a novel variant of the particle filter (PF) for state and parameter estimations of nonlinear engineering dynamical systems, modelled through stochastic differential equations (SDEs). The aim is to address a possible loss of accuracy in the estimates due to the discretization errors, which are inevitable during numerical integration of the SDEs. In particular, we adopt an explicit local linearization of the governing nonlinear SDEs and the resulting linearization errors in the estimates are corrected using Girsanov transformation of measures. Indeed, the linearization scheme via transformation of measures provides a weak framework for computing moments and this fits in well with any stochastic filtering strategy wherein estimates are themselves statistical moments. We presently implement the strategy using a bootstrap PF and numerically illustrate its performance for state and parameter estimations of the Duffing oscillator with linear and nonlinear measurement equations.     

Equations of state for strong compression

Abstract

Various higher order equations of state (EOS) are compared with theoretical predictions for strong compression as well as with different linearization schemes. One specially convenient EOS together with a conjugated linearization scheme is presented as a series expansions with the appropriate asymptotic behaviour at very strong compression. Tests for the uncertainties in EOS extrapolations are discussed. A comparison with literature data for highly compressible solids illustrates the advantages of the present EOS with respect to many other commonly used forms.     

A multi-symplectic numerical integrator for the two-component Camassa–Holm equation

A new multi-symplectic formulation of the two-component Camassa-Holm equation (2CH) is presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. Furthermore, this scheme preserves exactly two discrete versions of the Casimir functions of 2CH. Numerical experiments show that the proposed numerical scheme has good conservation properties.     

A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows

We extend [Shravan K. Veerapaneni, Denis Gueyffier, Denis Zorin, George Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, Journal of Computational Physics 228(7) (2009) 2334–2353] to the case of three-dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous flows. Although the main components of the algorithm are similar in spirit to the 2D case—spectral approximation in space, semi-implicit time-stepping scheme—the main differences are that the bending and viscous force require new analysis, the linearization for the semi-implicit schemes must be rederived, a fully implicit scheme must be used for the toroidal topology to eliminate a CFL-type restriction and a novel numerical scheme for the evaluation of the 3D Stokes single layer potential on an axisymmetric surface is necessary to speed up the calculations. By introducing these novel components, we obtain a time-scheme that experimentally is unconditionally stable, has low cost per time step, and is third-order accurate in time. We present numerical results to analyze the cost and convergence rates of the scheme. To verify the solver, we compare it to a constrained variational approach to compute equilibrium shapes that does not involve interactions with a viscous fluid. To illustrate the applicability of method, we consider a few vesicle-flow interaction problems: the sedimentation of a vesicle, interactions of one and three vesicles with a background Poiseuille flow.     

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.     

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.     

Failsafe flux limiting and constrained data projections for equations of gas dynamics

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L² projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L² projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.     

K(m,n,P)方程多-Compacton相互作用的数值研究

采用有限差分法对非线性色散K(m,n,p)方程的多-Compacton之间的相互作用进行了数值研究.该差分方法为二阶精度且线性意义下绝对稳定的无耗散格式,通过添加人工耗散项有效防止了数值解的爆破现象.首先对单-Compacton的长时间演化行为进行了数值模拟,验证了数值方法的有效性.然后对双-CompaCton和三-Compacton的碰撞过程进行了数值研究,发现多-Compacton碰撞之后基本保持碰撞之前的波形和波速,但在波后产生小振幅的Compacton-Anticompacton对.     

Auto-inflammation de substances explosives. Comparaison entre calcul analytique et numérique en vue d'une optimisation dans le domaine de la pyrotechnie

Self-ignition of energetic material was investigated in order to optimize safety in the field of pyrotechnic applications. Two approaches were used; the first one is relative to Frank-Kamenetskii stationary thermal explosion theory. The second approach consists of a choice of some numerical solutions of heat conduction equations in a non-stationary state. Comparison between these results was carried out in order to find the numerical scheme which is the most compatible with Frank-Kamenetskii stationary thermal explosion theory. Numerical data were used for three explosive substances. One of them was studied by the author. In all cases, the numerical stationary state is in agreement with the Frank-Kamenetskii stationary thermal explosion theory, more or less accurately. From this comparison, it may be concluded that it is preferable, for this kind of problem, to use an implicit scheme with linearization of the heat source term. Explicit numerical methods, with or without the addition of the heat term with the Zinn and Mader scheme are revealed to be less accurate and to need a greater optimization of spatial and temporal meshing.     

High-Precision Direct Method for the Radiative Transfer Problems

It is the main aim of this paper to investigate the numerical methods of the radiative transfer equation.Using the five-point formula to approximate the differential part and the Simpson formula to substitute for integral part respectively, a new high-precision numerical scheme, which has 4-order local truncation error, is obtained. Subsequently,a numerical example for radiative transfer equation is carried out, and the calculation results show that the new numerical scheme is more accurate.     

An improved high-order scheme for DNS of low Mach number turbulent reacting flows based on stiff chemistry solver

We present an improved numerical scheme for numerical simulations of low Mach number turbulent reacting flows with detailed chemistry and transport. The method is based on a semi-implicit operator-splitting scheme with a stiff solver for integration of the chemical kinetic rates, developed by Knio et al. [O.M. Knio, H.N. Najm, P.S. Wyckoff, A semi-implicit numerical scheme for reacting flow II. Stiff, operator-split formulation, Journal of Computational Physics 154 (2) (1999) 428–467]. Using the material derivative form of continuity equation, we enhance the scheme to allow for large density ratio in the flow field. The scheme is developed for direct numerical simulation of turbulent reacting flow by employing high-order discretization for the spatial terms. The accuracy of the scheme in space and time is verified by examining the grid/time-step dependency on one-dimensional benchmark cases: a freely propagating premixed flame in an open environment and in an enclosure related to spark-ignition engines. The scheme is then examined in simulations of a two-dimensional laminar flame/vortex-pair interaction. Furthermore, we apply the scheme to direct numerical simulation of a homogeneous charge compression ignition (HCCI) process in an enclosure studied previously in the literature. Satisfactory agreement is found in terms of the overall ignition behavior, local reaction zone structures and statistical quantities. Finally, the scheme is used to study the development of intrinsic flame instabilities in a lean H₂/air premixed flame, where it is shown that the spatial and temporary accuracies of numerical schemes can have great impact on the prediction of the sensitive nonlinear evolution process of flame instability.     

复杂地表条件下快速推进法地震波走时计算

为获得计算复杂地表条件下地震波走时的方法,对常规快速推进法(Fast marching method,简写为FMM)做了两点改进:①引入不等距差分格式,用于地表和界面处的局部走时计算;②增加新的网格节点类型,用于实现不规则边界条件下的窄带技术.通过对算法的计算精度、效率及实例的分析可得,算法计算精度高,其中反射波走时计算的精度高于初至波;不会因为处理不规则边界而引入过多额外的计算量;能灵活稳定地处理各种强起伏复杂地形、近地表及地下复杂介质等问题.计算结果满足复杂地表条件下地震波的传播规律.     

A high-order accurate hybrid scheme using a central flux scheme and a WENO scheme for compressible flowfield analysis

A high-order accurate hybrid central-WENO scheme is proposed. The fifth order WENO scheme [G.S. Jiang, C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228] is divided into two parts, a central flux part and a numerical dissipation part, and is coupled with a central flux scheme. Two sub-schemes, the WENO scheme and the central flux scheme, are hybridized by means of a weighting function that indicates the local smoothness of the flowfields. The derived hybrid central-WENO scheme is written as a combination of the central flux scheme and the numerical dissipation of the fifth order WENO scheme, which is controlled adaptively by a weighting function. The structure of the proposed hybrid central-WENO scheme is similar to that of the YSD-type filter scheme [H.C. Yee, N.D. Sandham, M.J. Djomehri, Low-dissipative high-order shock-capturing methods using characteristic-based filters, J. Comput. Phys. 150 (1999) 199–238]. Therefore, the proposed hybrid scheme has also certain merits that the YSD-type filter scheme has. The accuracy and efficiency of the developed hybrid central-WENO scheme are investigated through numerical experiments on inviscid and viscous problems. Numerical results show that the proposed hybrid central-WENO scheme can resolve flow features extremely well.     

On the roles of minimization and linearization in least-squares finite element models of nonlinear boundary-value problems

In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L₂ norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier–Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.     

A multilevel variational method for Au = λBu on composite grids

There are two existing general approaches to the numerical solution of differential eigen-problems by multigrid, one which uses a linear multigrid solver for the inner loop of a linearization technique (e.g., inverse iteration) and the other which integrates multigrid directly into the problem solution (e.g., FAS ). The present paper developes a third multigridlike approach, RQMG, that appeals directly to the variational problem of minimizing the Rayleigh quotient, applying easily to both global and local grid implementations. Numerical results are presented for a simplified single group diffusion problem.     

解流体力学方程组的一种隐式完全守恒差分格式

对Lagrange非守恒流体力学方程组给出了一种隐式完全守恒差分格式,既保证了质量、动量和总能量守恒的差分近似,又能满足内能与动能的平衡特性,提高了数值解的精度。并用该格式对两个可压缩理想流体模型进行了数值计算,并与其它差分格式作了比较。     

Multisymplectic Euler Box Scheme for the KdV Equation

We investigate the multisymplectic Euler box scheme for the Korteweg-de Vries （KdV） equation. A new completely explicit six-point scheme is derived. Numerical experiments of the new scheme with comparisons to the Zabusky-Kruskal scheme, the multisymplectic 12-point scheme, the narrow box scheme and the spectral method are made to show nice numerical stability and ability to preserve the integral invariant for long-time integration.     

A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.     

Broken-symmetry states of Dirac fermions in graphene with a partially filled high Landau level

We report on numerical study of the Dirac fermions in partially filled N=3 Landau level (LL) in graphene. At half-filling, the equal-time density-density correlation function displays sharp peaks at nonzero wave vectors +/-q*. Finite-size scaling shows that the peak value grows with electron number and diverges in the thermodynamic limit, which suggests an instability toward a charge density wave. A symmetry broken stripe phase is formed at large system size limit, which is robust against perturbation from disorder scattering. Such a quantum phase is experimentally observable through transport measurements. Associated with the special wave functions of the Dirac LL, both stripe and bubble phases become possible candidates for the ground state of the Dirac fermions in graphene with lower filling factors in the N=3 LL.