A simplified two-dimensional boundary element method with arbitrary uniform mean flow

To reduce computational costs, an improved form of the frequency domain boundary element method（BEM） is proposed for two-dimensional radiation and propagation acoustic problems in a subsonic uniform flow with arbitrary orientation. The boundary integral equation（BIE） representation solves the two-dimensional convected Helmholtz equation（CHE） and its fundamental solution, which must satisfy a new Sommerfeld radiation condition（SRC） in the physical space. In order to facilitate conventional formulations, the variables of the advanced form are expressed only in terms of the acoustic pressure as well as its normal and tangential derivatives, and their multiplication operators are based on the convected Green＇s kernel and its modified derivative. The proposed approach significantly reduces the CPU times of classical computational codes for modeling acoustic domains with arbitrary mean flow. It is validated by a comparison with the analytical solutions for the sound radiation problems of monopole,dipole and quadrupole sources in the presence of a subsonic uniform flow with arbitrary orientation.

SPACE-TIME FINITE ELEMENT METHOD FOR SCHRDINGER EQUATION AND ITS CONSERVATION

Energy conservation of nonlinear Schrodinger ordinary differential equation was proved through using continuous finite element methods of ordinary differential equation; Energy integration conservation was proved through using space-time continuous fully discrete finite element methods and the electron nearly conservation with higher order error was obtained through using time discontinuous only space continuous finite element methods of nonlinear Schrodinger partial equation. The numerical results are in accordance with the theory.

SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.

Topology Optimization of Sound-Absorbing Materials for Two-Dimensional Acoustic Problems Using Isogeometric Boundary Element Method

In this work,an acoustic topology optimizationmethod for structural surface design covered by porous materials is proposed.The analysis of acoustic problems is performed using the isogeometric boundary elementmethod.Taking the element density of porousmaterials as the design variable,the volume of porousmaterials as the constraint,and the minimum sound pressure or maximum scattered sound power as the design goal,the topology optimization is carried out by solid isotropic material with penalization(SIMP)method.To get a limpid 0–1 distribution,a smoothing Heaviside-like function is proposed.To obtain the gradient value of the objective function,a sensitivity analysis method based on the adjoint variable method(AVM)is proposed.To find the optimal solution,the optimization problems are solved by the method of moving asymptotes(MMA)based on gradient information.Numerical examples verify the effectiveness of the proposed topology optimization method in the optimization process of two-dimensional acoustic problems.Furthermore,the optimal distribution of sound-absorbingmaterials is highly frequency-dependent and usually needs to be performed within a frequency band.

Spectral Element Viscosity Methods for Nonlinear Conservation Laws on the Semi-Infinite Interval

In this paper we propose a spectral element: vanishing viscosity (SEW) method for the conservation laws on the semi-infinite interval. By using a suitable mapping, the problem is first transformed into a modified conservation law in a bounded interval, then the well-known spectral vanishing viscosity technique is generalized to the multi-domain case in order to approximate this trarsformed equation more efficiently. The construction details and convergence analysis are presented. Under a usual assumption of boundedness of the approximation solutions, it is proven that the solution of the SEW approximation converges to the uniciue entropy solution of the conservation laws. A number of numerical tests is carried out to confirm the theoretical results.

A modified discrete element method for concave granular materials based on energy-conserving contact model

The development of a general discrete element method for irregularly shaped particles is the core issue of the simulation of the dynamic behavior of granular materials.The general energy-conserving contact theory is used to establish a universal discrete element method suitable for particle contact of arbitrary shape.In this study,three dimentional(3D)modeling and scanning techniques are used to obtain a triangular mesh representation of the true particles containing typical concave particles.The contact volumebased energy-conserving model is used to realize the contact detection between irregularly shaped particles,and the contact force model is refined and modified to describe the contact under real conditions.The inelastic collision processes between the particles and boundaries are simulated to verify the robustness of the modified contact force model and its applicability to the multi-point contact mode.In addition,the packing process and the flow process of a large number of irregular particles are simulated with the modified discrete element method(DEM)to illustrate the applicability of the method of complex problems.

Guckenheimer structure of solution of Riemann problem with four pieces of constants in two space dimensions for scalar conservation laws

By using the generalized characteristic analysis method, the two-dimensional four-wave Riemann problem for scalar conservation laws, which is nonconvex along the y direction, was studied. Riemann solutions, which involve the Guckenheimer structure, were constructed.

CONSERVATION LAWS IN FINITE MICROCRACKING BRITTLE SOLIDS

This paper addresses the conservation laws in finite brittle solids with microcracks. The discussion is limited to the 2-D cases. First, after considering the combination of the Pseudo-Traction Method and the indirect Boundary Element Method, a versatile method for solving multi-crack interacting problems in finite plane solids is proposed, by which the fracture parameters （SIF and path-independent integrals） can be calculated with a desirable accuracy. Second, with the aid of the method proposed, the roles the conservation laws play in the fracture analysis for finite microcracking solids are studied. It is concluded that the conservation laws do play important roles in not only the fracture analysis but also the analysis of damage and stability for the finite microcracking system. Finally, the physical interpretation of the M-integral is discussed further. An explicit relation between the M-integral and the crack face area, i.e., M = GS, has been discovered using the analytical method, which can shed some light on the Damage Mechanics issues from a different perspective.

Continuous finite element methods for Hamiltonian systems

By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.

Finite Element Methods for Coupled Stokes and Darcy Problems

We derived and analyzed a new numerical scheme for the coupled Stokes and Darcy problems by using H（div） conforming elements in the entire domain. The approach employs the mixed finite element method for the Darcy equations and a stabilized H（div） finite element method for the Stokes equations. Optimal error estimates for the fluid velocity and pressure are derived. The finite element solutions from the new scheme not only feature a full satisfaction of the continuity equation, which is highly demanded in scientific computing, but also satisfy the mass conservation.

A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear,Dispersive Equations

The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.

CONVERGENCE OF AN EXPLICIT UPWIND FINITE ELEMENT METHOD TO MULTI-DIMENSIONAL CONSERVATION LAWS

Investigates the convergence of an explicit upwind finite element method, the edge-averaged finite element method to multidimensional scalar conservation laws. Discussion on the finite element scheme and main convergence theorem; Proofs of the technical lemmas.

The space time CE/SE method for solving one-dimensional batch crystallization model with fines dissolution

This article is concerned with the numerical investigation of one-dimensional population balance models for batch crystallization process with fines dissolution.In batch crystallization,dissolution of smaller unwanted nuclei below some critical size is of vital importance as it improves the quality of product.The crystal growth rates for both size-independent and size-dependent cases are considered.A delay in recycle pipe is also included in the model.The space–time conservation element and solution element method,originally derived for non-reacting flows,is used to solve the model.This scheme has already been applied to a range of PDEs,mainly in the area of fluid mechanics.The numerical results are compared with those obtained from the Koren scheme,showing that the proposed scheme is more efficient.

Numerical simulation of oxide nanoparticle growth characteristics under the gas detonation chemical reaction by space-time conservation element-solution element method

Under harsh conditions （such as high temperature, high pressure, and millisecond lifetime chemical reaction）, a long-standing challenge remains to accurately predict the growth characteristics of nanosize spherical particles and to determine the rapid chemical reaction flow field characteristics, The growth characteristics of similar spherical oxide nanoparticles are further studied by successfully introducing the space-time conservation element-solution element （CE/SE） algorithm with the monodisperse Kruis model. This approach overcomes the nanosize particle rapid growth limit set and successfully captures the characteristics of the rapid gaseous chemical reaction process. The results show that this approach quantitatively captures the characteristics of the rapid chemical reaction, nanosize particle growth and size distribution. To reveal the growth mechanism for numerous types of oxide nanoparticles, it is very important to choose a rational numerical method and particle physics model.

A three-dimensional Eulerian method for the numerical simulation of high-velocity impact problems

In the present paper, a three-dimensional （3D） Eulerian technique for the 3D numerical simulation of high-velocity impact problems is proposed. In the Eulerian framework, a complete 3D conservation element and solution element scheme for conservative hyperbolic governing equations with source terms is given. A modified ghost fluid method is proposed for the treatment of the boundary conditions. Numerical simulations of the Taylor bar problem and the ricochet phenomenon of a sphere impacting a plate target at an angle of 60~ are carried out. The numerical results are in good agreement with the corresponding experimental observations. It is proved that our computational technique is feasible for analyzing 3D high-velocity impact problems.

TWO-DIMENSIONAL FINITE ELEMENT METHOD FOR DETERMINING NONLINEAR WAVE FORCES ON LARGE SUBMERGED BODIES

In this paper, we first develop the far field asymptotic solutions of the second-order scattering waves for the vertical plane problem taking the second-order Stokes waves as the incident waves. The asymptotic solutions satisfy the Laplace equation, the sea bed and free surface boundary conditions and are the out-going waves. Then the radiation conditions of the second-order mattering waves are derived by using the asymptotic solutions. By using the two-dimensinal finite clement method with the radiation conditions imposed on the ar- tificial boundaries, the computer program, known as 'NWF2', for determining nonlinear wave forces on large submerged bodies has been written. As a numerical example, nonlinear wave forces on a semi-circu- lar cylinder lying on the sea bed arc presented.

SPECTRAL-FINITE ELEMENT METHOD FOR SOLVING TWO-DIMENSIONAL VORTICITY EQUATIONS

In this paper, we construct a spectral-finite element scheme for solving semi-periodical two-dimensional vorticity equations. The error between the genuine solution and approximate solutionis estimated strictly. The numerical results show the advantages of such a method. The techniqueused in this paper can be easily generalized to three-dimensional problems.

潜蚀的局部守恒有限元和有限体积交替解法

潜蚀的模拟需要计算土中的渗流过程以及土中细粒随孔隙水流的运移过程。常规有限元法在求解纯运移方程时存在稳定性问题,为此提出了一个采用有限元法求解渗流方程、有限体积法求解细粒运移方程的交替方法。鉴于常规有限元法不能给出满足有限体积法输入要求的流速场,给出了一个基于单元不平衡流量总体再平衡修正单元边界流速的算法。利用该算法对有限元计算的单元边界流速进行局部守恒处理,即可实现在同一有限元网格上利用有限体积法求解细粒运移方程,因此可以方便地和现有的有限元计算程序结合。算例验证表明,所提出的局部守恒处理算法及交替解法计算效率高且具有可接受的精度,是模拟潜蚀问题的一条简单实用的途径。

A second order explicit finite element scheme to multi-dimensional conservation laws and its convergence

A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation laws.L p convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently smooth solutions a second order error estimate inL 2 is proved under a stronger condition, Δt≤Ch 2/4

Tensile Shock Physics in Compressible Thermoviscoelastic Solid Medium

This paper addresses tensile shock physics in thermoviscoelastic (TVE) solids without memory. The mathematical model is derived using conservation and balance laws (CBL) of classical continuum mechanics (CCM), incorporating the contravariant second Piola-Kirchhoff stress tensor, the covariant Green’s strain tensor, and its rates up to order n. This mathematical model permits the study of finite deformation and finite strain compressible deformation physics with an ordered rate dissipation mechanism. Constitutive theories are derived using conjugate pairs in entropy inequality and the representation theorem. The resulting mathematical model is both thermodynamically and mathematically consistent and has closure. The solution of the initial value problems (IVPs) describing evolutions is obtained using a variationally consistent space-time coupled finite element method, derived using space-time residual functional in which the local approximations are in hpk higher-order scalar product spaces. This permits accurate description problem physics over the discretization and also permits precise a posteriori computation of the space-time residual functional, an accurate measure of the accuracy of the computed solution. Model problem studies are presented to demonstrate tensile shock formation, propagation, reflection, and interaction. A unique feature of this research is that tensile shocks can only exist in solid matter, as their existence requires a medium to be elastic (presence of strain), which is only possible in a solid medium. In tensile shock physics, a decrease in the density of the medium caused by tensile waves leads to shock formation ahead of the wave. In contrast, in compressive shocks, an increase in density and the corresponding compressive waves result in the formation of compression shocks behind of the wave. Although these are two similar phenomena, they are inherently different in nature. To our knowledge, this work has not been reported in the published literature.