Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.

A NEW DERIVATIVE FREE OPTIMIZATION METHOD BASED ON CONIC INTERPOLATION MODEL

In this paper, a new derivative free trust region method is developed based on the conic interpolation model for the unconstrained optimization. The conic interpolation model is built by means of the quadratic model function, the collinear scaling formula, quadratic approximation and interpolation. All the parameters in this model are determined by objective function interpolation condition. A new derivative free method is developed based upon this model and the global convergence of this new method is proved without any information on gradient.

A Flexible-Segment-Model-Based Dynamics Calculation Method for Free Hanging Marine Risers in Re-Entry

In re-entry, the drilling riser hanging to the holding vessel takes on a free hanging state, waiting to be moved from the initial random position to the wellhead. For the re-entry, dynamics calculation is often done to predict the riser motion or evaluate the structural safety. A dynamics calculation method based on Flexible Segment Model （FSM） is proposed for free hanging marine risers. In FSM, a riser is discretized into a series of flexible segments. For each flexible segment, its deflection feature and external forces are analyzed independently. For the whole riser, the nonlinear governing equations are listed according to the moment equilibrium at nodes. For the solution of the nonlinear equations, a linearization iteration scheme is provided in the paper. Owing to its flexibility, each segment can match a long part of the riser body, which enables that good results can be obtained even with a small number of segments. Moreover, the linearization iteration scheme can avoid widely used Newton-Rapson iteration scheme in which the calculation stability is influenced by the initial points. The FSM-based dynamics calculation is timesaving and stable, so suitable for the shape prediction or real-time control of free hanging marine risers.

On the interaction between bubbles and the free surface with high density ratio 3D lattice Boltzmann method

The bubbles rise up and burst at the free surface is a complex two-phase process.A free energy lattice Boltzmann method(LBM)model is adopted in this paper to study this phenomenon.The interface capturing technique[Zheng et al.,2006]is used to deal with the high density ratio problem.The Laplace law and the air-water interface capturing ability are validated for the multiphase model.The interaction between the single bubble or multiple bubbles and the free surface are studied by the multiphase model.The force acting on the bubble and the evolution of the free surface is studied.Meanwhile,effect of the initial distance between two adjacent bubbles on interaction effects of multiple bubbles is investigated as well.

Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates

This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.

Numerical and experimental investigation of rock breaking method under free surface by TBM disc cutter

To study the rock breaking method under the free surface induced by disc cutter,the rock breaking simulations were first conducted based on the discrete element method,and the dynamic process of rock breaking under the free surface was studied including stressed zone,crush zone,crack initiation and propagation.Then the crack propagation conditions,specific energy,etc.under different free surface distance(S)were also investigated combined with linear cutting experiments.The results show that the rock breaking process under the free surface induced by disc cutter is dominated by tension failure mode.There exists a critical S to promote crack propagation to free surface effectively.And this rock breaking method can improve the rock breaking force and breaking efficiency significantly when proper.

Numerical Simulation of Sloshing with Large Deforming Free Surface by MPS-LES Method

Moving particle semi-implicit （MPS） method is a fully Lagrangian particle method which can easily solve problems with violent free surface. Although it has demonstrated its advantage in ocean engineering applications, it still has some defects to be improved. In this paper, MPS method is extended to the large eddy simulation （LES） by coupling with a sub-particle-scale （SPS） turbulence model. The SPS turbulence model turns into the Reynolds stress terms in the filtered momentum equation, and the Smagorinsky model is introduced to describe the Reynolds stress terms. Although MPS method has the advantage in the simulation of the free surface flow, a lot of non-free surface particles are treated as free surface particles in the original MPS model. In this paper, we use a new free surface tracing method and the key point is ＂neighbor particle＂. In this new method, the zone around each particle is divided into eight parts, and the particle will be treated as a free surface particle as long as there are no ＂neighbor particles＂ in any two parts of the zone. As the number density parameter judging method has a high efficiency for the free surface particles tracing, we combine it with the neighbor detected method. First, we select out the particles which may be mistreated with high probabilities by using the number density parameter judging method. And then we deal with these particles with the neighbor detected method. By doing this, the new mixed free surface tracing method can reduce the mistreatment problem efficiently. The serious pressure fluctuation is an obvious defect in MPS method, and therefore an area-time average technique is used in this paper to remove the pressure fluctuation with a quite good result. With these improvements, the modified MPS-LES method is applied to simulate liquid sloshing problems with large deforming free surface. Results show that the modified MPS-LES method can simulate the large deforming free surface easily. It can not only capture the large impact pressure accurately on rolling tank wall but also can generate all physical phenomena successfully. The good agreement between numerical and experimental results proves that the modified MPS-LES method is a good CFD methodology in free surface flow simulations.

Prolonged simulation of near-free surface underwater explosion based on Eulerian finite element method

n the area of naval architecture and ocean engineering,the research about the underwater xplosion problem is of great significance.To achieve prolonged simulation of near-free surface underwater explosion,the underwater explosion transient numerical model is established in this paper based on compressible Eulerian finite element method(EFEM).Compared with Geers Hunter formula,EFEM is availably validated by simulating the free-field underwater xplosion case.Then,the bubble pulsation and flow field dynamic characteristics of the cases with different underwater explosive depth are compared in this work.Lastly,the height of the water hump and the pressure of flow flied are analyzed quantitatively through the simulation results.

Meshfree Method Numerical Simulation of Sheet Metal Forming Process

Metal forming plays an important role in manufacturing industry and is widely applied in industries.The tradi- tional finite element method(FEM)numerical simulation is commonly used to predict metal forming process.Conventional finite element analysis of metal forming processes often breaks down due to severe mesh distortion,therefore time-consuming remeshing is necessary.Meshfree methods have been developed since 1977 and can avoid this problem.This new generation of computational methods reduces time-consuming model generation and refinement effort,and its shape function has higher order connectivity than FEM’s.In this paper the velocity shape functions are developed from a reproducing kernel approximation that satisfies consistency conditions and is used to analyze metal tension rigid viscoplastic deforming and Magnesium Alloy(MB 15)sheet superplastic ten- sion forming.A meshfree method metal forming modeling program is set up,the partition of unity method is used to compute the integrations in weak form equations and penalty method is used to impose the essential boundary condition exactly.Metal forming examples,such as sheet metal superplastic tension forming and metal rigid viscoplastic tension forming,are analyzed to demon- strate the performance of mesh free method.

Free-surface Simulations of Newtonian and Non-Newtonian Fluids with the Lattice Boltzmann Method

This paper describes the application of a three-dimensional lattice Boltzmann method （LBM） to Newtonian and non-Newtonian （Bingham fluid in this work） flows with free surfaces. A mass tracking algorithm was incorporated to capture the free surface, whereas Papanastasiou＇s modified model was used for Bingham fluids. The lattice Boltzmann method was first validated using two benchmarks： Newtonian flow through a square cross-section tube and Bingham flow through a circular cross-section tube. Afterward, the dam-break problem for the Newtonian fluid and the slump test for Bingham fluid were simulated to validate the free-surface-capturing algorithm. The numerical results were in good agreement with analytical results, as well as other simulations, thereby proving the validity and correctness of the current method. The proposed method is a promising substitute for time-consuming and costly physical experiments to solve problems encountered in geotechnical and geological engineering, such as the surge and debris flow induced by a landslide or earthquake.

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectie eigenfunction expansion method.

Computational modeling of free-surface slurry flow problems using particle simulation method

The particle simulation method is used to solve free-surface slurry flow problems that may be encountered in several scientific and engineering fields.The main idea behind the use of the particle simulation method is to treat granular or other materials as an assembly of many particles.Compared with the continuum-mechanics-based numerical methods such as the finite element and finite volume methods,the movement of each particle is accurately described in the particle simulation method so that the free surface of a slurry flow problem can be automatically obtained.The major advantage of using the particle simulation method is that only a simple numerical algorithm is needed to solve the governing equation of a particle simulation system.For the purpose of illustrating how to use the particle simulation method to solve free-surface flow problems,three examples involving slurry flow on three different types of river beds have been considered.The related particle simulation results obtained from these three examples have demonstrated that:1) The particle simulation method is a promising and useful method for solving free-surface flow problems encountered in both the scientific and engineering fields;2) The shape and irregular roughness of a river bed can have a significant effect on the free surface morphologies of slurry flow when it passes through the river bed.

Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges

The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges.Based on the extended Hamilton＇s principle for the elastic dynamics system,the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method.Through numerical calculation,curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained,and three edges clamped and the other edge elastically restrained versus the spring constant,locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained.Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.

Dynamic Loads and Wake Prediction for Large Wind Turbines Based on Free Wake Method

With large scale wind turbines,the issue of aerodynamic elastic response is even more significant on dynamic behaviour of the system.Unsteady free vortex wake method is proposed to calculate the shape of wake and aerodynamic load.Considering the effect of aerodynamic load,inertial load and gravity load,the decoupling dynamic equations are established by using finite element method in conjunction of the modal method and equations are solved numerically by Newmark approach.Finally,the numerical simulation of a large scale wind turbine is performed through coupling the free vortex wake modelling with structural modelling.The results show that this coupling model can predict the flexible wind turbine dynamic characteristics effectively and efficiently.Under the influence of the gravitational force,the dynamic response of flapwise direction contributes to the dynamic behavior of edgewise direction under the operational condition of steady wind speed.The difference in dynamic response between the flexible and rigid wind turbines manifests when the aerodynamics/structure coupling effect is of significance in both wind turbine design and performance calculation.

一种用于深空网天线组阵的基于归一化权的Matrix-Free Power Method

对深空网天线组阵中的几种信号相关算法作简要介绍,重点研究无需形成矩阵的幂的方法——Matrix-Free Power Method。通过理论分析,提出一种改进算法,即基于归一化权的Matrix-Free Power Method。理论计算和软件仿真结果均表明,基于归一化权的Matrix-Free Power Method信噪比合成性能略优于原Matrix-Free Power Method。

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years.The method has yielded many new analytic solutions due to its rigorousness.In this study,the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge.The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied.The solution procedure incorporates separation of variables,symplectic eigen solution and superposition.The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems.The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use.The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.

Synthesis of Substituted u-Oxo-bis[tetra-phenyl porphyrinatoiron] Compounds from Free Base Porphyrins by a One-pot Method

A novel method for synthesis of substituted u-oxo-bis[tetraphenyl porphyrinatoiron] compotmds （[TRPPFe]20） based on the reaction among free base porphyrins （TRPPH2）, FeSO4·7H2O and H2O in one pot was proposed and investigated. Four kinds of [TRPPFe]2O were synthesized by this novel synthetic method, and their structures were characterized by elemental analysis, infrared spectra and UV-vis spectroscopy. The reaction conditions, including the effect of different iron salts on the formation of [TRPPFe]2O, the reaction time between FeSO4-7H2O and TRPPH2, the molar ratio of FeSO4-7H2O/TRPPH2 as well as the volume ratio of H2O/DMF, were investigated. The [TRPPFe]2O yield of 93% 98% could be obtained under the following optimal conditions： the reaction time of FeSOn-7H2O with TRPPH2 was about 10 h, the molar ratio of FeSOn-7H2O/TRPPH2 about 5 ： 1, and the volume ratio of H2O/DMF exceeded 2 ： 1. Further research indicated that only TRPPFeCI were formed once FeSOn-7H2O was replaced by FeCl2-4H2O, the reason of which might be that halogen ions in iron salts interfered the formation of [TRPPFe]2O, suggesting that halogen ions should be avoided in this novel synthetic method.

Buckling analysis of super-long rock-socketed filling piles in soft soil area by element free Galerkin method

In order to discuss the buckling stability of super-long rock-socketed filling piles widely used in bridge engineering in soft soil area such as Dongting Lake, the second stability type was adopted instead of traditional first type, and a newly invented numerical analysis method, i.e. the element-free Galerkin method (EFGM), was introduced to consider the non-concordant deformation and nonlinearity of the pile-soil interface. Then, based on the nonlinear elastic-ideal plastic pile-soil interface model, a nonlinear iterative algorithm was given to analyze the pile-soil interaction, and a program for buckling analysis of piles by the EFGM (PBAP-EFGM) and arc length method was worked out as well. The application results in an engineering example show that, the shape of pile top load-settlement curve obtained by the program agrees well with the measured one, of which the difference may be caused mainly by those uncertain factors such as possible initial defects of pile shaft and the eccentric loading during the test process. However, the calculated critical load is very close with the measured ultimate load of the test pile, and the corresponding relative error is only 5.6%, far better than the calculated values by linear and nonlinear incremental buckling analysis (with a greater relative error of 37.0% and 15.4% respectively), which also verifies the rationality and feasibility of the present method.