Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems

To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

Longitudinal vibration of pile in layered soil based on Rayleigh-Love rod theory and fictitious soil-pile model

The dynamic response of pile in layered soil is theoretically investigated when considering the transverse inertia effect.Firstly, the fictitious soil-pile model is employed to simulate the dynamic interaction between the pile and the soil layers beneath pile toe. The dynamic interactions of adjacent soil layers along the vertical direction are simplified as distributed Voigt models.Meanwhile, the pile and fictitious soil-pile are assumed to be viscoelastic Rayleigh-Love rods, and both the radial and vertical displacement continuity conditions at the soil-pile interface are taken into consideration. On this basis, the analytical solution for dynamic response at the pile head is derived in the frequency domain and the corresponding quasi-analytical solution in the time domain is then obtained by means of the convolution theorem. Following this, the accuracy and parameter value of the hypothetical boundaries for soil-layer interfaces are discussed. Comparisons with published solution and measured data are carried out to verify the rationality of the present solution. Parametric analyses are further conducted by using the present solution to investigate the relationships between the transverse inertia effects and soil-pile parameters.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.

Spline Fictitious Boundary Element Alternating Method for Edge Crack Problems with Mixed Boundary Conditions

The alternating method based on the fundamental solutions of the infinite domain containing a crack,namely Muskhelishvili’s solutions,divides the complex structure with a crack into a simple model without crack which can be solved by traditional numerical methods and an infinite domain with a crack which can be solved by Muskhelishvili’s solutions.However,this alternating method cannot be directly applied to the edge crack problems since partial crack surface of Muskhelishvili’s solutions is located outside the computational domain.In this paper,an improved alternating method,the spline fictitious boundary element alternating method(SFBEAM),based on infinite domain with the combination of spline fictitious boundary element method(SFBEM)and Muskhelishvili’s solutions is proposed to solve the edge crack problems.Since the SFBEM and Muskhelishvili’s solutions are obtained in the framework of infinite domain,no special treatment is needed for solving the problem of edge cracks.Different mixed boundary conditions edge crack problems with varies of computational parameters are given to certify the high precision,efficiency and applicability of the proposed method compared with other alternating methods and extend finite element method.

FICTITIOUS DOMAIN METHODS FOR NAVIER-STOKES EQUATIONS BASED ON PENALTY FACTOR ε

Abstract In this article we briefly studied the fictitious domain methods for steady and nonsteady Navier-Stokes equations based on Penalty factorεon the extended domain. The convergence u?→u in H01（Ω）d and L2（Ω）d（d=2,3） is given as well as p?→p in L02（Ω）.

ON FICTITIOUS DOMAIN METHOD FOR THE NUMERICAL SOLUTION TO HEAT CONDUCTION EQUATION WITH DERIVATIVE BOUNDARY CONDITIONS

This paper investigates the stability and convergence of some knowndifference schemes for the numerical solution to heat conduction equation withderivative boundary conditions by the fictitious domain method.The discrete vari-ables at the false mesh points are firstly eliminated from the difference schemes andthe local truncation errors are then analyzed in detail.The stability and convergenceof the schemes are proved by energy method.An improvement is proposed to obtainbetter schemes over the original ones.Several numerical examples and comparisonswith other schemes are presented.

On L ∞ Stability and Convergence of Fictitious Domain Method for the Numerical Solution to Parabolic Differential Equation with Derivative Boundary Conditions

This paper investigates some known difference schemes for the numerical solution to parabolic differential equation with derivative boundary conditions by the fictitious domain method.The stability and convergence in L ∞ are proven.

SEMI-ANALYTICAL FINITE ELEMENT METHOD FOR FICTITIOUS CRACK MODEL IN FRACTURE MECHANICS OF CONCRETE

Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to develop a novel analytical finite element for the fictitious crack model in fracture mechanics of concrete. The new analytical element can be implemented into FEM program systems to solve fictitious crack propagation problems for concrete cracked plates with arbitrary shapes and loads. Numerical results indicate that the method is more efficient and accurate than ordinary finite element method.

Fully Coupled Fluid-Structure Interaction Model Based on Distributed Lagrange Multiplier/Fictitious Domain Method

This paper, with a finite element method, studies the interaction of a coupled incompressible fluid-rigid structure system with a free surface subjected to external wave excitations. With this fully coupled model, the rigid structure is taken as ＂fictitious＂ fluid with zero strain rate. Both fluid and structure are described by velocity and pressure. The whole domain, including fluid region and structure region, is modeled by the incompressible Navier-Stokes equations which are discretized with fixed Eulerian mesh. However, to keep the structure＇ s rigid body shape and behavior, a rigid body constraint is enforced on the ＂fictitious＂ fluid domain by use of the Distributed Lagrange Multipher/Fictitious Domain （DLM/ FD） method which is originally introduced to solve particulate flow problems by Glowinski et al. For the verification of the model presented herein, a 2D numerical wave tank is established to simulate small amplitude wave propagations, and then numerical results are compared with analytical solutions. Finally, a 2D example of fluid-structure interaction under wave dynamic forces provides convincing evidences for the method excellent solution quality and fidelity.

Multigrid fictitious boundary method for incompressible viscous flows around moving airfoils

In this paper, an efficient multigrid fictitious boundary method （MFBM） coupled with the FEM solver package FEATFLOW was used for the detailed simulation of incompressible viscous flows around one or more moving NACA0012 airfoils. The calculations were carded on a fixed multigrid finite element mesh on which fluid equations were satisfied everywhere, and the airfoils were allowed to move freely through the mesh. The MFBM was employed to treat interactions between the fluid and the airfoils The motion of the airfoils was modeled by Newton-Euler equations. Numerical results of experiments verify that this method provides an efficient way to simulate incompressible viscous flows around moving airfoils.

Application of the Fictitious Domain Method for Navier-Stokes Equations

To apply the fictitious domain method and conduct numericalexperiments, a boundary value problem for an ordinary differential equation is considered. The results of numerical calculations for different valuesof the iterative parameter τ and the small parameter ε are presented. Astudy of the auxiliary problem of the fictitious domain method for NavierStokes equations with continuation into a fictitious subdomain by highercoefficients with a small parameter is carried out. A generalized solutionof the auxiliary problem of the fictitious domain method with continuationby higher coefficients with a small parameter is determined. After all theabove mathematical studies, a computational algorithm has been developedfor the numerical solution of the problem. Two methods were used to solvethe problem numerically. The first variant is the fictitious domain methodassociated with the modification of nonlinear terms in a fictitious subdomain.The model problem shows the effectiveness of using such a modification. Theproposed version of the method is used to solve two problems at once that arisewhile numerically solving systems of Navier-Stokes equations: the problem ofa curved boundary of an arbitrary domain and the problem of absence of aboundary condition for pressure in physical formulation of the internal flowproblem. The main advantage of this method is its universality in developmentof computer programs. The second method used calculation on a uniform gridinside the area. When numerically implementing the solution on a uniformgrid inside the domain, using this method it’s possible to accurately take intoaccount the boundaries of the curved domain and ensure the accuracy of thevalue of the function at the boundaries of the domain. Methodical calculationswere carried out, the results of numerical calculations were obtained. Whenconducting numerical experiments in both cases, quantitative and qualitativeindicators of numerical results coincide.

基于虚拟波场的可控源三维电磁法正演

根据扩散场和虚拟波场的变换关系,将频率域电磁扩散方程转换成虚拟域的波动方程,实现了虚拟波场的电磁场数值计算。在边界的处理上,通过引入复频率完全匹配层吸收边界条件,降低了内存的存储量。将空气层包含在计算域中,避免了地—空界面的复杂处理。通过与均匀半空间解析解对比,其相对误差在3.5%以内,验证了算法的有效性和正确性。最后通过典型地电模型的数值模拟表明:通过一次正演可以获得多个频率的三维电磁响应,提高了计算效率。虚拟波场计算的视电阻率对场源效应不敏感且对异常体的边界具有较好的识别能力。

ANALYSIS OF WAVEGUIDE PROBLEMS USING A RELAXED ITERATIVE DOMAIN DECOMPOSITION METHOD COMBINED WITH MULTIFRONTAL ALGORITHM

In this paper, an absorbing Fictitious Boundary Condition (FBC) is presented to generate an iterative Domain Decomposition Method (DDM) for analyzing waveguide problems.The relaxed algorithm is introduced to improve the iterative convergence. And the matrix equations are solved using the multifrontal algorithm. The resulting CPU time is greatly reduced.Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.

The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

壁面展向震荡诱导颗粒湍槽流减阻的直接数值模拟研究

对湍槽流的减阻研究具有科学意义和工程应用价值,已有大量研究表明向单相湍流中添加离散物质是一种有效的被动减阻方法.相比于被动减阻技术,主动减阻技术如壁面震荡减阻的可控性更高,近年来也得到广泛的关注,但对于壁面展向震荡诱导减阻的研究主要针对单相湍槽流,还未见有相关研究将这一手段用于含颗粒湍槽流的减阻.因此,文章采用直接数值模拟方法开展了壁面展向震荡诱导颗粒湍槽流减阻的机理研究.一方面关注壁面震荡对颗粒湍槽流的调制效果及机理.另一方面关注颗粒和震荡对单相湍槽流的耦合减阻效应.结果表明:壁面震荡可以达到有效减阻,存在最优震荡周期使减阻率达到最大,且最优震荡周期与单相流结果相近.在相同体积分数下,施加壁面震荡的小颗粒湍槽流减阻效果更好.相比于单相湍槽流,当震荡周期小于最优周期时,震荡和颗粒的耦合效应对减阻率的额外贡献较小且可能为负,当大于最优周期时额外贡献逐渐增大,对整体减阻率的占比最高可达10%左右.

破裂过程模拟的可变形圆化多边形离散单元法

为研究工程中岩体内部裂纹的萌生、拓展和贯通机制,模拟其破裂时接触碰撞的全过程,提出可变形圆化多边形离散单元法。该方法将有限单元法与圆化多边形离散单元法进行耦合,采用圆化多边形离散单元法计算非连续岩体间的相互接触作用,连续体内部的应力应变状态则通过有限单元法进行分析。为准确模拟裂纹演化过程,在有限单元间插入节理单元,采用虚拟裂缝模型和具有拉伸截断的摩尔-库仑准则描述节理单元在拉伸、剪切等不同破坏模式下的力学状态。最后模拟巴西圆盘开裂、单切口弯梁开裂及块体冲击破坏试验。研究结果表明:所提出方法能准确模拟岩体从连续体变化为非连续体的破裂过程。

基于虚拟区域法的黏弹性流体中微生物游动的数值模拟和应用

微生物是自然生态系统的重要组成部分,掌握微生物在复杂流体中的运动特性可以为微型器件的设计制造提供理论指导.壁面效应是微生物游动研究中的重要问题之一,已有研究表明微生物在壁面附近存在复杂的行为特征.然而已有研究大多集中于微生物在牛顿流体中的游动模拟,仅有少数涉及黏弹性流体等非牛顿流体.本文采用直接力虚拟区域法与乔列斯基分解相结合的数值方法,引入Squirmer微生物游动模型,研究了微生物在黏弹性流体中的游动问题.首先给出求解黏弹性流体本构方程的数值格式;并将该方法应用于研究微生物游动中的壁面效应.研究结果表明,游动方向是影响微生物颗粒壁面效应的重要因素.流体弹性应力会对微生物产生一个反向转矩,影响微生物的游动方向,从而阻碍微生物逃离壁面.微生物颗粒在黏弹性流体中与壁面作用时间较长,几乎达到牛顿流体的两倍以上.

岩石类材料拉裂过程模拟的三维可开裂拉格朗日元方法

本研究提出了适于模拟岩石类材料变形-拉裂过程的三维可开裂拉格朗日元方法,其实质是三维拉格朗日元方法与断裂力学中虚拟裂缝模型的耦合方法,主要适用于立方体单元。以三维拉格朗日元方法为基础,引入虚拟裂缝模型的具体过程主要包括3个关键步骤:首先,通过节点周围单元的应力获取节点的应力,进而利用节点的最大主应力和岩石单轴抗拉强度进行节点分离判断;然后,选择与节点最大主应力垂直平面最接近的单元边界作为裂缝扩展方向;最后,引入虚拟裂缝模型模拟裂缝的萌生和扩展过程。利用该方法模拟了单轴拉伸岩样和三点弯曲岩梁的变形-拉裂过程,并初步分析了Ⅰ型断裂能、岩样高度和单元尺寸的影响。同时,通过将本研究结果与单轴拉伸岩样峰值应力、临界位移的理论解和前人位移控制加载条件下三点弯曲岩梁的载荷-位移曲线的数值解进行对比,验证了本研究方法的正确性,这为将来的工作奠定了良好的基础。

颗粒群在燃烧环境中的曳力研究

对颗粒半径为100~500μm、体积分数为0.075~0.35的固定随机分布的颗粒群在303 K与1173K环境温度下的受力及相应的流场进行模拟.模拟工具为虚拟区域法.它采用统一形式的结构化网格描述颗粒与气体,并实现两相的相互作用,兼有较高的准确度与效率.相关结果显示,颗粒群在燃烧环境中形成的流场更稳定,这主要是由于气体在燃烧环境中的黏性更大,导致雷诺数更小.同时,颗粒群在燃烧环境中所受黏性力更大.在较低与较高的雷诺数下,颗粒群在燃烧环境中的曳力分别更大与更小.随着颗粒尺寸增大,颗粒群的燃烧强度与曳力均增大;随着颗粒体积分数增大,颗粒群的燃烧强度减小但是曳力增大.最后,本研究对于颗粒群的终端速度、曳力系数以及气-固动量交换系数提出了在燃烧环境中的修正.