Increment-Dimensional Scaled Boundary Finite Element Method for Solving Transient Heat Conduction Problem

An increment-dimensional scaled boundary finite element method (ID-SBFEM) is developed to solve the transient temperature field.To improve the accuracy of SBFEM,the effect of high frequency factor on dynamic stiffness is considered,and the first-order continued fraction technique is used.After the derivation,the SBFE equations are obtained,and the dimensions of thermal conduction,the thermal capacity matrix and the vector of the right side term in the equations are doubled.An example is presented to illustrate the feasibility and good accuracy of the proposed method.

Dynamic interaction numerical models in the time domain based on the high performance scaled boundary finite element method

Consideration of structure-foundation-soil dynamic interaction is a basic requirement in the evaluation of the seismic safety of nuclear power facilities. An efficient and accurate dynamic interaction numerical model in the time domain has become an important topic of current research. In this study, the scaled boundary finite element method （SBFEM） is improved for use as an effective numerical approach with good application prospects. This method has several advantages, including dimensionality reduction, accuracy of the radial analytical solution, and unlike other boundary element methods, it does not require a fundamental solution. This study focuses on establishing a high performance scaled boundary finite element interaction analysis model in the time domain based on the acceleration unit-impulse response matrix, in which several new solution techniques, such as a dimensionless method to solve the interaction force, are applied to improve the numerical stability of the actual soil parameters and reduce the amount of calculation. Finally, the feasibility of the time domain methods are illustrated by the response of the nuclear power structure and the accuracy of the algorithms are dynamically verified by comparison with the refinement of a large-scale viscoelastic soil model.

A scaled boundary node method applied to two-dimensional crack problems

A boundary-type meshless method called the scaled boundary node method （SBNM） is developed to directly evaluate mixed mode stress intensity factors （SIFs） without extra post-processing. The SBNM combines the scaled boundary equations with the moving Kriging （MK） interpolation to retain the dimensionality advantage of the former and the meshless attribute of the latter. As a result, the SBNM requires only a set of scattered nodes on the boundary, and the displacement field is approximated by using the MK interpolation technique, which possesses the 5 function property. This makes the developed method efficient and straightforward in imposing the essential boundary conditions, and no special treatment techniques are required. Besides, the SBNM works by weakening the governing differential equations in the circumferential direction and then solving the weakened equations analytically in the radial direction. Therefore, the SBNM permits an accurate representation of the singularities in the radial direction when the scaling center is located at the crack tip. Numerical examples using the SBNM for computing the SIFs are presented. Good agreements with available results in the literature are obtained.

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method （SBFEM） is a recently developed numerical method combining advantages of both finite element methods （FEM） and boundary element methods （BEM） and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors （SIFs） directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.

Dynamic Crack Propagation Analysis Using Scaled Boundary Finite Element Method

The prediction of dynamic crack propagation in brittle materials is still an important issue in many engineering fields. The remeshing technique based on scaled boundary finite element method(SBFEM) is extended to predict the dynamic crack propagation in brittle materials. The structure is firstly divided into a number of superelements, only the boundaries of which need to be discretized with line elements. In the SBFEM formulation, the stiffness and mass matrices of the super-elements can be coupled seamlessly with standard finite elements, thus the advantages of versatility and flexibility of the FEM are well maintained. The transient response of the structure can be calculated directly in the time domain using a standard time-integration scheme. Then the dynamic stress intensity factor(DSIF) during crack propagation can be solved analytically due to the semi-analytical nature of SBFEM. Only the fine mesh discretization for the crack-tip super-element is needed to ensure the required accuracy for the determination of stress intensity factor(SIF). According to the predicted crack-tip position, a simple remeshing algorithm with the minimum mesh changes is suggested to simulate the dynamic crack propagation. Numerical examples indicate that the proposed method can be effectively used to deal with the dynamic crack propagation in a finite sized rectangular plate including a central crack. Comparison is made with the results available in the literature, which shows good agreement between each other.

Acoustic reflection well logging modeling using the frequency-domain finite-element method with a hybrid PML

In this paper, we propose a hybrid PML （H-PML） combining the normal absorption factor of convolutional PML （C-PML） with tangential absorption factor of Mutiaxial PML （M-PML）. The H-PML boundary conditions can better suppress the numerical instability in some extreme models, and the computational speed of finite-element method and the dynamic range are greatly increased using this HPML. We use the finite-element method with a hybrid PML to model the acoustic reflection of the interface when wireline and well logging while drilling （LWD）, in a formation with a reflector outside the borehole. The simulation results suggests that the PS- and SP- reflected waves arrive at the same time when the inclination between the well and the outer interface is zero, and the difference in arrival times increases with increasing dip angle. When there are fractures outside the well, the reflection signal is clearer in the subsequent reflection waves and may be used to identify the fractured zone. The difference between the dominant wavelength and the model scale shows that LWD reflection logging data are of higher resolution and quality than wireline acoustic reflection logging.

Scaled Boundary Finite Element Analysis of Wave Passing A Submerged Breakwater

The scaled boundary finite element method （SBFEM） is a novel semi-analytical technique combining the advantage of the finite element method （FEM） and the boundary element method （BEM） with its unique properties. In this paper, the SBFEM is used for computing wave passing submerged breakwaters, and the reflection coeffcient and transmission coefficient are given for the case of wave passing by a rectangular submerged breakwater, a rigid submerged barrier breakwater and a trapezium submerged breakwater in a constant water depth. The results are compared with the analytical solution and experimental results. Good agreement is obtained. Through comparison with the results using the dual boundary element method （DBEM）, it is found that the SBFEM can obtain higher accuracy with fewer elements. Many submerged breakwaters with different dimensions are computed by the SBFEM, and the changing character of the reflection coeffcient and the transmission coefficient are given in the current study.

The Calculation of Stress-Strain State of Anisotropic Composite Finite-Element Area with Different Boundary Conditions on the Surface

The numerical analytic research approach of stress-strain state of anisotropic composite finite element area with different boundary conditions on the surface, is represented below. The problem is solved by using a spatial model of the elasticity theory. Differential equation system in partial derivatives reduces to one-dimensional problem using spline collocation method in two coordinate directions. Boundary problem for the system of ordinary higher-order differential equation is solved by using the stable numerical technique of discrete orthogonalization.

STRAIGHTFORWARD MULTI-SCALE BOUNDARY ELEMENT METHOD FOR GLOBAL/LOCAL MECHANICAL ANALYSIS OF ELASTIC HETEROGENEOUS MATERIAL

A straightforward multi-scale boundary element method is proposed for global and local mechanical analysis of heterogeneous material.The method is more accurate and convenient than finite element based multi-scale method.The formulations of this method are derived by combining the homogenization approach and the fundamental equations of boundary element method.The solution gives the convenient formulations to compute global elastic constants and the local stress field.Finally,two numerical examples of porous material are presented to prove the accuracy and the efficiency of the proposed method.The results show that the method does not require the iteration to obtain the solution of the displacement in micro level.

COMPUTER COMPUTATION OF THE METHOD OF MULTIPLE SCALES-DIRICHLET PROBLEM FOR A CLASS OF SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION ANDTHE ASYMPTOTICS OF SOLUTIONS (Ⅰ)

In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. Theseries solutions are constructed, the boundary layer effects are analysed and their asymptotics are proved.

A New Formulation of the Scaled Boundary Finite Element Method for Heterogeneous Media:Application to Heat Transfer Problems

The solution to heat transfer problems in two-dimensional heterogeneous media is attended based on the scaled boundary finite element method(SBFEM)coupled with equilibrated basis functions(EqBFs).The SBFEM reduces the model order by scaling the boundary solution onto the inner element.To this end,tri-lateral elements are emanated from a scaling center,followed by the development of a semi-analytical solution along the radial direction and a finite element solution along the circumferential/boundary direction.The discretization is thus limited to the boundaries of the model,and the semi-analytical radial solution is found through the solution of an eigenvalue problem,which restricts the methods’applicability to heterogeneous media.In this research,we first extracted the SBFEM formulation considering the heterogeneity of the media.Then,we replaced the semi-analytical radial solution with the EqBFs and removed the eigenvalue solution step from the SBFEM.The varying coefficients of the partial differential equation(PDE)resulting from the heterogeneity of the media are replaced by a finite series in the radial and circumferential directions of the element.A weighted residual approach is applied to the radial equation.The equilibrated radial solution series is used in the new formulation of the SBFEM.

Statistical second-order two-scale analysis and computation for heat conduction problem with radiation boundary condition in porous materials

This paper discusses a statistical second-order two-scale（SSOTS） analysis and computation for a heat conduction problem with a radiation boundary condition in random porous materials.Firstly,the microscopic configuration for the structure with random distribution is briefly characterized.Secondly,the SSOTS formulae for computing the heat transfer problem are derived successively by means of the construction way for each cell.Then,the statistical prediction algorithm based on the proposed two-scale model is described in detail.Finally,some numerical experiments are proposed,which show that the SSOTS method developed in this paper is effective for predicting the heat transfer performance of porous materials and demonstrating its significant applications in actual engineering computation.

An adaptive scaled boundary finite element method by subdividing subdomains for elastodynamic problems

The scaled boundary finite element method(SBFEM) is a semi-analytical numerical method,which models an analysis domain by a small number of large-sized subdomains and discretises subdomain boundaries only.In a subdomain,all fields of state variables including displacement,stress,velocity and acceleration are semi-analytical,and the kinetic energy,strain energy and energy error are all integrated semi-analytically.These advantages are taken in this study to develop a posteriori h-hierarchical adaptive SBFEM for transient elastodynamic problems using a mesh refinement procedure which subdivides subdomains.Because only a small number of subdomains are subdivided,mesh refinement is very simple and efficient,and mesh mapping to transfer state variables from an old mesh to a new one is also very simple but accurate.Two 2D examples with stress wave propagation were modelled.The results show that the developed method is capable of capturing propagation of steep stress regions and calculating accurate dynamic responses,using only a fraction of degrees of freedom required by adaptive finite element method.

Time-Domain Analysis of Underground Station-Layered Soil Interaction Based on High-Order Doubly Asymptotic Transmitting Boundary

Based on the modified scale boundary finite element method and continued fraction solution,a high-order doubly asymptotic transmitting boundary(DATB)is derived and extended to the simulation of vector wave propagation in complex layered soils.The high-order DATB converges rapidly to the exact solution throughout the entire frequency range and its formulation is local in the time domain,possessing high accuracy and good efficiency.Combining with finite element method,a coupled model is constructed for time-domain analysis of underground station-layered soil interaction.The coupled model is divided into the near and far field by the truncated boundary,of which the near field is modelled by FEM while the far field is modelled by the high-order DATB.The coupled model is implemented in an open source finite element software,OpenSees,in which the DATB is employed as a super element.Numerical examples demonstrate that results of the coupled model are stable,accurate and efficient compared with those of the extended mesh model and the viscous-spring boundary model.Besides,it has also shown the fitness for long-time seismic response analysis of underground station-layered soil interaction.Therefore,it is believed that the coupled model could provide a new approach for seismic analysis of underground station-layered soil interaction and could be further developed for engineering.

Stress Analysis of Wire Strands by Mesoscale Mechanics

Steel wire ropes have wide application in a variety of engineering fields such as ocean engineering and civil engineering.The stress calculation for steel wire ropes is of crucial importance when conducting strength and fatigue analyses.In this study,we performed a finite element analysis of single-strand steel wire ropes.For the geometric modeling,we used an analytic geometry of space method.We established helical line equations and used the coordinates of the contact points.The finite-element model was simplified using the periodic law.Periodic boundary conditions were used to simulate a wire strand of infinite length under tensile strain,for which we calculated the cross-sectional stresses and inner forces.The results showed that bending and torsion moments emerged when the wire strand was under tensile load.In some cases,the bending stress reached 18%of the tensile stress,and the torsion stress reached 29%of the tensile stress,which means that the total stress was higher than the nominal stress.Whereas in ear-lier studies,a conservative prediction of nominal stress was not possible,the results of our strength and fatigue analyses were more conservative.

MODE III 2-D FRACTURE ANALYSIS BY THE SCALED BOUNDARY FINITE ELEMENT METHOD

The scaled boundary finite element method （SBFEM） is a novel semi-analytical technique that combines the advantages of the finite element method and the boundary element method with unique properties of its own. This method has proven very efficient and accurate for determining the stress intensity factors （SIFs） for mode I and mode II two-dimensional crack problems. One main reason is that the SBFEM has a unique capacity of analytically representing the stress singularities at the crack tip. In this paper the SBFEM is developed for mode III （out of plane deformation） two-dimensional fracture anMysis. In addition, cubic B-spline functions are employed in this paper for constructing the shape functions in the circumferential direction so that higher continuity between elements is obtained. Numerical examples are presented at the end to demonstrate the simplicity and accuracy of the present approach for mode Ⅲ two-dimensional fracture analysis.

数据驱动的半无限介质裂纹识别模型研究

缺陷识别是结构健康监测的重要研究内容,对评估工程结构的安全性具有重要的指导意义,然而,准确确定结构缺陷的尺寸十分困难.论文提出了一种创新的数据驱动算法,将比例边界有限元法(scaled boundary finite element methods,SBFEM)与自编码器(autoencoder,AE)、因果膨胀卷积神经网络(causal dilated convolutional neural network,CDCNN)相结合用于半无限介质中的裂纹识别.在该模型中,SBFEM用于模拟波在含不同裂纹状缺陷半无限介质中的传播过程,对于不同的裂纹状缺陷,仅需改变裂纹尖端的比例中心和裂纹开口处节点的位置,避免了复杂的重网格过程,可高效地生成足够的训练数据.模拟波在半无限介质中传播时,建立了基于瑞利阻尼的吸收边界模型,避免了对结构全域模型进行计算.搭建了CDCNN,确保了时序数据的有序性,并获得更大的感受野而不增加神经网络的复杂性,可捕捉更多的历史信息,AE具有较强的非线性特征提取能力,可将高维的原始输入特征向量空间映射到低维潜在特征向量空间,以获得低维潜在特征用于网络模型训练,有效提升了网络模型的学习效率.数值算例表明:提出的模型能够高效且准确地识别半无限介质中裂纹的量化信息,且AE-CDCNN模型的识别效率较单CDCNN模型提高了约2.7倍.

基于SBFEM的剪切型裂纹动态开裂模拟

剪切型断裂是岩土工程中常见的破坏模式,了解剪切破坏机理并准确预测剪切型裂纹的萌生、扩展过程对保障工程结构的安全性与稳定性具有重要意义.文章建立了基于比例边界有限元法(scaled boundary finite element methods,SBFEM)和非局部宏-微观损伤模型的剪切型裂纹动态开裂模拟方法,定义了基于偏应变概念的物质点对的正伸长量,可作为预测剪切型裂纹扩展行为的动态开裂准则,一点的损伤定义为该点影响域范围内连接的物质键损伤的加权平均值,而物质键的损伤则与基于偏应变概念的物质点对的正伸长量相关联,并引入能量退化函数建立结构域几何拓扑损伤与能量损失之间的关系,将拓扑损伤与应力应变联系起来,通过能量退化函数修正了SBFEM的刚度系数矩阵,得到了子域在损伤状态下的刚度矩阵,推导了考虑结构损伤的SBFEM动力控制方程,采用Newmark隐式算法对控制方程进行时间离散.最后,通过3个典型算例验证了建议的模型可较好地模拟剪切型断裂问题,能够很好地捕捉剪切型裂纹的扩展路径,并得到较为准确的载荷-位移曲线.

考虑材料参数空间变异性的SBFEM开裂模拟

论文建立了基于比例边界有限元法(scaled boundary finite element methods,SBFEM)框架的非局部宏微观损伤模型,考虑材料细观物理参数的空间变异性,探讨了材料参数的空间变异性对结构开裂过程的影响。结果表明:考虑材料参数空间变异性后,裂纹扩展路径具有不确定性,建议的模型能够很好地反应材料内在的随机性;随着结构受力情况的复杂化和结构本体缺陷的增多,裂纹开裂模式的变异性也会增大。自相关长度和参数变异系数对结构开裂分析结果有重要影响。