Boundary element analysis of elastic contact problems using gap finite elements

Gap finite elements provide a practical approach for dealing with elastic contact problems, and it is not possible to derive something similar with boundary elements. This work introduces a simple technique for the analysis of elastic contact problems by coupling a gap finite element subregion with boundary element subregions. The developed algorithm proves to be accurate and reliable, combining the advantages within contact problems of both the gap finite element and the boundary element method.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.     

Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions. The model is formulated as a boundary value problem for the Helmholtz equation with a transparent boundary condition. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated Dirichlet-to-Neumann boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. A new adaptive finite element algorithm is proposed for solving the acoustic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.     

Boundary element software in engineering

The boundary element method starts to be a powerful tool for the analysis of continua. The main attraction of the method is that only the boundary/ surface of the domain needs to be discretised, thus the process of modelling is not so tedious as by the finite element method. The dimensionality of the problem analysed is reduced by one. Although the developments of the boundary element technique are still in early stages, the application of the method seems to acquire more and more users for its versatility and economy when compared with finite elements.The paper presented reviews applications of the boundary element method to the solution of a wide variety of problems in engineering. A review of boundary element software is given. Both categories of programs, general purpose and special purpose, are represented. Program capabilities include field problems, linear and nonlinear stress analysis, thermoelastic problems, etc. Programs where coupled finite element and boundary element techniques are implemented are also represented in the review.A comprehensive bibliography at the end of the paper includes program documentation, program applications and papers dealing with the boundary element method from the theoretical point of view.     

A 2.5D coupled FE–BE methodology for the dynamic interaction between longitudinally invariant structures and a layered halfspace

This paper presents a general 2.5D coupled finite element–boundary element methodology for the computation of the dynamic interaction between a layered soil and structures with a longitudinally invariant geometry, such as railway tracks, roads, tunnels, dams, and pipelines. The classical 2.5D finite element method is combined with a novel 2.5D boundary element method. A regularized 2.5D boundary integral equation is derived that avoids the evaluation of singular traction integrals. The 2.5D Green’s functions of a layered halfspace, computed with the direct stiffness method, are used in a boundary element method formulation. This avoids meshing of the free surface and the layer interfaces with boundary elements and effectively reduces the computational efforts and storage requirements. The proposed technique is applied to four examples: a road on the surface of a halfspace, a tunnel embedded in a layered halfspace, a dike on a halfspace and a vibration isolating screen in the soil.     

Mixed finite element method and high-order local artificial boundary conditions for exterior problems of elliptic equation

The mixed finite element method is used to solve the exterior elliptic problem with high-order local artificial boundary conditions. New unknowns are introduced to reduce the order of the derivatives to two. This leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary. Error estimates are obtained for some local artificial boundary conditions with prescribed order. Numerical examples are presented and the results demonstrate the effectiveness of this method.     

Optimality criterion approach using tapered finite elements with nodal averaging technique for two-dimensional problems

Use of tapered finite elements with three design variables per element and nodal averaging technique in the optimality criterion approach is studied in this paper. Minimum volume design of a uniformly heated square plate with a single temperature constraint is obtained using the proposed method. The present results with a lower order finite element mesh compare very well with those obtained with optimality criterion approach using constant thickness elements and with mathematical programming techniques using a higher order finite element mesh.     

Penalty resolution of the babuska circle paradox

If the solution for a simply-supported circular plate is approximated by a sequence of finite element solutions on successively refined polygonal domains the finite element approximations converge to the solution of a different problem. In the present study we present a finite element formulation with boundary penalty that produces valid approximations to the simply-supported plate. The penalty term is shown to require the use of reduced integration. The dependence of the penalty parameter on mesh size h is also examined. Numerical experiments confirm the validity of the method and determine rates of convergence. A second approach involving a modified corner condition is also considered and error estimates determined. This scheme is implemented also using a discrete penalty technique. The results and rates are compared with the boundary penalty method and their relative merits discussed.     

Finite elements using long vectors of the DAP

The Finite Element Method for solving partial differential equations using the long vector mode of the DAP is presented. This work was developed on a 32 × 32 version of the DAP attached to a Perq scientific workstation.

First, the implementation of finite elements using the long vector mode of the DAP is given, followed by the treatment of boundary conditions and the solution of the finite element equations using a parallel conjugate gradient method. Two solution procedures for the parallel conjugate gradient method, first without global matrix assembly and second with global matrix assembly, are presented and their advantages and disadvantages are discussed. Preconditioners for the conjugate gradient method based on iteration methods are also discussed and results include a 1-step point Jacobi preconditioner, a m-step point Jacobi preconditioner and a m-step multi-colour preconditioner. Finally long vector implementations for a larger system which stores multinodes per processor using a sliced mapping technique and domain decomposition are included.     

On the numerical analysis of compressible flow problems by the “modified FLIC method”

This paper presents a new computational technique for transonic flow problem analysis. This method, named Modified FLIC Method, is based on a time-marching technique of FLIC (fluid in cell) method and employs triangular elements conventionally used in finite element method. This technique can be applied to transonic flows with any complicated boundary shapes. Three problems were solved in this paper, the first was a supersonic flow around a circular cylinder, the second was a transonic flow between tubrine blade cascades and the last was an unsteady flow in a duct with a junction. The calculated results showed a good agreement with the experimental data.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Coons' surface method for formulation of finite element of plates and shells

In this paper, we apply the Coons' surface method and fitted function interpolation to fit boundary conditions of the finite element, and obtain different displacement functions of the plate and shell rectangular element, as well as the sectorial element, parallelogram element, ring element and quadrilateral element. The method is easily implemented and its geometric significance and mechanical conception are quite clear. Both computing technique and operational procedure are relatively unified. It is convenient to formulate conforming elements and high-precision elements, and also may be applied to formulate mixed and hybrid elements.     

等几何分析与有限元直接耦合法

等几何分析(IsoGeometric Analysis,IGA)具有几何模型精确,分析精度高和收敛速度快等优点,但积分效率不高、边界条件处理复杂.因此,提出一种IGA与有限元直接耦合的方法,将有限元与IGA交界处节点自由度用等几何控制节点直接表示出来,从而形成IGA与有限元的耦合单元.算例分析表明,该方法与常规有限元法相比具有精度高的特点,与IGA相比具有施加边界条件方便的优点.     

Duabechies小波基梁单元的构造及应用研究

以Daubechies小波尺度函数作为单元插值函数，构造了一类小波基梁单元，由于采用节点挠度和转角作为单元自由度，单元连接和约束条件处理能够像传统单元一样方便地进行，从而使该小波基梁单元可有效地用于变截面、局部承载等复杂梁弯曲问题分析，数值算例表明，本文构造的小波基梁单元对不同约束条件、不同结构形式梁弯曲问题均有较高分析精度，为有限元方法提出了一种新的研究途径．     

Looping behavior of arches using corotational finite element

Buckling of arches is studied using a corotational finite element model in conjunction with a modified Riks-Wempner technique. The corotational formulation allows for separation of rigid body displacements from deformation displacements of the element. The program can equally be applied to rigid or prestressed arches, as it takes into account the sequential fabrication of stressed (i.e. prebuckled) arches. Looping paths tracing nonlinear response of arches have been successfully obtained for several cases, including rigid and prestressed elastica arches with both symmetric and asymmetric modes of buckling. Comparisons are made with the results for prestressed arches using shooting method, and with the benchmark results for various rigid arches using discrete element method. When compared to a discrete element technique, the present method appears to be more cost-effective, as a finite element mesh with 60% fewer elements can result in virtually the same degree of accuracy.     

基于频率估计的多网格有限元相位展开方法

提出了一种基于频率估计的多网格有限元干涉SAR相位展开方法.为将可能的误差限制在噪声区域内,阻止相位误差在整个图像平面内传递.首先,介绍了一种新的局部频率估计方法:相似最大似然法;然后在较好的边值条件下利用最小二乘意义下的有限元法得出相位解缠等式;最后用多网格技术与超松弛迭代相结合的方法对等式进行求解,从而得到相位解缠结果.仿真和实测数据处理结果表明,方法很好地克服了最小二乘法对于相位坡度欠估计的缺点,且与其它方法相比具有较高的精度.     

Optimum shape design of rotating disks

This paper deals with optimum shape design of the rotating disks by nonlinear programming method. The shape of the cross section is defined by 5th degree polynomial which is completely determined by the boundary conditions and four design variables. The stress analysis of the disk is carried out by finite element method using isoparametric elements. The optimization technique used is with improved movelimit method of sequential linear programming. Progress of optimization is investigated with three different objective functions. After preliminary studies a weighted objective function is selected for detailed investigation. Optimum shapes are obtained for different speeds and for different fit pressures from hub. The results are presented in non-dimensionalised form.     

Generation of triangular mesh with specified size by circle packing

This paper describes an algorithm for the generation of a finite element mesh with a specified element size over an unbound 2D domain using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within the open domain. As soon as a circle is added to the generation front, triangular elements are directly generated by properly connecting frontal segments with the centre of the new circle. Circles are packed closely and in contact with the existing circles by an iterative procedure according to the specified size control function. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection between frontal segments. If the mesh generation of a physical object is required, the object boundary can be introduced. The boundary recovery procedure is fast and robust by tracing neighbours of triangular elements. The finite element mesh generated by circle packing can also be used through a mapping process to produce parametric surface meshes of the required characteristics. The sizes of circles in the pack are controlled by the principal surface curvatures. Five examples are given to show the effectiveness and robustness of mesh generation and the application of circle packing to mesh generation over curved surfaces.     

Treatment of singular integral caused by employing linear boundary elements for two-dimensional elastostatic problems

The boundary element method employing linear elements has the advantage of high precision in calculations. However, there exists a problem of singularity when the diagonal terms of the boundary influence coefficients are calculated. A method by which the singular terms are cancelled is proposed in this paper. A computer program for the solution of two-dimensional elastostatic problems using linear boundary elements is developed and verified through two examples. The technique dealing with the nodes at corners is discussed. The results show that the present method of solving singular integrals is credible.