Scaled boundary polygons with application to fracture analysis of functionally graded materials

This study presents a framework for the development of polygon elements based on the scaled boundary FEM. Underpinning this study is the development of generalized scaled boundary shape functions valid for any n‐sided polygon. These shape functions are continuous inside each polygon and across adjacent polygons. For uncracked polygons, the shape functions are linearly complete. For cracked polygons, the shape functions reproduce the square‐root singularity and the higher‐order terms in the Williams eigenfunction expansion. This allows the singular stress field in the vicinity of the crack tip to be represented accurately. Using these shape functions, a novel‐scaled boundary polygon formulation that captures the heterogeneous material response observed in functionally graded materials is developed. The stiffness matrix in each polygon is derived from the principle of virtual work using the scaled boundary shape functions. The material heterogeneity is approximated in each polygon by a polynomial surface in scaled boundary coordinates. The intrinsic properties of the scaled boundary shape functions enable accurate computation of stress intensity factors in cracked functionally graded materials directly from their definitions. The new formulation is validated, and its salient features are demonstrated, using five numerical benchmarks. Copyright © 2014 John Wiley & Sons, Ltd.     

A super‐element for crack analysis in the time domain

A super‐element for the dynamic analysis of two‐dimensional crack problems is developed based on the scaled boundary finite‐element method. The boundary of the super‐element containing a crack tip is discretized with line elements. The governing partial differential equations formulated in the scaled boundary co‐ordinates are transformed to ordinary differential equations in the frequency domain by applying the Galerkin's weighted residual technique. The displacements in the radial direction from the crack tip to a point on the boundary are solved analytically without any a priori assumption. The scaled boundary finite‐element formulation leads to symmetric static stiffness and mass matrices. The super‐element can be coupled seamlessly with standard finite elements. The transient response is evaluated directly in the time domain using a standard time‐integration scheme. The stress field, including the singularity around the crack tip, is expressed semi‐analytically. The stress intensity factors are evaluated without directly addressing singular functions, as the limit in their definitions is performed analytically. Copyright © 2004 John Wiley & Sons, Ltd.     

Use of higher‐order shape functions in the scaled boundary finite element method

The scaled boundary finite element method is a novel semi‐analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher‐order polynomial functions for the shape functions. Two techniques for generating the higher‐order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the p‐version of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p‐refinement are compared with the corresponding rates of convergence achieved when uniform h‐refinement is used, allowing direct comparison of the computational cost of the two approaches. The results show that it is advantageous to use higher‐order elements, and that higher rates of convergence can be obtained using p‐refinement instead of h‐refinement. Copyright © 2005 John Wiley & Sons, Ltd.     

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

A high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.     

Structural‐acoustic coupling on non‐conforming meshes with quadratic shape functions

Fully coupled finite element/boundary element models are a popular choice when modelling structures that are submerged in heavy fluids. To achieve coupling of subdomains with non‐conforming discretizations at their common interface, the coupling conditions are usually formulated in a weak sense. The coupling matrices are evaluated by integrating products of piecewise polynomials on independent meshes. The case of interfacing elements with linear shape functions on unrelated meshes has been well covered in the literature. This paper presents a solution to the problem of evaluating the coupling matrix for interfacing elements with quadratic shape functions on unrelated meshes. The isoparametric finite elements have eight nodes (Serendipity) and the discontinuous boundary elements have nine nodes (Lagrange). Results using linear and quadratic shape functions on conforming and non‐conforming meshes are compared for an example of a fluid‐loaded point‐excited sphere. It is shown that the coupling error decreases when quadratic shape functions are used. Copyright © 2012 John Wiley & Sons, Ltd.     

The scaled boundary finite element method in structural dynamics

The scaled boundary finite element method is extended to solve problems of structural dynamics. The dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution for the scaled boundary finite element equation. The inertial effect at high frequencies is modeled by high‐order terms of the continued fraction without introducing an internal mesh. By using this solution and introducing auxiliary variables, the equation of motion of the bounded domain is expressed in high‐order static stiffness and mass matrices. Standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain. Numerical examples for modal and direct time‐domain analyses are presented. Rapid convergence is observed as the order of continued fraction increases. A guideline for selecting the order of continued fraction is proposed and validated. High computational efficiency is demonstrated for problems with stress singularity. Copyright © 2008 John Wiley & Sons, Ltd.     

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two‐dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 John Wiley & Sons, Ltd.     

A scaled boundary finite element formulation for poroelasticity

This paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.     

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

This paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

On the use of bubble functions in the local buckling analysis of plate structures by the spline finite strip method

This paper augments bubble functions to the ordinary spline finite strip method in order to calculate the elastic local buckling coefficients of plates and plate structures. The results show that the use of bubble functions improves significantly the convergence of the spline finite strip method in terms of the strip subdivision, and therefore leads to smaller storage requirements for the global stiffness and stability matrices, and faster eigenvalue extraction. Benchmark numerical investigations are presented, including the study of plates with different boundary conditions under uniaxial and biaxial stresses, plates with different aspect ratios under shear, and a stiffened panel under combined shear and compression that has been studied elsewhere. These studies demonstrate that by implementation of the bubble functions, rapid convergence of the solution is obtained. The formulation is ideal for analysing local buckling under a variety of boundary and loading conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis of singular stress fields at multi‐material corners under thermal loading

The scaled boundary finite‐element method is extended to the modelling of thermal stresses. The particular solution for the non‐homogeneous term caused by thermal loading is expressed as integrals in the radial direction, which are evaluated analytically for temperature changes varying as power functions of the radial coordinate. When applied to model a multi‐material corner, only the boundary of the problem domain is discretized. The boundary conditions on the straight material interfaces and the side‐faces forming the corner are satisfied analytically without discretization. The stress field is expressed semi‐analytically as a series solution. The stress distribution along the radial direction, including both the real and complex power singularity and the power‐logarithmic singularity, is represented analytically. The stress intensity factors are determined directly from their definitions in stresses. No knowledge on asymptotic expansions is required. Numerical examples are calculated to evaluate the accuracy of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one‐dimensional polynomials expressed in barycentric co‐ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h‐Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C⁰ inter‐element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.     

多边形比例边界有限元非线性高效分析方法

比例边界有限元法作为一种高精度的半解析数值求解方法,特别适合于求解无限域与应力奇异性等问题,多边形比例边界单元在模拟裂纹扩展过程、处理局部网格重剖分等方面相较于有限单元法具有明显优势。目前,比例边界有限元法更多关注的是线弹性问题的求解,而非线性比例边界单元的研究则处于起步阶段。该文将高效的隔离非线性有限元法用于比例边界单元的非线性分析,提出了一种高效的隔离非线性比例边界有限元法。该方法认为每个边界线单元覆盖的区域为相互独立的扇形子单元,其形函数以及应变-位移矩阵可通过半解析的弹性解获得;每个扇形区的非线性应变场通过设置非线性应变插值点来表达,引入非线性本构关系即可实现多边形比例边界单元高效非线性分析。多边形比例边界单元的刚度通过集成每个扇形子单元的刚度获取,扇形子单元的刚度可采用高斯积分方案进行求解,其精度保持不变。由于引入了较多的非线性应变插值点,舒尔补矩阵维数较大,该文采用Woodbury近似法对隔离非线性比例边界单元的控制方程进行求解。该方法对大规模非线性问题的计算具有较高的计算效率,数值算例验证了算法的正确性以及高效性,将该方法进行推广,对实际工程分析具有重要意义。     

求解无限域动力刚度矩阵的双渐近算法

采用连分式算法可以有效地求解无限域动力刚度表示的比例边界有限元方程, 它具有收敛范围广、收敛速度快等优点. 该文在高频渐近连分式算法的基础上考虑了低频渐近, 发展了一种针对矢量波动方程的双渐近算法. 随着展开阶数的增加, 双渐近算法可以在全频域范围内快速逼近准确解. 引入了系数矩阵?X(i)来增强连分式算法的数值稳定性. 通过在高频极限、低频极限时满足动力刚度表示的比例边界有限元方程, 建立了递推关系以求得动力刚度矩阵. 通过二维半无限楔形体、三维均质弹性半空间数值算例表明, 双渐近算法比单渐近算法更稳定、优越.     

Polygon scaled boundary finite elements for crack propagation modelling

An automatic crack propagation modelling technique using polygon elements is presented. A simple algorithm to generate a polygon mesh from a Delaunay triangulated mesh is implemented. The polygon element formulation is constructed from the scaled boundary finite element method (SBFEM), treating each polygon as a SBFEM subdomain and is very efficient in modelling singular stress fields in the vicinity of cracks. Stress intensity factors are computed directly from their definitions without any nodal enrichment functions. An automatic remeshing algorithm capable of handling any n‐sided polygon is developed to accommodate crack propagation. The algorithm is simple yet flexible because remeshing involves minimal changes to the global mesh and is limited to only polygons on the crack paths. The efficiency of the polygon SBFEM in computing accurate stress intensity factors is first demonstrated for a problem with a stationary crack. Four crack propagation benchmarks are then modelled to validate the developed technique and demonstrate its salient features. The predicted crack paths show good agreement with experimental observations and numerical simulations reported in the literature. Copyright © 2012 John Wiley & Sons, Ltd.     

A scaled boundary finite element formulation for dynamic elastoplastic analysis

This study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.     

Quadratic serendipity finite elements over convex polyhedra

The conventional approach to construct quadratic elements for an n‐sided polygon will yield n(n+1)/2 shape functions, which increases the computational effort. It is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion. Floater and Lai proposed a systematic way to construct higher‐order approximations over arbitrary polygons using the generalized barycentric and triangular coordinates. This approach ensures 2n shape functions with nodes only on the boundary of the polygon. In this paper, we extend the polygonal splines approach to 3 dimensions and construct serendipity shape functions over hexahedra and convex polyhedra. This is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates. The quadratic shape functions possess Kronecker delta property and satisfy constant, linear, and quadratic precision. The accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests. The numerical results show that the quadratic serendipity elements pass the patch test, yield optimal convergence rate, and can tolerate extreme mesh distortion.     

Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

In this paper we present a new approach for finite element solution of time‐harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co‐ordinates. The system of FE equations is therefore reduced to a set of well‐known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available. Copyright © 2006 John Wiley & Sons, Ltd.