Singularity enrichment for complete sliding contact using the partition of unity finite element method

In this paper, the numerical modelling of complete sliding contact and its associated singularity is carried out using the partition of unity finite element method. Sliding interfaces in engineering components lead to crack nucleation and growth in the vicinity of the contact zone. To accurately capture the singular stress field at the contact corner, we use the partition of unity framework to enrich the standard displacement‐based finite element approximation by additional (enriched) functions. These enriched functions are derived from the analytical expression of the asymptotic displacement field in the vicinity of the contact corner. To characterize the intensity of the singularity, a domain integral formulation is adopted to compute the generalized stress intensity factor (GSIF). Numerical results on benchmark problems are presented to demonstrate the improved accuracy and benefits of this technique. We conduct an investigation on issues pertaining to the extent of enrichment, accurate numerical integration of weak‐form integrals and the rate of convergence in energy. The use of partition of unity enrichment leads to accurate estimations of the GSIFs on relatively coarse meshes, which is particularly beneficial for modelling non‐linear sliding contacts. Copyright © 2008 John Wiley & Sons, Ltd.     

An improved partition of unity finite element model for diffraction problems

The partition of unity finite element method (PUFEM) is explored and improved to deal with practical diffraction problems efficiently. The use of plane waves as an external function space allows an efficient implementation of an approximate exterior non‐reflective boundary condition, improving the original proposed by Higdon for general diffraction problems. A ‘virtually’ analytical integration procedure is introduced for multi‐dimensional high‐frequency problems which exhibits a dramatic decrease in the number of operations for a given error compared with standard integration methods. Suitable conjugate gradient type solvers for the whole range of wavenumbers are used, including such cases in which PUFEM can produce nearly singular matrices caused by ‘round‐off’ limits. Copyright © 2001 John Wiley & Sons, Ltd.     

A modified element‐free Galerkin method with essential boundary conditions enforced by an extended partition of unity finite element weight function

In this work we propose a method which combines the element‐free Galerkin (EFG) with an extended partition of unity finite element method (PUFEM), that is able to enforce, in some limiting sense, the essential boundary conditions as done in the finite element method (FEM). The proposed extended PUFEM is based on the moving least square approximation (MLSA) and is capable of overcoming singularity problems, in the global shape functions, resulting from the consideration of linear and higher order base functions. With the objective of avoiding the presence of singular points, the extended PUFEM considers an extension of the support of the classical PUFE weight function. Since the extended PUFEM is closely related to the EFG method there is no need for special approximation functions with complex implementation procedures, and no use of the penalty and/or multiplier method is required in order to approximately impose the essential boundary condition. Thus, a relatively simple procedure is needed to combine both methods. In order to attest the performance of the method we consider the solution of an analytical elastic problem and also some coupled elastoplastic‐damage problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Novel basis functions for the partition of unity boundary element method for Helmholtz problems

The BEM is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains. In the last decade, the partition of unity BEM, in which the approximation space is enriched with a linear combination of plane waves, has been developed; this significantly reduces the number of DOFs required per wavelength. It has been shown that the element ends are more susceptible to errors in the approximation than the mid‐element regions. In this paper, the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the Lagrangian shape function component of the basis functions. It is demonstrated, using numerical examples, that choosing trigonometric shape functions, rather than classical polynomial shape functions (quadratic in this case), provides accuracy benefits. Collocation schemes are investigated; it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the Fock domain. Copyright © 2012 John Wiley & Sons, Ltd.     

The partition of unity finite element method for short wave acoustic propagation on non‐uniform potential flows

A novel numerical method is proposed for modelling time‐harmonic acoustic propagation of short wavelength disturbances on non‐uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non‐uniform. Many wavelengths can be included within a single element leading to ultra‐sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid‐based schemes. Numerical results for lined uniform two‐dimensional ducts and for non‐uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.     

Moving particle finite element method with global smoothness

We describe a new version of the moving particle finite element method (MPFEM) that provides solutions within a C⁰ finite element framework. The finite elements determine the weighting for the moving partition of unity. A concept of ‘General Shape Function’ is proposed which extends regular finite element shape functions to a larger domain. These are combined with Shepard functions to obtain a smooth approximation. The Moving Particle Finite Element Method combines desirable features of finite element and meshfree methods. The proposed approach, in fact, can be interpreted as a ‘moving partition of unity finite element method’ or ‘moving kernel finite element method’. This method possesses the robustness and efficiency of the C⁰ finite element method while providing at least C¹ continuity. Copyright © 2004 John Wiley & Sons, Ltd.     

Generalized finite element method using proper orthogonal decomposition

A methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or simulated data. The approach is based on the proper orthogonal decomposition (POD) technique, which is used to generate low‐order representations of data that contain general information about the solution of partial differential equations. One of the main challenges in such enriched finite element methods is knowing how to choose, a priori, enrichment functions that capture the nature of the solution of the governing equations. POD produces low‐order subspaces, that are optimal in some norm, for approximating a given data set. For most problems, since the solution error in Galerkin methods is bounded by the error in the best approximation, it is expected that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. We demonstrate the potential of this approach through three numerical examples. Best‐approximation studies are conducted that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis. Copyright © 2009 John Wiley & Sons, Ltd.     

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X‐FEM) for three‐dimensional crack modelling is described. A discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three‐dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Dislocations by partition of unity

A new finite element method for accurately modelling the displacement and stress fields produced by a dislocation is proposed. The methodology is based on a local enrichment of the finite element space by closed form solutions for dislocations in infinite media via local partitions of unity. This allows the treatment of both arbitrary boundary conditions and interfaces between materials. The method can readily be extended to arrays of dislocations, 3D problems, large strains and non‐linear constitutive models. Results are given for an edge dislocation in a hollow cylinder and in an infinite medium, for the cases of a glide plane intersecting a rigid obstacle and an interface between two materials. Copyright © 2005 John Wiley & Sons, Ltd.     

基于拓展单位分解有限元法分析声波在多域场内的响应

基于拓展单位分解有限元方法,将平面波函数和贝塞尔函数作为基函数进行拓展。将亥姆霍兹方程离散,求解时不变情况下多域场内声波的响应,并分析基函数对求解精度的影响。将波动方程的时间导数利用二阶中心差分方法离散,得到方程的隐式表达式,划分时间步迭代求解时变情况下声波在多域场内的响应,分析迭代时间间隔对计算精度的影响,与典型算例的精确解进行比较,验证精确性。结果表明,平面波函数作为拓展基函数,利用二阶中心差分法离散时间导数,分析时不变以及时变情况下多域场内高波数声波的响应问题,具有较高的计算精度和计算效率。     

Mixed enrichment for the finite element method in heterogeneous media

Problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand, different enriched finite element methods such as the partition of unity, which proved to be very successful in treating similar problems, are developed and studied for homogeneous media. In this work, we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials. The idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method. Although presented within the partition of unity context, the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials. Copyright © 2014 John Wiley & Sons, Ltd.     

Partition of unity for localization in implicit a posteriori finite element error control for linear elasticity

The partition of unity for localization in adaptive finite element method (FEM) for elliptic partial differential equations has been proposed in Carstensen and Funken (SIAM J. Sci. Comput. 2000; 21 : 1465–1484) and is applied therein to the Laplace problem. A direct adaptation to linear elasticity in this paper yields a first estimator η_L based on patch‐oriented local‐weighted interface problems. The global Korn inequality with a constant C_Korn yields reliability for any finite element approximation u_h to the exact displacement u. In order to localize this inequality further and so to involve the global constant C_Korn directly in the local computations, we deduce a new error estimator µ_L. The latter estimator is based on local‐weighted interface problems with rigid body motions (RBM) as a kernel and so leads to effective estimates only if RBM are included in the local FE test functions. Therefore, the excluded first‐order FEM has to be enlarged by RBM, which leads to a partition of unit method (PUM) with RBM, called P₁+RBM or to second‐order FEMs, called P₂ FEM. For P₁+RBM and P₂ FEM (or even higher‐order schemes) one obtains the sharper reliability estimate . Efficiency holds in the strict sense of . The local‐weighted interface problems behind the implicit error estimators η_L and µ_L are usually not exactly solvable and are rather approximated by some FEM on a refined mesh and/or with a higher‐order FEM. The computable approximations are shown to be reliable in the sense of . The oscillations are known functions of the given data and higher‐order terms if the data are smooth for first‐order FEM. The mathematical proofs are based on weighted Korn inequalities and inverse estimates combined with standard arguments. The numerical experiments for uniform and adapted FEM on benchmarks such as an L‐shape problem, Cook's membrane, or a slit problem validate the theoretical estimates and also concern numerical bounds for C_Korn and the locking phenomena. Copyright © 2007 John Wiley & Sons, Ltd.     

The extended/generalized finite element method: An overview of the method and its applications

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non‐smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non‐smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi‐field problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

Mesh‐independent p‐orthotropic enrichment using the generalized finite element method

This paper is aimed at presenting a simple yet effective procedure to implement a mesh‐independent p‐orthotropic enrichment in the generalized finite element method. The procedure is based on the observation that shape functions used in the GFEM can be constructed from polynomials defined in any co‐ordinate system regardless of the underlying mesh or type of element used. Numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p‐orthotropic enrichment. Copyright © 2002 John Wiley & Sons, Ltd.     

On the use of the extended finite element method with quadtree/octree meshes

This paper describes the use of the extended finite element method in the context of quadtree/octree meshes. Particular attention is paid to the enrichment of hanging nodes that inevitably arise with these meshes. An approach for enforcing displacement continuity along hanging edges and faces is proposed and validated on various numerical examples (holes, material interfaces, and singularities) in both 2D and 3D. Copyright © 2010 John Wiley & Sons, Ltd.     

Allman's triangle,rotational DOF and partition of unity

A simplification of the 1984 Allman triangle (one of historically most important elements with rotational dofs) is presented. It is found that this old element takes a typical form of the partition of unity approximation. The Allman's rotation presented in the partition of unity form offers merits and convenience in formulation and practical applications. The stiffness matrix of the 1984 Allman triangle, which is originally computed from the linear strain triangular element (the 6 nodes quadratic triangle), can be obtained instead in a cheaper way from that of the constant strain triangular element. The constraint of the rotational terms during essential boundary treatment, which remains equivocal and ambiguous, is understood to be mandatory. The partition of unity notion enables a straightforward extension of the Allman's rotational dof to meshfree approximations. In numerical examples, we discuss suppression of spurious zero‐energy modes and patch tests. Standard benchmarks are carried out to assess performance of the newly formulated triangle and a meshfree approximation with the rotational dofs. Copyright © 2006 John Wiley & Sons, Ltd.     

A partition of unity FEM for time‐dependent diffusion problems using multiple enrichment functions

An enriched partition of unity FEM is developed to solve time‐dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here, the temporal decay in the solution is embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right‐hand side of the linear system of equations. The advantage is a significant saving in the computational effort where, previously, the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p‐version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of DoFs. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and the transient heat equation with multiple sources. The aim of such a method compared with the classical FEM is to solve time‐dependent diffusion applications efficiently and with an appropriate level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.