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.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Generalized finite element approaches for analysis of localized thermo‐structural effects

This work addresses computational modeling challenges associated with structures subjected to sharp, local heating, where complex temperature gradients in the materials cause three‐dimensional, localized, intense stress and strain variation. Because of the nature of the applied loadings, multiphysics analysis is necessary to accurately predict thermal and mechanical responses. Moreover, bridging spatial scales between localized heating and global responses of the structure is nontrivial. A large global structural model may be necessary to represent detailed geometry alone, and to capture local effects, the traditional approach of pre‐designing a mesh requires careful manual effort. These issues often lead to cumbersome and expensive global models for this class of problems. To address them, the authors introduce a generalized FEM (GFEM) approach for analyzing three‐dimensional solid, coupled physics problems exhibiting localized heating and corresponding thermomechanical effects. The capabilities of traditional hp‐adaptive FEM or GFEM as well as the GFEM with global–local enrichment functions are extended to one‐way coupled thermo‐structural problems, providing meshing flexibility at local and global scales while remaining competitive with traditional approaches. The methods are demonstrated on several example problems with localized thermal and mechanical solution features, and accuracy and (parallel) computational efficiency relative to traditional direct modeling approaches are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

An adaptive interface‐enriched generalized FEM for the treatment of problems with curved interfaces

An adaptive refinement scheme is presented to reduce the geometry discretization error and provide higher‐order enrichment functions for the interface‐enriched generalized FEM. The proposed method relies on the h‐adaptive and p‐adaptive refinement techniques to reduce the discrepancy between the exact and discretized geometries of curved material interfaces. A thorough discussion is provided on identifying the appropriate level of the refinement for curved interfaces based on the size of the elements of the background mesh. Varied techniques are then studied for selecting the quasi‐optimal location of interface nodes to obtain a more accurate approximation of the interface geometry. We also discuss different approaches for creating the integration sub‐elements and evaluating the corresponding enrichment functions together with their impact on the performance and computational cost of higher‐order enrichments. Several examples are presented to demonstrate the application of the adaptive interface‐enriched generalized FEM for modeling thermo‐mechanical problems with intricate geometries. The accuracy and convergence rate of the method are also studied in these example problems. Copyright © 2015 John Wiley & Sons, Ltd.     

The generalized finite element method: an example of its implementation and illustration of its performance

The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

This paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)‐scale problems. The local problems focus on the resolution of fine‐scale features of the solution in the vicinity of three‐dimensional cracks, while the global problem addresses the macro‐scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp‐GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three‐dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three‐dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

The Discontinuity‐Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity‐Enriched Finite Element Method (DE‐FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction‐free and cohesive crack examples. We show that DE‐FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.     

Numerical p‐version refinement studies for the regularized stress‐BEM

In the development of the boundary element method (BEM) and the finite element method (FEM) researchers have typically selected similar basis functions. That is, both methods typically employ low‐order interpolations such as piece‐wise linear or piece‐wise quadratic and rely on h‐version refinement to increase accuracy as required. In the case of the FEM, the decision to use low‐order elements is made for computational efficiency as an attractive compromise between local modeling accuracy and sparseness of the resulting linear system. However, in many BEM formulations, low‐order elements may be the only practical choice given the complexity of using analytic integration formulae in conjunction with special integral interpretations. Unlike their efficient use in the FEM, fine meshes of low‐order elements in the BEM are highly inefficient from a computational standpoint given the dense nature of BEM systems. Moreover, owing to singularities in the kernel functions, the BEM should be expected to benefit more so than the FEM from very high levels of local accuracy. Through the use of regularized algorithms which only require numerical integration, p‐version refinement in the BEM is easily extended to include any set of basis functions with no significant increase in programming complexity. Numerical results show that by using interpolations as high as 12th and 16th order, one can expect reductions in error by as many as five orders of magnitude over comparable algorithms based on similar system size. For two‐dimensional problems, it is also shown that, for a given level of error, one can expect reductions in system size by an order of magnitude, thus leading to a reduction in computational expense for conventional algorithms by three orders of magnitude. Copyright © 2003 John Wiley & Sons, Ltd.     

Clustered generalized finite element methods for mesh unrefinement,non‐matching and invalid meshes

In spite of significant advancements in automatic mesh generation during the past decade, the construction of quality finite element discretizations on complex three‐dimensional domains is still a difficult and time demanding task. In this paper, the partition of unity framework used in the generalized finite element method (GFEM) is exploited to create a very robust and flexible method capable of using meshes that are unacceptable for the finite element method, while retaining its accuracy and computational efficiency. This is accomplished not by changing the mesh but instead by clustering groups of nodes and elements. The clusters define a modified finite element partition of unity that is constant over part of the clusters. This so‐called clustered partition of unity is then enriched to the desired order using the framework of the GFEM. The proposed generalized finite element method can correctly and efficiently deal with: (i) elements with negative Jacobian; (ii) excessively fine meshes created by automatic mesh generators; (iii) meshes consisting of several sub‐domains with non‐matching interfaces. Under such relaxed requirements for an acceptable mesh, and for correctly defined geometries, today's automated tetrahedral mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user. A detailed technical discussion of the proposed generalized finite element method with clustering along with numerical experiments and some implementation details are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Parameter‐free,weak imposition of Dirichlet boundary conditions and coupling of trimmed and non‐conforming patches

We present a parameter‐free domain sewing approach for low‐order as well as high‐order finite elements. Its final form contains only primal unknowns; that is, the approach does not introduce additional unknowns at the interface. Additionally, it does not involve problem‐dependent parameters, which require an estimation. The presented approach is symmetry preserving; that is, the resulting discrete form of an elliptic equation will remain symmetric and positive definite. It preserves the order of the underlying discretization, and we demonstrate high‐order accuracy for problems of non‐matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p. We also demonstrate how the method may be used to model material interfaces, which may be curved and for which the interface does not coincide with the underlying mesh. This novel approach is presented in the context of the p‐version and B‐spline version of the finite cell method, an embedded domain method of high order, and compared with more classical methods such as the penalty method or Nitsche's method. Copyright © 2014 John Wiley & Sons, Ltd.     

Multi‐level hp‐adaptivity for cohesive fracture modeling

Discretization‐induced oscillations in the load–displacement curve are a well‐known problem for simulations of cohesive crack growth with finite elements. The problem results from an insufficient resolution of the complex stress state within the cohesive zone ahead of the crack tip. This work demonstrates that the hp‐version of the finite element method is ideally suited to resolve this complex and localized solution characteristic with high accuracy and low computational effort. To this end, we formulate a local and hierarchic mesh refinement scheme that follows dynamically the propagating crack tip. In this way, the usually applied static a priori mesh refinement along the complete potential crack path is avoided, which significantly reduces the size of the numerical problem. Studying systematically the influence of h‐refinement, p‐refinement, and hp‐refinement, we demonstrate why the suggested hp‐formulation allows to capture accurately the complex stress state at the crack front preventing artificial snap‐through and snap‐back effects. This allows to decrease significantly the number of degrees of freedom and the simulation runtime. Furthermore, we show that by combining this idea with the finite cell method, the crack propagation within complex domains can be simulated efficiently without resolving the geometry by the mesh. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of high‐order finite elements for convected wave propagation

In this paper, we examine the performance of high‐order finite element methods (FEM) for aeroacoustic propagation, based on the convected Helmholtz equation. A methodology is presented to measure the dispersion and amplitude errors of the p‐FEM, including non‐interpolating shape functions, such as ‘bubble’ shape functions. A series of simple test cases are also presented to support the results of the dispersion analysis. The main conclusion is that the properties of p‐FEM that make its strength for standard acoustics (e.g., exponential p‐convergence, low dispersion error) remain present for flow acoustics as well. However, the flow has a noticeable effect on the accuracy of the numerical solution, even when the change in wavelength due to the mean flow is accounted for, and an approximation of the dispersion error is proposed to describe the influence of the mean flow. Also discussed is the so‐called aliasing effect, which can reduce the accuracy of the solution in the case of downstream propagation. This can be avoided by an appropriate choice of mesh resolution. Copyright © 2013 John Wiley & Sons, Ltd.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

The extended finite element method with new crack‐tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X‐FEM) with new crack‐tip enrichment functions. In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and crack‐tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ? class and κ class, two classes of crack‐tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J‐integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 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.     

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.     

Conformal higher‐order remeshing schemes for implicitly defined interface problems

A new higher‐order accurate method is proposed that combines the advantages of the classical p‐version of the FEM on body‐fitted meshes with embedded domain methods. A background mesh composed by higher‐order Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher‐order accurate sub‐elements is realized. Therefore, the zero level sets are detected and meshed in a first step, which is called reconstruction. Then, based on the topological situation in the cut element, higher‐order sub‐elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub‐elements are not always well‐shaped. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparison of high‐order curved finite elements

Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points on the accuracy of the computation is also studied. Two‐dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBS‐enhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element method (FEM). Moreover, NEFEM outperforms Cartesian FEM and p‐FEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.