Methods for connecting dissimilar three‐dimensional finite element meshes

Two methods are presented for connecting dissimilar three‐dimensional finite element meshes. The first method combines the concept of master and slave surfaces with the uniform strain approach for finite elements. By modifying the boundaries of elements on a slave surface, corrections are made to element formulations such that first‐order patch tests are passed. The second method is based entirely on constraint equations, but only passes a weaker form of the patch test for non‐planar surfaces. Both methods can be used to connect meshes with different element types. In addition, master and slave surfaces can be designated independently of relative mesh resolutions. Example problems in three‐dimensional linear elasticity are presented. Published in 2000 by John Wiley & Sons, Ltd.     

Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods

In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patch‐based version of the DG formulation is developed, which decomposes the domain into enriched and unenriched subdomains. Continuity between patches is enforced with an internal penalty method. An element‐based form is also developed, where each element is considered a patch. The patch‐based DG is shown to have similar accuracy to the element‐based DG for a given discretization but requires significantly fewer degrees of freedom. The method is applied to material interfaces, cracks and dislocation problems. For the dislocations, a contour integral form of the internal forces that only requires integration over the patch boundaries is developed. A previously developed assumed strain (AS) method is also developed further and compared with the DG method for weak discontinuities and linear elastic cracks. The DG method is shown to be significantly more accurate than the standard XFEM for a given element size and to converge optimally, even where the standard XFEM does not. The accuracy of the DG method is similar to that of the AS method but requires less application‐specific coding. Copyright © 2007 John Wiley & Sons, Ltd.     

Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

Smooth finite element methods: Convergence,accuracy and properties

A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

On methods for stabilizing constraints over enriched interfaces in elasticity

Enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non‐smooth problems. Internal features, such as boundaries, are represented in such methods by using discontinuous enrichment of the standard finite element basis. Within such frameworks, however, imposition of interface constraints and/or constitutive relations can cause unexpected difficulties, depending upon how relevant fields are interpolated on un‐gridded interfaces. This work address the stabilized treatment of constraints in an enriched finite element context. Both the Lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields. We demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner. In a ‘bubble‐stabilized approach,’ fine‐scale degrees of freedom are added over elements supporting the interface. The variational form can be shown to have a similar form to a second approach we consider, Nitsche's method, with the exception that the stabilization terms follow directly from the bubble functions. In this work, we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two‐dimensional elasticity. We examine several benchmark problems in elasticity, and show that only Nitsche's method and the bubble‐stabilization approach produce stable traction fields over internal boundaries. We also demonstrate a novel difference between the penalty method and Nitsche's method in that the latter passes the patch test exactly, regardless of the stabilization parameter's magnitude. Results for more complicated geometries and triple interface junctions are also presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A transition element for uniform strain hexahedral and tetrahedral finite elements

A transition element is presented for meshes containing uniform strain hexahedral and tetrahedral finite elements. It is shown that the volume of the standard uniform strain hexahedron is identical to that of a polyhedron with 14 vertices and 24 triangular faces. Based on this equivalence, a transition element is developed as a simple modification of the uniform strain hexahedron. The transition element makes use of a general method for hourglass control and satisfies first‐order patch tests. Example problems in linear elasticity are included to demonstrate the application of the element. Copyright © 1999 John Wiley & Sons, Ltd. This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.     

Finite elements for materials with strain gradient effects

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C¹‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C⁰‐continuous shape functions and can achieve the same convergence as C¹ elements. These C⁰ elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

Exponential finite elements for diffusion–advection problems

A new finite element method for the solution of the diffusion–advection equation is proposed. The method uses non‐isoparametric exponentially‐varying interpolation functions, based on exact, one‐ and two‐dimensional solutions of the Laplace‐transformed differential equation. Two eight‐noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight‐ and 12‐noded polynomial elements. The exponential element based on two‐dimensional analytical solutions fails basic tests of convergence. The one based on one‐dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight‐noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions. Copyright © 2005 John Wiley & Sons, Ltd.     

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

An assumed strain formulation of efficient solid triangular element for general shell analysis

An efficient assumed strain triangular solid element is developed for the analysis of plate and shell structures. The finite element formulation is based on the two‐field assumed strain formulation with two independent fields of assumed displacement and assumed strain. The assumed strain field is carefully selected to alleviate the shear locking effect without triggering undesirable spurious kinematic modes. The curvilinear surface of shell structures is modelled with flat facet elements to obviate the membrane locking effect. The patch tests are successfully passed, and numerical test involving various example problems demonstrates the validity and efficiency of the present element. Copyright © 2000 John Wiley & Sons, Ltd.     

EAS concept for higher‐order finite shell elements to eliminate volumetric locking

The deficiency of volumetric locking phenomena in finite elements using higher‐order shell element formulations based on Lagrangean polynomials and a linear finite shell kinematics cannot be avoided by the existent enhanced assumed strain (EAS) concept established for low‐order elements. In this paper a consistent modification of the EAS concept is proposed to extend its applicability to higher‐order shell elements. This modification, affecting the transversal normal strain for polynomial orders p>1, eliminates pathological modes caused by volumetric locking. The efficiency of the proposed extended EAS method is demonstrated by means of eigenvalue analyses and two representative numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

A new finite element approach for solving three‐dimensional problems using trimmed hexahedral elements

In this paper, a novel finite element approach is presented to solve three‐dimensional problems using trimmed hexahedral elements generated by cutting a simple block consisting of regular hexahedral elements with a computer‐aided design (CAD) surface. Trimmed hexahedral elements, which are polyhedral elements with curved faces, are placed at the boundaries of finite element models, and regular hexahedral elements remain in the interior regions. Shape functions for trimmed hexahedral elements are developed by using moving least square approximation with harmonic weight functions based on an extension of Wachspress coordinates to curved faces. A subdivision of polyhedral domains into tetrahedral sub‐domains is performed to construct shape functions for trimmed hexahedral elements, and numerical integration of the weak form can be carried out consistently over the tetrahedral sub‐domains. Trimmed hexahedral elements have similar properties to conventional finite elements regarding the continuity, the completeness, the node–element connectivity, and the inter‐element compatibility. Numerical examples for three‐dimensional linear elastic problems with complex geometries show the efficiency and effectiveness of the present method. Copyright © 2015 John Wiley & Sons, Ltd.     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 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 practical large‐strain solid finite element for sheet forming

An alternative approach for developing practical large‐strain finite elements has been introduced and used to create a three‐dimensional solid element that exhibits no locking or hourglassing, but which is more easily and reliably derived and implemented than typical reduced‐integration schemes with hourglassing control. Typical large‐strain elements for forming applications rely on reduced integration to remove locking modes that occur with the coarse meshes that are necessary for practical use. This procedure introduces spurious zero‐energy deformation modes that lead to hourglassing, which in turn is controlled by complex implementations that involve lengthy derivations, knowledge of the material model, and/or undetermined parameters. Thus, for a new material or new computer program, implementation of such elements is a daunting task. Wang–Wagoner‐3‐dimensions (WW3D), a mixed, hexahedral, three‐dimensional solid element, was derived from the standard linear brick element by ignoring the strain components corresponding to locking modes while maintaining full integration (8 Gauss points). Thus, WW3D is easily implemented for any material law, with little chance of programming error, starting from programming for a readily available linear brick element. Surprisingly, this approach and resulting element perform similarly or better than standard solid elements in a series of numerical tests appearing in the literature. The element was also tested successfully for an applied sheet‐forming analysis problem. Many variations on the scheme are also possible for deriving special‐purpose elements. Copyright © 2005 John Wiley & Sons, Ltd.     

Analytical integrations in 2D BEM elasticity

In the context of two‐dimensional linear elasticity, this paper presents the closed form of the integrals that arise from both the standard (collocation) boundary element method and the symmetric Galerkin boundary element method. Adopting polynomial shape functions of arbitrary degree on straight elements, finite part of Hadamard, Cauchy principal values and Lebesgue integrals are computed analytically, working in a local coordinate system. For the symmetric Galerkin boundary element method, a study on the singularity of the external integral is conducted and the outer weakly singular integral is analytically performed. Numerical tests are presented as a validation of the obtained results. Copyright © 2001 John Wiley & Sons, Ltd.     

Arbitrary placement of local meshes in a global mesh by the interface‐element method (IEM)

A new method is proposed to place local meshes in a global mesh with the aid of the interface‐element method (IEM). The interface‐elements use moving least‐square (MLS)‐based shape functions to join partitioned finite‐element domains with non‐matching interfaces. The supports of nodes are defined to satisfy the continuity condition on the interfaces by introducing pseudonodes on the boundaries of interface regions. Particularly, the weight functions of nodes on the boundaries of interface regions span only neighbouring nodes, ensuring that the resulting shape functions are identical to those of adjoining finite‐elements. The completeness of the shape functions of the interface‐elements up to the order of basis provides a reasonable transfer of strain fields through the non‐matching interfaces between partitioned domains. Taking these great advantages of the IEM, local meshes can be easily inserted at arbitrary places in a global mesh. Several numerical examples show the effectiveness of this technique for modelling of local regions in a global domain. Copyright © 2003 John Wiley & Sons, Ltd.     

An effective hybrid displacement function element method for solving the edge effect of Mindlin–Reissner plate

In bending problems of Mindlin–Reissner plate, the resultant forces often vary dramatically within a narrow range near free and soft simply‐supported (SS1) boundaries. This is so‐called the edge effect or the boundary layer effect, a challenging problem for conventional finite element method. In this paper, an effective finite element method for analysis of such edge effect is developed. The construction procedure is based on the hybrid displacement function (HDF) element method [1], a simple hybrid‐Trefftz stress element method proposed recently. What is different is that an additional displacement function f related to the edge effect is considered, and its analytical solutions are employed as the additional trial functions for the first time. Furthermore, the free and the SS1 boundary conditions are also applied to modify the element assumed resultant fields. Then, two new special elements, HDF‐P4‐Free and HDF‐P4‐SS1, are successfully constructed. These new elements are allocated along the corresponding boundaries of the plate, while the other region is modeled by the usual HDF plate element HDF‐P4‐11 β [1]. Numerical tests demonstrate that the present method can effectively capture the edge effects and exactly satisfy the corresponding boundary conditions by only using relatively coarse meshes. Copyright © 2015 John Wiley & Sons, Ltd.