Analyzing static three-dimensional elastic domains with a new infinite boundary element formulation

A new two-dimensionally mapped infinite boundary element (IBE) is presented. The formulation is based on a triangular boundary element (BE) with linear shape functions instead of the quadrilateral IBEs usually found in the literature. The infinite solids analyzed are assumed to be three-dimensional, linear-elastic and isotropic, and Kelvin fundamental solutions are employed. One advantage of the proposed formulation over quadratic or higher order elements is that no additional degrees of freedom are added to the original BE mesh by the presence of the IBEs. Thus, the IBEs allow the mesh to be reduced without compromising the accuracy of the result. Two examples are presented, in which the numerical results show good agreement with authors using quadrilateral IBEs and analytical solutions.     

An advanced boundary element method for elasticity problems in nonhomogeneous media

Summary This paper presents a new boundary element method formulation and numerical implementation of elasticity problems in nonhomogeneous media. The fundamental solutions for elasticity in homogeneous media are employed and the nonsingular formulation is derived. A physically and mathematically meaningful elimination of internal degrees of freedom is proposed. The solution at an arbitrary point is expressed in terms of boundary displacements and tractions. The rank of the system matrix (for computation of relevant unknowns) is dependent only on the discretization of the boundary.     

A hybrid finite element method for heterogeneous materials with randomly dispersed rigid inclusions

A hybrid finite element approach is proposed for the mechanical response of two-dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes. In conventional finite element methods, many elements must be used to represent one inclusion. In this work, each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion. In numerically approximating stress and displacement distributions around the inclusion, classical elasticity solutions for a multiply-connected region are employed. A modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense. The accuracy and efficiency of the proposed method are demonstrated by two boundary value problems. In one example, the results based on the proposed method with only 64 hybrid elements (450 degrees of freedom) are shown to be almost identical to those based on the traditional method with 2928 conventional elements (5526 degrees of freedom).     

Finite element analysis of stiffened laminated plates under transverse loading

A finite element model is developed to study the behavior of stiffened laminated plates under transverse loadings. Transverse shear flexibility is incorporated in both beam and plate displacement fields. A laminated plate element with 45 degrees of freedom is used in conjunction with a laminated beam element having 12 degrees of freedom for the bending analysis of eccentrically-stiffened laminated plates. The validity of the formulation is demonstrated by comparing with the available solutions in the literature. The numerical results are presented for eccentrically-stiffened layered plates having various boundary conditions and with stiffeners varying in number.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

A general mass lumping scheme for the variants of the extended finite element method

A new strategy for the mass matrix lumping of enriched elements for explicit transient analysis is presented. It is shown that to satisfy the kinetic energy conservation, the use of zero or negative masses for enriched degrees of freedom of lumped mass matrix may be necessary. For a completely cracked element, by lumping the mass of each side of the interface into the finite element nodes located at the same side and assigning zero masses to the enriched degrees of freedom, the kinetic energy for rigid body translations is conserved without transferring spurious energy across the interface. The time integration is performed by adopting an explicit-implicit technique, where the regular and enriched degrees of freedom are treated explicitly and implicitly, respectively. The proposed method can be viewed as a general mass lumping scheme for the variants of the extended finite element methods because it can be used irrespective of the enrichment method. It also preserves the optimal critical time step of an intact finite element by treating the enriched degrees of freedom implicitly. The accuracy and efficiency of the proposed mass matrix are validated with several benchmark examples.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

A new infinite boundary element formulation combined to an alternative multi-region technique

The main objective of this work is to obtain an efficient three-dimensional boundary element (BE) formulation for the simulation of layered solids. This formulation is obtained by combining an alternative multi-region technique with an infinite boundary element (IBE) formulation. It is demonstrated that such a combination is straightforward and can be easily programmed. Kelvin fundamental solutions are employed, considering the static analysis of isotropic and linear-elastic domains. Establishing relations between the displacement fundamental solutions of the different domains, the alternative technique used in this paper allows analyzing all domains as a single solid, not requiring equilibrium or compatibility equations. It was shown in a previous paper that this approach leads to a smaller system of equations when compared to the usual multi-region technique and the results obtained are more accurate. The two-dimensionally mapped infinite boundary element (IBE) formulation here used is based on a triangular BE with linear shape functions. One advantage of this formulation over quadratic or higher order elements is that no additional degrees of freedom are added to the original BE mesh by the presence of the IBEs. Thus, the IBEs allow the mesh to be reduced without compromising the accuracy of the result. The use of IBEs improves the advantages of the alternative multi-region technique, contributing for the low computational cost and allowing a considerable mesh reduction. Furthermore, the results show good agreement with the ones given in other works, confirming the accuracy of the presented formulation.     

Hybrid trefftz plane elasticity elements with p -method capabilities

A family of p-method plane elasticity elements is derived based on the hybrid Trefftz formulation.¹ Exact solutions of the Lamé-Navier equations are used for the intra-element displacement field together with an independent displacement frame function field along the element boundary. The final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom at corner nodes and an optional number of hierarchic degrees of freedom associated with the mid-side nodes. Since the element matrices do not involve integration over the element area, the elements have a polygonal contour with an optional number of curved sides. The quadrilateral element has the same external appearance as the conventional p-method plane elasticity element.^2,3 But unlike in the conventional p-method approach, suitable special-purpose Trefftz functions are generally used to handle the singularity and/or stress concentration problems rather than a local mesh refinement. The practical efficiency of the new elements is assessed through a series of examples.     

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.     

Nonlinear rotation‐free three‐node shell finite element formulation

A new triangular thin‐shell finite element formulation is presented, which employs only translational degrees of freedom. The formulation allows for large deformations, and it is based on the nonlinear Kirchhoff thin‐shell theory. A number of static and dynamic test problems are considered for which analytical or benchmark solutions exist. Comparisons between the predictions of the new model and these solutions show that the new model accurately reproduces complex nonlinear analytical solutions as well as solutions obtained using existing, more complex finite element formulations. Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐reflecting boundary conditions for atomistic,continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non‐reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time‐history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non‐nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh‐free simulations. We illustrate the effectiveness of the method on a one‐dimensional model problem, and calculate the time‐history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.     

A quasi‐static discontinuous Galerkin configurational force crack propagation method for brittle materials

This paper presents a framework for r‐adaptive quasi‐static configurational force (CF) brittle crack propagation, cast within a discontinuous Galerkin (DG) symmetric interior penalty (SIPG) finite element scheme. Cracks are propagated in discrete steps, with a staggered algorithm, along element interfaces, which align themselves with the predicted crack propagation direction. The key novelty of the work is the exploitation of the DG face stiffness terms existing along element interfaces to propagate a crack in a mesh‐independent r‐adaptive quasi‐static fashion, driven by the CF at the crack tip. This adds no new degrees of freedom to the data structure. Additionally, as DG methods have element‐specific degrees of freedom, a geometry‐driven p‐adaptive algorithm is also easily included allowing for more accurate solutions of the CF on a moving crack front. Further, for nondeterminant systems, we introduce an average boundary condition that restrains rigid body motion leading to a determinant system. To the authors' knowledge, this is the first time that such a boundary condition has been described. The proposed formulation is validated against single and multiple crack problems with single‐ and mixed‐mode cracks, demonstrating the predictive capabilities of the method.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A coupled symmetric BE–FE method for acoustic fluid–structure interaction

A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamilton's principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C¹-continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.     

A hybrid element formulation for the three‐dimensional penalty finite element analysis of incompressible fluids

This work presents a hybrid element formulation for the three‐dimensional penalty finite element analysis of incompressible Newtonian fluids. The formulation is based on a mixed variational statement in which velocity and stresses are treated as independent field variables. The main advantage of this formulation is that it bypasses the use of ad hoc techniques such as selective reduced integration that are commonly used in penalty‐based finite element formulations, and directly yields high accuracy for the velocity and stress fields without the need to carry out smoothing. In addition, since the stress degrees of freedom are condensed out at an element level, the cost of solving for the global degrees of freedom is the same as in a standard penalty finite element method, although the gain in accuracy for both the velocity and stress (including the pressure) fields is quite significant. Copyright © 2007 John Wiley & Sons, Ltd.     

A FLAT TRIANGULAR SHELL ELEMENT WITH LOOF NODES

In the formulation of flat shell elements it is difficult to achieve inter-element compatibility between membrane and transverse displacements for non-coplanar elements. Many elements lack proper nodal degrees of freedom to model intersections making the assembly of elements troublesome. A flat triangular shell element is established by a combination of a new plate bending element DKTL and the well-known linear membrane strain element LST, and for this element the above-mentioned deficiencies are avoided. The plate bending element DKTL is based on Discrete Kirchhoff Theory and Loof nodes. The nodal configuration of the element is similar to the SemiLoof element, and the formulation is an improvement of a previous formulation. The element is used for both linear statics, linear buckling and geometrical non-linear analysis, and numerical examples are presented to show the robustness, accuracy and quick convergence of the element.     

Improved assumed-stress hybrid shell element with drilling degrees of freedom for linear stress,buckling and free vibration analyses

An improved 4-node quadrilateral assumed-stress hybrid shell element with drilling degrees of freedom is presented. The formulation is based on Hellinger–Reissner variational principle and the shape functions are formulated directly for the 4-node element. The element has 12 membrane degrees of freedom and 12 bending degrees of freedom. It has 9 independent stress parameters to describe the membrane stress resultant field and 13 independent stress parameters to describe the moment and transverse shear stress resultant field. The formulation encompasses linear stress, linear buckling and linear free vibration problems. The element is validated with standard test cases and is shown to be robust. Numerical results are presented for linear stress, buckling, and free vibration analyses.     

A family of simple and robust finite elements for linear and geometrically nonlinear analysis of laminated composite plates

A family of simple, displacement-based and shear-flexible triangular and quadrilateral flat plate/shell elements for linear and geometrically nonlinear analysis of thin to moderately thick laminate composite plates are introduced and summarized in this paper.

The developed elements are based on the first-order shear deformation theory (FSDT) and von-Karman’s large deflection theory, and total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from Timoshenko’s laminated composite beam functions, thus convergence can be ensured theoretically for very thin laminates and shear-locking problem is avoided naturally.

The flat triangular plate/shell element is of 3-node, 18-degree-of-freedom, and the plane displacement interpolation functions of the Allman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. The flat quadrilateral plate/shell element is of 4-node, 24-degree-of-freedom, and the linear displacement interpolation functions of a quadrilateral plane element with drilling degrees of freedom are taken as the in-plane displacements.

The developed elements are simple in formulation, free from shear-locking, and include conventional engineering degrees of freedom. Numerical examples demonstrate that the elements are convergent, not sensitive to mesh distortion, accurate and efficient for linear and geometric nonlinear analysis of thin to moderately thick laminates.     

A finite‐strain quadrilateral shell element based on discrete Kirchhoff–Love constraints

This paper improves the 16 degrees‐of‐freedom quadrilateral shell element based on pointwise Kirchhoff–Love constraints and introduces a consistent large strain formulation for this element. The model is based on classical shell kinematics combined with continuum constitutive laws. The resulting element is valid for large rotations and displacements. The degrees‐of‐freedom are the displacements at the corner nodes and one rotation at each mid‐side node. The formulation is free of enhancements, it is almost fully integrated and is found to be immune to locking or unstable modes. The patch test is satisfied. In addition, the formulation is simple and amenable to efficient incorporation in large‐scale codes as no internal degrees‐of‐freedom are employed, and the overall calculations are very efficient. Results are presented for linear and non‐linear problems. Copyright © 2005 John Wiley & Sons, Ltd.