A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

A super‐element for crack analysis in the time domain

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

A scaled boundary finite element formulation for poroelasticity

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

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.     

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.     

A scaled boundary finite element formulation for dynamic elastoplastic analysis

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

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 unified 3D‐based technique for plate bending analysis using scaled boundary finite element method

This paper presents a unified technique for solving the plate bending problems by extending the scaled boundary finite element method. The formulation is based on the three‐dimensional governing equation without enforcing the kinematics of plate theory. Only the in‐plane dimensions are discretised into finite elements. Any two‐dimensional displacement‐based elements can be employed. The solution along the thickness is expressed analytically by using a matrix function. The proposed technique is consistent with the three‐dimensional theory and applicable to both thick and thin plates without exhibiting the numerical locking phenomenon. Moreover, the use of higher order spectral elements allows the proposed technique to better represent curved boundaries and to achieve high accuracy and fast convergence. Numerical examples of various plate structures with different thickness‐to‐length ratios demonstrate the applicability and accuracy of the proposed technique. Copyright © 2012 John Wiley & Sons, Ltd.     

A novel scaled boundary finite element formulation with stabilization and its application to image‐based elastoplastic analysis

Digital images are increasingly being used as input data for computational analyses. This study presents an efficient numerical technique to perform image‐based elastoplastic analysis of materials and structures. The quadtree decomposition algorithm is employed for image‐based mesh generation, which is fully automatic and highly efficient. The quadtree cells are modeled by scaled boundary polytope elements, which eliminate the issue of hanging nodes faced by standard finite elements. A novel, simple, and efficient scaled boundary elastoplastic formulation with stablisation is developed. In this formulation, the return‐mapping calculation is only required to be performed at a single point in a polytope element, which facilitates the computational efficiency of the elastoplastic analysis and simplicity of implementation. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed technique for performing the elastoplastic analysis of high‐resolution images.     

A displacement‐based finite element formulation for general polyhedra using harmonic shape functions

A displacement‐based continuous‐Galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics. The polyhedra can have an arbitrary number of vertices or faces. The faces of the polyhedra can have an arbitrary number of edges and can be nonplanar. The polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex‐averaged centroid. Conforming shape functions are constructed using harmonic functions defined on the undeformed configuration, thus requiring the use of a total‐Lagrangian finite element formulation in large deformation applications. For nonlinear applications with computationally intensive constitutive models, it is important to minimize the number of element quadrature points. For this reason, an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices. As a first step toward verifying the element behavior in the general nonlinear setting, several linear verification examples are presented using both random Voronoi meshes and distorted hexahedral meshes. For the hexahedral meshes, results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation. The element behavior in the nearly incompressible regime is also examined. Copyright © 2013 John Wiley & Sons, Ltd.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

Polyhedral elements by means of node/edge‐based smoothed finite element method

The node‐based or edge‐based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces, and so retain a good geometric adaptability. The strain smoothing technique and implicit shape functions based on the linear point interpolation make the element formulation simple and straightforward. The resulting polyhedral elements are free from the excessive zero‐energy modes and yield a robust solution very much insensitive to mesh distortion. Several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior. The smoothed finite element method‐based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.     

A semi‐analytical curved element for linear elasticity based on the scaled boundary finite element method

This work introduces a semi‐analytical formulation for the simulation and modeling of curved structures based on the scaled boundary finite element method (SBFEM). This approach adapts the fundamental idea of the SBFEM concept to scale a boundary to describe a geometry. Until now, scaling in SBFEM has exclusively been performed along a straight coordinate that enlarges, shrinks, or shifts a given boundary. In this novel approach, scaling is based on a polar or cylindrical coordinate system such that a boundary is shifted along a curved scaling direction. The derived formulations are used to compute the static and dynamic stiffness matrices of homogeneous curved structures. The resulting elements can be coupled to general SBFEM or FEM domains. For elastodynamic problems, computations are performed in the frequency domain. Results of this work are validated using the global matrix method and standard finite element analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion

The scaled boundary finite element method, alias the consistent infinitesimal finite element cell method, is developed starting from the diffusion equation. Only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary, and thus no singular integrals need to be evaluated. Essential and natural boundary conditions on surfaces and conditions on interfaces between different materials are enforced exactly without any discretization. The solution of the function in the radial direction is analytical. This method is thus exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions. The semi‐analytical solution inside the domain leads to an efficient procedure to calculate singularities accurately without discretization in the vicinity of the singular point. For a bounded medium symmetric steady‐state stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption. Copyright © 1999 John Wiley & Sons, Ltd.     

A new finite element based on the theory of a Cosserat point—extension to initially distorted elements for 2D plane strain

This paper describes an improvement of the Cosserat point element formulation for initially distorted, non-rectangular shaped elements in 2D. The original finite element formulation for 3D large deformations shows excellent behaviour for sensitive geometries, large deformations, coarse meshes, bending dominated and stability problems without showing undesired effects such as locking or hourglassing, as long as the initial element shape resembles that of a rectangular parallelepiped. In the following, an extension of this element formulation for 2D plane strain is presented which has the same good properties also for the case of non-rectangular initial element shapes. Results of numerical tests are presented, that clearly show the advantages of the improved Cosserat point element compared to the standard displacement elements and the original version of the Cosserat point element. Copyright © 2007 John Wiley & Sons, Ltd.     

A new finite‐element formulation for electromechanical boundary value problems

In this paper, a new finite‐element formulation for the solution of electromechanical boundary value problems is presented. As opposed to the standard formulation that uses scalar electric potential as nodal variables, this new formulation implements a vector potential from which components of electric displacement are derived. For linear piezoelectric materials with positive definite material moduli, the resulting finite‐element stiffness matrix from the vector potential formulation is also positive definite. If the material is non‐linear in a fashion characteristic of ferroelectric materials, it is demonstrated that a straightforward iterative solution procedure is unstable for the standard scalar potential formulation, but stable for the new vector potential formulation. Finally, the method is used to compute fields around a crack tip in an idealized non‐linear ferroelectric material, and results are compared to an analytical solution. Copyright © 2002 John Wiley & Sons, Ltd.     

An element-based formulation for ES-FEM and FS-FEM models for implementation in standard solid mechanics finite element codes for 2D and 3D static analysis

Edge-based and face-based smoothed finite element methods (ES-FEM and FS-FEM, respectively) are modified versions of the finite element method allowing to achieve more accurate results and to reduce sensitivity to mesh distortion, at least for linear elements. These properties make the two methods very attractive. However, their implementation in a standard finite element code is nontrivial because it requires heavy and extensive modifications to the code architecture. In this article, we present an element-based formulation of ES-FEM and FS-FEM methods allowing to implement the two methods in a standard finite element code with no modifications to its architecture. Moreover, the element-based formulation permits to easily manage any type of element, especially in 3D models where, to the best of the authors' knowledge, only tetrahedral elements are used in FS-FEM applications found in the literature. Shape functions for non-simplex 3D elements are proposed in order to apply FS-FEM to any standard finite element.     

A modified scaled boundary finite element method for three‐dimensional dynamic soil‐structure interaction in layered soil

This paper is devoted to the analysis of elastodynamic problems in 3D‐layered systems which are unbounded in the horizontal direction. For this purpose, a finite element model of the near field is coupled to a scaled boundary finite element model (SBFEM) of the far field. The SBFEM is originally based on describing the geometry of a half‐space or full‐space domain by scaling the geometry of the near field / far field interface using a radial coordinate. A modified form of the SBFEM for waves in a 2D layer is also available. None of these existing formulations can be used to describe a 3D‐layered medium. In this paper, a modified SBFEM for the analysis of 3D‐layered continua is derived. Based on the use of a scaling line instead of a scaling centre, a suitable scaled boundary transformation is proposed. The derivation of the corresponding scaled boundary finite element (SBFE) equations in displacement and stiffness is presented in detail. The latter is a nonlinear differential equation with respect to the radial coordinate, which has to be solved numerically for each excitation frequency considered in the analysis. Various numerical examples demonstrate the accuracy of the new method and its correct implementation. These include rigid circular and square foundations embedded in or resting on the surface of layered homogeneous or inhomogeneous 3D soil deposits over rigid bedrock. Hysteretic damping is assumed in some cases. The dynamic stiffness coefficients calculated using the proposed method are compared with analytical solutions or existing highly accurate numerical results. Copyright © 2011 John Wiley & Sons, Ltd.     

On the enhancement of low‐order mixed finite element methods for the large deformation analysis of diffusion in solids

A stabilized scheme is developed for mixed finite element methods for strongly coupled diffusion problems in solids capable of large deformations. Enhanced assumed strain techniques are employed to cure spurious oscillation patterns of low‐order displacement/pressure mixed formulations in the incompressible limit for quadrilateral elements and brick elements. A study is presented that shows how hourglass instabilities resulting from geometrically nonlinear enhanced assumed strain methods have to be distinguished from pressure oscillation patterns due to the violation of the inf‐sup condition. Moreover, an element formulation is proposed that provides stable results with respect to both types of instabilities. Comparisons are drawn between material models for incompressible solids of Mooney–Rivlin type and models for standard diffusion in solids with incompressible matrices such as polymeric gels. Representative numerical examples underline the ability of the proposed element formulation to cure instabilities of low‐order mixed formulations. Copyright © 2015 John Wiley & Sons, Ltd.