A Comparative Study of Polygonal Element Formulations for Linear Elasticity

A combined virtual element method and the scaled boundary finite element method for linear elastic fracture mechanics

Convergence analysis of virtual element method for the electric interface model on polygonal meshes with small edges

ABSTRACTThis work explores the application of isogeometric scaled boundary method in the two-dimensional thermoelastic problems of irregular geometry. The proposed method inherits the advantages of both isogeometric analysis and scaled boundary finite element method and overcomes their respective disadvantages. In the proposed approach, the boundaries of the problem domain are discretized with nonuniform rational B-splines (NURBS) basis functions, while the temperature distributions inside the domain are represented by a sequence of power functions in terms of radial coordinate within the framework of scaled boundary finite element method. The resulting solution of the stress in radial direction can be computed analytically for the temperature changes. The construction of tensor product structure is circumvented for the two-dimensional problems as only the boundary information of the problem domain is required. Hence, the flexibility to represent the complex geometry can be significantly improved in the proposed method. Numerical examples are presented to validate the performance of the proposed method where it is seen that superior accuracy, efficiency, and convergence behavior can be achieved over the conventional scaled boundary finite element method.

A high-order finite element technique with automatic treatment of stress singularities by semi-analytical enrichment

The scaled boundary finite element method (SBFEM) allows mixed-mode stress intensity factors (SIFs) to be accurately determined directly from the definition because the stress solutions in the radial direction are analytical. This makes the SBFEM superior to the traditional finite element method and boundary element method when calculating SIFs, since the other approaches generally need special crack-tip treatment, such as refining the crack-tip mesh or using singular elements. In addition, anisotropic material behaviour can be handled with ease by the SBFEM. These advantages are exploited in this study, in which the new frequency-domain approach combining the Frobenius solution procedure [1] and the fast Fourier transform technique, as developed and validated in the accompanying paper [2], is applied to model transcient dynamic fracture problems. Two benchmark problems with isotropic and anistropic material behaviour are modeled with a small number of degrees of freedom. Excellent agreement is observed between the results of this study and those in published literature. The new frequency-domain approach thus provides a competitive alternative to model dynamic fracture problems with high accuracy and efficiency.

The scaled boundary finite element method (SBFEM) is a semianalytical computational scheme based on the characteristics of the finite element method (FEM) and boundary element method that combines their respective advantages. In this paper, the SBFEM and polygonal mesh technique are integrated into a new approach to solve steady-state and transient seepage problems. The proposed method is implemented in Abaqus employing a user-defined element (UEL). A detailed implementation of the procedure is presented in which the UEL element is defined, the internal variables RHS and AMATRX are updated, and the stiffness/mass matrix is solved using eigenvalue decomposition. Several benchmark problems are solved to verify the proposed implementation. The results show that the polygonal element of the polygonal SBFEM (PSBFEM) is more accurate than the standard FEM element of the same element size. For transient problems, the results for the PSBFEM and FEM are in excellent agreement. Hence, the proposed method is robust and accurate for solving steady-state and transient seepage problems. The developed UEL source code and the associated input files can be downloaded from GitHub.

Dynamic fracture simulations using the scaled boundary finite element method on hybrid polygon–quadtree meshes

The scaled boundary finite element method (SBFEM) is a semi-analytical method, 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 an efficient high-order polynomial element in the SBFEM to form the approximation in the circumferential direction. The governing equations are formulated from the classical linear elasticity theory via the SBFEM technique. The scaled boundary finite element equations are formulated within a general framework integrating the influence of the distributed body source, mixed boundary conditions, contributions the side face with either prescribed surface load or prescribed displacement. The position of scaling center is considered for modeling problem. The proposed method is evaluated by solving two-dimensional linear problem. A selected set of results is reported to demonstrate the accuracy and convergence of the proposed method for solving problems in general boundary conditions.

Seismic analysis of post-tensioned concrete gravity dams using scaled boundary finite elements implemented as ABAQUS UEL

SummaryThis article proposes an efficient approach for solving three‐dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)‐based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.

SummaryIn this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of ‐continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four‐node quadrilateral and eight‐node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright © 2015 John Wiley & Sons, Ltd.

We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.

The scaled boundary finite element method (SBFEM) has been used in many fields of engineering to solve the governing equations in bounded and unbounded 2D as well as 3D domains. In solid mechanics, the semi‐analytical solution strategy of the SBFE formulation (numerical in circumferential direction, analytical in radial direction) is based on the assumption of linear elastic material behavior and only small geometrical changes. However, a large group of materials (e.g. rubber) shows geometrical and physical nonlinearity at mechanical loading. In this contribution, the extension of the SBFEM to geometrical and physical nonlinearity is examined. A plane finite element is developed which uses the concept of shape functions constructed by the SBFEM in the framework of a nonlinear finite element analysis. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A blocked Hamiltonian Schur decomposition is herein proposed for the solution process of the Scaled Boundary Finite Element Method (SBFEM), which is demonstrated to comprise a robust simulation tool for Linear Elastic Fracture Mechanics (LEFM) problems. By maintaining Hamiltonian symmetry increased accuracy is achieved, resulting in higher rates of convergence and reduced computational toll, while the former need for adoption of a stabilizing parameter and, inevitably user-supervision, is alleviated. The method is further enhanced via adoption of superconvergent patch recovery theory in the formulation of the stress intensity factors. It is shown that in doing so, superconvergence, and in select cases ultraconvergence, is succeeded in the Stress Intensity Factors (SIFs) calculation. Based on these findings, a novel error estimator for the stress intensity factors within the context of SBFEM is proposed. To investigate and assess the performance of SBFEM in the context of linear elastic fracture mechanics, the method is contrasted against the Finite Element Method (FEM) and the eXtended Finite Element Method (XFEM) variants. The comparison, carried out in terms of computational toll and accuracy for a number of applications, reveals SBFEM as a highly performant method.

Scaled boundary finite element based two-level learning approach for structural flaw identification