Two-dimensional Linear Elastic Problems Research Articles

Fracture analysis in 2D plane strain problems for composite materials containing hard inclusions and voids using an extended consecutive-interpolation quadrilateral element

This paper investigates fracture mechanics in particle-reinforced composites by using the extended finite element method enhanced by the consecutive-interpolation quadrilateral element. These composite materials have discontinuous boundaries such as cracks, voids, holes, and soft inclusions. And the extended consecutive-interpolation quadrilateral element (XCQ4) is employed to model these boundaries in two-dimensional linear elastic deformation problems. XCQ4 combines the enrichment functions in the traditional extended finite element method with the consecutive interpolation on a 4-node quadrilateral element. This element uses both nodal values and averaged nodal gradients as interpolated conditions. In fracture analysis, the stress intensity factors (SIFs) are important parameters that must be defined. In this study, the values of SIFs at the crack tips are evaluated with the help of the interaction integrals approach. The obtained numerical results are compared with other reliable results showing high accuracy and convergence rate of the XCQ4 element.

Adaptive mesh refinements for analyses of 2D linear elasticity problems using the Kriging-based finite element method

Finite element analyses of irregular structures require adaptive mesh refinement to achieve more accurate results in an efficient manner. This is also true for a non-conventional finite element method with Kriging interpolation, called the Kriging-based finite element method (K-FEM). This paper presents a study of automatic adaptive meshing procedures for analyses of two-dimensional linear elasticity problems using the K-FEM. The Matlab Partial Differential Equation Toolbox was utilized for generating meshes with Delaunay triangulation. Three error indicators, namely, the strain energy error, the gradient of effective stresses, and the element-free Galerkin strain energy error, were employed for estimating the element errors. To find the most effective error indicator, the resulting total number of elements and configurations of the final meshes were compared. The results show that the resulting final meshes were affected by the initial mesh configurations, the refinement criteria, and the termination criteria. The gradient of effective stresses indicator was found to be the most effective error indicator for the K-FEM, as it can accurately estimate the element errors.

Virtual element method for modeling the deformation of multiphase composites

In this paper, we study applications of the virtual element method (VEM) for simulating the deformation of multiphase composites. The VEM is a Galerkin approach that is applicable to meshes that consist of arbitrarily-shaped polygonal and polyhedral (simple and nonsimple) elements. In the VEM, the basis functions are defined as the solution of a local elliptic partial differential equation, and are never explicitly computed in the implementation of the method. The stiffness matrix of each element is built by using the elliptic projection operator of the internal virtual work (bilinear form) and it consists of two terms: a consistency term that is exactly computed (linear patch test is satisfied) and a correction term (ensures stability) that is orthogonal to affine displacement fields and has the right scaling. The VEM simplifies mesh generation for a multiphase composite: a stiff inclusion can be modeled using a single polygonal or polyhedral element. Attributes of the virtual element approach are highlighted through comparisons with Voronoi-cell lattice models, which provide discrete representations of material structure. The comparisons involve a suite of two-dimensional linear elastic problems: patch test, axisymmetric circular inclusion problem, and the deformation of a three-phase composite. The simulations demonstrate the accuracy and flexibility of the virtual element method.

The method of fundamental solutions for two-dimensional elasticity problems based on the Airy stress function

This paper presents a new version of the method of fundamental solutions (MFS) for two-dimensional linear elasticity problems based on the stress function (Airy stress function), which is different from the MFS utilizing the fundamental solutions of displacement. The displacement compatibilities are derived by the single-valuedness of displacements in multiply-connected region. Based on the strain and rotation of the line element on the boundary, the displacement boundary conditions are deduced in forms of the Airy stress function. Furthermore, the displacement conditions exclude pure rigid body motion, and are in the derivative forms instead of the integral ones. The interpolation equations are also reconstructed by the single-valuedness conditions of the displacements and the approximate solutions in a multiply-connected region. The numerical examples show that the proposed method has good effect and keeps high accuracy in all kinds of boundary conditions.

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforcedviathe rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in theH(div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in theL2-norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.

The localized radial basis functions for parameterized level set based structural optimization

The current work is mainly about the implementation of the local radial basis functions (LRBFs) within the level set framework for structural optimization of two-dimensional linear elastic problems. The implicit representation of the structural geometry is accomplished through the LRBFs based level set function (LSF), and the geometry modification during the optimization process is carried out through an update of the LSF using a system of coupled ordinary differential equations (ODEs) instead of the Hamilton-Jacobi (HJ) type equation. This new implementation of LRBFs and level set method (LSM) allows essential topological changes automatically, i.e hole nucleation, hole merging with each other, and with the boundary to obtain the optimal structures. The coefficient matrix of the LRBFs is less sensitive to the values of shape parameter as well as the proposed method is mesh independent as explored during the numerical experiments performed. This new implementation is applied to distinct benchmark problems for minimum compliance with single and multiple load cases. The numerical experiments performed reveal significant insightful of the use of radial basis functions (RBFs) within the LSM framework, which may be considered as a step forward for further exploration of their intrinsic capabilities and improved performance for the solution of structural optimization problems.

An isogeometric-meshfree collocation approach for two-dimensional elastic fracture problems with contact loading

A strong form-based isogeometric-meshfree moving least-squares collocation (IMMLS-C) approach is developed for two-dimensional linear elastic fracture problems with contact loading. The IMMLS-C approach uses reproducing conditions to establish an equivalence between MLS shape functions and isogeometric basis functions. The advantages of this approach include the exact geometry representation, convenient crack modeling and flexible adaptive refinement. The IMMLS-C approach focuses on solving fracture problems based on strong formulations without requiring the numerical integration of Galerkin weak forms. Traction-free boundary conditions are enforced over a set of collocation points located on both sides of a crack surface. The displacement discontinuity along the crack surface is modeled by the visibility criterion and the singularity of near-tip stress fields is captured by adaptive mesh refinement without adding tip-enrichment functions, thereby reducing the degrees of freedom compared with the extended finite element method. Moreover, contact constraints are enforced by introducing a penalty algorithm to the strong formulations. The numerical results demonstrate that the adaptive refinement is able to achieve a high convergence rate at a low computational cost.

Boundary element analysis for elasticity problems using expanding element interpolation method

PurposeThis paper aims to propose a boundary element analysis of two-dimensional linear elasticity problems by a new expanding element interpolation method.Design/methodology/approachThe expanding element is made up based on a traditional discontinuous element by adding virtual nodes along the perimeter of the element. The internal nodes of the original discontinuous element are referred to as source nodes and its shape function as raw shape function. The shape functions of the expanding element constructed on both source nodes and virtual nodes are referred as fine shape functions. Boundary variables are interpolated by the fine shape functions, while the boundary integral equations are collocated on source nodes.FindingsThe expanding element inherits the advantages of both the continuous and discontinuous elements while overcomes their disadvantages. The polynomial order of fine shape functions of the expanding elements increases by two compared with their corresponding raw shape functions, while the expanding elements still keep independence to each other as the original discontinuous elements. This feature makes the expanding elements able to naturally and accurately interpolate both continuous and discontinuous fields.Originality/valueNumerical examples are presented to verify the proposed method. Results have demonstrated that the accuracy, efficiency and convergence rate of the expanding element method.

A polygonal XFEM with new numerical integration for linear elastic fracture mechanics

The extended finite element method (XFEM) has become a powerful and effective technique for modeling fracture problems without remeshing. In this paper, we introduce a novel and effective computational approach that is based on polygonal XFEM (named as PolyXFEM) for the analysis of two-dimensional (2D) linear elastic fracture mechanics problems. The PolyXFEM is equipped with a new numerical integration technique that uses the concept of Cartesian transformation method (CTM) over polygonal domains. The underlying idea of the CTM is to transform a domain integral into a boundary integral and a one-dimensional integral. This is computationally more efficient compared to the two-level mapping integration commonly used on polygons. Furthermore, the PolyXFEM is effective in modeling crack problems due to the local enrichment based on the partition of unity method. Our numerical results show that improvements in accuracy are reached thanks to the higher-order of the polygonal shape functions of the PolyXFEM. In addition, efficiency in computational time is achieved because of the usage of the CTM integration scheme, which appears to be quite suitable for polygonal domains. To demonstrate the accuracy and performance of the developed PolyXFEM approach, several numerical examples for 2D linear elastic fracture problems are considered, in which static stress intensity factors and crack propagation are investigated and compared with reference solutions. The convergence and mesh independence of the PolyXFEM for crack analysis is also analyzed.

An engineering perspective to the virtual element method and its interplay with the standard finite element method

In this manuscript we thoroughly study the behavior of the virtual element method (VEM) in the context of two-dimensional linear elasticity problems for an engineering audience versed in standard FEM. Through detailed convergence studies we show the accuracy and the convergence rates recovered by VEM, and we compare them to those obtained by the h- and p-versions of the finite element method (FEM). We also demonstrate the mixability between FEM and VEM; in particular, applications of VEM for coupling non-conforming discretizations and for local refinement are presented, showing higher versatility compared to FEM. Computer implementation aspects in displacement-based finite element codes are thoroughly explained, remarking on the main differences with respect to standard FEM.

Determination of Fiber-Structure for the Effective Reinforcement of Planar Composite-Material Constructions

To describe the behaviour of a composite material reinforced by fibers along curvilinear trajectories, a resolving system of differential equations has been obtained. I developed a structural model within the framework of the two-dimensional non-uniform linear elasticity problem. The reinforcement is performed on the basis of three approaches: according to the grid lines of the orthogonal coordinate system determined by the given conformal representation; according to trajectories isogonal to the given curves; according to the helical trajectories in the axisymmetric formulation of the problem. Particular solutions of the corresponding boundary problems have been obtained. Recommendations for the effective reinforcement of planar constructions have been presented.

Virtual element method for two-dimensional linear elasticity problem in mixed weakly symmetric formulation

We propose and analyze a virtual element method for two-dimensional linear elasticity problem in mixed weakly symmetric formulation, that is to say, stresses are not require to be symmetric, but only to satisfy a weaker condition based on Lagrange multipliers. The proposed method is well-posed, and the error bounds are shown to be uniform with the incompressibility parameter λ. Numerical tests confirm the convergence rate that is expected from the theory.

Mapped Displacement Discontinuity Method: Numerical Implementation and Analysis for Crack Problems

The displacement discontinuity method (DDM) is a kind of boundary element method aiming at modeling two-dimensional linear elastic crack problems. The singularity around the crack tip prevents the DDM from optimally converging when the basis functions are polynomials of first order or higher. To overcome this issue, enlightened by the mapped finite element method (FEM) proposed in Ref. [13], we present an optimally convergent mapped DDM in this work, called the mapped DDM (MDDM). It is essentially based on approximating a much smoother function obtained by reformulating the problem with an appropriate auxiliary map. Two numerical examples of crack problems are presented in comparison with the conventional DDM. The results show that the proposed method improves the accuracy of the DDM; moreover, it yields an optimal convergence rate for quadratic interpolating polynomials.

An Isogeometric Boundary Element Analysis of Two-Dimensional Linear Elasticity Problems

An Isogeometric Boundary Element Analysis of Two-Dimensional Linear Elasticity Problems

An Efficient Hybrid Implementation of MLPG Method

The computational implementation of moving least squares (MLS) shape functions is an important step to consider in some versions of the meshless local Petrov–Galerkin (MLPG) method for a variety of two-dimensional engineering problems. Here, the usage of conventional Gaussian quadrature in the MLPG may require an excessive number integration points to achieve acceptable accuracy. In addition, since for each integration point a search for nearby points contributing to the construction of the MLS shape functions is required, considerable increase in computational cost is often observed. Herein, an efficient hybrid implementation of CPU and GPU is proposed to accelerate the construction of MLS shape functions for MLPG. To this end, a new K-d-Tree (K-dimensional tree)-based data structure is introduced in order to accelerate the calculations involving computational geometry formulas such as MLS. The results are compared for implementations in CPU and [Formula: see text] using K-d-Tree with the traditional algorithm of brute force search of the neighboring points when dealing with two-dimensional linear elasticity problems. Finally, the performance gain in the numerical approximation is obtained when using the highlighted K-d-Tree.

A boundary element and level set based bi-directional evolutionary structural optimisation with a volume constraint

A new topology optimisation algorithm is implemented and presented for compliance minimisation of continuum structures using a volume preserving mechanism which effectively handles a volume constraint. The volume preserving mechanism is based on a unique combination of the level set method and a boundary element based bi-directional evolutionary structural optimisation approach using a bisectioning algorithm. The evolving structural geometry is implicitly represented with the level sets, efficiently handling complex topological shape changes, including holes merging with each other and with the boundary. Numerical results for two-dimensional linear elasticity problems suggest that the proposed adaptation provides smooth convergence of the objective function and a more robust, smoother geometrical description of the optimal design. Moreover, this new implementation allows for efficient material re-distribution within the design domain such that the objective function is minimised at constant volume. The proposed volume preserving mechanism can be easily extended to three-dimensional space.

Crack Tip Equation of Motion in Dynamic Gradient Damage Models

We propose in this contribution to investigate the link between the dynamic gradient damage model and the classical Griffith’s theory of dynamic fracture during the crack propagation phase. To achieve this main objective, we first rigorously reformulate two-dimensional linear elastic dynamic fracture problems using variational methods and shape derivative techniques. The classical equation of motion governing a smoothly propagating crack tip follows by considering variations of a space-time action integral. We then give a variationally consistent framework of the dynamic gradient damage model. Owing to the analogies between the variational ingredients of these two models and under some basic assumptions concerning the damage band structuration, one obtains a generalized Griffith criterion which governs the crack tip evolution within the non-local damage model. Assuming further that the internal length is small compared to the dimension of the body, the previous criterion leads to the classical Griffith’s law through a separation of scales between the outer linear elastic domain and the inner damage process zone.

A boundary element and level set based topology optimisation using sensitivity analysis

The structural topology optimisation method presented in this paper is based on the boundary element method, level set method and shape sensitivity analysis for two-dimensional linear elastic problems. The proposed method automatically nucleates holes within the design domain during the optimisation process using a topological derivative based hole insertion criterion. The level set method is used to provide an implicit description of the structural geometry, which is capable of automatically handling topological changes, i.e. holes merging with each other or with the boundary. During the optimisation process non-uniform rational b-splines are fitted through the zero level set contours, which links an implicit geometry representation to its structural model. In addition, this provides an optimal design in standard CAD format, and without intermediate material densities, which can be directly used in other design processes. The proposed optimisation method is tested against different benchmark examples and the optimal geometries generated are in close agreement those available in the literature of topology optimisation.

A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis

A three-node triangular element fitted to the numerical manifold method with continuous nodal stress called Trig3-CNS (NMM) element for accurately modeling two-dimensional linear elastic fracture problems is presented. By adopting two cover systems, namely, the mathematical cover and physical cover, the numerical manifold method (NMM) could easily solve continuous and discontinuous problems in a unified way. In contrast to the three-node triangular element (Trig3), the Trig3-CNS element has higher order of approximations, much better accuracy and continuous nodal stress. Moreover, it is free from the “linear dependence” which otherwise cripples many of the partition of unity based methods with high order approximations. The purpose of the present work is to synergize the advantages of both the recently developed Trig3-CNS element and the NMM to precisely model two-dimensional linear elastic fracture problems. A number of numerical examples indicate the accuracy and robustness of the present Trig3-CNS (NMM) element.

Improved non-singular method of fundamental solutions for two-dimensional isotropic elasticity problems with elastic/rigid inclusions or voids

In this work, an Improved Non-singular Method of Fundamental Solutions (INMFS) is developed for the solution of two-dimensional linear elasticity problems. The source points and field points are collocated on the physical boundary, while the conventional MFS requires a troublesome fictitious boundary outside the physical domain. In INMFS, the desingularization is, for complying with the displacement boundary conditions, achieved by replacement of the concentrated point sources by distributed sources over circular discs around the singularity, and for complying with the traction boundary conditions by assuming the balance of the forces. This procedure is much more efficient than the previously proposed procedure that involves two reference solutions and at the same time enables INMFS for solving problems with internal voids and inclusions. The method has been assessed by comparison with MFS, analytical solutions and previous desingularization technique. The method is easy to code, accurate, efficient, and straightforwardly extendable to three dimensions.