Linear Elastic Problems Research Articles

The generalized finite difference method for the inverse Cauchy problem in two-dimensional isotropic linear elasticity

In this study, the generalized finite difference method (GFDM) was used to stably and accurately solve two-dimensional (2D) inverse Cauchy problems in linear elasticity by using the Navier equations. In Cauchy problems, overdetermined boundary conditions are imposed on parts of the boundary, whereas there are missing boundary conditions on some parts of the boundary. In Cauchy problems, conventional numerical methods generally generate highly ill-conditioned matrices and thus provide unstable numerical solutions. Moreover, even if a slight noise is added in the boundary conditions, numerical errors are evidently magnified. The GFDM, one of the most promising meshless methods and an extension of the classical finite difference method, can avoid time-consuming tasks of mesh generation and numerical quadrature. The GFDM was applied in this study to stably solve the 2D Cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme. Moreover, the stability of the presented scheme for inverse Cauchy problems was proved by adding noise into the boundary conditions.

Weak and strong from meshless methods for linear elastic problem under fretting contact conditions

Abstract We present numerical computation of stresses under fretting fatigue conditions derived from closed form expressions. The Navier-Cauchy equations, that govern the problem, are solved with strong and weak form meshless numerical methods. The results are compared to the solution obtained from well-established commercial package ABAQUS, which is based on finite element method (FEM). The results show that the weak form meshless solution exhibits similar behavior as the FEM solution, while, in this particular case, strong form meshless solution performs better in capturing the peak in the surface stress. This is of particular interest in fretting fatigue, since it directly influences crack initiation. The results are presented in terms of von Mises stress contour plots, surface stress profiles, and the convergence plots for all three methods involved in the study.

A low-order nonconforming method for linear elasticity on general meshes

In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method of Di Pietro and Ern (2015), that requires the use of polynomials of degree k≥1 for stability. Specifically, we show that coercivity can be recovered for k=0 by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfilment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy- and the L2-norms of the error, that are shown to convergence, for smooth solutions, as h and h2, respectively (here, h denotes the meshsize). A thorough numerical validation on a complete panel of two- and three-dimensional test cases is provided.

Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

SummaryThis paper presents a new stress recovery technique for the generalized/extended finite element method (G/XFEM) and for the stable generalized FEM (SGFEM). The recovery procedure is based on a locally weighted L2 projection of raw stresses over element patches; the set of elements sharing a node. Such projection leads to a block‐diagonal system of equations for the recovered stresses. The recovery procedure can be used with GFEM and SGFEM approximations based on any choice of elements and enrichment functions. Here, the focus is on low‐order 2D approximations for linear elastic fracture problems. A procedure for computing recovered stresses at re‐entrant corners of any internal angle is also presented. The proposed stress recovery technique is used to define a Zienkiewicz‐Zhu (ZZ) a posteriori error estimator for the G/XFEM and the SGFEM. The accuracy, computational cost, and convergence rate of recovered stresses together with the quality of the ZZ estimator, including its effectivity index, are demonstrated in problems with smooth and singular solutions.

Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods

Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for example, steady-state heat conduction in random heterogeneous materials and stochastic elastostatics problems in linear elasticity. The adaptive process involves generating a sequence of surrogate models defined on a partition of the solution domain into regular subdomains and then, based on estimates of the error in the QoIs, assigning homogenized effective material properties to some subdomains and full random fine scale properties to others, to control the error so as to meet a preset tolerance. New model-based Multilevel Monte Carlo (mbMLMC) methods are presented that exploit the adaptive sequencing and are designed to reduce variances and thereby accelerate convergence of Monte Carlo sampling. Estimates of cost and mean squared error of the method are presented. The results of several numerical experiments are discussed that confirm that substantial saving in computer costs can be realized through the use of controlled surrogate models and the associated mbMLMC algorithms.

HDG-NEFEM for two dimensional linear elasticity

This paper proposes a new methodology for the solution of two dimensional linear elastic problems in domains with curved boundaries. The proposed method exploits the advantages of the hybridisable discontinuous Galerkin method to obtain an accurate approximation of both the displacement and the stress fields by solving a global problem that only involves the displacement field on the element edges as unknown. In addition, the methodology incorporates the exact boundary representation of the domain by means of the so-called NURBS-enhanced finite element method. Numerical examples are used to illustrate the three main advantages of the proposed method, namely the reproducibility of polynomials in domains with curved boundaries, the super-convergence of the solution even for linear approximation and the effectiveness and reliability of degree adaptive processes driven by displacement or stresses.

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size linear systems, especially when accurate results are sought for derived variables, like stress or deformation fields. Such a task represents the most time-consuming kernel, and motivates the development of robust and efficient linear solvers for these applications. On the one hand, direct solvers are robust and easy to use, but their computational complexity in the best scenario is superlinear, which limits applicability according to the problem size. On the other hand, iterative solvers, in particular those based on algebraic multigrid (AMG) preconditioners, can reach up to linear complexity, but require more knowledge from the user for an efficient setup, and convergence is not always guaranteed, especially in ill-conditioned problems. In this work, we present a novel AMG method specifically tailored for ill-conditioned structural problems. It is characterized by an adaptive factored sparse approximate inverse (aFSAI) method as smoother, an improved least-squared based prolongation (DPLS) and a method for uncovering the near-null space that takes advantage of an already existing approximation. The resulting linear solver has been applied in the solution of challenging linear systems arising from real-world linear elastic structural problems. Numerical experiments prove the efficiency and robustness of the method and show how, in several cases, the proposed algorithm outperforms state-of-the-art AMG linear solvers. Even more important, the results show how the proposed method gives good results even assuming a default setup, making it fully adoptable also for non-expert users and commercial software.

The families of nonconforming mixed finite elements for linear elasticity on simplex grids

We present a new family of nonconforming tetrahedral elements and a new family of nonconforming triangular elements for the stress-displacement system of linear elasticity problem. The local degrees of freedom of stress field only contain the normal moments on faces (sides) of element and the moments on element. The shape function spaces are simple, local, explicitly represented, and affine-equivalent. We also present two families simplified lowest-order finite elements by using the rigid motion model, and demonstrate our theory numerically in 2D area.

Membrane analogy for multi-material bars under torsion

Prandtl's membrane analogy for the torsion problem of prismatic homogeneous bars is extended to multi-material cross sections. The linear elastic problem is governed by the same equations describing the deformation of an inflated membrane, differently tensioned in regions that correspond to the domains hosting different materials in the bar cross section, in a way proportional to the inverse of the material shear modulus. Multi-connected cross sections correspond to materials with vanishing stiffness inside the holes, implying infinite tension in the corresponding portions of the membrane. To define the interface constrains that allow to apply such a state of prestress to the membrane, a physical apparatus is proposed, which can be numerically modelled with a two-dimensional mesh implementable in commercial finite-element model codes. This approach presents noteworthy advantages with respect to the three-dimensional modelling of the twisted bar.

IFEM benchmark problems for solid elements

Structural health monitoring (SHM) is a growing field of research, as it has the potential to simultaneously improve the reliability of structures and reduce their maintenance cost. SHM requires accurate stress and strain information, preferably for the entire structure. Unfortunately, it is often infeasible to instrument every part of the structure, making it necessary to estimate the stress and strain fields based on data from a limited number of sensors. One promising technique for making this estimate is the inverse finite element method (iFEM), which can be applied to any combination of geometry and loading conditions. In addition, it can also process several different types of sensor data. In this study, benchmark problems based on the MacNeal and Harder linear elastic problem set for FEM algorithms were extended to test the accuracy of iFEM algorithms. As the benchmarks use linear elastic materials, small displacements and strains, the iFEM implementation was also limited to these conditions. Accurate iFEM estimates can be obtained for the benchmark problems for which accurate FEM solutions can be obtained with solid elements, specifically 3-dimensional 20 node hexahedral elements with reduced integration (C3D20R), based on either displacement sensors, strain sensors, or both combined, and provided that a sufficient number of sensors is used. The iFEM algorithms generally produce more accurate estimates of displacements than of strains. The addition of Tikhonov regularization does not result in a significant increase in accuracy for either the displacement or strain distribution estimates and can even deteriorate the results in certain cases.

Symplectic Model Order Reduction with Non-Orthonormal Bases

Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.

Comparison of two projection methods for the solution of frictional contact problems

Frictional contact problems in linear elasticity are considered in this paper. The contact constraint is imposed in the weak sense using the fixed point method, which leads to a variational equation problem. For solving such a nonlinear variational problem, we study two projection methods using different self-adaptive rules. Based on the self-adaptive projection method, we propose a modified self-adaptive rule which is more effective to update the parameter. The methods can be implemented easily in conjunction with the boundary element method for the solution. Numerical experiments are reported to illustrate theoretical results.

Coupling of FEM and ordinary state-based peridynamics for brittle failure analysis in 3D

This paper presents a method to couple FEM meshes with ordinary state-based peridynamic grids. It generates a computational tool that is as efficient as computer methods based on classical continuum mechanics and as flexible as peridynamics in dealing with crack propagation. Moreover, it removes most of the problems due to the surface effects typical of nonlocal methods such as OSB-PD. The adaptive dynamic relaxation approach is used to solve linear elastic and crack propagation problems. The method is successfully applied to a complex crack propagation case in 3 D and is able to accurately reproduce the shape of the experimentally observed crack.

Rapid non-linear finite element analysis of continuous and discontinuous Galerkin methods in MATLAB

MATLAB is adept at the development of concise Finite Element (FE) routines, however it is commonly perceived to be too inefficient for high fidelity analysis. This paper aims to challenge this preconception by presenting two optimised FE codes for both continuous Galerkin (CG) and discontinuous Galerkin (DG) methods. Although this has previously been achieved for linear-elastic problems, no such optimised MATLAB script currently exists, which includes the effects of material non-linearity. To incorporate these elasto-plastic effects, the externally applied load is split into a discrete number of loadsteps. Equilibrium is determined at each loadstep between the externally applied load and the arising internal forces using the Newton–Raphson method. The optimisation of the scripts is primarily achieved using vectorised blocking algorithms, which minimise RAM-to-cache overheads and maximise cache reuse.The optimised codes yielded maximum speed gains of ×25.7 and ×10.1 when compared to the corresponding unoptimised scripts, for CG and DG respectively. It was identified that with increasing refinement of the mesh, the solver time begins to dominate the overall simulation time. This bottleneck has a greater disadvantage on the DG code, predominantly due the asymmetric nature of the global stiffness matrix. The implementation of an efficient solver would see further improvement to the overall run times, particularly for large problems.

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.

A two-step unconditionally stable explicit method with controllable numerical dissipations

A family of unconditionally stable direct integration algorithm with controllable numerical dissipations is proposed. The numerical properties of the new algorithms are controlled by three parameters α, β and γ. By the consistent and stability analysis, the proposed algorithms achieve the second-order accuracy and are unconditionally stable under the condition that α ≥ -0.5, β ≤ 0.5 and γ ≥ -(1+α)/2. Compared with other unconditionally stable algorithms, such as Chang’s algorithms and CR algorithm, the proposed algorithms are found to be superior in terms of the controllable numerical damping ratios. The unconditional stability and numerical damping ratios of the proposed algorithms are examined by three numerical examples. The results demonstrate that the proposed algorithms have a superior performance and can be used expediently in solving linear elastic dynamics problems.

Hybridized dual-mixed hp-finite element model for shells of revolution

A locking-free hp-version finite element is presented for linear elasticity problems of thin shells of revolution. The constructed hp-finite element is based on a hybridized dual-mixed variational formulation. The related theoretical model does not rely on the standard hypotheses used in the Naghdi- and Koiter shell theories, thus the unmodified three-dimensional constitutive equation can be applied. Nevertheless, since employing its inverse form, the hp-shell finite element is incompressibility locking-free. Besides, neither the thickness variation nor the membrane stress normal to the shell mid-surface is not eliminated from the shell formulation, thus it can be extended to much complicated (contact) problems of more complex (composite), extremely thin and moderately thick shell structures. The new hp-shell finite element is tested through some representative mixed and pure boundary value problems, namely bending- and membrane dominated situations, for singly and doubly curved shells of negative and positive Gaussian curvature. From the convergence behavior of the relative errors it follows that the developed hp-version shell finite element is insensitive to the decrease of the thickness value, i.e., membrane and shear locking-free, providing excellent numerical results not only for the displacement- but also for the stresses computations.

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.

Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity

SummaryThis paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic body is almost incompressible. Several alternative a posteriori error estimators for the energy norm of the finite element error are introduced and analysed. We establish upper and lower bounds for the energy error in terms of these error estimators and show that the constants in the bounds are independent of the Lamé coefficients: The proposed estimators are robust in the incompressible limit. We also consider the requirement for pressure stabilisation when using lowest‐order conforming approximation. Computational results are presented that validate the theoretical estimates. The software used to generate these results is available online.

Structural shape optimization with meshless method and swarm-intelligence based optimization

This paper presents a distinctive numerical approach for shape optimization by coupling meshless method with stochastic swarm intelligence based optimization technique for two dimensional linear elastic problems. Element free Galerkin method has been used here for structural analysis to circumvent frequently encountered issues with traditional grid-based technique like FEM in shape optimization such as heavy reliance on quality mesh for accurate solutions needing remeshing due to initial mesh distortion in case of large shape changes, discontinuous secondary field variables across element boundaries needing post-processing techniques and mesh optimization to minimize computational errors. Another distinguishing feature of present work is deployment of gradient-free particle swarm optimization technique for obtaining near optimal solution in shape optimization which eradicates computational efforts and errors associated with sensitivity computation. In this work, for design boundary representation Akima spline has been used due to its better stability against outlier points during shape optimization which generates natural looking boundaries. The performance of proposed technique is validated through numerical examples of shape optimization with behavior constraints on displacement and stress. To demonstrate effectiveness of present technique, results obtained through the proposed technique are compared with other techniques of past literature. To ensure acceptable levels of solution accuracy during shape optimization, h-refinement for initial problem geometry has been carried out as some shapes generated during the optimization process may have low field node density.