Low-order Elements Research Articles

A critical view on the use of Non‐Uniform Rational B‐Splines to improve geometry representation in enriched finite element methods

Enriched finite element methods have gained traction in recent years for modeling problems with material interfaces and cracks. By means of enrichment functions that incorporate a priori behavior about the solution, these methods decouple the finite element (FE) discretization from the geometric configuration of such discontinuities. Taking advantage of this greater flexibility, recent studies have proposed the adoption of Non-Uniform Rational B-Splines (NURBS) to preserve the interfaces' exact geometries throughout the analysis. In this article, we investigate NURBS-based geometries in the context of the Discontinuity-Enriched Finite Element Method (DE-FEM) based on linear field approximations. While optimal convergence is retained for problems with weak discontinuities without singularities, representing exact geometry via NURBS does not yield noticeable improvements when extracting stress intensity factors of cracked specimens. For low-order elements, we conclude that the benefits of exact geometry representation do not outweigh the increased complexity in formulation and implementation. The choice of linear FEs hinders the accuracy of the proposed formulation, suggesting that its full potential may only be unleashed by increasing the field representation order.

Vision-Based Human Detection Techniques: A Descriptive Review

Cameras are being used everywhere for the safety and security of citizens in different countries. Using a machine to detect humans in a photo or a video frame is a very complicated and challenging task. Various techniques have been developed for this purpose, which mainly rely on Artificial Intelligence. This article aims to provide a comprehensive review and analysis of the literatures from a descriptive perspective, which is its main differentiator from the existing survey papers in this area. Firstly, the vision-based human detection techniques and classifiers are elucidated in conjunction with the variants of feature extraction techniques. Secondly, various pros and cons of such techniques are discussed. Then, an investigation has been conducted and reported based on the state-of-the-art human detection descriptors (e.g. Log-Average Miss Rate and accuracy). Although techniques such as Viola-Jones and Speeded-Up Robust Features can detect objects in real-time and overcome Scale-Invariant Feature Transform (SIFT) limitations, they are still sensitive to illuminated conditions. Other techniques such as SIFT, Bag of Words, Orthogonal Moments, and Histogram of oriented Gradients provide other interesting benefits which include insensitivity to occlusion and clutters, simplicity, low-order element construction and invariance to illuminated conditions; nevertheless, they are computationally expensive and sensitive to image rotation. A meticulous review along similar lines revealed that the Deformable Part-based Model performs relatively better due to its ability to deal with particular pose variations and multiple views, occlusion handling (partial) and is application-free while its counterparts focus on only a single aspect. This article highlights and provides a brief description of each available data-sets for human detection research. Various use-cases of human detection systems are also elaborated. Finally, various conclusions are derived based on the conducted review followed by recommendations for future directions and possibilities to further improve the speed and accuracy of human detection systems.

Penny-shaped crack simulation with a single high order smooth boundary element

Based on Lagrange interpolation polynomials, the penny-shaped cracks placed on but not limited to flat surface are simulated with a single high order smooth boundary element in the present work. By taking advantage of geometrical features of circular shape such as the symmetry and periodicity, the smoothness within the element is realized by repeated use of real nodes for interpolation in both the radial and circumferential directions of the element using high order shape functions, so that the end node/line effects existing in conventional low order elements have been removed. In the application of the boundary element method (BEM) for crack problems, the hyper-singular integrals are treated with the aid of the technique of shape function manipulations. The near hyper-singular integrals are treated with the tangential distance transformations together with the subdivision techniques. The crack-related parameters including the crack opening displacements (COD), the stresses ahead of crack tip, the stress intensity factors (SIF) and the T-stresses are computed under various loads and compared with the analytical solutions in the numerical examples, showing the accuracy, efficiency and effectiveness of the proposed high order smooth crack element.

A Modified Strain-displacement Method for High Accuracy 8-Node Solid Finite Element

Higher-order three-dimensional solid elements are widely used for machine design and structural analyses. Although higher-order solid elements offer higher accuracy, the assembly routines often consume large amount of computational time and memory usage. In contrast, lower-order solid elements such as an 8-nod are simpler in programming implementation and consume less computational resources. However, they can produce problems of locking phenomena e.g. membrane and shear locking. Moreover, in a three-dimensional analysis using continuum solid elements, it is necessary to consider the stresses in the through-thickness direction, for example, in layered soil and foundation. This research aims to develop a modified strain-displacement finite element formulation that eliminates locking problems and generally applicable to both thick and thin threedimensional structures. The proposed formulation is based on the key concept of energy equivalence mapped between the global and natural curvilinear coordinates. The advantage of the proposed method is the ability to select a set of chosen strain functions that can be defined arbitrarily on the natural curvilinear coordinates.

A combined finite element–finite volume framework for phase-field fracture

Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. In particular, the use of linear finite elements to discretize both stress equilibrium and phase-field equations is widespread in the literature. However, the use of P1 Lagrange shape functions to model the phase-field is not optimal, as the latter contains cusps for fully developed cracks. These should in turn occur at locations corresponding to Gauss points of the associated FE model for the mechanics. Such a feature is challenging to reproduce accurately with low order elements, and element sizes must consequently be made very small relative to the phase-field regularization parameter in order to achieve convergence of results with respect to the mesh. In this paper, we combine a standard linear FE discretization of stress equilibrium with a cell-centered finite volume approximation of the phase-field evolution equation based on the two-point flux approximation constructed over the same simplex mesh. Compared to a pure FE formulation utilizing linear elements, the proposed framework results in looser restrictions on mesh refinement with respect to the phase-field length scale. This ability to employ coarser meshes relative to the traditional implementation allows for significant reductions on computational cost, as demonstrated in several numerical examples.

Incompatible Graded Finite Elements for Orthotropic Nonhomogeneous Media

Orthotropic materials or composites are challenging to model due to different elastic properties in two in-plane directions, which requires sufficient mesh refinement to attain desired accuracy. In this regard, an accurate and efficient incompatible graded element is developed for modeling orthotropic functionally graded materials. Properties of Graded finite elements such as Young’s moduli (E11, E22), shear modulus (G12) and Poisson’s ratio (v12) vary. A new incompatible graded element is developed using user subroutines in Abaqus and are compared to available exact solutions. In this study, performance (QM6) graded element is compared with lower-order and higher-order compatible elements (Q4 and Q8) as well as linear triangular (T3) and quadratic triangular (T6) elements with aid of several numerical examples. Additionally, orthotropic graded plate with carbon fibers properties as well as a circular disc with orthotropic properties under internal pressure is included. Furthermore, a curved beam with radially graded properties under the bending moment is studied. Additionally, the performance of various graded elements is compared with analytical solutions using Russell error.

Stable node-based smoothed extended finite element method for fracture analysis of structures

Based on low-order elements, a stable node-based smoothed extended finite element method (SNS-XFEM) is proposed for fracture analysis of structures in this study. For the proposed method, the problem domain is discretized using low-order elements, which can be easily generated for structures with complex shapes. The node-based smoothing domains are then generated to perform the strain smoothing technique, which can effectively avoid singular integration. The discontinuity caused by crack is modeled using enrichment functions and a stabilization term based on strain gradient is also taken into account to further improve the accuracy. Finally, some numerical cases are studied to fully investigate the performance of present method. The obtained results show that the proposed SNS-XFEM can perform much better than standard XFEM and NS-XFEM.

A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete $$H^2$$ norm are established for the corresponding finite element solutions. Error estimates in the $$L^2$$ norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence.

Anisotropic VMS solution of advection–diffusion problems by spectral approximation of sub-grid scales

In this article, we extend the Variational Multi-scale method with spectral approximation of the sub-scales to two-dimensional advection–diffusion problems. The spectral VMS method is cast for low-order elements as a standard VMS method with specific stabilized coefficients, that are anisotropic in the sense that they depend on two grid Péclet numbers, each associated to a component of the advection velocity. We compute the stabilized coefficients for grids of isosceles right triangles and right quadrilaterals, based upon the explicit computation of the eigen-pairs of the advection–diffusion operator with Dirichlet boundary conditions. To reduce the computing time, the stabilized coefficients are pre-computed at the nodes of a grid in an off-line step, and then interpolated by a fast procedure in the on-line computation.Finally, we present some numerical tests in order to compare our results with those provided by other stabilization coefficients, both isotropic and anisotropic ones. We consider tests for advection–diffusion equations with constant and anisotropic velocities, as well as tests in structured and unstructured meshes. We also test the method for the solution of Navier–Stokes equations. We observe a relevant accuracy gain for moderately large grid Péclet numbers for variable advection velocity.

Iterative solution of Helmholtz problem with high-order isogeometric analysis and finite element method at mid-range frequencies

Solving wave problems with isogeometric analysis has attracted a significant attention in the past few years. It is well known that keeping a fixed number of degrees of freedom per wavelength leads to an increased error as higher wavenumbers are considered. This behaviour often cited as the pollution error, improves significantly with isogeometric analysis when compared to the conventional finite element method. The improvement in handling pollution along with the ability to represent exact geometries has been the main reasons behind the attention that isogeometric analysis has received. Furthermore, using high order elements also presents major advantages over low order elements for this range of frequencies. However, it remains to be studied how iterative linear solvers, often necessary for solving high frequency wave problems, perform when using isogeometric analysis compared to the finite element method especially at high polynomial orders. This paper is one of the first studies in this direction.In this work we investigate the Generalised Minimal Residual method, a standard Krylov subspace iterative technique, for solving the linear system resulting form isogeometric analysis. Furthermore, we look into the use of some recently proposed preconditioners for Helmholtz problem, such as shifted Laplace or ILU with a complex shift preconditioners and how they perform with high order isogeometric analysis and finite element method. In general the results show improvement when using isogeometric analysis in terms of the number of iterations required for convergence compared to the finite element method for both preconditioned and non-preconditioned linear systems. We use eigenvalue spectra to understand this improvement.

An automated a priori knowledge-based p-adaptive three-dimensional finite element mesh improvement method for stress analysis of underground excavations with prismatic cross-sections

ABSTRACT With increasing modelling capabilities in geomechanics incorporating non-linear material behaviour, an increasing complexity of models representing excavation of tunnels can be analysed. Often, analysis is using finite elements, where problems size can be a controlling factor due to limited computational resources. An approach to analysing these problems is to optimise the discretisation of the differential equations by concentrating elements where solution accuracy counts the most. This paper describes a priori p-refinement method for mesh improvement for stress analysis of prismatic tunnels. The improvement results in a mesh with higher-order elements near regions of interest and lower-order elements elsewhere. The focus is on the automated insertion of transitional elements at the interface of the two regions, based on the zone of disturbance in the geomaterial. The method was incorporated into a finite element code, and its performance was tested on practical problems, where the global stiffness matrix size was reduced by 70%, and the calculation times were significantly shortened. The average difference compared to complex models without mesh improvement was less than 3%. It is anticipated that the presented method enables analysis of problems that would otherwise be beyond the current computing capabilities leading to safer and economical designs.

Analysis of Shear Wall Systems Using Plane Stress Elements

Finite element method takes advantage of robust methodology or analysis of different types of structures, particularly the tall buildings. Accordingly, a number of displacement and strain-based finite element approaches have been developed in various fields of civil engineering. However, many lower-order finite element methods still encounter analytical drawbacks dealing with analysis of tall buildings. The absence of an appropriate in-plane rotational stiffness in some of the lower-order finite element analysis methods and the existence of parasitic shear effects in the governing displacement functions are counted as the most effective factors which are likely to emerge within analysis of tall buildings. In this respect, finite element approach is presented in this paper, which is capable of incorporating the stiffness of all lateral load-resisting systems. Furthermore, as the horizontal strains are ignored in all elevation levels of the element, a uniform lateral displacement pattern is obtained for the proposed panel element. This panel-type element is a strain-based element including eight degrees of freedom which has been formulated according to the general behaviour of beam element. Consequently, in order to validate the accuracy and efficiency of the proposed approach, a number of numerical analyses are performed.

Recovery-based error estimator for the natural-convection problem based on penalized finite element method

Purpose This paper aims to proposes and analyzes a novel recovery-based posteriori error estimator for the stationary natural-convection problem based on penalized finite element method. Design/methodology/approach The optimal error estimates of the penalty FEM are established by using the lower-order finite element pair P1-P0-P1 which does not satisfy the discrete inf-sup condition. Besides, a new recovery type posteriori estimator in view of the gradient recovery and superconvergent theory to deal with the discontinuity of the gradient of numerical solution. Findings The stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations. Originality/value The provided reliability and efficiency analysis is shown that the true error can be effectively bounded by the recovery-based error estimator.

Mixed multiscale three-node triangular elements for incompressible elasticity

Purpose This study aims to introduce the hybrid finite element (FE) – meshfree method and multiscale variational principle into the traditional mixed FE formulation, leading to a stable mixed formulation for incompressible linear elasticity which circumvents the need to satisfy inf-sup condition. Design/methodology/approach Using the hybrid FE–meshfree method, the displacement and pressure are interpolated conveniently with the same order so that a continuous pressure field can be obtained with low-order elements. The multiscale variational principle is then introduced into the Galerkin form to obtain stable and convergent results. Findings The present method is capable of overcoming volume locking and does not exhibit unphysical oscillations near the incompressible limit. Moreover, there are no extra unknowns introduced in the present method because the fine-scale unknowns are eliminated using the static condensation technique, and there is no need to evaluate any user-defined stability parameter as the classical stabilization methods do. The shape functions constructed in the present model possess continuous derivatives at nodes, which gives a continuous and more precise stress field with no need of an additional smooth process. The shape functions in the present model also possess the Kronecker delta property, so that it is convenient to impose essential boundary conditions. Originality/value The proposed model can be implemented easily. Its convergence rates and accuracy in displacement, energy and pressure are even comparable to those of second-order mixed elements.

Parallel two-step finite element algorithm based on fully overlapping domain decomposition for the time-dependent natural convection problem

Purpose This paper aims to propose two parallel two-step finite element algorithms based on fully overlapping domain decomposition for solving the 2D/3D time-dependent natural convection problem. Design/methodology/approach The first-order implicit Euler formula and second-order Crank–Nicolson formula are used to time discretization respectively. Each processor of the algorithms computes a stabilized solution in its own global composite mesh in parallel. These algorithms compute a nonlinear system for the velocity, pressure and temperature based on a lower-order element pair (P1b-P1-P1) and solve a linear approximation based on a higher-order element pair (P2-P1-P2) on the same mesh, which shows that the new algorithms have the same convergence rate as the two-step finite element methods. What is more, the stability analysis of the proposed algorithms is derived. Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Findings Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Originality/value The novel parallel two-step algorithms for incompressible natural convection problem are proposed. The rigorous analysis of the stability is given for the proposed parallel two-step algorithms. Extensive 2D/3D numerical tests demonstrate that the parallel two-step algorithms can deal with the incompressible natural convection problem for high Rayleigh number well.

Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations

Purpose The purpose of this paper is to propose a local parallel finite element algorithm based on fully overlapping domain decomposition technique to solve the incompressible magnetohydrodynamic equations. Design/methodology/approach The algorithm uses a lower-order element pair to compute an initial approximation by the Oseen-type iteration and uses a higher-order element pair to solve a linear system in each processor. Findings Besides, the convergence analysis of local parallel finite element algorithm is given. Finally, numerical experiments are presented to verify the efficiency of the proposed algorithm. Originality/value Compared with the numerical solution of the common two-step method, this method is easy to realize and can produce a more accurate solution. And, this approach is executed in parallel, so it saves a lot of computational time.

Large elastic deformation of micromorphic shells. Part II. Isogeometric analysis

In Part I of this study, a variational formulation was presented for the large elastic deformation problem of micromorphic shells. Using the novel matrix-vector format presented for the kinematic model, constitutive relations, and energy functions, an isogeometric analysis (IGA)-based solution strategy is developed, which appropriately estimates the macro- and micro-deformation field components. Due to the capability of constructing exact geometries and the powerful mesh refinement tools, IGA can be successfully applied to solve the equilibrium equations with dominant nonlinear terms. It is known that different types of locking phenomena take place in the conventional finite element analysis of thin shells based on low-order elements. Non-standard finite element models with mixed interpolation schemes and additional degrees of freedom (DOFs) or the ones used the high-order Lagrangian shell elements which require high computational costs, are the available solutions to tackle locking issues. The present 16-DOFs IGA is found to be efficient because of possessing a good rate of convergence and providing locking-free stable responses for micromorphic shells. Such a conclusion is found from several comparative studies with available data in the well-known macro-scale benchmark problems based on the classical elasticity as well as the corresponding numerical examples studied in nano-scale beam-, plate-, cylindrical shell- and spherical shell-type structures on the basis of the micromorphic continuum theory.

Improving the bending response of four-node quadrilateral and eight-node hexahedral elements for Cosserat elasticity problems

PurposePoor bending response is a major shortcoming of lower-order elements due to excessive representation of shear stress/strain field. Advanced finite element (FE) formulations for classical elasticity enhance the bending response by either nullifying or filtering some of the symmetric shear stress/strain modes. Nevertheless, the stress/strain field in Cosserat elasticity is asymmetric; consequently any attempt to nullify or filter the anti-symmetric shear stress/strain modes may lead to failure in the constant couple-stress patch test where the anti-symmetric shear stress/strain field is linear. This paper aims at enhancing the bending response of lower-order elements for Cosserat elasticity problems.Design/methodology/approachA four-node quadrilateral and an eight-node hexahedron are formulated by hybrid-stress approach. The symmetric stress is assumed as those of Pian and Sumihara and Pian and Tong. The anti-symmetric stress components are first assumed to be completely linear in order to pass the constant couple-stress patch test. The linear modes are then constrained with respect to the prescribed body-couple via the equilibrium conditions.FindingsNumerical tests show that the hybrid elements can strictly pass the constant couple-stress patch test and are markedly more accurate than the conventional elements as well as the incompatible elements for bending problems in Cosserat elasticity.Originality/valueThis paper proposes a hybrid FE formulation to improve the bending response of four-node quadrilateral and eight-node hexahedral elements for Cosserat elasticity problems without compromising the constant couple-stress patch test.

Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -- in the sense that GMRES converges in a wavenumber-independent number of iterations -- for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.