Classical FEM Research Articles

A SBFEM formula for the mixed-order hexahedron interpolation based on serendipity elements

The hexahedron elements have been widely used in theoretical research and engineering applications because of their simple formulation and fine analytical performance. In this paper, a flexible mixed-order hexahedron interpolation is proposed based on the SBFEM theory, which is summarized as follows: (1) The interpolation functions are constructed for boundary surfaces by introducing the “Serendipity element”, which allows for a combination of linear and quadratic interpolation; (2) Two approaches for order conversion in hexahedron elements have been applied and investigated; (3) The accuracy and applicability of the proposed method are verified using several classical examples and engineering practice. In this way, the proposed method is capable of handling abundant patterns of different order combinations, thus the analysis performance of hexahedron elements has been improved. The precision of the proposed method is demonstrated by comparing it with the theoretical solution and the classical isoparametric FEM. By comparison, the computational efficiency can be improved by approximately 40–55 % with satisfactory accuracy. In general, the proposed method offers a new alternative channel for simulating particular problems, such as bending and stress concentration.

Method of matched sections as a beam-like approach for plate analysis

A new numerical method in application to the plate problem is suggested. It starts from consideration of the rectangular elements, each operating by 6 beam-like parameters: four bending parameters (displacement, angle of rotation, bending moment, transverse force) and two rotation parameters (angle of rotation and twisting moment). So, contrary to the classical FEM approach, the conjugation between the adjacent elements occurs between the adjacent sections rather than in polygon vertexes (nodes), and this gives the name to the method – method of matched sections, MMS. Technically, 6 left-side and 6 lower-side parameters are 12 inlet parameters, while 6 upper-side and right-side parameters are 12 outlet parameters; each element is defined by 24 parameters. Outlet parameters are related to inlet ones by 12 matrix relations, which are derived from the approximate solution of each differential equation (equilibrium, physical, and geometrical equations) of the plate theory. The matrix relation between inlet and outlet parameters is written in a form suitable for applying the transfer matrix method. The numerical examples for the thin and Mindlin plates show the high efficiency and accuracy of the method. In particular, the results for the Mindlin plate for minimal thickness give the same results as the thin plate (no shear locking); the method is insensitive to cases when, for adjacent elements, the ratio of dimensions or ratio of rigidities (elastic constants) differ by several orders of magnitude. The application of the method for the thermal as well as for the vibration problem is considered. The possible extension of the method to any curved geometry is discussed, too.

Structure Dynamic Analysis of C-9 Parachute Using Absolute Nodal Coordinate Formulation Thin Shell Element

In parachute fluid structure interaction (FSI) simulation, traditional parachute structure discrete methods including classic finite element method and mass-spring-damp method suffers from its large computational cost and deficiency in describing material property. This paper develops a generic shell element based on absolute nodal coordinate formulation (ANCF) to model parachute canopy to reduce computation resources and retain membrane strain/stress analyzing capability. We conduct multiple validation cases of ANCF plate and shell element and compare with current literature, in which large deformation configuration and stress distribution have been verified. A 1/4 C-9 parachute canopy inflation process by uniform pressure is simulated by discretizing into 7 ANCF shell and 84 shell element, for which the number of element discretization reduces to 1/10 than classic FEM and computational cost is considerably reduced using band sparse matrix solving technique.

Method of matched sections in application to thin-walled and Mindlin rectangular plates

The paper elaborates the principally new variant of finite element method in application to plate problem. It differs from classical FEM approach by, at least, three points. First, it uses the strong differential formulation rather than the weak one and suppose the approximate analytical solution of all differential equations. Second, it explicitly uses all geometrical and physical parameters in the procedure of solution, rather than some chosen ones, for example, displacement and angles of rotation as usually done in FEM formulation. Third, the conjugation between adjacent elements occurs between the adjacent sections rather than in polygon vertexes. These conditions require the continuity of displacements, angles, moments and forces. Each side of rectangular elements is characterized by 6 main parameters, so, at whole there are 24 parameters for each rectangular element. The right and upper sides’ parameters are considered as output ones, and they are related with lower and left sides ones by matrix equations, which allows to apply transfer matrix method for the compilation of the resulting system of equations for the whole plate. The numerical examples for the thin-walled and Mindlin plates show the high efficiency and accuracy of the method.

A 3D frequency domain finite element formulation for solving the wave equation in the presence of rotating obstacles

A frequency domain numerical method aimed at solving the propagation of sound waves in a three-dimensional domain enclosing rotating obstacles is presented. It relies on a domain decomposition whereby rotating parts are all embedded in a cylindrically-shaped domain. Following the classical Arbitrary Lagrangian Eulerian formalism, the governing equation, which is written in the local rotating frame, takes the form of a convected wave equation. The transmission conditions at the interface between the fixed and rotating domains is accomplished via the Frequency Scattering Boundary Conditions which, after classical FEM discretization, give rise to a series of coupled problems associated with a discrete set of frequencies. The performances of the method are demonstrated through several test cases involving rotating sources and/or obstacles inserted in circular and rectangular ducts.

On the simultaneous solution of structural membranes on all level sets within a bulk domain

A mechanical model and numerical method for structural membranes implied by all isosurfaces of a level-set function in a three-dimensional bulk domain are proposed. The mechanical model covers large displacements in the context of the finite strain theory and is formulated based on the tangential differential calculus. Alongside curved two-dimensional membranes embedded in three dimensions, also the simpler case of curved ropes (cables) in two-dimensional bulk domains is covered. The implicit geometries (shapes) are implied by the level sets and the boundaries of the structures are given by the intersection of the level sets with the boundary of the bulk domain. For the numerical analysis, the bulk domain is discretized using a background mesh composed by (higher-order) elements with the dimensionality of the embedding space. The elements are by no means aligned to the level sets, i.e., the geometries of the structures, which resembles a fictitious domain method, most importantly the Trace FEM. The proposed numerical method is a hybrid of the classical FEM and fictitious domain methods which may be labelled as “Bulk Trace FEM”. Numerical studies confirm higher-order convergence rates and the potential for new material models with continuously embedded sub-structures in bulk domains.

A hybrid analytical-FEM 3D approach including wear effects to simulate fretting fatigue endurance: Application to steel wires in crossed contact

Fretting fatigue causes failures in steel wire ropes. A crossed wires set-up was used to investigate the influence of displacement amplitude on fretting fatigue damage. The analysis confirms previous literature results, showing a beneficial effect of wear in gross slip regime, which extends the contact area and thus decreases the contact stresses. The computation of such combined cracking and wear processes is very expensive using a classical FEM strategy, particularly for 3D contact geometries. To palliate such limitations, a hybrid analytical-numerical strategy was developped. It consists in estimating pressure and shear stress surface fields using analytical formulations, then to transpose these fields to an FEM model. This allows to simulate contact geometry modifications in a fast and efficient way.

Exploring the numerical performance of node-based smoothed finite elements in coupled hydro-mechanical problems

Node-based smoothed finite element (NS-FEM) formulations for the coupled hydro-mechanical problem are now becoming popular in soil mechanics, but there has been limited comparative study of their performance. In this work, a node-based formulation is developed for the coupled hydro-mechanical problem (u−pw formulation) using low-order, equal-order interpolants and an implicit time marching scheme for the global problem. The numerical properties of the resulting formulation are then compared to different implementations of classical finite elements. The use of nodal smoothing does not render by itself the solution stable at the undrained limit, on the contrary, the range of undrained instability is increased. Stabilizing terms previously employed in FEM are equally successful in stabilizing NS-FEM solutions, but the computational cost of the stabilized solution is much higher for NS-FEM. It is also shown that NS-FEM does not present by itself any particular advantage in terms of mesh dependency over classic FEM: when strain localization is present the solution obtained is strongly dependent on mesh size.

Theoretical analysis and construction of numerical method for solving the Navier–Stokes equations in rotation form with corner singularity

In this paper, we introduced the notion of an Rν-generalized solution of the Oseen problem in rotation form in two-dimensional polygonal domain with a reentrant corner at the boundary. Its existence and uniqueness in the nonsymmetric variational formulation of the problem is established. An estimate related to the conservation of the energy balance of the approximation velocity field for the Navier–Stokes problem in the rotation form is obtained. A weighted finite element method is constructed. A series of numerical experiments of test examples based on the proposed method is carried out. A comparative analysis of the approach with the classical FEM is performed. The results of numerical experiments showed a significant advantage of the proposed approach.

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

<abstract><p>This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.</p></abstract>

Bimaterial interface crack analysis using an extended consecutive-interpolation quadrilateral element

A very important problem in the research of layer structures is the modeling of cracks on the material interface. Due to the complex physical and mechanical properties of this structure, the simulation of discontinuities such as cracks and material interface by traditional finite element methods requires a very fine mesh density. Furthermore, mesh smoothing requires a really large amount of computational resources. Therefore, the extended algorithm which does not require the remeshing technique was born to solve the crack problems. In this paper, the extended consecutive-interpolation finite element method (XCFEM) is employed to modeling the mix-mode interface cracks between two dissimilar isotropic materials. The XCFEM using 4-node consecutive-interpolation quadrilateral element (XCQ4) provides continuity of nodal gradient due to the concept of “consecutive-interpolation” so that the stress and strain fields derived from XCQ4 is smoother than that obtained by the classical FEM element. The accuracy and effectiveness of the method are demonstrated via various numerical examples and compared with other researches.

The application of FEM-BEM coupling method for steady 2D heat transfer problems with multi-scale structure

This paper presents a finite element method and boundary element method (FEM-BEM) coupling scheme to study the steady-state 2D heat transfer problem for structures with multi-scale features. Based on the geometric information or material properties, the considered model is divided into FE sub-domain and BE sub-domain. The closed BE sub-domain in the mixed FEM-BEM model is defined as a finite super-element for which the effective stiffness matrix and effective fluxes are obtained by the self written BE code and hence can be assembled to the FE systems by the user subroutine (UEL) provided in ABAQUS. This method offers a number of key improvements compared with current analysis methods available for multi-scale structures. These improvements are: (i) the flexibility and convenience by using the powerful pre/post-processing in ABAQUS; (ii) the higher accuracy of the solution over the classical FEM; and (iii) the computational cost and time can be highly reduced by using the coupling method. Numerical results are compared with reference solutions and demonstrate the accuracy and effectiveness of the proposed approach.

Simple, accurate, and efficient embedded finite element methods for fluid–solid interaction

This work presents a new approach to implementing a recently proposed optimal order Cut Finite Element Method (CutFEM) for problems with moving embedded solid structures in viscous incompressible flows. This new approach uses the notion of equivalent polynomials, introduced previously in the context of the eXtended Finite Element Methods (XFEM), to implement exact integration for terms involving products of polynomials with Heaviside and Dirac distributions. Combining CutFEM and equivalent polynomials results in a method for fluid–structure interaction that (1) has the same number of degrees of freedom as the underlying conforming Galerkin method on the fixed background mesh, which is independent of the configuration of non-conforming interfaces, (2) has the same element assembly structure as classical FEM on the background mesh—with standard quadrature rules, and (3) retains the convergence properties, indeed the precise theoretical structure, of the original CutFEM method. The result is a method that is robust, accurate, and efficient for interacting particles, and which provides a convenient approach for upgrading legacy finite element codes to include embedded boundaries with CutFEM or similar formulations that can retain the same asymptotic accuracy as the underlying boundary conforming finite element method.

A spline-based FE approach to modelling of high frequency dynamics of 1-D structures

In this paper a computational methodology leading to the development of a new class of FEs, based on the application of continuous and smooth approximation polynomials, being splines, has been presented. Application of the splines as appropriately defined piecewise elemental shape functions led the authors to the formulation of a new approach for FEM, named as spFEM, where contrary to the well-known NURBS approach, the boundaries of spFEs are well-defined, exactly as it is in the case of the traditional FEM.The current approach has been computationally verified by the authors it terms of high frequency dynamics including such problems as: spectra of natural frequencies, modes of natural vibrations as well as wave propagation problems, especially in the aspect of high frequency responses, all in the case of selected problems involving one- and two-mode theories of 1-D structural elements. The applicability of the proposed approach has been evaluated and compared, in terms of calculated dynamic responses, with the results obtained by the use of well-established FEM approaches: classical FEM as well as TD-SFEM.In all cases investigated by the authors the proposed spFEM approach turned out to be the most accurate approach, free from the main drawback of the other tested FEM approaches thanks to the class of differentiability of approximation polynomials, which guarantees the absence of frequency band gaps in calculated spectra of natural frequencies. A direct consequence of this feature of the proposed approach is that a larger part of the calculated spectra of natural frequency, the same as modes of natural vibrations, can effectively be used for more accurate calculations of dynamic reposes even in the case of multi-mode theories. This in contrast to the other tested FEM approaches.

Modeling of Thin Plate Flexural Vibrations by Partition of Unity Finite Element Method

This paper presents a conforming thin plate bending element based on the Partition of Unity Finite Element Method (PUFEM) for the simulation of steady-state forced vibration. The issue of ensuring the continuity of displacement and slope between elements is addressed by the use of cubic Hermite-type Partition of Unity (PU) functions. With appropriate PU functions, the PUFEM allows the incorporation of the special enrichment functions into the finite elements to better cope with plate oscillations in a broad frequency band. The enrichment strategies consist of the sum of a power series up to a given order and a combination of progressive flexural wave solutions with polynomials. The applicability and the effectiveness of the PUFEM plate elements are first verified via the structural frequency response. Investigation is then carried out to analyze the role of polynomial enrichment orders and enriched plane wave distributions for achieving good computational performance in terms of accuracy and data reduction. Numerical results show that the PUFEM with high-order polynomials and hybrid wave-polynomial combinations can provide highly accurate prediction results by using reduced degrees of freedom and improved rate of convergence, as compared with the classical FEM.

Study of the enriched mixed finite element method comparing errors and computational cost with classical FEM and mixed scheme on quadrilateral meshes

Mixed finite element formulations are used to approximate stress and displacement variables simultaneously for Poisson problems. The purpose of this article is to analyze a new discrete mixed approximation based on the application of the enriched version of classic Poisson-compatible spaces. With that purpose we decided to measure the computational cost of applying three formulations for two Poisson problems with known exact solution. The objective is not to compare which formulation is better, but rather to highlight characteristics of computational cost and the errors obtained for both the primal and dual variables. Weak formulations corresponding to the application of the classical H1-conforming, the mixed method and the enriched mixed method. In the developed algorithms, we computed the error of the primal variable and we measured the computational cost of the assembly and the solving processes. When analyzing these costs together with the errors obtained, we visualized that the cost of the enriched version of order k, is less expensive computationally than the non-enriched version of order k+1, however getting them the same approximation errors.

A posteriori error bounds for classical and mixed FEM’s for 4th-order elliptic equations with piece wise constant reaction coefficient having large jumps

We present guaranteed, robust and computable a posteriori error bounds for approximate solutions of the equation ΔΔu + κ 2 u = f by classical and mixed Ciarlet-Raviart finite element methods. We concentrate on the case when the reaction coefficient κ is subdomain (finite element) wise constant and chaotically varies between subdomains in the sufficiently wide range. It is proved that the bounds for the classical FEM’s are robust with respect to κ ∊ [0,ch −2], where c = const and h is the maximal size of finite elements, and possess additional useful features. The coefficients in fronts of two typical norms in their right parts only insignificantly worse than those for κ ≡ const, and the bounds can be calculated without resorting to the equilibration procedures. Besides, they are sharp at least for low order methods, if the testing moments and deflection in their right parts are found by accurate recovery procedures. The technique of derivation of the bounds is based on the approach similar to one used in our preceding papers for simpler problems.

Dynamic Stability Analysis of Slope Subjected to Surcharge Load considering Tensile Strength Cut-Off

Surcharge slopes are more vulnerable to instability under the effects of earthquake ground shaking, especially considering the tensile stress. In order to account for the adverse factors of seismic forces and tensile stress, the theory of soil with tensile strength cut-off is deduced and analyzed using the upper bound limit analysis method in this paper. Combined with the quasistatic analysis, the equation of critical acceleration expression for surcharge slope subjected to the dynamic conditions has been evaluated. By using the improved Newmark method, permanent displacements have been analyzed in the case of the classical earthquake ground motions. In addition, optimization algorithm has been undertaken, in which several influencing factors such as slope inclination, internal friction angle, surcharge factor, seismic load, and tension cut-off coefficient have been taken into account, and some results are verified with the classical solutions and FEM results. The results concluded the following: (1) The outcomes of verification results are accurate. (2) The critical acceleration of the slope is significantly affected by tension cut-off with the increasing of surcharge factor and seismic effects. (3) The permanent displacements of surcharge slope considering the tensile strength cut-off can be even 2 times of the traditional analysis; meanwhile, with more reduction of tensile strength, the cumulative displacements increase rapidly. Therefore, considering the influence of tensile strength cut-off is fundamental to the dynamic stability design of surcharge conditions.

Level Set-Based Extended Finite Element Modeling of the Response of Fibrous Networks Under Hygroscopic Swelling

Abstract Materials like paper, consisting of a network of natural fibers, exposed to variations in moisture, undergo changes in geometrical and mechanical properties. This behavior is particularly important for understanding the hygro-mechanical response of sheets of paper in applications like digital printing. A two-dimensional microstructural model of a fibrous network is therefore developed to upscale the hygro-expansion of individual fibers, through their interaction, to the resulting overall expansion of the network. The fibers are modeled with rectangular shapes and are assumed to be perfectly bonded where they overlap. For realistic networks, the number of bonds is large, and the network is geometrically so complex that discretizing it by conventional, geometry-conforming, finite elements is cumbersome. The combination of a level-set and XFEM formalism enables the use of regular, structured grids in order to model the complex microstructural geometry. In this approach, the fibers are described implicitly by a level-set function. In order to represent the fiber boundaries in the fibrous network, an XFEM discretization is used together with a Heaviside enrichment function. Numerical results demonstrate that the proposed approach successfully captures the hygro-expansive properties of the network with fewer degrees-of-freedom compared to classical FEM, preserving desired accuracy.

A unified finite strain theory for membranes and ropes

The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of governing equations. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classical ones as it does not rely on parametrizations implying curvilinear coordinate systems and the concept of co- and contravariant base vectors. This leads to the third unification: TDC-based models are applicable to two fundamentally different numerical approaches. On the one hand, one may use the classical Surface FEM where the geometry is discretized by curved one-dimensional elements for ropes and two-dimensional surface elements for membranes. On the other hand, it also applies to recent Trace FEM approaches where the geometry is immersed in a higher-dimensional background mesh. Then, the shape functions of the background mesh are evaluated on the trace of the immersed geometry and used for the approximation. As such, the Trace FEM is a fictitious domain method for partial differential equations on manifolds. The numerical results show that the proposed finite strain theory yields higher-order convergence rates independent of the numerical methodology, the dimension of the manifold, and the geometric representation type.