Standard Finite Element Research Articles

Polynomial eigenvalue solution for elastic wave prediction of piezoelectric shunting arrays

This paper presents a polynomial eigenvalue solution to predict the propagation behaviors of elastic wave in piezoelectric shunting arrays. Based on the Bloch theorem, one independent unit cell is selected to conduct the dynamic characteristic analysis instead of infinity. The reduced form of the discretized governing equations is first derived by the standard finite element procedures. To facilitate the subsequent acquisition of dispersion relationship, the dynamic stiffness matrix is then partitioned into a block matrix. Through applying the periodic boundary conditions, a polynomial eigenvalue equation concerning complex propagation constant is finally obtained. The wave propagation and attenuation characteristics in arbitrary directions are investigated using the above methodology. The results demonstrate that the present method can provide very accurate and reliable solutions for wave propagation prediction of piezoelectric shunting arrays.

Objectivity in quasi-brittle structural failure via adaptive formulation and mesh refinement

In this work, the combined implementation of Adaptive Formulation Refinement (AFR) and Adaptive Mesh Refinement (AMR) strategies is proposed for the efficient and mesh objective evaluation of quasi-brittle fracture.A critical issue when computing localized structural failure is the spurious mesh bias dependence of the crack trajectories and collapse mechanisms that ensues when adopting the standard displacement-based finite element (FE) formulation. This problem is resolved by the introduction of the mixed strain/displacement FE formulation, which allows for local convergence, producing mesh objective results.On the one hand, the proposed AFR approach allows to adaptively switch between the standard and mixed FE formulations during the numerical simulation depending on the level of accuracy required in different regions of the domain. Within this framework, the more accurate mixed formulation is activated only in the areas where the crack onset and evolution take place, while the standard formulation is adopted for the rest of the structure.On the other hand, the adopted octree-based AMR strategy allows to adaptively refine only certain areas of the domain, also in function of the level of accuracy required. Using this methodology, it is possible to start the simulation with an initially relatively coarse mesh and adaptively introduce several levels of mesh refinement only in the regions of interest, where cracks appear and propagate. This approach guarantees the necessary mesh resolution to capture the fracture phenomenon accurately while further increasing the computational efficiency of the simulation. The subsequent adaptive coarsening of the FE mesh is also allowed.Structural failure and fracture propagation are reproduced via a local isotropic damage model. For this reason, a refinement/coarsening criterion defined in function of the equivalent effective stress is conveniently adopted with both AFR and AMR strategies.To assess the capabilities of the proposed method, a set of numerical simulations featuring benchmark problems and experiments is included. Results show the mesh objectivity and cost-efficiency of the approach combining AFR and AMR in fracture problems. The proposed model is able to reproduce experimental crack paths, collapse mechanisms and force–displacement curves with accuracy and without spurious mesh bias dependence.The paper includes a comparative assessment of the performance of the proposed method and the standard FE formulation exclusively, showing the superior performance of the AFR methodology and its mesh objectivity. Convergence of the computed results when introducing different levels of AMR is also verified. Computational efficiency of the AFR and AMR techniques is specifically assessed. The combination of AFR and AMR approaches allows to provide accurate and mesh objective results in quasi-brittle fracture problems with very significant savings in computational cost.

Application of the gradient theory to interface crack between two dissimilar dielectric materials

In the present paper, the interface crack between two dissimilar dielectric materials under a mechanical load is investigated with including flexoelectricity effects. Flexoelectricity is a size dependent electro-mechanical coupling phenomenon, where the electric polarization is induced by a strain gradient in dielectrics. The strain gradients may potentially break the inversion symmetry in centrosymmetric crystals and polarization is observed even in all dielectric materials. The polarization is proportional to the strain gradients in the direct flexoelectricity. Layered composite structures are frequently utilized in microelectronics. Due to a poor adhesion of protection layer and basic material, the interface crack can be created there and for the prediction of failure of these structures it becomes essential to investigate distribution of the interfacial stress and strain fields. Governing equations in the gradient theory contain higher-order derivatives than in the standard continuum mechanics. Therefore, areliable computational tool is required to solve these boundary-value problems. The mixed finite element method (FEM) is developed, where the standard C0 continuous finite elements are utilized for independent approximations of displacements and strains. The constraints between the displacement gradients and strains are satisfied by collocation at Gaussian integration points inside elements. In numerical examples, a parametric study is performed with respect to flexoelectric and elastic coefficients for both material regions. The influence of these parameters on the crack opening displacement is discussed.

A fully non-invasive hybrid IGA/FEM scheme for the analysis of localized non-linear phenomena

This work undertakes to combine the interests of IsoGeometric Analysis (IGA) and standard Finite Element Methods (FEM) for the global/local simulation of structures. The idea is to adopt a hybrid global-IGA/local-FEM modeling, thereby benefiting from: (i) the superior geometric description and per-Degree-Of-Freedom accuracy of IGA for capturing global, regular responses, and (ii) the ability of FEM to compute local, strongly non-linear or even singular behaviors. For the sake of minimizing the implementation effort, we develop a coupling scheme that is fully non-invasive in the sense that the initial global spline model to be enriched is never modified and the construction of the coupling operators can be performed using conventional FE packages. The key ingredient is to express the FEM-to-IGA bridge, based on B{\'e}zier extraction, to transform the initial global spline interface into a FE one on which the local FE mesh can be constructed. This allows to resort to classic FE trace operators to implement the coupling. It results in a strategy that offers the opportunity to simply couple an isogeometric code with any robust FE code suitable for the modelling of complex local behaviors. The method also easily extends in case the users only have at their disposal FE codes. This is the situation that is considered for the numerical illustrations. More precisely, we only make use of the FE industrial software Code Aster to perform efficiently and accurately the hybrid global-IGA/local-FEM simulation of structures subjected locally to cracks, contact, friction and delamination.

Effect of flanges on the in-plane behavior of the masonry walls

In unreinforced masonry buildings, load bearing walls are not of only rectangular shape in the plan, but the cross walls connected to them make the walls of either I shape, T shape, C shape, etc. The presence of the cross walls can significantly change the behavior of the masonry walls in terms of strength and the failure modes of the walls. In this paper, a parametric study has been performed to investigate the effect of different parameters, namely pre-compressions, material properties, aspect ratios, and boundary conditions on the in-plane behavior of the masonry walls. The masonry walls have been modeled using the simplified micro-modeling approach in the standard finite element software Abaqus without any user-defined subroutine. It has been found that the strength and damage pattern of the flanged walls are different from the rectangular walls, which needs to be considered in the analysis of masonry walls using a numerical approach or while conducting experiments.

Numerical investigations of edge-based smoothed radial point interpolation method for transient wave propagations

It is known that the solutions of the standard finite element method for transient wave propagations are suffering from the numerical dispersion effect and the numerical anisotropy, resulting in the dispersive numerical waves along different propagation angles. In this work, the edge-based smoothed radial point interpolation method is combined with the Bathe time integration method to study transient wave propagations. By means of the generalized gradient smoothing approach, the smoothed radial point interpolation method can lead to the proper smoothed stiffness of the discrete model, thus greatly reducing the numerical dispersion effect and the numerical anisotropy. Besides, due to the appropriate high-frequency dissipative property, the Bathe time integration method can effectively depress the spurious high-frequency responses in transient wave solutions. Moreover, the wave velocity errors of several numerical methods for transient wave propagations are theoretically and numerically analyzed, revealing the superiority of the smoothed radial point interpolation method in comparison with the linear and quadratic finite element methods. Several numerical tests show that the smoothed radial point interpolation method combined with the Bathe time integration method can present the more accurate simulations of transient wave propagations than the standard finite element method.

Efficient formulation of a two‐noded geometrically exact curved beam element

AbstractThe article extends the formulation of a 2D geometrically exact beam element proposed by Jirásek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first‐order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier–Bernoulli hypothesis, with a possible extension to shear‐flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral‐shaped beam, and a spring‐like beam with a zig‐zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, that is, the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the Appendix.

An element‐based formulation for ES‐FEM and FS‐FEM models for implementation in standard solid mechanics finite element codes for 2D and 3D static analysis

AbstractEdge‐based and face‐based smoothed finite element methods (ES‐FEM and FS‐FEM, respectively) are modified versions of the finite element method allowing to achieve more accurate results and to reduce sensitivity to mesh distortion, at least for linear elements. These properties make the two methods very attractive. However, their implementation in a standard finite element code is nontrivial because it requires heavy and extensive modifications to the code architecture. In this article, we present an element‐based formulation of ES‐FEM and FS‐FEM methods allowing to implement the two methods in a standard finite element code with no modifications to its architecture. Moreover, the element‐based formulation permits to easily manage any type of element, especially in 3D models where, to the best of the authors' knowledge, only tetrahedral elements are used in FS‐FEM applications found in the literature. Shape functions for non‐simplex 3D elements are proposed in order to apply FS‐FEM to any standard finite element.

Automatic mesoscopic fracture modelling of concrete based on enriched SBFEM space and quad-tree mesh

• The methods by combining enriched SBFEM with quad-tree mesh are proposed to simulate crack propagation of concrete. • An improved random aggregate model is proposed to construct mesoscale concrete models. • The whole mesh generation process is highly efficient without any artificial interference. • Local remeshing is unnecessary as crack propagates. A novel approach for mesoscale modelling of concrete composites is proposed by combining enriched scaled boundary finite element methods with quad-tree mesh. The concrete meso -structures are comprised of randomly distributed aggregates, mortar matrix, and interface transition zone. An improved random aggregate generation technique is developed to construct digital images of mesoscale concrete models. Based on the quadtree decomposition algorithm, meshes can be generated automatically from the digital images of concrete mesostructure. The whole mesh generation process is highly efficient without any artificial interference and eliminates the issue of hanging nodes faced by standard finite elements. Additionally, local remeshing is unnecessary as crack propagates. Three numerical examples are modelled to demonstrate the performance of the proposed approach, and the effects of aggregate area fraction on the mechanical properties of concrete composites are also discussed.

Machine learning approach to diabetic foot risk classification with biomechanics data

Diabetic foot wound onset, recurrence and healing are globally considered a challenge [1]. In particular, the prevention topic is pursued through several recommendations, however, it is desirable to classify the risk of ulceration before their occurrence, to prevent severe ulcers and subsequent amputations [1]. With regard to the biomechanical properties of the diabetic foot, diabetic patients showed higher load on the push-off associated with lower dorsi-plantar flexion of the ankle during stance and the push-off phase [2]. The aim of this study was to automatically define a risk index of the diabetic foot ulceration by applying machine learning (ML) algorithms to a dataset of standard gait analysis, musculoskeletal modelling and finite element modeling data and of routine diabetic neuropathy screening scores [4]. The study included retrospectively the data of 80 subjects (26.3% healthy subjects, 37.5% diabetics without neuropathy and 36.3% diabetics with neuropathy). Diabetic patients were clinically assessed according to the ADA guidelines [3] as reported in [4]. Kinematic, kinetic and electromyographic (EMG) data were acquired through a modified version of the IORgait protocol [3,5] while subjects were walking several gait cycles at self selected speed. Muscle forces were estimated with the Opensim gait 2392 model and internal stresses were determined using a foot finite element model [6]. Joint angles and moments, ground reaction forces, plantar pressures, EMG envelopes and timing of activation, muscular forces and internal stresses were considered for the ML algorithm. For what regards the clinical features, presence or absence of peripheral and autonomic neuropathy, retinopathy, microalbuminuria, vasculopathy, knee-hallux-toes deformities, plantar callosity were selected for the ML algorithm. MongoDB was chosen as database program and data extractions were performed through the library Pandas. Regarding the ML algorithm, the values of different gaits were merged using an arithmetic mean, in order to obtain a narrower dataset and more stable values of the features. Due to the size of the dataset, ‘simple’ and linear models were adopted, and, among these, Logistic Regression, Linear Model and Perceptron were chosen. For completeness, some more complex classifiers like Kernel Support Vector Machine, Random Forest, Gradient Boosting Tree and eXtreme Gradient Boosting Tree were also adopted but showed lower performances. Logistic Regression with the adoption of 4 features achieved the best result: 83% of precision and 83% of accuracy (Fig. 1). Once the model was developed, a React Native mobile application was implemented with a relative Application Protocol Interface (API) to produce an environment to host and exploit the model. Diabetic foot risk classification according to ADA [3] was used as comparison for the results of the ML approach. The performance of the model is not considered to be optimal, but the results are encouraging from a diabetic foot risk point of view. Future development includes the expansion of databases and the exploration of other ML algorithms.

An extended GBT formulation for elastoplastic analyses of perforated thin-walled members

A physically and geometrically nonlinear beam finite element modelling of thin-walled members with arbitrarily complex holes is presented. The beam finite element approximation of the member configuration space is achieved by using the extended generalized beam theory kinematic description, which covers discontinuities in member cross-sections due to holes or cut-outs of various shapes. The proposed beam finite element deals with perforated members in a manner of treating them as non-perforated ones by filling the perforations with much softer materials. Then the level-set functions are used to represent the bi-material interfaces and also used to construct the enrichment functions which describe the discontinuities in the displacement fields. The enrichment functions are then integrated into the standard GBT-based beam finite element approximation by the Partition of Unity Method. The J2-plasticity model combined with isotropic hardening and associated flow rule for isotropic materials (e.g., soft steel) is used in the model. To show the potential of the proposed approach, four illustrative examples are provided, where results obtained by the presented model are compared with the shell finite element results. The proposed GBT model shows significant superiorities in the computational efficiency and structural clarity over the shell model.

An atomistic entropy based finite element multiscale method for modeling amorphous materials

A new concurrent multiscale method based on the maximum entropy statistical method is proposed for the analysis of amorphous materials. In addition to reducing the number of degrees of freedom, any irregular structure of amorphous materials can be accurately analyzed. The amorphous structure is generated from a solid crystalline structure by a heating/cooling process without the need for any specific independent technique to create such a random structure. The method is expected to perform efficiently because of its entropic and irregular intrinsic. Regions with moderate conditions are discretized by the entropy-based finite element method while the severe parts are simulated by the present atomistic-based multiscale technique. The new proposed approach allows for accurate analysis of amorphous structures across multiple scales and does not suffer from conventional complications such as the standard Cauchy Born rule and consistency of the molecular structure with the standard finite element geometries.The conventional Cauchy-Born rule cannot be directly used due to the non–crystalline microstructure of the material. A remedy is proposed based on the meshfree techniques by constructing a continuous atomic deformation field from the imposed macro deformation gradient. The resultant deformation gradient and the stress field remain consistent in micro scales. In addition, a genetic algorithm-based method, which has less sensitivity to the choice of initial point and number of parameters, is adopted for the maximization of the entropy function.The silicon amorphous structure is considered for MD simulations. It is obtained by quenching from a melted sample. The MD-obtained structure is further analyzed and the predicted displacements and stress contours, as well as the density and radial distribution functions are examined to assess the state of the material. Then, the proposed meshfree technique is applied to construct the continuous form of the deformation gradient on the MD model to improve the accuracy of the solution. The proposed concurrent multiscale method is verified and then employed to simulate an amorphous silicon specimen. Finally, the effects of sample size, strain rate and quenching speed on rupture stress and strain in different 3D tensile simulations are investigated by the proposed multiscale method.

A multi-point constraint unfitted finite element method

In this work a multi-point constraint unfitted finite element method for the solution of the Poisson equation is presented. Key features of the approach are the strong enforcement of essential boundary, and interface conditions. This, along with the stability of the method, is achieved through the use of multi-point constraints that are applied to the so-called ghost nodes that lie outside of the physical domain. Another key benefit of the approach lies in the fact that, as the degrees of freedom associated with ghost nodes are constrained, they can be removed from the system of equations. This enables the method to capture both strong and weak discontinuities with no additional degrees of freedom. In addition, the method does not require penalty parameters and can capture discontinuities using only the standard finite element basis functions. Finally, numerical results show that the method converges optimally with mesh refinement and remains well conditioned.

The Bubble Transform and the de Rham Complex

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh $${{\mathscr {T}}}$$ of the domain $$\varOmega $$ , we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in $$L^2$$ and also on the appropriate subspace consisting of k-forms with exterior derivatives in $$L^2$$ .

An all‐at‐once algebraic multigrid method for finite element discretizations of Stokes problem

AbstractWe consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transform‐then‐solve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite difference discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive over‐relaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that for stationary problems our method is competitive compared to a reference solver based on a block diagonal preconditioner and MINRES, and suggest that the transform‐then‐solve approach is also more robust. In particular, for problems with variable viscosity, the transform‐then‐solve approach demonstrates significant speed‐up with respect to the block diagonal preconditioner. The method is also particularly robust for time‐dependent problems whatever the time step size.

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

Modal localization in vibrations of circular cylindrical shells with geometric imperfections

The present study aims to investigate the effect of geometric imperfections in circular cylindrical shells on the vibration characteristics. Perfect circular shells are characterized by the presence of double shell-like modes, i.e., modes having the same frequency with modal shape shifted of a quarter of wave-length in the circumferential direction. However, in the presence of geometric imperfections, the double natural frequencies split into a pair of distinct frequencies, the splitting is proportional to the level of imperfection. In some cases, the imperfections cause an interesting phenomenon on the modal shapes, which present a strong localization in the circumferential direction. The present study has been carried out by means of a semi-analytical approach. Theoretical formulation were derived based on Sanders–Koiter thin shell theory. The analytical results have been compared with those of standard finite element analyses. The results corresponding to the analysis of modal localization are novel and can be used as a benchmark for further studies.

A Geometrically Exact Beam Finite Element for Non-Prismatic Strip Beams: The 2D Case

In this paper, a new finite element for 2D non-prismatic (curved and tapered) elastic beams with narrow rectangular cross-section is proposed. The element is geometrically exact, hence capable of handling large displacements and finite rotations, and allows for cross-section arbitrary warping through the inclusion of additional degrees of freedom (DOFs). These additional DOFs make it possible to capture the non-standard shear stress distributions that are developed in tapered members. All expressions required to implement the proposed finite element are derived and written in a simple vector/matrix form. To assess the accuracy and demonstrate the capabilities of the proposed element, several numerical examples are presented. For comparison purposes, results obtained with refined meshes of standard 2D finite elements are also shown. It is concluded that the proposed element leads to very accurate results with a small computational cost.

Stokes-Cahn-Hilliard formulation in sliding bi-periodic frames for the simulation of two-phase flows

A direct numerical simulation technique is proposed in this study for droplet emulsions in simple shear flow in Newtonian-Newtonian liquid systems. We combine the sliding bi-periodic frame with the diffuse interface method in the finite element framework to treat the multi-drop problem without wall effects. We employ the standard velocity-pressure formulation in a creeping regime and the diffuse interface method with the Galerkin weak formulation. Sliding bi-periodic frame constraints are implemented by Lagrangian multiplier. The results are presented for the morphological development of a single drop, two drops, and multiple drops in a sliding bi-periodic frame. To preserve the identical solution for a given relative configuration of droplets (with different droplet locations) within the context of the bi-periodic frame, a phase-translation method is proposed. The method corrects the inconsistency in the time integration, which yields discrepancy in flow solution particularly for problems with thin interfaces. By applying this procedure, accurate solutions for a thin interface are obtained regardless of location of the identical relative configuration of droplets. The effect of phase-field translation is also analyzed for the coalescence of two drops. For the first time, the sliding bi-periodic frame has been implemented with the combination of the standard finite element method and the diffuse interface method, which can be easily extended to more complex rheological liquids to solve industrially important problems.

Homogenization of discrete diffusion models by asymptotic expansion

AbstractDiffusion behaviors of heterogeneous materials are of paramount importance in many engineering problems. Numerical models that take into account the internal structure of such materials are robust but computationally very expensive. This burden can be partially decreased by using discrete models, however even then the practical application is limited to relatively small material volumes. This paper formulates a homogenization scheme for discrete diffusion models. Asymptotic expansion homogenization is applied to distinguish between (i) the continuous macroscale description approximated by the standard finite element method and (ii) the fully resolved discrete mesoscale description in a local representative volume element (RVE) of material. Both transient and steady‐state variants with nonlinear constitutive relations are discussed. In all the cases, the resulting discrete RVE problem becomes a simple linear steady‐state problem that can be easily pre‐computed. The scale separation provides a significant reduction of computational time allowing the solution of practical problems with a negligible error introduced mainly by the finite element discretization at the macroscale.