Efficient Finite Element Formulation Research Articles

Asymptotic [formula omitted]-FEM for nonlinear analysis of composite shells

Nonlinear shell analysis relies typically on Finite Element Methods (FEMs) and Iterative-Incremental Procedures (IIPs). These methodologies can become computationally expensive whenever high-fidelity meshes are required to capture very localized features or extremely nonlinear responses. Aim of this study is presenting a novel computational tool based on an efficient finite element formulation, the ps-FEM, and a rapid perturbation solution procedure, the Asymptotic-Numerical Method (ANM). The proposed approach adopts a polynomial space enrichment strategy, the p-refinement, and a mesh superposition technique, the s-refinement, to build numerical models with quasi-optimal accuracy-to-error ratios. The introduced asymptotic framework enhances the effectiveness of solving nonlinear problems compared to IIPs. A set of test cases and new benchmarks is presented to validate the tool and demonstrate its potential. The present results show that challenging problems involving bifurcations, jumps, snap-backs and anisotropy-induced localizations can be solved with excellent degree of accuracy and relatively small modeling/computational effort.

A selectively reduced degree basis for efficient mixed nonlinear isogeometric beam formulations with extensible directors

The effect of higher order continuity in the solution field by using NURBS basis function in isogeometric analysis (IGA) is investigated for an efficient mixed finite element formulation for elastostatic beams. It is based on the Hu–Washizu variational principle considering geometrical and material nonlinearities. Here we present a reduced degree of basis functions for the additional fields of the stress resultants and strains of the beam, which are allowed to be discontinuous across elements. This approach turns out to significantly improve the computational efficiency and the accuracy of the results. We consider a beam formulation with extensible directors, where cross-sectional strains are enriched to avoid Poisson locking by an enhanced assumed strain method. In numerical examples, we show the superior per degree-of-freedom accuracy of IGA over conventional finite element analysis, due to the higher order continuity in the displacement field. We further verify the efficient rotational coupling between beams, as well as the path-independence of the results.

Nonlinear analysis of reinforced concrete slabs using a quasi-3D mixed finite element formulation

This study presents an efficient quasi-3D mixed finite element (MFE) formulation for the materially nonlinear analysis of reinforced concrete (RC) slabs based on a refined layered global–local plate theory. The cross-section of the RC slab is divided into a series of concrete and steel layers. Each layer is treated as a plate structure and mechanical properties of the steel and concrete materials are individually applied to each layer. In addition to unknown variables of the displacement field, out-of-plane stress components are also assumed as independent field variables in the presented MFE formulation. It makes it possible to obtain the out-of-plane shear stresses at each node directly from constitutive equations with the enough accuracy. A four‐node rectangular element ensuring the C1‐continuity is used for discretizing the domain of the RC slab. The governing equations are derived based on a partially mixed-field variational principle. The accuracy and efficiency of the presented MFE have been investigated through some benchmark examples. Numerical tests show that the presented MFE yields results with sufficient accuracy at a low computational cost.

Stability Performance of Perforated Stiffened Laminated Composite Plates Under the Influence of Environmental and Operational Loading Conditions Using FE Approach

Eccentrically stiffened plates are widely used as a structural component in many engineering applications. However, these structures are exposed to complex loading conditions, resulting in strength reduction leading to failure. This research aims to examine the stability performance of the perforated stiffened laminated composite plate subjected to non-uniform edge load under the influence of environmental conditions. Towards this, a robust and computationally efficient finite element (FE) formulation has been developed, where the plate is modeled using a 9-noded heterosis element to avoid the shear locking of the plate element and a 3-noded isoparametric beam element is adopted to model the stiffener by applying a displacement compatibility condition at the skin-stringer interface. A torsion correction factor is introduced in the beam formulation to account for the twisting of the stiffener. A unique dynamic technique is utilized to obtain the buckling load by employing two types of boundary conditions due to the non-uniform stress distribution in the perforated plate under environmental and operational loading conditions. An analysis is performed on the unstiffened plate to determine a suitable cutout shape, loading pattern and lamina scheme based on its improved stability performance. Unlike previous studies, several stiffener configurations are considered to comprehend the influence of temperature and moisture on the stability behavior of centrally placed cutout plates and obtain a stiffener configuration with improved performance. It is noticed that when the perforated stiffened plate is exposed to thermal or hygroscopic load close to a critical temperature or moisture, the buckling capacity drops suddenly in the plate attached to a stiffener of greater depth. Moreover, it is observed that when the stiffened plate is exposed to higher intensity of hygrothermal load, the plate’s buckling capacity drops significantly as the thickness of the plate reduces.

On continuum modeling of cell aggregation phenomena

Cellular aggregates play a significant role in the evolution of biological systems such as tumor growth, tissue spreading, wound healing, and biofilm formation. Analysis of such biological systems, in principle, includes examining the interplay of cell-cell interactions together with the cell-matrix interaction. These two interaction types mainly drive the dynamics of cellular aggregates which is intrinsically out of equilibrium. Here we propose a non-linear continuum mechanics formulation and the corresponding finite element simulation framework to model the physics of cellular aggregate formation. As an example, we focus in particular on the process of bacterial colony formation as recently studied by Kuan et al. Thereby we describe the aggregation process as an active phase separation phenomenon. We develop a Lagrangian continuum description of the problem which yields a substantial simplification to the formulations of the governing equations. Due to the presence of spatial Hessian and Laplacian operators, a gradient-enhanced approach is required to incorporate C1 continuity. In addition, a robust and efficient finite element formulation of the problem is provided. Taylor-Hood finite elements are utilized for the implementation to avoid instabilities related to the LBB condition. Finally, through a set of numerical examples, the influence of various parameters on the dynamics of the cellular aggregate formation is investigated. Our proposed methodology furnishes a general framework for the investigation of the rheology and non-equilibrium dynamics of cellular aggregates.

Dynamic instability analysis of perforated stiffened laminated composite panels subjected to non-uniform in-plane edge load under hygrothermal condition

A robust and computationally efficient finite element (FE) formulation has been developed in an effort to understand the parametric instability behaviour of a perforated stiffened composite panel subjected to a pulsating non-uniform edge load under a hygrothermal environment. The panel is modelled using a 9-noded heterosis element to avoid shear locking, whereas the stringer uses a 3-noded isoparametric beam element with a displacement compatibility condition employed at the skin–stringer​ interface. Due to the non-uniform distribution of stress in the perforated panel as a result of exposure to environmental and mechanical loading conditions, a unique dynamic technique has been implemented to account for the instability performance by employing two types of boundary conditions. In the current investigation, an unstiffened panel is initially obtained based on the highest value of the dynamic instability opening (DIO) and the unstable excitation frequency of the panel for different lamina schemes, cutout shapes, and loading conditions. The obtained unstiffened panel is further investigated for various stiffener configurations to determine a highly unstable perforated stiffened panel based on the DIO value and instability zone frequency range. Detailed parametric studies are then carried out considering the cutout sizes, cutout orientation, panel’s aspect ratio, and stiffener depth to understand the parametric instability behaviour of the panel in a hygrothermal environment. It is found that moisture significantly influences the dynamic instability behaviour of the perforated stiffened panel compared to the temperature exposure by shifting the instability zone towards the lower frequency. The width of the instability region of the stiffened panel broadens abruptly as the magnitude of the applied static in-plane edge load enhances. Also, for certain combinations of the dynamic and static load factors, the lower bound of the instability region reduces to zero, resulting in fluttering. In general, it is observed that stiffer the panel, wider is the instability zone with a higher unstable excitation frequency range.

A computationally efficient C0 continuous finite element model for thermo-mechanical analysis of cross-ply and angle-ply composite plates in non-polynomial axiomatic framework

In the present article, the thermo-mechanical bending response of multi-layered composite plates is investigated in the framework of inverse-hyperbolic shear deformation theory using a generalized finite element model. The mathematical development is carried out under the assumptions of linear structural kinematics for the materials following generalized Hooke’s law. Energy-based finite element formulation and the principle of minimum potential energy are employed to develop the finite element governing equations. A computationally efficient C0 continuous finite element formulation is developed to examine the response of laminated composites subjected to constant, linear, and non-linear temperature change. Numerical analyses are carried out for composite laminates considering various lamination sequences (cross-ply as well as angle-ply), boundary conditions, loading conditions, span-thickness ratio, etc. The present results are compared with the existing analytical and numerical results and their agreement is observed. The effect of fiber orientation angle on bending response is analyzed to enable the optimal design of laminated composite structures under thermo-mechanical loading.

Computationally efficient finite element formulation for blood flow analysis in multi‐layered aorta modeled as viscoelastic material

AbstractIn the literature, blood flow study in an arterial segment/arterial network, using one‐dimensional wave propagation model, has been carried out considering flat/parabolic/logarithmic/different degree polynomial velocity profile functions. However, such assumptions are not capable of capturing actual blood and arterial wall interaction occurring while the blood flows through the arteries. In this study, a computationally efficient axisymmetric‐formulation of the Navier–Stokes equation is presented to eliminate the requirement of a priori assumed velocity profile function. Present formulation in terms of axial velocity (u), pressure (p), and domain radius (R) leads to the evolution of velocity profile as flow progresses with the time and space. We propose a multi‐layered structural model of the arterial wall with each layer idealized using four‐element Maxwell viscoelastic material model. Partial differential equations, mathematically representing the physical phenomena of blood flow in the complaint blood vessels, are discretized in the spatial domain by finite element method and in time domain by Galerkin time integration technique. A velocity boundary condition is prescribed at inlet and outlet boundary condition is prescribed in terms of three‐parameter based Windkessel model. Results of flow characteristics are found to be in excellent agreement with the three‐dimensional results available in the literature.

Influence of Large Amplitude Vibration on Geometrically Imperfect Sandwich Curved Panels Embedded with Gradient Metallic Cellular Core

This study investigates the influence of large amplitude vibration on geometrically imperfect sandwich curved panels embedded with gradient metallic cellular (GMC) core using an efficient nonlinear finite element formulation based on higher-order shear deformation theory (HSDT). The cores of the sandwich curved panels are assumed to have three distinct porosity distributions. The material properties of the sandwich curved panel’s GMC core layer vary in the thickness direction as a function of porosity coefficient and mass density. The present nonlinear finite element model is validated with limited results available in the open literature, and few new results are also computed that can be used as a benchmark solution. The influence of porosity coefficient, porosity distribution type, amplitude ratio, imperfection amplitude, and curvature ratio on the free vibration characteristics of the geometrically imperfect sandwich curved panels with the GMC core are studied in detail.

An Efficient Finite Element Formulation Based on Deformation Approach for Bending of Functionally Graded Beams

Finite element formulations based generally on classical beam theories such as Euler-Bernoulli or Timoshenko. Sometimes, these two formulations could be problematic expressed in terms of restrictions of Euler-Bernoulli beam theory, in case of thicker beams due to non-consideration of transverse shear; phenomenon that is known as shear locking characterized the Timoshenko beam theory, in case of thin beams; problem of slow of convergence in regards to the element of Timoshenko beam. In responding to this problematic, a new beam finite element model is developed to study the static bending of functionally graded beams. The originality of this model lies in the use of a deformation approach with the consideration of a central node positioned in the middle of the beam. The degrees of freedom of this node are subsequently eliminated by the method of static condensation. In addition, this model is suitable for all linear structures regardless of L/h ratio. Functionally graded material beams have a smooth variation of material properties due to continuous change in micro structural details. The mechanical properties of the beam are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. A simply supported beam subjected to uniform load for different length-to-thickness ratio has been chosen in the analysis. Finite element solutions obtained with the new finite element model are presented, and the obtained results are evaluated with the existing solutions to verify the validity of the present model.

Aspects of finite element formulations for the coupled problem of poroelasticity based on a canonical minimization principle

This work presents a new finite element treatment of the coupled problem of Darcy–Biot-type fluid transport in porous media undergoing large deformations, that is free from any stabilization techniques. The formulation bases on an incremental two-field minimization principle that is constrained by the equation of continuity for the fluid mass content and determines at a given state the deformation and the fluid mass flux vector. The big advantage of the minimization formulation over classical saddle point principles of poroelasticity is the omission of the inf-sup condition—a condition that makes the construction of stable and computationally efficient finite element formulations difficult. Due to the $$H(\hbox {Div}, {\mathcal B}_0)$$ variational structure of the minimization principle on the fluid side, lowest order Raviart–Thomas elements are used for the conforming approximation of the fluid mass flux. Furthermore, a standard nodal-based element using bilinear interpolation for both fields combined with a reduced numerical integration of the (volumetric) coupling term is analyzed and used for the solution of the minimization principle. Representative numerical examples demonstrate the performance of the proposed finite element designs of the minimization principle and clearly underline advantages over finite element formulations of the classical two-field saddle point principle formulated in deformation and fluid potential.

Influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates

Present study aims to investigate the influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates. Reddy's higher order shear deformation theory has been employed to model the displacement field. Variational approach has been used to derive the governing differential equations. Effect of material uncertainties in the formulation have been incorporated using first order perturbation technique (FOPT). An efficient stochastic finite element formulation (SFEM) have been used for the calculation of first and second order statics of natural frequency and transverse deflection. Validation of the results have been performed with the help of available literature and separately developed Monte Carlo formulation (MCS) algorithm. A large number of examples have been solved to quantify the effect of uncertainties on the vibration and bending characteristics of functionally graded skew sandwich plates.

An efficient finite element formulation of dynamics for a flexible robot with different type of joints

If two adjacent links of a flexible robot are connected via a revolute joint or a fixed prismatic joint, the relative motion of the next link will depend on both the joint motion and the elastic displacement of the distal end of the previous link. However, if the two adjacent links are connected via a sliding prismatic joint, the relative motion of the next link will depend additionally on the elastic deformation distributed along the previous link. Therefore, formulation of the motion equations for a multi-link flexible robot consisting of the revolute joints, the fixed prismatic joints and the sliding prismatic joints is challenging. In this study, the finite element kinematic and dynamic formulation was successfully developed and validated for the flexible robot, in which a transformation matrix is proposed to describe the kinematics of both the joint motion and the link deformation. Additionally, a new recursive formulation of the dynamic equations is introduced. As compared with the previous methods, the time complexity of the formulation is reduced by O(2η), where η is the number of finite elements on all links. The numerical examples and experiments were implemented to validate the proposed kinematic and dynamic modelling method.

A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff's plate theory

The strain gradient nonlocal theory is important to include the size effects of nanostructures in classical continuum theory with the corresponding development of computationally efficient numerical tool such as finite elements for the analysis of such structures with different boundary conditions. However, there is no literature on the finite element formulation of second-order strain gradient elastic plates. The weak form of the governing equation of motion of the Kirchhoff nanoplate using second-order positive/negative strain gradient nonlocal theories requires C2 continuity of transverse displacement. In this paper, a new computationally efficient nonconforming finite element formulation for the modelling of nanoplates using second-order positive/negative strain gradient nonlocal theories is presented. The performance of the developed finite element is compared with conforming finite element for rectangular isotropic Kirchhoff nanoplates with different boundary conditions. Analytical solution for static bending, free vibration, and buckling under biaxial in-plane compressive loading are also obtained for rectangular all edges simply supported isotropic Kirchhoff nanoplate for the comparison purpose. The nonconforming element is found to be computationally more efficient than the conforming element with better accuracy and convergence rate. The negative strain gradient model predicts results matching with the experimental results available in the literature.

Reduced order finite element formulations for vibration reduction using piezoelectric shunt damping

The present work proposes an original reduced order model for prediction of passive reduction of structural vibration by means of shunted piezoelectric patches. The problem consists of an elastic structure with surface-mounted piezoelectric patches. The piezoelectric elements are connected to a resonant shunt circuits in order to damp specific resonant frequencies of the structure. An efficient electromechanical finite element formulation for the dynamic analysis of the problem is first presented. The classical modal superposition techniques using the system eigenvectors with all patches short-circuited or open-circuited are then recalled. An advanced reduced order models using two new modal projection bases able to solve the problem at lower cost are developed: (i) the combined basis formed by both the short-circuited and open-circuited modes, and (ii) the coupled basis formed by the electromechanical modes that take into account the effect of the inductances of the electrical shunt circuits. Various numerical and experimental results are presented in order to validate and illustrate the efficiency of the proposed new finite element reduced order formulations.

An efficient finite element formulation for bending, free vibration and stability analysis of Timoshenko beams

In this paper, a new three-node element is proposed for analysis of beams with shear deformation effect. In each node of this element, there exist translation and rotation degrees of freedom. The element’s formulation is based on the first-order shear deformation theory. For this aim, the displacement field of the element is approximated by a fifth-order polynomial. The shear strain is varied as a quadratic function within the element. It is worth noting that the quadratic function can be used for axial displacement field as well. By employing curvature and shear strain relations of Timoshenko beam theory, the exact and explicit shape functions of the displacement fields are obtained. By utilizing these shape functions, the stiffness matrix and the geometric stiffness matrix of the element are calculated. The mass matrices of the proposed element are derived from kinetic energy relation of the beam. Finally, several numerical tests are performed to assess the robustness of the developed element. The results of the numerical tests prove the absence of the shear locking and demonstrate high accuracy and efficiency of the proposed element for bending, free vibration and stability analysis of Timoshenko beams.

Efficient finite-element formulation for analysis of whispering-gallery-mode optical resonators

This paper presents a new formulation for the fast and accurate three-dimensional vectorial analysis of dielectric whispering-gallery-mode (WGM) resonators using the finite element method (FEM). This method relies on two properties of the WGM resonators: (1) the axial symmetry to reduce the number of coordinate variables from three to two and (2) the use of the divergence equation for the magnetic field and its axial symmetry to express the azimuthal magnetic component of the field in terms of its transverse components. As a result, for our FEM analysis, only the transverse part of the magnetic Helmholtz equation is required, which is composed of two field variables and two coordinate variables. The method can use scalar elements without generating any spurious modes and without the need for a penalty function. This can substantially reduce the computational time and complexity. After verifying this FEM formulation through comparing the simulation results with the known analytical solutions for a few special resonators, a variety of WGM resonators in the silicon photonic platform are analyzed in terms of key parameters needed for designing integrated photonic structures.

On the numerical implementation of thermomechanically coupled distortional hardening

In contrast to the by now classic isotropic and kinematic hardening, the more general framework of distortional hardening characterised by an evolution of the yield surface's shape can capture the effect of texture evolution on the macroscopic response. Although different distortional hardening models can indeed be found in the literature, efficient numerical implementations of such models are still missing. This statement proves particularly true within a thermomechanically coupled framework which is important for most technologically relevant processes – such as for deep drawing. Accordingly, this paper deals with an efficient finite element formulation for distortional hardening within a thermomechanically coupled framework. As a first step towards this target, the recently advocated isothermal distortional hardening framework Shi et al. (2014) is extended to the thermomechanically coupled setting. In order to avoid an over-estimation of the temperature increase due to plastic deformation, the initial yield stress is decomposed into a classic dissipative part and a non-classic energetic part. By doing so, the restrictions imposed by thermodynamical principles are fulfilled and simultaneously realistic temperature predictions are obtained. For the resulting model, an efficient numerical implementation is proposed. By developing a suitable time integration scheme for the evolution equations of the fourth-order tensor describing the distortional hardening, a return-mapping scheme for updating the internal variables is derived which shows the same numerical complexity as a return-mapping scheme for purely isotropic hardening. This efficient return-mapping scheme is finally incorporated into a thermomechanically coupled finite element formulation, and the resulting set of equations is fully implicitly and monolithically solved by means of a Newton-type iteration. Several numerical complex examples demonstrate the capabilities of the distortional hardening model as well as the robustness and efficiency of the numerical formulation.

Formulation of solid-shell finite elements with large displacements considering different transverse shear strains approximations

This work presents a general formulation and implementation in solid-shell elements of the refined zigzag theory and the trigonometric shear deformation theory in an unified way. The model thus conceived is aimed for use in the analysis, design and verification of structures made of composite materials, in which shear strains have a significant prevalence. The refined zigzag theory can deal with composite laminates economically, adding only two nodal degrees of freedom, with very good accuracy. It assumes that the in-plane displacements have a piece-wise linear shape across the thickness depending on the shear stiffness of each composite layer. The trigonometric theory assumes a cosine variation of the transverse shear strain. A modification of this theory is presented in this paper allowing its implementation with C0 approximation functions. Two existing elements are considered, an eight-node tri-linear hexahedron and a six-node triangular prism. Both elements use a modified right Cauchy-Green deformation tensor C¯ where five of its six components are linearly interpolated from values computed at the top and bottom surfaces of the element. The sixth component is computed at the element center and it is enhanced with an additional degree of freedom. This basic kinematic is improved with a hierarchical field of in-plane displacements expressed in convective coordinates. The objective of this approach is to have a simple and efficient finite element formulation to analyze composite laminates under large displacements and rotations but small elastic strains. The assumed natural strain technique is used to prevent transverse shear locking. An analytic through-the-thickness integration and one point integration on the shell plane is used requiring hourglass stabilization for the hexahedral element. Several examples are considered on the one hand to compare with analytical static solutions of plates, and on the other hand to observe natural frequencies, buckling loads and the non-linear large displacement behavior in double curved shells. The results obtained are in a very good agreement with the targets used.

Probabilistic reliability of micro-resonators with thermoelastic coupling

Analyzing the probabilistic reliability of micro-resonators is essential for the design of microelectromechanical systems (MEMS). In order to investigate more complex structures and take into account the random variation of the design parameters, an efficient thermoelastic Finite Element (FE) formulation is derived from Hamilton’s variational principle. A generalized eigenvalue equation can be obtained through the formulation. The effects of thermoelastic coupling and the randomness of parameters on the natural frequency are studied by using a non-symmetric block Lanczos algorithm combined with the right and left eigenproblem. Considering the influence of thermoelastic coupling on temperature is little, the thermoelastic coupling term in heat conduction equation can be negligible, thus the heat conduction equation is simplified based on linearization. Depending on the eigenproblem for temperature field and thermal stress in micro-resonators, further analysis is conducted on the randomness of temperature and stress in micro-resonators. For reliability analysis, a method which considering both strength and frequency is proposed. The strength failure criterion is set based on fracture mechanics, and frequency reliability is taken into account the probabilistic of resonators when the natural frequency is close to the excitation frequency. With a simply supported model of micro-resonators as an example, the numerical solutions of quality factor have shown little difference with the analytical solutions proposed by Zener and Lifshitz, verifying the correctness of the model calculations. Finally, the natural frequency shift and the probabilistic reliability with temperature are calculated.