Finite Element Formulation Research Articles

Free and forced vibration analysis of piezolaminated plates via an isogeometric layerwise finite element

Isogeometric layerwise finite element (L-IGA) formulation is a recent state-of-the-art approach integrating Non-Uniform Rational B-spline (NURBS) basis functions into the quasi-static solution process of piezolaminated composite plates. This study extends the application of the L-IGA framework to encompass free, forced vibration, and displacement control analyses of laminated composite plates with straight/curvilinear fibers and piezoelectric layers. To this end, the NURBS basis functions, utilized in geometry definition, are employed to solve electromechanically coupled differential equations following Hamilton’s variational principle. The adoption of high-order continuous NURBS shape functions throughout the IGA discretization span both in-plane and through-thickness laminate dimensions. This effectively facilitates precise geometry representation directly from Computer-Aided Design (CAD). Besides, such a discretization accelerates the convergence of displacement and electric potential solution fields toward exact results. Various benchmark problems have been solved to verify the robustness and high accuracy of the proposed dynamic L-IGA method. These include comparative analyses between L-IGA dynamic solutions (i.e. employing the Newmark-Beta method), analytical solutions, and ANSYS-Solid 226 finite element results. All the results are compared across various span-to-thickness ratios, mechanical-potential loading scenarios, and fiber orientation angles. Remarkably, the L-IGA method attains almost excellently accurate time response of various fields (displacement, stress, electric potential) and modal results, with considerably fewer mesh elements than Solid 226 solutions. Overall, such an outcome reveals the high potential and practical merits of the proposed L-IGA formulation as a proficient finite element approach for the dynamic analysis of piezolaminated plates.

Development of improved finite element formulations for pile group behavior analysis under cyclic loading

AbstractThe effect of cyclic loading is an essential factor leading to progressive soil strength degradation. Therefore, a comprehensive analysis of the pile‐soil system behavior under cyclic loading is required to ensure the stability of pile group. There is room for improvement in the inherent constraint of the conventional numerical model in terms of approximating the soil resistance distribution along the pile by point loads at element nodes, necessitating a specific element that integrates considerations of pile group effect and cyclic loading within a unified framework. This study aims to develop a newly specific type of element for efficiently predicting nonlinear behavior within the pile‐soil system, addressing simulations involving nonlinear pile‐soil interaction, pile group effect, and cyclic loading. Modified element formulations based on soil stiffness matrices and soil resistance vectors specifically address pile group effect and consider parameters that influence pile behavior under cyclic lateral loading. The numerical solution procedure with Newton‐Raphson iteration allows the calculation of pile responses in geometric and material nonlinear analyses. The validation of the proposed method includes several examples, comparing it with existing numerical solutions and experimental tests of single piles and pile groups under cyclic loading. These comparisons further support the consistency of the proposed method with measured data and validate its accuracy in considering group effect and cyclic loading. The parametric study illustrates the ability of the proposed method to capture cyclic loading parameters while considering the influence of the number and magnitude of load cycles, the cyclic load direction, and the installation methods.

FUNDAMENTAL APPROACH TO PREDICT TIRE AIR PRESSURE LOSS OVER TIME

ABSTRACT Rubber material used in consumer products is vulcanized to improve its strength, viscoelasticity, and long and short-term durability. During the cure process, the morphology of the material evolves to form a highly crosslinked chain network, leading to a product that is less porous and less susceptible to air diffusivity. However, the formulation of the rubber compounds always create micropores or voids within them. Several laboratory tests are used to measure the material’s ability to resist the flow of air through it and to assess its porosity, especially under pressure. Residual-based integrated surface flux calculations are used to determine the evolution of air pressure as a function of time in an enclosed volume such as the cavity of an inflated tire. A finite element formulation was established for complex applications such as tires to predict the intracarcass pressure distribution. The capability of this simulation methodology was demonstrated for a tire with different inner liner compositions.

Modeling of textile composite using analytical network-averaging and gradient damage approach

In this contribution, we present a gradient damage model for anisotropic textile reinforcements including fiber inextensibility and fiber sliding. In contrast to previous works, the gradient damage formulation stems not from a numerical regularization basis but from the thermodynamics of internal variables. It results in a nonlocal term as the internal energy of fiber bending with measurable nonlocal parameter. Furthermore, to guarantee a priori that rotations and reflections determined by orthogonal tensors among the symmetry group do not affect the response function of the anisotropic constitutive law, a novel mesoscopic kinematic measure for the representative volume element of the fabric is defined on the basis of the analytical network-averaging concept. Such kinematic measure is of crucial importance for material modeling of damage-elastoplasticity in anisotropic textile reinforcements, and allows for analytical descriptions of inter- and intra-ply sliding of fibers. A mixed finite element formulation is then presented for textile reinforcements taking into account fiber inextensibility. The predictive capability of the computational model is demonstrated by comparing with multiple experimental datasets of dry textile fabrics.

Forced vibration characteristics of viscoelastic variable stiffness laminated composite plates using time and frequency domain approaches

The linear forced vibration characteristics of viscoelastic variable stiffness laminated composite plates (VVSLC) are studied using a generalized Maxwell model and finite element method. The integral form of the viscoelastic constitutive relation is converted to the incremental form for finite element formulation based on the Reissner–Mindlin plate theory. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response directly in the time domain is obtained using shooting technique coupled with Newmark’s time integration method. The implementation of shooting technique for Boltzmann integral-based viscoelasticity for curvilinear fibre composite plates is done for the first time in this study. For the comparison purpose, the response/resonance frequency/modal loss factor is also obtained using equivalent complex modulus based viscoelastic correspondence principle. It is observed that the variation in fibre orientation and boundary conditions leads to significant variations in response, stress/moment resultant amplitude and the damping factor of the VVSLC plates. Further, the present time domain based approach is capable of predicting damping factor at all forcing frequencies whereas complex eigenvalue analysis can predict damping factor only at discrete resonance frequency. Based on the detailed studies, it is found that the curvilinear fibre composite plates depict a significant reduction in response/moment resultants compared to straight fibre composite plates.

A new finite element formulation for the lateral torsional buckling analyses of orthotropic FRP-externally bonded steel beams

Abstract The present study develops a new finite element (FE) formulation for the lateral torsional buckling (LTB) analyses of steel beams exteriorly bonded with orthotropic fiber-reinforced polymer (FRP) layers. The formulation considers shear deformations, partial material interaction, local and global warping deformations, and orthotropic FRP properties. The buckling responses of multispan FRP-bonded steel beams predicted by the present solutions are excellently validated against experiment and numerical solutions. As observed, the FRP strengthening is highly effective for the LTB resistance of steel beams. However, the LTB responses are strongly dependent on the orthotropic properties and strengthening lengths of FRP layers. The effects of shear deformations, span ratios, and loading conditions on the LTB responses are also quantified in the present study.

Diffuse interface modeling of non‐isothermal Stokes‐Darcy flow with immersed transmissibility conditions

AbstractThe coupling between free and porous medium flows has received significant attention since it plays an important role in a wide range of problems from fluid‐soil interactions to biofluid dynamics. However, modeling this coupled process remains a difficult task as it often involves a domain decomposition algorithm in conjunction with a special treatment at the interface. The problem can become more challenging under non‐isothermal conditions because it requires the iterative procedure at every time step to simultaneously meet the transient mass continuity, force equilibrium, and energy balance for the entire system. This article presents a diffuse interface framework for modeling non‐isothermal Stokes‐Darcy flow and the corresponding finite element formulation that bypasses the need for explicitly splitting the domain into two, which enables the unified treatment for distinct regions with different hydrothermal flow regimes. To achieve this goal, we employ the Allen‐Cahn type phase field model to generate the diffuse geometry, where the solution field can be seen as a regularized approximation of the Heaviside indicator function, allowing us to transfer the interface conditions into a set of immersed boundary conditions. Our formulation suggests that the isothermal operator splitting strategy can be adopted without compromising accuracy if the heat and mass transfer processes are decoupled by assuming that the density and viscosity of the phase constituents are independent to the temperature. Numerical examples are also introduced to verify the implementation and to demonstrate the model capacity.

A reduced-order finite element formulation for the geometrically nonlinear dynamic analysis of viscoelastic structures based on the fractional-order derivative constitutive relation

A reduced-order finite element formulation for the geometrically nonlinear dynamic analysis of viscoelastic structures based on the fractional-order derivative constitutive relation

An optimally convergent Fictitious Domain method for interface problems

We introduce a novel Fictitious Domain (FD) unfitted method for interface problems associated with a second-order elliptic linear differential operator, that achieves optimal convergence without the need for adaptive mesh refinements nor enrichments of the Finite Element spaces. The key aspect of the proposed method is that it extends the solution into the fictitious domain in a way that ensures high global regularity. Continuity of the solution across the interface is enforced through a boundary Lagrange multiplier. The subdomains coupling, however, is not achieved by means of the duality pairing with the Lagrange multiplier, but through an L2 product with the H1 Riesz representative of the latter, thus avoiding gradient jumps across the interface. Thanks to the enhanced regularity, the proposed method attains an increase, with respect to standard FD methods, of up to one order of convergence in energy norm. The Finite Element formulation of the method is presented, followed by its analysis. Numerical tests on a model problem demonstrate its effectiveness and its superior accuracy compared to standard unfitted methods.

A cable finite element formulation based on exact tension field for static nonlinear analysis of cable structures

This paper presents a numerically exact cable finite element model for static nonlinear analysis of cable structures. The model derives the exact expression of the tension field using the geometrically exact beam theory coupled with the fundamental mechanical characteristics of cables. The equations for the cable elements are formulated by addressing the equilibrium conditions at the element boundaries and ensuring compatibility within the elements. Unlike previous studies that typically provide explicit expressions for cable models, this study develops a formulation that emphasizes numerical precision and broad applicability. It achieves this by deriving linearized equations with implicit expressions incorporating integrals. The present model accurately computes internal forces and deformation states, and determines the unstrained length of the cable. Additionally, it accounts for the variability in cross-sectional stiffness along the cable's length. The paper discusses solution implementations using the complete tangent matrix and element internal iterations. The effectiveness of the proposed cable elements is demonstrated through numerical examples.

A layered solid finite element formulation with interlaminar enhanced displacements for the modeling of laminated composite structures

A layered solid finite element formulation with interlaminar enhanced displacements for the modeling of laminated composite structures

High-thermal free vibration analysis of functionally graded microplates using a new finite element formulation based on TSDT and MSCT

Recent advancements in additive manufacturing (AM) have revolutionized the design and production of complex engineering microstructures. Despite these advancements, their mathematical modeling and computational analysis remain significant challenges. This research aims to develop an effective computational method for analyzing the free vibration of functionally graded (FG) microplates under high temperatures while resting on a Pasternak foundation (PF). This formulation leverages a new third-order shear deformation theory (new TSDT) for improved accuracy without requiring shear correction factors. Additionally, the modified couple stress theory (MCST) is incorporated to account for size-dependent effects in microplates. The PF is characterized by two parameters including spring stiffness () and shear layer stiffness (). To validate the proposed method, the results obtained are compared with those of the existing literature. Furthermore, numerical examples explore the influence of various factors on the high-temperature free vibration of FG microplates. These factors include the length scale parameter (), geometric dimensions, material properties, and the presence of the elastic foundation. The findings significantly enhance our comprehension of the free vibration of FG microplates in high thermal environments. In addition, the findings significantly enhance our comprehension of the free vibration of FG microplates in high thermal environments. In addition, the results of this research will have great potential in military and defense applications such as components of submarines, fighter aircraft, and missiles.

Macroscopic modeling of flexoelectricity-driven remanent polarization in piezoceramics

This paper presents a variational modeling framework for investigating the flexoelectricity-driven evolution of remanent polarization in piezoceramics. In small-scale electromechanical systems, strain gradients can exhibit polarization in dielectric materials via the direct flexoelectric effect. In ferroelectrics, it is reasonable to expect that a sufficiently large magnitude of the flexoelectricity leads to a switching of the domain structure and thus the material becomes remanently polarized. It is interesting to note that this means that poling would be able to occur in the absence of any external electrical source. This provides the motivation to gain a better understanding of this effect for a possible technical use in e.g. sensor applications. For this purpose, a macroscopic model is presented that couples flexoelectricity and ferroelectric domain switching processes. By embedding the model in the variational framework of the generalized standard materials (GSM), a minimum-type potential structure and thus a stable and efficient numerical treatment is obtained. A mixed finite element formulation based on the Helmholtz free energy is introduced to solve the higher-order flexoelectric boundary value problem. In order to realistically predict the flexoelectric material behavior, the model response is adapted to experimental results in literature obtained for a piezoceramic in a bending test. By simulations based on the adapted model the evolution of the flexoelectricity-driven remanent polarization in the vicinity of a notch is shown.

Accuracy and efficiency of finite element head models: The role of finite element formulation and material laws.

Traumatic brain injury is a significant problem worldwide. In the United States of America, around 1.7 million cases are documented annually, displaying the need for a deeper understanding of the effects on the human brain. The tests required for this assessment are very complex. Tests on cadavers may raise serious ethical questions, and in vivo crash tests are not viable. In this context, there is a great need to developing finite element head models (FEHM) to study the biomechanics of the tissues when submitted to a certain impact or acceleration/deceleration scenario. An excellent compromise between accuracy and CPU efficiency is always desirable for a FEHM, For this reason, this work focuses on the improvement of an existing head model, including the study of the behavior of the brain using distinct finite element types. The finite element type and formulation is of utmost importance for the general accuracy and efficiency of the models. Several validations were performed, comparing the simulation results against experimental data. The simulations with hexahedral elements, under specific conditions, obtained more accurate results with a lower computational cost. Using hexahedrals, a comparison was also performed using two material characterizations with more than 10 years apart, using the latest finite element head model validation experiment. Overall, the newer material model displays a less stiff response, although its implementation must always depend on the overall purpose of the model it is being applied to.

Discretisations of mixed-dimensional Thermo-Hydro-Mechanical models preserving energy estimates

In this study, we explore mixed-dimensional Thermo-Hydro-Mechanical (THM) models in fractured porous media accounting for Coulomb frictional contact at matrix fracture interfaces. The simulation of such models plays an important role in many applications such as hydraulic stimulation in deep geothermal systems and assessing induced seismic risks in CO2 storage. We first extend to the mixed-dimensional framework the thermodynamically consistent THM models derived in [19] based on first and second principles of thermodynamics. Two formulations of the energy equation will be considered based either on energy conservation or on the entropy balance, assuming a vanishing thermo-poro-elastic dissipation. Our focus is on space time discretisations preserving energy estimates for both types of formulations and for a general single phase fluid thermodynamical model. This is achieved by a Finite Volume discretisation of the non-isothermal flow based on coercive fluxes and a tailored discretisation of the non-conservative convective terms. It is combined with a mixed Finite Element formulation of the contact-mechanical model with face-wise constant Lagrange multipliers accounting for the surface tractions, which preserves the dissipative properties of the contact terms. The discretisations of both THM formulations are investigated and compared in terms of convergence, accuracy and robustness on 2D test cases. It includes a Discrete Fracture Matrix model with a convection dominated thermal regime, and either a weakly compressible liquid or a highly compressible gas thermodynamical model.

Finite element solution of coupled multiphysics reaction-diffusion equations for fracture healing in hard biological tissues

This study proposes a computational framework to investigate the multi-stage process of fracture healing in hard tissues, e.g., long bone, based on the mathematical Bailon-Plaza and Van der Meulen formulation. The goal is to explore the influence of critical biological factors by employing the finite element method for more realistic configurations. The model integrates a set of variables, including cell densities, growth factors, and extracellular matrix contents, managed by a coupled system of partial differential equations. A weak finite element formulation is introduced to enhance the numerical robustness for coarser mesh grids, complex geometries, and more accurate boundary conditions. This formulation is less sensitive to mesh quality and converges smoothly with mesh refinement, exhibiting superior numerical stability compared to previously available strong-form solutions. The model accurately reproduces various stages of healing, including soft cartilage callus formation, endochondral and intramembranous ossification, and hard bony callus development for various sizes of fracture gap. Model predictions align with the existing research and are logically coherent with the available experimental data. The developed multiphysics simulation clarifies the coordination of cellular dynamics, extracellular matrix alterations, and signaling growth factors during fracture healing.

Evaluation of the Transfemoral Bone-Implant Interface Properties Using Vibration Analysis.

Evaluating the bone-implant interface (BII) properties of osseointegrated transfemoral (TFA) implants is important for early failure detection and prescribing loads during rehabilitation. The objective of this work is to derive and validate a 1D finite element (FE) model of the Osseointegrated Prosthetic Limb (OPL) TFA system that can: (1) model its dynamic behaviour and (2) extract the BII properties. The model was validated by: (1) comparing the 1D FE formulation to the analytical and 3D FE solutions for a simplifiedcylinder, (2) comparing the vibration modes of the actual TFA geometry using 1D and 3D FE models, and (3) evaluating the BII properties for three extreme conditions (LOW, INTERMEDIATE, and HIGH) generated using 3D FE and experimental (where the implant was embedded, using different adhesives, in synthetic femurs) signals for additional validation. The modes predicted by the 1D FE model converged to the analytical and the 3D FE solutions for the cylinder. The 1D model also matched the 3D FE solution with a maximum frequency difference of 2.02% for the TFA geometry. Finally, the 1D model extracted the BII stiffness and the system's damping properties for the three conditions generated using the 3DFE simulations and the experimental INTERMEDIATE and HIGH signals. The agreement between the 1D FE and the 3D FE solutions for the TFA geometry indicates that the 1D model captures the system's dynamic behaviour. Distinguishing between the different BII conditions demonstrates the 1D model's potential usefor the non-invasive clinical evaluation of the TFA BII properties.

Modeling the mechanosensitive collective migration of cells on the surface and the interior of morphing soft tissues.

Cellular contractility, migration, and extracellular matrix (ECM) mechanics are critical for a wide range of biological processes including embryonic development, wound healing, tissue morphogenesis, and regeneration. Even though the distinct response of cells near the tissue periphery has been previously observed in cell-laden microtissues, including faster kinetics and more prominent cell-ECM interactions, there are currently no models that can fully combine coupled surface and bulk mechanics and kinetics to recapitulate the morphogenic response of these constructs. Mailand et al. (Biophys J 117(5):975-986, 2019) had shown the importance of active elastocapillarity in cell-laden microtissues, but modeling the distinct mechanosensitive migration of cells on the periphery and the interior of highly deforming tissues has not been possible thus far, especially in the presence of active elastocapillary effects. This paper presents a framework for understanding the interplay between cellular contractility, migration, and ECM mechanics in dynamically morphing soft tissues accounting for distinct cellular responses in the bulk and the surface of tissues. The major novelty of this approach is that it enables modeling the distinct migratory and contractile response of cells residing on the tissue surface and the bulk, where concurrently the morphing soft tissues undergo large deformations driven by cell contractility. Additionally, the simulation results capture the changes in shape and cell concentration for wounded and intact microtissues, enabling the interpretation of experimental data. The numerical procedure that accounts for mechanosensitive stress generation, large deformations, diffusive migration in the bulk and a distinct mechanism for diffusive migration on deforming surfaces is inspired from recent work on bulk and surface poroelasticity of hydrogels involving elastocapillary effects, but in this work, a two-field weak form is proposed and is able to alleviate numerical instabilities that were observed in the original method that utilized a three-field mixed finite element formulation.

Coupled higher-order layerwise mechanics and finite element formulations for laminated composite beams with active SMA layers

This study proposes a thermo-elastic coupled nonlinear finite element (FE) model for laminated composite beams with active SMA layers based on a higher-order layerwise theory and Brinson's model of SMA's phase transformation constitutive relations. The developed FE Model is further discretized by using the Newton-Raphson time integration scheme to update SMA's phase transformation progress step by step. This new model can be used for thermal-mechanical behavior analysis of hybrid SMA-composite laminated beams, which especially enables accurate prediction for static response of moderately thick to thick beams, large deformation analysis, through-thickness variations of interlaminar stresses and strains. The accuracy and effectiveness of the proposed approach are verified by several numerical examples, and the results are compared with those obtained from corresponding alternative solutions and experimental tests. Using the developed FEM, the effects of temperature, geometrical nonlinearity, pre-strain of SMA, and stacking sequence of composite laminates on the static response of hybrid SMA-composite beams are studied.

A computational study for simulating MHD duct flows at high Hartmann numbers using a stabilized finite element formulation with shock-capturing

The concern of this manuscript is stabilized finite element simulations of two-dimensional incompressible and viscous magnetohydrodynamic (MHD) duct flows governed by a coupled system of partial differential equations of the convection–diffusion type. In such flows, the high values of the Hartmann numbers (Ha) indicate that the convection process dominates the flow field. As is typical trouble encountered in convection-dominated flow simulations, classical discretization methods fail to function properly for MHD flows at high Hartmann numbers, yielding several numerical instabilities. In this study, the core computational tool we employ in order to overcome such challenges is the streamline-upwind/Petrov–Galerkin (SUPG) formulation. Beyond that, since the SUPG-stabilized formulation requires additional treatment where solutions experience rapid changes, we also enhance the SUPG-stabilized finite element formulation with a shock-capturing operator for achieving better solution profiles around sharp layers. The formulations are derived for both steady-state and time-dependent models. The proposed method, the so-called SUPG-YZβ formulation, techniques used, and in-house-developed finite element solvers are evaluated on a comprehensive set of numerical experiments. The test computations reveal that the SUPG-YZβ formulation yields quite good solution profiles near strong gradients without any significant numerical instabilities, even for quite challenging values of Ha, whereas the SUPG usually fails alone. Furthermore, by using only linear elements on relatively coarser meshes, the proposed formulation achieves this without the need for any adaptive mesh strategy, in contrast to most previously reported studies.