Nonlinear Finite Element Algorithm Research Articles

A numerical approach to bound the asymmetrically surface-damaged metallic strands response accounting for bending stiffness nonlinearity: Damage and structural integrity evaluation purposes

This study aims to propose an alternative numerical-based methodology to bound the response of asymmetrically surface-damaged metallic strand for damage and structural integrity evaluation purposes. A standard nonlinear finite element algorithm is extended and utilized in which a damaged strand is discretized into uniaxial two-noded nonlinear cable-beam elements in a 3-D space. This extension contemplates accounting for two limit cases regarding the contact regime of the strand wires: (i) stick regime obeying the Bernoulli’s kinematic hypothesis; and (ii) full slip regime leading to a nonlinear bending stiffness. The limit between both contact regimes is given by a critical curvature value associated to the bending response of the damaged strand induced by the asymmetry in surface damage distribution. In this way, the use of the proposed methodology gives rise to a solution space for each parameter of interest namely residual strength, residual linear stiffness, waviness, and fracture energy. The solution spaces are defined by the estimates associated to the purely stick contact regime (Stick model) and the ones that account for the nonlinearity of the bending stiffness (Stick-Slip model). 1 × 7 metallic strands with one and two surface wires cut with diameter values ranging from 1.5 mm to 14.3 mm are used to show the capabilities of the methodology which are validated by comparisons with measured data. Results suggest that curvature boundary layers at strands ends are developed due to the induced bending and that space solutions appropriately bound the measured data. Moderate and high effects on the residual linear stiffness and fracture energy, and waviness parameters due to wires slippage are observed respectively, relative to consider a complete stick contact regime.

Nonlinear finite element algorithm for solving fully coupled thermomechanical problems under strong aerothermodynamic environment

Nonlinear finite element algorithm for solving fully coupled thermomechanical problems under strong aerothermodynamic environment

Optimization of Fiber-Reinforced Polymer Bars for Reinforced Concrete Column Using Nonlinear Finite Element Algorithms

The ductility and strength of reinforced concrete (RC) columns could be noticeably improved by replacing steel bars with polymeric bars. Despite the previous research on RC columns, most of those studies focused only on the lateral load capacity of this structural member and were mainly costly experimental studies. However, this paper is concentrated on the previously occurred damages to the reinforced columns in the previous earthquakes. Subsequently, finite element analysis has been performed to examine 24 models including the various shapes of RC columns. In employing the plastic behavior of steel, carbon fiber-reinforced polymer (CFRP), and glass fiber reinforced polymer (GFRP) bars, the bilinear hardening has been considered. To capture both compressive and tensile behavior of the concrete, the concrete damage plasticity model has been implemented. Furthermore, the optimization technique is used for CFRP models to compare with other models. In this paper, the parameters of energy, seismic factor, stiffness, and ductility have been computed using the method proposed by the authors. This suggested method is considered to compare the results from each parameter. Finite element results of steel bars are compared with carbon and glass models. The results show the stiffness of models is improved by CFRP bars, while the energy absorption and ductility factor are enhanced with steel bars. Moreover, GFRP bars can enhance the seismic factor. The reduction of column stiffness to almost half would occur in some rectangular cross-section columns.

The nonlinear thermo-hyperelasticity wave propagation analysis of near-incompressible functionally graded medium under mechanical and thermal loadings

This article presents a non-Fourier thermo-hyperelastic model to investigate thermal and stress wave propagation phenomenon in a near-incompressible functionally graded medium (FGM) for various thermal and mechanical boundary conditions. A strain energy function is chosen to modify FGM’s hyperelastic equations considering the coupling effects of mechanical and thermal terms. By switching the strain tensor's invariants, equations are developed to estimate a near-incompressible model for rubber. The rubber is characterized by a gradual variation in the longitudinal direction. Therefore, the material properties of rubber mainly depend on coordinates in through an exponential function. The nonlinear governing equations are derived from the large displacement approach using Finite Strain Theory. To find an acceptable solution of nonlinear time-dependent thermo-hyperelastic equations, Newmark's time integration process and a nonlinear Hermitian finite element algorithm are employed. The final system’s responses to different boundary conditions such as input surface traction, heat flux and variable material properties are described schematically, and their influence on the wave propagation is calculated. It is shown that the amplitude of oscillation in a functionally graded hyperelastic medium is less than that of a medium with constant properties. The results also show that the wave travels through the medium faster than the FGM. Moreover, the modified Fourier law of heat conduction is applied and the impact of enhanced heat conduction model on the thermo-hyperelastic responses is discussed.

Design of crashworthy attenuator structures as a part of vehicle safety against impact: Application of waste aluminum can-based material

The impact attenuator is an essential system in both race cars and urban vehicles. The structure of an impact attenuator serves as a safety barrier between the impacted surface and the driver in an accident. Attenuator materials tend to have a high price; thus, alternative materials were explored in the current work, i.e., used cans from food and beverage containers. The study deployed a nonlinear finite element algorithm to calculate a series of impacts on the attenuator structures. The thickness of the cans and velocity of the impact were considered as the main parameters. Analysis results concluded that the attenuator's average energy was 16000 J for a can thickness of 1 mm. This value is more than two times the 0.5 mm thick used cans. The attenuator's new design was then matched with an attenuator regulation, and the results surpassed the standard value of 7350 J.

Dynamic load-bearing capacity analysis for the main Transmission Mechanism of Traction Machine in a Cargo Lift

This paper discusses the dynamic load-bearing capacity of the main transmission mechanism of worm pair traction machine in a cargo lift. By fully considering the mass, rotational inertia and friction for transmission of the worm pair, multi-body dynamics was adopted to calculate the worm gear teeth contact force and motion characteristics. Based on the nonlinear finite element algorithm for friction-contact and combined with the dynamics computation results, the finite element calculation for the teeth contact strength and bending strength were performed. The finite element calculation results and theoretical calculation results were compared to analyze their difference. The mechanism quality was evaluated from the running stability and mechanical strength, and the evaluation results show that the main transmission mechanism of traction machine in the cargo lift has good dynamic load-bearing capacity and high safety, and meets the design requirements.

Compliant structure under follower forces and any combined loading: Theoretical and experimental studies

A linear finite element and a nonlinear finite element solution scheme have been proposed to obtain the deflection of a compliant structure, modeled as a cantilever beam, under the action of a follower force acting at the end, in addition to any additional arbitrary loading. The developed techniques have been compared with a numerical technique available in literature, wherein, nonlinear Euler–Bernoulli beam theory is used in an iterative scheme. It was found that the results of the nonlinear finite element algorithm exactly match the solution from the available scheme; while on the other hand, the linear finite element gives fairly close results at a higher rate of convergence. Experiments are carried out to validate the two proposed solution techniques. Two example problems have been solved, which show the practical application of this numerical scheme, in the design of mechanisms actuated by smart materials like shape memory alloys (SMA) and macro fiber composites (MFC), in the process, obviating the limitations of the available techniques in solving such problems.

A general framework for the modeling of porous ferrogels at finite strains

Porous ferrogels are a new class of magneto-active composite materials that deform and alter their material characteristics under the influence of magnetic fields. In the future such materials could find a wide range of application in biomedicine and microfluidics. In this work we present a theoretical and computational framework for the macroscopic, continuum-based modeling of porous ferrogels at finite strains. Departing from the balance laws of continuum mechanics, we derive a thermodynamically consistent model based on Biot’s consolidation theory. Regarding constitutive modeling we limit our attention to isotropic materials. Furthermore we discuss details of the numerical implementation of the coupled three field problem within a nonlinear finite element algorithm. The modeling capabilities and algorithmic performance are demonstrated by means of two representative initial boundary value problems.

Correlative analysis of head kinematics and brain’s tissue response: a computational approach toward understanding the mechanisms of blast TBI

Upon impingement of blast waves on the head, stress waves generated at the interface of the skull are transferred into the cranium and the brain tissue and may cause mild to severe blast traumatic brain injury. The intensity of the shock front, defined by the blast overpressure (BoP), that is, the blast-induced peak static overpressure, significantly affects head kinematics as well as the tissue responses of the brain. While evaluation of global linear and rotational accelerations may be feasible, an experimental determination of dynamic responses of the brain in terms of intracranial pressure (ICP), maximum shear stress (MSS), and maximum principal strain (MPS) is almost impossible. The main objective of this study is to investigate possible correlations between head accelerations and the brain’s ICP, MSS, and MPS. To this end, three different blasts were simulated by modeling the detonation of 70, 200, and 500 g of TNT at a fixed distance from the head, corresponding to peak BoPs of 0.52, 1.2, and 2 MPa, respectively. A nonlinear multi-material finite element algorithm was implemented in the LS-DYNA explicit solver. Fluid–solid interaction between the blast waves and head was modeled using a penalty-based method. Strong correlations were found between the brain’s dynamic responses and both global linear and rotational accelerations at different blast intensities ( $$R^{2 }\ge $$ 98%), implying that global kinematic parameters of the head might be strong predictors of brain tissue biomechanical parameters.

Optimal vibration energy harvesting from nonprismatic piezolaminated beam

The present article encompasses a nonlinear finite element (FE) and genetic algorithm (GA) based optimal vibration energy harvesting from nonprismatic piezo-laminated cantilever beams. Three cases of cross section profiles (such as linear, parabolic and cubic) are modelled to analyse the geometric nonlinear effects on the output responses such as displacement, voltage, and power. The simultaneous effects of taper ratios (such as breadth and height taper) on the output power are also studied. The FE based nonlinear dynamic equation of motion has been solved by an implicit integration method (i.e., Newmark method in conjunction with the Newton-Raphson method). Besides this, a real coded GA based constrained optimization scheme has also been proposed to determine the best set of design variables for optimal harvesting of power within the safe limits of beam stress and PZT breakdown voltage.

Analysis of dynamic meshing characteristic of planetary gear transmission in wind power increasing gearbox

Dynamic behavior of planetary gear’s tooth contact surface in the different location can better conform operation condition comparing to the general gear pair. Nonlinear finite element algorithm was derived according to the basic control equation of contact dynamics. A finite element model of planetary gear transmission in wind power increasing gearbox was proposed considering different meshing locations based on nonlinear finite element solution. The characteristics of stress distribution at different meshing positions were analyzed. A simulation of the meshing process was conducted using finite element analysis. It was shown that node stresses of external meshing planetary gear varied significantly at different position. The analysis provides some useful insights into the performance of planetary gear’s tooth contact surface.

Geometric nonlinear finite element and genetic algorithm based optimal vibration energy harvesting from nonprismatic axially functionally graded piezolaminated beam

The present article deals with optimal vibration energy harvesting from an axially functionally graded (FG) non-prismatic piezolaminated beam using genetic algorithm (GA). Geometric nonlinear based finite element (FE) formulation using Newmark method in conjunction with Newton-Raphson method is formulated to solve the obtained governing equation. A two noded beam element with two degrees of freedom (DOF) at each node is used in the present FE formulation. The FG material (i.e. non-homogeneity) in the axial direction is considered which varies (continuously decreasing from root to tip of such cantilever beam) using a proposed power law formula. The various cross section profiles (such as linear, parabolic and cubic) are modelled using the Euler-Bernoulli beam theory. The effects of nonlinearity on the responses (such as displacement, voltage and output power) are deliberated with arbitrary power gradient index. The effects of tapers (both width and height in length directions) on the output power and voltage are analysed. A real-coded genetic algorithm based constrained optimization technique is also proposed to determine the best possible design variables for optimal power harvesting within the allowable limits of ultimate stress of beam and breakdown voltage of PZT sensor.

Nonlinear Numerical Simulation of Reinforced Concrete Columns Under Cyclic Biaxial Bending Moment and Axial Loading

A nonlinear finite element algorithm is proposed to analyze the reinforced concrete (RC) columns subjected to cyclic biaxial bending moment and axial loading. In the proposed algorithm, the following parameters are considered: uniaxial behavior of concrete and steel elements, the pseudo-plastic hinge produced in the critical sections, and global behavior of the columns. In the proposed numerical simulation, the column is discretized into two macro-elements located between the pseudo-plastic hinges at critical sections and the inflection point. The critical sections are discretized into fixed rectangular finite elements. The basic equilibrium is justified over a critical hypothetical cross section assuming the kinematics Navier’s hypothesis with an average curvature. The method used qualifies as a “strain plane control process” that requires the resolution of a quasi-static simultaneous equation system using a triple iteration process over the strains in each section. To reach equilibrium, three main strain parameters (the strains in the extreme compressive point, the strains in the extreme tensile point and the strains in another corner of the section) are used as the three main variables. The proposed algorithm has been validated by the results of tests carried out on full-scale RC columns. The application of the components effects combination method is also compared with the proposed simultaneous direct method. The results obtained show the necessity of applying SDM for the post-elastic phase, which occurs frequently during earthquake loading.

Numerical Investigation of the Role of Mill Scale Crevices on the Corrosion Initiation of Carbon Steel Reinforcement in Concrete

A numerical investigation was conducted to test the hypothesis that the composition of the pore solution in mill scale crevices on carbon steel rebar surfaces in concrete might be different from that of the bulk concrete pore solution, and this difference may create the necessary conditions for the premature breakdown of the passive film. The modeling was performed using a non-linear transient finite element algorithm, which involved the solution of coupled extended Nernst-Planck and Poisson's equations in a domain that represented typical mill scale crevices on carbon steel rebar. The numerical simulations showed that the chemistry of the pore solution, in particular pH and Cl−/OH−, within mill scale crevices provided more favorable conditions for depassivation than the bulk concrete pore solution. Local acidification and increase in Cl−/OH− within crevices were observed in all simulations, albeit to different degrees. Crevice geometry has been found to be the most important parameter affecting local acidification and the increase in Cl−/OH−. Simulations supported the hypothesis that the chemical composition of the pore solution within the crevices differs from that of the bulk solution through a process similar to the suggested mechanism of typical crevice corrosion.

Nonlinear numerical simulation of RC columns subjected to cyclic oriented lateral force and axial loading

A nonlinear Finite Element (FE) algorithm is proposed to analyze the Reinforced Concrete (RC) columns subjected to Cyclic Loading (CL), Cyclic Oriented Lateral Force and Axial Loading (COLFAL), Monotonic Loading (ML) or Oriented Pushover Force and Axial Loading (OPFAL) in any direction. In the proposed algorithm, the following parameters are considered: uniaxial behavior of concrete and steel elements, the pseudo-plastic hinge produced in the critical sections, and global behavior of RC columns. In the proposed numerical simulation, the column is discretized into two Macro-Elements (ME) located between the pseudo-plastic hinges at critical sections and the inflection point. The critical sections are discretized into Fixed Rectangular Finite Elements (FRFE) in general cases of CL, COLFAL or ML and are discretized into Variable Oblique Finite Elements (VOFE) in the particular cases of ML or OPFAL. For pushover particular case, a fairly fast converging and properly accurate nonlinear simulation method is proposed to assess the behavior of RC columns. The proposed algorithm has been validated by the results of tests carried out on full-scale RC columns.

Simulation of 3D parachute fluid–structure interaction based on nonlinear finite element method and preconditioning finite volume method

A fluid–structure interaction method combining a nonlinear finite element algorithm with a preconditioning finite volume method is proposed in this paper to simulate parachute transient dynamics. This method uses a three-dimensional membrane–cable fabric model to represent a parachute system at a highly folded configuration. The large shape change during parachute inflation is computed by the nonlinear Newton–Raphson iteration and the linear system equation is solved by the generalized minimal residual (GMRES) method. A membrane wrinkling algorithm is also utilized to evaluate the special uniaxial tension state of membrane elements on the parachute canopy. In order to avoid large time expenses during structural nonlinear iteration, the implicit Hilber–Hughes–Taylor (HHT) time integration method is employed. For the fluid dynamic simulations, the Roe and HLLC (Harten–Lax–van Leer contact) scheme has been modified and extended to compute flow problems at all speeds. The lower–upper symmetric Gauss–Seidel (LU-SGS) approximate factorization is applied to accelerate the numerical convergence speed. Finally, the test model of a highly folded C-9 parachute is simulated at a prescribed speed and the results show similar characteristics compared with experimental results and previous literature.

A multiplicative finite element algorithm for the inhomogeneous swelling of polymeric gels

The paper presents a nonlinear Finite Element (FE) algorithm to simulate the inhomogeneous steady swelling of polymeric gels. The algorithm is developed based on the multiplicative decomposition of the deformation gradient into an elastic part and a swelling part. The corresponding constitutive framework and linearization of the FE equation are elaborated, which leads to an equivalent body force driving the materials to swell. A staggered iterative procedure is designed to ensure the strongly coupled chemical and mechanical fields to reach the equilibrium state simultaneously. The constitutive equations are constructed in the framework of the volumetric–isochoric splitting of the free energy function, which is helpful both for a different treatment of the incompressible part in the FE equation and for a physical interpretation of the coupling effects between the chemical and mechanical fields. In the present work, an additive free energy function, coupling the contributions of the Flory–Huggins model and the non-Gauss statistical–mechanical model, is adopted and the corresponding consistent tangent modulus in the current configuration is also derived. To implement the FE algorithm, two two-dimensional (2D) elements and one three-dimensional (3D) element are developed, which are validated by some analytical solutions, such as free swelling of gels in 2D and 3D cases, squeezing a swollen gel with uniform pressure and the constrained swelling of a single-layer and a core–shell gel ring. The algorithm is also proved to be capable of handling the surface instabilities and some engineering applications such as hydrogel-based composite beams and thin films.

Moving Grid Method for Simulating Crack Propagation

A moving grid nonlinear finite element method was used in this study to simulate crack propagation. The relevant elements were split along the direction of principal stress within the element and thus automatic optimization processing of local mesh was realized. We discussed the moving grid nonlinear finite element algorithm was proposed, compiled the corresponding script files based on the dedicated finite element language of Finite Element Program Generator (FEPG), and generate finite element source code programs according to the script files. Analyses show that the proposed moving grid finite element method is effective and feasible in crack propagation simulation.

Behavior and modeling of superelastic shape memory alloy reinforced concrete beams

The re-centering phenomenon of superelastic Shape Memory Alloy (SMA) reinforced concrete is a unique characteristic that is appealing for structural applications, along with the ability to respond with stable hystereses and achieve similar strength and ductility to concrete reinforced with conventional deformed bars. The objective of this study was to investigate the structural performance of superelastic SMA reinforced concrete and to develop a preliminary constitutive model applicable to nonlinear finite element algorithms. Seven simply supported flexure-critical concrete beams, reinforced with either SMA bars in the critical region or conventional deformed reinforcement, were subjected to monotonic, cyclic, and reverse cyclic loading. The experiment results demonstrated the superior capacity of the SMA beams to recover inelastic displacements. The SMA beams sustained displacement ductility and strength capacity comparable to the conventional beams. Crack widths and crack spacing were larger in the SMA beams; however, upon removal of load, the crack openings were recovered. Energy dissipation was lower in the SMA beams, particularly when subjected to reverse cyclic loading. The constitutive model based on a trilinear backbone envelope response and linear unloading and reloading rules provided satisfactory simulations.

Comparison between neural network and response surface metamodels based on D-optimal designs

In this paper, two metamodelling techniques namely, the neural network and the response surface methodology are used and compared to approximate a multidimensional function to predict the springback amount of metallic sheets in the bending process. The training data required to train the two metamodelling techniques were generated using a verified non-linear finite element algorithm developed in this research. The algorithm is based on the updated Lagrangian formulation, which takes into consideration geometrical, material non-linearity, and contact. A neural network algorithm based on the back propagation algorithm has been developed. This research utilises computer generated D-optimal designs to select training examples for both metamodelling techniques so that a comparison between the two techniques can be considered as fair. Results from this research showed that the neural network metamodels outperform the response surface metamodels.