Nonlinear Finite Element Solution Research Articles

Optimal lamination angles via exact and efficient differentiation of the geometrically nonlinear finite element solution

Laminates allow tailoring the fiber orientations in the layers to obtain the desired mechanical response. The optimal layup design is a challenging task in the case of finite deformations and buckling. For an assigned design, an incremental-iterative finite element analysis is needed to compute the structural response. Gradient-based methods are very often the most efficient optimization tools. Their bottleneck is the gradient evaluation, generally possible only approximately by finite differences. This article shows how to compute the exact gradient of the geometrically nonlinear finite element solution with respect to the stacking sequence. The strategy relies on the implicit differentiation of the nonlinear discrete equations of a control equilibrium point corresponding to an assigned displacement. This provides the load factor gradient by a single fast-solution linear system with the already factorized tangent stiffness matrix, regardless of the number of design variables, and scalar products involving the partial derivatives of the discrete internal force vector, calculated in an exact and efficient way. Several applications demonstrate the efficiency and robustness of the approach.

A novel approach for generalized Green-Naghdi-type electro-magneto-thermo-hyperelasticity wave propagation and reflection investigations in near-incompressible layers under shock loads

A novel nonlinear coupled finite-strain electro-magneto-thermo-hyperelasticity (EMTHE) model is developed for the first- and second-sound wave propagation and reflection investigations in media that are exposed to electromagnetic fields and mechanical shocks. In contrast to the previous studies which studied behaviors of the finite-strain elastic or incompressible materials with small deformations, the current research proposes a much more complicated but much more accurate novel practical model for wave propagation and reflection analyses in near-incompressible finite-strain materials. Furthermore, to evaluate the effects of the electro-magneto-thermomechanical coupling, the strain energy density function of the hyperelastic material is expanded in a new way. The governing equations are obtained according to a nonlinear version of the Helmholtz free energy. The energy equations comprise the first- and second-order time rates of the temperature to enable the modeling of the finite-speed heat transfer; i.e., the establishment of a second-sound model. A nonlinear iterative finite element solution algorithm is proposed and implemented for the resulting coupled time-dependent generalized electro-magneto-thermo-hyperelasticity equations. The results show significant differences between the predicted wave propagation and reflection characteristics and behaviors of the near-incompressible and incompressible finite-strain models.

Large Deformation Analysis of Hyperelastic Continuum with Hexahedral Adaptive Finite Elements

The use of hyperelastic materials capable of large deformations, such as elastomeric bearings used to reduce seismic effects, is quite common in civil engineering. Such environments are, in most cases, addressed by numerical solution techniques such as the finite element method. In case of large deformations, nonlinear analysis is used in the solution. In the study presented here, large deformations of a hyperelastic continuum expressed by the Mooney-Rivlin material model are calculated using hexahedral adaptive finite elements. A code was written in MATLAB using the total Lagrangian formulation for the nonlinear adaptive finite element solution. Comparisons were made with Abaqus software to check the consistency of the results obtained from this program. It has been observed that local refinements in the adaptive element mesh occur in the regions where they are needed. Considering the variation of maximum displacement and maximum stress with the number of elements, it has been observed that mesh refinement creates a convergent solution.

A coupled nonlinear finite element scheme for anisotropic diffusion equation with nonlinear capacity term

A new nonlinear finite element method is studied to solve multi-dimensional anisotropic nonlinear diffusion equation with nonlinear capacity term, especially to secure a vivid simulation in the case of problems with transient physical quantities. In the scheme design, fully implicit two-layer coupled discretization is applied to assure high time accuracy, and finite element discretization is applied to assure high space accuracy. By introducing Ritz projection and new inductive reasoning methods, we develop the discrete functional analysis technique from that for finite difference schemes, and establish a new general analysis framework for nonlinear finite element schemes. Consequently, we overcome the difficulties arising from the strong coupling of the nonlinear finite element discretizations for the capacity term and diffusion operator, and prove the existence and boundedness of the nonlinear finite element solution, as well as its second-order time and optimal order space convergence. Numerical experiments and comparisons confirm the theoretical analysis results and demonstrate its high performance.

Correction factors to biaxial bending strength of thin silicon die in the ball-on-ring test by considering geometric nonlinearity and material anisotropy

Abstract The ball-on-ring test (BoR) is one of the standard tests for biaxial bending, suggested in ASTM F394-78. This test has been applied to determine the biaxial bending strength of silicon dies to avoid the die edge effect of the three-point bending tests. However, from the literature, when the relatively thin silicon dies are tested, this test suffers from a contact-nonlinearity effect, due to a maximum applied stress moving away from the loading pin center before the specimen failure, and thus results in overestimated maximum stress calculated by the theoretical linear solution. This study aims to investigate this mechanical issue experimentally, theoretically and numerically by taking into account the specimen material anisotropy and thickness effects on the maximum stresses and deflections, and then propose new correction factor equations to the theoretical linear solutions, based on the numerical fitting results of the geometric nonlinear finite element solutions. Those correction factor equations proposed in this study are material-property independent, but specimen thickness dependent, which can be estimated by an interpolation function. It has been proved that the BoR test using the conventional theory associated with the proposed correction factor equations can successfully determine the bending strength of the thin silicon dies on untreated surfaces, which mostly fails in the contact-nonlinear region.

Machine learning-enabled self-consistent parametrically-upscaled crystal plasticity model for Ni-based superalloys

This paper introduces a concurrent multiscale modeling framework for developing parametrically-upscaled crystal plasticity models (PUCPM) for crystalline metals that are characterized by multiple phases in their intragranular microstructure. Specifically, Ni-based superalloys with distributions of γ′ precipitates in the γ matrix in their sub-grain microstructure are modeled in this study. The interaction of the γ−γ′ phases with nano-scale dislocation mechanisms determine the elasto-plastic behavior of the material across multiple scales. The PUCPM is designed to explicitly account for the morphological and configurational statistics of these γ−γ′ intragranular microstructures in its crystal plasticity constitutive coefficients. This approach provides a thermodynamically-consistent foundation to enable microstructure-aware material simulation at the higher scale. Establishing this multiscale characteristic requires an automated toolchain of computational methods to generate and embed heterogeneous, statistically-equivalent representative volume elements (SERVEs) into the concurrent multiscale simulation domains for self-consistent homogenization. The self-consistency condition is enforced through an optimization strategy invoking a series of coupled nonlinear finite element solutions. Supervised and unsupervised machine learning methods are integrated with the physics-based modeling at all stages of model development to overcome computational limitations and to provide the final, connecting map between PUCPM constitutive coefficients and γ−γ′ microstructural descriptors. The resulting PUCPM benefits from orders of magnitudes of speedup compared to the equivalent explicit representation of the lower scale microstructure. This advantage enables unique model capabilities for the multiscale analysis of deformation and failure in materials and location-specific design.

An adaptive iterative linearised finite element method for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids

In this work, we further extend the theory in [C. Kreuzer, E. Süli, Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, ESAIM: Math. Model. Numer. Anal. 50 (5) (2016) 1333–1369] on the adaptive finite element analysis of implicitly constituted incompressible fluid flow problems by taking into account the approximation of the nonlinear finite element solutions by an iterative solver. Thereby we show that the computable sequence generated by the modified algorithm still has a weak limit point, which is a solution of the given problem. Our abstract theory can be applied to Bingham fluids, both with and without the convective term. The performance of the adaptive iterative linearised finite element algorithm in this context is illustrated by two numerical experiments.

Behavior of geopolymer concrete deep beams containing waste aggregate of glass and limestone as a partial replacement of natural sand

The geopolymer concrete has outstanding physical properties and positive impact on environment compared to conventional concrete. Present endeavor highlights a strategy for management of glass and limestone waste to recycle them in geopolymer concrete deep beams. In this study, 126 geopolymer concrete samples were made for investigating the mechanical properties of geopolymer concrete with the sand replacement ratio of 0–15% by weight in increments of 5%. In addition, seven Reinforced geopolymer concrete deep beams of 600,150,150 mm were casted with respect to the concrete mix and their performance was examined in four-point bending test as per ASTM C78/C78M considering sand replacing ratio of 0–15%. Moreover, proper material constitutive relationships were selected to simulate the performance of geopolymer concrete incorporating waste aggregate. Consequently, a nonlinear finite element solution in ANSYS 11.0 was developed using these constitutive models to predict the behavior of beams. Experimental observations revealed that the use of waste lime and waste glass aggregates reduced the compressive and tensile strength of geopolymer concrete. Furthermore, the loss in strength using waste glass aggregate was less than 15% when compared to geopolymer concrete with waste limestone. Whereas, 5–15% sand replacing with waste aggregate resulted in an average reduction of 19% in loading capacity of the deep beams. The degradation in the ultimate load of the beams with limestone aggregate was 46.6% lower than that for members containing waste glass aggregate. The strain energy of the deep beams containing waste limestone was enhanced by increasing the waste aggregate content from 5% to 15%. Whilst increasing the ratio of replacement for glass mixes led to a decrease in final compressive strain. The outputs of the numerical analysis, in terms of loading capacity of the geopolymer concrete deep beams, were in good matching with the experimental results.

TAM DERİNLİKLİ ESNEK ÜSTYAPILARIN KATMAN ÖZELLİKLERİNİN TAHMİNİ İÇİN LİG ŞAMPİYONASI ALGORİTMASI

This study proposes a backcalculation tool, based on the hybrid use of League Championship Algorithm (LCA) and Artificial Neural Network (ANN), in order to predict the stiffness related layer properties of full-depth asphalt pavements. The proposed algorithm, namely LCA-ANN, is composed of two main parts; (i) an ANN forward response model, which is developed with the nonlinear finite element solution, for computing the surface deflections, and (ii) LCA search algorithm which is employed to search and provide the best set of layer moduli to the ANN model. In order to evaluate the performance of the proposed method, a synthetically generated dataset and real field data are utilized. Moreover, to assess the searching ability of LCA, well-accepted metaheuristic algorithms; Simple Genetic Algorithm (SGA) and Particle Swarm Optimization (PSO) are employed for comparison purposes. Obtained results reveal that the proposed algorithm can predict the layer properties with a low order of error values and enables fast and reliable tool for backcalculation studies.

Behavior of hybrid Reinforced Concrete Beams made with two types of concrete (Normal Weight and Reactive Powder Concrete)

This research confirms the use of finite element modeling using volume 16 ANSYS package to represent the non pre stressed concrete beams reinforced with steel bars, reactive powder concrete beams and hybrid reinforced concrete beams. This work focused on two main purposes: the first one is for economic purpose to check the effects of inclusion of fresh reactive powder concrete with the fresh normal weight concrete beams body at compression and tension stress concentration regions to improve the ultimate load resistance and to control mid span deflection of simply supported beams or any great structural elements. The second purpose is to check effects of using steel fibers with different volume fractions (Vf = 0%, Vf = 1%, Vf = 1.5% and Vf = 2%) in reactive powder concrete beams with fixing all other mechanical properties of concrete to show clearly its effect on controlling of mid span deflection with applied loads or in other words, on beam ductility. The used beam was simple supported reinforced concrete beam with dimensions (110 X 200 X 1500)mm width, height and length, respectively, reinforced with bottom 2 main bars of 12mm diameter and 8mm diameter stirrups spaced at 50mm. Each beam was loaded up-to maximum load which makes failure under the effect of two concentrated parallel loads for beam used as a case study and one central concentrated load for parametric study beams to assign the load and deflection values at beams mid-span locations.The proposed nonlinear finite element solutions are compared with the experimental data obtained from other researcher [1], which gives a good agreement between the obtained results.Parametric studies are carried out by using the proved nonlinear ANSYS (version 16) model to investigate the effects of adding expensive RPC with cheap NWC in structural elements bodies to obtain an economic construction material and to reduce weight and volumes especially for great structural elements.

Failure Assessments for MQXF Magnet Support Structure With a Graded Approach

The high luminosity large hadron collider (LHC) upgrade requires new quadrupoles, MQXF, to replace the present LHC inner triplet magnets. The MQXFA magnet is the first prototype that has a 150-mm aperture and uses Nb <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> Sn superconducting technology in a 4.2-m magnetic length structure. The support structure design of the MQXFA magnet is based on the bladder-and-key technology, where a relatively low pre-stress at room temperature is increased to the final preload targets during the cool-down by the differential thermal contraction of the various components. The magnet support structure components experience different load levels from pre-load to cool-down and excitation. Consequently, a few parts experience high stresses that may cause localized plastic deformations or internal fracture development. The concept presented in this paper for the failure assessment of support structures integrates nonlinear finite-element (FE) analysis with detailed sub-models and fracture mechanics into an advanced engineering tool. The nonlinear FE solutions enable estimations of the structural response to the given loads, and the advanced fracture analysis with failure assessment diagram assesses the structure safety index of results obtained from the FE model. The paper describes how the MQXFA end-shell segments are being optimized based on the failure analyses.

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.

Reduced-order thermomechanical modeling of multibody systems using floating frame of reference formulation

In this study, a reduced-order thermomechanical coupling model, which accounts for the inertia coupling of the thermoelastic deformation and the large reference body motion, is proposed using the floating frame of reference formulation for the transient thermomechanical analysis of constrained multibody systems. In this approach, the reduced-order heat equations are fully embedded in the final form of the equations of motion. Accordingly, the transient thermal response as well as the resulting thermoelastic behavior of constrained multibody system can be predicted within the general multibody dynamics computer algorithm. It is demonstrated that appropriate selection of the thermal interface coordinates is crucial for describing the thermal modes (i.e. temperature distribution) induced by external heat sources using the Craig–Bampton component mode synthesis approach generalized for thermomechanical systems. Furthermore, a systematic procedure for imposing prescribed surface temperature given, for example, from thermal-fluid dynamics simulations is proposed for the thermomechanical floating frame of reference formulation. Using several numerical examples, simulation capabilities of the thermomechanical floating frame of reference formulation model are demonstrated for multibody dynamics applications. Numerical results show good agreement with the nonlinear thermomechanical finite element solutions considering the large rotational motion with substantial reduction in the model dimensionality and computational time.

Nonlinear finite element solutions of thermoelastic deflection and stress responses of internally damaged curved panel structure

• A novel numerical model is developed to study the deflection and stress values of the pre-damaged composite structure. • The internal damage has been modeled via sub-laminate approach including two kinematic theories. • The nonlinear FEM solutions are obtained under the combined thermomechanical loading. • The numerical solutions (deflection and stress) are evaluated for different panel geometries via the derived models. Thermoelastic deflection and corresponding stresses of the pre-damaged layered panel structure are investigated numerically in this article including the large deformation kinematics under the linearly varying temperature field. The composite structural deformation kinematics is derived using two different polynomial type of kinematic theories including the through-thickness stretching effect. The inter-laminar separation between the adjacent layers is incurred via the sub-laminate approach and Green–Lagrange strain to count the total structural deformation. Also, the intermittent displacement continuity conditions are imposed in the current mathematical model to establish the displacement continuity between the separated layers. The variational principle is adopted for the evaluation of the nonlinear structural equilibrium equations and solved via total Lagrangian approach . The convergence and the corresponding validity of the currently derived nonlinear finite element solutions are checked by solving different sets of numerical examples. Additionally, the comprehensive inferences are drawn from various numerical examples for the well-defined important input parameter including the size, position, and location of delamination .

Imperfection sensitivity analysis of steel columns at ultimate limit state

The present paper applies Sobol's variance-based global sensitivity analysis (SSA) to quantify the contribution of input imperfections to the load-carrying capacity (LCC) of an IPN 200 steel compressed member. LCC is evaluated using the geometrically and materially non-linear finite element solution with regard to the effects of initial random imperfections including residual stresses. Comparison of results of SSA for (i) buckling about the minor principal axis, (ii) buckling about the major principal axis and (iii) lateral–torsional buckling due to bending moment is performed on the non-dimensional slenderness interval of 0–2. SSA for (i) and (ii) is performed for steel grade (a) S235 and (b) S355, SSA for (iii) is performed only for steel grade S235. SSA found similarities in results (ia) and (ib), (iia) and (iib) and identified significant differences between results (ia) and (iiia), (iia) and (iiia), where sensitivity to the initial axial curvature is more than two times higher in (ia) than in (iiia). The relationships between the effects of initial imperfections on LCC and the design criteria of reliability of Eurocode 3 are discussed.

Nonlinear finite element solutions of thermoelastic flexural strength and stress values of temperature dependent graded CNT-reinforced sandwich shallow shell structure

This research article reported the nonlinear finite solutions of the nonlinear flexural strength and stress behaviour of nano sandwich graded structural shell panel under the combined thermomechanical loading. The nanotube sandwich structural model is derived mathematically using the higher-order displacement polynomial including the full geometrical nonlinear strain-displacement equations via Green-Lagrange relations. The face sheets of the sandwich panel are assumed to be carbon nanotube-reinforced polymer composite with temperature dependent material properties. Additionally, the numerical model included different types of nanotube distribution patterns for the sandwich face sheets for the sake of variable strength. The required equilibrium equation of the graded carbon nanotube sandwich structural panel is derived by minimizing the total potential energy expression. The energy expression is further solved to obtain the deflection values (linear and nonlinear) via the direct iterative method in conjunction with finite element steps. A computer code is prepared (MATLAB environment) based on the current higher-order nonlinear model for the numerical analysis purpose. The stability of the numerical solution and the validity are verified by comparing the published deflection and stress values. Finally, the nonlinear model is utilized to explore the deflection and the stresses of the nanotube-reinforced (volume fraction and distribution patterns of carbon nanotube) sandwich structure (different core to face thickness ratios) for the variable type of structural parameter (thickness ratio, aspect ratio, geometrical configurations, constraints at the edges and curvature ratio) and unlike temperature loading.

Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure

Abstract The nonlinear eigen frequency response of the functionally graded single-walled carbon nanotube reinforced sandwich structure is investigated numerically considering the Green-Lagrange nonlinear strain under uniform thermal environment. The mathematical model of the sandwich plate has been derived using the simple higher-order shear deformable kinematics including the temperature dependent properties of each constituent. The sandwich panel constitutes of graded carbon nanotube face sheets and homogeneous core. The desired nonlinear finite element solutions are obtained via the direct iterative method. Based on the necessary validation and convergence new results are computed for different design related parameters and discussed subsequently.

A robust Riks-like path following method for strain-actuated snap-through phenomena in soft solids

In this paper, a practical and robust Riks-like path following method is developed to automatically trace the nonlinear equilibrium path for initial-finite-strain-induced snap-through phenomena in soft solids. The deformation gradient is decomposed into an elastic part and an initial-strain part, where the initial strain is characterized through a prescribed pattern tensor and an unknown strain factor describing the magnitude. In the non-linear finite element (FE) solution scheme, an equivalent body force proportional to the increment of the strain factor is derived, which varies in each iteration step due to its dependence on deformed configuration and stress. Both the displacement and the strain factor increments are introduced as unknowns in the linearized equation, which permits to predict the critical equilibrium states before and after the snap-through. Several examples with various configurations and strain histories are constructed, including multilayer beams and film–substrate systems, which demonstrate the effectiveness and robustness of the algorithm to capture the strain-induced snap-through phenomena.

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.

Nonlinear Finite Element Solution of Post-buckling Responses of FGM Panel Structure under Elevated Thermal Load and TD and TID Properties

The nonlinear finite element solutions for the buckling and post-buckling responses of the functionally graded shell panel subjected to the non-uniform thermal environment have been presented in this article. The thermal fields are assumed as uniform, linear and nonlinear temperature rise across the thickness of shell panel and the properties of each constituent are considered to be temperature dependent. The effective material properties of the graded structure are evaluated using the Voigt’s micromechanical rule in conjunction with power-law distribution. For the analysis purpose, a general nonlinear mathematical model of the functionally graded shell panel has been developed based on the higher order shear deformation theory and Green-Lagrange type geometrical nonlinear strains. The system governing equation of the panel structure is derived using the variational principle. Further, suitable nonlinear finite element steps have been adopted to discretize the model for the computation of the desired responses in association with the direct iterative method. The convergence and the validation behavior of the present numerical model are initially tested to demonstrate its efficacy and significance. Finally, the effects of curvature, power law index and different support conditions on the buckling and post-buckling responses of the functionally graded shell panels are investigated and discussed in details.