Nonlinear Finite Element Method Research Articles

A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in R 2 {\mathbb {R}}^2 . It is based on an isoparametric C 0 C^0 finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented.

Effect of a gunite lining on the stability of salt caverns in bedded formations for hydrogen storage

In this paper, a novel methodology to improve the structural behaviour and durability of bedded salt caverns for hydrogen storage is proposed. The suggested technology consists in applying a gunite lining over the whole cavern surface by means of a pneumatic air blowing. This continuous flow projected at high speed onto the cavern surface produces a self-compacted gunite facing. The effect of adding a gunite lining on the structural performance of a bedded salt cavern is analysed. The nonlinear finite element method is then applied to solve the inherent nonlinear solid mechanics problem. It is shown that reinforcing salt caverns with a gunite lining improves the cavern behaviour, specially in the case of caverns located within a horizontal salt stratum confined between other strata. When dealing with these salt formations, the stratum geometry limits the cavern height. In general, wider caverns that exhibit larger displacements are required to achieve a huge storage capacity. Moreover, these caverns can be located at a considerable depth, where the temperature has an important role on the creep deformation process. Therefore, the proposition of stabilization techniques is required to ensure both their structural integrity and the ground subsidence. This technology allows the safe exploitation of salt caverns located at bedded-type stratified horizontal salt formations.

Study on fracture behaviour of pipeline girth welds based on curved wide plate specimens: Experimental and numerical analysis

Understanding the fracture characteristics and strain capacity of pipelines with girth welds under external loads is crucial for evaluating pipeline safety. The use of curved wide plate (CWP) specimens closely resembling full-scale (FS) pipelines in geometry provides a more realistic representation of crack tip constraint states compared to small-size fracture toughness test specimens in laboratory settings. This study aims to investigate the ductile fracture characteristics of center cracks on the outer surface of girth welds in high-grade steel pipelines under external loads through large-scale CWP tensile tests and advanced numerical simulations. Tensile tests were conducted on CWP specimens using a kiloton tensile testing machine, with double crack opening displacement (COD) gauge monitoring crack mouth opening displacement (CMOD), and high-precision displacement sensors and strain gauges tracking deformation and strains at various positions. The study yielded significant experimental results, and a finite element (FE) model of the CWP was developed using the nonlinear finite element method (FEM) to validate the experimental data against simulation results. The FE model was then used to investigate the influence of material properties on crack propagation resistance and driving force curves. A method for determining the ultimate tensile strain capacity (UTSC) of pipelines based on actual material fracture toughness was proposed. This research contributes valuable experimental data for further exploration of fracture behaviour in large-scale pipeline girth welds and offers insights for safety evaluations of such welds.

Nonlinear twofold saddle point-based mixed finite element methods for a regularized μ(I)-rheology model of granular materials

We propose and analyze new mixed finite element methods for a regularized μ(I)-rheology model of granular flows with an equivalent viscosity depending nonlinearly on the pressure and the Euclidean norm of the symmetric part of the velocity gradient. To this end, and besides the velocity, the pressure and the aforementioned strain rate, we introduce a modified stress tensor that includes the convective term, and the skew-symmetric vorticity, as auxiliary tensor unknowns, thus yielding a mixed variational formulation within a Banach spaces framework. Then, the pressure is obtained through an iterative postprocess suggested by the incompressibility condition of the fluid, which allows us to express this unknown in terms of the aforementioned stress and the velocity. A fixed-point strategy combined with a solvability result for a class of nonlinear twofold saddle point operator equations in Banach spaces, is employed to show, along with the classical Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. In particular, PEERS and AFW elements of order ℓ⩾0 for the stress, the velocity, and the skew-symmetric vorticity, and piecewise polynomials of degree ⩽ℓ+n (resp. ⩽ℓ+1) for the strain rate with PEERS (resp. with AFW), yield stable Galerkin schemes. Optimal a priori error estimates are derived and associated rates of convergence are established. Finally, numerical results confirming the latter and illustrating the good performance of the methods, are reported.

Validation of AISI Design of Cold-Formed Steel Beams Using Non-Linear Finite Element Analysis

In the local building industry of Pakistan, pre-engineered steel building manufacturers mainly employ their own self-developed software and Excel sheets. These systems are based on empirical formulas mentioned in the AISI manual. Under this scenario, a need was found to validate AISI flow charts using commercial software like CUFSM 5.04 and ABAQUS R2019x. This study presents a validation of the CUFSM software and the American Iron and Steel Institute (AISI) Direct Strength Method (DSM) results of channel section flexural members using the non-linear finite element method employing ABAQUS. In this study, eight standard cold-formed channel-section (C-section) steel members were modeled and analyzed using ABAQUS to simulate realistic behavior under four-point loading conditions. The non-linear finite element models incorporated material and geometric non-linearities to capture the actual response of the steel elements. The results obtained from ABAQUS were compared with those predicted by the CUFSM and DSM, focusing on critical parameters such as nominal strength, buckling modes, and deformation patterns. During this study, it was observed that out of the selected sections, the AISI charts predict conservative and even unsafe flexural capacities in some of the cases concerning other methods, with a maximum difference of 14.03%. The differences obtained using DSM and ABAQUS when compared with the results of the AISI charts varies on both the plus and minus sides. This study will not only affect the industry in terms of innovative designs for efficient structures but also the community in regards to low-budget construction.

An improved similarity method for stiffened plates subjected to combined loads of longitudinal compression and lateral pressure

Structures’ ultimate strength has been increasingly valued in the past few decades. The model test is an important and reliable way to estimate the ultimate strength. For large and complex structures, like ship hull structures, the test subject is usually scale models due to the limitation of loading capacity and space of laboratories. Therefore, the similarity method and the design of scale models are of great importance. The stiffened plate is the minimum unit that makes up thin-walled structures. The similarity method of the ultimate strength of stiffened plates is the base of the similarity method of large and complex structures. This paper is a further study of an existing similarity method proposed in our previous work. A stiffened plate from a real ship is taken as the subject. 22 combinations of scale ratios are used to design scale models. The 23 models (including the original model) are subjected to a combined load of longitudinal compression and lateral pressure. The non-linear finite element method is used to analyze these models. The results revealed that the previous method is only suitable for scale models with small degrees of geometrical distortion. It didn't consider the effect of geometrical distortions. An improved similarity method and scale model design method are proposed to solve these problems. The improved methods consider the geometrical distortion of scale models and are applicable to more geometrically distorted scale models. More numerical examples are performed to verify the correctness and advantage of the improved methods.

Experimental and numerical investigation of a steel yielding arc and ring damper

The use of yield steel dampers in structures improves seismic resilience by efficiently absorbing energy, reducing the impact of seismic forces, and enhancing overall safety and stability. This paper introduces Arc and Ring Dampers (ARDs), a replaceable structural fuse system with the goal of improving seismic performance of structures against earthquakes. The ARD configuration consists of a central ring surrounded by four arcs. For the first time, an experimental study is carried out on full-scale damper specimens to determine seismic performance and failure mechanisms under cyclic loading. Nonlinear finite element method (FEM) analysis is employed for numerical validation and parametric exploration, taking into account variables such as arc and ring thickness, ring and arc radius, and stiffener width. According to the results, by carefully selecting and designing the geometric characteristics of this damper, it would offer a remarkably efficient energy absorption capacity and a significant load-bearing ability, resulting in a 36 % increase in its equivalent damping ratio. This study also describes these geometric features in detail, including the arcs' and ring's width-to-thickness ratios. Furthermore, this damper maintains structural stiffness without degradation under cyclic loading conditions, showing good ductility up to a value of 5.5. The theoretical predictions here are in good agreement with the experimental outcomes.

Multi-Load Topology Optimization Design for the Structural Safety Maintenance of Low- and Intermediate-Level Radioactive Waste Packaging Containers in the Case of a Collision.

This paper presents an optimized design approach using nonlinear dynamic analysis and finite element methods to ensure the structural integrity of square-shaped containers made from ductile cast iron for intermediate- and low-level radioactive waste packaging. Ductile cast iron, with its spherical graphite structure, effectively distributes stress throughout the material, leading to a storage capacity increase of approximately 18%. Considering the critical need for containers that maintain integrity under extreme conditions like earthquakes, the design focuses on mitigating stress concentrations at the corners of square structures. Nonlinear dynamic analyses were conducted in five drop directions: three specified by ASTM-D5276 standards and two additional directions to account for different load patterns. Fractures were observed in four out of the five scenarios. For each direction where fractures occurred, equivalent loads causing similar displacement fields were applied to linear static models, which were then used for multi-load topology optimization. Three optimized models were derived, each increasing the volume by 1.4% to 1.6% compared to the original model, and the design that best met the structural integrity requirements during drop scenarios was selected. To further enhance the optimization process, weights were assigned to different load conditions based on numerical analysis results, balancing the impact of maximum stress, average stress, and plastic deformation energy. The final model, with its increased storage capacity and enhanced structural integrity, offers a practical solution for radioactive waste management, overcoming limitations in previous designs by effectively addressing complex load conditions.

Fabric soft pneumatic actuators with programmable turing pattern textures

This paper presents a novel computational design and fabrication method for fabric-based soft pneumatic actuators (FSPAs) that use Turing patterns, inspired by Alan Turing’s morphogenesis theory. These inflatable structures can adapt their shapes with simple pressure changes and are applicable in areas like soft robotics, airbags, and temporary shelters. Traditionally, the design of such structures relies on isotropic materials and the designer’s expertise, often requiring a trial-and-error approach. The present study introduces a method to automate this process using advanced numerical optimization to design and manufacture fabric-based inflatable structures with programmable shape-morphing capabilities. Initially, an optimized distribution of the material orientation field on the surface membrane is achieved through gradient-based orientation optimization. This involves a comprehensive physical deployment simulation using the nonlinear shell finite element method, which is integrated into the inner loop of the optimization algorithm. This continuous adjustment of material orientations enhances the design objectives. These material orientation fields are transformed into discretized texture patterns that replicate the same anisotropic deformations. Anisotropic reaction-diffusion equations, using diffusion coefficients determined by local orientations from the optimization step, are then utilized to create space-filling Turing pattern textures. Furthermore, the fabrication methods of these optimized Turing pattern textures are explored using fabrics through heat bonding and embroidery. The performance of the fabricated FSPAs is evaluated through three different deformation shapes: C-shaped bending, S-shaped bending, and twisting.

Buckling behavior analysis of corrugation reinforced composite pressure shell: Analysis of the influence of corrugation structural parameters

Based on CFRP(Carbon Fiber Reinforced Plastics) cylindrical pressure shell, the CFRP circumferential corrugation pressure shell is proposed in this study. The structure can further improve the buckling resistance capacity of CFRP pressure shell. Five CFRP circumferential corrugation pressure shells with varying corrugation structure parameters are designed and compared with CFRP cylindrical pressure shell. The buckling behaviors of the shells are analyzed in detail by nonlinear finite element method. The result show that reasonable corrugation structure can significantly improve the resist buckling capacity of CFRP cylindrical pressure shell. However, the increase of corrugation structure parameters will exacerbate the damage to the CFRP layers outside the pressure shell. Ultimately, pressure shell N2 with the best buckling resistance capacity is selected for hydrostatic pressure test, and the experimental results are found to align with the numerical simulation results. This study offers a novel approach for the design of deep-sea pressure shell structures.

A Data-Driven DNN Model to Predict the Ultimate Strength of a Ship’s Bottom Structure

Plates and curved plates are essential components in ship construction. In the design stage, the methods used to evaluate the ultimate strength required to confirm the structural safety of plates include prediction through analytical methods, finite-element analysis (FEA), and empirical formulas. However, with nonlinear buckling, the results of the empirical formula and the FEA differ for small flank angles (1~9). As a result, the prediction of the nonlinear ultimate strength of flank angle (1~9) plates still requires significant computation time and cost. To compensate for this, this study performed an ultimate strength prediction method utilizing a deep neural network together with the 4050 curved plate analysis. In addition, this paper presents the analysis results of the nonlinear finite-element method and the geometric shape and ratio of curved plates as training data. Based on the results of this study, designers can more efficiently design appropriate curved plate members by considering the ultimate strength.

A nonlinear interval finite element method for elastic–plastic problems with spatially uncertain parameters

This paper proposes a nonlinear interval finite element method for elastic–plastic analysis of structures with spatially uncertain parameters. The spatially uncertain parameters are described by the interval field, and the variation bounds of the elastic–plastic structural responses can be calculated effectively. Quantified by the interval field, the spatially uncertain parameters are represented by the interval Karhunen–Loève (K-L) expansion, based on which the nonlinear interval finite element equilibrium equation is formulated. An interval iterative method is then presented to solve the equilibrium equation and obtain an outer solution of the variation bounds of structural responses such as displacement. In this method, the Newton-Raphson iterative method is used to transform the nonlinear problem into a linear one, and then the interval iterative method is introduced to solve the interval linear equations. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method.

Simulation study on the thermal effect of continuous laser heating quartz materials.

The continuous development and application of laser technology, and the increasing energy and power of laser output have promoted the development of various types of laser optical systems. The optical components based on quartz materials are key components of high-power laser systems, and their quality directly affects the load capacity of the system. Due to the photothermal effect when the laser interacts with the quartz material and generates extremely high temperatures in a short period of time, it is impossible to experimentally solve the phenomena and physical mechanisms under extreme conditions. Therefore, it is very important to select a suitable method to investigate the thermal effect of intense laser interaction with quartz materials and explain the related physical mechanism. In this study, a three-dimensional quarter-symmetric laser heating quartz material geometry model by using nonlinear transient finite element method was established, and its transient temperature field distribution of the quartz material after being heated by a 1,064nm continuous laser was investigated. In addition, the influence of different laser parameters (laser spot radius, heat flux and irradiation time), material parameters (material thickness, material absorption rate of laser) on the thermal effect of heating quartz material were also studied. When the laser heat flux is 20W/cm2, the diameter of the laser spot is 10cm, the irradiation time is 600s and the thickness is 4cm, the temperature after laser heating can reach 940.18°C, which is far lower than the melting point. In addition, the temperature maximum probes were set at the overall model, spot edge and rear surface respectively, and their temperature rise curves with time were obtained. It is also found that there is a significant hysteresis period for the rear surface temperature change of the quartz material compared with the overall temperature change due to heat conduction. Finally, the method proposed can also be applied to the laser heating of other non-transparent materials.

Temperature Effects on GaN/AlN Heterojunction in a Piezomagnetic and Piezoelectric Semiconductor Composite Fiber

In this article, considering the thermal effects, a PN GaN/AlN heterojunction in a piezomagnetic and piezoelectric semiconductor composite fiber driven by dynamic magnetic loads is studied. The true nonlinear constitutive equations of the electric current density for the composite heterojunction are employed. Thus, a nonlinear finite‐element method (FEM) is adopted to obtain dynamic responses of the composite heterojunction, including the mechanical displacement, electric potential, and electron and hole concentrations. By comparing dynamic responses with those derived by COMSOL Multiphysics, the nonlinear FEM is verified. Based on numerical results, regulation effects of the temperature on distributions of the mechanical displacement, electron and hole concentrations, and temperature–current–voltage characteristics are discussed.

Behaviour of eccentrically loaded concrete-filled rectangular ferritic stainless steel tubular slender columns: Numerical modelling and design

Ferritic stainless steel achieves a lower and steady price level due to low nickel content compared to duplex and austenitic stainless steels. Accordingly, the combination of ferritic stainless steel tube and concrete will have better application prospects in corrosive marine environments. However, the structural application of concrete-filled ferritic stainless steel tube (CFFSST) members is still limited due to the absence of performance data and design guidelines. Accordingly, this paper aims to comprehensively investigate the behaviour of eccentrically loaded concrete-filled rectangular ferritic stainless steel tubular (CFRFSST) slender columns using the nonlinear finite element (FE) method, and develop a suitable design method. Two different material models were introduced to simulate corner and flat regions of rectangular ferritic stainless steel tube. Good consistency was found between the simulation and experimental results against the failure mode, ultimate load and moment, and axial load versus mid-height deflection curves for eccentrically loaded CFRFSST slender columns. This is followed by the mechanism analysis and parameter study of the studied CFRFSST columns, which revealed that the longitudinal stress and Mises stress show significant changes perpendicular to the loading line direction, while there is almost no change along the loading line direction. In addition, the initial stiffness and ultimate load increased by increasing the cross-sectional aspect ratio, yield stress and concrete strength, while they decreased as the eccentricity ratio, slenderness ratio and width-to-thickness ratio increased. The outputs of the FE analysis and existing experimental results were used to examine the applicability of current design codes for designing the studied CFRFSST columns. The results demonstrated an unexpected deviation generated in the design predictions. At the end, a new design method for eccentrically loaded CFRFSST slender columns was proposed which achieved better predictive consistency.

Multi-segmented fifth-order polynomial–shaped shells under hydrostatic pressure

The design and construction of submerged complex shells for new applications in offshore structures are increasingly popular. Therefore, the purpose of this study is to present the analytical model and Lagrange multipliers associated with the constraint equation for large displacement analysis of a fifth-order polynomial–shaped shell under hydrostatic pressure for the first time. The shell geometry can be computed using differential geometry with a fifth-order polynomial. The energy functional of the fifth-order polynomial–shaped shell is derived based on the principle of virtual work and written in the appropriate form. The nonlinear static responses of the fifth-order polynomial–shaped shell under hydrostatic pressure can be calculated using the nonlinear finite element method via the fifth-order polynomial shape function. This study develops the model using one-dimensional beam elements divided along the shell radius. To avoid the slope of a meridian curve at the equatorial plane approaching infinity, the shell is divided into two regions defined by different surface parameters. At the junction of two adjacent regions, the continuity requirements are established as the constraint conditions using Lagrange multipliers. The numerical results from the proposed methods are demonstrated and discussed, along with the effects of varied seawater depth, thickness, and elastic modulus on the deformed configuration and principal curvature at the deformed state. The results show that the nonlinear displacement is higher than the linear one in the case of the hydrostatic pressure, whereas the case of the internal pressure has an opposite result. For principal curvatures at the apex, the principal curvatures increase as the seawater depth increases, whereas the principal curvatures decrease when the thickness and elastic modulus increase.

Improved air gap permeance modelling for single-slice magnetic equivalent circuits of skewed induction motors

PurposeWhen rotor and stator teeth are close, the connecting air gap flux tube's cross-sectional area exceeds the tooth overlap area. This flux fringing effect is disregarded in the air gap permeance calculation of single-slice magnetic equivalent circuits (MECs) of electric motors with skewed rotors. This paper aims to extend an air gap permeance calculation method incorporating flux fringing for unskewed rotors to skewed and radially eccentric rotors.Design/methodology/approachAssuming axial independence, the unskewed air gap permeance is rotated according to the skew and integrated along the axial dimension, resulting in a first method. The integral is approximated analytically, resulting in a second method. Results are compared to a commonly used reference method and validated using a non-linear finite element method (FEM) simulation.FindingsThe proposed methods provide better alignment with the FEM validation compared to the reference method for skewed rotors and common rotor eccentricity, i.e. below 50% of the air gap length. The analytical method is shown to be competitive with the reference method regarding computational time cost.Originality/valueTwo novel air gap permeance methods are proposed for single-slice MECs with skewed rotors. Their characteristics are discussed and validated.

Robust design optimization using a non-intrusive second-order approximation of stochastic moments

This paper presents a new formulation of the second-order fourth-moment method (sometimes referred to as second-order perturbation method or second-order method of moments). The method allows to efficiently predict the stochastic moments of a response function and is therefore often used within robust design optimization. The new approach allows a non-intrusive implementation at the same cost as existing, highly intrusive formulations. Therefore, the new approach can be applied to any objective function without significant implementation effort. It is based on a few finite difference steps into special directions and hence is dependent on the corresponding step sizes. An automatic step size procedure is supplied beside a detailed convergence analysis. The advantages of the new formulation are demonstrated by robust design optimizations of a 2D and a 3D example using the geometrically nonlinear finite element method.

Phase field modeling of ductile fracture with isotropic hardening and radius return method

Phase field model has been investigated for brittle fracture in many static and dynamic scenarios, but its applications to ductile fracture is not as common as brittle fracture, especially implementing in software LS-DYNA with explicit scheme. In this study, an efficient LS-DYNA implementation of the phase field modeling of ductile fracture is presented and both with and without the split of elastic strain energy have been considered for the damage evolution. In more detail, plasticity formulation of ductile material with isotropic hardening is briefly presented first and then the governing equations of the classical phase field model are derived, which gives the displacement-phase coupled problem. For with the split of elastic strain energy, the shear component of elastic strain energy is considered for the damage evolution. The influence of degradation function on stress–strain curve is also investigated by using three kinds of function (polynomial function, algebraic fraction function and sigmoid function), which leads to linear and nonlinear finite element method (FEM) formulation of the phase field model and Newton–Raphson method is used to solve the nonlinear FEM formulation of the phase field model. A tensile bar test shows the influence of critical energy release rate and degradation function on stress–strain curve. Mode Ⅰ failure of three-point bending test, Mode Ⅱ failure of single-edge notched plate and mixed-mode failure of asymmetrical double-notched plate verify the proposed model in this study. From these simulations, with the split of elastic strain energy shows improvements on plastic deformation than without the split of elastic strain energy.

Stress and Strain Characteristics in Flexible Pavement Using Three-Dimensional Nonlinear Finite Element Analysis

AbstractA new numerical simulation was developed to obtain various structural parameters by employing a nonlinear finite element method using ABAQUS: stress, strain, and the displacement response of five layers on the flexible pavement. The primary purpose of this study was to develop a new simulation that can accurately and effectively characterize stress and strain problems in flexible pavement. New constitutive model equations were developed based on Hooke’s law the three-dimensional model for stress and strain. 40 kN wheel load was used to represent a set of dual tires, assuming a uniformly distributed contact area between the tire-pavement surfaced using linear viscoelastic and nonlinear viscoelastic materials. Four tire-inflation pressures were used 380, 490, 630, and 700 kPa. Results indicated that each flexible pavement layer created with a nonlinear viscoelastic material exhibited a 40% vertical plastic strain reduction compared to linear viscoelastic materials after 7000 s and repeated for 30,000 cyclic loadings. The implication of this newly developed constitutive model was verified against published experimental data. Analyses were completed to simplify the results model results considering the effects of stress and strain on flexible pavement layers. The newly developed constitutive model for solving stress and strain characterization problems can predict the effects on flexible pavement layers and various types of observed flexible pavement failure.