Nonlinear Mechanics Problems Research Articles

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.

2D numerical modeling of crack propagation using SFEMD method of a LEHI material

Numerical methods today play a useful and important role in solving various problems related to fracture mechanics including modeling crack propagation, fretting fatigue and cohesion... Furthermore, these methods have been widely used to solve problems in linear and nonlinear fracture mechanics in cases of elastic, plastic fracture problems. The evaluation of stress intensity factor in 2D and 3D geometries, thus these techniques widely used for non-standard crack configurations. The objective of this work is study the effects of the crack length and the number of structural elements on the two crack parameters such as the stress intensity factor KI and KII and the contour integral (J). As well as presenting a numerical modeling of crack propagation, for a LEHIM (Linear Elastic Homogeneous Isotopic Material) with mechanical characteristics by the method SFEMD (Stretching Finite Element Method Developed), the model chosen with quadratic elements with 4 nodes (CPE4). However, this method is based on the effect of the number of contours and the number of elements around the crack tip on the variation of the stress intensity factors. In addition, the crack propagation criterion (MCSC) was used, a computer program was created in FORTRAN language to develop and evaluate the stress intensity factors and the contour integral (J). Several examples of crack lengths a = 0.7, 1.4, 2.1, 2.8 and 3.5 mm were used. Additionally, the number of items has been changed several times. The stress intensity factors of modes I and II and direction angles (α) are calculated to solve the problem using the ABAQUS finite element code. The results obtained by the SFEMD method and the results of the analytical method are very close.

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Hybrid methods for solving structural geometric nonlinear dynamic equations: Implementation of fifth-order iterative procedures within composite time integration methods

This paper studies a class of hybrid methods that implement multi-point iterative procedures as the nonlinear solver within optimized composite implicit methods. These multi-point methods are employed to enhance convergence order and accuracy without requiring higher-order derivatives for solving structural geometric nonlinear dynamic equations. The promising approach provides an opportunity to delve deeper into its implications and explore the potential applications of multi-point techniques and composite methods across the analysis of dynamical systems. This study involves the utilization of two-point and five-point iterative methods with fifth-order convergence within three-step and four-step optimized composite implicit time integration methods. Moreover, a comprehensive hybrid scheme is introduced by integrating a family of optimized composite implicit methods with single-point and multi-point iterative methods. Then, the performance of the hybrid scheme is analyzed through structural examples, considering factors such as time step size, tolerance threshold, and the degrees of freedom of structures. The findings and numerical results illustrate that multi-point methods outperform the single-point method in terms of the number of iterations. Simultaneously, the four-step time integration method exhibits superior performance compared to the three-step time integration method, albeit with a marginal difference. It is observed that the hybrid method, consisting of the four-step time integration method and the five-point iterative method, performs better than other methods in terms of the number of iterations and errors for solving geometric nonlinear dynamic equations, especially in large structures. Thus, it stands out as a favorable choice for practical analyses and a possible candidate for generalization to other types of nonlinear problems in computational mechanics.

A hybrid radial basis function—Finite difference method for modelling two-dimensional thermo-elasto-plasticity, Part 2. Application to cooling of hot-rolled steel bars on a cooling bed

This paper represents Part 2 of the parallel paper Part 1, where the strong form hybrid RBF-FD method was developed for solving thermo-elasto-plastic problems. It addresses the industrial application of this novel meshless method to steel bars cooling on a cooling bed (CB) where the formation of residual stress is of primary interest. The study investigates the impact of the distance between the bars and the distance to the heat shield above the CB on radiative heat fluxes and, consequently, on thermo-mechanical response. The thermal model is solved on bars cross-section with a RBF-FD method where augmented polyharmonic splines are used for the local approximation. View factors, computed with a Monte-Carlo method, are included in radiative heat fluxes. The thermal solution is incrementally applied on a mechanical model that assumes a generalised plane strain state and captures bars bending. The study employs a hybrid RBF-FD method to resolve a nonlinear discontinuous mechanical problem successfully. The simulation of the process shows how different process parameters influence the thermo-mechanical response of the bars.

A quantum annealing-sequential quadratic programming assisted finite element simulation for non-linear and history-dependent mechanical problems

We propose a framework to solve non-linear and history-dependent mechanical problems based on a hybrid classical computer–quantum annealer approach. Quantum Computers are anticipated to solve particular operations exponentially faster. The available possible operations are however not as versatile as with a classical computer. Nevertheless, quantum annealers (QAs) are well suited to evaluate the minimum state of a Hamiltonian quadratic potential. Therefore, we reformulate the elasto-plastic finite element problem as a double-minimisation process framed at the structural scale using the variational updates formulation. In order to comply with the expected quadratic nature of the Hamiltonian, the resulting non-linear minimisation problems are iteratively solved with the suggested Quantum Annealing-assisted Sequential Quadratic Programming (QA-SQP): a sequence of minimising quadratic problems is performed by approximating the objective function by a quadratic Taylor’s series. Each quadratic minimisation problem of continuous variables is then transformed into a binary quadratic problem. This binary quadratic minimisation problem can be solved on quantum annealing hardware such as the D-Wave22D-Wave trademarks and registered trademarks used herein include D-Wave and Ocean. system. The applicability of the proposed framework is demonstrated with one- and two-dimensional elasto-plastic numerical benchmarks. The current work provides a pathway of performing general non-linear finite element simulations assisted by quantum computing.

"Analytical and numerical study of some problems in nonlinear mechanics"

"Analytical and numerical study of some problems in nonlinear mechanics"

A simple spectral representation of a second-order symmetric tensor and its variation

The spectral decomposition of a second-order, symmetric tensor is widely adopted in many fields of Computational Mechanics. As an example, in elasto-plasticity under large strain and rotations, given the Cauchy deformation tensor, it is a fundamental step to compute the logarithmic strain tensor.Recently, this approach has also been adopted in small-strain isotropic plasticity to reconstruct the stress tensor as a function of its eigenvalues, allowing the formulation of predictor–corrector return algorithms in the invariants space. These algorithms not only reduce the number of unknowns at the constitutive level, but also allow the correct handling of stress states in which the plastic normals are undefined, thus ensuring a better convergence with respect to the standard approach.While the eigenvalues of a symmetric, second-order tensor can be simply computed as a function of the tensor invariants, the computation of its eigenbasis can be more difficult, especially when two or more eigenvalues are coincident. Moreover, when a Newton–Raphson algorithm is adopted to solve nonlinear problems in Computational Mechanics, also the tensorial derivatives of the eigenbasis, whose computation is still more complicated, are required to assemble the tangent matrix.A simple and comprehensive method is presented, which can be adopted to compute a closed form representation of a second-order tensor, as well as their derivatives with respect to the tensor itself, allowing a simpler and numerically accurate implementation of spectral decomposition of a tensor in Computational Mechanics applications.

Sparse polynomial chaos expansion for high-dimensional nonlinear damage mechanics

Finite Element Simulations in solid mechanics are nowadays common practice in engineering. However, considering uncertainties based on this powerful method remains a challenging task, especially when nonlinearities and high stochastic dimensions have to be taken into account. Although Monte Carlo Simulation (MCS) is a robust method, the computational burden is high, especially when a nonlinear finite element analysis has to be performed behind each sample. To overcome this burden, several “model-order reduction” techniques have been discussed in the literature. Often, these studies are limited to quite smooth responses (linear or smooth nonlinear models and moderate stochastic dimensions).This paper presents systematic studies of the promising Sparse Polynomial Chaos Expansion (SPCE) method to investigate the capabilities and limitations of this approach using MCS as a benchmark. A nonlinear damage mechanics problem serves as a reference, which involves random fields of material properties. By this, a clear limitation of SPCE with respect to the stochastic dimensionality could be shown, where, as expected, the advantage over MCS disappears.As part of these investigations, options to optimise SPCE have been studied, such as different error measures and optimisation algorithms. Furthermore, we have found that combining SPCEs with sensitivity analysis to reduce the stochastic dimension improves accuracy in many cases at low computational cost.

A projection‐based reduced‐order model for parametric quasi‐static nonlinear mechanics using an open‐source industrial code

AbstractWe propose a projection‐based model order reduction procedure for a general class of parametric quasi‐static problems in nonlinear mechanics with internal variables. The methodology is integrated in the industrial finite element code codeaster. Model order reduction aims to lower the computational cost of engineering studies that involve the simulation to a costly high‐fidelity differential model for many different parameters, which correspond, for example to material properties or initial and boundary conditions. We develop an adaptive algorithm based on a POD‐Greedy strategy, and we develop an hyper‐reduction strategy based on an element‐wise empirical quadrature in order to speed up the assembly costs of the reduced‐order model by building an appropriate reduced mesh. We introduce a cost‐efficient error indicator which relies on the reconstruction of the stress field by a Gappy‐POD strategy. We present numerical results for a three‐dimensional elastoplastic system in order to illustrate and validate the methodology.

Structure‐preserving invariant interpolation schemes for invertible second‐order tensors

AbstractTensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data , followed by an individual interpolation of the two rotation tensors and and the positive definite diagonal eigenvalue tensor resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group , either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, and remain orthogonal tensors and remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application.

A novel machine learning-based approach for nonlinear analysis and in-situ assessment of masonry

Predicting the local and global mechanical response of masonry structures or estimating their in-situ properties are critical and challenging for the design or assessment of these structures. This article presents a fast prediction/assessment model, developed using a conditional generative adversarial neural network (cGAN) to address this challenge. For the first time, the model shows to be capable of establishing a relationship between masonry microstructural features and full-field local/global mechanical response, and vice-versa, overpassing the path dependency of the non-linear mechanical problems. The strain maps and reaction forces of masonry panels are predicted at any load level from images of masonry panels, which embedded information regarding the materials properties and loading scenarios only in colours, without having any prior knowledge of the material properties and constitutive laws and without the need to access these fields in previous loading levels. Also, it is shown that the mechanical properties of masonry constituents can be predicted from full-field strain maps and loading scenarios using the developed model. This promises to become a revolutionary metamodel for the expensive finite element simulations of masonry or to support in-situ/laboratory experimental testing in the identification of masonry material properties.

The effect analysis of the finite element approximation order on the protective polymer layer contact deformation

Problems implementation using the finite element method (FEM) is primarily associated with the proper selection of approximation order of the shape functions, material behavior models, the coupling nature analysis and the implementation techniques choice. The choice of numerical solution parameters affects the computational procedures. The paper solves problem of the FEM parameters selection and its realization methods in order to achieve a reasonable balance between the counting time and the solution quality. Simplex elements have shape functions with linear dependences on the coordinates. The FEM implementation with the use of simplex elements has proven itself during the last decades. However, one can note the description inaccuracy of curvilinear boundaries of research objects, which can introduce additional inaccuracies in the problem numerical solution. Second-order elements will allow describing curvilinear boundaries, but require a large expenditure of computational power especially for nonlinear mechanics problems. The development of numerical application software packages allows us to evaluate the influence of the FEM approximation degree on the problem solution with the use of grid methods and computer resources at the present time. The paper found that an increase in the numerical solution error is observed at the point of initial steel - polymer contact. The study considers the elements of order 1 and 2 to analyze the influence of the FEM approximation order on the problem numerical solution. The study is carried out within the framework of the classical Hertz problem. In addition the paper investigate obtained solutions for modeling different characters of contact surface coupling for the problem of contact between a spherical die and a steel half-space through a protective polymer coating 4 mm to 12 mm thick. The study reveal that the elements order has the least influence on the problem solution at ideal contact and the greatest one at frictional contact.

Nonlinear analysis of thermal–mechanical bending of laminated composite shallow arches reinforced with GPLs

The current research presents an analysis on the nonlinear thermal and mechanical bending problems of nanocomposite arches reinforced by graphene platelets. The graphene nanofillers are distributed through the thickness of the composite arch as the uniform or functionally graded patterns. Thermo-mechanical properties of the nanocomposite arch are evaluated using the Halpin–Tsai rule of mixtures approach. The multi-layered thick arch with shallow curvature is assumed to rest on an elastic nonlinear medium. Equilibrium equations of the assumed arch are obtained using the principle of virtual work and considering the third-order shear deformation theory with the von Kármán type of nonlinearity. The nonlinear governing equations are solved for the nanocomposite arch with immovable simply-supported and clamped–clamped boundaries by using a perturbation-based technique. Different numerical results are presented in this paper to evaluate the effects of curvature ratio, slenderness ratio, elastic medium, and the weight fraction and distribution pattern of GPLs on the thermal and mechanical bending behaviors of composite arches.

The Influence of the Boundary Conditions on the Buckling of Thin-walled Cans during Manufacturing

In this paper the effect of the boundary conditions on the stability of thin-walled aerosol cans under axial pressure is investigated. The main objective is to outline the main characteristics of this highly nonlinear mechanical problem and to present methods to simulate the buckling of cans with different boundary conditions. Due to the numerical difficulties coming from the contact between the can and different components of the machines, the effect of the different supports of the can is investigated on the crushing (or buckling) force at which the loss of stability occurs. The commercial finite element software Abaqus is used to solve the problems and to present the efficiency of FE codes in the design process of cans.

DeepFEM: A Novel Element-Based Deep Learning Approach for Solving Nonlinear Partial Differential Equations in Computational Solid Mechanics

In this paper, an element-based deep learning approach named DeepFEM for solving nonlinear partial differential equations (PDEs) in solid mechanics is developed to reduce the number of sampling points required for training the deep neural network. Shape functions are introduced into deep learning to approximate the displacement field within the element. A general scheme for training the deep neural network based on derivatives computed from the shape functions is proposed. For the sake of demonstrations, the nonlinear vibration, nonlinear bending, and cohesive fracture problems are solved, and the results are compared with those from the existing methods to evaluate the performance of the present method. The results demonstrate that DeepFEM can effectively approximate the solution of the nonlinear mechanics problems with high accuracy, while the shape functions can significantly improve the computational efficiency. Moreover, with the trained DeepFEM model, the solutions of nonlinear problems with different geometric or material properties can be obtained instantly without retraining. Finally, the proposed DeepFEM is employed in the identification of material parameters of composite plate. The results show that the longitudinal and transverse elastic moduli of the ply in the composite plates can be accurately predicted based on the nonlinear mechanical response of plates.

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components’ interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component – associated with the solution on interior boundaries – and a bubble component that vanishes at ports: since the bubble components are uniquely determined by the solution value at the corresponding port, we can recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-squares problem and can be solved using the Gauss–Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our method; we further discuss the well-posedness of the mathematical formulation and the a priori error analysis for linear coercive problems.

Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

We present a new Integrated Finite Element Neural Network framework (I-FENN), with the objective to accelerate the numerical solution of nonlinear computational mechanics problems. We leverage the swift predictive capability of neural networks (NNs) and we embed them inside the finite element stiffness function, to compute element-level state variables and their derivatives within a nonlinear, iterative numerical solution. This process is conducted jointly with conventional finite element methods that involve shape functions: the NN receives input data that resembles the material point deformation and its output is used to construct element-level field variables such as the element Jacobian matrix and residual vector. Here we introduce I-FENN to the continuum damage analysis of quasi-brittle materials, and we establish a new non-local gradient-based damage framework which operates at the cost of a local damage approach. First, we develop a physics informed neural network (PINN) to resemble the non-local gradient model and then we train the neural network offline. The network learns to predict the non-local equivalent strain at each material point, as well as its derivative with respect to the local strain. Then, the PINN is integrated in the element stiffness definition and conducts the local to non-local strain transformation, whereas the two PINN outputs are used to construct the element Jacobian matrix and residual vector. This process is carried out within the nonlinear solver, until numerical convergence is achieved. The resulting method bears the computational cost of the conventional local damage approach, but ensures mesh-independent results and a diffused non-local strain and damage profile. As a result, the proposed method tackles the vital drawbacks of both the local and non-local gradient method, respectively being the mesh-dependence and additional computational cost. We showcase through a series of numerical examples the computational efficiency and generalization capability of I-FENN in the context of non-local continuum damage, and we discuss the future outlook. The PINN training code and associated training data files have been made publicly available online, to enable reproducibility of our results.

Algorithms analysis for solving geometrically nonlinear mechanics problems in the scheme of the semi-analytical finite element method

The effectiveness of using the semi-analytical finite element method (SAFEM) to research geometrically nonlinear construction mechanics problems for 3D bodies of revolution under the arbitrary loads based on a basic ring finite element is considered. An estimate of the rational application parameters of algorithms for taking into account the geometric nonlinearity of a defined above structures class, which has been obtained.
 In the process of numerically solving spatial problems of the theory of elasticity and plasticity with finite displacements, the choice of rational algorithms for solving systems of nonlinear equations is of fundamental importance. It is conditioned by the need of determining the coordinates of the discrete model in the actual configuration and changing the metric characteristics of the finite elements, which, in its turn, leads to the necessity for multiple solutions of systems of nonlinear equations of high order. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation: the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFEM), it is possible, on the one hand, to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element, on the other hand- to maintain a high efficiency of spatial discretization. An important stage in the implementation of computer systems for solving spatial problems is the selection of optimal, from the point of the solution convergence speed and algorithms for solving equilibrium equations. The specificity of the algebraic equations of the SAFEM is conditioned by violation of the trigonometric function orthogonality in the space of the elasticity operator for bodies with variable stiffness and mass parameters along the guide. The clearly defined block structure of the stiffness matrix became the basis for using algorithms combining direct and iterative methods of solving. The reliability of the obtained results and the effectiveness of the approach are confirmed by the solution of a wide range of control examples under various boundary conditions and external loads.

Comparison between homotopy analysis method and homotopy renormalization method in fluid mechanics

In this paper, the homotopy renormalization method (HTR) is compared with the homotopy analysis method (HAM), an analytical approximation technique for highly nonlinear problems. Four problems in fluid mechanics, including the Blasius viscous flow, magnetohydrodynamic flow, free-convective boundary-layer flow, and Von Kármán swirling viscous flow, are used for detailed comparison. It is found that the HAM approximations are much more accurate than the HTR approximations at the same order. Furthermore, higher-order approximations with smaller errors can be obtained relatively simply by the HAM, while the HTR struggles in such scenarios. All of these confirm the superiority of the HAM for highly nonlinear problems in fluid mechanics.