Nonlinear Structural Mechanics Research Articles

Non-spherical axisymmetric deformations of hyperelastic shells

Spherical elastic shells commonly appear both in nature and man-made devices. Often, their functionality is governed by an incoming- or outgoing flux of fluid. The transient traction that the fluid exerts in the process causes the shell to depart from sphericity. Here, we develop a framework for determining non-spherical axisymmetric deformations, by combining tools from nonlinear continuum mechanics, structural mechanics, and asymptotic analysis. We apply our framework to analyze an exemplary problem of a Mooney–Rivlin shell that is filled by viscous fluid. Collectively, our framework and the insights gained from its application, promote the understanding of the mechanics of such fluid-filled deformable membranes and shells.

Spiking recurrent neural networks for neuromorphic computing in nonlinear structural mechanics

The present study aims to introduce an AI algorithm suitable for neuromorphic computing to solve Boundary Value Problems in Engineering Mechanics. Following the trend of sustainable Artificial Intelligence (AI), Spiking Neural Networks (SNNs), known as the third generation of neural networks, are proposed for developing surrogate models for mechanical tasks. SNNs are inherently recurrent and communicate through sparse signals consuming only a fraction of the energy needed by artificial neural networks (ANNs) of the second generation such as current deep learning methods. To take advantage of the energy-efficient spike models, an autoencoding strategy is proposed to convert the real-valued data at a certain time step to its binary spike representation which allows the model to be used for regression tasks. Further, we propose a hybrid model consisting of the spiking variant of the Legendre Memory Unit (LMU), spiking recurrent cells, and classical dense transformations to compute the nonlinear (physically and geometrically) response of shock wave-loaded plate elements. The inclusion of spikes enables the proposed model to be deployed on neuromorphic hardware, such as the Loihi chip. Thus, for a series of both training and validation experiments, a detailed comparison of the proposed hybrid model and its second-generation counterpart in terms of energy consumption and accuracy is presented.

Deep-HyROMnet: A Deep Learning-Based Operator Approximation for Hyper-Reduction of Nonlinear Parametrized PDEs

To speed-up the solution of parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projection-based ROMs such as the reduced-basis (RB) method, deep learning-based ROMs, as well as surrogate models obtained through machine learning techniques. Thanks to its physics-based structure, ensured by the use of a Galerkin projection of the full order model (FOM) onto a linear low-dimensional subspace, the Galerkin-RB method yields approximations that fulfill the differential problem at hand. However, to make the assembling of the ROM independent of the FOM dimension, intrusive and expensive hyper-reduction techniques, such as the discrete empirical interpolation method (DEIM), are usually required, thus making this strategy less feasible for problems characterized by (high-order polynomial or nonpolynomial) nonlinearities. To overcome this bottleneck, we propose a novel strategy for learning nonlinear ROM operators using deep neural networks (DNNs). The resulting hyper-reduced order model enhanced by DNNs, to which we refer to as Deep-HyROMnet, is then a physics-based model, still relying on the RB method approach, however employing a DNN architecture to approximate reduced residual vectors and Jacobian matrices once a Galerkin projection has been performed. Numerical results dealing with fast simulations in nonlinear structural mechanics show that Deep-HyROMnets are orders of magnitude faster than POD-Galerkin-DEIM ROMs, still ensuring the same level of accuracy.

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these methods, nonlinear elimination is an important ingredient to increase the convergence radius of Newton’s method. We prove standard globalization results for SQP-based globalization of Nonlinear FETI-DP, first for the case that the elimination set is empty. We then show how to combine nonlinear elimination and SQP-based globalization. The globalization preserves the block structure of the FETI-DP operator, which is the basis of the computational parallelism.Supporting numerical experiments using homogenous and heterogeneous model problems from nonlinear structural mechanics are provided. In the numerical experiments, we consider four standard choices of different elimination sets and different problem setups including stiff or almost incompressible inclusions in every subdomain. The numerical experiments illustrate that a good elimination set is important. However, the use of the SQP-based globalization approach presented here can improve the convergence of Nonlinear FETI-DP methods further, especially, if combined with a good choice of the elimination set.

Designing the nonlinear mechanical response of graphyne structures: A finite element structural mechanics approach

Over recent decades, the exploration of various allotropes of carbon at the nanoscale has attracted a significant amount of research interest. The demand for the synthesis of novel carbon-based nanostructures has resulted in the discovery of graphynes, i.e. one-atom-thick planar crystal lattice sheets of benzene rings connected by acetylene bonds. In this study, computer modelling and simulations were applied to characterise the full tensile and elastoplastic mechanical behaviour of such structures. Using a nonlinear finite element-based structural mechanics method, which is based on Morse potential functions, several parametric studies were performed, and various illustrative size-dependent diagrams were obtained, enabling the suitable and accurate design of graphyne-based material components based on key mechanical properties such as stiffness, tensile strength, and fracture strain. The four well-known types of graphyne sheets were numerically investigated: graphyne-1, also known as graphyne; graphyne-2, also known as graphdiyne; graphyne-3; and graphyne-4. The predictions of the mechanical properties of graphene were compared with other corresponding estimations to validate the research results.

NONLINEAR STRUCTURAL ANALYSIS BASED ON THE MODIFIED SEQUENTIAL LOAD METHOD

The article discusses ways to improve the accuracy of solving problems of nonlinear structural mechanics. It is shown that the combination of the method of sequential loading and the Newton-Kantorovich method can improve the accuracy of the solution and reduce the complexity of obtaining results. The solution of the given linear equations can be obtained by numerical and approximate methods known in the literature.

Roll-over stability as a problem of high-rise buildings’ designing

Roll-over stability of tall buildings under wind loads is considered. The nonlinear nature of the problem is taken into account, including geometric, physical, and structural non-linearity. The problem is solved on the base of a system of linearized incremental equations of structural mechanics that describes the behavior of a system tall building - foundation soil. Several methods are examined for solving nonlinear problems of roll-over stability, specifically: 1) deformation method of systems equilibrium states tracing; 2) method of linearization of nonlinear equations and systems equilibrium states tracing; 3) method of linearization of nonlinear physical relations of a systems with constructive, static, geometric nonlinearity; 4) method of linearization of nonlinear physical relations of a system with constructive nonlinearity based on nonlinear incremental structural mechanics; 5) method of the deformation process tracing for a physically nonlinear soil base, given the increase of discharge zones and constructive nonlinearity. Each of these methods is used to solve a model task. These tasks take into account roll-over stability of high structures under action of wind loads. In general, the problem of roll-over stability of a high object can be represented as repeatedly nonlinear one with various types of non-linearity. In this regard, in the practice of high-rise buildings designing, it is necessary to develop scientifically and methodically substantiated methods of assessing roll-over stability, considering non-linear factors. Taking these factors into account will make it possible to assess the roll-over stability of a high-rise object more accurate.

Evolution of methods for accounting nonlinearity in calculation of building structures

This article discusses the stages of development of disciplines that aim to calculate building structures. The prerequisites and some of the fundamental provisions that led to the formation of such a field of research as nonlinear structural mechanics are briefly described.

A simple extrapolated predictor for overcoming the starting and tracking issues in the arc-length method for nonlinear structural mechanics

This paper presents a simplified implementation of the arc-length method for computing the equilibrium paths of nonlinear structural mechanics problems using the finite element method. In the proposed technique, the predictor is computed by extrapolating the solutions from two previously converged load steps. The extrapolation is a linear combination of the previous solutions; therefore, it is simple and inexpensive. Additionally, the proposed extrapolated predictor also serves as a means for identifying the forward movement along the equilibrium path without the need for any sophisticated techniques commonly employed for explicit tracking. The ability of the proposed technique to successfully compute complex equilibrium paths in static structural mechanics problems is demonstrated using seven numerical examples involving truss, beam-column and shell models. The computed numerical results are in excellent agreement with the reference solutions. The present approach does not require prohibitively small increments for its success.

Data Augmentation and Feature Selection for Automatic Model Recommendation in Computational Physics

Classification algorithms have recently found applications in computational physics for the selection of numerical methods or models adapted to the environment and the state of the physical system. For such classification tasks, labeled training data come from numerical simulations and generally correspond to physical fields discretized on a mesh. Three challenging difficulties arise: the lack of training data, their high dimensionality, and the non-applicability of common data augmentation techniques to physics data. This article introduces two algorithms to address these issues: one for dimensionality reduction via feature selection, and one for data augmentation. These algorithms are combined with a wide variety of classifiers for their evaluation. When combined with a stacking ensemble made of six multilayer perceptrons and a ridge logistic regression, they enable reaching an accuracy of 90% on our classification problem for nonlinear structural mechanics.

Intelligent step-length adjustment for adaptive path-following in nonlinear structural mechanics based on group method of data handling neural network

A novel step-length adjustment method for adaptive path-following in geometrically nonlinear problems of solid mechanics is proposed in this paper. The core idea is how to predict the critical points on the equilibrium path properly, and then distribute the points on the path accordingly. We characterize the equilibrium path employing a scalar stiffness parameter. Here, we show how the stiffness parameter could be used to detect the critical points on the path. Then, we employ time-series forecasting based on group method of data handling (GMDH) to predict the stiffness parameter which in fact leads to the prediction of critical points. To adaptively adjust the step-length to the predicted critical point, we present a simple formulation to distribute the points on the equilibrium path. In this paper, a novel adaptive path-following algorithm is presented to trace and predict the path efficiently. The proposed algorithm, minimum residual displacement method (MRDM), arc length method (ALM) and generalized displacement method (GDM) are comparatively investigated for the geometrically nonlinear analysis of structures in both continuum and discrete problems (truss and cylindrical shell). Selected examples present large fluctuations in stiffness. We demonstrate with the numerical examples that the trained neural network can properly predict the critical points on the path. The results also illustrate that our adaptive path-following method is able to distribute converged points properly and trace the equilibrium path with a significantly reduced number of points compared to the analyses with the uniform step-length.

Tensile properties of microtubules: A study by nonlinear molecular structural mechanics modelling

A nonlinear molecular structural mechanics (MSM) model is proposed in this paper for studying the tensile properties of microtubules (MTs). In the nonlinear MSM models, the interactions between tubulin monomers in MTs are treated as nonlinear axial and torsional springs, whose stiffness coefficients are extracted from all-atom molecular dynamics simulations. The Young's modulus and fracture properties of MTs under tension extracted from the present nonlinear MSM models are found to agree well with the existing simulation and experiment results, which shows the efficiency and accuracy of the proposed nonlinear MSM models. In addition, the nonlinear MSM models are also extended to investigate the tensile properties including Young's modulus and fracture strain of MTs possessing lattice defects. The results obtained from nonlinear MSM models are utilized to develop a predictive equation for quickly predicting the tensile properties of MTs with different lattice defect levels.

A nonintrusive reduced order model for nonlinear transient thermal problems with nonparametrized variability

In this work, we consider a transient thermal problem, with a nonlinear term coming from the radiation boundary condition and a nonparametrized variability in the form complex scenarios for the initial condition and the convection coefficients and external temperatures. We use a posteriori reduced order modeling by snapshot Proper Orthogonal Decomposition. To treat the nonlinearity, hyperreduction is required in our case, since precomputing the polynomial nonlinearities becomes too expensive for the radiation term. We applied the Empirical Cubature Method, originally proposed for nonlinear structural mechanics, to our particular problem. We apply the method to the design of high-pressure compressors for civilian aircraft engines, where a fast evaluation of the solution temperature is required when testing new configurations. We also illustrate that when using in the reduced solver the same model as the one from the high-fidelity code, the approximation is very accurate. However, when using a commercial code to generate the high-fidelity data, where the implementation of the model and solver is unknown, the reduced model is less accurate but still within engineering tolerances in our tests. Hence, the regularizing property of reduced order models, together with a nonintrusive approach, enables the use of commercial software to generate the data, even under some degree of uncertainty in the proprietary model or solver of the commercial software.

A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems

We propose a non-intrusive reduced basis (RB) method for parametrized nonlinear partial differential equations (PDEs) that leverages models of different accuracy. From a collection of low-fidelity (LF) snapshots, parameter locations are extracted for the evaluations of high-fidelity (HF) snapshots to recover a reduced basis. Multi-fidelity Gaussian process regression (GPR) is employed to approximate the combination coefficients of the reduced basis. LF data is assimilated either via projection onto an LF basis or via an interpolation approach inspired by bifidelity reconstruction. The correlation between HF and LF data is modeled with hyperparameters whose values are automatically determined in the regression step. The proposed methods not only leverage the assimilated LF data to reduce the cost of the offline phase, but also allow for a fast evaluation during the online stage, independent of the computational cost of neither the low-nor the high-fidelity solution. Numerical studies demonstrate the effectiveness of the proposed approach on manufactured examples and problems in nonlinear structural mechanics. Clear benefits of using lower resolution models rather than reduced physics models are observed in both the basis selection and the regression step. An active learning scheme is used for additional snapshot selection at locations with high error. The speed-up in the online evaluation and the high accuracy of extracted quantities of interest makes the multifidelity RB method a powerful tool for outer-loop applications in engineering, as exemplified in uncertainty quantification.

Mechanics of High-Flexible Beams Under Live Loads

In this paper the mathematical formulation of the equilibrium problem of high-flexible beams in the framework of fully nonlinear structural mechanics is presented. The analysis is based on the recent model proposed by L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. In this model the complete three-dimensional kinematics of the beam is taken into account, both deformations and displacements are considered large and a nonlinear constitutive law in assumed. After having illustrated and discussed the peculiar mechanical aspects of this special class of structures, the criteria and methods of analysis have been addressed. A classification of the structures based on the degree of kinematic constraints has been proposed, distinguishing between isogeometric and hypergeometric structures. External static loads dependent on deformation (live loads) are also considered. The governing equations are derived on the basis of a moment-curvature relationship obtained in L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. The governing equations take the form of a highly nonlinear coupled system of equations in integral form, which is solved through an iterative numerical procedure. Finally, the proposed analysis is applied to some popular structural systems subjected to dead and live loads. The results are compared and discussed.

Optimal design of foundations by means of nonlinear calculation methods

This paper proposes using the defining equations from the theory of adaptive evolution of mechanical systems (which is based on the variational principles of nonlinear structural mechanics) to design the shape and size of foundations. It presents an expression for finding the potential energy of a system and the deformation energy density, as well as the variational Lagrange equation. The paper formulates a nonlinear boundary problem solved by finite-element analysis. The solution imposes a constraint on the modulus of elasticity to take into account the physico-mechanical properties of the materials. A calculation algorithm and an ADPL program are written for ANSYS. The paper also presents a solution to the problem of finding the rational foundation shaped for the case of plain strain. The solution-derived rational foundation shape is shown. The authors plot the stresses and energy densities as a function of evolution at the onset and finish of iterative processes. Note that the resulting foundation shape is more stable, more accurately positioned in the soil, and can carry a greater load compared to more conventional shapes.

A nonintrusive distributed reduced‐order modeling framework for nonlinear structural mechanics—Application to elastoviscoplastic computations

SummaryIn this work, we propose a framework that constructs reduced‐order models for nonlinear structural mechanics in a nonintrusive fashion and can handle large‐scale simulations. Three steps are carried out: (i) the production of high‐fidelity solutions by commercial software, (ii) the offline stage of the model reduction, and (iii) the online stage where the reduced‐order model is exploited. The nonintrusivity assumes that only the displacement field solution is known, and the proposed framework carries out operations on these simulation data during the offline phase. The compatibility with a new commercial code only needs the implementation of a routine converting the discretized solution into our in‐house data format. The nonintrusive capabilities of the framework are demonstrated on numerical experiments using commercial versions of Z‐set and Ansys Mechanical. The nonlinear constitutive equations are evaluated by using an external plugin. The large‐scale simulations are handled using domain decomposition and parallel computing with distributed memory. The features and performances of the framework are evaluated on two numerical applications involving elastoviscoplastic materials: the second one involves a model of high‐pressure blade, where the framework is used to extrapolate cyclic loadings in 6.5 hours, whereas the reference high‐fidelity computation would take 9.5 days.

Probing the Force Field to Identify Potential Energy

A small ball resting on a curve in a gravitational field offers a simple and compelling example of potential energy. The force required to move the ball, or to maintain it in a given position on a slope, is the negative of the vector gradient of the potential field: the steeper the curve, the greater the force required to push the ball up the hill (or keep it from rolling down). We thus observe the turning points (horizontal tangency) of the potential energy shape as positions of equilibrium (in which case the “restoring force” drops to zero). In this paper, we appeal directly to this type of system using both one- and two-dimensional shapes: curves and surfaces. The shapes are produced to a desired mathematical form generally using additive manufacturing, and we use a combination of load cells to measure the forces acting on a small steel ball-bearing subject to gravity. The measured forces, as a function of location, are then subject to integration to recover the potential energy function. The utility of this approach, in addition to pedagogical clarity, concerns extension and applications to more complex systems in which the potential energy would not be typically known a priori, for example, in nonlinear structural mechanics in which the potential energy changes under the influence of a control parameter, but there is the possibility of force probing the configuration space. A brief example of applying this approach to a simple elastic structure is presented.

Mechanical and mathematical research of local deformations of a steel roller shell with a variable geometry of contact surface

Abstract The article is devoted to solving the fundamental and applied problem of nonlinear structural mechanics of machines by introducing into the drum two additional stop cylinders with supporting rollers at the end and adjustable length, providing a given elliptical or circular shape of a flexible shell with a smoothly variable geometry in the area of its contact with compacted pavement material. Compaction of soil, gravel and asphalt concrete in the sphere of road is not only an integral part of the technological process of the roadbed, road foundation and surface construction, but it is actually the main operation to ensure their strength, stability and durability. The quality, cost and speed of road construction, the possibility of using fundamentally new technologies, structures and materials is largely determined by the availability of modern road machinery.

An Error Indicator-Based Adaptive Reduced Order Model for Nonlinear Structural Mechanics—Application to High-Pressure Turbine Blades

The industrial application motivating this work is the fatigue computation of aircraft engines’ high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and can reach values relatively close to the creep temperature: important nonlinear effects occur and the solution strongly depends on the used thermal loading. We consider a nonlinear reduced order model able to compute, in the exploitation phase, the behavior of the blade for a new temperature field loading. The sensitivity of the solution to the temperature makes the classical unenriched proper orthogonal decomposition method fail. In this work, we propose a new error indicator, quantifying the error made by the reduced order model in computational complexity independent of the size of the high-fidelity reference model. In our framework, when the error indicator becomes larger than a given tolerance, the reduced order model is updated using one time step solution of the high-fidelity reference model. The approach is illustrated on a series of academic test cases and applied on a setting of industrial complexity involving five million degrees of freedom, where the whole procedure is computed in parallel with distributed memory.