Asymptotic Numerical Method Research Articles

The asymptotic-numerical method for the post-buckling response of curvilinearly stiffened panels

Abstract In the present work, the asymptotic-numerical method is applied in conjunction with the Ritz method as a powerful mean for analysing the post-buckling response of panels with variable stiffness skin and curvilinear stringers. Main advantage of the proposed approach is the reduced computational time. The Ritz method guarantees an excellent ratio between accuracy and required degrees of freedom; the asymptotic-numerical method requires just one matrix inversion throughout the solution process. Moreover, the complete analytical representation of the non-linear equilibrium path is obtained, as opposed to the point-by-point representation of predictor-corrector algorithms. Several test cases are presented and compared with standard Newton-Raphson computations and commercial finite element simulations. The results show noticeable saving of computational time. For the test cases investigated, the asymptotic-numerical method requires about one third of the time required by a standard Newton-Raphson routine. These results demonstrate that the combination between Ritz and the asymptotic-numerical method is an excellent strategy for investigating the post-buckling response of innovative curvilinearly stiffened panels.

Theoretical and experimental analysis of a nonlinear vibro-acoustic energy sink

This study is concerned with the experimental and theoretical analysis of a vibroacoustic nonlinear system devoted to absorption application. Based on the nonlinear energy sink principle, the dynamics of the system is reduced to two coupled degrees of freedom, one of which exhibits nonlinear behavior due to a large displacement mechanism. The set-up allows for precisely characterizing the nonlinear function of the mechanical system, first uncoupled and then coupled with an acoustic and linear resonator. Continuation procedures are performed experimentally and numerically, respectively using a phase-locked loop (PLL) and the asymptotic numerical method implemented in the MANLAB software package. The comparison between theoretical predictions and experimental results obtained on the uncoupled system allows for a parameter identification of the nonlinear resonator. Measurements on the coupled system highlight the energy pumping phenomenon, which can be predicted from numerical predictions in which identified parameters are introduced. The procedure aims at proposing analytical models for achieving efficient level attenuation, and providing design guidelines for sound reduction at high levels in confined spaces.

Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

Investigating the bending and buckling behaviors of composite porous beams reinforced with carbon nanotubes and graphene platelets using a TRPIM path following mesh-free approach

The aim of the present work consists in investigating the nonlinear behavior of porous beams reinforced with graphene platelets (GPL) and supported carbon nanotubes (CNT), termed functionally graded graphene platelets reinforced composite beam (FG-GPLRC) and functionally graded nanotube carbon reinforced composite beam (FG-CNTRC), respectively. Notably, the distribution of GPL/CNT is explored in both uniform and non-uniform patterns across the beam's thickness. What sets this research apart is its utilization of a refined beam model as enhanced FSDT incorporating nonlinear shear terms which is a crucial advancement in accurately capturing the post-buckling response in certain boundary conditions, a feature lacking in the existing FSDT literature. Innovatively, the post-buckling load-deflection relationship is derived through the solution of governing equations incorporating cubic nonlinearity. This is achieved by employing Galerkin's method alongside a non-iterative high-order continuation technique based on the asymptotic numerical method coupled with the Tchebychev-radial point interpolation method (TRPIM), using a path-following where the solutions are obtained branch-by-branch by eliminating the need for iterative processes. In essence, this research underscores the pivotal role of porosity and GPL/CNT reinforcement in shaping the post-buckling configuration of both perfect and imperfect nanocomposite beams, thereby advancing our understanding of structural behavior in porous nanocomposite materials. The findings of this study illuminate the significant influence of parameters such as porosity coefficient, porosity distribution, GPL/CNT distribution, and GPL-weight/CNT-volume fraction on the nonlinear buckling behavior of porous beams.

Solving non-linearities of hyperelastic rubbery behaviour by the asymptotic numerical method: Highlighting with the case of a Hybrid Integral Approach (HIA) model

This paper presents mathematical modelling and simulation founded on an Asymptotic Numerical Method (ANM) for the rubber-like material hyperelastic behaviour. The basic models such as Mooney-Rivlin, and the Gent partial models and the Hybrid Integral Approach (HIA) model are considered. The HIA is a strong nonlinear model developed from the inverse Langevin function approximation is considered here for its efficiency. Explicit formulations of each model are presented as well as the associated asymptotic developments in the frame of the ANM. Various types of boundary conditions effects are analysed for each considered hyperelastic model. The effectiveness of the ANM compared to the Newton-Raphson method is demonstrated and particularly for the HIA model. We present as results stress-strain curves for the implementation of these different models where we specify the influence of the Imposed Displacement for tensile and compressive stresses when the structure is clamped and simply supported. In term of this study, we observe that the complete HIA model present the curves with the difference hardly perceptible when we vary results in term of stress-strain curves with different imposed Displacement. But when strain is high than 5%, the Mooney-Rivlin and the Gent partial models, presents the high disparities with different imposed Displacement. This observation justifying the limitations of partial models in term of characterisation of the behaviour law of hyperplastic materials.

Non-linear buckling analysis of thin-walled beams modeled with 7-parameter shell elements

The non-linear quasi-static buckling of thin-walled beams with arbitrary cross-sections is studied by developing coupling strategies between the composing plates (web, flanges). The element adopted for modeling the plates is a 7-parameter shell element using the Enhanced Assumed Strain concept (Büchter et al., 1994). Different nodal shell directors exist at the interface between plates for thin-walled beams with arbitrary cross-sections, requiring appropriate strategies. The coupling between plates first considers Lagrange multipliers at the interface, following the mortar method on the surface between the plates. Second a simplified pre-processing method is proposed by modifying the shell directors of the nodes close to the interface. In the case of the quasi-static buckling of thin-walled beams with L-shaped and I-shaped cross-sections, the two coupling strategies are thoroughly assessed by using three different solution procedures: Newton–Raphson, Newton–Riks and Asymptotic Numerical Method (ANM). The pre-processing method is simple and turns out to be robust and efficient.

Asymptotic [formula omitted]-FEM for nonlinear analysis of composite shells

Nonlinear shell analysis relies typically on Finite Element Methods (FEMs) and Iterative-Incremental Procedures (IIPs). These methodologies can become computationally expensive whenever high-fidelity meshes are required to capture very localized features or extremely nonlinear responses. Aim of this study is presenting a novel computational tool based on an efficient finite element formulation, the ps-FEM, and a rapid perturbation solution procedure, the Asymptotic-Numerical Method (ANM). The proposed approach adopts a polynomial space enrichment strategy, the p-refinement, and a mesh superposition technique, the s-refinement, to build numerical models with quasi-optimal accuracy-to-error ratios. The introduced asymptotic framework enhances the effectiveness of solving nonlinear problems compared to IIPs. A set of test cases and new benchmarks is presented to validate the tool and demonstrate its potential. The present results show that challenging problems involving bifurcations, jumps, snap-backs and anisotropy-induced localizations can be solved with excellent degree of accuracy and relatively small modeling/computational effort.

Combining the Asymptotic Numerical Method with the Harmonic Balance Method to capture the nonlinear dynamics of spur gears

The present work proposes the Asymptotic Numerical Method (ANM) combined to the Harmonic Balance Method (HBM) as a valuable approach to solve the nonlinear dynamics of gear pairs. The ANM is a continuation method based on high-order Taylor series expansion of the computed solution branch. The HBM is a periodic solution representation method based on high-order Fourier series. Thanks to a quadratic recast of the equation of motion, the Taylor and Fourier series can be computed in a very efficient way and each step produces a continuous representation of the solution branch making the continuation very robust. By employing this method, the periodic solutions may be easily expressed with respect to both the shaft rotation frequency and the gear mesh frequency as the adoption of a high number of harmonics has negligible effects on the computational burden. Effectiveness and reliability of the method are proven by comparing the numerical results with that obtained from the Runge–Kutta time integration scheme and experimental data from literature. Afterwards, a comparison in terms of computational efficiency is performed. Finally, some considerations are drawn in order to highlights the main differences between the two methods within gear dynamics computation.

Radial point interpolation‐Chebychev method with asymptotic numerical method for nonlinear buckling analysis of graphene oxide powder‐reinforced composite beams

SummaryThis study aims to analyze the buckling and post‐buckling behaviors of multilayer nanocomposite beams reinforced with graphene oxide powders (GOPs) at low concentrations. The GOPs are randomly oriented and evenly distributed throughout the composite layers, with their weight fraction varying in the thickness direction. The Halpin‐Tsai model is employed to estimate each layer's effective material properties. The nonlinear governing equations for the FG‐GOPRC beam are derived based on the first‐order shear deformation beam theory, and a nonlinear algebraic system is formulated using the principle of potential energy. In this research, a comprehensive parametric analysis is conducted to examine the influence of various factors on the buckling and post‐buckling behaviors of the composite beams. These factors include the distribution pattern and weight fraction of graphene platelets (GPLs) nanofillers, as well as the geometry, size, slenderness ratio, and boundary conditions of the beams. Through the analysis, the study highlights the significant strengthening effect of these factors on the buckling and post‐buckling performance of the composite beams. The findings from this investigation contribute to a deeper understanding of the behavior of multilayer nanocomposite beams reinforced with GOPs. This knowledge can be valuable in designing and optimizing of composite structures for enhanced buckling resistance and post‐buckling performance.

Asymptotic numerical method for hyperelasticity and elastoplasticity: a review

The literature about the asymptotic numerical method (ANM) is reviewed in this paper as well as its application to hyperelasticity and elastoplasticity. ANM is a generic continuation method based on the computation of Taylor series for solving nonlinear partial differential equations. Modern techniques of high-order differentiation provide simple tools for computing these power series, the corresponding algorithms for finite strain elasticity and elastoplasticity being summarized here. Taylor series is not only a computation tool, but it contains also useful information about the structure of the considered solution curve. The paper ends with a short historical account about the development of this numerical method.

Acceleration sensing based on the bifurcation dynamics of parametrically excited mode-localized resonators

A parametrically excited mode-localized accelerometer is designed using the bifurcation phenomenon to improve the robustness of the fluctuation of the driving voltage and damping while maintaining high sensitivity. A dynamic multi-physics model was established while considering both mechanical and electrostatic nonlinearities. The equation was solved by method of multiple scales and verified by harmonic balanced method coupled with the asymptotic numerical method. Two types of bifurcation exist in amplitude frequency response, namely Saddle-Node bifurcation and Supercritical Hopf bifurcation. By introducing Saddle-Node bifurcation, the response amplitude and measurement range can be improved by 100% and 1000%, respectively, while the sensitivity of the amplitude ratio is about 2 orders of magnitude higher than that based on the frequency ratio. At the Supercritical Hopf bifurcation point, a small acceleration will change the topological structure from Supercritical Hopf to Saddle-Node bifurcation. The variation in the amplitude ratio of the Supercritical Hopf point with acceleration is similar to the sign function, which leads to an extremely high sensitivity of 10000%/g in a dynamic range of ±0.001 g. Moreover, the Supercritical Hopf bifurcation point is not affected by the amplitude of the excitation voltage and damping coefficient, which provides a new method for improving the sensing robustness. Ethical Compliance: All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. Conflict of Interest declaration: The authors declare that they have NO affiliations with or involvement in any organization or entity with any financial interest in the subject matter or materials discussed in this manuscript.

Continuum damage mechanics of cyclic viscoplasticity using asymptotic numerical method

Conscious efforts on reduction of greenhouse gas emissions have led to an energy transition to renewable energy, however uncertainties of renewable energy production have resulted in higher thermal cycling demands from conventional power plants. Thermal load cycling at high temperature regions of steam turbine components leads to enhanced creep-fatigue damage accumulation. It is well established that such damage mechanism is numerically best predicted by unified constitutive modeling including damage as a variable as per the formalism of continuum damage mechanics at an expense of considerable computational efforts using finite element analysis. In this paper, the non-iterative Asymptotic Numerical Method (ANM), currently limited to partial cycle analysis with linear hardening plasticity model, is proposed for the first time to address cyclic viscoplasticity problems including damage capable of handling multiple cycles and most generalized loading conditions. Regularization techniques additionally necessary to implement loading-unloading-reloading criteria and advanced constitutive models etc. are presented. The constitutive model chosen for the formulation includes the non-linear multiple back stress variable modified Chaboche model to include damage combined with modified Chaboche-Rousselier isotropic hardening model to include damage, power law for viscoplasticity, Lemaitre’s damage potential and Kachanov-Rabotnov’s creep damage law. The method is verified with defined error measures and then applied to two high pressure steam turbine rotors, one with and another without thermal stress relief groove (TSRG) at the inlet under service type loading conditions to study the beneficial effect of the TSRG on creep-fatigue damage evolution. The accumulated errors of the proposed ANM and computational time are compared to a conventional Newton-Raphson solution.

A dimensionless analytical analysis for buckling and lateral buckling interaction of thin-walled beams with open cross sections

This paper investigates the interaction between buckling and lateral buckling of thin-walled beams with arbitrary open cross-section using nonlinear modeling in large rotation and with warping. First, the equilibrium equations have been transformed into dimensionless ones, and within a nonlinear stability model, various dimensionless parameters are introduced to control the interaction between buckling and lateral buckling. In comparison to a reference of literature, this work focuses on the case of arbitrary sections and includes a greater number of higher-order terms in our analysis. These additions have an impact on the coefficients of the stability formula by expanding its applicability to non- symmetric sections. Then, the effect of the different dimensionless parameters, thus introduced on the stability curve, has been studied. The proposed solutions are validated and compared to numerical solutions obtained by a general finite element package including the warping developed by the authors using Asymptotic Numerical Method (ANM) to solve the initial nonlinear equations.

Constitutive Model-Based Efficient Creep-Fatigue Damage Computation Technique for Steam Turbine Rotors to Enhance Flexible Operational Capabilities

Abstract In the new era of renewable energy, flexible operation of conventional power plants is inevitable, causing high creep-fatigue life consumption. Conventional life assessment methods are deemed to be conservative to address the current requirements. The use of unified constitutive models for the analysis of damage evolution in steam turbine rotors in the past decade has shown promising results and is closely related to experimental data. However, identification of large number of material constants and high computational time hinder the widespread use of such models. In this paper, the latter is addressed using a novel representative input cycle (RIC) concept together with the noniterative asymptotic numerical method (ANM). Various startup shutdown sequences of a typical steam turbine rotor are studied using a unified constitutive model based on Chaboche's kinematic hardening including the damage parameter, Chaboche–Rousselier's isotropic hardening model including the damage parameter, Norton's viscoplastic flow model, Lemaitre's damage potential function and Kachanov–Rabotnov's creep damage law. First, a conventional finite element method (FEM) technique is used and then the proposed RIC method is used to study the evolution of inelastic variables. The reduction in computational time and the compromise in accuracy using the proposed method are studied.

Asymptotic numerical method for cyclic elasto-plasticity and its application to a steam turbine rotor

Flexible operation of the conventional power plants is necessary for the integration of renewables in the energy mix. Steam turbine rotors are one of the most affected components due to the daily thermal load cycling requirements. In order to accurately study the effect of cycling on fatigue, it is necessary to solve cyclic 3D elasto-plasticity problems for a large number of cycles. The established methods are iterative in nature to arrive at the current yield surface for each elasto-plastic step and are computationally extensive. In this paper, we propose the non-iterative Asymptotic Numerical Method (ANM) based solution technique for 3D cyclic elasto-plasticity problems for the first time by a novel regularized Kuhn-Tucker condition. Regularization techniques used in previous works take care of the partial cycle only, i.e., elastic loading, elastic-plastic transition, elasto-plastic flow and elastic unloading for simple linear work hardening models of plasticity. In the present work, the proposed method extends the applicability of ANM by incorporating elastic-plastic transition and elasto-plasticity in reverse loading, second unloading and repeated loading for any number of cycles so that the computationally efficient fatigue analysis can be carried out. In addition, the extension also includes the application of ANM to non-linear kinematic hardening through the Chaboche plasticity model combined with isotropic hardening. For these advancements, additional regularizations are proposed and implemented in a Finite Element Code. The obtained results are then compared to the solutions obtained from the conventional Newton Raphson (NR) solution technique. The proposed ANM is applied to a steam turbine rotor undergoing a typical thermal load cycling and the cyclic elasto-plastic response is compared with that obtained using the conventional Newton Raphson solution technique.

Efficient resolution of incompressible Navier–Stokes equations using a robust high‐order pseudo‐spectral approach

AbstractAn accurate numerical tool is presented in this work to investigate the stationary incompressible Navier–Stokes equations. The proposed approach is based on a pseudo‐spectral method for discretizing the differential equations and the asymptotic numerical method to convert nonlinear systems into linear algebraic equations. The coupling of the spectral method with the asymptotic numerical method is considered as an efficient algorithm to solve any nonlinear differential equations. Their efficiency and robustness are examined here on the flow fluid in different canal with different geometries. These computational efficiency and performance have been analysed via several numerical and benchmark examples of incompressible fluid flow in lid‐driven cavity and vortex shedding over L‐Shaped cavity and fluid flow around a square obstacle. The validation of the proposed approach is made by comparison between the obtained results and those calculated using a finite element method or Ansys commercial code. This validation asserts that the presented numerical tool can be promise for solving fluid flow problems with high accuracy.

Bifurcation analysis of cantilever beams in channel flow

The flutter of cantilevered beams in channel flow is a benchmark example of flow-induced vibrations and its fundamental behaviour is found in numerous practical applications. Experiments have shown that such systems present a wide variety of complex nonlinear behaviour. However, the plethora of previous studies is mostly concerned with linear stability analysis. In this work, we provide an initial impulse for a comprehensive nonlinear study of these systems through bifurcation analysis. We consider a one-dimensional problem, where a cantilevered beam is treated in a modal framework and the surrounding flow is modelled by bulk-flow equations. The system is discretized in space and time via Galerkin procedures (modal, Tau and harmonic balance) and the continuation of periodic solutions is achieved using the asymptotic numerical method. Additionally, a numerical method for an “augmented” linear stability analysis is proposed, allowing the continuation of Hopf bifurcation branches, including their sub- or super-critical nature. The nonlinear dynamics are explored with respect to various dimensionless parameters. Results illustrate a number of behavioural trends: sub-critical bifurcations and hysteresis loops, internal resonances, grazing boundaries (separation between limit cycles with and without intermittent beam-wall impacts) as well as torus bifurcations and quasi-periodic oscillations.

Buckling of pressurized cylindrical shells using the Asymptotic Numerical Method

In this paper, we present a numerical analysis of the buckling of elastic thin structures by using Asymptotic Numerical Method (ANM) and Padé approximants. We consider the case of cylindrical shells submitted to external pressure and we show how the perturbation technique can be applied for conservative and non-conservative loadings. A shell formulation well adapted to thin structures and which uses the Enhanced Assumed Strain (EAS) concept and a three-dimensional constitutive law without condensation is considered in this study. Numerical examples are presented with comparison between conservative and non-conservative loading in term of critical pressure.

Material database construction for data-driven computing via a continuous path-following method

Data-driven computational homogenization has been proposed recently for the analyses of composite structures. Its basic idea is to construct an equivalent stress–strain database of composites via offline homogenization on the representative volume element and conduct online macroscopic simulation through distance-minimizing data-driven computing. Thanks to the scale separation of concurrent multiscale systems, this framework allows for improving online computational efficiency. However, high-density database construction in the offline stage remains a burdensome and time-consuming task. To this end, this work proposed an efficient approach that associates computational homogenization with the Asymptotic Numerical Method (ANM) to construct a high-density database. Being a reliable and efficient perturbation technique, the ANM allows for accurate tracking of the displacement–load paths and easily generates abundant equivalent stress–strain data on the paths. A fiber reinforced composite material with fiber buckling has been considered to demonstrate the accuracy and efficiency of the proposed method for the database construction of composites.

Global/local buckling analysis of thin-walled I-section beams via hierarchical one-dimensional finite elements

Buckling instability is one of the factors that limit the ultimate load-carrying capacity of I-section beams, and the coupling of global buckling and local buckling makes it difficult to predict the mechanical behavior of I-section beams. Hierarchical one-dimensional finite elements for the global/local buckling analysis of I-section beams are presented in this paper. Within the framework of Carrera’s Unified Formulation (CUF), a model is built by using Lagrange polynomials to describe the three-dimensional displacement field as an arbitrary order approximation by displacement variables over the beam cross-section. One-dimensional finite elements are obtained by discretizing the governing equation along the beam length. Due to its high efficiency and step length adaptability, the Asymptotic Numerical Method (ANM) is adopted to solve the nonlinear governing equations. Several typical buckling problems in I-section beams, including global buckling, local buckling and global–local coupled buckling, are investigated. The numerical results demonstrate the validity and efficiency of the proposed model. The critical buckling load and displacement field can be precisely predicted, and a good agreement is found in comparison with results obtained via three-dimensional finite element solutions.