A second‐order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

Fish, Jacob; Yang, Zhiqiang; Yuan, Zifeng

doi:10.1002/nme.6058

Cited by 28 publications

(14 citation statements)

References 89 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even though the equilibrium equation (21) has exactly the same form derived in Geus et al, 29 the main difference is that 0 is not a projection onto the function space V c (0). Indeed, we will point out that the Green function G = C 0 ∶ 0 is a projection operator onto the function space V c (0).…”

Section: Compatibility Projection Operator and Connection To The Exismentioning

confidence: 95%

“…As Equation (24) must hold true for arbitrary tensor-valued functions W, the equilibrium equation 21can be equivalently derived from this equation. In fact, Equations (21) and (24) are the strong form and weak form of the microscopic boundary value problem (10), respectively.…”

Section: Derivation Of Microequilibrium With Compatibility-enforcing mentioning

confidence: 99%

“…With appropriate boundary conditions applied to the RVE, the macrohomogeneity condition can be derived. The two‐scale computational procedure can be interpreted as a simple model derived by the asymptotic method because only the solution up to the first‐order approximation is considered and all the higher order terms are neglected in the computation (see Fish et al 21 ).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A surrogate model for computational homogenization of elastostatics at finite strain using high‐dimensional model representation‐based neural network

Nguyen‐Thanh

Nguyen

Rabczuk

et al. 2020

Numerical Meth Engineering

View full text Add to dashboard Cite

We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macroenergy density. This energy density is constructed by using a neural network architecture that mimics a high-dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary value problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two-scale computational procedure for nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. The finite element method is employed to solve the equilibrium equation at the macroscale. As for microscopic problems, an FFT-based collocation method is applied in tandem with the Newton-Raphson iteration and the conjugate-gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium. In this manner, the fixed-point iteration that might require quite strict numerical stability conditions in the nonlinear regime is avoided. In addition, we derive the projection operator used in the FFT-based method for homogenization of elasticity at finite strain. K E Y W O R D S computational homogenization, data-driven, FFT-based methods, nonlinear elasticity 1 INTRODUCTION Multiscale techniques are important for man-made and natural materials; one such approach is homogenization. Roughly speaking, homogenization is a rigorous version of what is known as averaging. It is a powerful tool to study the heterogeneous materials and composites. Based on the knowledge of the microstructure of materials, the objective is to Abbreviations: HDMR, high-dimensional model representation; FFT, fast Fourier transform; FEM, finite element method; NN, neural network.

show abstract

Section: Compatibility Projection Operator and Connection To The Exismentioning

confidence: 95%

Section: Derivation Of Microequilibrium With Compatibility-enforcing mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A surrogate model for computational homogenization of elastostatics at finite strain using high‐dimensional model representation‐based neural network

Nguyen‐Thanh

Nguyen

Rabczuk

et al. 2020

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…Bhattacharjee and Matouš 46 applied a manifold-based order reduction model to predict the macro performance of the hyperelastic heterogenous composites. Yuan, 47 Fish, 48 and Yang et al 49 constructed and applied the multiscale reduced models to evaluate any inelastic deformations of the heterogeneous composites, and could greatly save the computing time. Further, Yang et al, 50 expanded the multiscale reduced-order approach to compute the nonlinear axisymmetric structure with periodic configuration.…”

Section: Introductionmentioning

confidence: 99%

A second‐order reduced multiscale method for nonlinear shell structures with orthogonal periodic configurations

Yang

Liu

Sun

et al. 2021

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

This article develops an efficient second-order reduced multiscale (SORM) method to study the nonlinear shell structure with orthogonal periodic configurations. The heterogenous shell structure is periodically distributed in orthogonal curvilinear coordinate systems. At first, the nonlinear problems for the shell structure are introduced, and the detailed higher-order nonlinear multiscale formulas based on the asymptotic homogenization approach are given including microscale unit cell functions, effective material parameter and the homogenized equation. Also, since it requires a large number of computation cost to solve the nonlinear multiscale problems by the traditional high-order homogenization methods, the novel reduced order multiscale model is constructed.Further, according to the reduced-order multiscale models and higher-order nonlinear formulas, an effective SORM algorithm is provided for studying the nonlinear shell structures. The main characteristics of the proposed algorithm are that the novel reduced forms established to investigate the nonlinear shell structures and an efficient higher-order homogenized solution evaluated by postprocessing that does not need higher-order continuities of the homogenization solutions. Finally, according to some typical nonlinear examples including block structures, cylindrical shell and double-curved shallow shell, the availabilities of the SORM algorithm are confirmed.

show abstract

“…Some other recent application was to find the effect of damage amplification due to micro-crack interaction (see Markenscoff and Dascalu, 2012), to simulate thermoelectric composites (see Yang et al., 2013). It is noteworthy to mention that most of the analyses are based on O(1) expansion approaches which results in inaccuracy at regions of high gradients and for low-scale separation, higher order homogenisation has been developed for transient dynamics (see Hui and Oskay, 2014), elastic composites (see Ameen et al., 2018) and for elasto-plastic heterogeneity (see Fish et al., 2019).…”

Section: Introductionmentioning

confidence: 99%

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Bhattacharyya

Dureisseix

Faverjon

2020

International Journal of Damage Mechanics

View full text Add to dashboard Cite

This article deals with damage computation of heterogeneous structures containing locally periodic micro-structures. Such heterogeneous structure is extremely expensive to simulate using classical finite element methods, as the level of discretisation required to capture the micro-structural effects is too fine. The simulation time becomes even higher when dealing with highly non-linear material behaviour, e.g. damage, plasticity and such others. Therefore, a multi-scale strategy is proposed here that facilitates the simulation of non-linear heterogeneous material behaviour in a manner that is computationally feasible. Based on the asymptotic homogenisation theory, this multi-scale technique explores the micro–macro behaviour for elasto-(visco)plasticity coupled with damage. The theory inherently segregates the heterogeneous continua into a macroscopic homogeneous structure and an underlying heterogeneous microscopic periodic unit cell. Several heterogeneous structures have been simulated using the multi-scale method along with a one-dimensional verification with respect to a reference solution. Additionally, a reduced order modelling is used to prevent large memory requirement for storing micro-structural quantities of interest.

show abstract

A second‐order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

Cited by 28 publications

References 89 publications

A surrogate model for computational homogenization of elastostatics at finite strain using high‐dimensional model representation‐based neural network

A surrogate model for computational homogenization of elastostatics at finite strain using high‐dimensional model representation‐based neural network

A second‐order reduced multiscale method for nonlinear shell structures with orthogonal periodic configurations

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Contact Info

Product

Resources

About