Variational framework for distance-minimizing method in data-driven computational mechanics

Nguyen, Lu Trong Khiem; Rambausek, Matthias; Keip, Marc‐André

doi:10.1016/j.cma.2020.112898

Cited by 35 publications

(70 citation statements)

References 38 publications

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“…The double minimization problem ( 19) is a combination of continuous and discrete optimization problems, the former over the continuous manifold E n+1 , the latter in the discrete data-set D. It has a combinatorial complexity, since for each material point m contributing to the global distance-function (18), n dp points can be evaluated and the minimum should be chosen among those. To efficiently solve this computationally intensive combinatorial problem, following [28,26], a staggered solution scheme is adopted here which freezes the continuous minimization problem while solving the discrete one and vice-versa. It assumes at an iteration k the optimum point in the data-set * y k n+1 ∈ D to be known and finds a closest state z n+1 k+1 ∈ E n+1 to that data-set point.…”

Section: Distance Minimizing Data-driven Problemmentioning

confidence: 99%

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Data-driven reduced homogenization for transient diffusion problems with emergent history effects

Waseem

Heuzé

Geers

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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In this paper, we propose a data-driven reduced homogenization technique to capture diffusional phenomena in heterogeneous materials which reveal, on a macroscopic level, a history-dependent non-Fickian behavior. The adopted enriched-continuum formulation, in which the macroscopic history-dependent transient effects are due to the underlying heterogeneous microstructure is represented by enrichment-variables that are obtained by a model reduction at the micro-scale. The data-driven reduced homogenization minimizes the distance between points lying in a data-set and points associated with the macroscopic state of the material. The enrichment-variables are excellent pointers for the selection of the correct part of the data-set for problems with a time-dependent material state. Proofof-principle simulations are carried out with a heterogeneous linear material exhibiting a relaxed separation of scales. Information obtained from simulations carried out at the micro-scale on a unit-cell is used to determine approximate values of metric coefficients in the distance function. The proposed data-driven reduced homogenization also performs adequately in the case of noisy data-sets. Finally, the possible extensions to non-linear history-dependent behavior are discussed.

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Section: Distance Minimizing Data-driven Problemmentioning

confidence: 99%

“…In this work, following [13,16,26], a staggered distance-minimizing data-driven solver is adopted. It iteratively minimizes a quadratic distance function, defined on the material phasespace, while looking for a point in the data-set.…”

Section: Introductionmentioning

confidence: 99%

Data-driven reduced homogenization for transient diffusion problems with emergent history effects

Waseem

Heuzé

Geers

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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“…The RVE consists of two phases: (i) a circular inclusion and (m) a matrix surrounding the inclusion. Two phases are made of the Neo-Hookean materials that are characterized by the energy density (50) and the associated Young modulus and Poisson ratio. These material parameters are specialized to each phase…”

Section: Problem Settingmentioning

confidence: 99%

“…The microscopic BVPs are solved for 10 3 input macroscopic strain data

\overset{true‾}{ϵ}

that are randomly distributed in the range [0,2] to compute the output data of macroenergy density. As the microscopic BVPs can be solved analytically as in Nguyen et al, 50 the material data can be easily collected. Thus, we used 10 3 data points for the high resolution of the macroscopic solution, although much less data could provide high‐quality solution.…”

Section: Representative Numerical Examplesmentioning

confidence: 99%

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

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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.

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“…The stationary condition of Equation (12) yields the following data-driven variational equations (He and Chen, 2020;He et al, 2020b;Nguyen et al, 2020):…”

Section: Physical Solvermentioning

confidence: 99%

Physics-constrained local convexity data-driven modeling of anisotropic nonlinear elastic solids

He²,

Chen

et al. 2020

DCE

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As characterization and modeling of complex materials by phenomenological models remains challenging, data-driven computing that performs physical simulations directly from material data has attracted considerable attention. Data-driven computing is a general computational mechanics framework that consists of a physical solver and a material solver, based on which data-driven solutions are obtained through minimization procedures. This work develops a new material solver built upon the local convexity-preserving reconstruction scheme by He and Chen (2020) A physics-constrained data-driven approach based on locally convex reconstruction for noisy database. Computer Methods in Applied Mechanics and Engineering 363, 112791 to model anisotropic nonlinear elastic solids. In this approach, a two-level local data search algorithm for material anisotropy is introduced into the material solver in online data-driven computing. A material anisotropic state characterizing the underlying material orientation is used for the manifold learning projection in the material solver. The performance of the proposed data-driven framework with noiseless and noisy material data is validated by solving two benchmark problems with synthetic material data. The data-driven solutions are compared with the constitutive model-based reference solutions to demonstrate the effectiveness of the proposed methods.

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Variational framework for distance-minimizing method in data-driven computational mechanics

Cited by 35 publications

References 38 publications

Data-driven reduced homogenization for transient diffusion problems with emergent history effects

Data-driven reduced homogenization for transient diffusion problems with emergent history effects

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

Physics-constrained local convexity data-driven modeling of anisotropic nonlinear elastic solids

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