Effective response of heterogeneous materials using the recursive projection method

Peng, Xiaoyao; Nepal, Dhriti; Dayal, Kaushik

doi:10.1016/j.cma.2020.112946

Cited by 5 publications

(4 citation statements)

References 26 publications

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“…The first class is derived from the basic scheme, and operates on the displacement 8 (or, equivalently, on compatible strain fields). These solvers include Newton's method, [8][9][10] the linear and nonlinear conjugate gradient methods, 11,12 fast gradient methods, 13,14 the Barzilai-Borwein scheme, 15 the recursive projection method, 16 the Anderson-accelerated basic scheme, 17 and other Quasi-Newton methods. 18 The second class of solution methods are the polarization schemes, which operate on so-called polarization fields which are neither compatible nor equilibrated.…”

Section: Introductionmentioning

confidence: 99%

Lippmann‐Schwinger solvers for the computational homogenization of materials with pores

Schneider

2020

Numerical Meth Engineering

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We show that under suitable hypotheses on the nonporous material law and a geometric regularity condition on the pore space, Moulinec-Suquet's basic solution scheme converges linearly. We also discuss for which derived solvers a (super)linear convergence behavior may be obtained, and for which such results do not hold, in general. The key technical argument relies on a specific subspace on which the homogenization problem is nondegenerate, and which is preserved by iterations of the basic scheme. Our line of argument is based in the nondiscretized setting, and we draw conclusions on the convergence behavior for discretized solution schemes in FFT-based computational homogenization. Also, we see how the geometry of the pores' interface enters the convergence estimates. We provide computational experiments underlining our claims.

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Section: Introductionmentioning

confidence: 99%

Lippmann‐Schwinger solvers for the computational homogenization of materials with pores

Schneider

2020

Numerical Meth Engineering

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“…To conclude our discussion of Newton-type methods, let us remark that Volmer et al [152] proposed an improved initial guess for Newton-type solvers. Peng et al [153] investigated the recursive projection method, a variant of Newton's method, to take care of possible ill-conditioning of Lippmann-Schwinger solvers for porous materials and discretizations based on trigonometric polynomials.…”

Section: Newton and Quasi-newton Methodsmentioning

confidence: 99%

A review of nonlinear FFT-based computational homogenization methods

Schneider

2021

Acta Mech

150

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Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform.

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“…The decoder converts feature maps obtained from the Encoder into low-resolution von Mises stress fields using a series of transpose convolution layers, each performing transpose convolution operation, as shown in Equation (7),…”

Section: Decodermentioning

confidence: 99%

“…The stress fluctuations can lead to stress concentrations and regions of high stress, and it is critical to predict the peak stresses that are the dominant drivers of material failure. Although numerical methods for elasticity, such as the finite element method [3] or Fourier-based schemes [4–7], can solve for the stress field – and the peak stresses as subsequent post-processing – these methods are relatively expensive, particularly if applied to a large number of microstructures drawn from a random distribution. Therefore, machine-learning-based methods have recently been proposed as an alternative that can potentially be less expensive, particularly the process of prediction after training [8–13].…”

Section: Introductionmentioning

confidence: 99%

Predicting peak stresses in microstructured materials using convolutional encoder–decoder learning

Shrivastava

Liu

Dayal

et al. 2022

Mathematics and Mechanics of Solids

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This work presents a machine-learning approach to predict peak-stress clusters in heterogeneous polycrystalline materials. Prior work on using machine learning in the context of mechanics has largely focused on predicting the effective response and overall structure of stress fields. However, their ability to predict peak – which are of critical importance to failure – is unexplored, because the peak-stress clusters occupy a small spatial volume relative to the entire domain, and hence require computationally expensive training. This work develops a deep-learning-based convolutional encoder–decoder method that focuses on predicting peak-stress clusters, specifically on the size and other characteristics of the clusters in the framework of heterogeneous linear elasticity. This method is based on convolutional filters that model local spatial relations between microstructures and stress fields using spatially weighted averaging operations. The model is first trained against linear elastic calculations of stress under applied macroscopic strain in synthetically generated microstructures, which serves as the ground truth. The trained model is then applied to predict the stress field given a (synthetically generated) microstructure and then to detect peak-stress clusters within the predicted stress field. The accuracy of the peak-stress predictions is analyzed using the cosine similarity metric and by comparing the geometric characteristics of the peak-stress clusters against the ground-truth calculations. It is observed that the model is able to learn and predict the geometric details of the peak-stress clusters and, in particular, performed better for higher (normalized) values of the peak stress as compared to lower values of the peak stress. These comparisons showed that the proposed method is well-suited to predict the characteristics of peak-stress clusters.

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Effective response of heterogeneous materials using the recursive projection method

Cited by 5 publications

References 26 publications

Lippmann‐Schwinger solvers for the computational homogenization of materials with pores

Lippmann‐Schwinger solvers for the computational homogenization of materials with pores

A review of nonlinear FFT-based computational homogenization methods

Predicting peak stresses in microstructured materials using convolutional encoder–decoder learning

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