Abstract Thermodynamic modeling of the development of non-spherical inclusions as precipitates in alloys is an important topic in computational materials science. The precipitates may have markedly different properties compared to the matrix. Both the elastic contrast and the misfit eigenstrain may yield a remarkable generation of elastic strain energy which immediately influences the kinetics of the developing precipitates. The relevant thermodynamic framework has been mostly based on spherical precipitates. However, the shapes of actual particles are often not spherical. The energetics of such precipitates can be met by adapting the spherical energy terms with shape factors. The well-established Eshelby framework is used to evaluate the elastic strain energy of inclusions with ellipsoidal shapes (described by the axes a , b , and c ) that are subjected to a volumetric transformation strain. The outcome of the study is two shape factors, one for the elastic strain energy and the other for the interface energy. Both quantities are provided in the form of easy-to-use diagrams. Furthermore, threshold elastic contrasts yielding strain energy shape factors with the value 1.0 for any ellipsoidal shape are studied. 

/pdf/strain-and-interface-energy-of-ellipsoidal-inclusions-20kmxubr.pdf

Strain and interface energy of ellipsoidal inclusions subjected to volumetric eigenstrains: shape factors

Traditional equivalent inclusion method provides unreliable predictions of the stress concentrations of two spherical inhomogeneities with small separation distance. This paper determines the stres...

Multiple ellipsoidal/elliptical inhomogeneities embedded in infinite matrix by equivalent inhomogeneous inclusion method:

A novel mathematical formulation is presented for describing growth of phase in solid-to-solid phase transformations and it is applied for describing austenite to ferrite transformation. The formulation includes the effects of transformation eigenstrains, the local strains, as well as partitioning and diffusion. In the current approach the phase front is modelled as diffuse field, and its propagation is shown to be described by the advection equation, which reduces to the level-set equation when the transformation proceeds only to the interface normal direction. The propagation is considered as thermally activated process in the same way as in chemical reaction kinetics. In addition, connection to the Allen-Cahn equation is made. Numerical tests are conducted to check the mathematical model validity and to compare the current approach to sharp interface partitioning and diffusion model. The model operation is tested in isotropic two-dimensional plane strane condition for austenite to ferrite transformation, where the transformation produces isotropic expansion, and also for austenite to bainite transformation, where the transformation causes invariant plane strain condition. Growth into surrounding isotropic austenite, as well as growth of the phase which has nucleated on a grain boundary are tested for both ferrite and bainite formation. 

Full field model describing phase front propagation, transformation strains, chemical partitioning and diffusion in solid-solid phase transformations

/pdf/strain-and-interface-energy-of-ellipsoidal-inclusions-15zqrz3z.pdf

/pdf/full-field-model-describing-phase-front-propagation-2o9q6x6d.pdf

Cylindrical inclusions with constant cross section in an infinite isotropic matrix are usually treated as plane elasticity problems and solved by complex potential method without considering the longitudinal eigenstrains. This paper provides a closed-form solution for the Eshelby’s circular cylindrical inclusion with eigenstrains which are polynomial in transverse direction and uniform in longitudinal direction. The integrals of Green’s function are decomposed into the sum of customized L-integrals. Two sets of L-integrals for the regions inside and outside the circular cylindrical inclusion are evaluated by using the residue theorem. Further, the stress and strain fields inside and outside the inclusion resulted from the polynomial eigenstrains are obtained. Circular cylinder inclusions with uniform, linear, and quadric eigenstrains are, respectively, used as examples to illustrate the proposed solution. When the cylindrical inclusion only suffers transverse eigenstrains, the solution is appropriate for the circular inclusion with polynomial eigenstrains in plane elasticity. The proposed method has convenient formulae and simplifies the integrals of Green’s function with polynomial eigenstrains.

Eshelby’s circular cylindrical inclusion with polynomial eigenstrains in transverse direction by residue theorem

Fourier phase retrieval (FPR) is a challenging task widely used in various applications. It involves recovering an unknown signal from its Fourier phaseless measurements. FPR with few measurements is important for reducing time and hardware costs, but it suffers from serious ill-posedness. Recently, untrained neural networks have offered new approaches by introducing learned priors to alleviate the ill-posedness without requiring any external data. However, they may not be ideal for reconstructing fine details in images and can be computationally expensive. This paper proposes an untrained neural network (NN) embedded algorithm based on the alternating direction method of multipliers (ADMM) framework to solve FPR with few measurements. Specifically, we use a generative network to represent the image to be recovered, which confines the image to the space defined by the network structure. To improve the ability to represent high-frequency information, total variation (TV) regularization is imposed to facilitate the recovery of local structures in the image. Furthermore, to reduce the computational cost mainly caused by the parameter updates of the untrained NN, we develop an accelerated algorithm that adaptively trades off between explicit and implicit regularization. Experimental results indicate that the proposed algorithm outperforms existing untrained NN-based algorithms with fewer computational resources and even performs competitively against trained NN-based algorithms.

pdf/untrained-neural-network-embedded-fourier-phase-retrieval-3bxfrsuw.pdf

Untrained neural network embedded Fourier phase retrieval from few measurements

Fourier phase retrieval (FPR) is to recover a signal from its Fourier magnitude measurement. Due to the ill-posedness of the problem, it is often necessary to introduce prior information. Recently, replacing hand-crafted priors with data-learned priors, such as deep generative priors, has received a lot of attention in inverse problems. Note that the reconstruction performance of trained generative priors relies on a large amount of training data, in this paper we solve FPR problem with an untrained generative network, which approximates the unknown signal only with a fixed seed. We propose an algorithm that combines the alternating direction method of multipliers (ADMM) with an untrained generative network. Specifically, we model the problem as a constrained optimization problem, the ADMM is adopted to solve it alternately, and then an untrained generative network is embedded into the iterative process to constrain the estimated signal. The effectiveness of the proposed algorithm is demonstrated through experiments on grayscale images. Both PSNR, SSIM, and the visual quality of the reconstructed images are superior to that of the state-of-the-art algorithms, especially when the measurement is incomplete.

Fourier phase retrieval with untrained generative prior

Fourier phase retrieval(PR) is a severely ill-posed inverse problem that arises in various applications. To guarantee a unique solution and relieve the dependence on the initialization, background information can be exploited as a structural priors. However, the requirement for the background information may be challenging when moving to the high-resolution imaging. At the same time, the previously proposed projected gradient descent(PGD) method also demands much background information. In this paper, we present an improved theoretical result about the demand for the background information, along with two Douglas Rachford(DR) based methods. Analytically, we demonstrate that the background required to ensure a unique solution can be decreased by nearly $1/2$ for the 2-D signals compared to the 1-D signals. By generalizing the results into $d$-dimension, we show that the length of the background information more than $(2^{\frac{d+1}{d}}-1)$ folds of the signal is sufficient to ensure the uniqueness. At the same time, we also analyze the stability and robustness of the model when measurements and background information are corrupted by the noise. Furthermore, two methods called Background Douglas-Rachford (BDR) and Convex Background Douglas-Rachford (CBDR) are proposed. BDR which is a kind of non-convex method is proven to have the local R-linear convergence rate under mild assumptions. Instead, CBDR method uses the techniques of convexification and can be proven to own a global convergence guarantee as long as the background information is sufficient. To support this, a new property called F-RIP is established. We test the performance of the proposed methods through simulations as well as real experimental measurements, and demonstrate that they achieve a higher recovery rate with less background information compared to the PGD method. 

Phase Retrieval with Background Information: Decreased References and Efficient Methods

Fourier phase retrieval (FPR) is an inverse problem that recovers the signal from its Fourier magnitude measurement, it’s ill-posed especially when the sampling rates are low. In this paper, an untrained generative prior is introduced to attack the ill-posedness. Based on the alternating direction method of multipliers (ADMM), an algorithm utilizing the untrained generative network called Net-ADM is proposed to solve the FPR problem. Firstly, the objective function is smoothed and the dimension of the variable is raised to facilitate calculation. Then an untrained generative network is embedded in the iterative process of ADMM to project an estimated signal into the generative space, and the projected signal is applied to next iteration of ADMM. We theoretically analyzed the two projections included in the algorithm, one makes the objective function descent, and the other gets the estimation closer to the optimal solution. Numerical experiments show that the reconstruction performance and robustness of the proposed algorithm are superior to prior works, especially when the sampling rates are low.

pdf/admm-based-fourier-phase-retrieval-with-untrained-generative-7gy60jz9.pdf

ADMM based Fourier phase retrieval with untrained generative prior

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