Brittle solids are often toughened by adding a second-phase material. This practice often results in composites with material heterogeneities on the meso scale: large compared to the scale of the process zone but small compared to that of the application. The specific configuration (both geometrical and mechanical) of this mesoscale heterogeneity is generally recognized as important in determining crack propagation and, subsequently, the (effective) toughness of the composite. Here, we systematically investigate how dynamic crack propagation is affected by mesoscale heterogeneities taking the form of an array of inclusions. Using a variational phase-field approach, we compute the apparent crack speed and fracture energy dissipation rate to compare crack propagation under Mode-I loading across different configurations of these inclusions. If fixing the volume fraction of inclusions, matching the inclusion size to the K-dominance zone size gives rise to the best toughening outcome. Conversely, if varying the volume fraction of inclusions, a lower volume fraction configuration can lead to a better toughening outcome if and only if the inclusion size approaches from above the size of the K-dominance zone. Since the size of the K-dominance zone can be estimated \textit{a priori} given an understanding of the application scenario and material availability, we can, in principle, exploit this estimation to design a material's mesoscale heterogeneity that optimally balances the tradeoff between strength and toughness. This paves the way for realizing functional (meta-)materials against crack propagation in extreme environments.
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This paper studies time-dependent electromagnetic scattering from metamaterials that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.
Permutation tests enable testing statistical hypotheses in situations when the distribution of the test statistic is complicated or not available. In some situations, the test statistic under investigation is multivariate, with the multiple testing problem being an important example. The corresponding multivariate permutation tests are then typically based on a suitableone-dimensional transformation of the vector of partial permutation p-values via so called combining functions. This paper proposes a new approach that utilizes the optimal measure transportation concept. The final single p-value is computed from the empirical center-outward distribution function of the permuted multivariate test statistics. This method avoids computation of the partial p-values and it is easy to be implemented. In addition, it allows to compute and interpret contributions of the components of the multivariate test statistic to the non-conformity score and to the rejection of the null hypothesis. Apart from this method, the measure transportation is applied also to the vector of partial p-values as an alternative to the classical combining functions. Both techniques are compared with the standard approaches using various practical examples in a Monte Carlo study. An application on a functional data set is provided as well.
Batch effects are pervasive in biomedical studies. One approach to address the batch effects is repeatedly measuring a subset of samples in each batch. These remeasured samples are used to estimate and correct the batch effects. However, rigorous statistical methods for batch effect correction with remeasured samples are severely under-developed. In this study, we developed a framework for batch effect correction using remeasured samples in highly confounded case-control studies. We provided theoretical analyses of the proposed procedure, evaluated its power characteristics, and provided a power calculation tool to aid in the study design. We found that the number of samples that need to be remeasured depends strongly on the between-batch correlation. When the correlation is high, remeasuring a small subset of samples is possible to rescue most of the power.
Tactile sensing has recently been used in robotics for object identification, grasping, and material recognition. Most material recognition approaches use vibration information from a tactile exploration, typically above one second long, to identify the material. This work proposes a tactile multi-modal (vibration and thermal) material identification approach based on recursive Bayesian estimation. Through the frequency response of the vibration induced by the material and thermal features, like an estimate of the thermal power loss of the finger, we show that it is possible to identify materials in less than half a second. Moreover, a comparison between the use of vibration only and multi-modal identification shows that both recognition time and classification errors are reduced by adding thermal information.
Imaging through perturbed multimode fibres based on deep learning has been widely researched. However, existing methods mainly use target-speckle pairs in different configurations. It is challenging to reconstruct targets without trained networks. In this paper, we propose a physics-assisted, unsupervised, learning-based fibre imaging scheme. The role of the physical prior is to simplify the mapping relationship between the speckle pattern and the target image, thereby reducing the computational complexity. The unsupervised network learns target features according to the optimized direction provided by the physical prior. Therefore, the reconstruction process of the online learning only requires a few speckle patterns and unpaired targets. The proposed scheme also increases the generalization ability of the learning-based method in perturbed multimode fibres. Our scheme has the potential to extend the application of multimode fibre imaging.
Computing equilibrium shapes of crystals (ESC) is a challenging problem in materials science that involves minimizing an orientation-dependent (i.e., anisotropic) surface energy functional subject to a prescribed mass constraint. The highly nonlinear and singular anisotropic terms in the problem make it very challenging from both the analytical and numerical aspects. Especially, when the strength of anisotropy is very strong (i.e., strongly anisotropic cases), the ESC will form some singular, sharp corners even if the surface energy function is smooth. Traditional numerical approaches, such as the $H^{-1}$ gradient flow, are unable to produce true sharp corners due to the necessary addition of a high-order regularization term that penalizes sharp corners and rounds them off. In this paper, we propose a new numerical method based on the Davis-Yin splitting (DYS) optimization algorithm to predict the ESC instead of using gradient flow approaches. We discretize the infinite-dimensional phase-field energy functional in the absence of regularization terms and transform it into a finite-dimensional constraint minimization problem. The resulting optimization problem is solved using the DYS method which automatically guarantees the mass-conservation and bound-preserving properties. We also prove the global convergence of the proposed algorithm. These desired properties are numerically observed. In particular, the proposed method can produce real sharp corners with satisfactory accuracy. Finally, we present numerous numerical results to demonstrate that the ESC can be well simulated under different types of anisotropic surface energies, which also confirms the effectiveness and efficiency of the proposed method.
Stress prediction in porous materials and structures is challenging due to the high computational cost associated with direct numerical simulations. Convolutional Neural Network (CNN) based architectures have recently been proposed as surrogates to approximate and extrapolate the solution of such multiscale simulations. These methodologies are usually limited to 2D problems due to the high computational cost of 3D voxel based CNNs. We propose a novel geometric learning approach based on a Graph Neural Network (GNN) that efficiently deals with three-dimensional problems by performing convolutions over 2D surfaces only. Following our previous developments using pixel-based CNN, we train the GNN to automatically add local fine-scale stress corrections to an inexpensively computed coarse stress prediction in the porous structure of interest. Our method is Bayesian and generates densities of stress fields, from which credible intervals may be extracted. As a second scientific contribution, we propose to improve the extrapolation ability of our network by deploying a strategy of online physics-based corrections. Specifically, we condition the posterior predictions of our probabilistic predictions to satisfy partial equilibrium at the microscale, at the inference stage. This is done using an Ensemble Kalman algorithm, to ensure tractability of the Bayesian conditioning operation. We show that this innovative methodology allows us to alleviate the effect of undesirable biases observed in the outputs of the uncorrected GNN, and improves the accuracy of the predictions in general.
The flexoelectric effect, coupling polarization and strain gradient as well as strain and electric field gradients, is universal to dielectrics, but, as compared to piezoelectricity, it is more difficult to harness as it requires field gradients and it is a small-scale effect. These drawbacks can be overcome by suitably designing metamaterials made of a non-piezoelectric base material but exhibiting apparent piezoelectricity. We develop a theoretical and computational framework to perform topology optimization of the representative volume element of such metamaterials by accurately modeling the governing equations of flexoelectricity using a Cartesian B-spline method, describing geometry with a level set, and resorting to genetic algorithms for optimization. We consider a multi-objective optimization problem where area fraction competes with four fundamental piezoelectric functionalities (stress/strain sensor/ actuator). We computationally obtain Pareto fronts, and discuss the different geometries depending on the apparent piezoelectric coefficient being optimized. In general, we find competitive estimations of apparent piezoelectricity as compared to reference materials such as quartz and PZT ceramics. This opens the possibility to design devices for sensing, actuation and energy harvesting from a much wider, cheaper and effective class of materials.
We study the canonical momentum based discretizations of a hybrid model with kinetic ions and mass-less electrons. Two equivalent formulations of the hybrid model are presented, in which the vector potentials are in different gauges and the distribution functions depend on canonical momentum (not velocity). Particle-in-cell methods are used for the distribution functions, and the vector potentials are discretized by the finite element methods in the framework of finite element exterior calculus. Splitting methods are used for the time discretizations. It is illustrated that the second formulation is numerically superior and the schemes constructed based on the anti-symmetric bracket proposed have better conservation properties, although the filters can be used to improve the schemes of the first formulation.
This paper investigates the multiple testing problem for high-dimensional sparse binary sequences, motivated by the crowdsourcing problem in machine learning. We study the empirical Bayes approach for multiple testing on the high-dimensional Bernoulli model with a conjugate spike and uniform slab prior. We first show that the hard thresholding rule deduced from the posterior distribution is suboptimal. Consequently, the $\ell$-value procedure constructed using this posterior tends to be overly conservative in estimating the false discovery rate (FDR). We then propose two new procedures based on $\adj\ell$-values and $q$-values to correct this issue. Sharp frequentist theoretical results are obtained, demonstrating that both procedures can effectively control the FDR under sparsity. Numerical experiments are conducted to validate our theory in finite samples. To our best knowledge, this work provides the first uniform FDR control result in multiple testing for high-dimensional sparse binary data.
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