Tensor Transformation Research Articles

Localizing gradient damage model for anisotropic materials: Focusing on timber

This paper presents a new damage model for transversely isotropic materials, with special focus on timber. The model proposed in this study is formulated by employing the localized gradient form of continuum damage theory. The model accounts for anisotropic damage through two damage variables, delineating damage parallel and perpendicular to the grain, representing isotropic damage within the cross-sectional plane. The conditions assumed in this work pertain to loading scenarios that occur within the plane. An additional fourth-order tensor transformation is incorporated to facilitate the conversion between the global and local reference frames, allowing for arbitrary spatial orientations of the material matrix. In this work we concentrate on damage arising from tensile and shear stresses. The model is implemented as a consistent linearized coupled (monolithic) problem.The performance verification of the model relies on numerous numerical examples, incorporating experimental benchmark problems. The outcomes demonstrate that the suggested damage model adeptly captures both the initiation of failure and the propagation of damage.

STile: Searching Hybrid Sparse Formats for Sparse Deep Learning Operators Automatically

Sparse operators, i.e., operators that take sparse tensors as input, are of great importance in deep learning models. Due to the diverse sparsity patterns in different sparse tensors, it is challenging to optimize sparse operators by seeking an optimal sparse format, i.e., leading to the lowest operator latency. Existing works propose to decompose a sparse tensor into several parts and search for a hybrid of sparse formats to handle diverse sparse patterns. However, they often make a trade-off between search space and search time: their search spaces are limited in some cases, resulting in limited operator running efficiency they can achieve. In this paper, we try to extend the search space in its breadth (by doing flexible sparse tensor transformations) and depth (by enabling multi-level decomposition). We formally define the multi-level sparse format decomposition problem, which is NP-hard, and we propose a framework STile for it. To search efficiently, a greedy algorithm is used, which is guided by a cost model about the latency of computing a sub-task of the original operator after decomposing the sparse tensor. Experiments of two common kinds of sparse operators, SpMM and SDDMM, are conducted on various sparsity patterns, and we achieve 2.1-18.0× speedup against cuSPARSE on SpMMs and 1.5 - 6.9× speedup against DGL on SDDMM. The search time is less than one hour for any tested sparse operator, which can be amortized.

On simplified methods for determining the elastic constants of layered polymer composites

The paper considers the standard layer-by-layer method for calculating composite structures. The main problems of the method are highlighted and the approaches which allow to simplify the calculation are analyzed. It is shown that the transition to trace can be used to restore the elastic constants of a monolayer on the basis of only one experiment. Simplified tensor transformations for rotating the coordinate system are considered. The conclusion about the limits of applicability of the simplified relations for different materials is made.

An algorithm for calculating spectral radius of $ s $-index weakly positive tensors

<abstract><p>In this paper, we introduced $ s $-index weakly positive tensors and discussed the calculation of the spectral radius of this kind of nonnegative tensors. Using the diagonal similarity transformation of tensor and Perron-Frobenius theory of nonnegative tensor, the calculation method of the maximum $ H $-eigenvalue of $ s $-index weakly positive tensors was given. A variable parameter was introduced in each iteration of the algorithm, which is equivalent to a translation transformation of the tensor in each iteration to improve the calculation speed. At the same time, it was proved that the algorithm is linearly convergent for the calculation of the spectral radius of $ s $-index weakly positive tensors. The final numerical example shows the effectiveness of the algorithm.</p></abstract>

Non-Invasive Imaging of Mechanical Properties of Cancers In Vivo Based on Transformations of the Eshelby's Tensor Using Compression Elastography.

Knowledge of the mechanical properties is of great clinical significance for diagnosis, prognosis and treatment of cancers. Recently, a new method based on Eshelby's theory to simultaneously assess Young's modulus (YM) and Poisson's ratio (PR) in tissues has been proposed. A significant limitation of this method is that accuracy of the reconstructed YM and PR is affected by the orientation/alignment of the tumor with the applied stress. In this paper, we propose a new method to reconstruct YM and PR in cancers that is invariant to the 3D orientation of the tumor with respect to the axis of applied stress. The novelty of the proposed method resides on the use of a tensor transformation to improve the robustness of Eshelby's theory and reconstruct YM and PR of tumors with high accuracy and in realistic experimental conditions. The method is validated using finite element simulations and controlled experiments using phantoms with known mechanical properties. The in vivo feasibility of the developed method is demonstrated in an orthotopic mouse model of breast cancer. Our results show that the proposed technique can estimate the YM and PR with overall accuracy of (97.06 ± 2.42) % under all tested tumor orientations. Animal experimental data demonstrate the potential of the proposed methodology in vivo. The proposed method can significantly expand the range of applicability of the Eshelby's theory to tumors and provide new means to accurately image and quantify mechanical parameters of cancers in clinical conditions.

A Multidimensional Tensor Low Rank Method for Magnetic Resonance Image Denoising.

In this paper, we present the Magnetic Resonance Image (MRI) denoising method via nonlocal multidimensional low rank tensor transformation constraint (NLRT). We first design a nonlocal MRI denoising method by non-local low rank tensor recovery framework. Furthermore, a multidimensional low rank tensor constraint is used to obtain low-rank prior information combined with 3-dimensional structure feature of MRI image cubes. Our NLRT can achieve denoising by retaining more image detail information. The optimization and updating process of the model is solved via the alternating direction method of multipliers (ADMM) algorithm. Several state-of-the-art denoising methods are selected for comparative experiments. In order to reflect the performance of the denoising method, Rician noise with different levels is added to the experiment to analyze the results. The experimental results prove that our NLTR has more outstanding denoising ability and can obtain better MRI images.

Boundary dependent physics-informed neural network for solving neutron transport equation

The boundary dependent physics-informed neural network (BDPINN) is proposed for solving the neutron transport equation (NTE). Developed based on physics-informed neural network (PINN), BDPINN eliminates errors caused by boundary conditions, effectively avoids the curse of dimensionality and reduces ray effects caused by solid angle discretization. Original constructions of trial functions are proposed for common boundary conditions in NTE to improve calculation accuracy. Three techniques are introduced to improve BDPINN: third-order tensor transformation, rearranging the training set, and result reconstruction in high order. These techniques reduce computational cost and improve accuracy. In addition, two existing techniques, area decomposition and multi-group iteration are applied to solve multi-group and complex area problems. The accuracy of BDPINN is verified through benchmark comparisons with a nodal method and the mechanism for reducing the ray effect is analyzed. This work provides novel perspectives and techniques for solving NTE.

CAD-integrated stiffener sizing-topology design via force flow members (FFM)

Stiffener design plays a crucial role in the lightweight design of stiffened panels. We propose a CAD-integrated stiffener design approach for shells. The problem is partitioned into two steps: (1) generating clear force flow members (FFM) based on the direction of principal stress, and (2) introducing a nodal variable-based sizing-topology optimization for further lightweight design enhancement. Specifically, we generate dense principal stress trajectory families using the iterative tracing technique, and extract FFM through hierarchical clustering. Generating high-precision principal stress trajectories on free-form surfaces is a challenging task. Our method differs from previous approaches in that we employ tensor transformations to convert principal stress directions from the 3D physical space into the 2D parameter space. We then create NURBS curve-based FFM in the 2D parameter space. The NURBS-based free-form deformations (NFFD) technique is utilized to map FFM from the 2D parameter space to the 3D physical space, ensuring FFMs are precise and watertight constructions of geometries. We propose a nodal variable-based sizing-topology method for the lightweight design of FFM. The heights of FFM control points are designated as design variables. Steadily eliminating pseudo-heights is achieved by using the Sigmoid function to penalize the design variables. Moreover, we introduce an NFFD-based isogeometric analysis method for stiffened panels modeled by NFFD-based FFM to realize the integration of design and analysis. The effectiveness and robustness of the presented method are demonstrated through various benchmarks and examples with industrial backgrounds. The proposed optimization framework exhibits strong application potential in the lightweight design of stiffened panels.

Robust interpolation for dispersed gas‐droplet flows using statistical learning with the fully Lagrangian approach

SummaryA novel methodology is presented for reconstructing the Eulerian number density field of dispersed gas‐droplet flows modelled using the fully Lagrangian approach (FLA). In this work, the nonparametric framework of kernel regression is used to accumulate the FLA number density contributions of individual droplets in accordance with the spatial structure of the dispersed phase. The high variation which is observed in the droplet number density field for unsteady flows is accounted for by using the Eulerian‐Lagrangian transformation tensor, which is central to the FLA, to specify the size and shape of the kernel associated with each droplet. This procedure enables a high level of structural detail to be retained, and it is demonstrated that far fewer droplets have to be tracked in order to reconstruct a faithful Eulerian representation of the dispersed phase. Furthermore, the kernel regression procedure is easily extended to higher dimensions, and inclusion of the droplet radius within the phase space description using the generalised fully Lagrangian approach (gFLA) additionally enables statistics of the droplet size distribution to be determined for polydisperse flows. The developed methodology is applied to a range of one‐dimensional and two‐dimensional steady‐state and transient flows, for both monodisperse and polydisperse droplets, and it is shown that kernel regression performs well across this variety of cases. A comparison is made against conventional direct trajectory methods to determine the saving in computational expense which can be gained, and it is found that times fewer droplet realisations are needed to reconstruct a qualitatively similar representation of the number density field.

Determination of fiber orientation model parameters for injection molding simulations via automated metamodel optimization

Injection molded short fiber reinforced components reveal a sound light weight potential with moderate costs and thus are widely used in many demanding engineering applications. The accurate determination of the fiber orientation (FO) is essential for predicting the overall mechanical behavior of discontinuous (short or long, with varying aspect ratio) fiber reinforced composites. The simulation of the FO requires a proper modeling of the hydrodynamics, the closure transformation of the FO tensor and optionally the application of specific correction functions. The determination of parameters for the fiber orientation models commonly used in injection molding simulations is a challenging task because they cannot be determined directly in experiments. Hence, a novel way is shown in our paper to derive these parameters faster, more efficiently and accurately by the usage of an automated metamodel optimization. For this, injection molding simulations were performed iteratively by an optimization program until a minimal deviation error of the simulated parameters was reached. The optimization was performed based on proper computed tomography FO data of selected regions of interest. The new approach was tested for a rotationally symmetric Venturi tube geometry made from short glass fiber reinforced polyamide (PA-GF). The fiber orientation distribution models chosen were the iARD-RPR equation with 3 parameters and the novel anisotropic IISO equation with 5 parameters. It was shown that the optimization method is feasible for the calibration of fiber orientation models. Furthermore, the IISO equation with its 2 additional parameters allowed a more accurate prediction of the fiber orientation distribution, especially of the core layer of the injection molded part.

Lorentz transformation of three dimensional gravitational wave tensor

Recently there has been more and more interest in the gravitational wave (GW) of moving sources. This paper introduces a Lorentz transformation problem of GWs. Although the Bondi-Metzner-Sachs (BMS) theory has in principle already included the Lorentz transformation of GWs, the transformation of the three-dimensional GW tensor has not been explicitly calculated before. Within four-dimensional spacetime, GWs have the properties of ‘boost weight zero’ and ‘spin weight 2’. This fact makes the Lorentz transformation of GWs difficult to understand. In the current paper, we adopt the traditional three-dimensional tensor description of a GW. Such a transverse-traceless tensor describes the GW freedom directly. We derive the explicit Lorentz transformation of the GW tensor. The transformation is similar to the Lorentz transformation for an electric field vector and a magnetic field vector which are three-dimensional vectors. Based on the deduced Lorentz transformation of the GW three-dimensional tensor, we can construct the gravitational waveform of a moving source with high speed if only the waveform of the corresponding rest waveform is given.

Study of Local Fatigue Methods (TCD, N-SIF, and ESED) on Notches and Defects Related to Numerical Efficiency

The fatigue strength of structural components is strongly affected by notches and imperfections. Both can be treated similarly, as local notch fatigue strength methods can also be applied to interior defects. Even though Murakami’s √area approach is commonly used in the threshold-based fatigue design of single imperfections, advanced concepts such as the Theory of Critical Distances (TCD), Notch Stress Intensity Factors (N-SIF), or Elastic Strain Energy Density (ESED) methods provide additional insight into the local fatigue strength distribution of irregularly shaped defects under varying uniaxial load vectors. The latter methods are based on the evaluation of the elastic stress field in the vicinity of the notch for each single load vector. Thus, this work provides numerically efficient methods to assess the local fatigue strength by means of TCD, N-SIF, and ESED, targeting the minimization of the required load case count, optimization of stress field evaluation data points, and utilization of multi-processing. Furthermore, the Peak Stress Method (PSM) is adapted for large opening angles, as in the case of globular defects. In detail, two numerical strategies are devised and comprehensively evaluated, either using a sub-case-based stress evaluation of the defect vicinity with an unchanged mesh pattern and varying load vector on the exterior model region with optimized load angle stepping or by the invocation of stress and strain tensor transformation equations to derive load angle-dependent result superposition while leaving the initial mesh unaltered. Both methods provide numerically efficient fatigue post-processing, as the mesh in the evaluated defect region is retained for varying load vectors. The key functions of the fatigue strength assessment, such as the evaluation of appropriate planar notch radius and determination of notch opening angle for the discretized imperfections, are presented. Although the presented numerical methods apply to planar simulation studies, the basic methodology can be easily expanded toward spatial fatigue assessment.

T-Linear Tensor Subspace Learning for Robust Feature Extraction of Hyperspectral Images

Subspace learning has been widely applied for feature extraction of hyperspectral images (HSIs) and achieved great success. However, the current methods still leave two problems that need to be further investigated. First, those methods mainly focus on finding one or multiple projection matrices for mapping the high-dimensional data into a low-dimensional subspace, which can only capture the information from each direction of high-order hyperspectral data separately. Second, the performance of feature extraction is barely satisfactory when the hyperspectral data is severely corrupted by noise. To address these issues, this article presents a t-linear tensor subspace learning (tLTSL) model for robust feature extraction of HSIs based on t-product projection. In the model, t-product projection is a new defined tensor transformation way similar to linear transformation in vector space, which can maximally capture the intrinsic structure of tensor data. The integrated tensor low-rank and sparse decomposition can effectively remove the noise corruption and the learned t-product projection can directly transform the high-order hyperspectral data into a subspace with information from all modes comprehensively considered. Moreover, a proposition related to tensor rank is proofed for interpreting the meaning of the tLTSL model. Extensive experiments are conducted on two different kinds of noise (i.e., simulated and real noise) corrupted HSI data, which validate the effectiveness of tLTSL.

Significantly anisotropic Raman response of the (110) surface in quasi-one-dimensional system (TaSe4)2I by polarized Raman spectroscopy.

Quasi-one-dimensional materials are usually characterized by optical response spectroscopy methods, which show significant polarization dependence. Herein, we report a systematical investigation of polarized Raman scattering on the (110) crystal surface of the layered (TaSe4)2I compound. Taking into account group theory analysis of the crystal structure and the Raman tensor transformation technique, the vibrational mode of the Raman peaks can be differentiated by the angular dependence of the Raman peak intensity in parallel and vertical polarization Raman scattering tests. Moreover, density functional perturbation theory (DFPT) calculation confirmed the form of the Raman tensor of the (110) crystal surface, which was consistent with the result of the Raman tensor transformation technique, and the Raman spectrum and phonon dispersion curve calculations were also performed based on the Vienna ab initio simulation package (VASP). This new method provides useful insight for accurately identifying the lattice vibration behavior in new 2D layered structures.

A Full Tensor Decomposition Network for Crop Classification with Polarization Extension

The multisource data fusion technique has been proven to perform better in crop classification. However, traditional fusion methods simply stack the original source data and their corresponding features, which can be only regarded as a superficial fusion method rather than deep fusion. This paper proposes a pixel-level fusion method for multispectral data and dual polarimetric synthetic aperture radar (PolSAR) data based on the polarization extension, which yields synthetic quad PolSAR data. Then we can generate high-dimensional features by means of various polarization decomposition schemes. High-dimensional features usually cause the curse of the dimensionality problem. To overcome this drawback in crop classification using the end-to-end network, we propose a simple network, namely the full tensor decomposition network (FTDN), where the feature extraction in the hidden layer is accomplished by tensor transformation. The number of parameters of the FTDN is considerably fewer than that of traditional neural networks. Moreover, the FTDN admits higher classification accuracy by making full use of structural information of PolSAR data. The experimental results demonstrate the effectiveness of the fusion method and the FTDN model.

A Method for Local Contrast Enhancement of Endoscopic Images Based on Color Tensor Transformation into a Matrix of Color Vectors’ Modules Using a Sliding Window

A new method aimed at endoscopic color images’ local contrast enhancement is proposed, based on local sliding histogram equalization with adaptive threshold limitation, color distortions correction, and image brightness preservation. For this, the original RGB image, represented as a tensor of size M × N × 3, is transformed into a matrix of size M × N, composed by the color vectors’ modules. As a result of local contrast enhancement, the obtained color vectors are symmetrical in respect of the input ones, because they satisfy the requirement for invariance after rotation. To enhance the local contrast, recursive local histogram equalization with adaptive thresholding is applied to each matrix element. This threshold divides the histogram into two regions of equal areas. A new metric for local contrast enhancement evaluation based on the mean square difference entropy is proposed. The presented new method is characterized by low computational complexity, due to the lack of direct and inverse color conversion and the possibility for adaptive local contrast enhancement, which is essential for accurate medical diagnosis based on endoscopic images analysis. In addition, the presented method performs both the correction of color distortions and the brightness preservation of each pixel.

Anisotropic behavior in the lithiation of a silicon nanopillar

There are reports on experimental observation of anisotropic expansion/swelling in the lithiation of crystal silicon (Si) nanopillars/nanowires. Such anisotropic expansion/swelling can cause the cracking of Si electrodes, leading to the capacity fading in Si-based lithium-ion batteries (LIBs). In this paper, we propose an anisotropic chemo-mechanical model that takes into account anisotropic characteristics of the physical properties of crystalline Si, including stiffness tensor (matrix), diffusion tensor and partial-molar volume tensor. Using direction-cosine matrix for the tensor transformation, we determine the orientation-dependent elastic constants, diffusion coefficients and partial molar volumes in the anisotropic chemo-mechanical model instead of using phenomenal parameters associated with experimental results. For the purpose of demonstration, we analyze the lithiated Si phase of Li15Si4, which exhibits cubic structure, and illustrate geometrical profiles of the lithiated Si nanopillars of [110] and [111] in longitudinal direction at Li15Si4 state. The numerical results reveal that the cross section of the [111]-oriented Si nanopillar remain circular and the cross section of the [110]-oriented Si nanopillar evolves to elliptical shape. Such results are in good accordance with the experimental observation reported by Lee et al. [P. Natl. Acad. Sci. 109(11) (2012) 4080–4085]. The numerical results also show that the anisotropic expansion likely reduces von-Mises stress in the [110]-oriented Si nanopillar, retarding mechanical degradation of the Si-based electrodes. This work provides a new insight into the anisotropic expansion and stress generation in crystalline electrodes.

PH-Net: Parallelepiped microstructure homogenization via 3D Convolutional Neural Networks

Microstructures are attracting academic and industrial interest because of the rapid development of additive manufacturing. The numerical homogenization method has been well studied for analyzing mechanical behaviors of microstructures; however, it is too time-consuming to be applied to online computing or applications requiring high-frequency calling, e.g., topology optimization. Data-driven homogenization methods are considered a more efficient choice but the microstructures are limited to cubic shapes, therefore are unsuitable for periodic microstructures with a more general shape, e.g., parallelepipeds. This paper introduces a fine-designed 3D convolutional neural network (CNN) for fast homogenization of parallelepiped microstructures, named PH-Net. Superior to existing data-driven methods, PH-Net predicts the local displacements of microstructures under specified macroscopic strains instead of direct homogeneous material, empowering us to present a label-free loss function based on minimal potential energy. For dataset construction, we introduce a shape-material transformation and voxel-material tensor to encode microstructure type, base material and boundary shape together as the input of PH-Net, such that it is CNN-friendly and enhances PH-Net on generalization in terms of microstructure type, base material, and boundary shape. PH-Net predicts homogenized properties hundreds of times faster than numerical homogenization and even supports online computing. Moreover, it does not require a labeled dataset and thus the training process is much faster than current deep learning methods. Because it can predict local displacement, PH-Net provides both homogeneous material properties and microscopic mechanical properties, e.g., strain and stress distribution, and yield strength. We also designed a set of physical experiments using 3D printed materials to verify the prediction accuracy of PH-Net.

Number-state preserving tensor networks as classifiers for supervised learning

We propose a restricted class of tensor network state, built from number-state preserving tensors, for supervised learning tasks. This class of tensor network is argued to be a natural choice for classifiers as 1) they map classical data to classical data, and thus preserve the interpretability of data under tensor transformations, 2) they can be efficiently trained to maximize their scalar product against classical data sets, and 3) they seem to be as powerful as generic (unrestricted) tensor networks in this task. Our proposal is demonstrated using a variety of benchmark classification problems, where number-state preserving versions of commonly used networks (including MPS, TTN and MERA) are trained as effective classifiers. This work opens the path for powerful tensor network methods such as MERA, which were previously computationally intractable as classifiers, to be employed for difficult tasks such as image recognition.

Iterative tensor eigen rank minimization for low-rank tensor completion

By minimizing the tensor ranks, recent methods exploit the tensor spatial correlation between the tensor entries for the low-rank tensor completion (TC) problem. However, these methods suffer from the difficulty in sufficiently exploiting the tensor spatial correlation, which reduces the TC performance. To tackle such difficulty, this paper defines a novel tensor rank called the tensor eigen rank for the low-rank TC problem. Specifically, the novel tensor rank is obtained from a tensor transformation, which directly transforms an arbitrary-order tensor into a tensor pile. By minimizing the novel tensor rank, we subsequently establish a low-rank TC model. Within the framework of the iterative shrinkage and thresholding scheme, an algorithm named iterative tensor eigen rank minimization (IterMin) is further proposed to address the established model. To verify the effectiveness of the proposed method, we investigate the TC observing probability, model recovery error, algorithm convergence, and algorithm complexity. The experiments on the synthetic and real-world datasets show that the proposed method outperforms several state-of-the-art methods in terms of the quantitative and visual results.