Relevant bibliographies by topics / Fourth-order Tensors

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Academic literature on the topic 'Fourth-order Tensors'

Author: Grafiati
Published: 6 September 2023
Last updated: 7 September 2023

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Journal articles on the topic "Fourth-order Tensors"

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Wang, Gang, Linxuan Sun, and Lixia Liu. "M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors." Complexity 2020 (January 8, 2020): 1–8. http://dx.doi.org/10.1155/2020/2474278.

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M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.
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Betten, Josef. "Irreducible invariants of fourth-order tensors." Mathematical Modelling 8 (1987): 29–33. http://dx.doi.org/10.1016/0270-0255(87)90535-5.

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Chen, Zhongming, Yannan Chen, Liqun Qi, and Wennan Zou. "Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor." Mathematics and Mechanics of Solids 24, no. 10 (March 8, 2019): 3092–102. http://dx.doi.org/10.1177/1081286519835246.

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The elasticity tensor is one of the most important fourth-order tensors in mechanics. Fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith–Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.
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He, Jun, Yanmin Liu, Junkang Tian, and Zhuanzhou Zhang. "New Sufficient Condition for the Positive Definiteness of Fourth Order Tensors." Mathematics 6, no. 12 (December 5, 2018): 303. http://dx.doi.org/10.3390/math6120303.

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In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.
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Desmorat, Boris, and Rodrigue Desmorat. "Tensorial Polar Decomposition of 2D fourth-order tensors." Comptes Rendus Mécanique 343, no. 9 (September 2015): 471–75. http://dx.doi.org/10.1016/j.crme.2015.07.002.

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Zhao, Jianxing. "E-eigenvalue Localization Sets for Fourth-Order Tensors." Bulletin of the Malaysian Mathematical Sciences Society 43, no. 2 (April 27, 2019): 1685–707. http://dx.doi.org/10.1007/s40840-019-00768-y.

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Ishida, Akio, Takumi Noda, Jun Murakami, Naoki Yamamoto, and Chiharu Okuma. "Calculation of Fourth-Order Tensor Product Expansion by Power Method and Comparison of it with Higher-Order Singular Value Decomposition." Applied Mechanics and Materials 444-445 (October 2013): 703–11. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.703.

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Higher-order singular value decomposition (HOSVD) is known as an effective technique to reduce the dimension of multidimensional data. We have proposed a method to perform third-order tensor product expansion (3OTPE) by using the power method for the same purpose as HOSVD, and showed that our method had a better accuracy property than HOSVD, and furthermore, required fewer computation time than that. Since our method could not be applied to the tensors of fourth-order (or more) in spite of having those useful properties, we extend our algorithm of 3OTPE calculation to forth-order tensors in this paper. The results of newly developed method are compared to those obtained by HOSVD. We show that the results follow the same trend as the case of 3OTPE.
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Papenfuss, Christina. "Maximum Entropy Closure Relation for Higher Order Alignment and Orientation Tensors Compared to Quadratic and Hybrid Closure." Journal of Modeling and Simulation of Materials 5, no. 1 (December 31, 2022): 39–52. http://dx.doi.org/10.21467/jmsm.5.1.39-52.

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A closure relation expresses the fourth order orientation tensor as a function of the second order one. Two well-known closure relations, the hybrid closure and the maximum entropy closure, are compared in the case of a rotation symmetric orientation distribution function. The maximum entropy closure predicts a positive fourth order parameter in the whole range of the second order parameter, whereas the hybrid closure results in negative fourth order parameters for small values of the second order one. For the maximum entropy closure quadratic fit polynomials are presented. For a general distribution without rotation symmetry, the expression for the entropy is exploited to derive an explicit form for the maximum entropy distribution. Lowest order approximation of this distribution function leads to simple closure forms for the fourth order alignment tensor and also for higher order alignment tensors.
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Xiao, B., and J. Feng. "Higher order elastic tensors of crystal structure under non-linear deformation." Journal of Micromechanics and Molecular Physics 04, no. 04 (December 2019): 1950007. http://dx.doi.org/10.1142/s2424913019500073.

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The higher-order elastic tensors can be used to characterize the linear and non-linear mechanical properties of crystals at ultra-high pressures. Besides the widely studied second-order elastic constants, the third- and fourth-order elastic constants are sixth and eighth tensors, respectively. The independent tensor components of them are completely determined by the symmetry of the crystal. Using the relations between elastic constants and sound velocity in solid, the independent elastic constants can be measured experimentally. The anisotropy in elasticity of crystal structures is directly determined by the independent elastic constants.
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Desmorat, R., N. Auffray, B. Desmorat, B. Kolev, and M. Olive. "Generic separating sets for three-dimensional elasticity tensors." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2226 (June 2019): 20190056. http://dx.doi.org/10.1098/rspa.2019.0056.

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We define a generic separating set of invariant functions (a.k.a. a weak functional basis ) for tensors. We then produce two generic separating sets of polynomial invariants for three-dimensional elasticity tensors, one consisting of 19 polynomials and one consisting of 21 polynomials (but easier to compute), and a generic separating set of 18 rational invariants. As a by-product, a new integrity basis for the fourth-order harmonic tensor is provided.
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Dissertations / Theses on the topic "Fourth-order Tensors"

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Agrawal, Manish. "Efficient Simulation Strategies for Electromechanical Systems, Contact Mechanics and Time Finite Elements, within the Framework of Hybrid Finite Elements." Thesis, 2017. http://etd.iisc.ac.in/handle/2005/4228.

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The purpose of this thesis is to provide robust numerical schemes for various challenging nonlinear large-deformation elasticity problems. The presence of geometric and material nonlinearities poses interesting challenges in the numerical implementation of time finite element methods, multiphysics and contact problems, and it is the goal of this thesis to address these challenges. It is well known that displacement based elements show overstiff behavior known as ‘locking’ for a large class of practical problems with thin structures, structures with high curvature, and almost incompressible material. To alleviate the locking problem associated with displacement based elements, shell elements based on mixed formulations and hybrid elements have been proposed in the literature. However, due to the kinematic assumptions involved in the development of shell elements, the applicability of these elements is limited. Hybrid elements, based on two-field variational principle are hexahedral elements, with no kinematic assumptions involved in their development. Further, hybrid elements have been shown to provide excellent coarse mesh accuracy for a large class of problems. In this thesis, we further develop the hybrid finite element method for the following new class of problems: Developing a quadratic time finite element method for elastodynamics systems For chaotic systems, the quadratic transient time finite element strategy is significantly more efficient and robust as compared to the linear transient scheme. Motivated by this, in the first part of the thesis, we propose a quadratic time finite element strategy for large-deformation elastodynamics systems. The proposed method is a modified time finite element strategy that conserves linear and angular momenta exactly, and energy in an approximate sense, in the fully discrete setting. Further, to obtain good coarse mesh accuracy, the proposed transient strategy is extended to the hybrid element framework. Analysis of electromechanical systems Due to the strong coupling between electromagnetic and displacement fields, there is a need to develop a robust fully-coupled scheme for modeling electromechanical phenomenon. To achieve this, we present a fully-coupled monolithic numerical scheme for modeling electromechanical systems. Further, in view of the superiority of hybrid elements, we extend our monolithic formulation to the hybrid finite element framework. We perform a consistent linearization of the eddy current and structural equations in the reference configuration to ensure a quadratic rate of convergence. Contact analysis using mortar methods In the final part of the thesis, we attempt the classical problem of contact between two bodies under the large-deformation elasticity framework. We use mortar elements to satisfy the geometric constraint of non-penetrability, and provide a detailed expla-nation for using two 2-node linear elements on the edge of a 9-node two-dimensional quadrilateral element for the interpolation of the Lagrangian multiplier. We further propose a new projection technique for enhancing the accuracy of the predicted con-tact pressure. We then extend the above displacement-based formulation to the hybrid framework so as to be able to model the contact of shell-type as well as ‘chunky’ geometries efficiently. 2
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Books on the topic "Fourth-order Tensors"

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Bernard, Seth. A Cost Analysis of the Republican Circuit Walls. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780190878788.003.0004.

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This chapter analyzes the social and economic effects of the single largest construction project of pre-Imperial Rome, the circuit of walls built in the first half of the fourth century BCE. I employ an “energetics” approach (Abrams 1994), quantifying the labor-cost of the walls’ construction by means of comparative data and close study of the monument’s remains. The resulting model provides a better idea not only of the walls’ total cost, but of the workforce’s composition, the balance of skilled and unskilled labor, and the schedule of construction. The costs are then input into a general model of the Roman economy in order to assess the broader impact of the walls’ costs. Archaeological and textual evidence indicate the walls were built with compulsory labor. This further underscores the likelihood that the project widened economic inequality and worsened existing socioeconomic tensions.
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Book chapters on the topic "Fourth-order Tensors"

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Itskov, Mikhail. "Fourth-Order Tensors." In Tensor Algebra and Tensor Analysis for Engineers, 123–35. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-98806-1_5.

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Itskov, Mikhail. "Fourth-Order Tensors." In Tensor Algebra and Tensor Analysis for Engineers, 121–33. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-16342-0_5.

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Itskov, Mikhail. "Fourth-Order Tensors." In Tensor Algebra and Tensor Analysis for Engineers, 107–19. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-30879-6_5.

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Itskov, Mikhail. "Fourth-Order Tensors." In Tensor Algebra and Tensor Analysis for Engineers, 103–14. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-93907-8_5.

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Betten, J. "Invariants of Fourth-Order Tensors." In Applications of Tensor Functions in Solid Mechanics, 203–26. Vienna: Springer Vienna, 1987. http://dx.doi.org/10.1007/978-3-7091-2810-7_11.

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Qi, Liqun, Haibin Chen, and Yannan Chen. "Fourth Order Tensors in Physics and Mechanics." In Advances in Mechanics and Mathematics, 249–84. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-10-8058-6_8.

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Ghosh, Aurobrata, and Rachid Deriche. "Fourth Order Symmetric Tensors and Positive ADC Modeling." In Mathematics and Visualization, 163–88. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-54301-2_7.

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Herberthson, Magnus, Evren Özarslan, and Carl-Fredrik Westin. "Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions." In Mathematics and Visualization, 3–22. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56215-1_1.

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AbstractCalculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor $$R_{abcd}$$ R abcd . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor $$R_{abcd}$$ R abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors $$R_{abcd}$$ R abcd and $$\widetilde{R}_{abcd}$$ R ~ abcd . In terms of components, such an equivalence means that components $$R_{ijkl}$$ R ijkl of the first tensor will transform into the components $$\widetilde{R}_{ijkl}$$ R ~ ijkl of the second tensor for some change of the coordinate system.
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Chemin, Alexandre, François Henrotte, Jean-François Remacle, and Jean Van Schaftingen. "Representing Three-Dimensional Cross Fields Using Fourth Order Tensors." In Lecture Notes in Computational Science and Engineering, 89–108. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13992-6_6.

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Moakher, Maher. "The Algebra of Fourth-Order Tensors with Application to Diffusion MRI." In Mathematics and Visualization, 57–80. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-88378-4_4.

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Conference papers on the topic "Fourth-order Tensors"

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Jack, David A., and Douglas E. Smith. "Assessing the Use of Tensor Closure Methods With Orientation Distribution Reconstruction Functions." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-42828.

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Orientation tensors are widely used to describe fiber distri-butions in short fiber reinforced composite systems. Although these tensors capture the stochastic nature of concentrated fiber suspensions in a compact form, the evolution equation for each lower order tensor is a function of the next higher order tensor. Flow calculations typically employ a closure that approximates the fourth-order orientation tensor as a function of the second order orientation tensor. Recent work has been done with eigen-value based and invariant based closure approximations of the fourth-order tensor. The effect of using lower order tensors tensors in process simulations by reconstructing the distribution function from successively higher order orientation tensors in a Fourier series representation is considered. This analysis uses the property that orientation tensors are related to the series expansion coefficients of the distribution function. Errors for several closures are investigated and compared with errors developed when using a reconstruction from the exact 2nd, 4th, and 6th order orientation tensors over a range of interaction coefficients from 10−4 to 10−1 for several flow fields.
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Hergl, Chiara, Thomas Nagel, Olaf Kolditz, and Gerik Scheuermann. "Visualization of Symmetries in Fourth-Order Stiffness Tensors." In 2019 IEEE Visualization Conference (VIS). IEEE, 2019. http://dx.doi.org/10.1109/visual.2019.8933592.

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Koochakzadeh, Ali, and Piya Pal. "Canonical Polyadic (CP) Decomposition of Structured Semi-Symmetric Fourth-Order Tensors." In 2019 IEEE Data Science Workshop (DSW). IEEE, 2019. http://dx.doi.org/10.1109/dsw.2019.8755549.

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Wang, Yuan, Xueyong Ding, and Liansheng Wang. "Research of the Microscopic Polarizability Tensors for the Third and Fourth-Order Nonlinear Spectroscopy." In 5th International Conference on Information Engineering for Mechanics and Materials. Paris, France: Atlantis Press, 2015. http://dx.doi.org/10.2991/icimm-15.2015.69.

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Gerolymos, Georges A., and Isabelle Vallet. "Contribution to Single-Point-Closure Reynolds-Stress Modelling of Inhomogeneous Flows." In ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/fedsm2003-45346.

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The purpose of this paper is to present recent advances on the development of fully single-point-closure Reynolds-stress models, for flows with strong inhomogeneities, such as solid-wall effects or strong streamwise gradients (eg. shockwave/turbulent-boundary-layer-interaction). As a starting point it is shown that several recently developed wall-normal-free (wall-topology-free) RSMs, using gradients of turbulence length-scale and of anisotropy-invariants to replace geometric normals, can be interpreted as a generalization of well-known redistribution closures but with coefficients that are not scalars but fourth-order tensors. These tensorial coefficients are function of anisotropy-invariants and of their gradients (which indicate the direction of inhomogeneity). In view of the above result, it is suggested that the theory of the redistribution tensor closure should be revisited, with emphasis on inhomogeneity effects. Four baseline sets of coefficient values are given, and the proposed models are applied for various flows (developing flow in a square duct, 2-D and 3-D separated flows).
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Xirui Zhang, Zhiwen Liu, Yougen Xu, and Xiaofeng Gong. "Adaptive fourth-order tensor beamformer." In 2011 3rd International Conference on Awareness Science and Technology (iCAST). IEEE, 2011. http://dx.doi.org/10.1109/icawst.2011.6163176.

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Nobre, Ricardo, Aleksandar Ilic, Sergio Santander-Jiménez, and Leonel Sousa. "Tensor-Accelerated Fourth-Order Epistasis Detection on GPUs." In ICPP '22: 51st International Conference on Parallel Processing. New York, NY, USA: ACM, 2022. http://dx.doi.org/10.1145/3545008.3545066.

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Zhou, Shiwei, and Michael Yu Wang. "The Generalized Cahn-Hilliard Equations of Multiphase Transition for Structural Topology Optimization." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84751.

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This paper describes a generalized Cahn-Hilliard model for the topology optimization of multi-material structure. Unlike the traditional Cahn-Hilliard model applied to spinodal separation which only has bulk energy and interface energy, the generalized model couples the elastic energy into the total free energy. As a result, the morphology of the small phase domain during phase separation and grain coarsening process is not random islands and zigzag web-like objects but regular truss structure. Although disturbed by elastic energy, the Cahn-Hilliard system still keeps its two most important properties: energy dissipation and mass conservation. Therefore, it is unnecessary to compute the Lagrange multipliers for the volume constraints and make great effort to minimize the elastic energy for the optimization of structural topology. Furthermore, this model also makes the simple interpolation of stiffness tensors reasonable for multi-material structure in real simulation. To resolve these fourth-order nonlinear parabolic Cahn-Hilliard equations coupled with elastic energy, we developed a powerful mutigrid algorithm. Finally, we demonstrate that this new method is effective in optimizing the topology of multi-material structure through several 2-D examples.
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Gomes, Paulo R. B., Andre L. F. de Almeida, and Joao Paulo C. L. da Costal. "Fourth-order tensor method for blind spatial signature estimation." In ICASSP 2014 - 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2014. http://dx.doi.org/10.1109/icassp.2014.6854149.

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Park, Jonghyun, Nguyen Trung Kien, Hyesuk Kim, and Gueesang Lee. "Corrupted Region Restoration Using Second Order Tensors and Segmentation." In Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007). IEEE, 2007. http://dx.doi.org/10.1109/fskd.2007.231.

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Reports on the topic "Fourth-order Tensors"

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Busche, M. J. K, the fourth order coefficient tensor used in ALE3D's quadratic generalized von mises yield function, in five easy steps. Office of Scientific and Technical Information (OSTI), August 2000. http://dx.doi.org/10.2172/15013130.

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