Tensor Research Articles

Load Deflection Analysis of Beam-Column using Total Potential Energy (TPE) Principle

A modified TPE approach to perform finite deflection analysis of slender beam-column elements has been developed. The proposed approach utilizes the energy principal method and takes into account the geometric nonlinearity including the effects of axial force on bending stiffness, the end moments on axial stiffness (bowing), and the initial imperfection. A new equation of the deformation curve that approaches to the exact solution is used in the strain-displacement relation to obtain a more accurate beam-column response. The derived formulation of displacement of the beam-column under axial compressive load with single curvature bowing is presented with initial imperfection and different end eccentricities. The Green strain tensor equation is developed to consider higher-order bowing term. Nonlinear analysis of central finite deflection is carried out using Newton-Raphson iteration that includes high order terms of total potential energy (TPE). The beam-column stability is verified by computing the hessian determinant of the total potential energy. The validity of the new approach is established by comparing the numerical results obtained using the proposed equations against data previously published in the literature. Outputs from the analysis indicate that the proposed approach is capable of capturing the deflection of the beam-column with enhanced accuracy, ranging from 8.5% for e = 0.025 and up to 23.5% when e = 0.125.

Pseudospectrum of Kronecker product and stability analysis of tensor equations

Pseudospectrum of Kronecker product and stability analysis of tensor equations

Geodesic path planning characteristics of the reconfigurable 1-S robot workspaces for hyperbolic, elliptical, and Euclidean geometries

The path-planning strategies are implemented by establishing the Riemann curvature tensor and geodesic equations of the 1-S robot workspace. This paper’s originality lies in formulation of the parametric 1-S robotworkspace for path planning, which is based on the differential geometry of the geodesic and Riemann curvature equations. The novel results in defining the path plan with diffeomorphic and expandable trajectories with zero and negative sectional curvatures are encouraging, as shown in the research article’s result sections. The constant negative, constant positive, and zero sectional curvatures produce hyperbolic, elliptical, and Euclidean geometries. The workspace equation, derived using Lie algebra, defines the parameters of ��1, ��2, ��3, and ��4 to obtain the shortest distances in path planning. The geodesic equations determine the shortest distances in the context of Riemann curvature tensor equations. These parameters from the workspace equation (��1, ��2, ��1, ��1) are used in the geodesic and Riemann curvature tensor equations. The results show that one needs to choose the most convenient parameters of the mechanism for path-planning capabilities. Both the topology of the mechanism, which is 1-S herein and the parameters of the workspaces should be selected for the pre-defined trajectories of the path planning, as shown in the results. The reconfigurable robots have many mechanism topologies to transform.

A Family of Iterative Methods Without Inversion to Solve a System of Nonlinear Tensor Equations with Einstein Product

A Family of Iterative Methods Without Inversion to Solve a System of Nonlinear Tensor Equations with Einstein Product

Randomized extended average block Kaczmarz method for inconsistent tensor equations under t-product

Randomized extended average block Kaczmarz method for inconsistent tensor equations under t-product

A System of Tensor Equations over the Dual Split Quaternion Algebra with an Application

In this paper, we propose a definition of block tensors and the real representation of tensors. Equipped with the simplification method, i.e., the real representation along with the M-P inverse, we demonstrate the conditions that are necessary and sufficient for the system of dual split quaternion tensor equations (A∗NX,X∗SC)=(B,D), when its solution exists. Furthermore, the general expression of the solution is also provided when the solution of the system exists, and we use a numerical example to validate it in the last section. To the best of our knowledge, this is the first time that the aforementioned tensor system has been examined on dual split quaternion algebra. Additionally, we provide its equivalent conditions when its Hermitian solution X=X∗ and η-Hermitian solutions X=Xη∗ exist. Subsequently, we discuss two special dual split quaternion tensor equations. Last but not least, we propose an application for encrypting and decrypting two color videos, and we validate this algorithm through a specific example.

Fractional mimetic dark matter: A fractional action-like variational approach

In this paper, we propose a nonlocal extension of the mimetic dark matter model based on the FALVA implementation of fractional calculus. Our primary objective to explore how certain properties of dark matter can be modeled within the FALVA framework rather than formulate a phenomenological model. We begin by constructing the action functional of the cosmological extension of the mimetic dark model and deriving the nonlinear equations of motion in the general case. Next, we focus on a fractional-power homogeneous mimetic dark field with an exponential expansion factor. We derive the equation of motion for the lapse field and obtain its general solution. Analytical solutions are then obtained for small time intervals and arbitrary values of the fractionality parameter. These solutions enable us to establish the physical line element of spacetime, incorporating the mimetic dark matter field. From this line element, we derive the Ricci tensor, Ricci scalar, and geodesic equations.

On the viscoelastic-electromagnetic-gravitational analogy

The analogy between electromagnetism and gravitation was achieved by linearizing the tensorial gravitational equations of general relativity and converting them into a vector form corresponding to Maxwell’s electromagnetic equations. On this basis, we use the equivalence with viscoelasticity and propose a theory of gravitational waves. We add a damping term to the differential equations, which is equivalent to Ohm’s law in electromagnetism and Maxwell’s viscosity in viscoelasticity, to describe the attenuation of the waves. The differential equations in viscoelasticity are those of cross-plane shear waves, commonly referred to as SH waves. A plane-wave analysis gives the phase velocity, the energy velocity, the quality factor and the attenuation factor of the field as well as the energy balance. To obtain these properties, we use the analogy with viscoelasticity; the properties of electromagnetic and gravitational waves are similar to those of shear waves. The presence of attenuation means that the transient field is generally a composition of inhomogeneous (non-uniform) plane waves, where the propagation and attenuation vectors do not point in the same direction and the phase velocity vector and the energy flux (energy velocity) are not collinear. The polarization of cross-plane field is linear and perpendicular to the propagation-attenuation plane, while the polarization of the field within the plane is elliptical. Transient wave fields in the space–time domain are analyzed with the Green function (in homogeneous media) and with a grid method (in heterogeneous media) based on the Fourier pseudospectral method for calculating the spatial derivatives and a Runge–Kutta scheme of order 4 for the time stepping. In the examples, wave propagation at the Sun–Earth and Earth–Moon distances using quadrupole sources is considered in comparison to viscoelastic waves. The Green and grid solutions are compared to test the latter algorithm. Finally, an example of propagation in heterogeneous media is presented.

On applying deflation and flexible preconditioning to the adaptive Simpler GMRES method for Sylvester tensor equations

Due to their multidimensional structure, the use of tensors when solving linear matrix equations has been one of the most explored topics over the last decade. In particular, the development of Krylov subspace methods for tensor equations is currently attracting a great deal of interest. In this work, we develop the adaptive simpler GMRES method based on tensor format (Ad-SGMRES−BTF) for solving Sylvester tensor equations, in order to address the numerical instability of the simpler version of GMRES based on tensor format (SGMRES−BTF). In addition, an efficient Krylov subspace method, formed by the integration of adaptive deflation and preconditioning and called FAd-SGMRES-D−BTF(m,k) -in which m is the restart parameter and k is the number of harmonic Ritz pairs-, is proposed. Note that the idea behind the FAd-SGMRES-D−BTF(m,k) method is to reuse certain key information, which can be obtained from the current tensor Krylov subspace in the next cycle of the FAd-SGMRES-D−BTF(m) method. It is also important to note that FAd-SGMRES-D−BTF(m,k) represents the first attempt to synchronously combine flexible preconditioning and deflation techniques, with the aim of improving the convergence rate and reducing the computational cost of Ad-SGMRES−BTF(m). Finally, the computational cost of each cycle of the proposed methods when applied to a three-dimensional Sylvester tensor equation is estimated. Moreover, numerical experiments on synthetic and real-world applications demonstrate the potential of Ad-SGMRES−BTF and FAd-SGMRES-D−BTF(m) to solve Sylvester tensor equations.

Viscoelasticity, logarithmic stresses, and tensorial transport equations

We introduce models for viscoelastic materials, both solids and fluids, based on logarithmic stresses to capture the elastic contribution to the material response. The matrix logarithm allows to link the measures of strain, that naturally belong to a multiplicative group of linear transformations, to stresses, that are additive elements of a linear space of tensors. As regards the viscous stresses, we simply assume a Newtonian constitutive law, but the presence of elasticity and plastic relaxation makes the materials non‐Newtonian. Our aim is to discuss the existence of weak solutions for the corresponding systems of partial differential equations in the nonlinear large‐deformation regime. The main difficulties arise in the analysis of the transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, we only need to consider the equation for the left Cauchy–Green tensor, while for the fluid model, we add an evolution equation for the elastically‐relaxed strain. Due to the tensorial nature of the fields, available techniques cannot be applied to the analysis of such transport equations. To cope with this, we introduce the notion of charted weak solution, based on non‐standard a priori estimates, that lead to a global‐in‐time existence of solutions for the viscoelastic models in the natural functional setting associated with the energy inequality.

A covariant formulation for finite strain modelling of orthotropic elasticity and orthotropic plasticity with plasticity-induced evolution of orthotropy: Application to natural fibres

We introduce a rate-independent model for orthotropic elastic and orthotropic plastic material behaviour in a hyper-elasto-plastic framework at finite strains. The model is based on the postulate of covariance and does not rely on a multiplicative decomposition of the deformation gradient. Furthermore, a plastic-deformation-induced evolution of orthotropy is considered, similar to the notion of plastic spin. We propose that the orthotropic strain energy function and the orthotropic yield criterion are guided by identical structural tensors that evolve with plasticity. The modelled material behaviour is significant for natural fibres such as flax, hemp, or pulp fibres. Our formulation has three findings. Firstly, the covariant formulation of plasticity provides rate equations for the plastic variables and the structural tensors suitable for reproducing stress–strain diagrams of natural fibres. Secondly, the introduction of plastic-deformation-induced evolution of orthotropy in the proposed covariant setting results in a non-associative plasticity algorithm. Thirdly, the covariant setting allows the incorporation of suitable constitutive equations for the structural tensors to evolve orthotropy. The latter successfully models the stiffness increase in stress–strain diagrams of cyclic tensile tests of natural fibres.

Unveiling correlated two-dimensional topological insulators through fermionic tensor network states—classification, edge theories and variational wavefunctions

The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations andnon-chiraltopological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

The BiCG Algorithm for Solving the Minimal Frobenius Norm Solution of Generalized Sylvester Tensor Equation over the Quaternions

In this paper, we develop an effective iterative algorithm to solve a generalized Sylvester tensor equation over quaternions which includes several well-studied matrix/tensor equations as special cases. We discuss the convergence of this algorithm within a finite number of iterations, assuming negligible round-off errors for any initial tensor. Moreover, we demonstrate the unique minimal Frobenius norm solution achievable by selecting specific types of initial tensors. Additionally, numerical examples are presented to illustrate the practicality and validity of our proposed algorithm. These examples include demonstrating the algorithm’s effectiveness in addressing three-dimensional microscopic heat transport and color video restoration problems.

Krylov subspace methods based quaternion tensor form for generalized Sylvester quaternion tensor equation with application to color video restoration

In this paper, we consider Krylov subspace methods based quaternion tensor form for a class of generalized Sylvester quaternion tensor equation. A novel quaternion tensor product and a global quaternion Arnoldi process based on the tensor form are proposed. Then, the quaternion tensor Krylov subspaces and their orthogonal bases are constructed via the proposed global quaternion Arnoldi process. The quaternion full orthogonalization method based on tensor format (denoted by QFOM−BTF) and the quaternion generalized minimal residual method based on tensor format (denoted by QGMRES−BTF) are established to solve the quaternion tensor equation. The theoretically analyze results of the proposals are discussed on the quaternion tensor form. Finally, we demonstrate the feasibility of QFOM−BTF and QGMRES−BTF methods on some numerical examples including the color video restoration.

Alternative Arnoldi process for ill-conditioned tensor equations with application to image restoration

Alternative Arnoldi process for ill-conditioned tensor equations with application to image restoration

On sign function of tensors with Einstein product and its application in solving Yang–Baxter tensor equation

On sign function of tensors with Einstein product and its application in solving Yang–Baxter tensor equation

Predefined-Time Zeroing Neural Networks With Independent Prior Parameter for Solving Time-Varying Plural Lyapunov Tensor Equation.

As an extension of the Lyapunov equation, the time-varying plural Lyapunov tensor equation (TV-PLTE) can carry multidimensional data, which can be solved by zeroing neural network (ZNN) models effectively. However, existing ZNN models only focus on time-varying equations in field of real number. Besides, the upper bound of the settling time depends on the value of ZNN model parameters, which is a conservative estimation for existing ZNN models. Therefore, this article proposes a novel design formula for converting the upper bound of the settling time into an independent and directly modifiable prior parameter. On this basis, we design two new ZNN models called strong predefined-time convergence ZNN (SPTC-ZNN) and fast predefined (FP)-time convergence ZNN (FPTC-ZNN) models. The SPTC-ZNN model has a nonconservative upper bound of the settling time, and the FPTC-ZNN model has excellent convergence performance. The upper bound of the settling time and robustness of the SPTC-ZNN and FPTC-ZNN models are verified by theoretical analyses. Then, the effect of noise on the upper bound of settling time is discussed. The simulation results show that the SPTC-ZNN and FPTC-ZNN models have better comprehensive performance than existing ZNN models.

Transpose-free quasi-minimal residual method based on tensor format for generalized coupled Sylvester tensor equations

Transpose-free quasi-minimal residual method based on tensor format for generalized coupled Sylvester tensor equations

Couple Stress-Based Flexoelectric Theory in Orthogonal Curvilinear Coordinates and its Application to an Infinitely Long Dielectric Cylinder

In this paper, the formulations of the couple stress based flexoelectric theory in orthogonal curvilinear coordinates are presented. By following the two rules of transforming tensor equations from Cartesian coordinates to curvilinear ones: replacing the partial differentiation symbol with the covariant differentiation symbol and ensuring that the identical indices are positioned diagonally, the governing equations, constitutive equations and boundary conditions in orthogonal curvilinear coordinates are obtained. Then the couple stress based flexoelectric theory is specified for cylindrical coordinates. The flexoelectric responses of an infinitely long dielectric cylinder under a uniform electric field are analyzed. Numerical results show that the radial displacement increases rapidly as the mechanical length scale parameter [Formula: see text] increases. When [Formula: see text] is larger than 5 [Formula: see text]m, the circumferential displacement decreases rapidly. [Formula: see text] has a large effect on the induced displacements but a small effect on the induced electric potential, while the electrical length scale parameter [Formula: see text] has a small effect on both the induced displacements and the induced electric potential.

The Dirac equation as a linear tensor equation for one component

The Dirac equation is one of the most fundamental equations of modern physics. It is a spinor equation, but some tensor equivalents of the equation were proposed previously. Those equivalents were either nonlinear or involved several components of the Dirac field. On the other hand, the author showed previously that the Dirac equation in electromagnetic field is equivalent to a fourth-order equation for one component of the Dirac spinor. The equivalency is used in this work to derive a linear tensor equivalent of the Dirac equation for just one component. This surprising result can be used in applications of the Dirac equation, for example, in general relativity or for lattice approximation of the Dirac field, and can improve our understanding of the Dirac equation.