Control of the elastic properties of a shell working at stability

This examines a shell with elastic properties varying across the coordinates, which are prescribed by means of scalar functions of the invariants of the elasticity tensor. The basis of the arrangement of the tensor for the elasticity consists of q linear-independent tensors of the fourth range (q is the number of linear-independent components of the elasticity tensor) which are obtained by multiplying and turning the first tensor of the surface and the tensor characterizing the class of symmetry of the medium. The invariants of the elasticity tensor present in the stability equation and their derivatives are taken to be the equations and parameters for the state of the system (shell), and the problem is thus reduced to a problem of optimum equations. As an example we shall examine an orthotropic cylindrical shell with a model varying over the length under the action of external pressure.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 93–100, January–February, 1974.     

Generic and typical ranks of multi-way arrays

The concept of tensor rank was introduced in the 20s. In the 70s, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini’s lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries.     

Tensorial properties of multiple view constraints

We define and derive some properties of the different multiple view tensors. The multiple view geometry is described using a four‐dimensional linear manifold in ℝ^3m, where m denotes the number of images. The Grassman co‐ordinates of this manifold build up the components of the different multiple view tensors. All relations between these Grassman co‐ordinates can be expressed using the quadratic p‐relations. From this formalism it is evident that the multiple view geometry is described by four different kinds of projective invariants; the epipoles, the fundamental matrices, the trifocal tensors and the quadrifocal tensors. We derive all constraint equations on these tensors that can be used to estimate them from corresponding points and/or lines in the images as well as all transfer equations that can be used to transfer features seen in some images to another image. As an application of this formalism we show how a representation of the multiple view geometry can be calculated from different combinations of multiple view tensors and how some tensors can be extracted from others. We also give necessary and sufficient conditions for the tensor components, i.e. the constraints they have to obey in order to build up a correct tensor, as well as for arbitrary combinations of tensors. Finally, the tensorial rank of the different multiple view tensors are considered and calculated. Copyright © 2000 John Wiley & Sons, Ltd.     

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Kinematics of finite elastoplastic deformations

In this paper we review various approaches to the decomposition of total strains into elastic and nonelastic (plastic) components in the multiplicative representation of the deformation gradient tensor. We briefly describe the kinematics of finite deformations and arbitrary plastic flows. We show that products of principal values of distortion tensors for elastic and plastic deformations define principal values of the distortion tensor for total deformations. We describe two groups of methods for decomposing deformations and their rates into elastic and nonelastic components. The methods of the first group additively decompose specially built tensors defined in a common basis (initial, current, or “intermediate”). The second group implies a certain relation connecting tensors that describe elastic and plastic deformations. We adduce an example of constructing constitutive relations for elastoplastic continuums at large deformations from thermodynamic equations.     

A complete steady state model of solute and water transport in the kidney

The purpose of this paper is to incorporate a detailed model, along with an optimized set of parameters for the proximal tubule, into J. L. Stephenson's current central core model of the nephron. In this model a set of equations for the proximal tubule are combined with Stephenson's equations for the remaining four tubules and interstitium, to form a complete nonlinear system of 34 ordinary differential and algebraic equations governing fluid and solute flow in the kidney. These equations are then discretized by the Crank-Nicholson scheme to form an algebraic system of nonlinear equations for the unknown concentrations, flows, hydrostatic pressure, and potentials. The resulting system is solved via factored secant update with a finite-difference approximation to the Jacobian. Finally, numerical simulations performed on the model showed that the modeled behavior approximates, in a general way, the physiological mechanisms of solvent and solute flow in the kidney.     

The tensor t‐function: A definition for functions of third‐order tensors

A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.     

Tensor inversion and its application to the tensor equations with Einstein product

Recently, the inverse of an even-order square tensor has been put forward in [Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl. 2013;34(2):542–570] by means of the tensor group consisting of even-order square tensors equipped with the Einstein product. In this paper, several necessary and sufficient conditions for the invertibility of a tensor are obtained, and some approaches for calculating the inverse (if it exists) are proposed. Furthermore, the Cramer's rule and the elimination method for solving the tensor equations with the Einstein product are derived. In addition, the tensor eigenvalue problem mentioned in [Qi L-Q. Theory of tensors (hypermatrices). Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University; 2014] can also be addressed by using the elimination method mentioned above. By the way, the LU decomposition and the Schur decomposition of matrices are extended to tensor case. Numerical examples are provided to illustrate the main results.     

A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors

The mixed finite element method for approximately solving flow equations in porous media has received a good deal of attention in the literature. The main idea is to solve for the head/pressure and fluid velocity (Darcy velocity) simultaneously to obtain a higher order approximation of the fluid velocity. In the case of a diagonal transmissivity tensor the algebraic equations resulting from the discretization can be reduced to a system of algebraic equations for the head/pressure variable alone. This reduction results in a smaller number of unknows to be solved for in an iterative method such as preconditioned conjugate gradient method. The fluid velocity is then obtained from an algebraic relationship. In the case of full transmissivity tensor, the algebraic reduction is more difficult. This paper investigates some algorithms resulting from the modification of the mixed finite element that take advantage of the mixed finite element method for the diagonal tensor case. The resulting schemes are more efficient implementations that maintain the same order of accuracy as the original schemes. © 1993 John Wiley & Sons, Inc.     

On a group approach to studying the Einstein and maxwell equations

Based on the classical problem for decomposition of the tensor product of representations into irreducible components, which is considered in the elementary representation theory for orthogonal groups, a partial classification of the Einstein equations is carried out. A new class of Maxwell equations for relativistic electrodynamics is singled out and studied. Pointwise-irreducible decompositions for the energy-momentum and electromagnetic field tensors are obtained and a physical interpretation of the decomposition components is given. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 32–43, April, 1997.     

On Tensor Completion via Nuclear Norm Minimization

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher-order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors into matrices and then apply techniques for matrix completion. We show here that such matricization fails to exploit the tensor structure and may lead to suboptimal procedure. More specifically, we investigate a convex optimization approach to tensor completion by directly minimizing a tensor nuclear norm and prove that this leads to an improved sample size requirement. To establish our results, we develop a series of algebraic and probabilistic techniques such as characterization of subdifferential for tensor nuclear norm and concentration inequalities for tensor martingales, which may be of independent interests and could be useful in other tensor-related problems.     

Non-linear deformation properties of materials with stochastically distributed anisotropic inclusions

In the present contribution, the problem of non-linear deformation of materials with stochastically distributed anisotropic inclusions is considered on the basis of the methods of mechanics of stochastically non-homogeneous media. The homogenization model of materials of stochastic structure with physically non-linear components is developed for the case of a matrix which is strengthened by unidirectional ellipsoidal inclusions. It is assumed that the matrix is isotropic, deforms non-linearly; inclusions are linear-elastic and have transversally-isotropic symmetry of physical and mechanical properties. Stochastic differential equations of physically non-linear elasticity theory form the underlying equations. Transformation of these equations into integral equations by using the Green's function and application of the method of conditional moments allow us to reduce the problem to a system of non-linear algebraic equations. This system of non-linear algebraic equations is solved by the Newton-Raphson method. On the analytical as well as the numerical basis, the algorithm for determination of the non-linear effective characteristics of such a material is introduced. The non-linear behavior of such a material is caused by the non-linear matrix deformations. On the basis of the numerical solution, the dependences of homogenized Poisson's coefficients on macro-strains and the non-linear stress-strain diagrams for a material with randomly distributed unidirectional ellipsoidal pores are predicted and discussed for different volume fractions of pores. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant tensors and partial differential equations

We consider tensors with coefficients in a commutative differential algebra A. Using the Lie derivative, we introduce the notion of a tensor invariant under a derivation on an ideal of A. Each system of partial differential equations generates an ideal in some differential algebra. This makes it possible to study invariant tensors on such an ideal. As examples we consider the equations of gas dynamics and magnetohydrodynamics.     

Generalized inverses of tensors via a general product of tensors

We define the {i}-inverse (i = 1, 2, 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.     

On the constitutive inequalities and acceleration waves in micropolar media

Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor are derived. The condition for acceleration wave's propagation is formulated as an algebraic spectral problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Biot Stress and Strain in Thin-Plate Theory for Large Deformations

We propose a theory of nonlinear deformation of a plate on the basis of an energetically conjugate pair of the Biot stress tensors and the right stretch tensor. When the dimensionality of the problem is reduced from three to two, the classical Kirchhoff conjectures are used, the linear part is retained in the expansion of the right stretch tensor with respect to a degenerate coordinate, and no additional simplifications are assumed. Connection is obtained between the asymmetric and symmetric components of the Biot tensor; the equivalence is demonstrated of the virtual work principle with the equilibrium equations, the natural boundary conditions, and additional conditions for the dependence of asymmetric stress moment resultants on symmetric moments.     

Integral geometry of tensor valuations

We prove a complete set of integral geometric formulas of Crofton type (involving integrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger's general integral geometric theorem, the Crofton formulas yield also kinematic formulas for Minkowski tensors. The explicit calculations of integrals over affine Grassmannians require several integral geometric and combinatorial identities. The latter are derived with the help of Zeilberger's algorithm.     

New Zonal,Spectral Solutions for the Navier‐Stokes Layer and Their Applications

New zonal, spectral forms for the axial, lateral and vertical velocity's components, density function and absolute temperature inside of compressible three‐dimensional Navier‐Stokes layer (NSL) over flattened, flying configurations (FC), are here proposed. The inviscid flow over the FC, obtained after the solidification of the NSL, is here used as outer flow. If the spectral forms of the velocity's components are introduced in the partial differential equations of the NSL and the collocation method is used, the spectral coefficients are obtained by the iterative solving of an equivalent quadratical algebraic system with slightly variable coefficients.     

Tensor ranks on tangent developable of Segre varieties

We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon's conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.     

On a Quasi-Linear Partial Differential Algebraic System of Equations

Computational Mathematics and Mathematical Physics - A mixed problem for a quasi-linear first-order partial differential algebraic system of equations of index $$(1,0)$$ with two independent...