Invariants Of Tensor Research Articles

Unified three-dimensional finite elements for large strain analysis of compressible and nearly incompressible solids

This work proposes a displacement-based finite element model for large strain analysis of isotropic compressible and nearly-incompressible hyperelastic materials. Constitutive law is written in terms of invariants of the right Cauchy-Green tensor; coupled and decoupled formulations of strain energy functions are presented, whereas a penalty function is used to impose an incompressibility constraint. Based on a total Lagrangian formulation, the nonlinear governing equations are thus obtained by employing the principle of virtual displacements. Analytic expression of both internal forces vector and tangent matrix of linear and high-order hexahedral finite elements are derived by adopting a three-dimensional formalism based on the Carrera Unified Formulation. Popular benchmark problems in hyperelasticity are analyzed to establish the capabilities of the present implementation of fully-nonlinear solid elements in the case of compressible and nearly-incompressible beams, cylindrical shells, and curved structures.

Heavy inertial particles in rotating turbulence: Distribution of particles in flow and evolution of Lagrangian trajectories.

We revisit the problem of heavy particles suspended in homogeneous box turbulence flow subjected to rotation along the vertical axis, which introduces anisotropy along the vertical and horizontal planes. We investigate the effects of the emergent structures due to rotation, on the spatial distribution and temporal statistics of the particles. The distribution of particles in the flow are studied using the joint probability distribution function (JPDFs) of the second and third principle invariants of the velocity gradient tensor, Q and R. At high rotation rates, the JPDFs of Lagrangian Q-R plots show remarkable deviations from the well-known teardrop shape. The cumulative probability distribution functions for times during which a particle remains in vortical or straining regions show exponentially decaying tails except for the deviations at the highest rotation rate. The average residence times of the particles in vortical and straining regions are also affected considerably due to the addition of rotation. Furthermore, we compute the temporal velocity autocorrelation and connect it to the Lagrangian anisotropy in presence of rotation. The spatial and temporal statistics of the particles are determined by a complex competition between the rotation rate and inertia of the particle.

The Hypoplastic Constitutive Model for Sandy Soil Considering the Rotation of the Principal Stress Axis

Considering the influence of fabric on the critical state of sandy soil, an anisotropic hypoplastic constitutive model is developed by introducing the anisotropic critical state line that takes into account the rotation of the principal stress axes. With the introduction of new anisotropic state variables defined by the joint invariants of the stress tensor and fabric tensor, the critical state equation of sandy soil is established to describe the effects of three factors, namely, anisotropic parameters, stress states, and the relationship between principal stresses and fabric directions, on the critical state. The mechanical response of sandy soil under different deposition angles can be described by considering the rotation of principal stresses relative to the fabric. The application range of Wu et al.’s isotropic hypoplastic model (2017) is extended by incorporating the effect of principal stress rotation on the stress–strain relationship of sandy soil. Based on a series of Toyoura sand plane strain tests, the effects of void ratio, confining pressure, and principal stress axis rotation angle on anisotropic strength and deformation characteristics are simulated under low confining pressure. Furthermore, a comparison with Wu’s transversely isotropic hypoplastic model (1998) is made regarding their simulation performances. The proposed model exhibits a balanced performance when simulating the variation of anisotropy in both strength and deformation with respect to the rotation angle, without being overestimated within a certain range of rotation angles. The prediction results demonstrate, to a certain degree, the validity and effectiveness of the proposed model.

Ефективна пластична поведінка пористих матеріалів зі структурою інверсного опалу

Based on the theoretical principles of the mechanics of composites, the effective plastic behavior of a porous material with a periodic inverse opal structure under uniaxial loading was studied in detail by means of finite element modeling. The creation of such materials is based on the inversion of pores and skeleton of partially sintered dense packing of polystyrene spheres. Electrodeposited nickel was used as the skeleton of the porous material. According to the macroscopic uniaxial loading or unloading, was finding a stress-strain state at the meso-level. For this, equilibrium equations were solved at the meso-level using special boundary conditions for a periodic unit cell. Such boundary conditions relate the problem of equilibrium at the meso-level with the "effective" deformations of the composite. This made it possible to calculate macroscopic residual strains after a cycle of uniaxial loading and unloading and iteratively find the value of effective stress corresponding to residual strains of 0.2%. In this way, the yield strength of inverse opal for uniaxial loading is calculated. At the same time, as a result of finite-element calculations, the transverse deformations coefficient (plastic Poisson ratio) is determined. This coefficient, in turn, makes it possible to approximate the general plastic behavior of the metamaterial by an elliptic yield curve in the plane of invariants of the stress tensor. Invariants mean average pressure and von Mises stress. These calculations were performing for several cases of the inverse opal structure, both with and without an additional coating. Yield stresses under any type of material loading are very sensitive to porosity. In particular, the application of an additional coating, even with a thickness less than 0,05 of the diameter of the spherical pores (initial polymer particles), causes an increase in the yield strength several times. Keywords: metamaterials, inverse opal, porous plasticity model, micromechanics, theory of plasticity.

Two novel alternative integration schemes for multi-invariants dependent isotropic finite deformation plasticity

Multi-invariants dependent isotropic yield function in terms of first invariant I1 of stress tensor, second J2 and third J3 invariants of deviatoric stress tensor is required for accurate description of plasticity of polycrystalline solids. In the context of multi-invariants dependent multiplicative hyperelasto-plasticity, two integration schemes (IS-1 and IS-2) are formulated and their computational costs are compared for the first time in this paper. IS-1 is fully-implicit in which backward Euler difference approximation of equivalent plastic strain rate is coupled with the exponential approximation of return mapping in principal Kirchhoff stress space and accordingly principal Kirchhoff stresses, equivalent plastic strain and plastic Lagrange multiplier need to be updated iteratively. IS-2 is semi-implicit in which plastic Lagrange multiplier evaluated at current time step and semi-implicit forward Euler difference approximation of equivalent plastic strain rate is coupled with logarithmic approximation of return mapping in Kirchhoff stress space and accordingly only plastic Lagrange multiplier needs to be updated iteratively. For J2 dependent plasticity, computation time involved in numerical simulations for both the integration schemes is nearly the same for same accuracy. For IS-1, computation time involved in simulation for multi-invariants dependent plasticity is significantly greater than that for J2 dependent plasticity due to the computation of the Jacobian matrix in return mapping involving lengthy terms of second order partial derivatives of multi-invariants dependent yield function. For IS-2, computation time involved in simulation for multi-invariants dependent plasticity is nearly the same as that for J2 dependent plasticity. This is due to the fact that the computation of the Jacobian matrix in IS-2 is not required and return mapping involves only first partial derivatives of yield function. Thus, IS-2 is computationally cost effective than IS-1.

Properties of the turbulent/non-turbulent layer of a turbulent Boussinesq plume: A study using direct numerical simulation

The turbulent/non-turbulent layer (TNTL) in a turbulent Boussinesq plume is analyzed using direct numerical simulations. The Reynolds number (Re) used in the simulation, based on the scales defined at the bottom hot patch, is 2000, and the Reynolds number (Reλ) based on the Taylor microscale (λ) is 114.4 in the self-similar region. The flow is sufficiently resolved till the Kolmogorov scale. The outer edge of the TNTL is detected using the vorticity magnitude. Conditional statistics of various quantities are calculated with reference to the outer edge of the TNTL, which is referred to as the irrotational boundary (IB). The profiles of conditional vorticity magnitude are used to identify the TNTL and determine its thickness. The presence of viscous superlayer (VSL) and turbulent sublayer (TSL) within the TNTL is revealed by exploring the conditional profiles of the enstrophy transport equation. The baroclinic torque, which is a source of vorticity, has been shown to be inconsequential in determining the width of the VSL. The widths of the TNTL and the VSL are determined to be δTNTL≈15η and δVSL≈3.12η, respectively, where η is the Kolmogorov length scale. This gives the width of the TSL as δTSL=δTNTL−δVSL≈13.88η. The invariants of the velocity gradient tensor have been analyzed across the TNTL. The joint probability density function of the invariants Q and R shows a teardrop shape within the turbulent core. The teardrop shape is not fully developed within the TNTL. The TNTL and the turbulent core have a mixed tendency for irrotational dissipation, vortex sheets, and vortex tubes. When normalized by Kolmogorov length and velocity scales, the conditional statistics of the TNTL of the plume are similar to other turbulent flow types, and hence, this provides additional evidence for the universality of small-scale motion within/around the TNTL of various turbulent flows.

Efficient calculation of three-dimensional tensor networks

We have proposed an efficient algorithm to calculate physical quantities in the translational invariant three-dimensional tensor networks, which is particularly relevant to the study of the three-dimensional classical statistical models and the ($2+1$)-dimensional quantum lattice models. In the context of a classical model, we determine the partition function by solving the dominant eigenvalue problem of the transfer matrix, whose left and right dominant eigenvectors are represented by two projected entangled simplex states. These two projected entangled simplex states are not Hermitian conjugate to each other but are appropriately arranged so that their inner product can be computed much more efficiently than in the usual prescription. For the three-dimensional Ising model, the calculated internal energy and spontaneous magnetization agree with the published results in the literature. The possible improvement and extension to other models are also discussed.

Entanglement entropy of local gravitational quenches

We study the time dependence of Rényi/entanglement entropies of locally excited states created by fields with integer spins s ≤ 2 in 4 dimensions. For spins 0, 1 these states are characterised by localised energy densities of a given width which travel as a spherical wave at the speed of light. For the spin 2 case, in the absence of a local gauge invariant stress tensor, we probe these states with the Kretschmann scalar and show they represent localised curvature densities which travel at the speed of light. We consider the reduced density matrix of the half space with these excitations and develop methods which include a convenient gauge choice to evaluate the time dependence of Rényi/entanglement entropies as these quenches enter the half region. In all cases, the entanglement entropy grows in time and saturates at log 2. In the limit, the width of these excitations tends to zero, the growth is determined by order 2s + 1 polynomials in the ratio of the distance from the co-dimension-2 entangling surface and time. The polynomials corresponding to quenches created by the fields can be organized in terms of their representations under the SO(2)T × SO(2)L symmetry preserved by the presence of the co-dimension 2 entangling surface. For fields transforming as scalars under this symmetry, the order 2s + 1 polynomial is completely determined by the spin.

Phenomenological model for the γγ→π+π−π0 reaction

We present a phenomenological model for the $\gamma\gamma \to \pi^+\pi^-\pi^0$ reaction by including the production mechanism of the $a_2(1320)$ resonance, as well as the contributions from the $\sigma/f_0(500)\, \pi^0$ and $f_2(1270)\, \pi^0 $ production channels. Furthermore, the $\gamma\gamma\to\rho^{\pm} \pi^{\mp}\to\pi^{+}\pi^-\pi^0$ channel, which is essential for a description in the low-energy region, is investigated carefully by introducing the complete set of gauge invariant and Lorentz-covariant tensors for the $\gamma\gamma\to\rho^\pm \pi^\mp$ subprocess. The full amplitude is constructed to yield a correct high-energy Regge behavior. Within our model, we achieve a very reasonable description of ARGUS and L3 data of the total cross section, as well as of the $\pi^\pm\pi^0$ and $\pi^+\pi^-$ invariant mass distributions. We also predict the invariant mass distributions in the $\gamma\gamma$ center-of-mass energy range from $0.8$ GeV to $2.0$ GeV, which will be studied by the forthcoming data of the BESIII collaboration.

Numerical simulation of bluff body turbulent flows using hybrid RANS/LES turbulence models

Many engineering applications involve turbulent flows around bluff bodies. Because of their intrinsically unsteady dynamics, bluff body characteristic flows feature unique turbulence-related phenomena, which makes their numerical modeling challenging. Accordingly, accounting for a circular bluff body flow configuration, three different turbulence modeling approaches are investigated in this work, (i) Reynolds-averaged Navier–Stokes (RANS), (ii) large eddy simulation (LES), and (iii) hybrid RANS/LES. Regarding the hybrid approaches, two variants of the detached eddy simulation (DES) one, delayed DES (DDES) and improved delayed DES (IDDES), are studied. As RANS model, the $$\mathrm{k}-\mathrm{\omega SST}$$ is utilized here. This RANS model is also used as the background one for both DDES and IDDES. Wall-adaptive local eddy viscosity (WALE) is used in turn as the sub-grid scale (SGS) model for LES. The velocity two-point correlation function is used to assess the mesh size requirements. When compared to experimental data, the obtained numerical results indicate that RANS overestimates the recirculating bubble length by over 18% and is not capable of describing the turbulent kinetic energy and the flow anisotropy in agreement with the experimental data. In contrast, LES, DDES, and IDDES are all within 1% of the recirculating bubble length while predicting both the Reynolds stress tensor components and the corresponding flow anisotropy in agreement with the measurements. Besides, normalized anisotropy tensor invariants maxima in the shear layer were reproduced by all scale resolving models studied here, but they failed to yield the local extrema measured within the wake recirculation region. A comparative analysis of the anisotropic Reynolds stress tensor invariances underscores the adequacy of the scale resolving models.

Invariant Theory and wheeled PROPs

We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra V=T(V)⊗T(V⋆), where T(V) is the tensor algebra on an n-dimensional vector space over a field K of characteristic 0. First we classify all the ideals of the initial object Z in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of V are exactly subalgebras of the form VG where G is a closed, reductive subgroup of the general linear group GL(V). When V is a finite dimensional Hilbert space, a similar description of invariant tensors for an action of a compact group was given by Schrijver. We also generalize the theorem of Procesi that says that trace rings satisfying the n-th Cayley-Hamilton identity can be embedded in an n×n matrix ring over a commutative algebra R. Namely, we prove that a wheeled PROP can be embedded in R⊗V for a commutative K-algebra R if and only if it satisfies certain relations.

On the role of tensor interpolation in solving high-WI viscoelastic fluid flow

The high Weissenberg number (Wi) problem (HWNP) has long been a challenge of viscoelastic fluid flow simulation. This Letter points out that the tensor interpolation method during solving the differential constitutive equations is the main origin of the loss of symmetric positive-definite (SPD) property of the conformation tensor, which is the trigger of the HWNP. Instead of component-based interpolation, we propose a tensor-based interpolation method for the conformation tensor, which is essentially SPD, and the results show that this method is very effective in dealing with the HWNP by significantly improving the numerical accuracy on the invariants of conformation tensor as well as greatly improving the SPD property of the conformation tensor. Moreover, the high-order total variation diminishing schemes can also be easily constructed and applied to solve high-Wi viscoelastic fluid flow under the proposed framework without adding artificial diffusion.

Hydrocarbon reservoir depth identification in the eastern black sea basin: Implications from euler deconvolution and gravity tensor invariants

The Black Sea Basin (BSB) consisting of Eastern Black Sea (EBS) and Western Black Sea (WBS) basins which separated by the Mid Black Sea High (MBSH) is formed by the collision of Eurasia and Arabian-African Plates. Petroleum and gas hydrate contents of the BSB have been discussed by using different geological and geophysical research. In this study, the gravity gradient tensor and invariant data based on the Bouguer gravity anomaly of the EBS basin were evaluated to define the hydrocarbon reservoirs. Afterwards, Euler deconvolution technique is applied to the invariant to reveal the depth of subsurface bodies. Furthermore, the significant invariant anomalies are compared with known oil seepage and petroleum wells. The depth values of oil trace closest to the known Sürmene and Hopa drill holes were computed as 4809 m, and 4455 m according to the structural index values of SI = 2.

Modelling and computation of gravitational attraction, gradient tensors, rotational and horizontal invariants of Asteroid Bennu (101955), Itokawa (25143) and Eros (433) via 2D Non-Uniform FFT

The internal structure and mass distribution of the terrestrial objects are yet unknown. The 2D gravity model with a constant density of the terrestrial objects can shed light on the surficial or textural heterogeneity due to topographic variations of the terrestrial objects. Three different asteroids, which are Bennu (101955), Itokawa (25143) and Eros (433) are modelled in this study. During the modelling phase, a different number of edges, elements, nodes, and faces are used to describe the 3D models of Bennu, Itokawa, and Eros. These 3D models are used in 2D Non-Uniform Fast Fourier Transform (NU-FFT) applications to obtain gravitational attraction with a constant density polyhedron model. Tensor gradients and tensor invariants of the modelled gravity anomaly are calculated. Three major outcomes are interpreted from gradient tensors and tensor invariants. Firstly, textural heterogeneity due to relatively low topography is detected in the central part of Bennu. Secondly, considerably different properties which can be related to surface variations between the two lobes of Itokawa are observed. Lastly, directional surficial heterogeneities were detected in Eros.

No invariant perfect qubit codes

Perfect tensors describe highly entangled quantum states that have attracted particular attention in the fields of quantum information theory and quantum gravity. In loop quantum gravity, the natural question arises whether SU(2) invariant tensors, which are fundamental ingredients of the basis states of spacetime, can also be perfect. In this work, we present a number of general constraints for the layout of such invariant perfect tensors (IPTs) and further describe a systematic and constructive approach to check the existence of an IPT of given valence. We apply our algorithm to show that no qubit encoding of valence 6 can be described by an IPT and close a gap to prove a no-go theorem for invariant perfect qubit encodings. We also provide two alternative proofs for the non-existence of 4-valent qubit IPTs which has been shown in [1, 2].

Two-loop QCD corrections to the V → qoverline{q}g helicity amplitudes with axial-vector couplings

We compute the two-loop corrections to the helicity amplitudes for the coupling of a massive vector boson to a massless quark-antiquark pair and a gluon, accounting for vector and axial-vector couplings of the vector boson and distinguishing isospin non-singlet and singlet contributions. A new four-dimensional basis for the decomposition of the amplitudes into 12 invariant tensor structures is introduced. The associated form factors are then computed up to two loops in QCD using dimensional regularization. After performing renormalization and infrared subtraction, the finite parts of the renormalized non-singlet vector and axial-vector form factors are shown agree with each other, and to reproduce the previously known two-loop amplitudes. The singlet axial-vector amplitude receives a contribution from the axial anomaly from two loops onwards. This amplitude is computed for massless and massive internal quarks. Our results provide the last missing two-loop amplitudes entering the NNLO QCD corrections of vector-boson-plus-jet production at hadron colliders.

Statistics of geometrical invariants of magnetic field gradient tensors in the turbulent magnetosheath based on magnetospheric multiscale mission

Geometrical invariants of magnetic field gradient tensors are used to classify the topological structures of a magnetic field. This study presents a statistical analysis on the geometrical invariants of magnetic field gradient based on high-quality data measured by magnetospheric multiscale mission in turbulent magnetosheath. The method for the classification of velocity field topologies cannot be applied to magnetic field with strong intensity directly because the magnetic field cannot be transformed to zero by selecting a co-moving reference frame in which the velocity is zero. During a strong magnetic field, flux ropes and tubes are the most possible magnetic structures. Statistics in the plane formed by geometrical invariants show that about 23% are force-free structures comprised of 20.5% flux tubes and 79.5% flux ropes. The remaining actively evolved structures are comprised of 30% flux tubes and 70% flux ropes. Moreover, the conditional average of current density and Lorentz force decomposition in geometrical invariants plane are investigated. The results show that flux ropes carried more current density than flux tubes for the same geometrical invariants, and flux ropes tend to associate with magnetic pressure force and flux tubes tend to associate with magnetic tension.

Evolution of the velocity gradient invariants in homogeneous isotropic turbulence with an inverse energy cascade

An investigation of topological features of homogeneous isotropic turbulence with an inverse energy cascade is performed by using a direct numerical simulation. The inverse energy cascade is induced by reversing the velocity field (i.e., under the transformation from ui to −ui) of a freely decaying isotropic turbulence. In the backward energy transfer process, the joint probability density function (PDF) of Q and R exhibits a novel shape, which seems to be symmetric with the well-known teardrop shape about the Q-axis (Q and R are the second and third invariants of the velocity gradients tensor, respectively). The predominance of the top-right (R > 0 and Q > 0) and the bottom-left (R < 0 and Q < 0) quadrants is observed in the backward energy transfer process. This observation is different from that of the forward energy cascade, in which the top-left (R < 0 and Q > 0) and the bottom-right (R > 0 and Q < 0) quadrants are dominant. The unexpected shape of the joint PDF of Q and R indicates that the turbulence with an inverse energy cascade is dominated by vortex compression and tube-like structures. The PDF of the intermediate eigenvalue of the strain-rate tensor in the inverse energy cascade is negatively skewed, which is opposite to the universal feature of the forward energy cascade. Nevertheless, the preferential alignment of vorticity with the intermediate eigenvector is rather robust, no matter whether the direction of the energy transfer is forward or backward. In addition, it is universal that the vorticity is mostly perpendicular to the eigenvector, which corresponds to a strain-rate eigenvalue with the maximum absolute value among the three eigenvalues. Since the velocity gradient invariants are closely related to the local flow topology, the numerical results reported in this work are expected to shed light on the intrinsic dynamics and mechanisms of inverse energy cascade.

Numerical investigation of noise suppression and amplification in forced oscillations of single and tandem cylinders in high Reynolds number turbulent flows

The noise generated by a three-dimensional turbulent wake behind an oscillating single or tandem cylinder(s) is studied using detached eddy simulation and the Ffowcs Williams-Hawkings (FW-H) method. This hybrid methodology is validated using some experimental data of a turbulent flow past a single or tandem stationary cylinder(s) and some numerical data (obtained from a direct numerical simulation) of a laminar flow past tandem oscillating cylinders. For an oscillating cylinder at a fixed Reynolds number Re=120,000, the sound energy is suppressed compared to that of a stationary cylinder provided the value of oscillation frequency approaches 57% of the original vortex-shedding frequency of the stationary cylinder. This noise reduction arises from the structural oscillations suppressing the noise generated by the vortex shedding from the stationary cylinder—the additional source of noise stemming from the structural oscillations is too small to compensate for the mechanism responsible for the noise suppression. The noise generated by the cylinder motion includes contributions from the loading and thickness noise, whereas the noise generated by vortex shedding from the stationary cylinder only includes the loading noise. The flow past tandem stationary cylinders exhibits a larger number of spectral peaks in the sound pressure power spectrum and the peaks occur at a lower frequency compared to the flow past a single cylinder. The drag force fluctuations result in the propagation of the sound pressure in the in-line direction, whereas the lift force fluctuations result in the propagation of the sound pressure in the transverse direction. Five different scenarios of stationary and/or oscillating tandem cylinders are studied. The second and third invariants of the velocity gradient tensor are used to visualize the distinct vortex structures in the wake for the five different scenarios of tandem cylinders. A single cylinder oscillating in the transverse direction exhibits a larger sound pressure level (SPL) than the streamwise oscillation of the same cylinder. In contrast, the streamwise (in-line) oscillation of one or the other cylinder in a tandem array results in a larger SPL than the transverse oscillation of one or the other cylinder. Finally, the SPL is more sensitive to the oscillations of the rear (downstream) cylinder than the front (upstream) cylinder.

Criteria for opening an axial cavity in cross-wedge rolling

The application of empirical, phenomenological deformation and energy theories of metal fracture in relation to the process of cross-wedge rolling is considered. A distinctive feature of cross-wedge rolling in comparison with other metal pressure treatment processes is the complex stress-strain state in the deformation focus and the possibility of opening the axial cavity. The existing empirical criteria of destruction are given. The scheme of a new calibration test based on crosswedge rolling and the fracture criterion proposed by Zbigniew Pater are considered. It is shown that the Zbigniew Pater’s empirical criterion for the destruction of Zbigniew Pater provides a correct determination of the moment of opening the cavity during cross-wedge rolling. The improved phenomenological deformation theory is based on the theory of fracture developed by V. L. Kolmogorov. It is proposed to evaluate the plasticity of the metal depending on the stress state in the form of two independent invariants of the stress tensor: the average stress and the parameter of the third invariant of the stress tensor. The dependence of the limit value of accumulated deformation, at which destruction occurs, on the stress state is constructed in the form of a plasticity surface. The phenomenological deformation criterion of metal destruction during plastic deformations allows us to determine the moment of opening the cavity during cross-wedge rolling and the plasticity resource of the metal.