Joint Invariants Research Articles

Hypoplastic constitutive model of inherently anisotropic sand

The modeling of anisotropic sand has always been a challenging and prominent issue in geotechnical constitutive research. This paper presents a hypoplastic constitutive model for inherently anisotropic sand. An anisotropic state variable A, which is defined by the joint invariants of the stress and fabric tensor, is introduced to account for the fabric anisotropy. This form of A can be referred to as a re-arrangement that allows for easier calibration. And this modification leads to the anisotropic contraction of the failure surface. The softening parameter in the critical state line is revised from a fixed to a state-dependent type, for achieving unified simulations of monotonic sand mechanical behaviour under different drainage conditions and within a wide range of stress levels and densities. Simulations of various monotonic element tests show that the salient features of inherently anisotropic sand, including barotropy, pyknotropy, strain softening, and dilatancy, can be well captured. Particularly, we make the first attempt to simulate pure stress-controlled true triaxial tests using hypoplastic constitutive under both drained and undrained conditions. Moreover, we provide a method to further optimize the flow direction by introducing an exponential formula.

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.

Geometric modeling with small defects of over-constrained Parallel Kinematic Machine

In the case of over-constrained mechanisms, the system can only be assembled or move under strict geometric conditions. A crucial step is to identify these conditions from the geometric model. Classically, the geometric model of a robot is computed from the Denavit–Hartenberg formalism. However, when imperfect mechanisms are studied, this formalism does not introduce exhaustively small geometric defects. In this article, a formalism based on kinematic joint invariants is preferred to describe the geometric behavior. The efficiency of this formalism is demonstrated for the accuracy improvement of a serial robot. A stationarity analysis of the geometric model is then performed to determine the geometric constraints induced by the over-constrained systems and to reduce the number of geometric parameters initially introduced. The methodology is first illustrated on an over-constrained slider–rod–crank system. Then, it is applied to the over-constrained mechanism of the Tripteor X7, a Parallel Kinematic Machine-tool. The benefit of our methodology is validated by a comparison of the geometric model obtained with a CAD model and a previous geometric model proposed in the literature. Finally, the identification of the parameters of the defined geometric model is conducted in order to quantify the potential accuracy benefit.

Geometrical defect identification of a SCARA robot from a vector modeling of kinematic joints invariants

This article introduces a new geometric vector modeling method of serial kinematic robot consistent with the identification process. This method is based on the definition of position and orientation of the robot joint invariants. For example, the invariant of the rotational joint is a straight-line (rotational joint axis). Thus, only independent geometrical parameters are introduced to model the joint axis position and orientation in space. Note that, the orientation is not constrained as in the Denavit–Hartenberg (DH) formalism. This article presents the methodology to define these geometrical parameters and the geometrical model. In this context, the identification method relies on "Circle Point Analysis". The points are measured with a laser tracker. Indeed, with a relevant processing of the measured points, we directly identify the invariants of joints. This method is applied to a SCARA robot geometric modeling. After an identification process, this methodology allows improving inverse kinematic error compared to the classical DH geometrical model with first and second-order defects. Moreover, the obtained residual error mean value is close to the accuracy of the measurement process.

Joint Invariants of Linear Symplectic Actions

We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants.

Nonlinear Tensor Functions of Two Arguments and Some “Orthogonal Effects” of the Stress–Strain State

The general representation of a symmetric isotropic tensor-function of the second rank in three-dimensional space, depending on two tensor arguments of the second rank, is investigated. This representation includes eight scalar material functions of ten invariants of dependent tensors, including four joint invariants. The tensor function is interpreted as the constitutive relation of an isotropic nonlinear material, which expresses the relationship between deformations and stresses and stress rates. A certain type of stress state is selected, corresponding to a combination of axial tension of the rod by a constant force and (θz) -torsion by a time-periodic moment. It is shown that with a proper selection of the above-mentioned eight material functions as functions of the stress state invariants, it is possible to describe a much stronger change in axial deformations under joint tension and cyclic torsion than under separate tension without torsion. Such an “orthogonal effect” of the stress-strain state is close in form to the ratchetting and vibration creep known in experimental mechanics.

Characterizing stress variability within granular samples upon liquefaction

A tensor-based approach is adopted to investigate the evolution of stress variability within granular media based on DEM simulations of both cyclic and static liquefactions. The bulk stress dispersion is evaluated using the effective variance of the entire stress tensor field and the local stress fluctuation is quantified by the Euclidean distance between the stress tensor of individual particles and the global stress tensor. The orientation discrepancies between the local stress tensor and global fabric tensor, and between the local and global stress tensors are examined by their joint invariants. We observe that the bulk stress dispersion decreases but the discrepancy of local stress fluctuation increases upon liquefaction. The initially deviatoric stress-dominated stress variability gradually becomes hydrostatic stress-dominated, and returns to deviatoric stress-dominated again after liquefaction. The liquefaction process is characterized by the degradation of coaxiality between the local stress tensor and the global fabric tensor and that between the local and global stress tensors. There exists a range of threshold values for such coaxiality, below which liquefaction occurs. Particles with coaxiality higher than the threshold value and particles with coaxality lower than the threshold value exhibit different roles in sustaining external loads. However, the discrepancy becomes neutral upon liquefaction.

Joint differential invariants of binary and ternary forms

We use moving frames to construct and classify the joint invariants and joint differential invariants of binary and ternary forms. In particular, we prove that the differential invariant algebra of ternary forms is generated by a single third order differential invariant. To connect our results with earlier analysis of Kogan, we develop a general method for relating differential invariants associated with different choices of cross-section.

Invariant discrete flows

AbstractIn this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.

Hilbert polynomials of the algebras of $SL_ 2$-invariants

We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group $SL_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in algebraic geometry. It is well known that the Hilbert function of the algebra $SL_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form $\mathcal{C}_{d}= \mathbb{C}[V_d\oplus \mathbb{C}^2]_{SL_2}$ (here $V_d$ is the $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was generalized to the algebra of joint invariants for $n$ binary forms. But the Cayley-Sylvester formula is not expressed in terms of polynomials.In our article we consider the problem of computing the Hilbert polynomials for the algebras of joint invariants and joint covariants of $n$ linear forms and $n$ quadratic forms. We express the Hilbert polynomials $\mathcal{H} \mathcal{I}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i, \mathcal{H}(\mathcal{C}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i,$ $\mathcal{H}(\mathcal{I}^{(n)}_2,i)=\dim(\mathcal{I}^{(n)}_2)_i, \mathcal{H}(\mathcal{C}^{(n)}_2,i)=\dim(\mathcal{C}^{(n)}_2)_i$ of those algebras in terms of quasi-polynomial. We also present them in the form of Narayana numbers and generalized hypergeometric series.

Group foliation of finite difference equations

Using the theory of equivariant moving frames, a group foliation method for invariant finite difference equations is developed. This method is analogous to the group foliation of differential equations and uses the symmetry group of the equation to decompose the solution process into two steps, called resolving and reconstruction. Our constructions are performed algorithmically and symbolically by making use of discrete recurrence relations among joint invariants. Applications to invariant finite difference equations that approximate differential equations are given.

3-D DEM simulations of drained triaxial tests on inherently anisotropic granulates

A series of triaxial tests were simulated by the discrete element method (DEM) to study the effect of inherent anisotropy on the mechanical behaviours of granulates. In this study, clumps consisting of a collection of rigid spherical balls were used to simulate irregular-shaped particles, and the multi-layer under-compaction method was employed to prepare the anisotropic specimens with different initial density and fabric. The macroscopic responses including the strength characteristic, the volumetric response, the critical state and the stress–dilatancy relationship were carefully examined. In addition, two microscopic descriptors, the particle orientation (PO) fabric and contact normal (CN) fabric, were mainly monitored and investigated to shed light on the underlying micro-mechanics. The results show that comparing to the experimental results on natural sands, the typical macro-mechanical behaviours of anisotropic granulates can be successfully captured in the DEM simulations. Specimens with smaller inclination angle are more dilative. Nevertheless, the influence of the inherent anisotropy on the shear–dilatancy relation seems to be limited. The joint invariants NP representing the relative orientation between the particle fabric direction and stress direction can be taken as an indicator of critical state. The rearrangement of the CN is more sensitive to the applied force than that of PO.

Rotation invariants from Gaussian-Hermite moments of color images

The topic of the paper is recognition of objects and patterns in color images regardless of their position, orientation, and scale. Gaussian–Hermite moment invariants designed especially for color images are introduced in this paper. We extend the existing invariants for graylevel images and show that in the case of color images there exist additional independent invariants, which can be constructed as joint invariants from cross-channel moments and/or from new non-trivial low-order moments. The experiments on real data confirmed that the new invariants improve the recognition rate.

Poincare series for the algebras of joint invariants and covariants of $n$ quadratic forms

We consider one of the fundamental problems of classical invariant theory - the research of Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincare series at the point $z = 1$ give us important information about the structure of the algebra $\mathcal{I}_{d}.$ It was derived by Hilbert for the algebra ${\mathcal{I}_{d}=\mathbb{C}[V_d]^{\,SL_2}}$ of invariants for binary $d-$form (by $V_d$ denote the vector space over $\mathbb{C}$ consisting of all binary forms homogeneous of degree $d$). Springer got this result, using explicit formula for the Poincare series of this algebra. We consider this problem for the algebra of joint invariants $\mathcal{I}_{2n}=\mathbb{C}[\underbrace{V_2 \oplus V_2 \oplus \cdots \oplus V_2}_{\text{n times}}]^{SL_2}$ and the algebra of joint covariants $\mathcal{C}_{2n}=\mathbb{C}[\underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{n times}}{\oplus}\mathbb{C}^2 ]^{SL_2}$ of $n$ quadratic forms. We express the Poincare series $\mathcal{P}(\mathcal{C}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{C}_{2n})_{j}\, z^j$ and $\mathcal{P}(\mathcal{I}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{I}_{2n})_{j}\, z^j$ of these algebras in terms of Narayana polynomials. Also, for these algebras we calculate the degrees and asymptotic behavious of the degrees, using their Poincare series.

A Minimal Integrity Basis for the Elasticity Tensor

We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity tensor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.

Admitted region of the joint invariants of the strain and rotation rate tensors

The admitted region of five joint irreducible invariants of the strain and rotation rate tensors calculated as the traces of the products of these tensors is determined.

A note on the relation between joint and differential invariants

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie’s finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allows to compute differential invariants from joint invariants.

On computing joint invariants of vector fields

A constructive version of the Frobenius integrability theorem -- that can be programmed effectively -- is given. This is used in computing invariants of groups of low ranks and recover examples from a recent paper of Boyko, Patera and Popoyvich \cite{BPP}.

Invariant discretization of partial differential equations admitting infinite-dimensional symmetry groups

This paper is concerned with the invariant discretization of differential equations admitting infinite-dimensional symmetry groups. By way of example, we first show that there are differential equations with infinite-dimensional symmetry groups that do not admit enough joint invariants preventing the construction of invariant finite difference approximations. To solve this shortage of joint invariants we propose to discretize the pseudo-group action. Computer simulations indicate that the numerical schemes constructed from the joint invariants of discretized pseudo-group can produce better numerical results than standard schemes.

Symmetry classification and joint invariants for the scalar linear (1 + 1) elliptic equation

The equations for the classification of symmetries of the scalar linear (1+1) elliptic partial differential equation (PDE) are obtained in terms of Cotton’s invariants. New joint differential invariants of the scalar linear elliptic (1+1) PDE in two independent variables are derived in terms of Cotton’s invariants by application of the infinitesimal method. Joint differential invariants of the scalar linear elliptic equation are also deduced from the basis of the joint differential invariants of the scalar linear (1+1) hyperbolic equation under the application of the complex linear transformation. We also find a basis of joint differential invariants for such type of equations by utilization of the operators of invariant differentiation. The other invariants are functions of the basis elements and their invariant derivatives. Examples are given to illustrate our results.