Finite Transformation Research Articles

On constraint-conforming numerical discretizations in constitutive material modeling

AbstractFor the modelling of complex materials, internal variables are usually introduced which characterize the microstructural state. Then, evolution equations describe the change of the internal variables due to varying external loading conditions. These equations can be derived, for instance, on the basis of variational principles. The consideration of characteristic observations, such as the preservation of the volume during a change in the microstructural state, can significantly improve the accuracy of the evolution equations. We present a Hamilton principle that provides a unique way to derive evolution equations that obey holonomic constraints and opens up new possibilities for their algorithmic treatment. This is demonstrated for isochoric finite plasticity and phase transformation based on Backward-Euler time discretization. The models presented are efficient and are characterized by simple implementation compared to the exponential map, for example, without suffering a loss of accuracy due to unfulfilled constraints.

Subshifts of finite symbolic rank

Abstract The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper, we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank- $1$ subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts, we also show that the rank-1 subshifts are dense but not generic. Finally, we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank $2$ , and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.

Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Let TX be the full transformation monoid over a finite set X, and fix some a∈TX of rank r. The variant TXa has underlying set TX, and operation f⋆g=fag. We study the congruences of the subsemigroup P=Reg(TXa) consisting of all regular elements of TXa, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product of Cong(Tr), and the full equivalence relation lattices of certain combinatorial systems of subsets and partitions. We use this to give an explicit classification of the congruences themselves, and we also give a formula for the height of the lattice.

Dynamic behavior of the orthotropic hollow cylinder under a magnetic field: a thermoelastic analysis

The present article focuses on the complete analytical solution of an orthotropic hollow cylinder using a proposed magneto-thermoelastic model and finite Hankel transformation. The study considers the Lord–Shulman and Green–Lindsay models of hyperbolic generalized thermoelasticity within the context of thermoelasticity theory. The analytical expressions of the field variables of the cylinder in a magnetic field were obtained under thermo-mechanical loadings. The study also offered closed-form field functions and measured the thermoelastic behavior concerning the magnetic field intensity of the cylinder based on thermoelastic theories involving a second sound. The inclusion of hyperbolic-type heat conduction models enhanced the visualization of the reflection of thermal waves into the cylinder. The study also investigated the impacts of orthotropic material properties on the considered medium and compared the results with the existing literature.

Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation.

An image encryption scheme based on the four-dimensional chaotic system and the mealy finite state machine

This paper proposes a four-dimensional hyperchaotic system to overcome the defect of weak encryption effect due to the simple complexity of low-dimensional chaotic systems in chaos-based image encryption schemes, which is verified to have better chaotic properties by dynamics analysis, sensitivity analysis, and randomness test. In addition, this paper proposes an image encryption scheme by combining the proposed chaotic system with the Mealy finite state machine (MFSM) to overcome the problem that some schemes are not resistant to statistical analysis. Firstly, the chaotic sequence is applied in the Knuth-Durstenfeld shuffling method to scramble the original image efficiently; secondly, according to the different information contained in the bit-plane, the pixel is decomposed and cyclically shifted at the bit level to achieve bit-level scrambling and diffusion; thirdly, the chaotic sequence is applied to select the rules for DNA encoding, and the diffusion process is achieved by Mealy finite state machine transformation; lastly, decoding the diffused image by selecting DNA rules randomly to get cipher image. This paper gives the experimental results, demonstrating that the scheme is highly secure. It can improve the encryption scheme’s sensitivity to plaintext and resist attacks such as differential attacks and select attacks.

Coupled Axisymmetric Thermoelectroelasticity Problem for a Round Rigidly Fixed Plate

Introduction. To describe the operation of temperature piezoceramic structures, the theory of thermoelectroelasticity is used, in which the mathematical model is formulated as a system of nonself-adjoint differential equations. The complexity of its integration in general leads to the study of problems in an unrelated formulation. This does not allow us to evaluate the effect of electroelastic fields on temperature. The literature does not present studies on these problems in a three-dimensional coupled formulation in which closed solutions would be constructed. At the same time, conducting such studies allows us to understand the interaction picture of mechanical, thermal and electric fields in a structure. To solve this problem, a new closed solution of a coupled problem for a piezoceramic round rigidly fixed plate has been constructed in this research. It provides for qualitative assessment of the cross impact of thermoelectroelastic fields in this electroelastic system.Materials and Methods. The object of the study is a piezoceramic plate. The case of unsteady temperature change on its upper front surface is considered, taking into account the convection heat exchange of the lower plane with the environment (boundary conditions of the 1st and 3rd kind). The electric field induced as a result of the thermal strain generation is fixed by connecting the electrodated surfaces to the measuring device. The thermoelectroelasticity problem includes the equations of equilibrium, electrostatics, and the unsteady hyperbolic heat equation. It is solved by the generalized method of finite biorthogonal transformation, which makes it possible to construct a closed solution of a nonself-adjoint system of equations.Results. A new closed solution of the coupled axisymmetric thermoelectroelasticity problem for a round plate made of piezoceramic material was constructed.Discussion and Conclusion. The obtained solution to the initial boundary value problem made it possible to determine the temperature, electric and elastic fields induced in a piezoceramic element under arbitrary temperature axisymmetric external action. The calculations performed provided determining the dimensions of solid electrodes, which made it possible to increase the functionality of piezoceramic transducers. Numerical analysis of the results enabled us to identify new connections between the nature of external temperature action, the deformation process, and the value of the electric field in a piezoceramic structure. This can validate a proper program of experiments under their designing and significantly reduce the volume of field studies.

Nonlocal symmetries, soliton-cnoidal wave solution and soliton molecules to a (2+1)-dimensional modified KdV system

A (2+1)-dimensional modified KdV (2DmKdV) system is considered from several perspectives. Firstly, residue symmetry, a type of nonlocal symmetry, and the Bäcklund transformation are obtained via the truncated Painlevé expansion method. Subsequently, the residue symmetry is localized to a Lie point symmetry of a prolonged system, from which the finite transformation group is derived. Secondly, the integrability of the 2DmKdV system is examined under the sense of consistent tanh expansion solvability. Simultaneously, explicit soliton-cnoidal wave solutions are provided. Finally, abundant patterns of soliton molecules are presented by imposing the velocity resonance condition on the multiple-soliton solution.

Nonlocal symmetries and solutions of the multi-dimensional integrable long water wave equations

In this paper, the (2+1)-dimensional integrable long water wave equations (LWWs) are constructed for the first time using the conservation law of the (1+1)-dimensional LWWs. The new (1+1)-dimensional LWWs can be obtained by introducing a constraint to the (2+1)-dimensional LWWs. This new (1+1)-dimensional LWWs are studied by using nonlocal symmetry methods for the first time. The closed system corresponding to nonlocal symmetry is established by the lax pairs of equations and the potential function determined using conservation laws. Exact solutions of the equations are obtained by finite symmetry transformation and symmetry approximation of this closed system. The dynamic behavior of the equations is studied by means of figures of the exact solutions.

Nonlocal symmetry and group invariant solutions of dissipative (2+1)-dimensional AKNS equation

In this paper, the nonlocal symmetry of dissipative (2+1)-dimensional AKNS equation is constructed by auxiliary equation method. In order to better study new group invariant solutions of AKNS equation with the help of nonlocal symmetry, we introduce an new auxiliary dependent variable, the (2+1) dimensional AKNS equation is extended to a new closed prolonged system. Therefore, the nonlocal symmetry of the original equation is the Lie point symmetry of the new prolonged system. Then, by solving the initial-value problems, the finite symmetry transformation for the prolonged system is derived. Furthermore, by using symmetry reduction method to the prolonged system, the interaction solution between cnoidal waves and solitary waves is given.

Finite transforms with applications to Bessel differential equations of order higher than two

A finite transformation method is introduced. This method is equivalent to the Z transform method to a certain extent but generalizes it. By applying the presented method to the Bessel functions, it is possible to solve related ordinary differential equations of order higher than 2 with given initial conditions.

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Time-space formulation of a conservative string subject to finite transformations

This paper investigates power-balanced descriptions of the conservative vibrating string. First, the hypotheses of continuum mechanics are recalled for the case of infinitesimal transformations. The string is described by classical partial differential equations (PDEs) and reformulated as a port Hamiltonian system (PHS). Second, the case of finite (possibly large) transformations is considered, for which the time variation of the elastic energy appears to be no longer the elastic power. To naturally solve this difficulty and ensure the invariance of the elastic power with respect to the superposition of rigid body motion, the problem is here addressed in the framework of a time-space formulation. Eulerian and Lagrangian conservative formulations are proposed in the context of non relativistic velocities. This work also yields perspectives for a time-space representation of a port-Hamiltonian vibrating string invariant to the change of observer in a relativistic context.

Equivalence of computed tomography problem with the problem of recovery of functions by finite convolutions in a scheme of computational (numerical) diameter

The equivalence of the norms of deviations of the desired density of a body from operators such as finite density transformation with specially constructed elements and the Radon transformation from it is stated. It is shown how Computer Science, previously established in the theory of Computational (Numerical) diameter, immediately leads to non-trivial results in Computed Tomography.

NORMAL SUBMONOIDS AND CONGRUENCES ON A MONOID

AbstractA notion of normal submonoid of a monoid M is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf {NorSub}(M)$ of normal submonoids of M is a complete lattice. Joins are explicitly described and the lattice is computed for the finite full transformation monoids $T_n$ , $n\geq ~1$ . It is also shown that $\mathsf {NorSub}(M)$ is modular for a specific family of commutative monoids, including all Krull monoids, and that it, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice $\mathsf {Cong}(M)$ of congruences on M. This leads to a new strategy for computing $\mathsf {Cong}(M)$ consisting of computing $\mathsf {NorSub}(M)$ and the so-called unital congruences on the quotients of M modulo its normal submonoids. This provides a new perspective on Malcev’s computation of the congruences on $T_n$ .

A novel finite spectral entropy: Gated term memory unit recursive network integrated with Ladybug Beetle Optimization algorithm for epileptic seizure detection.

Professional medical experts use a visual electroencephalography (EEG) signal for epileptic seizure detection, although this method is time-consuming and highly subject to bias. The majority of previous epileptic detection techniques have poor efficiency, detection performance and also which are unsuited to handle large datasets. In order to solve the aforementioned issues and to assist medical professionals with an advanced technology, a computerized epileptic seizure detection system is essential. Therefore, the proposed work intends to design an automated detection tool for predicting an epileptic seizure from EEG signals. For this purpose, a novel non-linear feature analysis and deep learning algorithms are deployed in this work. Initially, the signal decomposition, filtering and artifacts removal operations are carried out with the use of finite Haar wavelet transformation technique. After that, the finite spectral entropy (FSE) based feature extraction model has been used to extract the time, frequency, and time-frequency features from the normalized signal. Consequently, the novel gated term memory unit recursive network (GTRN) model is employed to predict the given EEG signal as whether healthy or seizure affected including the class with high accuracy. During this process, the recently developed Ladybug Beetle Optimization (LBO) algorithm is used to compute the logistic sigmoid function based on the solution. The purpose of using this algorithm is to simplify the process of classification with increased seizure prediction accuracy and performance. Moreover, the standard and popular benchmark EEG datasets are used to validate and test the results of the proposed FSE-GTRN-LBO mechanism. By leveraging the finite Haar wavelet transformation and FSE-based feature extraction, we can efficiently process EEG signals. The utilization of the GTRN model enables accurate classification of healthy and seizure-affected EEG data. To optimize the classification process further, we integrate the LBO algorithm, streamlining the computation of the logistic sigmoid function. Through comprehensive validation on standard EEG datasets, our proposed FSE-GTRN-LBO mechanism achieves outstanding seizure prediction accuracy and performance, surpassing existing state-of-the-art techniques.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

The symmetric breathers and lumps of the Boussinesq equation using the Alice–Bob transformation and Hirota’s bilinear derivative method

Many two-place physical problems can be explicitly presented as related events model named Alice–Bob systems. In this paper, an integrable Alice–Bob Boussinesq system is introduced via the Boussinesq equation with parameters, which may meet the symmetry transformation of Psˆx (parity with a shift) and Tdˆt (time reversal with a delay). After constructing an Bäcklund transformation, the system has rich symmetry solutions with the aid of auxiliary functions. The structures of obtained soliton solutions, such as the breathers, lumps and their hybrids, are all satisfied the Pˆsx or Tˆdt symmetry. To illustrate the symmetric characteristic, some lower-order solutions and the related dynamic structures are explicitly presented. The residual symmetry and its finite transformation for this system are also verified.