Uncoupled Problems Research Articles

Applied Theory of Flexural Vibrations of a Piezoactive Bimorph in the Framework of an Uncoupled Boundary-Value Problem of Thermoelectroelasticity

Applied Theory of Flexural Vibrations of a Piezoactive Bimorph in the Framework of an Uncoupled Boundary-Value Problem of Thermoelectroelasticity

Sun–Jupiter–Saturn System May Exist: A Verified Computation of Quasiperiodic Solutions for the Planar Three-Body Problem

In this paper, we present evidence of the stability of a model of our Solar System when taking into account the two biggest planets, a planar (Newtonian) Sun–Jupiter–Saturn system with realistic data: masses of the Sun and the planets, their semiaxes, eccentricities and (apsidal) precessions of the planets close to the real ones. (We emphasize that our system is not in the perturbative regime but for fixed parameters.) The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasiperiodic motion (on an invariant torus) with the appropriate frequencies. To do so, we first use KAM numerical schemes to compute translated tori to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is an invariant torus. Second, we use KAM numerical schemes for invariant tori to refine the solution giving the desired torus. Lastly, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM–iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem.

Energy Loss in Frictional Hertzian Contact Subjected to Two-Dimensional Cyclic Loadings

We investigate the effect of three different harmonically varying loads as a function of the friction coefficient on energy loss in a three-dimensional discrete uncoupled frictional contact problem. Three loading cases include (1) a normal force is constant and a tangential force varies, (2) normal and tangential forces both vary, but the loading and unloading curves are identical, and (3) normal and tangential forces both vary, but the loading and unloading curves are different. For a higher coefficient of friction, three loading cases show different characteristics. If a normal force is constant and a tangential force varies, there is always some slip, but dissipation tends asymptotically to zero at large coefficient of friction. If normal and tangential forces both vary, but the loading and unloading curves are identical, there is no slip and no dissipation above a critical coefficient of friction. If the loading and unloading curves are different, dissipation occurs for all values of the coefficient of friction, and we expect that the dissipation is asymptotic to the relaxation damping value as the coefficient of friction approaches infinity. For lowering coefficient of friction, the three loading cases show similar behavior. Dissipation increases and reaches a maximum just before a state where gross slip is possible.

Uncoupled thermoelasticity problem for a finite rectangular composite

Uncoupled thermoelasticity problem for a finite rectangular composite

On applying polyharmonic radial basis functions to solve 3D uncoupled anisotropic thermoelasticity problems using Boundary Element and Dual Reciprocity Method

On applying polyharmonic radial basis functions to solve 3D uncoupled anisotropic thermoelasticity problems using Boundary Element and Dual Reciprocity Method

Thermal stress analysis of variable angle tow composite plates through high-order structural models

This work focuses on thermo-elastic analysis of Variable Angle Tow (VAT) composite plates. Emphasis is given to the through-the-thickness 3D stress component distributions as a consequence of constant temperature profiles. High-order structural models are used, in order to take into account the volumetric and the distortion contributions. The last contribute arise from boundary conditions or asymmetric staking sequences. Governing equations of the uncoupled thermo-elastic problem are derived in the domain of the Carrera Unified Formulation (CUF), which in turn is coupled with a classical finite element method to obtain opportune numerical solutions. The advantage of using CUF lies in the fact that different theories of structures can be implemented automatically and a critical study on the use of standard to high-order plate finite elements can be performed with ease. In this manner we classify models based on their efficiency and depending on the degree of accuracy needed, both in terms of displacements and stresses. Different numerical problems are considered and it is demonstrated that layerwise approximations are needed whenever shear stresses trends are of interest. Boundary Conditions influence more the choice of the adequate expansion theory otherwise.

LOCAL MODEL OF THE THERMOELASTIC STATE OF A POROUS MATERIAL

In the paper, a local stationary model of the thermoelastic state of a porous material is constructed. The model is based on uncoupled boundary value problems for stationary equations of heat conduction and thermoelasticity for a space with two spherical cavities. The temperature field is determined by the constant temperature on the surfaces of the cavities, which are considered free of forces. The problem was solved by the generalized Fourier method (GFM), for which the paper presents its further development for a certain class of thermoelasticity problems. For this purpose, systems of equally directed spherical coordinates were introduced, the origins of which are connected with the centers of the cavities. In the work, a new set of axisymmetric basic solutions of the Lamé equation for a sphere is constructed and addition theorems are proved for it and for solutions of the vector biharmonic equation in the given coordinate systems. The GFM formalism made it possible to reduce the boundary value problems to algebraic solvable systems with Fredholm operators in the space l2 under the condition that the spherical surfaces do not intersect. The reduction method was used for the numerical solution of the systems. Graphs of stresses on the surface of one of the cavities and stresses on the axis of symmetry of the problem between the cavities at different relative sizes of the cavities and at different temperatures of their heating were obtained. The obtained results agree with those known for one cavity. The convergence of the reduction method is verified numerically.

Stochastic collocation for optimal control problems with stochastic PDE constraints by meshless techniques

In this paper, we investigate the stochastic collocation method with meshless techniques for an optimal control problem of elliptic equations with random inputs. The proposed method does not depend on the mesh partition and only requires the solution of uncoupled deterministic problem at each interpolation point, which can be competitive for high-dimensional random inputs and handle easily a wider range of situations such as random variables that are unbounded and input data that depend nonlinearly on the random variables. A priori error estimates are provided for the control, state and co-state variables. The stochastic collocation method based on gradient projection is presented, and some numerical examples are carried out to illustrate the performance of the new scheme.

Неосесимметричная нестационарная задача термоэлектроупругости для длинного пьезокерамического цилиндра

A new closed solution to the uncoupled non-axisymmetric problem of thermoelectroelasticity was constructed for a long hollow piezo-ceramic cylinder for the case of non-stationary temperature alteration on its inner surface, taking into account the convective heat transfer between the outer surface and the environment. Cylindrical surfaces were electroded and connected to a measurement device with large input resistance (idle mode), while the internal surface was grounded. The Fourier—Kirchhoff non-stationary heat conduction equation was considered without taking into account the influence of alteration in the body dimensions and the electric field on the temperature field. The closed solution to the heat conduction problem was constructed by the finite integral transformations (FIT) method. The quasi-static coupled electroelasticity problem at a certain temperature field was solved without taking into account the cylinder inertial properties by the FIT method. The calculated relations obtained made it possible to determine the temperature field, the stress-strain state, as well as the electric field in the long piezo-ceramic cylinder under non-stationary non-axisymmetric action in the form of a function of the temperature alteration. Numerical analysis of the results made it possible to determine the cylinder wall thickness and the region of the temperature effect alteration, where deformation could most efficiently transform into the electrical pulse.

Applied theory of flexural vibrations of a piezoactive bimorph in the framework of an uncoupled boundary-value problem of thermoelectroelasticity

In this paper, we consider transverse steady oscillations of a piezoactive bimorph in the formulation of a plane deformation. The problem is solved within the framework of linear thermoelectroelasticity, while the temperature problem is solved separately and the temperature distribution is taken into account in the constitutive relations of electroelasticity. On the basis the Kirchhoff-Love type hypothesis for mechanical quantities and a symmetric quadratic distribution of the electric potential, an approximate theory for calculating bimorph vibrations is constructed. Numerical experiments have been carried out for various cases of pinning and excitation of vibrations. The results of these experiments were compared with calculations made using the finite element method in the COMSOL package and showed the adequacy of the constructed theory in the low-frequency region.

On the stability of transient bimaterial contact in the presence of dry friction and slip

An analytic formulation for the study of planar elastodynamic contact in bimaterial interfaces subject to dry friction and slip is presented. Using the Wiener–Hopf technique, explicit analytical expressions for the elastic waves scattered by a planar contact interface are derived. Two cases are studied: (1) the uncoupled problem where it is assumed that normal and tangential loads cause, respectively, no tangential and vertical displacements; (2) the fully coupled problem, where they do. In both cases, an analytic formulation for determining the magnitude of the regions of stick and slip expected at the interface is offered. This provides a complete analytic account of the interfacial tractions and, as a results, serves to model the elastic wave scattering by contact interfaces, a problem of interest in fields as disparate as geophysics, non-destructive testing, and fracture mechanics. The uncoupled problem is shown to be inconsistent: the magnitude of the reciprocal displacements caused by the normal/tangential loads is non-negligible, and of the same magnitude as the interfacial slip distribution itself. The coupled problem is shown to lead to a matricial Wiener–Hopf problem the scattering kernel of which is non-commutative; an Abrahams approximation reliant on the Padé approximants is used to study the problem. It is shown that in many circumstances, the bimaterial contact interface is bound to detach locally, even when the far-field, steady-state solutions would predict otherwise. This is brought about by interfacial loading mismatches brought about by the coupling, and is shown to be affected by various factors: weak pressure loads, the friction coefficient, the disparity in the elastic constants of the media under contact. An useful analytic criteria for detachment is offered.

Numerical simulation of buoyant plumes using a fixed point iterative method

A numerical simulation of unsteady plumes for driven thermal high convection dominated flows is presented in this work. The problem is of practical interest for example in the cooling of electronic devices and heat transfer from pipes in heat exchange systems. This kind of thermal flows may be modeled using the unsteady Boussinesq approximation in the Stream function-vorticity formulation. Results are obtained with a simple numerical method previously reported for isothermal/natural and mixed convection flows. The numerical method is based on a fixed point iterative process to solve the non-linear elliptic system that results after an appropriate discretization in time. The iterative process leads us to the solution of uncoupled, well-conditioned and symmetric linear elliptic problems for which very efficient solvers are known to exist regardless of the space discretization.

An uncoupled limit model for a high-contrast problem in a thin multi-structure

We investigate a degenerating elliptic problem in a multi-structure \Omega_\varepsilon of \mathbb{R}^3 , in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, \Omega_\varepsilon consists of a fixed basis \Omega^- surmounted by a thin cylinder \Omega_\varepsilon^+ with height 1 and cross-section with a small diameter of order \varepsilon . Moreover, \Omega^+_\varepsilon contains a cylindrical core, always with height 1 and cross-section with diameter of order \varepsilon , with conductivity of order 1 , surrounded by a ring with conductivity of order \varepsilon^2 . Also \Omega^- has conductivity of order \varepsilon^2 . By assuming that the temperature is zero on the top and on the bottom of the boundary of \Omega_\varepsilon , while the flux is zero on the remaining part of the boundary, under a suitable choice of the source term we prove that the limit problem, as \varepsilon vanishes, boils down to two uncoupled problems: one in \Omega^- and one in \Omega^+_1 , and the problem in \Omega^+_1 is nonlocal. Moreover, a corrector result is obtained.

Optimization of two-beam laser cleavage of silicate glass

Subject of study. The values of technological parameters ensuring the effective two-beam laser cleavage of silicate glasses are determined in this study based on numerical modeling. Method. Multi-objective optimization of two-beam laser cleavage of glass plates is performed using the multi-objective genetic algorithm (MOGA) implemented in the DesignXplorer module of Ansys Workbench software. Main results. The temperatures and thermoelastic stresses are calculated using the finite element method in a quasistatic setting within the uncoupled thermoelasticity problem using the Ansys Parametric Design Language. A regression model of two-beam laser cutting of glass is obtained using a face-centered variant of the central composite design of the experiment. The machining speed, the power of a laser with a wavelength of 10.6 µm, the radius of the radiation spot of the beam with a wavelength of 10.6 µm, the power of a laser with a wavelength of 1.06 µm, and the radius of the radiation spot of the beam with a wavelength of 1.06 µm are used as the variable factors. The maximum temperatures and tensile stress values in the laser machining area are used as the responses. The regression model is tested on a test dataset. The obtained results indicate that the regression model is consistent with the finite element analysis data. The effect of the technological parameters of machining on the maximum values of temperature and tensile stress in the laser machining area is evaluated. We establish that the machining speed and the parameters of the laser beam with a wavelength of 10.6 µm have the biggest effect on the maximum temperatures, whereas all the variable factors have a significant effect on the maximum values of tensile stress. Two-beam laser cleavage of glass is optimized for two variants of the problem setting: either using the criterion of the maximum tensile stress or the criteria of the maximum tensile stress and machining speed. The parameters obtained after optimization and through finite element modeling were compared. The maximum relative error of the results obtained using the MOGA do not exceed 5% for the determination of maximum temperatures and 9% for the determination of the maximum values of thermoelastic stress in the area of the laser machining. Practical significance. The machining parameters that ensure significant increase in the productivity and reliability of two-beam laser cutting when applied in practice are determined using modeling.

On the Bingham Flow in a Thin Y-Like Shaped Structure

We consider the steady Bingham flow in a two-dimensional thin Y-like shaped structure, with no-slip boundary conditions and under the action of given external forces. After passage to the limit with respect to a small parameter related to the thickness of the domain, we obtain three uncoupled problems. Each of these problems describes an anisotropic flow, corresponding to a lower-dimensional “Bingham-like” constitutive law. These results are in accordance with the engineering models.

Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains.

We demonstrate how to apply the tensor-train format to solve the time-independent Schrödinger equation for quasi-one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fröhlich-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low-rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach that directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in the case of homogeneous systems, the tensor-train ranks of state vectors only marginally depend on the chain length, which results in a linear growth of the storage consumption. However, the central processing unit time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton-phonon coupling strength, although only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich-Holstein type Hamiltonians.

Preface

The 7th International Workshop on Numerical Modelling of High Temperature Superconductors (HTS 2020) was held online, hosted by Dr Kévin Berger (University of Lorraine, Nancy, France), on the 22nd and 23rd June 2021. In fact, the HTS 2020 workshop was expected to be held in Nancy in May 2020 but has been postponed due to COVID-19.The workshop received 95 abstracts submitted for presentation and was attended by 115 delegates from around the world. The conference programme covered the latest research and developments in modelling methods and tools to facilitate the accelerated development of superconducting science and technology – including modelling fundamentals, coupled and uncoupled multiphysics problems and applications – with an emphasis on high-temperature superconducting materials.Selected papers presented at HTS 2020 have been published separately in a Special Focus Issue of the journal Superconductor Science and Technology (SUST). This present volume in the Journal of Physics: Conference Series (JPCS) collects together the remainder of the (eight) contributed papers submitted to the HTS 2020 proceedings in JPCS.List of Editors, Scientific committees are available in the pdf

Boundary Value Problems in Thermo Viscoplasticity

In this work, we study two uncoupled quasistatic problems for thermo viscoplastic materials. In the model of the equation of generalised thermo viscoplasticity, both the elastic and the plastic rate of deformation depend on a parameter $theta $ which may be interpreted as the absolute temperature. The boundary conditions considered here as displacement-traction conditions as well as unilateral contact conditions. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem, reducing the isotherm problem to an ordinary differential equation in a Hilbert space.

Low-Depth Mechanisms for Quantum Optimization

One of the major application areas of interest for both near-term and fault-tolerant quantum computers is the optimization of classical objective functions. In this work, we develop intuitive constructions for a large class of these algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. This is connected to effects from wavefunction confinement, phase randomization, and shadow defects lurking in the objective far away from the ideal solution. As an example, we explore the surprising deficiency of many quantum methods in solving uncoupled spin problems and how this is both predictive of performance on some more complex systems while immediately suggesting simple resolutions. Further examination of canonical problems like the Hamming ramp or bush of implications show that entanglement can be strictly detrimental to performance results from the underlying mechanism of solution in approaches like QAOA. Kinetic energy and graph Laplacian perspectives provide new insights to common initialization and optimal solutions in QAOA as well as new methods for more effective layerwise training. Connections to classical methods of continuous extensions, homotopy methods, and iterated rounding suggest new directions for research in quantum optimization. Throughout, we unveil many pitfalls and mechanisms in quantum optimization using a physical perspective, which aim to spur the development of novel quantum optimization algorithms and refinements.

Uncoupled Quasistatic Problem of Thermoelasticity for a Two-Layer Hollow Thermally Sensitive Cylinder Under the Conditions of Convective Heat Exchange

We determine the unsteady temperature field depending on the radial coordinate and the thermoelastic state caused in a two-layer hollow thermally sensitive cylinder both by this field and the action of external force loads. The nonlinear boundary-value problem of heat conduction with discontinuous coefficients is reduced by the integro-interpolation method to the Cauchy problem for a system of ordinary differential equations, which is solved numerically. The stressed state is determined from the integral equations of the second kind and integral conditions obtained as a result of direct integration of the problem of quasistatic thermoelasticity in stresses. We study the influence of the temperature dependence of the thermal and mechanical characteristics of the layers of materials on the values and character of distributions of temperature and stresses caused by this dependence in a two-layer cylinder.