Stationary Equation Research Articles

On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space

We establish blow-up results for a system of semilinear hyperbolic inequalities in an exterior domain of the half-space. The considered system is investigated under an inhomogeneous Dirichlet-type boundary condition depending on both time and space variables. In certain cases, an optimal criterium of Fujita-type is derived. Our results yield naturally sharp nonexistence criteria for the corresponding stationary wave system and equation.

A Semi-Analytical Approach for the Linearized Vibration of Clamped Beams with the Effect of Static and Thermal Load

The geometric nonlinearity due to static and thermal load can significantly alter the vibration response of structures. This study presents a semi-analytical approach to illustrate the nonlinear vibration of clamped-clamped beams under static and thermal loads. The von Karman strain and Hamilton’s principle are employed to derive the nonlinear static equilibrium equation and nonlinear governing equation. The vibration equation’s coefficient is variable. The transfer-matrix method and local homogenization are used to solve the equation. The proposed method’s accuracy is validated by commercial software and literature. The numerical results indicate that uniform stress caused by thermal load only reduces the structural mode frequencies. The geometric nonlinearity of the structural static deformation affects both the mode frequencies and mode shapes. And the mode shapes cannot be approximated by harmonic functions. When the static deformation is significant, the structure’s local RMS response is substantially affected. The combined loads have a more significant impact on the acceleration response than the superposition of individual load effects.

Modeling Catalyst-Free Growth of III-V Nanowires: Empirical and Rigorous Approaches

Catalyst-free growth of III-V and III-nitride nanowires (NWs) by the self-induced nucleation mechanism or selective area growth (SAG) on different substrates, including Si, show great promise for monolithic integration of III-V optoelectronics with Si electronic platform. The morphological design of NW ensembles requires advanced growth modeling, which is much less developed for catalyst-free NWs compared to vapor-liquid-solid (VLS) NWs of the same materials. Herein, we present an empirical approach for modeling simultaneous axial and radial growths of untapered catalyst-free III-V NWs and compare it to the rigorous approach based on the stationary diffusion equations for different populations of group III adatoms. We study in detail the step flow occurring simultaneously on the NW sidewalls and top and derive the general laws governing the evolution of NW length and radius versus the growth parameters. The rigorous approach is reduced to the empirical equations in particular cases. A good correlation of the model with the data on the growth kinetics of SAG GaAs NWs and self-induced GaN NWs obtained by different epitaxy techniques is demonstrated. Overall, the developed theory provides a basis for the growth modeling of catalyst-free NWs and can be further extended to more complex NW morphologies.

Finite-Time Convergent Modified Davidenko Method for Dynamic Nonlinear Equations

Neurodynamic methods, especially zeroing neural networks (ZNN), are widely used for solving nonlinear equations. Being simpler than the ZNN, the Davidenko method is efficient for static nonlinear equations. However, it has been shown that there are large lagging errors when the Davidenko method is applied to dynamic nonlinear equations (DNEs). In this brief, we address this issue by proposing a modified Davidenko method via normalization, making it suitable for solving DNEs with unknown derivatives. The proposed method converges in finite time for DNEs, regardless of additive noise. Both theoretical guarantees and numerical examples of the proposed method are provided.

Improved overturning moment calculation method for large caliber artillery based on dynamics analysis

The effect of launch load on rotating parts is very complicated during firing of large caliber artillery. In the design of artillery firepower system, the traditional method to derive artillery overturning moment by taking the elevating part as the analysis object and using the static equilibrium equation cannot accurately reflect the real state of artillery firing. Therefore, taking the whole gun dynamics simulation as the analysis method and combining with the shooting test data, the force state of the elevating mechanism is obtained. The dynamic simulation results, the calculation results of the traditional formula, the measured results of the shooting test and the attitude change of the gun monitored by high-speed camera are compared and analyzed. The result shows that the turret race shaking is an important factor affecting the accuracy of design and simulation calculation. When the turret race shaking is considered, the simulation results are consistent with the experimental results, but inconsistent with the calculation results of the traditional design formula; The simulation results without considering the turret race shaking are consistent with the results of the traditional design formula, but they are different from the test results. Considering the influence of the angular acceleration of the rotating part given by the turret race shaking, the overturning moment formula is improved, and the force analysis formulas of the rotating part and the elevating part are carried out respectively and the calculation results of the improved formula are in good agreement with the test results. The research complements the calculation method of the overturning moment of the gun, which can be extended to the design of key components such as elevating mechanism and gun carriage.

Динаміка підресореної частини КТЗ і стійкість руху

The article investigates the influence of longitudinal-angular vibrations of the sprung mass of wheeled vehicles (WV) with nonlinear characteristics of the suspension system and kinematic parameters of move-ment along the curved sections of the road on the stability of WV. The main results of the work were obtained using the equations of kinetic statics, a mathematical model of relative nonlinear longitudinal-angular oscillations of the sprung mass (SM), based on the analysis of the quantitative and qualitative influence of the main force parameters of the suspension system (SS) and the determining parameters of longitudinal-angular oscillations of the SM on the level of interaction of the tires with the road coating. The critical speed of steady movement of the vehicle is determined as a function of the geometric and power parameters of the SS, the characteristics of the road surface, and the amplitude of longitudinal and angular oscillations. The numerical analysis of obtained ratios has shown that the critical values of steady motion speed for the skidding of the WV along the curved sections of the road are different: a) a larger value of the amplitude of SM oscillations corresponds to a smaller value of the critical speed of steady motion; b) it is significantly overestimated if the longitudinal and angular fluctuations of the sprung mass are not taken into account; c) the value is bigger for WV with the progressive force characteristic of the shock absorbers at small amplitudes of oscillations compared to the linear force characteristic (under the condition of equality of static deformation), and vice versa for the “large” ones; d) the value for the regressive force characteristic of elastic shock absorbers at small amplitudes of oscillations is smaller than for the linear characteristic of the shock absorbers, and the opposite for the “large” ones; e) the critical value of the speed of steady motion is bigger for the progressive characteristic of elastic shock absorbers at larger values of the static deformation of the SS and, on the contrary, it is smaller for the regressive one. The obtained results can be used for the designing or modernization of suspensions and well as a basis for development of a new software product for the controlled suspension systems. They coincide in the limiting case with those known in the scientific literature, which confirms their credibility. Keywords: Oscillations of sprung mass, amplitude, frequency, critical speed of steady motion.

Vibration characteristics of smart laminated carbon nanotube-reinforced composite cylindrical shells resting on elastic foundations with open circuit

This study analyzes the vibration of smart laminated carbon nanotube (CNT)-reinforced composite cylindrical shells integrated with piezoelectric materials and resting on elastic foundations. The effect of open circuit electrical boundary condition on vibration characteristics is investigated when a quadratic variation is considered for the electrical potential. Furthermore, the elastic foundations are included in the mathematical modeling to see their effects when the open circuit electrical boundary condition is applied. The first-order shear deformation shell theory and Maxwell’s static electricity equation are employed to derive the governing equations, and natural frequencies are calculated by solving an eigenvalue problem. Vibration characteristics with the open circuit electrical boundary condition are compared with those of closed (short) circuit electrical boundary condition for various mechanical boundary conditions and shell and piezoelectric geometrical parameters. The results demonstrate that the open circuit electrical boundary condition leads to higher frequencies than the closed circuit condition and it must be noticed in the design of smart composite structures with embedded or surface-bonded piezoelectric materials.

Deep learning soliton dynamics and complex potentials recognition for 1D and 2D [formula omitted]-symmetric saturable nonlinear Schrödinger equations

In this paper, we firstly extend the physics-informed neural networks (PINNs) to learn data-driven stationary and non-stationary solitons of 1D and 2D saturable nonlinear Schrödinger equations (SNLSEs) with two fundamental PT-symmetric Scarf-II and periodic potentials in optical fibers. Secondly, the data-driven inverse problems are studied for PT-symmetric potential functions discovery rather than just potential parameters in the 1D and 2D SNLSEs. Particularly, we propose a modified PINNs (mPINNs) scheme to identify directly the PT potential functions of the 1D and 2D SNLSEs by the solution data. And the inverse problems about 1D and 2D PT-symmetric potentials depending on propagation distance z are also investigated using mPINNs method. We also identify the potential functions by the PINNs applied to the stationary equation of the SNLSE. Furthermore, two network structures are compared under different parameter conditions such that the predicted PT potentials can achieve the similar high accuracy. These results illustrate that the established deep neural networks can be successfully used in 1D and 2D SNLSEs with high accuracies. Moreover, some main factors affecting neural networks performance are discussed in 1D and 2D PT Scarf-II and periodic potentials, including activation functions, structures of the networks, and sizes of the training data. In particular, twelve different nonlinear activation functions are in detail analyzed containing the periodic and non-periodic functions such that it is concluded that selecting activation functions according to the form of solution and equation usually can achieve better effect.

A hybrid high-order scheme for the stationary, incompressible magnetohydrodynamics equations

Abstract We propose and analyse a hybrid high-order scheme for the stationary incompressible magnetohydrodynamics equations. The scheme has an arbitrary order of accuracy and is applicable on generic polyhedral meshes. For sources that are small enough, we prove error estimates in energy norm for the velocity and magnetic field, and $L^2$-norm for the pressure; these estimates are fully robust with respect to small faces, and of optimal order with respect to the mesh size. Using compactness techniques, we also prove that the scheme converges to a solution of the continuous problem, irrespective of the source being small or large. Finally, we illustrate our theoretical results through 3D numerical tests on tetrahedral and Voronoi mesh families.

Stability and instability of radial standing waves to NLKG equation with an inverse-square potential

In this paper, we consider radial standing waves to a nonlinear Klein-Gordon equation with a repulsive inverse-square potential. It is known that there exist “radial” ground states to the stationary problem of the nonlinear Klein-Gordon equation. Here, “radial” ground states are non-trivial solutions having least energy among all non-trivial radial solutions to the stationary equation. We deal with the stability and instability of “radial” ground state standing waves.

Projection Methods for Neural Field Equations

Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields and derive a priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice depends on the choice of the projector. Using an elementary time stepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector and one from the time discretization. We give examples of concrete projection methods: two collocation schemes (piecewise linear and spectral collocation) and two Galerkin schemes (finite element and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests.

Thermal thin shell approximation towards finite element quench simulation

Superconducting electromagnets commonly exhibit thin layers with high aspect ratio such as insulation layers or turn-to-turn contacts. A finite element (FE) analysis of these devices can lead to unfavorable meshes in these thin layers, either because of a high number of degrees of freedom or mesh elements of poor quality which decrease the accuracy of the simulation results. To mitigate these issues when conducting a thermal FE analysis solving the heat equation, this work proposes to collapse thin volume layers into surfaces by using a thermal thin shell approximation (TSA). The proposed method uses one-dimensional Lagrange elements across the thickness of the thin layer and can handle a variety of interface conditions, multi-layered structures, heat sources, nonlinear material behavior or coupling to physics other than heat transfer. The efficiency of the proposed approximation is highlighted by comparison with a reference model with a conventionally meshed insulation for a model problem exhibiting a brick wall structure where a stationary heat equation is solved. The formulation is then verified against reference models with meshed insulation solving a transient heat equation for an insulated high-temperature superconductor pancake coil exhibiting a local defect which causes a thermal runaway. The benefit of using the model with the TSA is studied by analyzing pancake coils with different ratios of the insulation layer to the coated conductor thickness. It is shown that the smaller the ratio, the shorter the solution time and the lower the number of unknowns of the thin shell model when compared to the conventionally meshed insulation in order to reach the same numerical accuracy. The method is implemented in an open-source FE framework and a reference implementation for a simple model problem is shared alongside this paper.

Existence of solutions to steady Navier-Stokes equations via a minimax approach

Our objective in this paper is to develop and utilize a minimax principle for proving the existence of symmetric solutions for the stationary Navier-Stokes equations. Notwithstanding its application to symmetric solutions in this paper, our minimax principle is broad enough to capture other types of solutions provided the equation and the external force are compatible under a family of operations including but not limited to being invariant by compact groups. The subset of functions compatible under this family of operations is not required to be a linear subspace, and being a closed convex set suffices for our purpose.

On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability

On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability

Numerical investigations of a pivoted point absorber wave energy converter integrated with breakwater using CFD

Wave energy resources are enormous and widely distributed worldwide and evaluated as resources to replace fossil fuels. One of the methods to collect wave energy is using a point absorber device, which is a device designed to react sensitively to the movement of the water surface for harnessing wave energy. This paper is about the influence of a fixed breakwater on a Wavestar-shaped pivoted point absorber wave energy converter (WEC) using computational fluid dynamics (CFD). The fixed breakwater may generate a stationary wave and this phenomenon helps to enlarge the movement of the water surface. The numerical model based on CFD was validated against the available published data and verified to prove the accuracy of the numerical solution. It was found that the present numerical results have a good agreement with the available experimental results. A parametric study was conducted to understand the influence of the fixed breakwater on the WEC. The gap between the fixed breakwater and a buoy of the WEC has been changed according to wave period and the results depending on the gap have been compared. The results suggest a better hydrodynamic performance of the motion responses of the WEC, compared to those without the fixed breakwater. A stationary wave equation considering the gap was derived as a trigonometric function in order to investigate the correlation between a stationary wave and the motion response.

Bending information and dynamic stability of the laminated axisymmetric nanoplate on the linear and torsional elastic substrate based on zigzag theory

In the current work, for the first time, bending and vibration information of the axisymmetric thick annular nanoplate made of laminated composite layers is studied. The structure is placed on Winkler-Pasternak substrate with elastic and tortional coefficients. Initial stress as an external load is also considered in the composite structure. For modeling the mechanical deformation in layer-wise composite materials, zigzag theory that its functions themselves are continues in the interface of layers is presented. By coupled quasi-3D new refined theory (Q3D-NRT) and nonlocal strain gradient theory are used for obtaining the nonclassical governing equations a general boundary condition of the current composite nanoplate. Also, for obtaining the static and dynamic equations of the current nanostructure, Hamilton’s, and Minimum total potential energy principles are used, respectively. Via differential quadrature method with weighting factors and nth-order derivative, the nonclassical governing equations with the aid of size-dependent boundary conditions are solved. By comparing the current results with the outputs of previous reports, the accuracy of the results is shown.

Effect of crack characteristics on the vibration behavior of post-buckled functionally graded plates

This study investigates the free vibration analysis of a cracked post-buckled functionally graded plate under a uniaxial compressive load. The crack is assumed to be open-edged and modeled using a linear, rotational spring. The material is distributed through its thickness gradually. The nonlinear differential equations of motion are derived using the Mindlin plate theory for an initially imperfect plate. First, the nonlinear static differential equations are converted to algebraic equations using the differential quadrature method and solved by an arc-length strategy. The solution of the post-buckling equations gives the equilibrium state of the vibration. Next, the results were implemented into the motion equations to derive the vibrational differential equations. Then, the solution of the standard linearized eigenvalue problem, using the differential quadrature method, results in the plate's natural frequencies and mode shapes. The study compares the effects of various plate parameters on its vibrational behavior before and after buckling. It can be an excellent reference to consider the effect of crack characteristics on vibrations of the functionally graded plates before and after buckling. The presented results agree well with those available in the literature and those computed from the finite element method.

Unrestricted Douglas-Rachford algorithms for solving convex feasibility problems in Hilbert space

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted and extended the algorithm with the aim of solving CFPs and even more general problems, such as finding zeros of the sum of two maximally monotone operators. Many developments which implement various concepts concerning this algorithm have occurred during the last decade. We introduce an unrestricted DR algorithm, which provides a general framework for such concepts. Using unrestricted products of a finite number of strongly nonexpansive operators, we apply this framework to provide new iterative methods, where, inter alia, such operators may be interlaced between the operators used in the scheme of our unrestricted DR algorithm.

Parameter estimation with maximal updated densities

A recently developed measure-theoretic framework solves a stochastic inverse problem (SIP) for models where uncertainties in model output data are predominantly due to aleatoric (i.e., irreducible) uncertainties in model inputs (i.e., parameters). The subsequent inferential target is a distribution on parameters. Another type of inverse problem is to quantify uncertainties in estimates of “true” parameter values under the assumption that such uncertainties should be reduced as more data are incorporated into the problem, i.e., the uncertainty is considered epistemic. A major contribution of this work is the formulation and solution of such a parameter identification problem (PIP) within the measure-theoretic framework developed for the SIP. The approach is novel in that it utilizes a solution to a stochastic forward problem (SFP) to update an initial density only in the parameter directions informed by the model output data. In other words, this method performs “selective regularization” only in the parameter directions not informed by data. The solution is defined by a maximal updated density (MUD) point where the updated density defines the measure-theoretic solution to the PIP. Another significant contribution of this work is the full theory of existence and uniqueness of MUD points for linear maps with Gaussian distributions. Data-constructed Quantity of Interest (QoI) maps are also presented and analyzed for solving the PIP within this measure-theoretic framework as a means of reducing uncertainties in the MUD estimate. We conclude with a demonstration of the general applicability of the method on two problems involving either spatial or temporal data for estimating uncertain model parameters. The first problem utilizes spatial data from a stationary partial differential equation to produce a MUD estimate of an uncertain boundary condition. The second problem utilizes temporal data obtained from the state-of-the-art ADvanced CIRCulation (ADCIRC) model to obtain a MUD estimate of uncertain wind drag coefficients for a simulated extreme weather event near the Shinnecock Inlet located in the Outer Barrier of Long Island, NY, USA.

Modelling of Tendon-Driven Continuum Robot Based on Constraint Analysis and Pseudo-Rigid Body Model

Quasi-static models of tendon-driven continuum robots (TDCR) require consideration of both the kinematic and static conditions simultaneously. While the Pseudo-Rigid Body (PRB-3R) model has been demonstrated to be efficient, existing works ignore the mechanical effect of the tendons such as elongation. In addition, the static equilibrium equations for the partially constrained tendons have been expressed in different forms within the literature. This leads to inconsistent simulation results which have not been validated by experimental data when external loads are applied. Furthermore, the inverse problem for solving the required inputs for a prescribed end effector pose has not been studied for the PRB-3R model. In this work, we introduce a new modelling approach based on constraint analysis (CA) of a multi-body system and Lagrange multipliers to systematically derive all the relevant governing equations required for a planar TDCR. This method can include tendon mechanics and efficiently solve for the direct and inverse kinetostatic models with either forces or displacements as the actuation inputs. We validate the proposed CA method using numerical simulation of a benchmark model and experimental data.