Search Results (14)

This study delves into the spatial characteristics of solutions for a specific class of evolution equations that incorporate biharmonic operators. The process begins with the construction of an energy function. Subsequently, by employing an integro-differential inequality method, it is deduced that this energy function satisfies an integro-differential inequality. Resolving this inequality enables us to establish an estimate for the spatial decay of the solution. Ultimately, the finding affirms that the spatial exponential decay is reminiscent of Saint-Venant-type estimates. Full article

Material defects resulting from manufacturing and processing can significantly affect material properties. Voids and dislocations are material defects considered in this study, in which a numerical solution of the 3D stress field of a dislocation line (infinite or finite) outside a cylindrical void (either parallel to the cylinder axis or not) is developed using the collocation point method. The collocation point method is utilized to solve ordinary differential equations, partial differential equations, differential-algebraic equations, and integral equations by enforcing the solution at a set of spatial collocation points. Analytical solutions for such three-dimensional (3D) problems, e.g., a dislocation line or segment near an internal void of any shape, were not found. Therefore, a numerical solution for this problem has been constructed in this paper. The numerical solution developed is verified using an existing two-dimensional analytical solution. The numerical results and the 2D analytical solution are in perfect agreement as long as the cylindrical void is sufficiently long and the Saint-Venant’s principle is followed. Full article

The present work investigates the spatial evolution characteristics of solutions to the Moore–Gibson–Thompson heat equation within a three-dimensional cylindrical geometry. By constructing an integral-differential inequality framework, we establish rigorous estimates demonstrating the exponential spatial decay of the solution as the axial distance from the inlet boundary increases without bound. This finding aligns with a generalized interpretation of the Saint-Venant principle, demonstrating its applicability under the present asymptotic conditions. The integral-differential inequality method proposed in this paper can also be used for the study of the Saint-Venant principle for other equations. Full article

Non-prismatic slender continua are the prototypical models of many structural elements used in engineering applications, such as wind turbine blades and towers. Unfortunately, closed-form expressions for stresses and strains in such continua are much more difficult to find than in prismatic ones, e.g., the de Saint-Venant’s cylinder, for which some analytical solutions are known. Starting from a suitable mechanical model of a tapered slender continuum with one dimension much larger than the other tapered two, a variational principle is exploited to derive the field equations, i.e., the set of partial differential equations and boundary conditions that govern its state of stress and strain. The obtained equations can be solved in closed form only in a few cases. Paradigmatic examples in which analytical solutions are obtainable in terms of stresses, strains, or related mechanical quantities of interest in engineering applications are presented and discussed. Full article

The effect of high contrast on the multiscale behaviour of elastic laminates is studied. Mathematical modelling in this area is of significant interest for a variety of modern applications, including but not limited to advanced sandwich structures and photovoltaic panels. As an example, the antiplane shear of a symmetric, three-layered plate is considered. The problem parameters expressing relative thickness, stiffness and density are assumed to be independent. The high contrast may generally support extra length and time scales corresponding to degenerated boundary layers and propagating long-wave low-frequency vibration modes. The main focus is on the relation between these two phenomena. The developed multiparametric approach demonstrates that those do not always appear simultaneously. The associated explicit estimates on contrast parameters are established. In addition, the recent asymptotic extension of the classical Saint-Venant’s principle is adapted for calculating the contribution of the degenerate boundary layer or long-wave low-frequency propagation mode. The peculiarity of the limiting absorption principle in application to layered media is also addressed. Full article

Non-point source pollution inflow is one of the main causes of water quality decline in urban river networks. In this paper, aiming at the problem of non-point source pollutant transport in river network, the lateral outflow term in the Saint-Venant equation is improved from the previous constant to the time-varying flow process, and a mathematical model considering the time-varying source and sink term is established. Based on the initial rainfall intensity, surface confluence and non-point source pollutant concentration, a method for calculating the time-varying lateral pollutant input of nodes and tributaries with linear increase and exponential decay in the initial rainfall period is proposed. Based on the principle of proximity, the watershed is divided into districts. According to the principle of elevation, the non-point source pollutants are allocated to the calculation nodes of adjacent rivers in a certain proportion and incorporated into the model calculation so as to improve the mathematical model of river network water quality and apply it to the simulation of river network water quality in Maozhou River Basin. Verified by the measured data, the NSE values of the improved model are 0.805 and 0.851, respectively, indicating that the model has reliable hydrodynamic and water quality simulation accuracy, indicating that the model can be applied to the calculation of non-point source pollutants in the basin. Based on the improved model, the variation of COD concentration in the Maozhou River of Shenzhen before and after optimized water replenishment was calculated, and the time variation and spatial distribution law of the sudden drop of water quality in the river network caused by the inflow of non-point source pollution in the initial rainfall runoff and the rapid recovery after optimized water replenishment were revealed. Full article

We develop a general approach for the computation of the effective properties of nonlinear viscoelastic composites. For this purpose, we employ the asymptotic homogenisation technique to decouple the equilibrium equation into a set of local problems. The theoretical framework is then specialised to the case of a strain energy density of the Saint-Venant type, with the second Piola–Kirchhoff stress tensor also featuring a memory contribution. Within this setting, we frame our mathematical model in the case of infinitesimal displacements and employ the correspondence principle which results from the use of the Laplace transform. In doing this, we obtain the classical cell problems in asymptotic homogenisation theory for linear viscoelastic composites and look for analytical solutions of the associated anti-plane cell problems for fibre-reinforced composites. Finally, we compute the effective coefficients by specifying different types of constitutive laws for the memory terms and compare our results with available data in the scientific literature. Full article

The antiplane shear of a semi-infinite multi-layered elastic strip with traction free faces and edges subject to prescribed stress is studied. A high contrast is assumed in the stiffnesses of two types of homogeneous isotropic layers. Explicit conditions on the edge load are derived, ensuring the decay of stress components at the distance of order strip thickness. One of these conditions corresponds to the canonical Saint-Venant’s principle, manifesting the self-equilibrium of the load. The rest of the decay conditions consider the presence of high contrast and are of an asymptotic nature, in contrast to the exact former condition. The number of asymptotic conditions is the same as that of soft layers. An example of the implementation of the proposed decay conditions for calculating the solution for the interior (outside of a boundary layer zone) domain of a three-layered semi-strip, considering geometric asymmetry, is presented. Full article

Racks retain debris in wastewater treatment plants and shield sensitive machinery in numerous engineering applications. In open-channel flows, racks impound the upstream water level by posing local obstacles in the flow. Based on experimental investigations, empirical approaches usually predict the flow resistance by relying on Bernoulli’s energy principle. Since this principle does not correctly consider downstream conditions such as submerged flows, we present a more accurate workflow to determine the flow resistance based on the Saint-Venant equations. We will demonstrate how the loss coefficient and the hydraulic head loss are determined more reliably without adding complexity to the engineering realisation. In contrast to relying solely on two cross-sections with Bernoulli’s energy principle, applying the Saint-Venant equations enables determining the flow depth profile and the flow velocity in the entire channel. This workflow additionally allows predicting the channel’s hydraulic capacity and freeboard in arbitrary applications. Full article

In this paper, we consider the Brinkman equations pipe flow, which includes the salinity and the temperature. Assuming that the fluid satisfies nonlinear boundary conditions at the finite end of the cylinder, using the symmetry of differential inequalities and the energy analysis methods, we establish the exponential decay estimates for homogeneous Brinkman equations. That is to prove that the solutions of the equation decay exponentially with the distance from the finite end of the cylinder. To make the estimate of decay explicit, the bound for the total energy is also derived. Full article

The spatial properties of solutions for a class of thermoelastic plate with biharmonic operator were studied. The energy method was used. We constructed an energy expression. A differential inequality which the energy expression was controlled by a second-order differential inequality is deduced. The $Phragm\acute{e}n$-$Lindel\ddot{o}f$ alternative results of the solutions were obtained by solving the inequality. These results show that the Saint-Venant principle is also valid for the hyperbolic–hyperbolic coupling equations. Our results can been seen as a version of symmetry in inequality for studying the $Phragm\acute{e}n$-$Lindel\ddot{o}f$ alternative results. Full article

The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this estimate, the uniqueness theorems were obtained for solutions of the first boundary value problem for third order equations in unlimited domains. The energy estimates are illustrated on two examples. Full article

This paper presents an extended Model Predictive Control scheme called Multi-objective Model Predictive Control (MOMPC) for real-time operation of a multi-reservoir system. The MOMPC approach incorporates the non-dominated sorting genetic algorithm II (NSGA-II), multi-criteria decision making (MCDM) and the receding horizon principle to solve a multi-objective reservoir operation problem in real time. In this study, a water system is simulated using the De Saint Venant equations and the structure flow equations. For solving multi-objective optimization, NSGA-II is used to find the Pareto-optimal solutions for the conflicting objectives and a control decision is made based on multiple criteria. Application is made to an existing reservoir system in the Sittaung river basin in Myanmar, where the optimal operation is required to compromise the three operational objectives. The control objectives are to minimize the storage deviations in the reservoirs, to minimize flood risks at a downstream vulnerable place and to maximize hydropower generation. After finding a set of candidate solutions, a couple of decision rules are used to access the overall performance of the system. In addition, the effect of the different decision-making methods is discussed. The results show that the MOMPC approach is applicable to support the decision-makers in real-time operation of a multi-reservoir system. Full article

In the last decades, microelectromechanical systems have been increasing their number of degrees of freedom and their structural complexity. Hence, most recently designed MEMSs have required higher mobility than in the past and higher structural strength and stability. In some applications, device thickness increased up to the order of tens (or hundred) of microns, which nowadays can be easily obtained by means of DRIE Bosch process. Unfortunately, scalloping introduces stress concentration regions in some parts of the structure. Stress concentration is a dangerous source of strength loss for the whole structure and for comb-drives actuators which may suffer from side pull-in. This paper presents an analytical approach to characterize stress concentrations in DRIE micro-machined MEMS. The method is based on the linear elasticity equations, the de Saint-Venant Principle, and the boundary value problem for the case of a torsional state of the beam. The results obtained by means of this theoretical method are then compared with those obtained by using two other methods: one based on finite difference discretization of the equations, and one based on finite element analysis (FEA). Finally, the new theoretical approach yields results which are in accordance with the known value of the stress concentration factor for asymptotically null radius notches. Full article