Additional Boundary Conditions Research Articles

Outlier-free spline spaces for isogeometric discretizations of biharmonic and polyharmonic eigenvalue problems

We present outlier-free isogeometric Galerkin discretizations of eigenvalue problems related to the biharmonic and the polyharmonic operator in the univariate setting. These are Galerkin discretizations in certain spline subspaces that provide accurate approximations of all eigenfrequencies and eigenmodes, without the occurrence of spurious outliers. The theoretical cornerstone is the use of spline subspaces that are optimal in the sense of L2 Kolmogorov n-widths for special function classes related to the eigenvalue problems under investigation. This work is a continuation and extension of a similar study, recently presented for the Laplace operator, towards higher-order problems. As for the Laplace operator, the considered optimal spline spaces are identified by additional homogeneous boundary conditions and by specific sequences of break points. For fourth- and higher-order problems, however, optimal spline spaces are not known for all degrees and the optimal break points are not explicitly given. A careful analysis of the properties of optimal spaces allows us to determine spline subspaces of practical interest that are outlier-free. In particular, in the biharmonic case, we are able to construct such outlier-free spline spaces, for any odd and even degree, by considering break points that are uniform, or just a minor modification of the uniform ones. This improves upon the (numerical) results available in the literature. The theoretical findings are validated by a selection of numerical tests.

Peridynamics with strain gradient for modeling carbon nanotube under static and dynamic loading

This study couples peridynamic (PD) theory with strain gradient elasticity (SGE) theory to investigate the combined effect of PD and SGE length scale parameters on size effect. The SGE theory introduces a length scale parameter in the stress–strain relations to account for size effect. The solution to the classical form of SGE equation of motion requires two additional non-classical boundary conditions arising from the presence of fourth-order spatial derivatives. The wave dispersions level off as the wave number increases as observed in real materials. The PD theory allows for nonlocal interactions of a material point within its horizon which serves as the PD length scale parameter. The equation of motion emerges in the form of an integral equation and the internal forces are expressed through nonlocal interactions (bonds) between the material points within a continuous body. It is a reformulation of classical continuum mechanics which is free of spatial derivatives. The numerical results concern a previously considered carbon nanotube (CNT). Under quasi-static axial loading, the results indicate an increased strengthening effect along the length of the tube. Under dynamic loading, its longitudinal vibration response is captured through explicit time integration. The numerical predictions capture the reference solutions corresponding to the classical SGE equation.

The Casimir interaction in a nonlocal electrolyte slab

In this paper, the Casimir force per unit area across a nonlocal slab of an electrolyte between two semispaces of ordinary materials has been studied through the scattering approach. Because of the insufficiency of Maxwell’s boundary conditions, additional boundary conditions are used as well. The numerical calculations show that for polystyrene semispaces at small separations, ascending the concentration causes intensifying the Casimir pressure. However, it is illustrated that in the case of silver substrates the Casimir pressure has the same amount for two electrolyte concentrations and this intensification does not occur by increasing the concentration. At small separations for polystyrene semispaces, the Hamaker coefficient has two different magnitudes corresponding to 90 and 0.9[Formula: see text]mM concentrations, but for silver substrates the Hamaker coefficient starts from the same value for both concentrations.

Improving streamflow simulation in Dongting Lake Basin by coupling hydrological and hydrodynamic models and considering water yields in data-scarce areas

Study regionThe Dongting Lake Basin is a typical regional study in humid southern China with data-scarce areas. Study focusThis study improved the streamflow simulation by coupling hydrological and hydrodynamic models and considering water yields in data-scarce areas. We constructed a soil and water assessment tool (SWAT) hydrological model of the Dongting Lake Basin to simulate the streamflow in the data-scarce areas, which was further coupled into the MIKE21 hydrodynamic model as additional boundary conditions. New hydrological insightsThe results showed that the relative error of streamflow simulation was reduced from 24.64 % to 10.50 % in the coupled hydrological-hydraulic model over the singular hydrodynamic model, which also indirectly verified the results of streamflow simulation in the data-scarce area. Based on the coupled model, the annual water yields in the data-scarce areas were estimated to be 38.95 × 109 m3, representing 15.13 % of the yearly water yields in the basin. The water yields in the data-scarce areas showed a seasonal variation, which was concentrated from April to July. The monthly water balance error of the Dongting Lake Basin was significantly reduced (57.42 %) when considering the water yields in data-scarce areas. The model-coupling approach in this study can be applied to other data-scarce areas to improve streamflow simulation.

Extreme Refractive-, Diffractive- and Hybrid-Hyperchromats: Minimizing the Equivalent Abbe Number of a Two-Lens System

This work provides a comprehensive analysis of the maximum chromatic axial split of two-element hyperchromats, with the distance between the two lenses being a key variable. Purely refractive and diffractive systems are considered, as well as hybrid layouts combining refractive and diffractive elements. In order to achieve extreme chromatic axial splitting and accordingly a minimum equivalent Abbe number for lens combinations, a three-step procedure was used. In the first paraxial step, purely optical quantities such as focal lengths of the lenses, inter-lens distances and dispersion properties of the lenses were investigated. In the second step, which also takes place in the paraxial domain, additional geometric boundary conditions such as the radii, diameters and thicknesses of the lenses are taken into account. The results of this step serve as an input for the final optimization using optical design software, which derives practical solutions for minimum equivalent Abbe numbers with diffraction-limited image quality. As a significant result, the comparison with directly cemented lens doublets shows that the introduction of a distance between the elements allows for a much stronger chromatic decomposition for refractive, diffractive and also hybrid combinations. Quantitatively, the minimum equivalent Abbe number for refractive systems is reduced from 2.5 (without spacing) to 1.79 (with spacing). For hybrid combinations, a corresponding reduction from 0.4 to 0.29 is achieved.

Numerical simulation of wave propagation in ice-covered ocean environments based on the equivalent-source method

Accurate modeling of sound propagation in ice-covered ocean environments can help with interpreting discrepancies between predictions and experimental observations in the changing Arctic Ocean; this is advantageous for environmental conservation, resource exploration, and naval applications. Building on the recent development of the equivalent-source (ES) method (ESM), herein, an ESM-based sub-ice model (ESM-SUBICE) is presented for wave propagation in an ice-covered ocean acoustic environment. The presented model solves exact governing equations for acoustic–elastic propagation in an ice-covered waveguide by expressing the wave solution in terms of a field superposition produced by several sets of ESs. Their unknown amplitudes are solved by strictly enforcing additional ice-layer boundary conditions. ESM-SUBICE achieves high efficiency using a water–seabed Green's function to automatically satisfy the boundary conditions at this interface. By further dividing the ocean environment into layers, ESM-SUBICE is extended for more general situations including stratified sound-speed structures and seabed range dependencies. ESM-SUBICE is benchmarked against a finite-element model, and it is found to produce high-quality solutions with high efficiency. Transmission-loss predictions for elastic, fluid, and free-surface ice representations in different ocean environments are compared to examine the effect of ice elasticity on propagation and scattering. The results suggest that the fluid representation is adequate for deep-water environments where the seabed is soft and the surface duct effect is insignificant; otherwise, for accurate predictions, the ice elasticity should be considered.

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation.
 Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written.
 The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

Current trends and nonlinear effects in multilayered metamaterials—FF-1:IL01

We revisit a few aspects of the unusual features arising with metamaterials and metasurfaces in the context of their electromagnetic properties. We consider the properties of multilayered structures and indicate that a strong spatial dispersion as well as complex additional boundary conditions arise for a one-dimensional assembly of dielectric layers. With that basis, we analyse nonlinear properties of multilayered structures and illustrate the implications of spatial dispersion for nonlinear properties. Further on, we briefly review the implications of hyperbolic dispersion, readily available with multilayered structures, for nonlinear processes.

Бикубическая ленточная поверхность

A bicubic ribbon is a surface of constant width extended along the Ox-axis and formed by a set of rectangular bicubic portions connected to each other with smoothness C1 (continuity of gradient between portions) or C2 (continuity of curvature). Each portion is limited by cubic parabolas lying in vertical planes x=const, y=const. The article presents algorithms for calculating a bicubic band based on the use of boundary curve equations as the main boundary conditions. The «flat corners» conditions are accepted as additional boundary conditions. The proposed approach makes it possible to reduce the size of the characteristic matrix of a system of linear equations with respect to the coefficients included in the equations of bicubic portions. For example, the calculation of 16 coefficients of the equation of a bicubic portion passing through fixed boundary curves reduces to solving a system of four linear equations. Criteria for smooth joining of bicubic portions are formulated (in the form of theorems). Theorem 1 formulates and proves the continuity conditions for the gradient. Theorem 2 contains conditions for the continuity of curvature. Examples of calculation and visualization of C1 and C2- smooth ribbon surfaces, consisting of two or three bicubic portions, are presented.

Finite element analysis of screw fixation durability under multiple boundary and loading conditions for a custom pelvic implant.

Despite showing promising functional outcomes for pelvic reconstruction after sarcoma resection, custom-made pelvic implants continue to exhibit high complication rates due to fixation failures. Patient-specific finite element models have been utilized by researchers to evaluate implant durability. However, the effect of assumed boundary and loading conditions on failure analysis results of fixation screws remains unknown. In this study, the postoperative stress distributions in the fixation screws of a state-of-the-art custom-made pelvic implant were simulated, and the risk of failure was estimated under various combinations of two bone-implant interaction models (tied vs. frictional contact) and four load cases from level-ground walking and stair activities. The study found that the average weighted peak von Mises stress could increase by 22-fold when the bone-implant interactions were modeled with a frictional contact model instead of a tied model, and the likelihood of fatigue and pullout failure for each screw could change dramatically when different combinations of boundary and loading conditions were used. The inclusion of additional boundary and loading conditions led to a more reliable analysis of fixation durability. These findings demonstrated the importance of simulating multiple boundary conditions and load cases for comprehensive implant design evaluation using finite element analysis.

Modelling the effect of vibrations on the surface tension of a liquid droplet using meshless methods

Application of vibration impacts for purposeful influence on such processes as drop formation, melt bath formation and crystallization of welding bead allows to control heat and mass transfer in liquid, crystallization process and shape of bead in technological processes of welding. Impact of vibration influences on nature of motion of liquid in the drop, which is reflected in the change of value of surface tension coefficient, is considered in the article. The mathematical model of the liquid flow considering surface tension force in formalism of smoothed particles hydrodynamics method is offered. This method allows direct consideration of the vibration effect by introducing additional boundary conditions. Verification of developed mathematical model is conducted in comparison with in-situ experiments, in which dependence of surface tension coefficient value on amplitude of speed of vibration influences was determined. To determine surface tension coefficient two methods were implemented: pending drop method and stalagmometric method. The implemented model satisfactorily describes the effect of decreasing surface tension coefficient for water. A series of numerical experiments for determining the effect of vibration influences on the value of surface tension coefficient for 12X18H10T steel grade was carried out. It was found that at vibration with speed amplitude equal to 2.0 m/s the decrease of surface tension coefficient value by 30 % is observed. Decrease in surface tension coefficient should facilitate the realization of continuous flowing of metal from the wire, which may positively influence the formation of metal during wire surfacing. Thus, the proposed mathematical model can clearly simulate the effect of vibration effects on the value of the surface tension coefficient and will allow the effect of vibration effects in additive manufacturing to be investigated in the future.

Scattering solution to the problem of additional boundary conditions

Maxwell's boundary conditions (MBCs) were long known insufficient to determine the optical responses of spatially dispersive medium. Supplementing MBCs with additional boundary conditions (ABCs) has become a normal yet controversial practice. Here the problem of ABCs is solved by analyzing some subtle aspects of a physical surface. A generic theory is presented for handling the interaction of light with the surfaces of an arbitrary medium and applied to study the traditional problem of exciton polaritons. We show that ABCs can always be adjusted to fit the theory but they can by no means be construed as intrinsic surface characteristics, which are instead captured by a \textit{surface response function} (SRF). Unlike any ABCs, a SRF describes essentially non-local boundary effects. Methods for experimentally extracting the spatial profile of this function are proposed.

A projection method for Navier–Stokes equations with a boundary condition including the total pressure

We consider a projection method for time-dependent incompressible Navier-Stokes equations with a total pressure boundary condition. The projection method is one of the numerical calculation methods for incompressible viscous fluids often used in engineering. In general, the projection method needs additional boundary conditions to solve a pressure-Poisson equation, which does not appear in the original Navier-Stokes problem. On the other hand, many mechanisms generate flow by creating a pressure difference, such as water distribution systems and blood circulation. We propose a new additional boundary condition for the projection method with a Dirichlet-type pressure boundary condition and no tangent flow. We demonstrate stability for the scheme and establish error estimates for the velocity and pressure under suitable norms. A numerical experiment verifies the theoretical convergence results. Furthermore, the existence of a weak solution to the original Navier-Stokes problem is proved by using the stability.

Minlos–Faddeev Regularization of Zero-Range Interactions in the Three-Body Problem

To regularize the three-body problem, Minlos and Faddeev suggested a modification of zero-range model, which diminishes interaction at the triple-collision point. The analysis reveals that this regularization results in four alternatives depending on the regularization parameter $ \sigma $. Explicitly, Efimov or Thomas effects remain for $ \sigma < \sigma_c $, the additional boundary conditions of two types should be imposed at the triple-collision point for $ \sigma_c \le \sigma < \sigma_e $ and $ \sigma_e < \sigma < \sigma_r $, and the problem is regularized for $ \sigma \ge \sigma_r $. Critical values $ \sigma_c < \sigma_e < \sigma_r $ separating different alternatives are determined both for a two-component three-body system and for three identical bosons.

Thermodynamic framework for variance-based non-local constitutive models

The present work deals with the development of a framework dedicated to the construction of constitutive models with non-local internal variables. Such internal variables allow considering the impact of microstructural changes on the current state of a material point. Non-locality is introduced by considering, not only the spatial average, but also the spatial variance of an internal variable in constitutive relations. The proposed framework relies on continuum thermodynamics to construct the set of constitutive equations. Such a framework allows including some information regarding the spatial distribution of internal variables when constructing non-local models for thermomechanical applications. In contrast with gradient-type models, this strategy does not require additional equilibrium equations and boundary conditions. For the purpose of illustration, some numerical examples are presented. According to the numerical results, the proposed framework can be used to circumvent the difficulties associated with excessive spatial localization or to consider size effects.

Energy dissipation and shock isolation using novel metamaterials

Dissipating kinetic energy from shock and vibration is an urgent requirement for various applications in aerospace to mechanical engineering. This paper proposes a series of innovative metamaterials with planar and cylindrical patterns for elastic energy dissipation and shock isolation. The planar unit cell with two axes of reflectional symmetry mainly consists of two V-shaped regions, four identical right triangle regions, and narrow regions. The stereometric and cylindrical unit cells are obtained through rotating and convolving methods. The mechanics of energy dissipation and shock isolation are systematically investigated with finite element analysis (FEA), and experiments. Results show that various mechanical responses of unit cells are obtained through tailoring combined geometric parameters and additional boundary conditions, and the planar unit cell has stronger bistability than the cylindrical one. Then, the designed multilayers metamaterials exhibit considerable energy dissipation via the snap-through induced hysteric force-displacement behaviors, whose performances are influenced by geometric parameters and the number of layers. Lastly, the designed metamaterials could effectively suppress the acceleration responses via the snap-through behaviors induced by elastic instability. The shock response process and corresponding deformation mechanics are investigated experimentally and numerically. The designed metamaterials have potential in shock and vibration engineering.

A model for the electric field-driven flow and deformation of a drop or vesicle in strong electrolyte solutions

A model is constructed to describe the flow field and arbitrary deformation of a drop or vesicle that contains and is embedded in an electrolyte solution, where the flow and deformation are caused by an applied electric field. The applied field produces an electrokinetic flow, which is set up on the charge-up time scale $\tau _{*c}=\lambda _{*} a_{*}/D_{*}$ , where $\lambda _{*}$ is the Debye screening length, $a_{*}$ is the inclusion length scale and $D_{*}$ is an ion diffusion constant. The model is based on the Poisson–Nernst–Planck and Stokes equations. These are reduced or simplified by forming the limit of strong electrolytes, for which dissolved salts are completely ionised in solution, together with the limit of thin Debye layers. Debye layers of opposite polarity form on either side of the drop interface or vesicle membrane, together forming an electrical double layer. Two formulations of the model are given. One utilises an integral equation for the velocity field on the interface or membrane surface together with a pair of integral equations for the electrostatic potential on the outer faces of the double layer. The other utilises a form of the stress-balance boundary condition that incorporates the double layer structure into relations between the dependent variables on the layers’ outer faces. This constitutes an interfacial boundary condition that drives an otherwise unforced Stokes flow outside the double layer. For both formulations relations derived from the transport of ions in each Debye layer give additional boundary conditions for the potential and ion concentrations outside the double layer.

A novel frequency-domain integrated sensor for in-situ estimating unsaturated soil hydraulic conductivity

Unsaturated soil hydraulic conductivity (Ku) plays a significant role in agricultural, hydrological and ecological applications. However, the existing measurement technologies are failed to consecutively measure soil hydraulic conductivity in-situ, high cost or destructive to soil structure. Here we designed a novel frequency-domain integrated sensor that is combined with Richards’ equation to in-situ estimate Ku. The novel sensor has three perforated cylinder coaxial (PCC-1, PCC-2, PCC-3) probes and a stainless steel pin to measure soil water potentials at up (PCC-1), middle (PCC-2) and bottom (PCC-3) layers and soil water content at middle layer, respectively. After sensor calibration, drying experiments were conducted with sandy loam and clay loam. The Richards’ equation was used to in-situ estimate dynamics of Ku combining finite difference method with measurements of the novel sensor under certain assumptions, and thus neither additional parameters nor boundary condition was required. In order to assess the accuracy of the in-situ estimated Ku, the actual Ku were determined in laboratory using van Genuchten and Campbell hydraulic models as references. Then, a field experiment was conducted at an experimental farm to test the performance of the novel integrated sensor. The results showed that the in-situ estimated Ku in unsaturated range had a good agreement with the model calculated Ku in laboratory. The RMSEs (in log10) were 0.707 cm h−1 (van Genuchten) and 0.426 cm h−1 (Campbell) for sandy loam and 0.749 cm h−1 (van Genuchten) and 0.587 cm h−1 (Campbell) for clay loam. In near-saturated range, the in-situ estimated Ku were somewhat underestimated in comparison with the model calculated Ku. This may attribute to the limitation of accuracy of soil water potential sensor (TEROS 21) used for calibration, the insensitivity of drying method to the estimation of Ku or the effect of structural macropores or discontinuity of soil intrinsic hydraulic characteristics in near-saturated range. The overall trend of in-situ estimated Ku in the field is similar to that in laboratory. The consistent between the in-situ estimated and model calculated Ku in laboratory verified the availability and feasibility of the novel integrated sensor and the estimation of Ku in the field was achieved. In addition, the novel integrated sensor is low cost and reduces sensor-to-sensor uncertainties when multiple sensors were used, which has distinct advantages and potential application under field conditions.

The localized method of fundamental solutions for 2D and 3D inhomogeneous problems

In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multi-dimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme.

Applicability and efficacy of Galerkin-based approximation for solving the buckling and dynamics of nanobeams with higher-order boundary conditions

In this research effort, the general nonlocal elasticity theory is applied to an axially-loaded Euler-Bernoulli nanobeam with varying boundary conditions. In applying the general nonlocal elasticity theory, a modified version of Eringen's nonlocal elasticity theory, a sixth-order partial differential equation is obtained for the transverse governing equation of motion. Obtaining the sixth-order equation created the need for additional non-classical beam boundary conditions. These boundary conditions are determined using the weighted residual approach and are physically interpreted as higher-order force and moment terms. This study investigates the buckling and dynamic responses of nanobeams using a Galerkin-based approximation with different trial mode shapes, as it relates to the exact solution. As such, comparisons are made between using three distinct types guess functions in the approximation, namely, the exact unloaded classical mode shapes, exact unloaded Eringen nonlocal mode shapes, and exact unloaded general nonlocal mode shapes. Unloaded mode shapes refer to the mode shapes in which the applied axial load is neglected or set to zero. Additional studies are conducted focusing on direct comparison between Eringen nonlocal and general nonlocal beams for both equal and unequal nonlocal parameters. It is shown that selection of an appropriate guess function that satisfies all boundary conditions is necessary for the approximate solution. Additionally, the response of Eringen and general nonlocal nanobeams with equal nonlocal parameters and hinged-hinged boundary conditions is identical using both the exact and approximate solutions. Finally, it is demonstrated that the general nonlocal dynamic response follows a different trajectory than the Eringen nonlocal dynamic response as a function of the applied axial load. The findings of this investigation may be used by other researchers to simplify analysis procedures and to better understand the fundamental relationships existing in the size-dependent governing equations and boundary conditions for nanobeam-based systems.