Natural Boundary Conditions Research Articles

Nonlinear free vibration and flexural analysis of hyperelastic beam utilizing a meshless method based on radial basis function

This paper investigates nonlinear free and flexural analysis of hyperelastic beams. The constitutive relations of the hyperelastic beam were derived using the neo-Hookean strain energy function and Timoshenko beam theory. Also, the nonlinear governing equations and nonlinear natural boundary conditions were derived using Hamilton’s principle. The meshless collocation method based on the multiquadric radial basis function (MQ-RBF) was utilized to discretize the nonlinear governing equations. Also, the arc-length algorithm was used to solve the nonlinear system of equations. To validate the results of the meshless method, different boundary conditions (clamped–clamped, simply supported–simply supported, clamped–simply supported, and clamped-free) were examined, and the results obtained from the meshless method were compared with those of the finite element method in Abaqus finite element software. The results show that the maximum and minimum differences between meshless and finite element methods occur in clamped and free-boundary conditions, respectively. The results also show that the meshless method based on the MQ-RBF has good accuracy compared to the finite element method for bending and free vibration analysis of hyperelastic beams.

A meshless fragile points method for the solution of the monodomain model for cardiac electrophysiology simulation

Meshless methods for in silico modeling and simulation of cardiac electrophysiology are gaining more and more popularity. These methods do not require a mesh and are more suitable than the Finite Element Method (FEM) to simulate the activity of complex geometrical structures like the human heart. However, challenges such as numerical integration accuracy and time efficiency remain, which limit their applicability. Recently, the Fragile Points Method (FPM) has been introduced in the meshless methods family. It uses local, simple, polynomial, discontinuous functions to construct trial and test functions in the Galerkin weak form. This allows for accurate integration and improved efficiency while enabling the imposition of essential and natural boundary conditions as in FEM. In this work, we consider the application of FPM for cardiac electrophysiology simulation. We derive the cardiac monodomain model using the FPM formulation and we solve several benchmark problems in 2D and 3D. We show that FPM leads to solutions of accuracy and efficiency similar to FEM while alleviating the need for a mesh. Additionally, FPM demonstrates comparable convergence to FEM in the considered benchmarks.

Implementation of point interpolation meshless method for failure analysis of laminated composite beams

The present study develops computationally efficient meshless model for the first-ply failure analysis and predicts the weakest ply, based on various failure criteria via one-dimensional, higher-order beam theory. The effective property of lamina is obtained with the help of micromechanics based Eshelby’s-Mori-Tanaka model. The governing equation is obtained with the help of point interpolation method based on polynomial basis function. The effects of different essential and natural boundary conditions, aspect ratio, lamination angle and volume fraction on the first-ply failure load of composite beam is studied.

Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

This paper focuses on the so-called weighted inertia-dissipation-energy variational approach for the approximation of unsteady Leray–Hopf solutions of the incompressible Navier–Stokes system. Initiated in (Ortiz et al 2018 Nonlinearity 31 5664–82), this variational method is here extended to the case of non-Newtonian fluids with power-law index r ⩾ 11/5 in three space dimension and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimisation. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition . This provides the first theoretical explanation for the onset of such boundary conditions.

Mathematical modelling of salt ion transfer in the three-dimensional desalting channel of an electrodialysis apparatus

A new 3D model of 1:1 salt ion transfer in the desalting channel of an electrodialysis apparatus is presented and investigated in this paper. For the first time a three-dimensional mathematical model of salt ion transfer in the desalting channel taking into account the electroconvection based on the system of Nernst – Planck, Poisson and Navier – Stokes equations with the electric force and the natural boundary conditions is proposed. To solve the boundary value problem, the finite element method is used in the cross-platform numerical analysis software COMSOL Multiphysics in combination with the method of successive approximations, when the electrochemical and hydrodynamic parts of the problem are solved one by one on the current layer. In turn, the electrochemical and hydrodynamic parts of the problem are solved by Newton’s method. As a result of numerical analysis, the fundamental regularities of salt ion transfer in a three-dimensional channel, the emergence and development of electroconvective vortices, including the discovery of new three-dimensional spiral forms of salt ions, are established for the first time. It is shown that electroconvective vortices exist in the form of clusters, within which vortex bifurcations can occur. Thus, the currently existing simplified view of the structure of electroconvective vortices is clarified and developed.

Uniform convergence guarantees for the deep Ritz method for nonlinear problems

We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the p-Laplace equation or the Modica–Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of right-hand sides.

New approach for imposing the nonlinear floating potential boundary condition in the Poisson-continuity coupled system

PurposeThe purpose of this study is to develop a new approach for simulating the ion flow field with the floating conductors based on the existing inversion method.Design/methodology/approachThe space charge keeps charging the floating conductors until the dynamic balance is reached. The corresponding floating potential boundary condition is linearized as the combination of the equipotential constraints and natural boundary condition, which can be directly imposed in the inversion process.FindingsThe numerical results demonstrate that the modified inversion algorithm performs well in the ion flow field with the floating conductors and the floating potential converges within five steps’ iterations. The proposed approach is applied to investigate the effect of the grounded line to the floating conductors in the practical high voltage direct current (HVDC) transmission lines.Originality/valueThe modified inverse algorithm provides a general way to deal with the Kapzov hypothesis and the floating boundary condition simultaneously within one loop, which can be applied to the multiphysics systems with different kinds of irregular boundary conditions.

Configurational forces and geometrically exact formulation of sliding beams in non-material domains

The paper presents a geometrically exact, ALE finite element formulation for dynamic problems of nonlinear sliding beams. Previous researches on finite element methods for sliding beams have not taken into account the equation of configurational momentum and boundary condition of equilibrium of configuration forces at the sliding interfaces. In this paper, dual quaternions are used to represent the exact kinematics of rigid-section motion of the beam. Hamilton’s principle of variation of action is applied to derive the equations of mechanical and configurational momentum, and also boundary conditions of equilibrium of mechanical and configurational forces. The material and mesh motions of the cross-sections, material and mesh variations of motions, and also variation of the non-material domain due to sliding motion are correctly considered to get the complete set of governing equations and boundary conditions. In the finite element implementation, proper integration by parts is employed to weakly enforce natural boundary conditions of mechanical and configurational forces, and also to avoid terms of higher order spatial derivatives in the inertial forces. The configurational variables of internal nodes are associated with the arclength coordinates at the beam’s tips to avoid redundancy of the configurational momentum equation. These proposed strategies lead to robust ALE elements for geometrically exact beams. Numerical examples of increasing complexities are presented to validate correctness of the proposed approach.

Anatomy of Photoevaporation Base: Linking the Property of the Launched Wind to Irradiation Flux

Ultraviolet and X-rays from radiation sources disperse nearby gas clumps by driving winds due to heating associated with the photochemical processes. This dispersal process, photoevaporation, constrains the lifetimes of the parental bodies of stars and planets. To understand this process in a universal picture, we develop an analytical model that describes the fundamental physics in the vicinity of the wind-launching region. The model explicitly links the density and velocity of photoevaporative winds at the launch points to the radiation flux reaching the wind-launching base, using a jump condition. The model gives a natural boundary condition for the wind-emanating points. We compare the analytical model with the results of radiation hydrodynamic simulations, where a protoplanetary disk is irradiated by the stellar extreme-ultraviolet, and confirm good agreement of the base density and velocity, and radial profiles of mass-loss rates. We expect that our analytical model is applicable to other objects subject to photoevaporation not only by extreme-ultraviolet but by far-ultraviolet/X-rays with suitable modifications. Future self-consistent numerical studies can test the applicability.

Lagrangian mean curvature flow with boundary

We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck and the self-shrinking Clifford torus, and demonstrate long-time existence and convergence of the flow in the first instance and of the rescaled flow in the second.

Point forces in elasticity equation and their alternatives in multi dimensions

Deep dermal wounds induce skin contraction as a result of the traction forcing exerted by (myo)fibroblasts on their immediate environment. These (myo)fibroblasts are skin cells that are responsible for the regeneration of collagen that is necessary for the integrity of skin We consider several mathematical issues regarding models that simulate traction forces exerted by (myo)fibroblasts. Since the size of cells (e.g. (myo)fibroblasts) is much smaller than the size of the domain of computation, one often considers point forces, modelled by Dirac Delta distributions on boundary segments of cells to simulate the traction forces exerted by the skin cells. In the current paper, we treat the forces that are directed normal to the cell boundary and toward the cell centre. Since it can be shown that there exists no smooth solution, at least not in H1 for solutions to the governing momentum balance equation, we analyse the convergence and quality of approximation. Furthermore, the expected finite element problems that we get necessitate to scrutinize alternative model formulations, such as the use of smoothed Dirac Delta distributions, or the so-called smoothed particle approach as well as the so-called ‘hole’ approach where cellular forces are modelled through the use of (natural) boundary conditions. In this paper, we investigate and attempt to quantify the conditions for consistency between the various approaches. This has resulted into error analyses in the L2-norm of the numerical solution based on Galerkin principles that entail Lagrangian basis functions. The paper also addresses well-posedness in terms of existence and uniqueness. The current analysis has been performed for the linear steady-state (hence neglecting inertia and damping) momentum equations under the assumption of Hooke’s law.

Experimental Verification of Thermal Insulation in Timber Framed Walls.

Current environmental crisis calls for sustainable solutions in the building industry. One of the possible solutions is to incorporate timber-framed constructions into designs. Among other benefits, these structures are well established in many countries, originating in traditional building systems. This paper focuses on experimental timber-frame walls. Different wall assemblies vary in thermal insulation materials and their combinations. We investigated ten experimental wall structures that have been exposed to natural external boundary conditions since 2015. The emphasis was on their state in terms of visual deterioration, mass moisture content, and thermal conductivity coefficient. We detected several issues, including defects caused by inappropriate realization, causing local moisture increase. Material settlement in loose-fill thermal insulation was another issue. Concerning was a significant change in the thermal conductivity of wood fiber insulation, where the current value almost doubled in one case compared to the design value determined by the producer.

A new ϕ‐FEM approach for problems with natural boundary conditions

AbstractWe present a new finite element method, called ‐FEM, to solve numerically elliptic partial differential equations with natural (Neumann or Robin) boundary conditions using simple computational grids, not fitted to the boundary of the physical domain. The boundary data are taken into account using a level‐set function, which is a popular tool to deal with complicated or evolving domains. Our approach belongs to the family of fictitious domain methods (or immersed boundary methods) and is close to recent methods of CutFEM/XFEM type. Contrary to the latter, ‐FEM does not need any nonstandard numerical integration on cut mesh elements or on the actual boundary, while assuring the optimal convergence orders with finite elements of any degree and providing reasonably well conditioned discrete problems. In the first version of ‐FEM, only essential (Dirichlet) boundary conditions was considered. Here, to deal with natural boundary conditions, we introduce the gradient of the primary solution as an auxiliary variable. This is done only on the mesh cells cut by the boundary, so that the size of the numerical system is only slightly increased. We prove theoretically the optimal convergence of our scheme and a bound on the discrete problem conditioning, independent of the mesh cuts. The numerical experiments confirm these results.

Analytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory

In the present work, the equations of motion of a thin orthotropic nanoplate were obtained based on the new modified couple stress theory and the third-order shear deformation plate theory. The nanoplate was considered as a size-dependent orthotropic plate. The governing equations were derived using the dynamic version of Hamilton’s principle and natural boundary conditions were formulated. An analytical solution in the form of a double Fourier series was obtained for a simply supported rectangular nanoplate. The eigenvalue problem was set and solved. It was analytically shown that the displacements of the median surface points in the plane of the plate do not depend on the material length scale parameters in the same directions; these in-plane directional displacements depend on the material length scale parameter in the out-of-plane direction only. On the other hand, the out-of-plane directional displacement depends on the length scale parameter in the plane directions only. The cross-section rotation angles depend on all length scale parameters. It was shown that the size-dependent parameters only have a noticeable effect on the deformed state of the plate if their order is not less than the order (plate height)-1.

Hygrothermal effects on buckling behaviors of porous bi-directional functionally graded micro-/nanobeams using two-phase local/nonlocal strain gradient theory

Rapid development of technology puts forward higher performance requirements for advanced materials with multi-field coupling stability. The porous functionally graded materials, which is well developed with mature design, tools, can be customized according to specific application through nanostructures. In this work, the influences of hygro-thermo-mechanical coupling loadings on buckling behaviors of porous bi-directional functionally graded (2D FG) Timoshenko nanobeam is investigated through the numerical study. The governing equation and boundary conditions are derived on the basis of two phase local/nonlocal strain gradient theory employing the principle of virtual work, and a processing method for Clamped-Free (C–F) boundary condition is proposed. By employing generalized differential quadrature method (GDQM), the theoretical solution of buckling behavior of the Timoshenko nanobeam is obtained, with the implementation of comparison and convergence studies to demonstrate the accuracy and effectiveness. On the best of the authors’ knowledge, it is novel to investigate the buckling response of 2D FG Timoshenko nanobeam under hygrothermal environment using the two local/nonlocal strain gradient theory. Herein, the influence of nonlocal parameter, strain gradient theory, aspect ratio, axial gradient index, thickness gradient index, hygrothermal loadings and local volume fraction on the critical buckling load under various natural boundary conditions are acquired and discussed. The numerical results indicate that the present model can capture the buckling response of porous 2D FG Timoshenko nanobeams under hygrothermal environment, exhibiting consistent softening/hardening effects under all boundary conditions and modes.

A high-order accurate multidomain Legendre–Chebyshev spectral method for 2D Maxwell’s equations in inhomogeneous media with discontinuous waves

A multidomain Legendre–Chebyshev spectral method is developed for solving two dimensional Maxwell’s equations in inhomogeneous media with discontinuous electromagnetic waves. The method keeps spectral accuracy being not affected by the discontinuity of solutions. We construct a reasonable weak form which deals with the interface conditions similar to the natural boundary condition. Polynomial spaces of different degrees are used to approximate the electric and magnetic fields so that they can be decoupled in computation, and the optimal error estimate is obtained which improves the previous results. Numerical examples confirm the higher accuracy compared with other related method.

A non-standard class of variational problems of Herglotz type

<p style='text-indent:20px;'>In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function <inline-formula><tex-math id="M1">\begin{document}$ x $\end{document}</tex-math></inline-formula> and its derivative and an unknown functional <inline-formula><tex-math id="M2">\begin{document}$ z $\end{document}</tex-math></inline-formula>, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.</p>

Fusion-UDCGAN: Multifocus Image Fusion via a U-Type Densely Connected Generation Adversarial Network

Multifocus image fusion has attracted considerable attention because it can overcome the physical limitations of optical imaging equipment and fuse multiple images with different depths of the field into one full-clear image. However, most existing deep learning-based fusion methods concentrate on the segmentation of focus&#x2013;defocus regions, resulting in the loss of the details near the boundaries. To address the issue, this article proposes a novel generation adversarial network with dense connections (Fusion-UDCGAN) to fuse multifocus images. More specifically, the encoder and the decoder are first composed of dense modules with dense long connections to ensure the generated image&#x2019;s quality. The content and clarity loss based on the <inline-formula> <tex-math notation="LaTeX">$L1$ </tex-math></inline-formula> norm and the novel sum-modified-Laplacian (NSML) is further embedded to provide the fused images retaining more texture features. Considering that the previous dataset-making approaches may lose the relation between the overall structure and the information near the boundaries, a new dataset, which is uniformly distributed and can simulate natural focusing boundary conditions, is constructed for model training. Subjective and objective experimental results indicate that the proposed method significantly improves the sharpness, contrast, and detail richness compared to several state-of-the-art methods.

Vibrations of the underground pipeline in the vertical plane considering the rotational inertia and transversal shear

The subject of the study is the underground pipeline vibration in a vertical plane, considering the inertia of rotation and transverse shear. A system of differential equations of motion of an underground pipeline with natural boundary and initial conditions under spatial seismic loading is derived in the article. From the formulas obtained, it is necessary to determine the internal forces in the sections of the underground structure. Research methods are based on the variational equation of seismic loading of underground structures in a matrix form; methods of mathematical modeling and numerical methods for their solutions An algorithm and a program for the numerical solution of the resulting system of equations are compiled. Having determined the displacements, the internal forces in the sections of the underground pipeline are defined. The influence of the inertia of rotation of the section and the transverse shear on the values of displacements and force factors of the pipe is studied. Calculation results are given.

Soil-water strong coupled ISPH based on [formula omitted] formulation for large deformation problems

This paper is dedicated to the introduction of a strong coupled soil–water interaction formulation based on an incompressible smoothed particle hydrodynamics (ISPH) framework. The method is based on the u−w−p Biot’s formulation and adapted to a semi-implicit projection method for incompressibility condition of pore water and soil grains. The SPH Lagrangian particles move according to the soil velocity, while water variables are embedded into such soil particles. This allows to solve the pressure Poisson equation in a strong coupling way, in addition to enable to update the Darcy’s drag force implicitly. A simple boundary treatment on natural boundary conditions for soil particle is proposed to take into account both non-penetration and friction effects. The proposed method was verified and validated through a series of numerical tests resulting in good agreements with both theoretical and experimental results. Finally, we show the applicability of the proposed method in the famous Selborne experiment, a full-scale slope failure problem.