Linear Boundary Conditions Research Articles

General boundary conditions for a Boussinesq model with varying bathymetry

AbstractThis paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.

Numerical analysis for nonlinear wave equations with boundary conditions: Dirichlet, Acoustics and Impenetrability

In this article, we present an error estimation in the L2 norm referring to three wave models with variable coefficients, supplemented with initial and boundary conditions. The first two models are nonlinear wave equations with Dirichlet, Acoustics, and nonlinear dissipative impenetrability boundary conditions, while the third model is a linear wave equation with Dirichlet, Acoustics, and linear dissipative impenetrability boundary conditions. In the field of numerical analysis, we establish two key theorems for estimating errors to the semi-discrete and totally discrete problems associated with each model. Such theorems provide theoretical results on the convergence rate in both space and time. For conducting numerical simulations, we employ linear, quadratic, and cubic polynomial basis functions for the finite element spaces in the Galerkin method, in conjunction with the Crank-Nicolson method for time discretization. For each time step, we apply Newton's method to the resulting nonlinear problem. The numerical results are presented for all three models in order to corroborate with the theoretical convergence order obtained.

Walkin’ Robin: Walk on Stars with Robin Boundary Conditions

Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes the walk on stars (WoSt) algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt produces bounded-variance estimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as the walk on boundary algorithm. We also develop bidirectional and boundary value caching strategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation.

Wave drag and wave patterns by ships moving in a single-file formation

To minimize energy expenditure for each individual, animals adopt distinctive formations, such as fish schooling, “V” formation by flying birds, and single-file formation by waterfowls. The phenomenon of ducklings following their mothers in a single-file configuration has been revealed by the mechanisms of wave-riding and wave-passing. Drawing inspiration from this phenomenon, an investigation is undertaken on ships moving in a single-file formation. The objective is to quantify how much energy can be saved in different configurations. In this study, a three-dimensional boundary element method incorporating linear free-surface boundary conditions is used to obtain the wave drag and wave patterns. It is found that when constructive wave interference occurs in a two-ship formation, the wave resistance of the trailing ship increases and the leading ship experiences a decrease in its wave drags, especially when the two ships are in close proximity. Mutual benefit arises when destructive wave interference occurs between two ships. In addition, increasing the size of the trailing vessel facilitates the effect of wave-riding by the leading ship, but this effect becomes less pronounced as the speed increases. In a multi-ship formation configuration, changing the size of the leading ship will have a localized effect on the wave-passing, but the fleet will eventually tend to a dynamic equilibrium. When the position of the first trailing vessel is changed, there is similarly a localized effect on the wave-passing. Adjusting the first trailing ship to the position of the constructive wave interference is not favorable to reducing its own drag but enhances the wave-riding effect of its close follower. Finally, to achieve wave-passing, the trailing ship does not necessarily have to occupy an optimum position. This can still be accomplished if the trailing ship moves backward by an integer multiple of wavelength.

Linear Spectral Method for Simulating the Generation of Regular Waves by a Moving Bottom in a 3-dimensional Space

In this study, we introduce a linear spectral method capable of simulating wave generation and transformation caused by a moving bottom in a 3-dimensional space. The governing equations are linear dynamic free-surface boundary conditions and linear kinematic free-surface boundary conditions, which are solved in Fourier space. Solved velocity potential and free-surface displacement should satisfy continuity equation and kinematic bottom boundary condition. For numerical analysis, a 4th order Runge-Kutta method was utilized to analyze the time integral. The results obtained in Fourier space can be converted into velocity potential and free-surface displacement in a real space using inverse Fourier transform. Regular waves generated by various types of moving bottoms were simulated with the linear spectral method. Additionally, obliquely generated regular waves using specified bottom movements were simulated. The results obtained from the spectral method were compared to analytical solutions, showing good agreement between the two.

Multiscale modelling of composite laminates with voids through the direct FE2 method

A concurrent multiscale method, namely the direct FE2, is developed to analyse the mechanical response of fibre-reinforced composites with several void fractions in a nested manner, one at the macro and another one at the microscale. The FE2 is developed through the finite element (FE) mesh at the macro scale, wherein there is a representative volume element (RVE) at every Gauss point where the strains and stresses are evaluated. The RVEs, located at macroscopic FE mesh, with linear boundary conditions applied to the boundary nodes, are linked to macro-scale nodes via Multi-Point Constraints (MPCs). This procedure enables solving a single equilibrium problem, with boundary conditions and loadings at the macro scale and provides data for all defined RVEs. The methodology is applied to a composite laminate under transverse tension. Initially, the method is evaluated to compare the results from the FE2 with those from analyses where the entire domain is discretised (conventional FE full-scale modelling). The stresses obtained from the FE2 method match traditional FE analyses but utilise a significantly lower number of finite elements, resulting in substantial computational cost savings. It follows a parametric study of void distribution effects within the RVE. Afterwards, the stress distribution within the RVE is assessed and then failure of the composites is predicted at the macro scale through the stresses and strains concurrently calculated at the micro-scale.

Linear Displacement Boundary Condition Replacing Periodic Boundary Condition to Easily and Efficiently Homogenize the Elastic Properties of Composites

Periodic boundary condition is commonly used in the FE homogenization method for predicting the thermo-mechanical properties of composites. However, its existing numerical algorithms are generally inefficient. Linear displacement boundary condition is much easier and more efficient to implement for the Representative Volume Elements (RVEs) with conformal and nonconformal surface meshes. Aiming at replacing the periodic boundary condition by the linear displacement boundary condition to easily and efficiently predict the thermo-mechanical properties of composites, this paper compares the implementation efficiency and prediction accuracy of the periodic and linear displacement boundary conditions. Regarding the prediction of the elastic properties of composites, the balance of micro-energy and macro-energy of the RVEs imposed the periodic and linear displacement boundary conditions is checked, and the results demonstrate that these two boundary conditions guarantee the Hill lemma (i.e. the balance of micro-energy and macro-energy of the RVEs). Moreover, comparing with the experimental test and the multi-step micro-mechanical method, the effectiveness of the periodic and linear displacement boundary conditions to predict the elastic properties of composites is validated.

Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

Initial–boundary value problems for a linear diffusion–advection–reaction equation are considered, with general nonhomogeneous linear boundary conditions and general linear nonlocal boundary conditions. Analytical solutions are obtained using an embedding method. The solutions are expressed in terms of time‐varying functions that satisfy coupled linear Volterra integral equations of the first kind. A boundary element method is applied to numerically solve the integral equations. Three examples are given to demonstrate the accuracy of the numerical solutions when compared with the analytical solutions. The embedding method is applicable to problems with bounded and unbounded spatial domains.

Математичні моделі локального нагрівання елементів електронних пристроїв

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media sub-jected to internal and external local heat load, have been developed. In the case of nonlinear boundary-value problems, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equations and nonlinear boundary conditions were linearized, and as a result, linearized second-order differential equations with partial derivatives and a discontinu-ous right-hand side and partially linearized boundary conditions were obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the ap-proximation of the temperature according to one of the spatial coordinates on the boundary sur-faces of the inclusion was performed by piecewise constant functions. To solve linear bounda-ry-value problems, as well as obtained linearized boundary-value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in this envi-ronment were obtained. Numerical analysis of temperature behavior as a function of spatial co-ordinates for given values of geometric and thermophysical parameters was performed. The in-fluence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal and external heat load, software tools were developed, using which a ge-ometric image of the temperature distribution depending on the spatial coordinates was made.

Existence and multiplicity of solutions of Stieltjes differential equations via topological methods

In this work, we use techniques from Stieltjes calculus and fixed point index theory to show the existence and multiplicity of solution of a first order non-linear boundary value problem with linear boundary conditions that extend the periodic case. We also provide the Green’s function associated to the problem as well as an example of application.

SUTTERBY NANOFLUID FLOW WITH MICROORGANISMS AROUND A CURVED EXPANDING SURFACE THROUGH A POROUS MEDIUM: THERMAL DIFFUSION AND DIFFUSION THERMO IMPACTS

This study anticipates examining a slip bioconvective movement of a non-Newtonian Sutterby nanofluid (SF) layer with motile microorganisms, where the fluid layer flows over a curved stretching surface. The movement is taken across a permeable medium under the influence of thermal diffusion, diffusion thermo, an unchanged vertical magnetic field (MF), joule heating, thermal radiation, and chemical reactions. The mathematical construction comprises momentum, energy, nanoparticles volume fraction, and microorganism concentration equations along with linear slip velocity and applicable boundary conditions (BCs). The motivation of the problem concerns recent progress in curved electronics and microchip technology, which made a growing development of the remarkable weaknesses of traditional planar electronics, which concerns the importance of the current work. Furthermore, the implication of this work emerges from the participation of microorganisms in the flow over a curved surface and shares with the temperature, velocity, and nanoparticle system of equations. This prototype is widely applicable in some manufacturing and engineering mechanisms like conduits, sports balls, combustion, inflated broadcast, and flow-structure contact between hydrodynamics and aerodynamics. The configuration of nonlinear partial differential equations (PDEs) is converted into ordinary differential equations (ODEs) by consuming suitable similarity transformations. The resulting equations are numerically analyzed via the fourth-order Runge-Kutta (RK-4) in concurrence with the shooting technique. The graphical construction of the targeted distributions is analyzed to recognize the effects of the relevant material coefficients. As key outcomes, it is noted that the greater the curvature of the surface, the greater the temperature, velocity, microorganisms, and nanoparticle distributions. Correspondingly, the Soret and Dufour impacts are found to be the improvement coefficients of the heat and dampness of both nanoparticle and microorganism condensation. Additionally, heat transmission develops with almost all relevant parameters, which is a noteworthy finding that can benefit potential applications.

Analysis of a hybrid fractional coupled system of differential equations in $ n $-dimensional space with linear perturbation and nonlinear boundary conditions

<abstract><p>In this paper, we investigated $ n $-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in $ 3 $-dimensional space at the end of the paper.</p></abstract>

Solvability of nonlinear boundary value problems for non-sloping Timoshenko-type isotropic shells of zero principal curvature

We study the solvability of boundary value problem for a system of second order partial differential equations under boundary given conditions describing the equilibrium of elastic non-sloping isotropic inhomogeneous shells with free boundary in the framework of the Timoshenko shear model. The base of the study method are the integral representations for generalized motions involving arbitrary functions, which also involve arbitrary holomorphic functions. The arbitrary functions ate determined so that the generalized motions satisfy a linear system of equations and linear boundary conditions extracted from the original boundary value problem. The holomorphic functions are sought as Cauchy type integrals with real densities. The integral representations allow us to reduce the initial boundary value problem to a nonlinear operator equations for generalized motions in the Sobolev space. While studying the solvability of this operator equation, the most essential point is to invert it with respect to the linear part. As a result, the work is reduced to an equation, the solvability of which is established on the base of the contracting mapping principle.

Modeling Stiffness and Stress in Serpentine Flexures for Use in a Compliant Bone Plate

Abstract Serpentine flexures offer several advantages for use in linear motion mechanisms, including distributed compliance to reduce stress and increase range of motion. In this work, we develop an analytical model for predicting the moment, vertical deflection, and maximum stress experienced in serpentine flexures in response to an input vertical force. Two classes of serpentines are introduced and modeled with linear motion boundary conditions enforced. Finite element analysis demonstrates a mean model error of 0.86% for these metrics across many flexure topologies. Experimental testing is performed to validate the force–deflection response of three steel serpentine compliant mechanisms. The model is able to predict the experimental stiffness data with a mean error at yield of 5.3%, compared to 6.5% with finite element analysis. Large displacement simulations show the model could remain below 10% error for deflections 3–7 times beyond the mechanisms’ deflection at yield. Finally, the model’s utility is demonstrated in the design of a novel single-piece compliant fracture fixation plate that leverages serpentine flexures to deliver controlled axial motion for long bone secondary healing. Model-derived stress-equivalent flexures are compared in their transverse and torsional rigidity. The proposed model and specific findings can be leveraged to design linear motion mechanisms that incorporate serpentine flexures across a wide range of applications.

Study of the Stability Properties for a General Shape of Damped Euler–Bernoulli Beams under Linear Boundary Conditions

We study in this paper a general shape of damped Euler–Bernoulli beams with variable coefficients. Our main goal is to generalize several works already done on damped Euler–Bernoulli beams. We start by studying the spectral properties of a particular case of the system, and then we determine asymptotic expressions that generalize those obtained by other authors. At last, by adopting well-known techniques, we establish the Riesz basis property of the system in the general case, and the exponential stability of the system is obtained under certain conditions relating to the feedback coefficients and the sign of the internal damping on the interval studied of length 1 .

A multiscale approach in modeling of chemically reactive porous media

Chemical reactions occur in geotechnical, biological and synthetic porous media. Swelling and shrinkage phenomena can be observed in clays, shales and gels, when they are in close contact with water. In petroleum engineering, wellbore-stability problems are caused by swelling or shrinking of the wellbore. Thus, it is significant to model the Chemo-Hydro-Mechanical (CHM) processes in porous media that include different physical and geometrical heterogeneities. In this paper, a multiscale computational homogenization approach is developed for the analysis of CHM problems. In this manner, the first-order homogenization technique is adopted for the two-scale formulation. The heterogeneities are assumed in the microscale level and a homogeneous domain is considered for the macroscale level, where the constitutive behavior is derived from the microscopic level. The governing equations and constitutive equations of the CHM processes are presented in the coupled manner. The primary variables are taken as the displacement, pore pressure, and solute concentration fields. Moreover, the transient terms are included in the microscale level to increase the accuracy of computations. Consequently, a generalized form of Hill-Mandel principle of macro-homogeneity is defined between the two scales. Appropriate microscopic boundary conditions, i.e. the linear and periodic boundary conditions, are employed to satisfy the averaging constraints. Finally, the accuracy and efficiency of the proposed computational multiscale method are illustrated through several numerical examples.

Математичні моделі температурного поля у елементах електронних пристроїв із чужорідним включенням

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic media with semi-transparent foreign inclusions that are also subject to external heat load, have been developed. To solve the nonlinear boundary-value problem, a linearizing function was introduced, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients relative to the linearizing function and partially linearized boundary conditions was obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the approximation of the temperature according to one of the spatial coordinates on the boundary surfaces of the inclusion was performed by piecewise constant functions. To solve the obtained linear boundary value problem, the Hankel integral transformation method was used, as a result of which an analytical solution was obtained, which determines the introduced linearizing function. For the numerical analysis of temperature behavior and heat exchange processes caused by external heat load, software tools have been developed, which are used to create a geometric image of temperature distribution depending on spatial coordinates. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The developed models make it possible to analyze environments with foreign elements under external thermal loads regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual elements, but also the entire structure.

Existence of solutions of nonlinear systems subject to arbitrary linear non-local boundary conditions

In this paper, we obtain an explicit expression for the Green’s function of a certain type of systems of differential equations subject to non-local linear boundary conditions. In such boundary conditions, the dependence on certain parameters is considered. The idea of the study is to transform the given system into another first-order differential linear system together with the two-point boundary value conditions. To obtain the explicit expression of the Green’s function of the considered linear system with non-local boundary conditions, it is assumed that the Green’s function of the homogeneous problem, that is, when all the parameters involved in the non-local boundary conditions take the value zero, exists and is unique. In such a case, the homogeneous problem has a unique solution that is characterized by the corresponding Green’s function g. The expression of the Green’s function of the given system is obtained as the sum of the function g and a part that depends on the parameters involved in the boundary conditions and the expression of function g. The novelty of our work is that in the system to be studied, the unknown functions do not appear separated neither in the equations nor in the boundary conditions. The existence of solutions of nonlinear systems with linear non-local boundary conditions is also studied. We illustrate the obtained results in this paper with examples.

Boundary and Contact Conditions of Higher Order of Accuracy for Grid-Characteristic Schemes in Acoustic Problems

Seismic wave propagation through geological media is described by linear hyperbolic systems of equations. They correspond to acoustic, isotropic, and anisotropic linear elastic porous fluid-saturated models. They can be solved numerically by applying grid-characteristic schemes, which take into account propagation of solution discontinuities along characteristics. An important property of schemes used in practice is their high order of accuracy, due to which signal wavefronts can be clearly resolved. Previously, much attention was given to this property at interior points of the computational domain. In this paper, we study the order of a scheme up to the boundary of the domain inclusive. An approach is proposed whereby arbitrary linear boundary and contact conditions can be set up to high accuracy. The presentation is given for the system of one-dimensional acoustic equations with constant coefficients.

Non-linear dynamics of cantilevered circular cylindrical shells with thickness stretch, containing quiescent fluid with small-amplitude sloshing

Studies on nonlinear vibrations of circular cylindrical shells containing fluid are mostly limited to thin simply supported shells, such that no results regarding the behavior of thick cantilevered shells with shear and thickness deformations can be found in the literature. In this article, for the first time, thin and thick circular cylindrical shells with clamped-free boundary conditions are modeled according to the third-order shear deformation theory with thickness stretch. The contained liquid satisfies the Laplace equation and linearized boundary conditions are imposed at the free surface and at the contact with the shell. Using Lagrange equations, the governing differential equations of the system in terms of shell-fluid interaction are derived. In linear free vibration analysis, it was found that increasing the fluid level inside the shell and decreasing the shell thickness, cause a significant rise in the fluid free-surface wave elevation. This increase is such that limits the application of linear sloshing theory. It was observed that the fluid presence can change the nonlinear behavior from softening to hardening type and intensifies the shell thickness deformation. In addition, the presence of contained liquid reduces the circumferential dynamic contraction of the shell caused by large amplitude vibrations.