Nonlinear Boundary Conditions Research Articles

A coupled model of the fluid flow with nonlinear slip Tresca boundary condition

In this article, we consider the finite element discretization of the time-dependent Stokes equations written under the Boussinesq approximation, coupled with the convection-diffusion equation and provided with Tresca's Friction Law. Existence and uniqueness of a solution are established. To solve Stokes variational inequality, we suggest an alternating direction method of multiplier. We derive some a priori estimates and validate the obtained theoretical results by numerical simulations.

Nonlinear vibration of microbeams subjected to a uniform magnetic field and rested on nonlinear elastic foundation

Abstract This study investigates the nonlinear vibration motions of the Euler–Bernoulli microbeam on a nonlinear elastic foundation in a uniform magnetic field based on Modified Couple Stress Theory (MCST). The effect of size, foundation, and magnetic field on the nonlinear vibration motion of microbeam has been examined. The governing equations related to the nonlinear vibration motions of the microbeam are obtained by using Hamilton’s Principle, and the Multiple Time Scale Method was used to obtain the solutions for the governing equations. The linear natural frequencies of microbeam are presented in the table according to nonlinear parameters and boundary conditions. The linear and nonlinear natural frequency ratio graphs are shown. The present study results are also compared with previous work for validation. It is observed that length scale parameters and magnetic force have a more significant effect on the natural frequency of microbeams. It is seen that when the linear elastic foundation coefficient, the Pasternak foundation and the magnetic force effects increase, the ratio of nonlinear and linear natural frequency decreases.

An equivalent spring model for seam-clip connections of high-vertical standing seam metal cladding systems

The mechanical behavior of the seam-clip connections plays a pivotal role in the effective wind resistance design of the extensively used high vertical standing seam metal cladding system (SSMCS). An equivalent spring model is developed to represent the mechanical behavior of these connections to simplify the contact problems in the general finite element model, which is time-consuming due to the highly nonlinear contact boundary conditions. The developed model is composed of continuous horizontal and rotational springs as well as dispersed vertical rigid connections. A series of tensile experiments are first conducted to investigate the mechanical behavior of seam-clip connections to calibrate the spring parameters in the developed model. Subsequently, the equivalent spring model is validated by the structural response of a double-span sheet module (DSSM) using experimental investigation and finite element analyses. It is found that the shell finite element model incorporating the developed equivalent spring model can achieve an acceptable structural response while remarkably reducing memory requirements and computational time to about 0.3 % and 0.4 ‰, correspondingly, compared to the contact boundary condition analysis. It highlights the advancement in evaluating the structural response of high-vertical SSMCS in practical engineering applications using this developed model. Furthermore, the developed equivalent spring model has proved to be effective in predicting wind-induced seam-clip pullout failures according to the established failure criteria derived from the structural response of the system.

Global asymptotic stability for Gurtin-MacCamy’s population dynamics model

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker’s type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter.

Heat transfer characteristics of rectangular/triangular/convex-shaped fins filled with hybrid nanoparticles under dry and wet conditions

Numerous researchers have been interested in nanofluids because of their improved thermal characteristics and heat transmission capabilities. Recently, it has been possible to create a novel nanofluid with exceptional thermal properties by combining ternary nanoparticles of various shapes. In this respect, it is believed that the thickness of the fin will change with the length of the fin and that the impacts of thermal radiation, convection on a heat transfer mechanism, and internal heat production in a fin wetted with ternary hybrid nanofluid will depend on the length of the fin. As a result, several fin profiles, including triangular, convex, and rectangular, have been taken into consideration. This study also investigates the comparison of heat and thermal energy fluctuations in both wet and dry conditions. In order to examine the porous nature, Darcy's model is required. With the aid of the Maple computer program, the resultant nonlinear partial differential equation and boundary conditions are non-dimensionalized and numerically resolved using the implicit finite difference approach, the graphic explanation of fin efficiency, and transient thermal response for different values of the essential parameters. The investigation yielded the novel discovery that the effectiveness of the fins is enhanced by the presence of a ternary hybrid nanofluid. Three fins with varied shapes have been compared in both wet and dry circumstances. The study has discovered that triangular fins have a quicker rate of temperature decline, whereas rectangular fins have a greater efficiency. The investigation's results have a significant impact on improving heat transmission in industrial operations.

Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition

We consider the linear wave equation on with initial conditions and a nonlinear Neumann boundary condition at x = 0. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where f is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at x = 0 have enough regularity to solve the initial value problem with their own initial data.

On a new method for the state and output feedback exponential stabilization of a cascaded wave PDE-ODE system with nonlinear boundary condition

Abstract This study presents a novel approach for the state and output feedback exponential control of cascaded unstable ODE-wave equations with nonlinear boundary condition. The new approach can be also used to investigate the state and output feedback stabilization of other cascaded ODE-partial differential equation (PDE) systems with nonlinear boundary condition. The interconnection between the ODE and wave PDE with the nonlinear boundary is bi-directional at two points. First, an exponential controller is planned and the well-posed and exponential stability of the closed-loop system is derived. Next, an exponential observer is designed, whereby the exponential stability of the overall error system is shown. Then, an observer-based output feedback control that causes the overall real system to be exponentially stable is built. The existence and exponential stability of the solution to the overall closed-loop system are further deduced. Finally, simulation results are given to verify the effectiveness, feasibility and validity of the proposed technique.

Isogeometric boundary element formulation for cathodic protection of amphibious vehicles

In this study, we propose an isogeometric boundary element formulation for the cathodic protection (CP) modeling for amphibious vehicles which includes the treatment of non-linear boundary conditions. Half-space Green’s functions are utilized which leads to the discretization of the hull surface only. Non-Uniform Rational B-splines (NURBS) are employed to represent both geometry and field variables to obtain higher accuracy where discontinuous collocation points are utilized to make multi-patch implementation easier. Variable condensation technique is applied to manipulate system matrices in a such way that the solution is iterated only on the surfaces where non-linear boundary conditions are assigned which results in reduced computational cost. The computational performance of the formulation is assessed with different solvers for a representative hull geometry.

Numerical analysis for electromagnetic scattering with nonlinear boundary conditions

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic fields along the boundary. Based on time-dependent jump relations of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretizing the nonlinear system of boundary integral equations with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.

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.

Simulation of Nonlinear Free Surface Waves using a Fixed Grid Method

The simulation of nonlinear surface waves is of significant importance in safety studies of fluid containers and reservoirs. In this paper, nonlinear free surface flows are simulated using a fixed grid method which employs local exponential basis functions (EBFs). Assuming the flow to be inviscid and irrotational, the velocity potential Laplace’s equation is spatially discretized and solved by considering the nonlinear Bernoulli’s equation for irrotational flow as the boundary condition on the free surface. The nonlinear boundary conditions are imposed through a semi-implicit iterative time marching. The fixed grid feature of the method, based on a Lagrangian description of fluid flow, allows for retaining the portion of the discretization performed in the first time step for the bulk of the fluid. Thus, the portion which pertains to the regions near the moving boundaries is reprocessed during the time marching. The accuracy and efficiency of the existing solution is shown by simulating various problems such as liquid sloshing induced by external excitation of the reservoir or initial deformed shape of liquid, seiche phenomena and solitary wave propagation in a basin with constant depth or with a step, and comparing the results with those which are analytically available or those from available codes such as Abaqus. The proposed method shows far better stability of the results when compared with those of Abaqus which sometimes exhibit divergence after a relatively large number of time steps. For instance, in the propagation of the considered solitary wave in an infinite-like domain problem, the wave height is calculated by the maximum error of 1.6% and 9% using the present method and Abaqus, respectively.

Nonlinear gravity waves in the channel covered by broken ice

The two-dimensional nonlinear problem of a steady flow in a channel covered by broken ice with an arbitrary bottom topography including a semi-circular obstruction is considered. The mathematical model is based on the velocity potential theory accounting for nonlinear boundary conditions on the bottom of the channel and at the interface between the liquid and the layer of the broken ice, which are coupled through a numerical procedure. A mass loading model together with a viscous layer model is used to model the ice cover. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. The coupled problem is reduced to a system of integral equations in the unknown velocity magnitude at the interface, which is solved numerically using a collocation method. Case studies are conducted both for the subcritical and for the supercritical flow regimes in the channel. For subcritical flows, it is found that the ice cover allows for generating waves with amplitudes larger than those that may exist in the free surface case; the ice cover prevents the formation of a cusp and extends the solution to larger obstruction heights on the bottom. For supercritical flow regimes, the broken ice significantly affects the waveform of the soliton wave making it gentler. The viscosity factor of the model apparently governs the wave attenuation.

On a class of singular double phase problems with nonnegative weights whose sum can be zero

We study the existence, nonexistence and multiplicity of positive solutions for a class of one-dimensional double phase problems with nonnegative weights whose sum can be zero, singular nonlinearities and nonlinear boundary conditions. Such a weight condition is interesting in that it weakens the regularities of solutions, making it challenging to construct solution operators corresponding to the problems. We use a fixed point theorem to establish the existence and multiplicity of positive solutions according to the behaviors of the nonlinearity near 0 and ∞. In particular, we find intervals that guarantee at least three positive solutions for the problems with positive nonlinearities and at least two positive solutions for the problems with sign-changing nonlinearities.

From ductile damage to unilateral contact via a point-wise implicit discontinuity

Ductile damage models and cohesive laws incorporate the material plasticity entailing the growth of irrecoverable deformations even after complete failure. This unrealistic growth remains concealed until the unilateral effects arising from the crack closure emerge. We address this issue by proposing a new strategy to cope with the entire process of failure, from the very inception in the form of diffuse damage to the final stage, i.e. the emergence of sharp cracks. To this end, we introduce a new strain field, termed discontinuity strain, to the conventional additive strain decomposition to account for discontinuities in a continuous sense so that the standard principle of virtual work applies. We treat this strain field similar to a strong discontinuity, yet without introducing new kinematic variables and nonlinear boundary conditions. In this paper, we demonstrate the effectiveness of this new strategy at a simple ductile damage constitutive model. The model uses a scalar damage index to control the degradation process. The discontinuity strain field is injected into the strain decomposition if this damage index exceeds a certain threshold. The threshold corresponds to the limit at which the induced imperfections merge and form a discrete crack. With three-point bending tests under pure mode I and mixed-mode conditions, we demonstrate that this augmentation does not show the early crack closure artifact which is wrongly predicted by plastic damage formulations at load reversal. We also use the concrete damaged plasticity model provided in Abaqus commercial finite element program for our comparison. Lastly, a high-intensity low-cycle fatigue test demonstrates the unilateral effects resulting from the complete closure of the induced crack.

Observability and observer design for a class of hyperbolic PDEs with van de Pol type boundary conditions

This paper focuses on observability and observer design for nonlinear complex dynamical systems described by a class of hyperbolic partial differential equations (PDEs) with nonlinear van de Pol type boundary conditions. The systems exhibit complex dynamics due to its imbalance of energy flows. Both the exact observability and approximate observability of the systems with different boundary output measurements are shown by using methods of characteristics and boundary nonlinear reflections. Motivated by the approximate observability of the systems, a PDE state observer by using the boundary displacement measurement only is designed, and a sufficient condition to guarantee the estimation error systems to be exponentially stable is given. Theoretical results are proved rigorously, with some numerical simulations performed to validate the effect of the proposed observer.

Global existence of solutions to the chemotaxis system with logistic source under nonlinear Neumann boundary conditions

We consider classical solutions to the chemotaxis system with logistic source, au−μu2, under nonlinear Neumann boundary conditions ∂u∂ν=|u|p with p>1 in a smooth convex bounded domain Ω⊂Rn, where n≥2. This paper aims to show that if p<32, and μ>0, n=2, or μ is sufficiently large when n≥3, then the parabolic-elliptic chemotaxis system admits a unique nonnegative global-in-time classical solution that is bounded in Ω×(0,∞). The similar result is also true if p<32, n=2, and μ>0 or p<75, n=3, and μ is sufficiently large for the parabolic-parabolic chemotaxis system.

Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries

AbstractIn this work, we analyze the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating(N+1)\left(N+1)-dimensional thin domains (i.e., a family of bounded open sets fromRN+1{{\mathbb{R}}}^{N+1}, with corrugated bounder, which degenerates to an open bounded set inRN{{\mathbb{R}}}^{N}). We also allow monotone nonlinear boundary conditions on the rough border whose magnitude depends on the squeezing of the domain. According to the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition, we obtain different regimes establishing effective homogenized limits inNN-dimensional open bounded sets. In order to do that, we combine monotone operator analysis techniques and the unfolding method used to deal with asymptotic analysis and homogenization problems.

New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems

In this paper we present new embedding results for Musielak–Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of W^{1,mathcal {H}}(Omega ) into L^{mathcal {H}_*}(Omega ), where mathcal {H}_* is the Sobolev conjugate function of mathcal {H}, we present much stronger embeddings as known in the literature. Based on these results, we consider generalized double phase problems involving such new type of growth with Dirichlet and nonlinear boundary condition and prove appropriate boundedness results of corresponding weak solutions based on the De Giorgi iteration along with localization arguments.

The Role of Quadratic-Linearly Radiating Heat Source with Carreau Nanofluid and Exponential Space-Dependent Past a Cone and a Wedge: A Medical Engineering Application and Renewable Energy

A mathematical analysis for a steady, incompressible, laminar Carreau nanofluid flow over a porous cone and wedge is considered. The novelty of this research work is to scrutinize an exponential space-dependent Cattaneo–Christov heat flux and quadratic Rosseland approximation effects. Two-phase nanofluid model, i.e., Brownian motion, and thermophoretic effects are included. This study also employs non-linear thermal radiation and convective surface boundary conditions to investigate heat transport phenomena. The Carreau nanofluid boundary layer equations are derived using the standard boundary layer approximations. By applying the similarity transformations, the managing set of partial differential equations (PDEs) becomes a system of connected non-linear ordinary differential equations (ODEs). The resultant non-linear ODEs are solved numerically utilizing the numerical method, particularly the bvp4c function in MATLAB. The significances of the fluid motion are visualized by sketching several Carreau nanofluid flow’s relevant parameters using graphic outputs. The effects of a wide range of thermophysical variables on liquid properties including velocity, temperature, concentration, skin-friction, Nusselt number, and Sherwood number are investigated and discussed. This study’s findings are compared to previously announced results within a limited range, where strong validation was seen. The results demonstrate that the opposite mechanism is observed with repercussion of Weissenberg number (We) on velocity profile and concentration, the enhancement of We increases the momentum boundary layer while it reduces the concentration boundary layer. The outcomes could be applied to the cooling of equipment, electronics, and various industrial units.

A comparison of FEM results from the use of different governing equations in a galvanic cell part I: In the presence of a supporting electrolyte

The use of reduced-order models for simulating potential, current density, and species distributions in an electrolyte are attractive due to the high computational costs required to solve the intricate set of highly nonlinear partial differential equations and boundary conditions characteristic of electrochemistry problems. The loss in accuracy of the reduced-order models is assessed in this work by the comparison of the results obtained by these reduced-order models to those from the most general model (i.e., using the Nernst-Planck-Poisson as the governing equation). The models were performed using the governing equations available on COMSOL Multiphysics®. For electrolyte concentrations typical of most environments of interest in corrosion and electrochemistry, the reduced-order models saved substantial computational time without significant loss in accuracy.