Nonlinear Boundary Conditions Research Articles

A local meshless method for transient nonlinear problems: Preliminary investigation and application to phase-field models

Transient nonlinear problems play an important role in many engineering problems. Phase-field equations, including the well-known Allen-Cahn and Cahn-Hilliard equations, fall in this category, and have applications in cutting-edge technologies such as modeling the diffusion of lithium (Li) ions in two-phase electrode particles of Li-ion batteries. In this paper, a local meshless method for solving this category of partial differential equations (PDEs) is proposed. The Newton-Kantorovich scheme is employed to transform the nonlinear PDEs to an iterative series of linear ones which can be solved with the proposed method. The accuracy and performance of the method are examined in various linear and nonlinear problems, such as Laplace equation, three dimensional elasticity as well as some abstract mathematical equations with linear or nonlinear boundary conditions. The main focus of the work is on applying the proposed method in solution of the phase-field equations, including the Allen-Cahn and Cahn-Hilliard equations. In addition to homogeneous Neumann boundary condition which has been widely examined in the literature, we also employ a practical nonlinear, inhomogeneous Neumann boundary condition formulation specialized for modeling the diffusion of lithium ions in electrode particles of Li-ion batteries. The generalized- α method is used for time integration of diffusion-type equations to overcome the intrinsic stiffness of the phase-field equations. It is shown that the method is capable of capturing the main features of the phase-field models i.e. phase separation, coarsening and energy decay in closed systems.

Non-intrusive estimation of model error and discrepancy in dynamics models

Errors in formulating the system physics model, due to inability to rigorously account for nonlinear behavior, boundary conditions, and multi-component and multi-physics interactions result in discrepancies between model predictions of system responses and the corresponding experimentally measured values. In our previous work, we have developed a Bayesian state estimation-based framework for the estimation of these model errors, and the subsequent prediction of model discrepancies at unmeasured locations, for untested inputs, and for untested configurations. In this approach, we represent model errors as external inputs in the dynamic system model, and estimate them using Bayesian state estimation. However, this approach is intrusive in the sense that it requires access to the governing system of equations, the ability to modify the system model by introducing an external system input, and the ability to represent system inputs and responses as probabilistic quantities. In this paper, we develop semi-intrusive and non-intrusive approaches to extend the model error estimation to black-box models. The semi-intrusive method does not need access to the governing equations, but requires the ability to interrupt the simulation at any time instant, update the system responses at that moment, and resume the simulation using the updated value of system responses. For situations where this is not permitted, we propose a fully non-intrusive approach where, a surrogate model is constructed and used in the Bayesian state estimation procedure. The proposed approaches are illustrated with three examples: structural deformation of a Timoshenko beam, heat transfer across a rigid bar, and deformation of a Timoshenko beam subjected to heating.

Existence of solutions for a nonlinear problem at resonance

Abstract In this work, we are interested at the existence of nontrivial solutions for a nonlinear elliptic problem with resonance part and nonlinear boundary conditions. Our approach is variational and is based on the well-known Landesman-Laser-type conditions.

USING THE MEASURE OF NONCOMPACTNESS TO STUDY A NONLINEAR IMPULSIVE CAUCHY PROBLEM WITH TWO DIFFERENT KINDS OF DELAY

We apply the noncompactness measure by Mawhin to establish sufficient conditions for the existence and uniqueness (EU) of solution to a nonlinear Cauchy problem. The aforesaid problem is investigated under impulsive and nonlinear integral boundary conditions. Further, the problem under consideration contains Caputo-type fractional-order derivative while two different kinds of delay are also involved. After establishing the existence results, we derive some adequate results for the stability analysis of Hyers–Ulam (HU) type. Here, it should be kept in mind that the proposed tools can reduce the strong compact condition by weaker compactness. For justification, we provide some examples also.

A Nonfield Analytical Method for Solving Some Nonlinear Problems in Heat Transfer

Abstract This paper presents an extension of the nonfield analytical method—known as the method of Kulish—to some nonlinear problems in heat transfer. In view of the fact that solving nonlinear problems is very complicated in general, the extension of the method is presented in the form of several important illustrative examples. Two classes of problems are considered: first are the problems, in which the heat equation contains nonlinear terms, while the second type of problems includes some problems with nonlinear boundary conditions. From the practical viewpoint, the case considering asymptotic solutions is of the greatest interest: it is shown that, for complex heat transfer problems, where applications of the nonfield method are practically impossible due to a large volume of necessary computations, it is still possible to analyze the solution behavior and automatically determine similarity criteria for the limiting values of the parameters. Wherever possible the obtained solutions are compared with known solutions obtained by other methods. The practical advantages of the nonfield method over other analytical methods are emphasized in each case.

Differential equations involving homeomorphism with nonlinear boundary conditions

We establish existence results for differential equations of the form: where is continuous and is a homeomorphism, with nonlinear boundary conditions: where are continuous and are continuous functionals. Our methods of proofs are based on the extension of Mawhin's continuation theorem for quasi‐linear operators.

Multiplicity results for p(x)-biharmonic equations with nonlinear boundary conditions

In this paper, we are interested in the existence of multiple weak solutions of the following fourth-order nonlinear elliptic problem with a -biharmonic operator where is a bounded domain, is the operator of fourth order called the -biharmonic operator, and , are non-negative weight functions with compact support in Ω. Our analysis mainly relies on variational arguments and some recent theory on the generalized Lebesgue–Sobolev spaces and .

Boundary value problems for second-order causal differential equations

This paper focuses on second-order differential equations involving causal operators with nonlinear two-point boundary conditions. By applying the monotone iterative technique in the presence of upper and lower solutions, with a new comparison theorem, we obtain the existence of extremal solutions. This is an extension of classical theory of second-order differential equations. Finally, we present two examples to show the usefulness of our results.

Unified primal-dual active set method for dynamic frictional contact problems

In this paper, we propose a semi-smooth Newton method and a primal-dual active set strategy to solve dynamical contact problems with friction. The conditions of contact with Coulomb’s friction can be formulated in the form of a fixed point problem related to a quasi-optimization one thanks to the semi-smooth Newton method. This method is based on the use of the primal-dual active set (PDAS) strategy. The main idea here is to find the correct subset mathcal{A} of nodes that are in contact (active) opposed to those which are not in contact (inactive). For each case, the nonlinear boundary condition is replaced by a suitable linear one. Numerical experiments on both hyper-elastic problems and rigid granular materials are presented to show the efficiency of the proposed method.

Simulation of water entry into shallow water based on boundary element method

The problem of a two-dimensional wedge entering into shallow water is investigated with the boundary element method based on the potential flow theory, and fully nonlinear boundary conditions in time domain are imposed. A stretched coordinate system is used at the initial stage of water entry to avoid the numerical problems originated from extremely small wetted surface. An auxiliary function is used to calculate the pressure on the body surface. The free surface is updated through the fourth order Runge-Kutta scheme. Particularly, a jet cut model is used to avoid the numerical difficulties when the thin and high jet forms. Detailed results through the free surface, pressure distribution and hydrodynamic load at constant speed and prescribed time-varying speed are provided, the comparison with the results from experiments, Wagner theory, semi-analytic method are made and effects of water depth are investigated in detail. It is found that the bottom of the shallow water domain would exert a noticeable effect on the pressure distribution and force of the wedge, especially when the tip of wedge is close to the domain bottom.

Non-isothermal non-Newtonian flow problem with heat convection and Tresca's friction law

We consider an incompressible non-isothermal fluid flow with non-linear slip boundary conditions governed by Tresca's friction law. We assume that the stress tensor is given as where θ is the temperature, π is the pressure, u is the velocity and is the strain rate tensor of the fluid while p is a real parameter. The problem is thus given by the p-Laplacian Stokes system with subdifferential type boundary conditions coupled to a elliptic equation describing the heat conduction in the fluid. We establish first an existence result for a family of approximate coupled problems where the coupling term in the heat equation is replaced by a bounded one depending on a parameter , by using a fixed point technique. Then we pass to the limit as δ tends to zero and we prove the existence of a solution to our original coupled problem in Banach spaces depending on p for any .

Extension of lower and upper solutions approach for generalized nonlinear fractional boundary value problems

Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results.

A three-step stabilized algorithm for the Navier-Stokes type variational inequality

In the light of two-grid finite element discretization, this paper is concerned with a three-step stabilized algorithm for the Navier-Stokes equations with nonlinear slip boundary conditions of friction type. The proposed algorithm proceeds three steps: the first step solves a stabilized nonlinear variational inequality problem on a coarse grid, the second step solves a stabilized linear variational inequality problem based on Newton iteration on a fine grid, and the third step solves the same stabilized linear problem with only different right-hand sides on a fine grid. We analyze the stability of the presented algorithm, and derive error estimate of the approximate solutions in L2 norms of the velocity gradient and pressure. Finally, we demonstrate the effectiveness of the present algorithm by two numerical tests.

Numerical Investigation on Higher-Order Harmonic Waves Induced by a Submerged Inclined Plate

In this paper, a two-dimensional time-domain numerical flume has been established to model and investigate nonlinear interactions between nonlinear surface waves and a submerged inclined thin plate. The model solves the Laplace equation and the fully nonlinear free surface boundary conditions within the framework of potential flow theory based on the high-order boundary element method. The mixed Euler–Lagrangian method is applied to update the water surface at each time step, and the fourth-order Runge–Kutta method for time stepping. A so-called four-point method was employed to separate the second-order harmonic free and bounded wave that has the same wave frequency but different wave celerity in front of and behind the submerged plate. It is found that the amplitude of the second-order harmonic free wave increases with the inclination angle of the submerged plate, and the level/amount of the increase is larger for a larger wave steepness. In addition, the amplitudes of both the second-order reflected and transmitted waves are found to increase with the wave steepness, and their empirical relationships are derived for potential use in practical engineering.

Computational modeling of capillary perfusion and gas exchange in alveolar tissue

Gas exchange is an essential function of the respiratory system that couples fundamentally with perfusion in respiratory alveoli. Current mathematical formulations and computational models of these two phenomena rely on one-dimensional approximations that neglect the intricate volumetric microstructure of alveolar structures. In this work, we introduce a coupled three-dimensional computational model of pulmonary capillary perfusion and gas exchange that conforms to alveolar morphology. To this end, we derive non-linear partial differential equations and boundary conditions from physical principles that govern the behavior of blood and gases in arbitrary alveolar domains. We numerically solve the resulting formulation by proposing and implementing a non-linear finite-element scheme. Further, we carry out several numerical experiments to validate our model against one-dimensional simulations and demonstrate its applicability to morphologically-inspired geometries. Numerical simulations show that our model predicts blood pressure drops and blood velocities expected in the pulmonary capillaries. Moreover, we replicate partial pressure dynamics of oxygen and carbon dioxide reported in previous studies. This overall behavior is also observed in three-dimensional alveolar geometries, providing more detail associated with the spatial distribution of fields of interest and the influence of the shape of the domain. We envision that this model opens the door for enhanced in silico studies of gas exchange and perfusion on realistic geometries, coupled models of respiratory mechanics and gas exchange, and multi-scale analysis of lung function; furthering our understanding of lung physiology and pathology. Codes are available at https://github.com/comp-medicine-uc/num-model-alv-perfusion-gas-exchange.

Математическое моделирование динамики тепловых ударных волн в нелинейных локально-неравновесных средах

We performed a mathematical simulation of heat transfer in a local non-equilibrium medium whose transfer characteristics are functions of the temperature distribution. A homogeneous polynomial of arbitrary degree represents nonlinearities in thermal conductivity and thermal diffusivity. The mathematical model consists of a hyperbolic nonlinear heat transfer wave equation, initial conditions and nonlinear boundary conditions of the second and first kind. To solve this problem, we used a conservative homogeneous finite-difference scheme along the upper time grid line (implicitly). We then used the tridiagonal matrix algorithm of the second order in the spatial variable and of the first order in time to solve the resulting system of linearised algebraic equations. A periodic series of rectangular temperature or heat flux pulses form the boundary conditions of the first and second kind. Computation results reveal ultimate propagation rates of temperature and heat fronts featuring pronounced first-kind discontinuities with attenuating magnitudes. As the process unfolds, the initial pulses heat the region between the boundary and the heat wave front, while the subsequent pulses traverse this region at a higher velocity due to thermal diffusivity being a function of temperature, their fronts "catching up" with the previous fronts, increasing the discontinuity magnitude at the initial pulse front, that is, forming a thermal shock wave front similar to that of a shock wave in gas dynamics. We obtained such thermal shock waves for boundary conditions of both the first and the second kind. We also analysed kinematic and dynamic characteristics of thermal waves

General decay for weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions

In this paper, we consider the general decay of solutions for the weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions. By using suitable energy and Lyapunov functionals, we prove the general decay for the energy, which depends on the behavior of both σ and k.

Blow‐up of solutions of wave equation with a nonlinear boundary condition and interior focusing source of variable order of growth

In this paper, being investigated an initial‐boundary value problem for a one‐dimensional wave equation with a nonlinear source of variable order and nonlinear dissipation at the boundary. The existence of a local solution of the problem under consideration is proved. Then the question of the absence of global solutions is investigated. Depending on the relationship between the order of growth of the nonlinear source and the nonlinear boundary dissipation, different results are obtained on the blow‐up of weak solutions in a finite time interval.

Strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape

We characterize the homogenization limit of the solution of a Poisson equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a Holder nonlinear \(\sigma(u)\). We consider the case in which the diameter of the perforations (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a "strange" nonlinear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on \(L^2(0,T)\), independently of the regularity of \(\sigma\).

Similarity criterion for the nonlinear thermal analysis of the soil freezing process: considering the dual effect of nonlinear thermal parameters and boundary conditions

Similarity criterion for the nonlinear thermal analysis of the soil freezing process: considering the dual effect of nonlinear thermal parameters and boundary conditions