Nonhomogeneous Boundary Conditions Research Articles

Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Poro-elastic systems have been used extensively in modeling fluid flow in porous media in petroleum and earthquake engineering. Nowadays, they are frequently used to model fluid flow through biological tissues, cartilages, and bones. In these biological applications, the fluid-solid mixture problems, which may also incorporate structural viscosity, are considered on bounded domains with appropriate non-homogeneous boundary conditions. The recent work in [ 12 ] provided a theoretical and numerical analysis of nonlinear poro-elastic and poro-viscoelastic models on bounded domains with mixed boundary conditions, focusing on the role of visco-elasticity in the material. Their results show that higher time regularity of the sources is needed to guarantee bounded solution when visco-elasticity is not present. Inspired by their results, we have recently performed local sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the boundary source of traction associated with the elastic structure [ 3 ]. Our results show that the solution is more sensitive to boundary datum in the purely elastic case than when visco-elasticity is present in the solid matrix. In this article, we further extend this work in order to include local sensitivities of the solution of the coupled system to the boundary conditions imposed on the Darcy velocity. Sensitivity analysis is the first step in identifying important parameters to control or use as control terms in these poro-elastic and poro-visco-elastic models, which is our ultimate goal.

A HYBRID METHOD FOR STIFF REACTION-DIFFUSION EQUATIONS.

The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction-diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order hybrid IIF-ETD method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction-diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method.

Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications

We consider the Schrodinger equation on a half space in any dimension with a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and the so-called transparent boundary conditions. Building upon recent local in time Strichartz estimates (for Dirichlet boundary conditions), we obtain global Strichartz estimates for initial data in H s , 0 ¤ s ¤ 2 and boundary data in a natural space H s. For s ¥ 1{2, the issue of compatibility conditions requires a thorough analysis of the H s space. As an application we solve nonlinear Schrodinger equations and construct global asymptotically linear solutions for small data. A discussion is included on the appropriate notion of scattering in this framework, and the optimality of the H s space.

Comparative Study of Wellhole Surrounding Rock under Nonuniform Ground Stress

The stress analysis of the wellhole surrounding rock and the regular failure of the wellhole has always been a concern for the well builders. Firstly, the Hamilton canonical equations are obtained by using the Hamiltonian variational principle in the sector domain, and the zero eigensolution and nonzero eigensolutions of the homogeneous equation are solved. According to the Hamiltonian operator matrix with the orthogonal eigenfunction system, the special solution form of the nonhomogeneous boundary condition equation is obtained. Then, according to the principle of the same coefficient being equal, the relationship equation between the direction eigenvalue and the angle coefficient is obtained, from which the specific expression of the special solution of the equation can be determined. Furthermore, the analytical solution of the wellhole surrounding rock problem under nonuniform ground stress is obtained by using the linear elastic accumulative principle. Finally, a concrete example is given to compare the finite element method and the symplectic algorithm. The results are consistent, which ensures the accuracy and the reliability of the symplectic algorithm. The relationship between the circumferential stress distribution around the hole and the lateral pressure coefficient is further analyzed.

Stokes and Navier–Stokes equations with Navier boundary condition

In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω⊂R3 of class C1,1 from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in W1,p(Ω) (and W2,p(Ω)) to the linear problem for all 1<p<∞ considering minimal regularity of α, using some inf–sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large α, which enables us to obtain the strong convergence of Stokes solutions with Navier slip boundary condition to the one with no-slip boundary condition as α→∞. Finally, we discuss the same questions for the non-linear system.

Control of the radiative heating of a semi-transparent body

In a preceding paper, we have studied the radiative heating of a semi-transparent body $$\varOmega $$ (e.g., glass) by a black radiative source S surrounding it, black source at absolute uniform temperature u(t) at time t between time 0 and time $$t_\mathrm{f}$$ , the final time of the radiative heating. This problem has been modeled by an appropriate coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. In the present paper, u being considered as the control variable, we want to adjust the absolute temperature distribution $$(x,t) \mapsto T(x,t)$$ inside the semi-transparent body $$\varOmega $$ near a desired temperature distribution $$T_\mathrm{d}(\cdot ,\cdot )$$ during the time interval of radiative heating $$]0,t_\mathrm{f}[$$ by acting on u, the purpose being to deform $$\varOmega $$ to manufacture a new object. In this respect, we introduce the appropriate cost functional and the set of admissible controls $$U_\mathrm{ad}$$ , for which we prove the existence of optimal controls. Introducing the state space and the state equation, a first-order necessary condition for a control $$u:t \mapsto u(t)$$ to be optimal is then derived in the form of a Variational Inequality by using the implicit function theorem and the adjoint problem. We close this paper by some numerical considerations.

A regularized Fourier sine series solution of a 3D backward heat conduction problem with extremal long time span

In the article, we solve an extremal long time span backward heat conduction problem (BHCP) of a 3D nonhomogeneous heat conduction equation with nonhomogeneous boundary conditions in a cuboid. We first derive a time-dependent 3D homogenization function, such that in terms of the new variable by a variable transformation we can find the expansion coefficients in a closed form by using the Fourier sine series method. After a simple regularization technique, a stable analytic series solution of the 3D BHCP is available. We also develop a regularized Fourier sine series solution of temperature in the whole space-time domain. Numerical tests for the BHCPs in a large space-time domain reveal that the present method is very accurate to recover the initial temperature and the whole solution.

Some non-local logistic population model with non-zero boundary condition

In this paper, we study some type of equations which may model the behavior of species inhabiting in some habitat. For our purpose, using a priori bounded techniques, we obtain a positive solution to a family of non-local partial differential equations with non-homogeneous boundary conditions.

Dissipative solution to the Ericksen–Leslie system equipped with the Oseen–Frank energy

We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state.

Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

This paper deals with the randomized heat equation defined on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.

Lp theory for Boussinesq system with Dirichlet boundary conditions

ABSTRACTWe consider the stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain of class with a possibly disconnected boundary. We prove the existence of weak solutions in , strong solutions in and very weak solutions in of the stationary Boussinesq system by assuming that the fluxes of the velocity are sufficiently small. Finally, as it is expected, we obtain the uniqueness of the solution by considering small data.

Two-Dimensional Exact Subdomain Technique of Switched Reluctance Machines with Sinusoidal Current Excitation

This paper presents a two-dimensional (2D) exact subdomain technique in polar coordinates considering the iron relative permeability in 6/4 switched reluctance machines (SRM) supplied by sinusoidal waveform of current (aka, variable flux reluctance machines). In non-periodic regions (e.g., rotor and/or stator slots/teeth), magnetostatic Maxwell’s equations are solved considering non-homogeneous Neumann boundary conditions (BCs). The general solutions of magnetic vector potential in all subdomains are obtained by applying the interface conditions (ICs) in both directions (i.e., r- and θ-edges ICs). The global saturation effect is taken into account, with a constant magnetic permeability corresponding to the linear zone of the nonlinear B(H) curve. In this investigation, the magnetic flux density distribution inside the electrical machine, the static/dynamic electromagnetic torques, the magnetic flux linkage, the self-/mutual inductances, the magnetic pressures, and the unbalanced magnetic forces (UMFs) have been calculated for 6/4 SRM with two various non-overlapping (or concentrated) windings. One of the case studies is a M1 with a non-overlapping all teeth wound winding (double-layer winding with left and right layer) and the other is a M2 with a non-overlapping alternate teeth wound winding (single-layer winding). It is important to note that the developed semi-analytical model based on the 2D exact subdomain technique is also valid for any number of slot/pole combinations and for non-overlapping teeth wound windings with a single/double layer. Finally, the semi-analytical results have been performed for different values of iron core relative permeability (viz., 100 and 800), and compared with those obtained by the 2D finite-element method (FEM). The comparisons with FEM show good results for the proposed approach.

Analytical solutions of some internal boundary value problems of elasticity for domains with hyperbolic boundaries

An analytical solution of two-dimensional problems of elasticity in the region bounded by a hyperbola in elliptic coordinates is constructed using the method of separation of variables. The stress–strain state of a homogenous isotropic hyperbolic body and that with a hyperbolic cut is studied when there are non-homogenous (non-zero) boundary conditions given on the hyperbolic boundary. The graphs for the numerical results of some test problems are presented.

Non-homogeneous boundary value problems of the fifth-order KdV equations on a bounded interval

We investigate well-posedness of initial boundary value problem for the fifth-order KdV equation (or Kawahara equation) posed on a finite interval ∂ t u − ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 with general non-homogeneous boundary conditions. Firstly, all possible boundary conditions are found while searching enough dissipative effects to the initial boundary value problem. Then, boundary smoothing effect is established through estimating explicit solution established by using Laplacian transformation. Finally, based on smoothing effect, local well-posedness of the initial boundary value problem was proved by applying Banach's fixed point theorem.

Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form div[A(x,u,∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, degenerate or both in x in the sense that they behave like some weight function μ, which is in the A2 class of Muckenhoupt weights. Global and interior weighted W1,p(Ω,ω)-regularity estimates are established for weak solutions of these equations with some other weight function ω. The results obtained are even new for the case μ = 1 because of the dependence on the solution u of A. In case of linear equations, our W1,p-regularity estimates can be viewed as the Sobolev’s counterpart of the Holder’s regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.

Improved receding horizon H∞ temperature spectrum tracking control for Debye media in microwave heating process

This paper considers the problem of tracking control for the coupled thermal-electromagnetic model with temperature-dependent permittivity in microwave heating process, where the incident electrical field is subject to the asymmetrical saturation. Due to the nonhomogeneous Neumann boundary condition and inherent infinite-dimensional characteristics for microwave heating process, it is difficult for us to readily design and implement tracking controller. With the help of improved spectral Galerkin method, the traditional partial differential equation (PDE) model is transformed into the linear finite-dimensional ordinary differential equation (ODE) model which can be subsequently used for controller design. Then, the receding horizon H∞ control strategy is improved to stabilize the derived uncertain ODE error model with asymmetrical input saturation and bounded external disturbance. The outcome of improved algorithm is formulated as a linear matrix inequalities (LMIs) optimization problem, which can be on-line updated by applying the finite-different time-domain (FDTD) method. Finally, the improved control strategy is successfully applied in one-dimensional cavity model for tracking temperature spectrum.

Unsteady micropolar fluid flow in a thin domain with Tresca fluid–solid interface law

We consider a micropolar fluid flow in a two-dimensional domain. We assume that the velocity field satisfies a non-linear slip boundary condition of friction type on a part of the boundary while the micro-rotation field satisfies non-homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of a solution. Then motivated by lubrication problems we assume that the thickness and the roughness of the domain are of order 0<ε<<1 and we study the asymptotic behaviour of the flow as ε tends to zero. By using the two-scale convergence technique we derive the limit problem which is totally decoupled for the limit velocity and pressure (v0,p0) on one hand and the limit micro-rotation Z0 on the other hand. Moreover we prove that v0, p0 and Z0 are uniquely determined via auxiliary well-posed problems.

Isogeometric configuration design optimization of shape memory polymer curved beam structures for extremal negative Poisson’s ratio

Using a continuum-based design sensitivity analysis (DSA) method, a configuration design optimization method is developed for curved Kirchhoff beams with shape memory polymers (SMP), from which we systematically synthesize lattice structures achieving target negative Poisson’s ratio. A SMP phenomenological constitutive model for small strains is utilized. A Jaumann strain, based on the geometrically exact beam theory, is additively decomposed into elastic, stored, and thermal parts. Non-homogeneous displacement boundary conditions are employed to impose mechanical loadings. At each equilibrium configuration, an additional nonlinear analysis is performed to calculate the Poisson’s ratio and its design sensitivity of the SMP material. The design objectives are twofold: for purely elastic materials, lattice structures are designed to achieve prescribed Poisson’s ratios under finite compressive deformations. Also, SMP-based lattice structures are synthesized to possess target Poisson’s ratios in specified temperature ranges. The analytical design sensitivity of the Poisson’s ratio is verified through comparison with finite difference sensitivity. Several configuration design optimization examples are demonstrated.

A mixed virtual element method for a pseudostress-based formulation of linear elasticity

In this paper we introduce and analyze a mixed virtual element method (mixed-VEM) for a pseudostress-displacement formulation of the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. We follow a previous work by some of the authors, and employ a mixed formulation that does not require symmetric tensor spaces in the finite element discretization. More precisely, the main unknowns here are given by the pseudostress and the displacement, whereas other physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through simple postprocessing formulae in terms of the pseudostress variable. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, we utilize a well-known local projector onto a suitable polynomial subspace to define a calculable version of our discrete bilinear form, whose continuous version requires information of the variables on the interior of each element. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Babuška–Brezzi theory. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solutions as well as for the fully computable projections of them. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. In addition, this postprocessing formula can also be applied to the postprocessed stress tensor. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

Efficient compact finite difference method for variable coefficient fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions in conservative form

In this paper, a high-order compact finite difference scheme is derived for the time fractional sub-diffusion equation with nonhomogeneous Neumann boundary conditions in conservative form. Based on the L2- $$1_{\sigma }$$ approximation formula of the time fractional derivative, a high-order efficient compact finite difference method is developed. The unconditional stability of the resulting scheme and its convergence of second order in time and fourth order in space are rigourously proved by a discrete energy analysis method. The theoretical results are illustrated by examples.