Nonhomogeneous Neumann Boundary Research Articles

Boundary regularity results for minimisers of convex functionals with (p, q)-growth

Abstract We prove improved differentiability results for relaxed minimisers of vectorial convex functionals with ( p , q ) \left(p,q) -growth, satisfying a Hölder-growth condition in x x . We consider both Dirichlet and Neumann boundary data. In addition, we obtain a characterisation of regular boundary points for such minimisers. In particular, in case of homogeneous boundary conditions, this allows us to deduce partial boundary regularity of relaxed minimisers on smooth domains for radial integrands. We also obtain some partial boundary regularity results for non-homogeneous Neumann boundary conditions.

Lower Bounds of Blow Up Time for a Class of Slow Reaction Diffusion Equations with Inner Absorption Terms

In this paper, a class of slow reaction-diffusion equations with nonlocal source and inner absorption terms are studied. By using the technique of improved differential inequality, the lower bounds of blow up time for the system under either homogeneous Dirichlet or nonhomogeneous Neumann boundary conditions are obtained.

Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The paper provides an elementary analysis to derive the Hadamard shape derivative formula for complicated problem settings that allow non-homogeneous Neumann boundary conditions. Moreover, an algorithm to rigorously evaluate the Hadamard formula is proposed by utilizing the guaranteed computation for both eigenvalues and eigenfunctions of differential operators, which is recently developed by the second author. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix–Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.

An improved partial differential equation filter scheme for topology optimization of additively manufactured coated structure

In this paper, an improved partial differential equation (PDE) filter is presented for density-based topology optimization of coated structures with both regular and irregular design domains. The proposed filter efficiently addresses the boundary effect problems that causes identification failure of the material interface for modeling coating layer during topology optimization by replacing the original Dirichlet boundary conditions with non-homogeneous Neumann boundary conditions. One attractive feature of the proposed approach is that it can serve as an ideal alternative to the common domain extension approach, particularly for designing geometrically complex engineering structures with irregular design domain meshed by unstructured mesh. Based on the improved PDE filter, a new topology optimization design framework for coated structures with complete coating layer and controllable coating thickness is achieved. Examples of designing regular and irregular two-dimensional and three-dimensional coated structures are provided to demonstrate the effectiveness of the proposed approach.

Guaranteed Lower Eigenvalue Bounds for Steklov Operators Using Conforming Finite Element Methods

Abstract For the eigenvalue problem of the Steklov differential operator, an algorithm based on the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed lower eigenvalue bounds utilize the a priori error estimation for FEM solutions to non-homogeneous Neumann boundary value problems, which is obtained by constructing the hypercircle for the corresponding FEM spaces and boundary conditions. Numerical examples demonstrate the efficiency of our proposed method.

ADI+MDA orthogonal spline collocation for the pressure Poisson reformulation of the Navier–Stokes equation in two space variables

A numerical method for solving a pressure Poisson reformulation of the Navier–Stokes equation in two space variables is presented. The method discretizes in space using orthogonal spline collocation with splines of order r. The velocity terms are obtained through an alternating direction implicit extrapolated Crank –Nicolson scheme applied to a Burgers’ type equation and the pressure term is found by applying a matrix decomposition algorithm to a Poisson equation satisfying non-homogeneous Neumann boundary conditions at each time level. Numerical results suggest that the scheme exhibits convergence rates of order r in space in the H1 norm and semi-norm for the velocity and pressure terms, respectively, and is order 2 in time. Finally, the scheme is applied to the lid-driven cavity problem and is compared to standard benchmark values.

Analysis of fractures in linear viscoelastic media using a generalized finite element method and the elastic–viscoelastic correspondence principle

A methodology for an efficient analysis of 3-D fracture mechanics problems in linear viscoelastic media is presented. It combines the Elastic–Viscoelastic Correspondence Principle (EVCP), a Generalized Finite Element Method (GFEM), and an inverse Laplace transform (ILT) method to compute Energy Release Rate (ERR) and Crack Mouth Opening Displacement (CMOD) for viscoelastic fracture problems in the time domain. The method is denoted EVCP-GFEM-ILT and consists of solving a reference elastic problem in the Laplace domain using a quadratic p-hierarchical GFEM, computing quantities of interest in this domain, and transforming them to the time domain with the aid of a numerical ILT method. One of the key features of the method is that a single elastic solution is required, regardless of the time interval of the target viscoelastic problem. A detailed verification of the methodology against analytical and reference numerical solutions is performed. Problems subjected to either non-homogeneous Neumann or Dirichlet boundary conditions of increasing complexity are considered. Two ILT methods are studied and the one based on Fourier series is shown to deliver accurate results for all tested boundary conditions, as well as non-trivial viscoelastic material models. The computational cost of the method applied to a 3-D problem is compared with the one for Abaqus which solves the problem directly in the time domain using a FEM mesh with quarter-point elements. A fully 3-D mixed-mode fracture problem with a non-trivial viscoelastic material model is solved to highlight the applicability of the proposed methodology. Finally, a problem of practical interest is analyzed in order to show how the methodology proposed in this work can be applied to efficiently handle large-scale simulations.

A numerical approach to a 2D porous-medium mathematical model: Application to an atherosclerosis problem

This work deals with the study and implementation of a Local Space-Time DG ADER approach, in the finite volume framework with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space, in the context of 2D porous media problems. The particular application performed concerns a biomedical problem dealing with the first stages of atherosclerosis disease, where the artery is considered as a porous medium, although the methodology described can be applied to other situations. The mathematical model on which this research is based is given by a system of two-dimensional nonlinear reaction-diffusion equations with a nonlinear source term in one of the equations, which is a variant of the model proposed originally in N. El Khatib, S. Genieys & V. Volpert (2007) Math. Model. Nat. Phenom., vol 2(2), pp. 126–141 [1]. The 2D model considered in this work has two main features: (i) the artery is taken as a porous medium and (ii) a nonlinear non-homogeneous Neumann boundary condition is incorporated, aimed to account for the recruitment of immune cells through the upper boundary, as a response to the production of cytokines. Certain theoretical properties of the stationary solutions and the evolution solution are stated and proved. Some of them are also verified with the numerical simulation results.

Homogenization of a nonlinear drift–diffusion system for multiple charged species in a porous medium

We consider a nonlinear drift–diffusion system for multiple charged species in a porous medium in 2D and 3D with periodic microstructure. The system consists of a transport equation for the concentration of the species and Poisson’s equation for the electric potential. The diffusion terms depend nonlinearly on the concentrations. We consider non-homogeneous Neumann boundary condition for the electric potential. The aim is the rigorous derivation of an effective (homogenized) model in the limit when the scale parameter ϵ tends to zero. This is based on uniform a priori estimates for the solutions of the microscopic model. The crucial result is the uniform L∞-estimate for the concentration in space and time. This result exploits the fact that the system admits a nonnegative energy functional which decreases in time along the solutions of the system. By using weak and strong (two-scale) convergence properties of the microscopic solutions, effective models are derived in the limit ϵ→0 for different scalings of the microscopic model.

Multiplicity of solutions for a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the p ( x )-Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao (1999) [12] and Winkert-Zacher (2012) [47] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the C1,α-regularity follows immediately.

Existence and stability of solutions to nonlinear parabolic problems with perturbed gradient and measure data

In this paper we prove the existence of an entropy solution to nonlinear parabolic equations with diffuse Radon measure data which does not charge the sets of zero p(⋅)-capacity and nonhomogeneous Neumann boundary condition. By a time discretization technique we analyze existence, the uniqueness and the stability questions. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

Stationary solutions in thermodynamics of stochastically forced fluids

We study the full Navier–Stokes–Fourier system governing the motion of a general viscous, heat-conducting, and compressible fluid subject to stochastic perturbation. The system is supplemented with non-homogeneous Neumann boundary conditions for the temperature and hence energetically open. We show that, in contrast with the energetically closed system, there exists a stationary solution. Our approach is based on new global-in-time estimates which rely on the non-homogeneous boundary conditions combined with estimates for the pressure.

Gradient flows in metric random walk spaces

Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each xin X, which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.

A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation

AbstractWe consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above $1/2$ in space and $1/4$ in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two-dimensional domain.

An improved model for performances calculation in spoke-type permanent-magnet machines considering magnetization orientation and finite soft-magnetic material permeability

PurposeIn semi-analytical modeling of spoke-type permanent-magnet (PM) machines (STPMM), the saturation effect is usually neglected (i.e. iron parts are considered to be infinitely permeable) and the PM magnetization is assumed tangential (i.e. magnetization pattern is considered to be tangential-parallel). This paper aims to present an improved two-dimensional (2D) subdomain technique for STPMM with the PM magnetization orientation in quasi-Cartesian coordinates by using hyperbolic functions considering non-homogeneous Neumann boundary conditions (BCs) in non-periodic regions and by applying the interfaces conditions (ICs) in both directions (i.e. t- and θ edges ICs).Design/methodology/approachThe polar coordinate system is transformed into a quasi-Cartesian coordinate system. The rotor and stator regions are divided into primary subdomains, and a partial differential equation (PDE) is assigned to each subdomain. In the PM region, the magnetization orientation is considered in the equations. By applying BCs, the general solution of the equations is determined, and by applying the ICs, the corresponding coefficients are determined.FindingsUsing the proposed coordinate system, the general solution of PDEs and their coefficients can mathematically be simplified. The magnetic field and non-intrinsic unbalanced magnetic forces (UMF) calculations have been performed for three different values of iron core relative permeability (200, 800 and ∞), as well as different magnetization orientation values (135 and 80 degrees). The semi-analytical model based on the subdomain technique is compared with those obtained by the 2D finite-element analysis (FEA). Results disclose that the PM magnetization angle can affect directly the performance characteristics of the STPMM.Originality/valueA new model for prediction of electromagnetic performances in the STPMM takes into account magnetization direction, and soft magnetic material relative permeability in a pseudo-Cartesian coordinate system by using subdomain technique is presented.

A high-order compact difference method on fitted meshes for Neumann problems of time-fractional reaction–diffusion equations with variable coefficients

This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L2-norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

Fluctuations Around a Homogenised Semilinear Random PDE

We consider a semilinear parabolic partial differential equation in mathbf{R}_+times [0,1]^d, where d=1, 2 or 3, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic homogenised parabolic PDE, which is a form of law of large numbers. Our main interest is in the associated central limit theorem. Namely, we study the limit of a properly rescaled difference between the initial random solution and its LLN limit. In dimension d=1, that rescaled difference converges as one might expect to a centred Ornstein–Uhlenbeck process. However, in dimension d=2, the limit is a non-centred Gaussian process, while in dimension d=3, before taking the CLT limit, we need to subtract at an intermediate scale the solution of a deterministic parabolic PDE, subject (in the case of Neumann boundary condition) to a non-homogeneous Neumann boundary condition. Our proofs make use of the theory of regularity structures, in particular of the very recently developed methodology allowing to treat parabolic PDEs with boundary conditions within that theory.

Lucid characterization of unsteady heat conduction in large, square crossbars and cubes subject to uniform surface heat flux by way of “quasi–steady” Helmholtz equations

The technical paper addresses unsteady heat conduction in large, square crossbars and cubes heated continually with uniformly surface heat flux. The uniform surface heat flux is customarily provided by electrical, radiative and pool fire heating in engineering applications. This type of heating fall under the category of non-homogeneous Neumann boundary conditions. The existing traditional method for solving the unsteady, heat conduction equation in two and three dimensions with constant thermo-physical properties and subject to non-homogeneous uniform heat flux boundary conditions is the principle of superposition of partial solutions of simpler unsteady, heat conduction equations. For the latter, the exact, analytical temperature distributions are customarily expressed by infinite series regardless of the solution method. For the numerical evaluation of spatio-temporal temperatures, the main inconvenience of the infinite series lies in the poor convergence when the time is small. To prevent the impeding obstacle, the Transversal Method Of Lines (TMOL) is implemented in this work to transform the unsteady, two- and three-dimensional heat conduction equations into degraded “quasi steady”, two- and three-dimensional heat conduction equations, namely the homogeneous Helmhotz equations. Thereby, the exact, analytical solutions of the converted partial differential equations of elliptic type and homogeneous were determined by the standard method of separation of variables. The resulting approximate, semi-analytical two- and three-dimensional temperature distributions for large, square crossbars and cubes do not contain elaborate infinite series. Instead, the temperature disdtributions contain hyperbolic functions elucidating worthy compactness and superb quality, regardless of the time, either “small time”, “moderate time” or “large time”.

A spectral method for solving nonhomogeneous Neumann boundary value problems on quadrilaterals

In this paper, we develop a spectral method with essential imposition of nonhomogeneous Neumann boundary condition on quadrilaterals. Results on irrational orthogonal approximation are established, which facilitate the numerical solutions of partial differential equations of the sort. As applications, we provide spectral schemes for two model problems, and prove their spectral accuracy. Efficient numerical implementations are described. Numerical results demonstrate high efficiency of the suggested algorithms.