Dirichlet Boundary Conditions Research Articles

Topological wave equation eigenmodes in continuous 2D periodic geometries

In this paper, we address the topological characterization of the wave equation solutions in continuous two-dimensional (2D) periodic geometries with Neumann or Dirichlet boundary conditions. This characterization is relevant in the context of 2D vibrating membranes and our approach allows one to understand the topological behavior recently observed in acoustic three-dimensional artificial lattices. In particular, the dependence of the topological behavior on the experimental positioning of the coupling channels is explained using simple arguments and a simple method of construction of an equivalent effective tight-binding Hamiltonian is presented.

Master–slave elimination scheme for arbitrary smooth nonlinear multi-point constraints

AbstractNonlinear multi-point constraints are essential in modeling various engineering problems, for example in the context of (a) linking individual degrees of freedom of multiple nodes to model nonlinear joints, (b) coupling different element types in finite element analysis, (c) enforcing various types of rigidity in parts of the mesh and (d) considering deformation-dependent Dirichlet boundary conditions. One method for addressing constraints is the master–slave elimination, which offers the benefit of reducing the problem dimension as opposed to Lagrange multipliers and the penalty method. However, the existing master–slave elimination method is limited to linear constraints. In this paper, we introduce a new master–slave elimination method for handling arbitrary smooth nonlinear multi-point constraints in the system of equations of the discretized system. We present a rigorous mathematical derivation of the method. Within this method, new constraints can be easily considered as an item of a “constraint library”; i.e. no case-by-case-programming is required. In addition to the theoretical aspects, we also provide helpful remarks on the efficient implementation. Among others, we show that the new method results in a reduced computational complexity compared to the existing methods. The study also places emphasis on comparing the new approach with existing methods via numerical examples. We have developed innovative benchmarks which encompass all relevant computational properties, and provide analytical and reference solutions. Our findings demonstrate that our new method is as accurate, robust and flexible as the Lagrange multipliers, and more efficient due to the reduction of the total number of degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.

Modeling heat and mass transfer in metal hydride hydrogen storage systems: Impact of operating parameters and reactor geometry

This work introduces a transient 2D axis-symmetrical model for hydrogen absorption in a cylindrical LaNi5 reactor employing finite volume method with a fully implicit Euler's time integration scheme, coupling equations for heat, mass, and momentum transport. The validated model is used to investigate the impacts of pressure, cooling fluid temperature, and variations in reactor geometry and size on temperature and reacted fraction profiles. The findings reveal that the charging pressure affects both peak temperature and reaction kinetics, whereas cooling fluid temperature predominantly impacts the absorption kinetics. A comparative analysis of two models, one incorporating Darcy's velocity and one without, demonstrates that while Darcy's law introduces numerical instability in the coupled equations, its impact on the model outcomes is negligible. The effect of changing the non-homogeneous Neumann to Dirichlet boundary condition is also demonstrated to anticipate the utilization of phase change materials (PCM) instead of the cooling fluid.

Insensitizing controls for the micropolar fluids

In this paper, we investigate the existence of insensitizing controls for the micropolar fluids in a bounded domain with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. The study of insensitizing controls is essential to solve a stability problem, which means that we look for controls such that some functionals of the velocity fields (the so‐called sentinels) are insensitive to the small perturbations of initial data. The problem of insensitizing controls is transformed into a suitable controllability problem for a cascade system. Our proof relies on a new global Carleman inequality and the inverse mapping theorem.

Entanglement entropies of an interval for the massless scalar field in the presence of a boundary

We study the entanglement entropies of an interval for the massless compact boson either on the half line or on a finite segment, when either Dirichlet or Neumann boundary conditions are imposed. In these boundary conformal field theory models, the method of the branch point twist fields is employed to obtain analytic expressions for the two-point functions of twist operators. In the decompactification regime, these analytic predictions in the continuum are compared with the lattice numerical results in massless harmonic chains for the corresponding entanglement entropies, finding good agreement. The application of these analytic results in the context of quantum quenches is also discussed.

Hopf bifurcation in a delay-diffusion housefly equation with Dirichlet boundary conditions

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate

This paper is about Hölder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time . In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

Hamiltonian for the Hilbert-Pólya conjecture

Abstract We introduce a Hamiltonian to address the Hilbert-Pólya conjecture. The eigenfunctions of the introduced Hamiltonian, subject to the Dirichlet boundary conditions on the positive half-line, vanish at the origin by the nontrivial zeros of the Riemann zeta function. Consequently, the eigenvalues are determined by these nontrivial Riemann zeros. If the Riemann hypothesis (RH) is true, the eigenvalues become real and represent the imaginary parts of the nontrivial zeros. Conversely, if the Hamiltonian is self-adjoint, or more generally, if it admits only real eigenvalues, then the RH follows. In our attempt to demonstrate the latter, we establish the existence of a similarity transformation of the introduced Hamiltonian that is self-adjoint on the domain specified by an appropriate boundary condition, which is satisfied by the transformed eigenfunctions through the vanishing of the Riemann zeta function. Our result can be extended to a broader class of functions whose zeros lie on the critical line.

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

Fermions, quantum gravity, and holography in two dimensions

We study a model comprising N flavors of Kähler Dirac fermion propagating on a triangulated two-dimensional disk which is constrained to have a negative average bulk curvature. Dirichlet boundary conditions are chosen for the fermions. Quantum fluctuations of the geometry are included by summing over all possible triangulations consistent with these constraints. We show in the limit N→∞ that the partition function is dominated by a regular triangulation of two-dimensional hyperbolic space. We use strong coupling expansions and Monte Carlo simulation to show that in this limit boundary correlators of the fermions have a power law dependence on boundary separation as one expects from holography. However, we argue that this behavior breaks down for any finite number of massive fields in the thermodynamic limit and quantum fluctuations of the bulk geometry drive the theory into a nonholographic phase. In contrast, for massless fermions, we find evidence that the boundary is conformal even for finite N. This is consistent with theoretical results in quantum Liouville theory. Published by the American Physical Society 2024

Two solutions for Dirichlet double phase problems with variable exponents

Abstract This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Maximum principles for elliptic operators in unbounded Riemannian domains

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

Numerical simulation of Rashba spin–orbit coupling effect on quantum Hall resistivity within tight-binding approximation

A numerical simulation of the effect of Rashba spin–orbit coupling (RSO) on quantum transport properties of electrons confined to a two-dimensional electron gas (2DEG) in the integer quantum Hall regime is presented. The mechanism of quantization and the dependence of the Hall resistivity (HR) on the system built-in and external parameters are studied using the non-equilibrium Green’s function (NEGF) technique. Relevant Green’s functions are computed numerically using spectral decomposition method in combination with recursive techniques. The tight-binding-like representation of the system Hamiltonian including magnetic field is obtained within the finite difference method, using Dirichlet boundary conditions. Corresponding formula linking the Hall resistivity with the transmission function of the system has been derived analytically within the NEGF formalism in a general form, in the sense that the formula can also be used beyond the linear response limit and for nonzero temperatures. In particular, it was shown explicitly that in the presence of the RSO coupling the even- and odd plateaus emerging in the magnetic-field dependence of the quantum Hall resistivity are related to different physical mechanisms.

Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions

Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions

Solving forward and inverse problems of contact mechanics using physics-informed neural networks

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues λ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (n)$$\\end{document} of shapes with n edges that are of the form λ(n)∼x∑k=0∞Ck(x)nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (n) \\sim x\\sum _{k=0}^{\\infty } \\frac{C_k(x)}{n^k}$$\\end{document} where x is the limiting eigenvalue for n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document}. Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order Ck(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_k(x)$$\\end{document} and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary).

Transient Heat Transfer Analysis in Metal Plates with Variable Thickness

Nonlinear transient heat transfer via conduction–radiation is a dynamic topic of long-standing interest with applications ranging from aeronautical and mechanical engineering to industrial and civil security. To gain a better understanding of the performances of materials having thermal proprieties that change during nonlinear heat transfer, several studies using the finite element method (FEM) have been conducted. Such studies apply nonlinear thermal material characteristics to describe the complete system under different loading conditions in each region by adjusting the temperature values for the other three edges and the thickness parameter with Dirichlet boundary conditions. As a result, while modeling and simulating temperature distributions for such situations, nonlinearities generated by temperature-dependent thermal conductivity must be considered. In this work, we focus on the analysis of coupled transient heat transfer through two metal plates with temperature-dependent thermal characteristics in which the temperature is fixed along the bottom edge and heat is transferred from both the top and bottom faces of the two plates. FEM is employed to solve the nonlinear heat equation and compute the temperature as a function of time for variable thickness. The study examines the effect of modifying the thickness parameter values on the temperature distribution over time for various edge values over 5,000 s.

Three-dimensional hygro-thermo-mechanical bending analysis of anisotropic skewed parallelogram plates on a tri-parameter foundation using a wavelet method

In the paper, a brand-new three-dimensional hygro-thermo-mechanical bending model of anisotropic skewed parallelogram plates with extreme skew angle in large deformation supported by a tri-parameter nonlinear elastic foundation is proposed. By investigating the mutual coupled interaction of hygrothermal effects between in-plane distribution and thickness variation, nine kinds of spatially analytical hygrothermal fields in parallelogram domain are established by analyzing the Dirichlet, Neumann and Robin thermal boundary conditions of sinusoidally induced heat conduction at top and bottom surfaces. With coordinate transformation in oblique system considering the Von Kármán membrane strains, the highly coupled and strongly nonlinear governing partial differential equations have been derived. A wavelet solving procedure has been developed with the convergence process validated and precision of obtained solutions verified in good accordance with published results. Parametric studies have been carried out to investigate the large-deflection behavior of simply supported or clamped skew plates on Winkler-Pasternak foundation under combined mechanical and hygrothermal loads. The influence of skew angle, material orthotropy and foundation contact nonlinearity is examined, which reveals the geometrically nonlinear responses of plates are highly affected by large oblique angle, principle orientation of material orthotropy and foundation parameters.

Numerical solutions of boundary problems in partial differential equations: A deep learning framework with Green's function

This work introduces Green Networks (GreenNets), a novel class of neural network architectures designed to address a wide spectrum of boundary problems in partial differential equations (PDEs). Demonstrated versatility includes solving the Poisson equation, convection-diffusion equation, and unsteady incompressible Navier-Stokes equations. GreenNets comprises two specialized variants: Boundary Transformation Green Network (BT-GreenNet), focusing on transforming Dirichlet boundary conditions into source terms, and Boundary Integral Green Network (BI-GreenNet), considering boundary integral terms with shared weights between the boundary and source integrals. Comparisons with existing deep learning and traditional numerical methods underscore their accuracy and efficiency. BT-GreenNet excels in efficiency, and BI-GreenNet offers enhanced flexibility. The research pinnacle, Boundary Unified Green Network (BU-GreenNet), merges BT-GreenNet and BI-GreenNet to address coupled PDEs, specifically unsteady incompressible Navier-Stokes equations. BU-GreenNet stands out by surpassing traditional numerical approaches, offering remarkable computational efficiency while faithfully capturing the intricate evolution of fluid dynamics phenomena. This research redefines the means of solution to boundary problems in PDEs through innovative neural network architectures, emphasizing practical utility and computational efficiency.