Well-posed Boundary Conditions Research Articles

ON WELL-POSED BOUNDARY CONDITIONS AND ENERGY STABLE FINITE-VOLUME METHOD FOR THE LINEAR SHALLOW WATER WAVE EQUATION

Abstract We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite-volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

Flow physics of a subcritical carbon dioxide jet in a multiphase ejector

Existing literature on ejectors, different researchers utilized different turbulence models by their choice, neither analyzing the physics of turbulence nor providing any sufficient optical diagnostics of the flow field. In this article, the physics captured by the Unsteady Reynolds Averaged Navier–Stokes (URANS) and Large Eddy Simulation (LES) are compared based on well-posed boundary conditions obtained experimentally from a carbon dioxide (CO2) vapor compression cycle with an error ≤2.5%. The experimental results utilized for this study are at Reynold’s number (Re) 1.2e5 at the high-pressure inlet (motive flow) maintaining CO2 in subcritical state. The low-pressure inlet (suction flow) is maintained at Re = 9.8e4 with CO2 in vapor state, providing a pressure lift ratio of 1.2 at the diffuser outlet. Comparison of flow topologies with different URANS models reveals that only standard, Shear Stress Transport (SST) k−ω, and Reynold’s Stress Model-Shear Stress Gradient (RSM-SSG) are the suitable URANS models capturing a linearly increasing anisotropy tensor component with increasing strain in the flow. LES also predicted the similar physics. However, standard k−ω overpredicted the suction pressure beyond the acceptable uncertainty limits, which made it inapplicable. For the first time, a spatio-temporal representation of the flow topology with LES revealed the actual jet morphology of a CO2 ejector. Specifically, finger-like structures are key features to entrain more surrounding warm fluid into the colder dense core fluid at the cost of destabilizing and breaking up the jet by stretching the streamwise vortices and obstructing the increase in pressure-lift ratio.

New Well-Posed boundary conditions for semi-classical Euclidean gravity

We consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet conditions do not yield a well-posed elliptic system, and Anderson has suggested boundary conditions that do. Here we point out that there exists a one-parameter family of boundary conditions, parameterized by a constant p, where a suitably Weyl rescaled boundary metric is fixed, and all give a well-posed elliptic system. Anderson and Dirichlet boundary conditions can be seen as the limits p → 0 and ∞ of these. Focussing on static Euclidean solutions, we derive a thermodynamic first law. Restricting to a spherical spatial boundary, the infillings are flat space or the Schwarzschild solution, and have similar thermodynamics to the Dirichlet case. We consider smooth Euclidean fluctuations about the flat space saddle; for p > 1/6 the spectrum of the Lichnerowicz operator is stable — its eigenvalues have positive real part. Thus we may regard large p as a regularization of the ill-posed Dirichlet boundary conditions. However for p < 1/6 there are unstable modes, even in the spherically symmetric and static sector. We then turn to Lorentzian signature. For p < 1/6 we may understand this spherical Euclidean instability as being paired with a Lorentzian instability associated with the dynamics of the boundary itself. However, a mystery emerges when we consider perturbations that break spherical symmetry. Here we find a plethora of dynamically unstable modes even for p > 1/6, contrasting starkly with the Euclidean stability we found. Thus we seemingly obtain a system with stable thermodynamics, but unstable dynamics, calling into question the standard assumption of smoothness that we have implemented when discussing the Euclidean theory.

Existence and uniqueness of well-posed fractional boundary value problem.

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.

On Well-Posed Boundary Conditions for the Linear Non-Homogeneous Moment Equations in Half-Space

On Well-Posed Boundary Conditions for the Linear Non-Homogeneous Moment Equations in Half-Space

Continuity of families of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.

Linear Moment Models to Approximate Knudsen Layers

We propose a well-posed Maxwell-type boundary condition for the linear moment system in half-space. As a reduction of the Boltzmann equation, the moment equations are available to model Knudsen layers near a solid wall, where proper boundary conditions should be prescribed. Utilizing an orthogonal decomposition, we separate the part with a damping term from the system and then impose a new class of Maxwell-type boundary conditions on it. Due to the new boundary condition, we show that the half-space boundary value problem admits a unique solution with explicit expressions. Instantly, the well-posedness of the linear moment system is achieved. We apply the procedure to classical flow problems with the Shakhov collision term, such as the velocity slip and temperature jump problems. The model can capture Knudsen layers with very high accuracy using only a few moments.

Performance estimation of multi-stage cooled axial flow turbines under choked conditions

Development of a fast and accurate tool to estimate the performance of axial flow turbines in all range of working conditions including choked region is of great interest in the design steps of turbines. Mean-line (one-dimensional) analysis is one of the solutions that is computationally cost-effective along with producing accurate results. However, the development of a mean-line model that able to solve the flow in a choked turbine (one-row chocking and multi-row choking) is a challenging problem. This study presents a new algorithm for mean-line analysis of axial flow turbines with a set of well-posed boundary conditions, i.e., the inlet total pressure, the inlet total temperature, inlet flow angle, and outlet static pressure and calculates the flow condition through the turbine row-by-row. The proposed algorithm is implemented in an object-oriented C++ code named Axial Turbine Simulator (ATSim). The results of ATSim program is validated and verified under various working conditions, e.g., low and high-pressure ratios (choking condition), for four different turbine configurations including two uncooled single-stage turbines, an uncooled two-stage turbine, and a cooled three-stage turbine. ATSim provide good consistency with previous experimental and numerical studies. Moreover, the results show that ATSim is capable of estimating the performance of the turbine in a very short time over a wide range of working conditions, including choking (one-row choking or multi-row choking) region.

Irreducible representation of surface distributions and Piola transformation of external loads sustainable by third gradient continua

In this paper Piola transformations are found that relate the Eulerian and Lagrangian external loads which third gradient continua can sustain. As shown by Gabrio Piola and Paul Germain, the most effective postulation scheme in mechanics is based on the principle of virtual work and therefore continuum mechanics must be mathematically founded based on the theory of distributions and on differential geometry. Using the principle of virtual work, the set of admissible external loads sustainable by third gradient continua is seen to include: i) volume force density, ii) surface density of contact force, iii) surface density of contact double force, iv) surface density of contact triple force, v) line density of edge contact forces, vi) line density of contact edge double forces and vii) contact forces concentrated on wedge points. Following the nomenclature introduced by Paul Germain, forces are dual in virtual work of virtual displacements, surface and line double forces are dual of the derivatives of virtual displacements in the normal direction(s) of the surfaces and edges constituting the boundary of the continuum, and surface triple forces are dual of the second normal derivatives of virtual displacements. Volume and surface forces transform as in first gradient Cauchy continua. Moreover we find that: a) the virtual work expended by Eulerian surface triple force, when transformed into the Lagrangian description, must be represented as the work expended by all the kinds of external Lagrangian loads listed in i)-vii); b) Eulerian surface double force transforms into Lagrangian surface double force, surface contact force and edge contact line force; c) Eulerian edge contact line double force transforms into Lagrangian edge contact line double forces, edge line forces and point concentrated wedge forces; d) Eulerian edge and wedge contact line forces transforms into their Lagrangian counterpart only. The Piola transformation formulas deduced in this paper depend on the first, second and third gradients of placement. The presented results allow for the formulation of well-posed boundary condition problems for third gradient continua in the Lagrangian description, and are relevant in computational mechanics. In view of the obtained Piola transformation formulas, the concept of dead loads needs to be modified. We believe to have given an example of how the Mechanics in the French Style, as developed on the ideas by D’Alembert and Lagrange, is still a fertile tool of invention.

A comparative study of two different shallow water formulations using stable summation by parts schemes

This study provides numerical solutions to the two-dimensional linearized shallow water equations (SWE) using a high-order finite difference scheme in Summation By Parts (SBP) form. In addition to the SBP operators for the discretizations, penalty terms, Simultaneous Approximation Terms (SAT) are applied to impose well-posed open boundary conditions. The conventional SWE with height and velocities as the prognostic variables, and a new type of the vorticity–divergence SWE with wave height gradients, vorticity and divergence as the prognostic variables were investigated. It was shown that the solution in all numerical tests enter and exit the domain without instabilities. The convergence rates were correct for all orders of the SBP operators in both the entrance and exit tests. Interestingly, the error norm of the wave height were orders of magnitude lower in the vorticity–divergence solutions compared to the conventional SWE solutions.

A Decoupling and Coupling Approach for the Bi-harmonic Equation

This paper is devoted to the numerical solution of the bi-harmonic problem: Δ2 w = p . The operator is split into two harmonic ones. The finite difference scheme of the harmonic operator is used to get discretized operators, coupling those leads to a global system , this system is to be solved with any well-posed boundary conditions. In this way, boundary conditions need not be specified with the decoupled equations, but rather with the global system. Numerical test problems are provided. The extension of this technique to poly harmonic equations is straightforward.

Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach.

In this paper, a coherent boundary value problem to model metamaterials' behaviour based on the relaxed micromorphic model is established. This boundary value problem includes well-posed boundary conditions, thus disclosing the possibility of exploring the scattering patterns of finite-size metamaterial specimens. Thanks to the simplified model's structure (few frequency- and angle-independent parameters), we are able to unveil the scattering metamaterial's response for a wide range of frequencies and angles of propagation of the incident wave. These results are an important stepping stone towards the conception of more complex large-scale meta-structures that can control elastic waves and recover energy. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Boundary and interface conditions in the relaxed micromorphic model: Exploring finite-size metastructures for elastic wave control

In this paper, we establish well-posed boundary and interface conditions for the relaxed micromorphic model that are able to unveil the scattering response of fully finite-size metamaterial samples. The resulting relaxed micromorphic boundary value problem is implemented in finite-element simulations describing the scattering of a square metamaterial sample whose side counts nine unit cells. The results are validated against a direct finite-element simulation encoding all the details of the underlying metamaterial’s microstructure. The relaxed micromorphic model can recover the scattering metamaterial’s behavior for a wide range of frequencies and for all possible angles of incidence, thus showing that it is suitable to describe dynamic anisotropy. Finally, thanks to the model’s computational performances, we can design a metastructure combining metamaterials and classical materials in such a way that it acts as a protection device while providing energy focusing in specific collection points. These results open important perspectives for the short-term design of sustainable structures that can control elastic waves and recover energy.

Lyapunov-type inequality for a Riemann-Liouville type fractional boundary value problem with anti-periodic boundary conditions

In this article, we establish a Lyapunov-type inequality for a two-point Riemann-Liouville type fractional boundary value problem associated with well-posed anti-periodic boundary conditions. As an application, we estimate a lower bound for the eigenvalue of the corresponding fractional eigenvalue problem.

A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form

We present a stable discontinuous Galerkin (DG) method with a perfectly matched layer (PML) for three and two space dimensional linear elastodynamics, in velocity-stress formulation, subject to well-posed linear boundary conditions. First, we consider the elastodynamics equation, in a cuboidal domain, and derive an unsplit PML truncating the domain using complex coordinate stretching. The hyperbolic structure of the underlying system enables the construction of continuous energy estimates, in the time domain for the elastic wave equation, and in the Laplace space for a sequence of PML model problems, with variations in one, two and three space dimensions, respectively. They correspond to PMLs normal to boundary faces, along edges and in corners. Second, we develop a DG numerical method for the linear elastodynamics equation using physically motivated numerical flux and penalty parameters, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes in all directions, by construction, our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Third, to ensure numerical stability of the discretization when PML damping is present, it is necessary to extend the numerical DG fluxes, and the numerical inter-element and boundary procedures, to the PML auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical solutions are evolved in time using the high order arbitrary derivative (ADER) time stepping scheme of the same order of accuracy with the spatial discretization. By combining the DG spatial approximation with the high order ADER time stepping scheme and the accuracy of the PML we obtain an arbitrarily high-order accurate wave propagation solver in the time domain. Numerical experiments are presented in two and three space dimensions corroborating the theoretical results.

Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles

The Lattice Boltzmann method (LBM) has been applied for the simulation of lid-driven flows inside cavities with internal two-dimensional circular obstacles of various diameters under Reynolds numbers ranging from 100 to 5000. With the LBM, a simplified square cross-sectional cavity was used and a single relaxation time model was employed to simulate complex fluid flow around the obstacles inside the cavity. In order to made better convergence, well-posed boundary conditions should be defined in the domain, such as no-slip conditions on the side and bottom solid-wall surfaces as well as the surface of obstacles and uniform horizontal velocity at the top of the cavity. This study focused on the flow inside a square cavity with internal obstacles with the objective of observing the effect of the Reynolds number and size of the internal obstacles on the flow characteristics and primary/secondary vortex formation. The current LBM has been successfully used to precisely simulate and visualize the primary and secondary vortices inside the cavity. In order to validate the results of this study, the results were compared with existing data. In the case of a cavity without any obstacles, as the Reynolds number increases, the primary vortices move toward the center of the cavity, and the secondary vortices at the bottom corners increase in size. In the case of the cavity with internal obstacles, as the Reynolds number increases, the secondary vortices close to the internal obstacle become smaller owing to the strong primary vortices. In contrast, depending on the sizes of the obstacles ( R / L = 1/16, 1/6, 1/4, and 2/5), secondary vortices are induced at each corner of the cavity and remain stationary, but the secondary vortices close to the top of the obstacle become larger as the size of the obstacle increases.

Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition

For arbitrary two-point boundary condition, which makes the corresponding linear uniform problem well-posed, we obtain an existence and uniqueness result for the boundary value problem of finite beam deflection resting on arbitrary nonlinear non-uniform elastic foundation. The difference between the desired solution and the corresponding linear uniform one in L^{infty} sense is bounded explicitly in terms of given inputs of the problem. Our results seamlessly unify linear uniform and nonlinear non-uniform problems and lead to an iteration algorithm for uniformly approximating the desired deflection.

Lyapunov-Type Inequalities for Riemann-Liouville Type Fractional Boundary Value Problems with Fractional Boundary Conditions

In this article, we establish Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with well-posed fractional boundary conditions. To illustrate the applicability of established results, we estimate lower bounds for eigenvalues of the corresponding eigenvalue problems and deduce criteria for the nonexistence of real zeros of certain Mittag-Leffler functions.

On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation

In this paper, we develop a provably energy stable discontinuous Galerkin spectral element method (DGSEM), approximating the perfectly matched layer (PML) for the three and two space dimensional (3D and 2D) linear acoustic wave equations, in first order form, subject to well-posed linear boundary conditions. First, using the well-known complex coordinate stretching, we derive an efficient un-split modal PML for the 3D acoustic wave equation, truncating a cuboidal computational domain. Second, we prove asymptotic stability of the continuous PML by deriving energy estimates in the Laplace space, for the 3D PML in a heterogeneous acoustic medium, assuming piece-wise constant PML damping. In the time-domain, the energy estimate translates to a bound for the solutions in terms of the initial data. Third, we develop a DGSEM for the wave equation using physically motivated numerical flux, with penalty weights, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yields an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Fourth, to ensure numerical stability of the discretization when PML is present, it is necessary to systematically extend the numerical fluxes, and the inter-element and boundary procedures, to the PML auxiliary differential equations. This is critical for deriving discrete energy estimates analogous to the continuous energy estimates. Finally, we propose a procedure to compute PML damping coefficients such that the PML error converges to zero, at the optimal convergence rate of the underlying numerical method. Numerical solutions are evolved in time using the high order Taylor-type time stepping scheme of the same order of accuracy of the spatial discretization. By combining the DGSEM spatial approximation with the high order Taylor-type time stepping scheme and the accuracy of the PML we obtain an arbitrarily accurate wave propagation solver in the time domain. Numerical experiments are presented in 2D and 3D corroborating the theoretical results.

Drag Decomposition Using Partial-Pressure Fields in the Compressible Navier–Stokes Equations

The static-pressure field in the steady and compressible Navier–Stokes equations is decomposed into Euler (inviscid) and dissipative (viscous) partial-pressure fields as a generalization of the incompressible pressure decomposition previously reported by Schmitz and Coder (“Inviscid Circulatory-Pressure Field Derived from the Incompressible Navier–Stokes Equations,” AIAA Journal, Vol. 53, No. 1, 2015, pp. 33–41). The primary purpose of partial-pressure fields is to provide a means of dissecting local drag contributors over a lifting body. The analysis shows that the integral of the Euler pressure over the surface of a lifting body of thickness recovers the Kutta–Joukowski theorem for lift, and results in Maskell’s formula for the vortex-induced drag in the limit of high Reynolds number; the combined integral of the dissipative pressure and wall shear stress results in a generalized form of Oswatitsch’s formula for entropy-flux drag. Transport equations are derived with well-posed boundary conditions for both the Euler and dissipative partial-pressure fields for implementation in computational fluid dynamics codes as a complement to far-field and volumetric methods for drag decomposition of complex aircraft configurations.