Integral Equation Formulation Research Articles

Differential dynamic programming for finite‐horizon zero‐sum differential games of nonlinear systems

AbstractIn this article, we present an iterative algorithm based on differential dynamic programming (DDP) for finite‐horizon two‐person zero‐sum differential games. The technique of DDP is used to expand the Hamilton–Jacobi–Isaacs (HJI) partial differential equation into higher‐order differential equations. Using value function and saddle point approximations, the DDP expansion is transformed into algebraic matrix equation in integral form. Based on the algebraic matrix equation, a DDP iterative algorithm is developed to learn the solution to the differential games. Strict proof is proposed to guarantee the iterative convergences of the value function and saddle point. The new algorithm is fundamentally different from existing results, in the sense that it overcome the technical obstacle to address the time‐varying behavior of HJI partial differential equation. Simulation examples are given to demonstrate the effectiveness of the proposed method.

Isentropic Swirling Flow Through Supersonic Nozzles - Part I (free Vortex Flow)

An analytical model has been developed and an approximate solution for the free vortex swirling supersonic flow through nozzles. Earlier work by Mager, is potential flow based. In the present model swirl number has been redefined to confine the flow properties within the valid range. Flow properties are always determined by the total conditions (P0 and a0), nozzle profile and swirl intensity as in the case of one-dimensional flow. Integral form of the continuity equation has been used in order to satisfy the boundary conditions, since radial velocity component is ignored. For free vortex case, it has been shown that there exists a minimum radius along the length of the nozzle at which the density is zero and flow seems to be void causing reasonable blockage to the flow at throat. In defining the non-dimensional swirl number (β), axial velocity at each section was employed for non-dimensionalization as compared to Mager’s definition of using a global constant to find swirl number (α). The variations of performance parameters of the nozzle such as co-efficient of discharge (Cd) thrust efficiency (ηs) and impulse efficiency (ηI) and flow field parameters such as density, pressure, velocity etc. are presented as a function of swirl numbers (α, β) and compared with the available experimental data. The swirl numbers (α, β) are also correlated.

THE APPLICATION OF THE EXTENDED CELLS METHOD TO SIMULATE THE FLOW OF COMBUSTION GASES IN THE LPRE CHAMBER


 
 
 Abstract. The numerical modeling of the process of two-dimensional axisymmetric flow of combustion gases in the chamber of a liquid rocket engine is considered in this study. In general, when solving such problems, meshes are used which lines coincide with the boundaries of the computational domain. However, an alternative solution is proposed here, which is to apply the extended cells method. It allows using rectangular Cartesian grids, which lines do not coincide with the boundaries of the computational domain, without reducing the stability of the numerical solution due to the fractional finite volumes. This also simplifies the setting of boundary conditions in such volumes. The advantage of the proposed approach over the generally accepted one is the absence of the global geometric transformations during the entire modelling process, which leads to a reduction in its duration. To perform the numerical modelling, an inviscid ideal compressible gas of constant chemical composition was chosen as a basic model of a continuum. It is described by a system of the unsteady Euler equations in integral form, which was closed by the Mendeleev-Clapeyron equation of state. For the numerical solution of this system, the finite volume method was used with the reconstruction of the flow parameters by the WENO algorithm of the third order of accuracy. The solution of the Riemann problem was carried out using the Lax-Friedrichs relations. Time integration of the system of equations was performed using the explicit Runge-Kutta method of the third order of accuracy. All calculations were carried out on a uniform rectangular Cartesian mesh, which lines did not coincide with the boundaries of the computational domain. The results were compared with the solution of the same problem using the ANSYS Fluent on an unstructured mesh coinciding with the boundaries of the computational domain. The value of the relative error obtained as a result of comparing both solutions did not exceed 0.05.
 
 

A numerical investigation of unsteady space–time dependent coefficients anisotropic-diffusion convection reaction equation

Two dimensional transient problems for anisotropic FGMs governed by a diffusion convection reaction (DCR) equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation which can be written in forms of boundary-only integral equations and solved using a standard BEM for the solutions. These BEM solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials with compressible or incompressible flow are solved. The numerical solutions obtained are quite accurate.

Current Based Automated Design of Realizable Metasurface Antennas With Arbitrary Pattern Constraints

We present a 3-D method to numerically design a realizable metasurface, which transforms a given incident field into a radiated field that satisfies mask-type (inequality) constraints. The method is based on an integral equation formulation, with local impedance boundary condition (IBC) approximation. The procedure yields the spatial distribution of the impedance, yet the process involves the synthesis of the equivalent current only. This current is constrained to correspond to a realizable surface impedance, i.e., passive, lossless, and with reactance values bounded by practical realizability limits. The current-based design avoids any solution to the forward problem, and the impedance is obtained from the synthesized current only at the end of the process. The procedure is gradient-based, with the gradient expressed in closed form. This allows handling large metasurfaces, with full spatial variability of the impedance in two dimensions. The method requires no a priori information, and all relevant operations in the iterative process can be evaluated with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol { O}(N~\log~{N})$ </tex-math></inline-formula> complexity. Application examples concentrate on the case of on-surface excitation and far-field (FF) pattern specifications; they show designs of circular and rectangular metasurface antennas of 20 wavelengths in size, with pencil- and shaped-beam patterns, and for both circular and linear polarizations.

An Iterative Nonconformal Domain Decomposition Surface Integral Equation Method for Scattering Analysis of PEC Objects

Domain decomposition methods (DDMs) can effectively solve electromagnetic scattering problems induced by multiscale perfectly electrically conducting (PEC) objects. However, boundary conditions between adjacent subdomains usually need to be carefully treated, especially when nonconformal meshes exist. To mitigate this issue, an iterative nonconformal domain decomposition surface integral equation (IDD-SIE) formulation is proposed. Each subdomain is independently solved with scattering fields generated from other subdomains as additional excitations. The continuity of current densities across touching interfaces is guaranteed by iteratively solving coupled fields among subdomains. Therefore, troublesome boundary transmission conditions can be avoided and nonconformal meshes are intrinsically supported. Numerical results show that the proposed IDD-SIE formulation is accurate and efficient, and has excellent convergent properties in terms of iteration numbers.

On Preconditioners of the FFT-JVIE for Inhomogeneous Dielectric Objects

An improved block-diagonal preconditioner (IBDP) is developed for the FFT accelerated the volume integral equation formulation based on equivalent volumetric currents (FFT-JVIE). A comparative study on the realistic models is carried out among the IBDP, BDP, the previously developed analytical preconditioner (AnP) and the circulant-based preconditioners (CBP).

Automatic CHIEF Point Selection for Finite Element–Boundary Element Acoustic Backscattering

Computing the backscattering of harmonic acoustic waves from underwater elastic targets of arbitrary shapes is a challenging problem of considerable practical significance. The finite element method is well suited for the discretization of the target, while the boundary element method addresses the radiation boundary condition at infinity. A disadvantage of the boundary integral method is that it yields non-unique solutions at certain wavenumbers. This failure is associated with the existence of eigensolutions of the Helmholtz equation in the interior of the complement of the fluid domain (acoustic modes). The combined Helmholtz integral equation formulation (CHIEF) credited to Schenk is employed to combine the surface Helmholtz boundary integral with equations of the interior Helmholtz relation written down at selected points within the cavity of the scatterer (i.e., in the complement of the fluid domain).The difficulty associated with this approach has always been the lack of guidance on the necessary number of interior points and on their locations. The solution to this problem proposed here is to compute the acoustic modes using the finite element method to complement of the fluid domain and to identify locations of the peaks.This novel approach aids the decision as to how many points should be employed and where they should be located. Our numerical experiments demonstrate the robustness of the proposed automatic selection of the CHIEF points’ numbers and locations.

An Excitation-Aware and Self-Adaptive Frequency Normalization for Low-Frequency Stabilized Electric Field Integral Equation Formulations

The accurate solution of quasi-Helmholtz decomposed electric field integral equations (EFIEs) in the presence of arbitrary excitations is addressed: Depending on the specific excitation, the quasi-Helmholtz components of the induced current density do not have the same asymptotic scaling in frequency, and thus, the current components are solved for with, in general, different relative accuracies. In order to ensure the same asymptotic scaling, we propose a frequency normalization scheme of quasi-Helmholtz decomposed EFIEs which adapts itself to the excitation and which is valid irrespective of the specific excitation and irrespective of the underlying topology of the structure. Specifically, neither an ad-hoc adaption nor <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a-priori</i> information about the excitation is needed as the scaling factors are derived based on the norms of the right-hand side (RHS) components and the frequency. Numerical results corroborate the presented theory and show the effectiveness of our approach.

A Generalized Scalar Potential Integral Equation Formulation for the DC Analysis of Conductors

The electrostatic modeling of conductors is a fundamental challenge in various applications, including the prediction of parasitic effects in electrical interconnects, the design of biasing networks, and the modeling of biological, microelectromechanical, and sensing systems. The boundary element method (BEM) can be an effective simulation tool for these problems because it allows modeling three-dimensional objects with only a surface mesh. However, existing BEM formulations can be restrictive because they make assumptions specific to particular applications. For example, capacitance extraction formulations usually assume a constant electric scalar potential on the surface of each conductor and cannot be used to model a flowing current, nor to extract the resistance. When modeling steady currents, many existing techniques do not address mathematical challenges such as the null space associated with the operators representing the internal region of a conductor. We propose a more general BEM framework based on the electric scalar potential for modeling conductive objects in various scenarios in a unified manner. Restrictive application-specific assumptions are not made, and the aforementioned operator null space is handled in an intuitive and rigorous manner. Numerical examples drawn from diverse applications confirm the accuracy and generality of the proposed method.

The Regularization Technique in Modeling of the Plane E-Polarized EM Wave Scattering by Coplanar System of Electrically Conducting Flat Strips

The integral equations technique has been successfully modified for studying the plane E-polarized electromagnetic wave scattering by multilevel coplanar systems of zero-thickness impedance strips. Reformulation of the scattering problem in the form of the second-kind regular integral equations has been realized on the base of the Carleman regularization technique. Two novel and original classes of specific Cantor functions have been presented and analyzed. Using new Cantor functions, it is easy to create a lot of non-classical orderings for specific multilevel coplanar strips systems. That sort of system can be useful for modeling irregular natural processes and zones in the future. Considerable attention was focused on the plane E-polarized electromagnetic wave scattering by sparsely filled coplanar systems of electrically narrow impedance strips. An explicit solution of the scattering problem has been obtained for such case of strips system.

Efficient spectral implementation of ODE wall model and the extension of integral wall model to unstructured LES solvers

While wall modeling has become an indispensable tool for scale-resolving simulation of high Reynolds numbers wall-bounded turbulent flows, discussion of efficient implementation of wall models in parallel unstructured-grid solvers is lacking. In this paper, we discuss physical and numerical improvements to two zonal wall models based on integral form of the boundary layer differential equations within an unstructured-grid cell-centered finite-volume LES solver. The first model is a novel implementation of the ODE equilibrium wall model, where the velocity profile is expressed in the integral form using the constant shear-stress layer assumption and the integral is evaluated using a spectral quadrature method, resulting in a local and algebraic (grid-free) formulation. The second model is the modified form of the integral wall model of Yang et al. (2015) [14], which is based on the vertically-integrated thin-boundary-layer PDE along with a prescribed composite velocity profile in the wall-modeled region. A modification to the inner layer velocity profile which fixes the coordinate invariance issue of the original model is proposed. Additionally, several numerical challenges unique to the implementation of these integral models in unstructured mesh environments, such as the exchange of wall quantities between wall faces and LES cells, and the computation of surface gradients, are identified and possible remedies are proposed. The computational performance of the wall models is assessed both in a priori and a posteriori settings against the traditional finite-volume based ODE equilibrium wall model, showing a comparable computational cost for the integral wall model, and superior cost scaling for the spectral implementation over the finite-volume based approach. Load imbalance among the processors in parallel simulations seems to severely degrade the parallel efficiency of finite-volume based ODE wall model, whereas the spectral implementation is remarkably agnostic to these effects. The integral wall model, however, is even more prone to parallel efficiency degradation than the finite-volume based ODE wall model due to communication overhead among processors in highly parallelized settings.

Semi-Analytical Solution of Transverse Vibration of Cylinders with Non-Circular Cross-Section Partially Submerged in Water

The free transverse vibration of a surface-piercing, vertical cylinder partially submerged in water was studied. The cylinder had an arbitrary non-circular, but symmetric, cross-section in the vibration direction. The water was assumed to be an incompressible and inviscid fluid. The effect of the surface waves of water was neglected in the analysis. The exact solution of velocity potential of water was derived by the method of separation of variables. The unknown coefficients in the solution of the velocity potential were expressed in the form of integral equations, including the dynamic deformation of the beam. Then, the governing differential equation of bending vibration of the cylinder under the hydrodynamic pressure was obtained. The Galerkin method was used to obtain the eigenvalue equation by expanding the wet modes of the cylinder into a series of dry modes. The elliptical cylinders partially submerged in water were taken as the numerical example. The accuracy of the proposed method was evaluated by the convergence studies. As a consequent result, the non-dimensional added virtual mass incremental (NAVMI) factor solutions were compared to the present Galerkin solutions, which can be used as a benchmark test for more sophisticated numerical simulations of computational fluid dynamics.

The inapplicability of variational methods in the cohesive crack problem for initially rigid traction‐separation relation and its solution using integral equations

AbstractA cohesive crack problem with initially rigid traction‐separation relation has been considered here. It has been shown that the total potential energy is not differentiable at the crack‐tip. This makes the application of a variational operator over this total potential energy theoretically incorrect in the general sense. It further implies that variational methods are not applicable in this situation. One way to overcome this problem is to introduce a penalty of initial stiffness in the traction‐separation relationship. However, this means modifying the material property of a solid. To avoid variational operator, an integral equation form of this problem has been numerically solved. The results thus obtained conform to the present understanding of cohesive stresses. This method is theoretically more accurate to solve a cohesive crack problem whenever the total potential energy is non‐differentiable.

SOLVABILITY OF QUADRATIC HADAMARD-TYPE FRACTIONAL INTEGRAL EQUATIONS IN ORLICZ SPACES

We demonstrate some properties of Hadamard fractional operators such as boundedness, monotonicity, continuity, and acting conditions in Orlicz spaces Lφ. We apply these properties with a proper measure of noncompactness (MNC) to inspect the existence of monotonic solutions of some general, but abstract, form of quadratic Hadamard-type fractional integral equations in Lφ. We also discuss the uniqueness of the solution.

Cosmic voids and the kinetic analysis

We consider a principal problem, that of the possible dominating role of self-consistent gravitational interaction in the formation of cosmic structures: voids and their walls in the local Universe. It is in the context of the Hubble tension as a possible indication of the difference in the descriptions of the late (local) and early (global) Universe. The kinetic Vlasov treatment enables us to consider the evolution of gravitating structures where the fundamental role has the modified gravitational potential with a cosmological constant, leading to the prediction of a local flow with a Hubble parameter that is nonidentical to that of the global Hubble flow. The Poisson equation for a potential with an additional repulsive term, including an integral equation formulation, is analyzed, and we predict the appearance of multiply connected two-dimensional gravitating structures and voids in the local Universe. The obvious consequence of the developed mechanism is that the cosmological constant poses a natural scaling for the voids, along with the physical parameters of their local environment, which can be traced in observational surveys.

An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems

The use of boundary elements in the analysis of exterior acoustic problems poses challenges at specific frequencies, since fictitious eigenfrequencies may arise at the internal resonances of cavities, leading to inaccurate results or even unstable behavior. To filter out these fictitious eigenfrequencies, a scheme based on the combined Helmholtz integral equation formulation (CHIEF) can be used to prevent the so-called non-uniqueness problem, although it requires additional equations and points. The BEM formulation final accuracy will, however, depend on the correct choice of these points. Here, a strategy to help in defining good approximations for the position and number of such points is proposed, based on an optimization process which maximizes the system matrix’s smallest singular value. The accuracy of the method for exterior radiation problems is investigated using different examples. With low computational cost and simple implementation, the two proposed algorithms automatically circumvent the non-uniqueness problem, aiding the implementation of more stable BEM codes.

A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method

This research focuses on the analysis of the competitive model used in the banking sector based on the stochastic fractional differential equation. For the approximate solution, a pseudospectral technique is utilized for the proposed model based on the stochastic Lotka–Volterra equation using a wide range of fractional order parameters in simulations. Conditions for stable and unstable equilibrium points are provided using the Jacobian. The Lotka–Volterra equation is unstable in the long term and can produce highly fluctuating dynamics, which is also one of the reasons that this equation is used to model the problems arising in finance, where fluctuations are important. For this reason, the conventional analytical and numerical methods are not the best choices. To overcome this difficulty, an automatic procedure is used to solve the resultant algebraic equation after the discretization of the operator. In order to fully use the properties of orthogonal polynomials, the proposed scheme is applied to the equivalent integral form of stochastic fractional differential equations under consideration. This also helps in the analysis of fractional differential equations, which mostly fall in the framework of their integrated form. We demonstrate that this fractional approach may be considered as the best tool to model such real-world data situations with very reasonable accuracy. Our numerical simulations further demonstrate that the use of the fractional Atangana–Baleanu operator approach produces results that are more precise and flexible, allowing individuals or companies to use it with confidence to model such real-world situations. It is shown that our numerical simulation results have a very good agreement with the real data, further showing the efficiency and effectiveness of our numerical scheme for the proposed model.

Closed‐form solution of fluid flow in and around a crack disk embedded in a 3D porous medium

AbstractThis paper considers the fluid flow through a porous medium containing intersecting fractures and presents three main analytical findings, namely: (1) mass exchange between fractures and surrounding matrix at the fracture intersection; (2) fluid potential solution (pressure field) within the whole domain under the form of a single singular integral equation; and (3) closed‐form solutions of fluid flow in and around a crack disc under a far field pressure gradient. The crack is represented mathematically by a 2D smooth surface (i.e., zero thickness) within a 3D porous medium, while physically by a constant aperture. The fluid flow within the crack obeys Poisseuille's law, while Darcy's law is used to represent the fluid flow in the surrounding matrix. The general solution of pressure field for the general case of multiple intersecting cracks is firstly derived under a singular integral equation form. The mass exchange between the porous matrix and the crack, as well as the mass conservation at the intersection between cracks are the keys to obtaining this general solution. Then, the general solution is written for the case of a single crack. Rigorous derivation of the latter equation allows obtaining a closed‐form solution of flow through a single crack. Introducing this solution of flow into the general equation gives the pressure field around the crack. The solution derived in this paper for a crack disk with Poisseuille's flow is slightly different from the well‐known Eshelby's solution for the case of flattened inclusion in which the flow obeys Darcy's law.

Theory of Edge Effects and Conductance for Applications in Graphene-Based Nanoantennas

In this paper, we present a theory of edge effects in graphene for its applications to nanoantennas in the THz, infrared, and visible frequency ranges. The novelty of the presented model is reflected in its self-consistency, which is reached due to the formulation in terms of dynamical conductance instead of ordinary surface conductivity. The physical model of edge effects is based on using the concept of the Dirac fermion and the Kubo approach. In contrast with earlier well-known and widely used models, the surface conductance becomes non-homogeneous and non-local. The numerical simulations of the spatial behavior of the surface conductance were performed in a wide range of values, known from the literature, for the graphene ribbon widths and electrochemical potential. It is shown that if the length exceeds 800 nm, our model agrees with the classical Drude conductivity model with a relatively high degree of accuracy. For rather short lengths, the conductance exhibits a new type of spatial oscillations, which are not present in the ordinary conductivity model. These oscillations modify the form of effective boundary conditions and integral equations for electromagnetic field at the surface of graphene-based antenna. The developed theory opens a new way for realizing electrically controlled nanoantennas by changing the electrochemical potential via gate voltage. The obtained results may be applicable for the design of different carbon-based nanodevices in modern quantum technologies.