Exterior Problems Research Articles

Analysis of Slitted Parallel‐Plate Waveguides Using the Method of Kobayashi Potential

AbstractA new solution for the analysis of slitted parallel‐plate waveguide is proposed using the method of Kobayashi Potential (KP). The interior and exterior problems are studied for the TE and TM polarizations. Simplified forms of Weber‐Schafheitlin (WS) integrals are utilized for problem formulation. The Fourier function space is exploited to construct the governing linear system of equations. A simple strategy is suggested for the evaluation of the required integrals. Numerical results are validated through convergence analysis and available reports based on the Moment Method (MM). It is shown that the KP is superior to the conventional MM in the sense of tracking edge singularities, especially when the radiation problem is of interest.

Concepts for Frequency Sweeps and Efficient Repeated Analysis in the Context of Vibroacoustic Optimization and Uncertainty Quantification

This paper aims at passive noise control for vibroacoustic problems, which are analyzed by finite and boundary element techniques. The author distinguishes interior and exterior problems mainly because of the quantities used as the objective function to assess the acoustic quality. For interior problems, it is common to use local quantities such as the sound pressure at a field point or, in rare cases, energy density at a field point. The situation is different for exterior problems where the radiated sound power accounts for a suitable and global quantity to assess the emission from a vibrating structure. For most engineering purposes, the assessment requires frequency sweeps in which the problem needs to be solved at many discrete frequencies. In vibroacoustic optimization and in sampling based uncertainty quantification, it is very common that structural parameters are varied, while the acoustic field remains the same throughout the entire process. We will review concepts and recent developments of efficient frequency sweeps and repeated analysis with unmodified fluid domain. For many practical cases, the situation for interior problems is rather simple to survey. Either the authors have applied a modal analysis and used a modal superposition for the frequency sweep and repeated analysis, or the concept of unmodified acoustic transfer vectors is applied. Both concepts are quite successful as long as certain conditions are fulfilled. For exterior problems, a modal superposition is possible but, so far, only for a limited number of cases practically applicable as discussed herein. The concept of using acoustic transfer vectors becomes inefficient since the evaluation of the radiated sound power asanintegral over a closed enveloping surface would require an excessively high amount of storage capacity. Therefore, other concepts are being followed. For frequency sweeps, a number of methods are using a frequency interpolation based on a limited number of discrete sample frequencies. Often, these techniques are used together with Krylov–subspace model order reduction techniques. Further recently published approaches investigate low-rank approximations, greedy algorithms, and deflated Krylov subspace techniques. A completely different kind of approach is based on multi–fidelity models and Gaussian processes. The field of efficient repeated analysis shows some interesting developments, which can be easily applied to sampling based uncertainty quantification but do not seem to be easily and generally applied to optimization.

Spectral properties of the Dirichlet-to-Neumann operator for spheroids.

We study the spectral properties of the Dirichlet-to-Neumann operator and the related Steklov problem in spheroidal domains ranging from a needle to a disk. An explicit matrix representation of this operator for both interior and exterior problems is derived. We show how the anisotropy of spheroids affects the eigenvalues and eigenfunctions of the operator. As examples of physical applications, we discuss diffusion-controlled reactions on spheroidal partially reactive targets and the statistics of encounters between the diffusing particle and the spheroidal boundary.

Finite element analysis of a generalized rotating FGM vessel subjected to thermo-mechanical loadings: Effect of Poisson ratio and inhomogeneity parameters

Cylinders and thick walled cylindrical shells are commonly utilized in several industries to transport and store fluids under certain pressure and temperature conditions. In the present paper, a numerical solution is developed in order to investigate displacement, temperature and stress fields in a rotating pressure vessel made of generalized functionally graded material (FGM) subjected to different thermo-mechanical boundary conditions. The aim is to investigate the effect of Poisson ratio, internal pressure and temperature and inhomogeneity parameters on the stress and deformation distributions of the rotating pressure vessel. The material is considered isotropic nonhomogeneous and linearly elastic with its properties varying along the radial direction. Additionally, certain conditions, such as exterior or interior problems where r → ∞ or r → 0, respectively, are impossible to resolve using the variation of attributes as a power-law distribution. An approach to the spatial Young modulus distribution that is more broad has been suggested in the literature which can be applied to such physical challenges. The rotation of the pressure vessel is considered in the analysis, and the temperature distribution is assumed to be non-uniform. Since an analytical solution to the differential equation is not accessible, the conventional Galerkin discretization approach of the Finite Element Method (FEM) is applied, nowadays is considered one of the main numerical tools for solving Boundary Value Problems (BVP). It is addressed how stress, strain, and displacement are affected by the inhomogeneity parameter, rotation speed, pressure, temperature, and Poisson ratio. The examination of the various findings indicates that changes in the temperature profile, rotation, and inhomogeneity parameter on the thermoelastic field have a substantial impact on the stress and strain in the FGM cylinder. The findings indicate that the Poisson ratio and inhomogeneity parameters have a significant impact on the stress and deformation distributions. According to the results, the above-mentioned parameters can be adapted to control the thermoelastic filed in a FGM cylinder. The present research offers significant perspectives on the development and enhancement of rotating FGM pressure vessels intended for high-temperature applications.

Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain

In this study, the finite propogation speed properties investigated for a two dimensional exterior problem defined by nonlinear Klein-Gordon equation. Under some assumptions on the initial data and the nonlinearity, the solution is shown to have a finite propogation speed. Furthermore, it is demonstrated that the problem has a unique solution, and accurate numerical solutions have been produced by the use of the dual reciprocity boundary element approach with linear radial basis functions.

On the Exterior Problem for Parabolic Hessian Quotient Equations

On the Exterior Problem for Parabolic Hessian Quotient Equations

Modal reduction for exterior acoustic and vibroacoustic problems

Modal analysis, mode superposition and modal reduction are considered standard approaches for interior acoustic problems as well as for pure vibration problems. Things are different for exterior problems. Only a few methods are known to formulate a linearized eigenvalue problem for unbounded acoustic problems. Even fewer techniques on modal superposition and reduction are found in the literature. This talk will review such techniques which are either based on conjugate Astley-Leis infinite elements or on boundary elements for the unbounded region. While the results appear to be promising, a number of open questions need to be solved in this context. Among others, it is yet unclear which modes are actually required for a modal reduction and how to find a method to just evaluate these modes and the related eigenvalues. The situation is different for vibroacoustic radiation problems with clear resonances. In such cases, a modal reduction is able to substantially accelerate the solution process. Future work might lead to modal reduction using modes from non-linear eigenvalue problems which are known from the numerous papers on the boundary element methods published over the last decade.

Wave scattering problems in exterior domains with the method of fundamental solutions

The method of fundamental solutions has been mainly applied to wave scattering problems in bounded domains and to our knowledge there have not been works addressing density results for general shapes, or addressing the calculation of the complex resonance frequencies that occur in exterior problems. We prove density and convergence of the fundamental solutions approximation in the context of wave scattering problems, with and without a priori knowledge of the frequency, which is of particular importance to detect resonance frequencies for trapping domains. We also present several numerical results that illustrate the good performance of the method in the calculation of complex resonance frequencies for trapping and non trapping domains in 2D and 3D.

A Boundary Node Method for Solving Interior and Exterior Acoustic Problems

The Boundary Node Method (BNM) is a meshfree scheme for solving Boundary Integral Equations (BIE). BNM simplifies the problem at two levels. Primarily, as BNM aims to solve the integral form of the governing equation, dimensionality of the problem is reduced by one order. Additionally, the mesh-free scheme eliminates the need for meshing, proving particularly beneficial for problems involving complex geometries. In this study, BNM is formulated using Element Free Galerkin (EFG) scheme to solve interior and exterior problems in acoustics. A 3-dimensional linear basis is used to construct moving least square shape functions. This eliminates the need to define additional local or parametric coordinate systems, making the method easily applicable to any arbitrary geometry. To illustrate the capabilities of the proposed method, four example problems are solved. An open pipe at resonance and a simple expansion muffler are analyzed to validate BNM’s performance in solving interior problems. Radiation from a pulsating sphere and radiation from a sphere with harmonic velocity excitation are analyzed as examples of exterior problems. Results from all four sample problems indicate that the BNM scheme proposed for solving acoustic problems is accurate and robust.

3D isogeometric indirect BEM solution based on virtual surface sources on the boundaries of Helmholtz acoustic problems

A solution for 3D Helmholtz acoustic problems is introduced based on an indirect boundary element method (indirect BEM) coupled with isogeometric analysis (IGA). The novelty of this work arises from using virtual surface sources placed directly on the scatterer boundaries, producing robust results. These virtual surface sources are discretized by the same Non-Uniform Rational B-Splines (NURBS) approximating the scatterer CAD model. This allows modeling of general irregular geometries. The proposed solution has the same features of BEM approaches, which do not need any domain discretization or truncation boundaries at the far-field. It shows an additional merit by arranging the linear system of equations directly depending on a single coefficient matrix, consuming less computational time compared to other BEM methods. A Greville abscissae collocation scheme is proposed with offsets at C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document}-continuities. This collocation scheme allows for easy evaluation for both free-terms and normals at the collocation points. The performance of the proposed solution is discussed on 3D numerical exterior problems and compared against other BEM methods. Then, the practical interior muffler problem with internal extended thin tubes is studied and the obtained results are compared against other numerical methods in addition to the available experimental data, showing the capability of the proposed solution in handling thin-walled geometries.

A comprehensive review of numerical simulation methods for hydraulic fracturing

AbstractHydraulic fracturing unlocks previously inaccessible hydrocarbons in unconventional reservoirs by creating artificial pathways in the unconventional reservoir. Numerical simulation expands the scope of hydraulic fracturing design for various reservoir conditions. This review paper explores the synergy between numerical simulation and hydraulic fracturing modeling, focusing on critical elements like geomechanical behavior, geological conditions, and fluid dynamics. Analytical models in hydraulic fracturing design are discussed to underscore their foundational importance. The assumption of constant fracture height limits the application of Perkins–Kern–Nordgren model (PKN) and Kristianovich‐Geertsma‐de Klerk model (KGD). Radial models assume 3D flow but lack reliability in nonradial settings, and the Pseudo 3D model (P3D) shares PKN's assumptions with variable fracture height, sacrificing some details for efficiency. Planar 3D model (PL3D) enhances accuracy by discarding PKN's elastic response assumption but requires extended computation. Unconventional fracture model is effective for complex scenarios but relies on DFN modeling parameters for accuracy. The choice of numerical simulation method in hydraulic fracturing depends on the specific aspect studied, each with its strengths and limitations. For instance, boundary element methods are efficient for exterior problems, finite element modeling suits 3D nonplanar fractures, and the extended finite element method excels in hydraulic and natural fracture interactions. Peridynamics shows potential but needs further development for cost‐effectiveness. PFC and UDEC/3DEC‐based simulations can explore microscopic mechanisms. Combining these methods with other approaches provides a comprehensive study of realistic reservoir conditions. This review guides the selection of a suitable numerical simulation methodology based on the study's scope.

The scalar Oseen problem in two-dimensional exterior domains

In the present paper, we investigate the scalar Oseen equation in a two-dimensional exterior domain Ω, with Dirichlet boundary condition. This problem represents a linearized form of the steady Navier-Stokes equations. In order to better control the behavior at infinity of functions, the use of weighted Sobolev spaces as a functional framework turns out to be adequate. The established results are related to the existence and the uniqueness of weak solutions on Lp-theory for any real 1<p<∞. Our analysis is mainly based on the principle that linear exterior problems can be solved by combining their properties in bounded domains which are also established in this paper and their properties well known in the whole space R2.

An image reconstruction algorithm for spectral exterior CT problem based on edge-preserving diffusion and edge-preserving smoothing

The exterior CT problem is a special x-ray CT imaging problem, which is often used for nondestructive inspections of large tubular samples. It can image a pipeline wall with a relatively small detector. However, exterior imaging is challenging owing to the incomplete data obtained. In this paper, we propose a regularization model for the exterior CT problem, in which an edge-preserving diffusion and edge-preserving smoothing regularizer is employed. In addition, the polychromatic characteristic of the emitted x-rays is also considered so that the model is closer to the actual physical process. Using the linearization strategy, a corresponding solution algorithm is derived. The proposed model and the algorithm are verified with both simulated data and real data. It is shown that the two type of artifacts, exterior problem artifacts and beam-hardening artifacts, can be effectively suppressed.

An efficient Fourier-Laguerre spectral-Galerkin method for exterior problems of two-dimensional complex obstacles

In this paper, we propose a Fourier-Laguerre spectral method for exterior problems of two-dimensional complex obstacles based on the mapping method. We first use a polar coordinate transformation to convert the exterior domains of complex obstacles into the exterior domain of the unit disk. Then we apply the polar coordinate transformation to the exterior problem of an elliptic equation, derive its weak formulation and prove the existence and uniqueness of the weak solution. On this basis, we construct the Fourier-Laguerre spectral Galerkin scheme, describe its numerical implementation and analyze the convergence of the numerical solution under the H1-norm. Numerical results indicate that our spectral Galerkin method is easy to implement and possesses high-order accuracy.

Acoustic sensitivity analysis for 3D structure with constant cross-section using 2.5D singular boundary method

In this paper, a 2.5D singular boundary method (SBM) in conjunction with the direct differentiation method (DDM) and adjoint variable method (AVM) are formulated for the sensitivity analysis of 3D longitudinally invariant structures. The assumption of a constant cross-section in the longitude direction enables it possible to decouple the 3D system into 2D problems at every wavenumber, and thus makes it a 2.5D-problem. The 2D sensitivity problem is solved by a boundary-type method, SBM. The SBM solves a problem with a linear combination of the fundamental solutions with respect to boundary collocation points. The fundamental solution makes the SBM available for exterior problems without the artificial truncated boundary. The source singularity issue of fundamental solution is overcome by a simple analytical formula and its derivatives. Numerical experiments present the accuracy, efficiency and feasibility of the proposed methods, and indicate that the proposed methods can be considered as an effective alternative in 3D problems with a longitudinally invariant cross-section.

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in R3. We assume that the flow is periodic in x3-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. This reveals the relation between the integrability of ∇v and the decay rate of ω near the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

Topology optimization of a rotating electric machine by the topological derivative

AbstractWe consider the topology optimization of a rotating electric machine in magnetoquasistatic operation in two space dimensions. This amounts to a topology optimization problem subject to a parabolic PDE constraint on a moving domain which we intend to solve by means of the topological derivative concept. For that purpose, we consider a topological perturbation of the materials in the space‐time cylinder along a trajectory given by the rotation of the machine. Using a Lagrangian approach, we derive the topological derivative formula, which depends on the solution of an exterior problem that is bounded in time and unbounded in the space directions. We use this sensitivity information for the design optimization of an electric motor by means of a level set method.

Short-time diffusive fluxes over membrane receptors yields the direction of a signalling source.

An essential ability of many cell types is to detect stimuli in the form of shallow chemical gradients. Such cues may indicate the direction that new growth should occur, or the location of a mate. Amplification of these faint signals is due to intra-cellular mechanisms, while the cue itself is generated by the noisy arrival of signalling molecules to surface bound membrane receptors. We employ a new hybrid numerical-asymptotic technique coupling matched asymptotic analysis and numerical inverse Laplace transform to rapidly and accurately solve the parabolic exterior problem describing the dynamic diffusive fluxes to receptors. We observe that equilibration occurs on long timescales, potentially limiting the usefulness of steady-state quantities for localization at practical biological timescales. We demonstrate that directional information is encoded primarily in early arrivals to the receptors, while equilibrium quantities inform on source distance. We develop a new homogenization result showing that complex receptor configurations can be replaced by a uniform effective condition. In the extreme scenario where the cell adopts the angular direction of the first impact, we show this estimate to be surprisingly accurate.

On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group

Abstract We investigate the existence and nonexistence of nonnegative radial solutions to exterior problems of the form Δ H m u ( q ) + λ ψ ( q ) K ( r ( q ) ) f ( r 2 − Q ( q ) , u ( q ) ) = 0 {\Delta }_{{{\mathbb{H}}}^{m}}u\left(q)+\lambda \psi \left(q)K\left(r\left(q))f\left({r}^{2-Q}\left(q),u\left(q))=0 in B 1 c {B}_{1}^{c} , under the Dirichlet boundary conditions u = 0 u=0 on ∂ B 1 \partial {B}_{1} and lim r ( q ) → ∞ u ( q ) = 0 {\mathrm{lim}}_{r\left(q)\to \infty }u\left(q)=0 . Here, λ ≥ 0 \lambda \ge 0 is a parameter, Δ H m {\Delta }_{{{\mathbb{H}}}^{m}} is the Kohn Laplacian on the Heisenberg group H m = R 2 m + 1 {{\mathbb{H}}}^{m}={{\mathbb{R}}}^{2m+1} , m &gt; 1 m\gt 1 , Q = 2 m + 2 Q=2m+2 , B 1 {B}_{1} is the unit ball in H m {{\mathbb{H}}}^{m} , B 1 c {B}_{1}^{c} is the complement of B 1 {B}_{1} , and ψ ( q ) = ∣ z ∣ 2 r 2 ( q ) \psi \left(q)=\frac{| z{| }^{2}}{{r}^{2}\left(q)} . Namely, under certain conditions on K K and f f , we show that there exists a critical parameter λ ∗ ∈ ( 0 , ∞ ] {\lambda }^{\ast }\in \left(0,\infty ] in the following sense. If 0 ≤ λ &lt; λ ∗ 0\le \lambda \lt {\lambda }^{\ast } , the above problem admits a unique nonnegative radial solution u λ {u}_{\lambda } ; if λ ∗ &lt; ∞ {\lambda }^{\ast }\lt \infty and λ ≥ λ ∗ \lambda \ge {\lambda }^{\ast } , the problem admits no nonnegative radial solution. When 0 ≤ λ &lt; λ ∗ 0\le \lambda \lt {\lambda }^{\ast } , a numerical algorithm that converges to u λ {u}_{\lambda } is provided and the continuity of u λ {u}_{\lambda } with respect to λ \lambda , as well as the behavior of u λ {u}_{\lambda } as λ → λ ∗ − \lambda \to {{\lambda }^{\ast }}^{-} , are studied. Moreover, sufficient conditions on the the behavior of f ( t , s ) f\left(t,s) as s → ∞ s\to \infty are obtained, for which λ ∗ = ∞ {\lambda }^{\ast }=\infty or λ ∗ &lt; ∞ {\lambda }^{\ast }\lt \infty . Our approach is based on partial ordering methods and fixed point theory in cones.

Global well-posedness to the compressible nematic liquid crystal flows with large oscillations and vacuum in 3D exterior domains

This paper aims at extending our previous work (Y. Liu and X. Zhong, arXiv:2204.06227) to the exterior problem. We show the global well-posedness of strong solutions to a simplified Ericksen–Leslie model of compressible nematic liquid crystal flows in a three-dimensional exterior domain when the initial total energy is sufficiently small. In order to obtain necessary a priori estimates on the boundary, some new analytic techniques are used.