Problems In Multiply Connected Domains Research Articles

A non-boundary-fitted-grid method, based on compact integrated-RBF approximations, for solving differential problems in multiply-connected domains

This paper presents a new non-boundary-fitted-grid numerical technique for solving partial differential equations (PDEs) in multi-hole domains. A multiply-connected domain is converted into a simply-connected domain of rectangular or non-rectangular shape that is discretised using a Cartesian grid. Compact radial basis function (RBF) stencils, which are constructed through integration rather than the conventional differentiation, are used to discretise the field variables. The imposition of inner boundary conditions is conducted by means of body forces that are derived from satisfying the governing equations and prescribed boundary conditions in small subregions. Salient features of the proposed method include: (i) simple pre-processing (Cartesian grid), (ii) high rates of convergence of the solution accuracy with respect to grid refinement achieved with compact integrated-RBF stencils, where both nodal function and derivative values are included in the approximations, (iii) the system matrix kept unchanged for the case of moving holes, and (iv) no interpolation between Lagrange and Euler meshes required. Several linear and nonlinear problems, including rotating-cylinder flows and buoyancy-driven flows in eccentric and concentric annuli, are simulated to verify the proposed technique.

Numerical solutions of two-dimensional Laplace and biharmonic equations by the localized Trefftz method

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated.

Dynamic Green's functions for multiple elliptical inclusions with imperfect interfaces

The problem of an unbounded elastic solid with multiple elliptical inclusions subjected to a time-harmonic anti-plane concentrated force is semi-analytically solved by using the collocation multipole method. The displacement of matrix and inclusion are represented by angular and radial Mathieu functions. The imperfect interface between the matrix and the inclusion is characterized as a linear spring model with vanishing thickness. It is the derivative that the imperfect condition is involved. The addition theorem of Mathieu function is frequently used to solve multiply-connected domain problems in the traditional multipole method. An alternate here is a direct computation. The associated normal derivative with respect to a non-local elliptical coordinate system is developed by means of directional derivative. Besides simple computation, no truncation error is caused. The displacement field is determined by using the imperfect interface conditions through collocating points along the interface. Several numerical experiments are done to investigate the effects of the driving frequency of the concentrated force, imperfect interface and the convexity of elliptical inclusions on the dynamic Green's functions. © 2020 The Authors. Published by Elsevier Ltd.

Mathematical modeling of coupled heat transfer on cooled gas turbine blades

The paper presents a physico-mathematical model for determining the heat transfer parameters between viscous gasdynamic flotations and cooled gas turbine blades made using the technology of composite permeable membranes (CPM). The mathematical model includes equations of dynamic and thermal boundary layers taking into account injection, three-dimensional transient heat conduction in CPM cooled frames, complex hydraulic flow and cooling inside the channels of a cooled blade with air taken from the power plant compressor. A complex problem was solved to determine the thermal state of microrocket engines, where a method was proposed for immersing a multiply-connected domain in a region of the simplest form. A new economical, absolutely stable method is proposed for numerically solving of spatial nonstationary heat transfer problems in multiply-connected domains. The results obtained showed greater cooling efficiency at lower cooling medium costs.

Dynamic Green's functions for multiple circular inclusions with imperfect interfaces using the collocation multipole method

This paper presents a semi-analytical approach to solve anti-plane dynamic Green's functions for an elastic infinitely extended isotropic solid (matrix) containing multiple circular inclusions with imperfect interfaces. A linear spring model with vanishing thickness is employed to character the imperfect interface. The multipole expansions of anti-plane displacement of the matrix and inclusion, induced by a time-harmonic anti-plane line force located in the matrix or in the inclusion, are expanded by using Hankel and Bessel functions, respectively. The imperfect interface condition is satisfied by uniformly collocating points along the interface of each inclusion. For the imperfect interface condition, the normal derivative of the anti-plane displacement with respect to a non-local polar coordinate system is developed without any truncation error for multiply-connected domain problems. For the case of one circular inclusion, the proposed quasi-static stress field matches well with the analytical static solution. The proposed quasi-static stress fields containing two and three circular inclusions are critically compared with those calculated by static analysis using the finite element method. Finally, extensive studies are presented to investigate the effects of the frequency of excitation, imperfect interface and separation between inclusions on the dynamic Green's functions.

An iterative algorithm for the stable solution of inverse heat conduction problems in multiply-connected domains

The Cauchy problem for inverse heat conduction in a multiply-connected domain Ω was solved in this paper. To obtain this, an iterative algorithm which minimized a functional comprising the homogeneity of the outer domain surface temperature and the temperature gradient within the domain Ω was developed. Calculations were made for the known distribution of the heat transfer coefficient and surrounding temperature on the outer boundary of the domain, disturbed by the random error δ[%] = 0, 1, 5, 10. The influence of temperature gradient on time and accuracy of calculations was investigated. Taking into consideration the temperature gradient in the functional being minimized in the analytical process shortens the time of calculations and reduces oscillations of temperature and heat flux distributions on the inner boundary of the multiply-connected domain. An example of application of the algorithm presented in this paper can be the optimization problem in cooling of gas turbine blades.

Acoustic eigenproblems of elliptical cylindrical cavities with multiple elliptical cylinders by using the collocation multipole method

Abstract Acoustic eigenproblems of 3-D elliptical cylindrical cavities having multiple elliptical cylinders are solved by means of a 3-D semi-analytical formulation based on a multipole expansion, directional derivative, collocation technique and singular value decomposition (SVD). The multipole expansion for the acoustic pressure is formulated in terms of angular and radial Mathieu functions due to elliptical boundaries considered here. The boundary conditions are satisfied by uniformly collocating points along the boundaries. When considering sound-hard or Neumann boundary conditions, the normal derivative of acoustic pressure with respect to non-local elliptic coordinates is derived using the directional derivative for multiply-connected domain problems. By truncating the multipole expansion, a finite linear algebraic system is acquired. The direct searching approach is applied to identify the natural frequencies by using the SVD. Several numerical examples are examined, including those of an elliptical cylindrical cavity, a confocal elliptical annulus cylindrical cavity and an elliptical cylindrical cavity with two elliptical cylinders. Convergence and comparison studies are done to demonstrate the validation and accuracy of the present method. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy and fast rate of convergence are the main features of the present method thanks to its semi-analytical character.

The axially symmetric flow-through problem for the Navier-Stokes equations in variables “vorticity-stream function”

The axially symmetric steady-state motion of a viscous incompressible fluid in a region of the type of a spherical layer is considered. The boundary problem for the Navier-Stokes equations is formulated in terms of the stream function and vorticity. The solvability of this problem was previously established provided that the fluid flow rate through each layer boundary is sufficiently small. The main result of this study is in proving the solvability of the axially symmetric flow-through problem without limitations on the magnitude of the flow rate. The proof is based on the a priori Dirichlet estimate of the flow velocity field. The asymptotic properties of the solution near the symmetry axis are established conditionally. The proof procedure admits the generalization to the case of axially symmetric problems in multiply-connected domains.

A symmetric Galerkin BEM variational framework for multi-domain interface problems

In this paper we discuss an energy-based variational framework for the solution of interior problems in multiply-connected domains comprising multiple piecewise homogeneous subdomains, using exclusively boundary integral equations. The primary goal is to provide a unified variational setting that lends itself naturally to symmetric Galerkin boundary element formulations in terms of Dirichlet-type variables only. The approach hinges on the explicit imposition of the normal derivative of the classical integral representation of the interior solution on each subdomain via Lagrange multipliers in the augmented Lagrangian of the system. We use Maue-type identities to resolve the hypersingular kernels, leading to a scheme that requires only standard single- and double-layer evaluations. In addition, the usual difficulty with multi-valued normals at subdomain corners is treated here within the same variational framework, by incorporating into the variational formulation the constraint equation between the limiting normal derivatives at either side of the corner. The resulting scheme remains fully symmetric. The numerical implementation avoids the explicit presence of Neumann-type unknowns on the boundaries, through condensation at the subdomain level. In all integral evaluations, three- or four-point Gauss quadrature rules are sufficient for accurate results. We describe the theory and present illustrative examples for thermal and acoustic problems governed by Laplace and Helmholtz equations, respectively. This technique, however, can be applied without essential modification to more general problems.

An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation

The boundary integral equation approach has been shown to suffer a nonunique solution when the geometry is equal to a degenerate scale for a potential problem. In this paper, the degenerate scale problem in boundary element method for the two-dimensional Laplace equation is analytically studied in the continuous system by using degenerate kernels and Fourier series instead of using discrete system using circulants [Engng Anal. Bound. Elem. 25 (2001) 819]. For circular and multiply-connected domain problems, the rank-deficiency problem of the degenerate scale is solved by using the combined Helmholtz exterior integral equation formulation (CHEEF) concept. An additional constraint by collocating a point outside the domain is added to promote the rank of influence matrix. Two examples are shown to demonstrate the numerical instability using the singular integral equation for circular and annular domain problems. The CHEEF concept is successfully applied to overcome the degenerate scale and the error is suppressed in the numerical experiment.

The walk on boundary algorithms for solving boundary value problems in multiply-connected domains

Article The walk on boundary algorithms for solving boundary value problems in multiply-connected domains was published on January 1, 1990 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 5, issue 6).