Exterior Boundary Value Problems Research Articles

Isogeometric Fast Multipole Boundary Element Method Based on Burton-Miller Formulation for 3D Acoustic Problems

An isogeometric boundary element method is applied to simulate wave scattering problems governed by the Helmholtz equation. The NURBS (non-uniform rational B-splines) widely used in the CAD (computer aided design) field is applied to represent the geometric model and approximate physical field variables. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The singular integrals existing in Burton-Miller formulation are evaluated directly and accurately using Hadamard’s finite part integration. Fast multipole method is applied to accelerate the solution of the system of equations. It is demonstrated that the isogeometric boundary element method based on NURBS performs better than the conventional approach based on Lagrange basis functions in terms of accuracy, and the use of the fast multipole method both retains the accuracy for isogeometric boundary element method and reduces the computational cost.

The vector Helmholtz equation revisited: Inverse obstacle scattering

The vector Helmholtz equation, from a mathematical point of view, provides a generalization of the time-harmonic Maxwell equations for the propagation of time-harmonic electromagnetic waves. After reviewing some classic results on the two main exterior boundary value problems for the vector Helmholtz equation, i.e., the so-called electric boundary condition and the magnetic boundary condition, we prove reciprocity results for scattering of plane waves and point sources. Then we make use of them for obtaining uniqueness results for the inverse obstacle scattering problem to determine the shape of the scatterer from knowing the far field pattern of the scattered waves and results on the so-called far field operator for the two scattering problems. These results are generalizations of corresponding results for the Maxwell equations and analogues to results for the Dirichlet and Neumann boundary condition for the scalar Helmholtz equation. We also briefly consider the extension of the so-called DB boundary condition from the Maxwell equations to the vector Helmholtz equation.

Solution of the Exterior Boundary Value Problem for the Helmholtz Equation Using Overlapping Domain Decomposition

Solution of the Exterior Boundary Value Problem for the Helmholtz Equation Using Overlapping Domain Decomposition

A Boundary Element Procedure for 3D Electromagnetic Transmission Problems with Large Conductivity

We consider the scattering of time-periodic electromagnetic fields by metallic obstacles, or the eddy current problem. In this interface problem, different sets of Maxwell equations must be solved both in the obstacle and outside it, while the tangential components of both electric and magnetic fields are continuous across the interface. We describe an asymptotic procedure, applied for large conductivity, which reflects the skin effect in metals. The key to our method is a special integral equation procedure for the exterior boundary value problems corresponding to perfect conductors. The asymptotic procedure leads to a great reduction in complexity for the numerical solution, since it involves solving only the exterior boundary value problems. Furthermore, we introduce a FEM/BEM coupling procedure for the transmission problem and consider the implementation of Galerkin’s elements for the perfect conductor problem, and present numerical experiments.

The calculation of the parameters of disc-shaped hydraulic fracturing crack propagating in hard rock roofs of a coal seam near the in-seam working

The model of geomechanical state of the disc-shaped hydraulic fracturing crack that propagates in hard rocks near the in-seam working was developed. The peculiarity of the model is in non-homogeneous stress field which is conditioned by the presence of the working. This field plays an important role in calculating the trajectory of the crack propagation and under certain conditions it drastically changes the direction of the initial crack. The problem of the rock massif stress state in the vicinity of the in-seam working is narrowed to the second exterior boundary value problem of the elasticity theory for integrated singular equation. It can be reached by substituting limit stress coal seam marginal zones by their acting stresses. Analyzing the results it is defined that when the ratio of the working liquid pressure to the gravitational hard rock pressure equals four units the trajectory of the crack is linear and coincides with the direction of the initial crack. If the ratio of the working liquid pressure to the gravitational pressure changes from three to four units the trajectory of the crack looks as a gently sloping curve that insignificantly changes its direction towards the initial crack.

Direct boundary method toolbox for some elliptic problems in FreeHyTE framework

FreeHyTE Direct Boundary Method Toolbox is a new computational framework for the solution of interior and exterior boundary value problems in two dimensions using three classes of direct methods: the Boundary Element Method, the Method of Fundamental Solutions and the Trefftz-Herrera Method. The toolbox, currently including solvers for Laplace and Helmholtz boundary value problems, is straightforward to use, featuring a simple graphical user interface and automatic mesh generators, and amenable to extension, as it provides modular computational procedures, directly applicable to other types of boundary elements and differential equations. The toolbox supports the definition of simply or multiply-connected domains, boundary elements of any order, complex wavenumbers, and Dirichlet, Neumann and Robin boundary conditions. FreeHyTE Direct Boundary Method Toolbox is freely distributed under the GNU General Public License and supported by manuals to quickly get new users started.

Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

Http://www.mining-science.ru/download/2020/3/04.pdf

Introduction. Reliable forecasts of pillars geomechanical state are required to ensure rhythmic and safe work when mining a coal bed. Research aim is to construct a state model of the coal pillar located between the headways, based on the fundamental methods of elasticity theory and mechanics of a granular media, carry out a computational experiment within the model, and analyse the results. Methodology. The stress field in the coal pillar has been constructed in the course of solving the elastoplastic problem. By replacing the ultimately stressed marginal zone of the bed with the stresses which act within the zone, the problem has been reduced to the second exterior boundary value problem of elasticity theory and has been solved by the boundary element method. Ordinary and special Coulomb–Mohr criteria simultaneously fulfilled for the coal bed and rock mass contact are the criterion of the limit state onset. Actual pillar load is determined by integrating the vertical stress curve along the bed roof, which has been obtained from elastoplastic problem solution, while the ultimate load is determined from the condition that the whole pillar is in ultimately stressed state. Results. The dependence between the safety factor of the pillar between two identical headways, determined by V. D. Shevyakov method, and the growth of its width represents a graph in the form of a monotonically increasing curve. The curve flattens as soon as the depth increases. Summary. The results from the developed model of coal rock mass geomechanical state can be successfully used as coal pillar strength forecasts.

Existence and integral representation of solutions for plane deformations of a micropolar elastic solid with surface elasticity

AbstractWe consider a linear theory of elastic boundary reinforcement of a micropolar elastic solid subjected to plane‐strain deformations. The reinforcement consists of a thin micropolar elastic coating bonded to part of the boundary of the solid. The elastic properties of the coating incorporate both classical and micropolar bending, extension and twisting effects. Interior and exterior mixed boundary problems are formulated and analyzed using the boundary integral equation method. The boundary value problems are reduced to systems of singular integro‐differential equations to which Noether‐type theorems are shown to apply. We consider also the corresponding boundary value problems based on an alternative lower‐order shell model of the reinforcement. Finally, existence and uniqueness results are presented for the corresponding interior and exterior boundary value problems in the appropriate classical function spaces.

Evaluation of the hypersingular boundary integral equation for acoustic wave-propagation simulations

We have developed a hypersingular boundary integral equation (BIE) approach to simulate wave scattering by inclusions such as karstic cavities and gypsum salts. The solution of conventional BIEs for exterior boundary value problems is of nonuniqueness and numerical instability at characteristic wavenumbers (corresponding to fictitious resonant frequencies). With the hypersingular boundary integral operator, we can handle this problem, but with the difficult-to-calculate hypersingular kernels. Our method to evaluate the singular part in the operator is more efficient than other singularity-elimination techniques. Numerical examples validate the method by comparing with the analytical solution.

Twofold Domain Decomposition Method for the Analysis of Multiscale Composite Structures

A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.

A Method for Solving an Exterior Boundary Value Problem for the Laplace Equation by Overlapping Domain Decomposition

We propose a numerical method for solving an exterior three-dimensional boundary value problem for the Laplace equation based on the overlapping decomposition of the computational domain. The initial boundary value problem is reduced to solving an operator equation for the sought values of the function on an auxiliary sphere enclosing the interior boundary. This equation is approximated by a system of linear algebraic equations which is solved by iterative methods in the Krylov subspaces. A series of numerical experiments for model problems with known solutions demonstrates not only the convergence of the method and the attained accuracy of the calculations but also a sufficiently short runtime.

Inverse transmission scattering problem via a Dirichlet-to-Neumann map

In this paper, we consider an inverse scattering problem of a time-harmonic wave from a penetrable obstacle. A Dirichlet-to-Neumann map is defined to convert this problem into an exterior boundary value problem with an impedance-like boundary condition. The problem is then solved without the need of solving an interior problem which is inevitable in the usual way. This process reduces the size of the problem and thus enables an efficient treatment of this inverse scattering problem.

A method for solving an exterior boundary value problem for the Laplace equation by overlapping domain decomposition

A method for solving an exterior boundary value problem for the Laplace equation by overlapping domain decomposition

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.

New blow-up results for nonlinear boundary value problems in exterior domains

We consider exterior boundary value problems with nontrivial boundary conditions, including three types of semilinear equations: heat equation, wave equation and damped wave equation. Using a test function approach with a choice of a suitable family of test functions, we obtain the critical exponents in the sense of Fujita for the considered problems. Our obtained results close some open questions from the literature.

An isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution

A study of structural shape optimization and absorbing material distribution optimization of noise barrier structures based on the recently proposed isogeometric analysis method with exact geometric definitions is presented. The acoustic scattering is approximated using the basis functions that represent the geometry. A fast multipole method is applied to accelerate the solution of the boundary element method (BEM). The Burton–Miller formulation is used to overcome the fictitious frequency problem when a single Helmholtz boundary integral equation is used for the exterior boundary–value problem. The strongly singular integrals in the Burton–Miller formulation using the isogeometric BEM are evaluated explicitly and directly, particularly for the sensitivity formulation with a hyper-singular integral. The optimality criteria method is used for two types of optimization analyses, shape optimization and material distribution topology optimization. For the shape optimization, the design variables can be set to the locations of the control points because the control points determine the shape of structure. For the second optimization, a new material interpolation scheme for acoustic problems based on the solid isotropic material with penalization (SIMP) method is given, where the interpolation variable is not the real structural density used in a conventional SIMP, but a fictitious material density that determines the normalized surface admittance. Several examples are presented to demonstrate the validity and efficiency of the proposed algorithm.

English

The Galerkin method is used to numerically solve the exterior boundary value problem for the radiosity equation for the spherical quatrefoil. The radiosity equation is a mathematical model for the brightness of a collection of one or more surfaces when their reflectivity and emissivity are given. On planet Mars the surface emissivity is closely related to its surface temperature. The radiosity of a surface is the rate at which the energy leaves that surface; it includes the energy emitted by a surface as well as the energy reflected from other surfaces. Key words: Radiosity equation, Galerkin method, spherical quatrefoil.

Exact solution of some exterior boundary value problems of elasticity in parabolic coordinates

The present work, by using the method of the separation of variables, states and analytically (exactly) solves the external boundary value problems of elastic equilibrium of the homogeneous isotropic body bounded by the parabola, when normal or tangential stresses are given on a parabolic border. Using MATLAB software, the numerical results and constructed graphs of the mentioned boundary value problems are obtained.

Transmission scattering problem via a DtN map

In this paper, we consider the scattering problem of time-harmonic waves from a penetrable obstacle. A Dirichlet-to-Neumann map is defined to convert this problem into an exterior boundary value problem which mimics an impedance problem. This process reduces the size of the problem and thus enables an elegant treatment of the scattering problem.