A Higher-Order Poly-Region Boundary Element Method for Steady Thermoviscous Fluid Flows

2004 ◽  
Author(s):  
M. . M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Boundary Integral Equations ◽  
Fluid Flows ◽  
Boundary Elements ◽  
Higher Order ◽  
Boundary Integral ◽  
Iterative Solvers ◽  
Boundary Element Methods ◽  
Element Formulation ◽  
Thermoviscous Fluid

In this presentation, we re-visit the poly-region boundary element methods (BEM) proposed earlier for the steady Navier-Stokes [1] and Boussinesq [2] flows, and develop a novel higher-order BEM formulation for the thermoviscous fluid flows that involves the definition of the domains of kernel influences due to steady Oseenlets. We introduce region-by-region implementation of the steady-state Oseenlets within the poly-region boundary element fequatramework, and perform integration only over the (parts of) higher-order boundary elements and volume cells that are influenced by the kernels. No integration outside the domains of the kernel influences are needed. Owing to the properties of the convective Oseenlets, the kernel influences are very local and propagate upstream. The localization becomes more prominent as the Reynolds number of the flow increases. This improves the conditioning of the global matrix, which in turn, facilitates an efficient use of the iterative solvers for the sparse matrices [3]. Here, we consider quartic boundary elements and bi-quartic volume cells to ensure a high level resolution in space. Similar to the previous developments [4–6], coefficients of the discrete boundary integral equations are evaluated with the sufficient precision using semi-analytic approach to ensure exceptional accuracy of the boundary element formulation. To demonstrate the attractiveness of the poly-region BEM formulation, we consider a numerical example of the well-known Rayleigh-Benard problem governed by the Boussinesq equations.

Regularization Techniques Applied to Boundary Element Methods

1994 ◽  
Vol 47 (10) ◽  
pp. 457-499 ◽  
Author(s):  
Masataka Tanaka ◽  
Vladimir Sladek ◽  
Jan Sladek
Keyword(s):  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Scalar Potential ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Element Formulation ◽  
Hypersingular Integrals ◽  
Regularization Techniques

This review article deals with the regularization of the boundary element formulations for solution of boundary value problems of continuum mechanics. These formulations may be singular owing to the use of two-point singular fundamental solutions. When the physical interpretation is irrelevant for this topic of computational mechanics, we consider various mechanical problems simultaneously within particular sections selected according to the main topic. In spite of such a structure of the paper, applications of the regularization techniques to many mechanical problems are described. There are distinguished two main groups of regularization techniques according to their application to singular formulations either before or after the discretization. Further subclassification of each group is made with respect to basic principles employed in individual regularization techniques. This paper summarizes the substances of the regularization procedures which are illustrated on the boundary element formulation for a scalar potential field. We discuss the regularizations of both the strongly singular and hypersingular integrals, occurring in the boundary integral equations, as well as those of nearly singular and nearly hypersingular integrals arising when the source point is near the integration element (as compared to its size) but not on this element. The possible dimensional inconsistency (or scale dependence of results) of the regularization after discretization is pointed out. Finally, we discuss the numerical approximations in various boundary element formulations, as well as the implementations of solutions of some problems for which derivative boundary integral equations are required.

A Boundary Element Method for Three-Dimensional Steady Convective Heat Diffusion

2004 ◽  
Author(s):  
M. M. Grigoriev ◽  
G. F. Dargush
Keyword(s):  
Boundary Element ◽  
Convective Heat ◽  
Sparse Matrices ◽  
Three Dimensional ◽  
Higher Order ◽  
Iterative Solvers ◽  
Boundary Element Methods ◽  
Heat Diffusion ◽  
Element Formulation ◽  
Definition Of

Higher-order boundary element methods (BEM) are presented for three-dimensional steady convective heat diffusion at high Peclet numbers. An accurate and efficient boundary element formulation is facilitated by the definition of an influence domain due to convective kernels. This approach essentially localizes the surface integrations only within the domain of influence which becomes more narrowly focused as the Peclet number increases. The outcome of this phenomenon is an increased sparsity and improved conditioning of the global matrix. Therefore, iterative solvers for sparse matrices become a very efficient and robust tool for the corresponding boundary element matrices. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the higher-order BEM formulations for high Peclet numbers in the range from 1,000 to 100,000. The bi-quartic boundary elements included in this study are shown to provide very efficient and extremely accurate solutions, even on a single engineering workstation.

Dual Boundary Element Analysis for Time-Dependent Fracture Problems in Creeping Materials

2008 ◽  
Vol 383 ◽  
pp. 109-121 ◽  
Author(s):  
E. Pineda ◽  
M.H. Aliabadi
Keyword(s):  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Creep Behaviour ◽  
Boundary Integral ◽  
Element Formulation ◽  
Secondary Creep ◽  
Element Analysis ◽  
Power Law Creep ◽  
Element Method

This paper presents the development of a new boundary element formulation for analysis of fracture problems in creeping materials. For the creep crack analysis the Dual Boundary Element Method (DBEM), which contains two independent integral equations, was formulated. The implementation of creep strain in the formulation is achieved through domain integrals in both boundary integral equations. The domain, where the creep phenomena takes place, is discretized into quadratic quadrilateral continuous and discontinuous cells. The creep analysis is applied to metals with secondary creep behaviour. This is con…ned to standard power law creep equations. Constant applied loads are used to demonstrate time e¤ects. Numerical results are compared with solutions obtained from the Finite Element Method (FEM) and others reported in the literature.

Boundary Element Formulation for Numerical Surface Wave Modelling in Poroviscoelastisity

2016 ◽  
Vol 685 ◽  
pp. 172-176 ◽  
Author(s):  
Leonid Igumnov ◽  
S.Yu. Litvinchuk ◽  
A.A. Belov ◽  
A.A. Ipatov
Keyword(s):  
Boundary Element ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Dynamic Responses ◽  
Element Formulation ◽  
Weakly Singular Kernel ◽  
Viscoelastic Parameters ◽  
Viscoelastic Effects ◽  
Biot’S Equations

Problems of the poroviscodynamics are considered. Theory of poroviscoelasticity is based on Biot’s equations of fluid saturated porous media under assumption that the skeleton is viscoelastic. Viscoelastic effects of solid skeleton are modeled by mean of elastic-viscoelastic correspondence principle, using such viscoelastic models as a standard linear solid model and model with weakly singular kernel. The fluid is taken as original in Biot’s formulation without viscoelastic effects. Boundary integral equations method is applied to solve three-dimensional boundary-value problems. Boundary-element method with mixed discretization and matched approximation of boundary functions is used. Solution is obtained in Laplace domain, and then Durbin’s algorithm of numerical inversion of Laplace transform is applied to perform solution in time domain. An influence of viscoelastic parameters on dynamic responses is studied. Numerical example of the surface waves modelling is considered.

scholarly journals Adaptive boundary-element methods for transmission problems

1997 ◽  
Vol 38 (3) ◽  
pp. 336-367 ◽  
Author(s):  
Carsten Carstensen ◽  
Ernst P. Stephan
Keyword(s):  
Boundary Element ◽  
Boundary Integral Equations ◽  
Integral Operators ◽  
Adaptive Algorithms ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
A Posteriori ◽  
Multiplicative Constant ◽  
Transmission Problems ◽  
Weakly Singular

AbstractIn this paper we present an adaptive boundary-element method for a transmission prob-lem for the Laplacian in a two-dimensional Lipschitz domain. We are concerned with an equivalent system of boundary-integral equations of the first kind (on the transmission boundary) involving weakly-singular, singular and hypersingular integral operators. For the h-version boundary-element (Galerkin) discretization we derive an a posteriori error estimate which guarantees a given bound for the error in the energy norm (up to a multiplicative constant). Then, following Eriksson and Johnson this yields an adaptive algorithm steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically good triangulations and are efficient.

scholarly journals Boundary Element Modeling of Dynamic Bending of a Circular Piezoelectric Plate

2018 ◽  
Vol 183 ◽  
pp. 01025
Author(s):  
Leonid Igumnov ◽  
Ivan Markov ◽  
Alexandr Konstantinov
Keyword(s):  
Boundary Element ◽  
Mechanical Response ◽  
Initial Conditions ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Element Formulation ◽  
Laplace Domain ◽  
Piezoelectric Plates

Accurate modelling of a coupled dynamic electro-mechanical response of circular piezoelectric plates under various loading conditions is of particular importance. Piezoelectric plates are not only basic structural elements, but with certain considerations can be conveniently fit for numerical simulation of piezoelectric sensors and transducers. In this work, a Laplace domain direct boundary element formulation is applied for dynamic analysis of three-dimensional linear piezoelectric moderately thick circular plates. Zero initial conditions, vanishing body forces and the absence of the free electrical charges are assumed. Weakly singular expressions of Laplace domain boundary integral equations for the generalized displacements are employed. Spatial discretization is based on the nodal collocation method. Mixed boundary elements are implemented. The geometry of the elements, generalized displacement and generalized tractions are represented with different shape functions: quadratic, linear and constant, accordingly. Integral expressions of the three-dimensional Laplace domain piezoelectric displacement fundamental solutions are used. After solving the problem on a set of Laplace transform parameter values, time-domain solutions are retrieved from the corresponding Laplace domain solutions by employing a numerical inversion routine. Numerical example is provided to show reliability and accuracy of the proposed boundary element formulation.

scholarly journals Boundary element methods for acoustic scattering by fractal screens

2021 ◽  
Vol 147 (4) ◽  
pp. 785-837 ◽  
Author(s):  
Simon N. Chandler-Wilde ◽  
David P. Hewett ◽  
Andrea Moiola ◽  
Jeanne Besson
Keyword(s):  
Boundary Element ◽  
Boundary Integral Equations ◽  
Sufficient Conditions ◽  
Acoustic Scattering ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Fractal Boundary ◽  
Time Harmonic ◽  
Simulation Strategy ◽  
Theoretical Results

AbstractWe study boundary element methods for time-harmonic scattering in $${\mathbb {R}}^n$$ R n ($$n=2,3$$ n = 2 , 3 ) by a fractal planar screen, assumed to be a non-empty bounded subset $$\Gamma $$ Γ of the hyperplane $$\Gamma _\infty ={\mathbb {R}}^{n-1}\times \{0\}$$ Γ ∞ = R n - 1 × { 0 } . We consider two distinct cases: (i) $$\Gamma $$ Γ is a relatively open subset of $$\Gamma _\infty $$ Γ ∞ with fractal boundary (e.g. the interior of the Koch snowflake in the case $$n=3$$ n = 3 ); (ii) $$\Gamma $$ Γ is a compact fractal subset of $$\Gamma _\infty $$ Γ ∞ with empty interior (e.g. the Sierpinski triangle in the case $$n=3$$ n = 3 ). In both cases our numerical simulation strategy involves approximating the fractal screen $$\Gamma $$ Γ by a sequence of smoother “prefractal” screens, for which we compute the scattered field using boundary element methods that discretise the associated first kind boundary integral equations. We prove sufficient conditions on the mesh sizes guaranteeing convergence to the limiting fractal solution, using the framework of Mosco convergence. We also provide numerical examples illustrating our theoretical results.

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