Block multipole method for boundary value problems in complex‐shaped domains

Власов, В. И.; Volkov-Bogorodsky, Dmitriy B.

doi:10.1002/zamm.199807815122

Cited by 6 publications

(6 citation statements)

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“…With d = t = 1, the formula (20) turns to (19). According to the formula (20), grid impaction toward the upper side of the rectangle C causes the node (2,2) to shift toward the upper point of the semiellipse.…”

Section: A Horseshoementioning

confidence: 98%

“…According to the formula (20), grid impaction toward the upper side of the rectangle C causes the node (2,2) to shift toward the upper point of the semiellipse. With d = 0, t = 1, and b = 10, we have y 2,2 = 8.875, see Fig.…”

Section: A Horseshoementioning

confidence: 99%

“…In literature, it is called a "horseshoe" domain. On this domain, an etalon solution was obtained by Vlasov and Bezrodnykh using the multipole method [20]. We consider it as an analytical solution.…”

Section: A Horseshoementioning

confidence: 99%

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On 2D structured mesh generation by using mappings

Azarenok

2010

Numer. Methods Partial Differential Eq.

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A grid generation problem in two-dimensional domains is considered by using a quasi-conformal mapping of the parametric domain with a given square mesh onto the physical domain where the grid is required. To this end, a harmonic mapping is first applied, which, by the Radó theorem, is a diffeomorphism subject to some known conditions. However, the discrete harmonic mapping may produce folded meshes on a nonconvex domain with a strongly bent boundary. We demonstrate that it is caused by the truncation error. With the aim of controlling grid node location, an additional mapping is used. The Dirichlet problem for the universal elliptic partial differential equations is solved to construct the mapping. This allows to produce unfolded grids with a prescribed cell shape.

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Section: A Horseshoementioning

confidence: 98%

Section: A Horseshoementioning

confidence: 99%

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On 2D structured mesh generation by using mappings

Azarenok

2010

Numer. Methods Partial Differential Eq.

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“…The block analytic-numerical method [4]- [5] is applied to modeling of stress-strain distribution and for an estimation of energy in elementary cell of composite. It is based on dividing of initial domain onto more simple subdomains called blocks and on representation of the solution in each block as expansion on system of the special functions named multipoles.…”

Section: Numerical Modeling Of Stress-strain State In Interphase Layementioning

confidence: 99%

Multiscale Modeling of the Interphase Interactions in the Mechanics of Materials

Lurie¹,

Volkov-Bogorodsky

Tuchkova³

2005

Proc Appl Math and Mech

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The consistent and correct model of media taking into account scale effects (cohesion and adhesive interactions) is constructed as a special case of the Cosserat's pseudocontinuum model. The variant of the interphase layer theory is elaborated, which includes the following moments: formal mathematical statement, physical constitutive equations, numerical estimations of an interphase layer influence on the stress state and energy density distribution in a composite. The mathematical statement of interphase layer theoryRecently, in the papers [1]- [2], the generalized continuum model with kept dislocations was developed. Generally, the model presented allows to describe local-cohesion interactions [2] and superficial effects [1]. The interphase layer theory allows to study the specific local interactions which determining features of the properties of contacting phases and material as whole: cohesion fields and the internal interactions associated to them; interfacial adhesive interactions. The marked types of interactions are characterized by essential localness, small area of interaction, concentrate about defects, borders, interfaces.This mathematical statement is completely determined by following equation for the Lagrange functional L and the following variational equation:on any plane with a normal n i on the boundary surface the vector of forces T i is determinedand moment vectorHere ∇ 2 is the Laplace operator, R i are components of the displacement vector, l 2 0 = µ/C, γ ij and θ are the components of the deviator of strain and spherical deformation tensor, ω k are the components of the rotation vector, E ijk is the Levi-Civita tensor, δ ij is the Kronecker delta, µ, λ are the Lame coefficient, C is the physical constant that determine the cohesion interactions, D ijṘiṘj = A n i n jṘiṘj + B (δ ij − n i n j )Ṙ iṘj is energy density associated with changing of defectiveness of a contact surface due to modified a surface, P V i is the vector of density of the external loads over the body volume, P F i is the vector of density of the surface loads, F is the boundary surface and L ij (...) is the operator of the classical theory of elasticity.New physical constants A and B determine the surface effects associated with the normal to the surface of the body, and the superficial effects in the tangent plane respectively. Note that the ideal adhesive interaction influences only for a local state and does not change classical boundary conditions. Using variational formulation of the problem we can formulate it's differential representation as Euler equations of the functional:where S is surface of inclusion in composite. To understand the physical sense let's define displacement of cohesion field u i and separate general displacements R i on two components satisfied classical and non-classical equation of theory of elasticity:

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“…This opportunity is given by the analytic-numerical multipole method [88,90,91], which enables one to effectively solve boundary-value problems for the Laplace equation with precision guaranteed by an a posteriori estimate in a uniform norm with respect to the domain and with exponential convergence rate. Thus, this method opens new possibilities in the constructive theory of harmonic mappings.…”

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confidence: 99%

On a Problem of the Constructive Theory of Harmonic Mappings

Безродных

Власов

2014

J Math Sci

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The problem of irremovable error appears in finite difference realization of the Winslow approach in the constructive theory of harmonic mappings. As an example, we consider the well-known Roache-Steinberg problem and demonstrate a new approach, which allows us to construct harmonic mappings of complicated domains effectively and with high precision. This possibility is given by the analytic-numerical method of multipoles with exponential convergence rate. It guarantees effective construction of a harmonic mapping with precision controlled by an a posteriori estimate in a uniform norm with respect to the domain.

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Block multipole method for boundary value problems in complex‐shaped domains

Abstract: New efficient analytic‐numerical method for solving elliptic boundary‐value problems in 2D and 3D complex‐shaped domains is presented. The case of domains with rounded cones is considered in detail.

Cited by 6 publications

References 1 publication

On 2D structured mesh generation by using mappings

On 2D structured mesh generation by using mappings

Multiscale Modeling of the Interphase Interactions in the Mechanics of Materials

On a Problem of the Constructive Theory of Harmonic Mappings

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