scholarly journals Modification of the fictitious absorption method for solving integral equations of mixed problems for arbitrary simply connected regions

2020 ◽  
Vol 1709 ◽  
pp. 012024
Author(s):  
I S Telyatnikov ◽  
A V Pavlova ◽  
M S Kapustin
Keyword(s):  
Integral Equations ◽  
Absorption Method ◽  
Mixed Problems ◽  
Simply Connected

scholarly journals An Alternating Potential-Based Approach To The Cauchy Problem For The Laplace Equation In A Planar Domain With A Cut

2008 ◽  
Vol 8 (4) ◽  
pp. 315-335 ◽  
Author(s):  
R. CHAPKO ◽  
B.T. JOHANSSON
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Neumann Boundary ◽  
Normal Derivative ◽  
Mixed Problems ◽  
Simply Connected Domain ◽  
The Cauchy Problem ◽  
The Laplace Operator ◽  
Simply Connected ◽  
Direct Problems

Abstract We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.

Periodic Contact and Mixed Problems of the Elasticity Theory (Review)

2021 ◽  
pp. 22-33
Author(s):  
Dmitrii A. Pozharskii
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Half Space ◽  
Three Dimensional ◽  
Contact Problems ◽  
Mixed Problems ◽  
Periodic Problems ◽  
Straight Line ◽  
Periodic Contact ◽  
Space Boundary

Results are reviewed collected in the investigations of periodic contact and mixed problems of the plane, axially symmetric and spatial elasticity theory. Among mixed problems, cut (crack) problems are focused integral equations of which are connected with those for contact problems. The periodic contact problems stimulate research of the discrete contact of rough (wavy) surfaces. Together with classical elastic domains (half-plane, half-space, plane and full space), we consider periodic problems for cylinder, layer, cone and spatial wedge. Most publications including fun-damental ones by Westergaard and Shtaerman deals with plane periodic problems of the elasticity theory. Here, one can mention approaches based on complex variable functions, Fourier series, Green’s functions and potential func-tions. A fracture mechanics approach to the plane periodic contact problem was developed. Methods and approaches are considered which allow us to take friction forces, adhesion and wear into account in the periodic contact. For spatial periodic and doubly periodic contact and properly mixed problems, we describe such methods as the localiza-tion method, the asymptotic methods, the method of nonlinear boundary integral equations, the fast Fourier trans-form. The half-space is the simplest model for elastic solids. But for the simplest straight-line periodic punch system, some three-dimensional contact problems (normal contact or tangential contact for shifted cohesive coatings) turn out to be incorrect because their integral equations contain divergent series. Considering three-dimensional periodic problems, I.G. Goryacheva disposes circular punches in special way (circular orbits, polar coordinated are used for centers of the punches), in this case one can prove convergence of the series in the integral equation (it is important that the punches are circular). For the periodic problems for an elastic layer, V.M. Aleksandrov has shown that the series in integral equations converge but the kernels become more complicated. In the present paper, we demonstrate that for the straight-line periodic punch system of arbitrary form the contact problem for a half-space turns out to be correct in case of more complicated boundary conditions. Namely, it can be sliding support or rigid fixation of a half-plane on the half-space boundary, the half-plane boundary should be parallel to the straight-line (the punch system axis) for arbitrary finite distance between the parallel lines. On this way, for sliding support, the kernel of the period-ic problem integral equation kernel is free of integrals, it consists of single convergent series (normal contact, the kernel is given in two equivalent forms). We consider classical percolation (how neighboring contact domains pene-trate one to another, investigated by K.L. Johnson, V.A. Yastrebov with co-authors) for the three-dimensional periodic contact amplification as well as percolation for the straight-line punch system. A similar approach is suggested for the case of periodic tangential contact (coatings system cohesive with a half-space boundary shifted along its axis or perpendicular to it). Here, one can separate out unique solutions of auxiliary problems because the line of changing boundary conditions on the half-space boundary can provoke non-uniqueness. The method proposed opens possibility to consider more complicated three-dimensional periodic contact problems for straight-line punch systems with changing boundary conditions inside the period.

scholarly journals A comparison between current-based integral equations approaches for eddy current problems

2021 ◽  
Vol 2090 (1) ◽  
pp. 012137
Author(s):  
F Lucchini ◽  
N Marconato
Keyword(s):  
Integral Equations ◽  
Eddy Current ◽  
Large Scale ◽  
Low Rank ◽  
Loop Current ◽  
Problem Size ◽  
Low Rank Approximation ◽  
Approximation Techniques ◽  
Current Formulation ◽  
Simply Connected

Abstract In this paper, a comparison between two current-based Integral Equations approaches for eddy current problems is presented. In particular, the very well-known and widely adopted loop-current formulation (or electric vector potential formulation) is compared to the less common J-φ formulation. Pros and cons of the two formulations with respect to the problem size are discussed, as well as the adoption of low-rank approximation techniques. Although rarely considered in the literature, it is shown that the J-φ formulation may offer some useful advantages when large problems are considered. Indeed, for large-scale problems, while the computational efforts required by the two formulations are comparable, the J-φ formulation does not require any particular attention when non-simply connected domains are considered.

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