S. E. Mikhailov

Localized boundary-domain integral formulations for problems with variable coefficients

Abstract Specially constructed localized parametrixes are used in this paper instead of a fundamental solution to reduce a boundary value problem with variable coefficients to a localized boundary-domain integral or integro-differential equation (LBDIE or LBDIDE). After discretization, this results in a sparsely populated system of linear algebraic equations, which can be solved by well-known efficient methods. This make the method competitive with the finite element method for such problems. Some methods of the parametrix localization are discussed and the corresponding LBDIEs and LBDIDEs are introduced. Both mesh-based and meshless algorithms for the localized equations discretization are described.

A functional approach to non-local strength conditions and fracture criteria. I: Body and point fracture

A general form of non-local strength condition based on a nonlinear space strength functional is proposed, and its relation with some known non-local strength conditions is discussed. The strength functional is associated with the supremum of a positive factor by which a given stress field may be multiplied to obtain a non-fracturing stress field. Mathematical constraints on the functional form caused by the demand of functional boundedness on admissible stress fields are explored. The notions of strength homogeneity, strength isotropy and finite non-locality for non-local strength conditions are introduced.

A functional approach to non-local strength conditions and fracture criteria—II. Discrete fracture

Abstract Some new notions of discrete fracture mechanics are introduced: fracture quantum, homogeneous and isotropic fracture quanta sets, and fracture quanta sets that are weakly sensitive to the boundary. Non-local functional strength conditions for a fracture quantum and for a body are considered. Definitions are given for strength homogeneity, isotropy, finite non-locality, and weak sensitivity to the boundary for a discretely fracturable body. Quantum-point dualism in a fracture description is analyzed, which allows one to go from the continuous point of view to the discrete one and vice versa. Presented approaches are applied to generalizations of some known non-local strength conditions on near-boundary points.

Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains

For functions from the Sobolev space Hs(Ω), 1 2 < s < 3 2 , definitions of non-unique generalised and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain Ω, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the co-normal derivative inherent non-uniqueness are given. Some new facts about trace operator estimates, Sobolev spaces characterisations, and solution regularity of PDEs with non-smooth coefficients are also presented.

Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics

Mapping and invertibility properties of some parametrix-based surface and volume potentials are studied in Bessel-potential and Besov spaces. These results are then applied to derive regularity and asymptotics of the solution to a system of boundary-domain integral equations associated with a mixed BVP for a variablecoefficient PDE, in a vicinity of the curve of change of the boundary condition type.

Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed bvps in exterior domains

Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.

Analysis of some localized boundary-domian integral equations

Some direct segregated localized boundary-domain integral equation (LBDIE) systems associated with the Dirichlet and Neumann boundary value problems (BVP) for a scalar ”Laplace” PDE with variable coefficient are formulated and analysed. The parametrix is localized by multiplication with a radial localizing function. Mapping and jump properties of surface and volume integral potentials based on a localized parametrix and constituting the LBDIE systems are studied in a scale of Sobolev (Bessel potential) spaces. The main results established in the paper are the LBDIEs equivalence to the original variable-coefficient BVPs and the invertibility of the LBDIE operators in the corresponding Sobolev spaces.

Finite-dimensional perturbations of linear operators and some applications to boundary integral equations

Finite-dimensional perturbing operators are constructed using some incomplete information about eigen-solutions of an original and/or adjoint generalized Fredholm operator equation (with zero index). Adding such perturbing operator to the original one reduces the eigen-space dimension and can, particularly, lead to an unconditionally and uniquely solvable perturbed equation. For the second kind Fredholm operators, the perturbing operators are analysed such that the spectrum points for an original and the perturbed operator coincide except a spectrum point considered, which can be removed for the perturbed operator. A relation between resolvents of original and perturbed operators is obtained. Efiective procedures are described for calculation of the undetermined constants in the right-hand side of an operator equation for the case when these constants must be chosen to satisfy the solvability conditions not written explicitly. Implementation of the methods is illustrated on a boundary integral equation of elasticity.

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with a variable coefficient

In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.

Theoretical Backgrounds of Durability Analysis by Normalized Equivalent Stress Functionals

Generalized durability diagrams and their properties are considered for a material under a multiaxial loading given by an arbitrary function of time. Material strength and durability under such loading are described in terms of durability, safety factor and normalized equivalent stress. Relations between these functionals are analysed. We discuss some material properties including time and load stability, self-degradation (ageing), and monotonic damaging. Phenomenological strength conditions are presented in terms of the normalized equivalent stress. It is shown that the damage based durability analysis is reduced to a particular case of such strength conditions. Examples of the reduction are presented for some known durability models. The approach is applicable to the strength and durability description at creep and impact loading and their combination.