Block Element Method Research Articles

To the study of block lithospheric structures of seismically vulnerable zones in the region

The paper describes a method for using regional seismic data based on processing the results of experiments carried out using a tiltmeter to study dangerous seismic conditions and clarify the structural boundaries of lithospheric units in the region. The use of experimental methods will make it possible to determine the true parameters of the block structure, which are necessary for the practical application of the block element method in the study of regional seismicity.

К исследованию влияния дефекта покрытия на поверхностный волновой процесс

The paper presents an analytical and numerical factorization method for studying the interaction of coating plates in the contact area on a rectilinear fault. Using the method of eigenfunctions and the block element method, we obtained new analytical representations of the solution for the problem, which makes it possible to estimate the behavior of the surface of the medium. We have calculated the wave field on the surface of a homogeneous elastic layer with a pinched bottom face and a composite coating with varying physical and mechanical characteristics of the plates and substrate. We have also illustrated the results of the numerical implementation for the developed algorithm and given a brief discussion about the influence of the fault boundary conditions, as well as the physical and mechanical properties of the inhomogeneous coating and substrate on the wave field formed under the influence of a vertical localized surface source.

About Earthquakes in Subduction Zones with the Potential to Cause a Tsunami

The problem of occurrence of starting earthquakes in subduction zones is considered. Subduction is the phenomenon of movement of the oceanic lithospheric plate under the continental one. The oceanic lithospheric plate at a certain depth melts from below and can slide. The paper considers the occurrence of starting earthquakes under the assumption that lithospheric plates have different contact conditions, being on a rigid base in the subduction zone. A molten lithospheric plate has no tangential contact stresses, while the other, oceanic, is rigidly connected to the base. The block element method is used to study the occurrence of the starting earthquake and the peculiarity of its consequences. The conditions to generate of tsunamis as a result of such earthquakes are being studied. Solutions to boundary value problems that are constructed precisely, rather than approximatively, allow us to reveal the mechanisms of destruction of the environment that were not previously known. In particular, the results obtained allowed us to detect a new type of crack that was not previously described. They destroy the environment in a different way than Griffiths cracks, which is demonstrated in this paper and is important in engineering practice.

Study of the tension of a heterogeneous hydroelastic system

The purpose of the article is to analyze seismic-acoustic wave fields and stress-strain state in a complex heterogeneous seismogenic structure of the contact area of the continental and oceanic crust, taking into account two types of impact: technogenic and pressure of the oceanic water column. The study uses the theory of block structures and the block element method, developed on the basis of a topological approach. It is shown that the contact stress can be considered as a superposition of stress waves occurring at the interface between the liquid and elastic media. The number of waves, their amplitude and frequency are determined by the physical and geometric characteristics of the volume of the liquid, the elastic semi-bounded layer, the elastic foundation, as well as the oscillation frequency of the external harmonic load. The model can be useful in the design of a transport network, production facilities on the coastal territory, since it allows one to assess seismic risks before the start of the process, as well as the risks of associated hazardous exogenous phenomena - landslides, tsunamis, and volcanism activation.

On the properties of topological discretization of solutions to boundary value problems

This paper seems to show for the first time that Packed block elements used in solving a boundary value problem by the block element method are elements of a discrete topological space. Since the solutions to boundary value problems belong to a discrete topological space instead of equations, it is possible to obtain a solution in a new coordinate system without having to study the boundary value problem. The block element method used in problems of continuum mechanics, which has a connection with topology, can become more effective in applications if the General properties of topological spaces studied in detail are used more deeply. One of the important properties of topology is the existence of discrete topological spaces. Their characteristic property is that an element that represents the Union of any set of elements in a topological space belongs to a discrete topological space. Belonging of discrete block elements to a topological space means that it is possible to completely cover any area with a piecewise smooth boundary and, thus, an exact solution of the boundary problem in it. In topological space, continuous geometric transformations and transitions to new coordinate systems are possible. In this paper, it is shown that Packed block elements generated by the boundary value problem for the Helmholtz equation, are elements of a discrete topological space. Given that scalar solutions of the Helmholtz equation can describe solutions to a fairly wide set of vector boundary value problems, this property also applies to solutions of more complex boundary value problems. The constructions made for the homogeneous Helmholtz equation remain valid for the inhomogeneous one.

On the role of coating defects in the form of cracks for the destruction of the pre-landslide structure

In this paper, the block element method is used to construct a model of the pre-landslide state of a block structure consisting of a water-saturated medium and an elastic coating, taking into account the formation of a crack in the coating. The spatial pre-landslide structure occupies an unlimited cylindrical region, in the cross section of which is the third quadrant. It is filled with a medium described by the anisotropic Helmholtz equation, which is extremely fluid among other water-saturated media. Taking into account the physical and mechanical properties of the pre-landslide structure, it is a vertical deformable containment wall with a deformable horizontal coating, called the sarcophagus of the landslide environment. To construct a model that is adequately formulated, we consider the boundary value problem for the three-dimensional Helmholtz equation in the specified domain, taking into account the presence of deformable walls and coverings. The block element method is used to construct an exact solution of the boundary value problem for the accepted coatings on the membrane boundary. The properties of the crack formed in the sarcophagus membrane and the consequences of its development for the destruction of the pre-landslide structure in the constructed model are investigated. All the results constructed for the Helmholtz equations, thanks to the approach outlined in the authors' latest publication in the Reports of the Russian Academy of Sciences, are transferred to materials of various rheological properties.

Investigation of the Three-Dimensional Helmholtz Equation for a Wedge Using the block Element Method

Investigation of the Three-Dimensional Helmholtz Equation for a Wedge Using the block Element Method

Block element method in solving vector boundary problems using scalar

Block element method in solving vector boundary problems using scalar

The Block-Element Method in Expansion of the Solutions of Complex Boundary-Value Problems in Mechanics

An exact solution in the first quadrant of a plane boundary-value problem of the second kind for the dynamic Lame elasticity equations is constructed, probably for the first time, and is expanded in terms of solutions of boundary-value problems for the Helmholtz equation. These solutions are presented in the form of packed block elements.

Analysis and Assessment of Seismic Hazard for the Azov-Black Sea Recreation Area

One of the geological structures accountable for the implementation of seismic potential of the region is the largest vertical faults in the Earth’s interior, where earthquake foci are usually located. This article is aimed at developing a better method for calculation of stresses and strains that occur in such seismogenic areas. According to the results of the analysis of data collected during the expeditionary work, the geophysical medium is modeled by a block structure in the form of a half-space with a cut rectangular parallelepiped, which is divided into five block elements. The state of material in the geological medium is described in each block by motion equations for a homogeneous, isotropic elastic medium in the Lamé form. Following the block element method, the algorithm of the differential factorization method is implemented in each block. Based on the numerical analysis results, the main trends in contact stresses and dynamics of displacement amplitudes were determined depending on the mechanical property values of the block material and the geometric parameters of the structure.

On features of starting earthquakes at rigid contacts of lithospheric plates with the base

The paper reports the results of a study of the strength of fragments of a lithospheric plate with a fault under the assumption that a granite lithospheric plate is rigidly connected to a basalt base along the Conrad border. In this case, one vertical and two horizontal contact stress components arise in the contact zone. Two types of faults are considered. In the first case, the fault banks are at a certain distance, whereas in the second case they are in contact. When solving the boundary value problem by the block element method, the problem of differential factorization of a third-order matrix function is solved exactly. In the process of solving the posed boundary problem, it is possible to accurately implement all the steps of the block element method, which allows us to obtain an exact analytical solution of the boundary problem. As a result, the problem of concentration of contact stresses for two types of faults is accurately solved. In the first case of nonzero distance between the plates, the concentration of contact stresses results in a finite energy of the system. In the second case of contact of the plates, the stress concentrations in each component of the vector of contact stresses leads to infinite energy in the system that is treated as an earthquake. Calculation of the Earth surface displacement in the epicenter zone in the case of a starting earthquake shows that surface motion occurs simultaneously in three directions. This explains the fact that such earthquakes cause greatest seismic damage on the Earth surface.

Топологическая дискретизация решений граничных задач механики сплошных сред

This paper seems to show for the first time that Packed block elements used in solving a boundary value problem by the block element method are elements of a discrete topological space. Since the solutions to boundary value problems belong to a discrete topological space instead of equations, it is possible to obtain a solution in a new coordinate system without having to study the boundary value problem. The block element method used in problems of continuum mechanics, which has a connection with topology, can become more effective in applications if the General properties of topological spaces studied in detail are used more deeply. One of the important properties of topology is the existence of discrete topological spaces. Their characteristic property is that an element that represents the Union of any set of elements in a topological space belongs to a discrete topological space. Belonging of discrete block elements to a topological space means that it is possible to completely cover any area with a piecewise smooth boundary and, thus, an exact solution of the boundary problem in it. In topological space, continuous geometric transformations and transitions to new coordinate systems are possible. In this paper, it is shown that Packed block elements generated by the boundary value problem for the Helmholtz equation, are elements of a discrete topological space. Given that scalar solutions of the Helmholtz equation can describe solutions to a fairly wide set of vector boundary value problems, this property also applies to solutions of more complex boundary value problems.

Cracks of a New Type and Models of Some Nano Materials

It is proved in the article that the recently revealed cracks of a new type, which have a different mechanism of destruction of the medium than the Griffith cracks, cannot be obtained as a result of the convergence of the sides of the wedge-shaped cavity in the plate. Thus, cracks of a new type are obtained as a result of the virtual convergence of the lateral sides of a rectangular cavity in the plate. The block element method is used for proof. The same method is used to simulate some types of nanoparticles, materials of mosaic structure and to study the formation of new cracks in such materials.

Application of the Block Element Method in a Certain Boundary Value Problem of Academician I.I. Vorovich

By the example of the boundary value problems—the vector problem (set of differential equations) and the scalar problem (single equation)—the features of their solutions by the block element method are demonstrated. In this method, the solutions can be presented in packed and unpacked forms. Here, these properties are explained in comparison by the example of two problems, the vector and scalar boundary value problems, solved for the first time exactly by the block element method. The vector boundary value problem was formulated previously by Academician I.I. Vorovich to a student, and its exact solution is given in this study.

Об упрощении сложных уравнений применением блочных элементов

There are several approaches aimed at simplifying complex partial differential equations or their systems involved in the formulation of boundary value problems by introducing simpler, but in a larger number of differential equations. Their solutions allow us to describe solutions to complex boundary value problems. However, to implement this approach, it is necessary to construct solutions of simplified boundary value problems for arbitrary boundary conditions in solvability spaces boundary value problem. In some cases, this can be done using the block element method. The block element method, which has a topological basis, reveals both global and local properties of solutions to boundary value problems for partial differential equations. At the same time, it can be used to study and solve more complex boundary value problems by applying relations that describe certain equations of the continuum by means of relatively simple equations, for example, Helmholtz. To do this, we need to construct solutions of the Helmholtz equations that satisfy boundary conditions that contain completely arbitrary values, rather than partial values, set at the boundary of functions. In relation to the Helmholtz equations, this is achieved using the block element method. Examples of constructing solutions to boundary value problems for Helmholtz equation for Dirichlet and Neumann problems and a comparative analysis of solutions are given in this article.

The Block Element Method in the Theory of a New Type of Cracks

The principal features of cracks of a new type discovered recently by the authors in an investigation of initial earthquakes are described. The equations describing the stress–strain state of this new type of cracks are built, and the problems of their relation to the Griffith–Irwin cracks are discussed.

On Bearing Resources and on the Mechanics of Subduction Processes

The article considers two problems, one of which is related to the engineering direction, devoted to the study of the resources of bearings with coatings that have received a certain defect. The second relates to the mechanics of natural phenomena and is associated with the phenomenon of subduction. Subduction is the process of moving oceanic lithospheric plates under the continental in the coastal zone. It is the phenomena of subduction that lead to earthquakes that generate tsunamis. It is estimated that the length of the coastal zones where the subduction process takes place is 45 thousand kilometers. Subduction zones are zones of intense coastal marine and oceanic earthquakes that are characterized by faults in lithospheric plates. The authors managed to notice that both of these problems are mathematically described by the same boundary-value problems. To study and solve them, the block element method is used, which allows one to precisely solve these problems. The study showed that in the zones of defects and faults, high stress concentrations occur, leading to destruction of the coatings. The consequences of such destruction are investigated.

К исследованию состояния системы множественных штолен под действием нормальных и сдвиговых нагрузок

The method for studying the boundary value problems modeling multilayer structures with multiple interlayer bulkheads, developed in the works of scientists of the SSC RAS and KubSU, is generalized in this paper to the vector case of a general spatial problem.A complementary approach is presented, allowing analyzing the result of vertical and shearing effectson partitionsof ore formation. As a substrate and a coating layer, we consider homogeneous elastic layers that possess the same physical and mechanical properties but have different thicknesses.The boundary of the coating layer is considered free of stress. Linear static equations for the Kirchhoff plate system are used to describethe state of the ore formation. The block element method used in the work allowed us to reduce the boundary value problem for two elastic layers separated by a plate with a set of parallel linear holes to a system of integral equations of the Fredholm type solvableby the integral factorization method. The described approach can be applied to the study of different-sized block structures on other scales.

ОЦЕНКА ГЕОЭКОЛОГИЧЕСКИХ ПОСЛЕДСТВИЙ ДОБЫЧИ УГЛЕВОДОРОДОВ НА ШЕЛЬФЕ

Time analysis of the level of seismicity in the territories of intensive hydrocarbon production shows a clear trend of increasing seismic activity. At the same time, the maximum magnitude of registered earthquakes is noted after the start of production activities. The paper presents methods for assessing the tension of the geological massif before the production process (natural tension), after the extraction of hydrocarbon raw materials from the reservoir and after pumping water into the reservoir. The study is carried out using the block element method, with the introduction of a block element in the form of a cut-out ball for reservoir modeling. Expressions for the functional and pseudo-differential equations and the integral representation of the solution are obtained. Model calculations were performed, and the main trends in tension changes were identified. The results obtained can be used in monitoring systems for assessing the geoecological consequences of mining activities to develop preventive measures that allow performing a prospective analysis of man-made stresses in the earth’s crust and assessing the risk of a seismic event before the start of the production process.

Об одном обобщенном подходе в проблеме оценки прочности подземных сооружений, параллельных штолен

The work is devoted to studying the characteristics of the stress-strain state of underground structures with multiple partitions, for example, thin deposits exposed by a system of parallel horizontal adits. The structure is modeled by a system of Kirchhoff plates located between a linearly elastic layer located on top and a deformable foundation on the bottom. The latter can be modeled both by the elastic layer and by the Winkler foundation. We consider a method for studying boundary value problems for two deformable layers separated by a plate that has a finite number of strip cavities, based on the application of the block element method using the integral factorization method. The described approach allows us to reduce the boundary-value problem to a system of integral equations. In the work we propose a method of approximate factorization of matrix functions with two complex variables, including polynomial ones, in respect to one of the variables with fixed real values of the other. This method can be used to solve the systems of integral equations to which the problems under consideration are reduced, thus finding contact stresses on supports and sagging of adit roofs.