Atomic Functions: the History of the Formation, Development and Practical Application

Atomic functions are infinitely differentiable compactly supported solutions of functional differential equations of a special type. After the first successful building of the functions performed by VL Rvachev and VA Rvachev in the 70s of the previous century, different classes of the atomic functions of one and several variables were studied, which have found application in the solution of various problems of mathematical analysis and mathematical modeling of practical problems. Generalization of atomic functions to the case of several variables associated with the expansion of their possible application to solving boundary value problems in partial derivatives had been considered, in particular, and the development of new methods for the numerical solution of such tasks. Mathematical tools based on atomic functions of several variables have the necessary properties of universality and locality, to be requested in the practice of numerical solutions of boundary value problems. The study of functional differential equations, which are used for their formation other differential operators, fo rexample, Laplace, Helmholtz, biharmonic operators et al., leads to the construction of the special form of atomic functions. The atomic functions form the classes radial basis functions that allow you to develop on their basis meshless scheme of solving boundary value problems. In comparison with the known radial basis functions atomic radial basis functions have advantages, namely, are infinitely smooth, satisfy the functional-differential equation, effectively computable, have explicit formulas for the calculation of the Fourier transform

СИСТЕМЫ РАЗЛОМОВ В ВЕРХНЕЙ КОРЕ ФЕННОСКАНДИНАВСКОГО ЩИТА ВОСТОЧНО-ЕВРОПЕЙСКОЙ ПЛАТФОРМЫ

Directions of 683 faults located in the southeastern part of the Fennoscandian (Baltic) shield were statistically analyzed, and three orthogonal associations of fault systems were identified in the study area. According to the dynamic analysis of the fault systems and their associations, the main NW-striking faults belong to the fault network originating mainly from the early Paleoproterozoic. These faults functioned in the Paleoproterozoic during four main deformation stages: D1 – sinistral shear transtension and asymmetric rift genesis (2.1–1.9 Ga); D2 – sinistral shear transpression under oblique accretion and convergence (1.9 Ga); D3 – sinistral shear transpression under oblique collision (1.89–1.80 Ga); D4 – dextral strike-slip displacements at the background of complex escape tectonics of the late collision stage (1.80–1.78 Ga). The regional stress field changed as follows: D1 – northeast- or east-trending extension; D2 – northeast compression; D3 – sub-latitudinal compression; D4 – sub-meridian compression. Changes in dynamic loading conditions led to multiple kinematic inversions of the fault networks. Widespread transtension and transpression settings in the southeastern parts of the Baltic Shield give evidence of asymmetric rifting, oblique accretion and collision in the Paleoproterozoic, which must be taken in to account in geodynamic reconstructions.Статистический анализ направлений 683 разломов юго-восточной части Фенноскандинавского (Балтийского) щита позволил выделить три ортогональные ассоциации систем дизъюнктивных нарушений. Динамический анализ систем разломов и их ассоциаций показал, что главные структурообразующие разломы территории, имеющие северо-западное простирание, принадлежат сети разломов, которая была создана преимущественно в раннем палеопротерозое. В палеопротерозое они функционировали на протяжении четырех главных этапов деформаций: D1 – левосдвиговая транстенсия и асимметричный рифтогенез (2.2–1.9 млрд лет), D2 – левосдвиговая транспрессия в обстановке косой аккреции и конвергенции (1.9 млрд лет), D3 – левосдвиговая транспрессия в условиях косой коллизии (1.89–1.80 млрд лет), D4 – правый сдвиг на фоне сложной коллажной тектоники позднеколлизионного этапа (1.80–1.78 млрд лет). Региональное поле напряжений в процессе эволюции нарушений менялось следующим образом: D1 – растяжение в северо-восточном (или ВСВ) направлении, D2 – сжатие в северо-восточном направлении, D3 – сжатие в субширотном направлении, D4 – сжатие в субмеридиональном направлении. Изменения динамических условий нагрузки обусловили многократную кинематическую инверсию сети разрывных нарушений. Широкое распространение обстановок транстенсии и транспрессии на юго-востоке Балтийского щита свидетельствует о проявлении асимметричного рифтинга, косой аккреции и коллизии в палеопротерозое, что необходимо учитывать при геодинамических реконструкциях

ТЕКТОНИКА И МОДЕЛЬ ФОРМИРОВАНИЯ ОНЕЖСКОГО СИНКЛИНОРИЯ В ПАЛЕОПРОТЕРОЗОЕ

Consideration is being given to the Onega Paleoproterozoic structure (Onega synclinorium, OS) as a tectonotype of intraplate negative structures, which experience intermittent subsidence over a long period of time. The paper presents a model of the OS and discusses its tectonic evolution. The model is based on the geological and structural data, already published and collected so far by the authors, as well as on the data concerning the OS deep structure, particularly on the interpretation of the 1-EV seismic profile and potential fields. The proposed model illustrates an example of conjectured interaction between different geodynamic factors and explains reasons for the development of the OS throughout the Paleoproterozoic, including the periods of intense subsidence and magmatism, inversions of local basins comprising the Onega trough, and deformations of the Paleoproterozoic strata. An important role in the formation of the OS was played by shear dislocations within an imbricate fan of its controlling Central-Karelian shear zone. The shear dislocations were accompanied by rotation of a large block located to the west of the OS, which led to the rotational-indentational interaction between adjacent blocks and to compensated coexistence among transtensional and transpressional regimes along their separating shear zone. Compensatory dynamic mechanism also manifested itself in crustal layers at the base of the OS. Horizontal flow of the mid-crustal masses and their outflow from the depression were compensated by the development of deep-seated thrust duplexes and uplifts around the depression as well as by the upper crustal extension associated with low-angle dilatant normal faulting. Successive propagation of these faults, dynamically related to shear dislocations within an imbricate fan of the Central Karelia zone, controlled the formation features and southward migration of the OS-contained basins as well as magmatic and syllogenesis-related occurrences. Multilayered subhorizontal flow of low-viscosity rocks at the base and inside the OS section against the background of shear dislocations gave rise to the occurrence of crest-like and diapir-like folding. The processes of OS formation occurred amid the development and localization of active mantle plumes and asthenospheric diapirs. One of the factors of their development and localization were the phenomena of relative decompression within the imbrication fan of the Central Karelian shear zone.Охарактеризовано строение и разработана модель тектонической эволюции Онежской палеопротерозойской структуры (синклинория, ОС), представляющей собой тектонотип внутриплитных отрицательных структур, испытывавших периодическое прогибание на протяжении длительного времени. Модель разработана на основе обобщения опубликованных и авторских геолого-структурных материалов, а также сведений о глубинном строении ОС, в частности интерпретации сейсмического разреза 1-ЕВ и потенциальных полей. Модель иллюстрирует пример сопряженного взаимодействия различных геодинамических факторов и объясняет причины длительного формирования ОС на протяжении всего палеопротерозоя, включая периоды интенсивного прогибания и магматизма, инверсии составляющих Онежский прогиб локальных бассейнов и деформации палеопротерозойских толщ. При формировании ОС большое значение имели сдвиговые дислокации, проявленные в пределах имбрикационного веера Центрально-Карельской зоны сдвига, контролирующего позицию этой структуры. Сдвиговые перемещения были сопряжены с вращением крупного блока, расположенного западнее ОС, что привело к ротационно-инденторному взаимодействию смежных блоков и компенсационному сосуществованию областей транспрессии и транстенсии вдоль разделяющей их зоны сдвига. Компенсационный динамический механизм проявился и в коровых слоях основания ОС. Горизонтальное течение и отток среднекоровых масс из области депрессии компенсировались формированием глубинных надвиговых дуплексов и поднятий в обрамлении депрессии, а также растяжением верхней коры с развитием систем пологих дилатансионных сбросов. Последовательная пропагация этих сбросов, динамически сопряженных со сдвиговыми нарушениями имбрикационного веера Центрально-Карельской зоны, контролировала особенности формирования и миграцию бассейнов ОС в южном направлении, а также проявления магматизма и силлогенеза. Многоярусное субгоризонтальное течение маловязких пород в основании и внутри разреза ОС, проявившееся на фоне сдвиговых дислокаций, привело к развитию гребневидной и диапироподобной складчатости. Процессы формирования ОС проходили на фоне высокой активности мантийных плюмов и астеносферных диапиров. Одним из факторов их развития и локализации были явления относительной декомпрессии в пределах имбрикационного веера Центрально-Карельской зоны сдвига

FAULT SYSTEMS IN THE UPPER CRUST OF THE FENNOSCANDIAN SHIELD, THE EAST EUROPEAN PLATFORM

Directions of 683 faults located in the southeastern part of the Fennoscandian (Baltic) shield were statistically analyzed, and three orthogonal associations of fault systems were identified in the study area. According to the dynamic analysis of the fault systems and their associations, the main NW-striking faults belong to the fault network originating mainly from the early Paleoproterozoic. These faults functioned in the Paleoproterozoic during four main deformation stages: D1 – sinistral shear transtension and asymmetric rift genesis (2.1–1.9 Ga); D2 – sinistral shear transpression under oblique accretion and convergence (1.9 Ga); D3 – sinistral shear transpression under oblique collision (1.89–1.80 Ga); D4 – dextral strike-slip displacements at the background of complex escape tectonics of the late collision stage (1.80–1.78 Ga). The regional stress field changed as follows: D1 – northeast- or east-trending extension; D2 – northeast compression; D3 – sub-latitudinal compression; D4 – sub-meridian compression. Changes in dynamic loading conditions led to multiple kinematic inversions of the fault networks. Widespread transtension and transpression settings in the southeastern parts of the Baltic Shield give evidence of asymmetric rifting, oblique accretion and collision in the Paleoproterozoic, which must be taken in to account in geodynamic reconstructions

The active working shoot modeling profile of the centrifugal microturbine

For the small distributed power, microturbines of electric 30-500 kW capacity are used: gas piston engines, gas and steam turbines, each of which has certain advantages and disadvantages. In the work, the active working blade profile is simulated for a single-stage, two-stream, centripetal microturbine, in order to determine the optimum profile design satisfying the reliability conditions and economy. The basis is a humidsteam microturbine of a horizontal electric version with a capacity of 30 kW. The initial data for software simulation were the microturbineactive stage characteristics, determined by the steam turbines calculation traditional methods

The active working shoot modeling profile of the centrifugal microturbine

For the small distributed power, microturbines of electric 30-500 kW capacity are used: gas piston engines, gas and steam turbines, each of which has certain advantages and disadvantages. In the work, the active working blade profile is simulated for a single-stage, two-stream, centripetal microturbine, in order to determine the optimum profile design satisfying the reliability conditions and economy. The basis is a humidsteam microturbine of a horizontal electric version with a capacity of 30 kW. The initial data for software simulation were the microturbineactive stage characteristics, determined by the steam turbines calculation traditional methods