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Many interesting findings were obtained during the INDEPTH geophysical surveys but the results of these surveys were not sufficient to explore the deep crustal relations among the Himalayan Mega Thrust namely MCT, STD and GCT. In this study, 2-D finite element method was used to analyze the state of stress and deep crustal relation between MCT, STD and GCT. Elasto-plastic, plane strain model constrained by the northward convergent displacement boundary condition was used to simulate the stress field. Modelling results reveal that thrust/fault geometry, their deep crustal relations and how do they terminate had effect on the stress field, displacement vector, shear stress and exhumation. The lateral variation of the stress orientation and surface exposure of fault/thrust and lithologic units may be the expression of the deep crustal relations between the major structures, their geometry and how do they terminate at the depth. So the along strike variation in the Himalaya may be the manifestation of structural geometry and their deep crustal relationships.

The Shillong Plateau, NE India is reported to be one of the most seismically active ‘pop-up’ structures lying between the Dauki fault and recently reported Oldham fault in northeast India. Dauki fault lies down faulted Bengal basin to the south and Oldham fault, the Assam valley, the Himalayan foothills region, lesser Himalaya and Butan and Tibet further north. We examined convergent displacement due to northward movement of the Indian plate with respect to the Eurasian plate and also consider uplift of the Plateau simultaneously. Four models with different boundary conditions under both plane strain and plane stress have been selected. The maximum compressive stress shows a preferred orientation that helps us to explain the tectonic environment as well as the fault pattern in plane strain condition. The realistic regional maximum horizontal compressive stress (σ_Hmax) orientation is observed within the best-fit model under plane stress condition. The best-fit model result under plane strain condition shows that the compressive stress regime is dominant in the study area, except for the uppermost part of the crust where the tensional stress is dominant. These tensional stress regimes support normal faulting and structural discontinuities. With increase progressive convergent displacement (under plane strain condition), the principal stresses rotate anticlockwise around the fault zones, and the upper crust of the Bengal basin and the Assam valley, on the other hand, behave thrust faulting. Rock properties (density, Poisson's ratio, Young's modulus, cohesion, angle of internal friction) and Mohr-Coulomb failure criterion are used to calculate failure and faulting pattern. The simulated results have significant explanation on the proposed Oldham fault as well as on the geologically evidenced Dauki fault. Our preferred model explains deformation and seismic activity of the area. Our plane stress simulation results show that the Dauki fault is the major controlling eleme

The present-day stress field in the Earth's crust is important and provides insights into mechanisms that drive plate motions. In this study, an elastic plane stress finite element modeling incorporating realistic rock parameters have been used to calculate the stress field, displacement field and deformation of the plate interactions in Anatolia. Modeled stress data for the African-Arabian-Anatolian plate interactions with fixed Eurasian platform correlate well with observed stress indicator from the world stress map (WSM) and focal mechanism of earthquakes; while displacement field agree qualitatively well with GPS vectors and sense of motion indicated by focal mechanisms for large crustal earthquakes (M＞6) and plate motion models. Modeling result shows the direction of maximum horizontal compressive stress (σ_) toward the direction of absolute motion of these plates. Large perturbations in σ_ orientations are shown to occur in and around tectonic boundaries between those plates. It is observed that, although the African plate acts mostly as indenter, which transmits the collisional motion from the Arabian plate to the Anatolian plate, in the current situation the far-field stress probably from the Hellenic subduction is needed to satisfy the present-day stress field in Anatolia.

F E modeling under plane stress condition is used to analyze the state of stress in and around the San Andreas Fault (SAF) System taking whole area of California. In this study we mainly focus on the state of stress at the general seismogenic depth of 12 km, imposing elastic rheology. The purpose of the present study is to simulate the regional stress field, displacement vectors and failures. Stress perturbation due to major fault, its geometry and major branches are analyzed. Depthwise variation from 20 km to 0.5 km is considered for the fault type analysis. Series of calculations are performed with the parametrical variations of domain properties and are applied for the strong/weak SAF. All the simulated results are finally utilized for the implication of present day plate kinematics. Although in nature there is lateral and vertical variation in rheology within single domain and different domains (not considered in this study), our simulated results are comparable with the observed data. The imposed boundary condition (fixed North American plate, Pacific plate motion along N34°W vector up to northern terminus of the San Andreas faults and N50°E vector motion for the subducting Gorda and Juan de Fuca plates) had simulated the present day regional σHmax orientation and displacement vector. Simulated results show some local effect on the stress field and displacement vector by the main strand of the fault and probably its geometry and branch (Garlock Fault). Probably the low angle σHmax orientation to the strike of the SAF system in the South California (although there are some high angle σHmax orientation in southeastern California) compare to the central and northern California is due to the big bend and the Garlock Fault. This modeling had also beautifully simulated the left lateral strike slip movement on the Garlock Fault. Failure analysis performed based on Byerlee's law directly test the strong SAF and variation of physical parameters in the Mohr-Coulomb failu

Finite element (FE) modeling has designed to investigate dynamics of Southern Ryukyu Arc based on synthesized stress-strain, velocity, geology and seismicity data. Result of horizontal minimum stress (σHmin) orientation is compared to present stress distribution. We constructed 1300 km X 600 km model for Southern Ryukyu Arc. FE model was bounded along the Ryukyu Trench. Studied region is divided into 3 domains: arc region, Okinawa Trough and continental margin. As displacement velocity is 9.0 cm/yr in Taiwan, we set up NW boundary displacement of 2000 m that correspond to c.a. 20,000 year on Ryukyu Trench to Taiwan. Four boundary conditions under 1, 10 and 30 km depth are considered. In addition, single domain model is also constructed. Total 16 experiments are performed. Result of σHmin orientation shows good agreement with T-axis distribution in Taiwan and Ryukyu Arc (Kubo and Fukuyama, 2003; Otsubo and Hayashi, 2003). In addition, large stress concentrates on eastern area of Taiwan while in Okinawa trough, computed σHmin orientation is arc-parallel which is inconsistent with observed σHmin orientation. Depth affects greatly to the stress magnitude but not to stress orientations in the Ryukyu Arc. It is found that orientation of σHmin is principally controlled by rock domain properties of major structural zones and the direction of convergence of the Philippine Sea plate relative to Eurasia. Direction of displacement vector is good agreement with that of observed velocity field in Taiwan, while direction of displacement vector is trending NW or W in Okinawa trough and Ryukyu Arc, indicating that this model does not fit in this point.

3D FE elastic program was included to my FE software package in 1989, though the theory and FE formulation of this 3D FE elastic program was not described. The functionals for 3D and 2D elasticity are explained in this paper regarding the variational principle that is more sophisticated theoretical basis of FE formulation than the principle of virtual work. The FE formulation of 3D elasiticity is explained here based on the principle of virtual work, because the 3D elastic program including my FE software package was developed using the principle of virtual work.

This paper describes about field study as well as laboratory test performed for the geotechnical properties of rock and soil along the Tinpiple-Banchare Dada road alignment. The purpose of the present study was to determine the geotechnical properties for both rocks and soils along the road alignment. Detailed geotechnical soil/rock field study sheets were used to record geotechnical parameters in the field and to know the site condition. Dynamic cone penetration test was performed in the in situ soil. Laboratory test were performed for the classification of the soil, and to determine the optimum moisture content. Results reveal that most part of the road is aligned through the completely weathered sandstone, its residual soil (classified as SM) along with the gneisses. The results of the Dynamic Cone Penetration Test show that minimum CBR % varies from 18% at 10 cm depth to 2% at 58 cm. These values of the CBR % are considered good so no capping is required. The results of the compaction test show that maximum dry density for most of the samples were obtained at 10-15 % optimum moisture content. The results of the joint analysis revealed that chainage 2+ 115, 2+475, 5+125 to 5+ 175 and 7+220 km has probability of lateral, lateral / central, central and central wedge failure respectively. Small landslides were initiated in upslope side due to road cutting and can be stabilized by simple bioengineering techniques. The results of this study are very important for the construction of a sound road and should be considered during the full phase onstruction of the road.

2D elastoplastic FE program has been developed in 2002 (Hayashi, 2002). Improved FE program was applied to the Umaoi hills in Hokkaido, Japan to estimate plate convergence rate (Morishita and Hayashi, 2005; Hayashi and Morishita, 2006). This FE elastoplastic software is derived from 2D von Mises' plastic theory. The detail of elastoplastic theory and FE formulation are explained in this paper.

The Main Central Thrust (MCT) zone is one of the most important tectonic zones of the Himalayas along which the inverted metamorphism has been observed. The MCT zone around Annapurna have been surveyed to analyse 3D strain by adopting the least square method (Hayashi, 1994,2001). The problem which is encountered in treating 3D strain analysis, is the precision of strain. The consistent value (c-value) was derived to estimate how the resultant value of strain is consistent with true value (Hayashi, 1995). Setting the c-value properly, invalid samples are excluded from the strain analysis. The paper summarized as its important result that the foliation plane and XY plane are almost parallel together. Although this result has been recognized as a general agreement by many geologists, there has been no direct proof using 3D strain analysis up to now. Shear heating model is considered as the reasonable cause of inverted metamorphism along the MCT from the intermediate intensity of strain. The 3D strain analysis introduced here has benefits that any rock which is composed of grains can be used as marker rock and we can find out such rocks ubiquitously throughout the Himalayas.

The immense Himalayan arc is evolved as a consequence of the collision between Indian and Eurasian landmasses some 50 million year ago. The Indian plate converges northward at the average rate of 20.5 ± 2 mm/year (Bilham et al., 1988), and is under-thrusting beneath Tibet. This continuous northward penetration of India under Eurasia has produced the broad zone of active crustal deformation, shortening, slicing and surface uplift of the northern margin of the Indian continent; and build up the Himalaya is under very strong compressive strain that made the entire Himalayan region one of the most seismotectonically dynamic intercontinental regions of the world. In the present study, an approach has been made to model a NE-SW cross-section (Ram et al., 2005) extending from the Gangetic Plain to the Tethys Himalaya including potentially active major faults by means of FE method (Hayashi, 2008) considering an elastic rheology under plane strain condition with convergent boundary environment in the fold-and-thrust belt of the Garhwal Himalaya. The present study provides an opportunity to understand the neotectonic stress distribution, style of deformation and present day shortening rate in the Himalayan front. Simulation results reveal that the compressive stress of stress developed in the northern part and tensional stress fields developed in the southern part of the Garhwal Himalaya, and consequently, thrust are developed in the north and normal faults are developed in southern regions respectively, which is consistent with the characteristics of the fold and thrust belt. Furthermore, modeling results show good agreement with the Quaternary deformation, active faulting, microseismicity and focal mechanism solution of the fold-and-thrust belt of the Garhwal Himalaya.

Theoretical basis of a FE software package is explained in term of the theory of finite element method, 2D elastic FE problem, fault analysis, physical property of rocks. Flow chart and function of the FE software package is added to see the perspective view among many programs. The aim of the paper is to show the soundness of the FE software package which is used in the Simulation Tectonic Laboratory.

An understanding of orientation of contemporary maximum horizontal compressive stress (σ_) is important to many aspects of earth sciences, e.g. seismicity, neotectonics, and plate driving mechanisms. Comparison of recent stress observations and the results of stress modeling provide a powerful approach to refine our understanding of geodynamics processes. This is especially important for complex area like Himalayan-Tibet orogen, a continental collision zone between the Indian and Eurasian Plates. The frequently occurring earthquakes and other tectonic stress indicator have provided vast set of database on maximum horizontal compressive stress (σ_) that can be useful to study contemporary stress sources, plate kinematics and ongoing geodynamics. In this contribution, taking advantage of elastic plane stress finite element modeling (FEM), and observed data on σ_, several models are presented to reproduce stress field. Simulated models show that the convergence normal to the orogen is essential to reproduce observed σ_, which in turn controls the magnitude and orientation of σ_. The kinematics equivalent to east-west tectonic escape did not reproduce the observed stress field. Therefore, the best-fit model of present day stress field is obtained only in three domains model with southeastward tectonic escape of the Tibetan crust rather than eastward extrusion. There is, however, significant increase in σ_ magnitude with increasing crustal depth because of stress amplification. Incorporation of suture zones in the model did not change orientation of σ_, significantly. Considering these facts, 'continuum tectonic model' is more preferable than the 'block tectonic model' for the active deformation of the Tibetan Plateau. Contemporary stress field deduced from several tectonic stress markers reconciles with the predicted one giving insights on their sources and ongoing plate kinematics of the continental collision zone between India

The Main Central Thrust (MCT) zone is one of the most important tectonic zones of the Himalayas along which the inverted metamorphism has been observed. The MCT zone around Annapurna have been surveyed to analyse 3D strain by adopting the least square method (Hayashi, 1994,2001). The problem which is encountered in treating 3D strain analysis, is the precision of strain. The consistent value (c-value) was derived to estimate how the resultant value of strain is consistent with true value (Hayashi, 1995). Setting the c-value properly, invalid samples are excluded from the strain analysis. The paper summarized as its important result that the foliation plane and XY plane are almost parallel together. Although this result has been recognized as a general agreement by many geologists, there has been no direct proof using 3D strain analysis up to now. Shear heating model is considered as the reasonable cause of inverted metamorphism along the MCT from the intermediate intensity of strain. The 3D strain analysis introduced here has benefits that any rock which is composed of grains can be used as marker rock and we can find out such rocks ubiquitously throughout the Himalayas.

The immense Himalayan arc is evolved as a consequence of the collision between Indian and Eurasian landmasses some 50 million year ago. The Indian plate converges northward at the average rate of 20.5 ± 2 mm/year (Bilham et al., 1988), and is under-thrusting beneath Tibet. This continuous northward penetration of India under Eurasia has produced the broad zone of active crustal deformation, shortening, slicing and surface uplift of the northern margin of the Indian continent; and build up the Himalaya is under very strong compressive strain that made the entire Himalayan region one of the most seismotectonically dynamic intercontinental regions of the world. In the present study, an approach has been made to model a NE-SW cross-section (Ram et al., 2005) extending from the Gangetic Plain to the Tethys Himalaya including potentially active major faults by means of FE method (Hayashi, 2008) considering an elastic rheology under plane strain condition with convergent boundary environment in the fold-and-thrust belt of the Garhwal Himalaya. The present study provides an opportunity to understand the neotectonic stress distribution, style of deformation and present day shortening rate in the Himalayan front. Simulation results reveal that the compressive stress of stress developed in the northern part and tensional stress fields developed in the southern part of the Garhwal Himalaya, and consequently, thrust are developed in the north and normal faults are developed in southern regions respectively, which is consistent with the characteristics of the fold and thrust belt. Furthermore, modeling results show good agreement with the Quaternary deformation, active faulting, microseismicity and focal mechanism solution of the fold-and-thrust belt of the Garhwal Himalaya.

Theoretical basis of a FE software package is explained in term of the theory of finite element method, 2D elastic FE problem, fault analysis, physical property of rocks. Flow chart and function of the FE software package is added to see the perspective view among many programs. The aim of the paper is to show the soundness of the FE software package which is used in the Simulation Tectonic Laboratory.

FEM analysis for stress determination of fold-and-thrust belts of the South Central High Andes (SCRA) was carried out on Paleozoic to Quaternary rocks from the Coastal Cordillera to Precordillera regions of the Chile and Argentinean Andes, between 30^o and 33^o south latitudes. A two-dimensional vertical cross-section including 8 layers through the Andean crust up to Moho has been represented by a finite element model composed of an assembly of 2699 elements and 1456 nodes in a state of plane strain conditions. The model is assumed to be multilayered lithospheric crustal block undergoing convergence. and we choose to demonstrate our results by applying horizontal displacement rate (average velocity of 6.50 cm/yr) of descending Nazca plate. The failure of elements is defined by adopting the concept of Mohr-Coulomb failure criterion. The assigned rock rheology and physical properties of each layer in simulation make the model behavior as elastic body in which well defined failure location gives a hint of thrust development. Studies of lithospheric deformation of the fold-and-thrust belts in this zone where structural style varies from thinskinned to thick-skinned fold-and-thrust belts have revealed various compressional states of stress. The maximum compressional stress (σ_1) was generally oriented in an E-W direction. Along the westernmost part of the Coastal Cordillera, the strain deformation is extensional in some elements. which can be explained by a co-seismic crustal bending readjustment. Overall study results imply that most of the basement-involved deep thrusts (about 20 km) and deformation. occurred within the upper crustal parts that lie in. Layers-4, 5 and 6, are more likely related to crustal anisotropy.

The Red Sea is a tectonic rift that was formed in the late oligocene-early Miocene when the originally connected African and Arabian land masses broke apart. At first it was a continental rift, then, as Arabia drifted away, developed into an intercontinental system that today separates the independent Arabian plate from the African plate. The Red Sea rift is part of an extensive global system of faults running approximately north to south. In the present study, numerical modeling on the Saudi Arabian seismic reflection profiles is carried out to examine the neotectonic stress field in the south western Red Sea-Arabian plate margin to reveal a kinetics of active fault system using two-dimensional elastic finite element method (FEM) under plane strain condition. The Mohr-Coulomb failure criterion has been adopted to analyze the relationship between stress distribution and fault formation.A Saudi Arabian reflection profile (Mooney et al, 1985; Prodehl, 1985) is adopted for the modeling and extensional displacement boundary condition is imposed along NE-SW direction. Our result shows the extensional displacement and physical properties of rock layer control the distribution, orientation, magnitude and intensity of the stress and fault development. According to the calculated stress patterns of failure elements, normal faults develop in the Red Sea and Arabian Plate margin. The results from our simulation are in good agreement with those of the seismicity, focal mechanism solution of earthquakes and active faulting in the Red Sea.

FEM analysis for stress determination of fold-and-thrust belts of the South Central High Andes (SCRA) was carried out on Paleozoic to Quaternary rocks from the Coastal Cordillera to Precordillera regions of the Chile and Argentinean Andes, between 30^o and 33^o south latitudes. A two-dimensional vertical cross-section including 8 layers through the Andean crust up to Moho has been represented by a finite element model composed of an assembly of 2699 elements and 1456 nodes in a state of plane strain conditions. The model is assumed to be multilayered lithospheric crustal block undergoing convergence. and we choose to demonstrate our results by applying horizontal displacement rate (average velocity of 6.50 cm/yr) of descending Nazca plate. The failure of elements is defined by adopting the concept of Mohr-Coulomb failure criterion. The assigned rock rheology and physical properties of each layer in simulation make the model behavior as elastic body in which well defined failure location gives a hint of thrust development. Studies of lithospheric deformation of the fold-and-thrust belts in this zone where structural style varies from thinskinned to thick-skinned fold-and-thrust belts have revealed various compressional states of stress. The maximum compressional stress (σ_1) was generally oriented in an E-W direction. Along the westernmost part of the Coastal Cordillera, the strain deformation is extensional in some elements. which can be explained by a co-seismic crustal bending readjustment. Overall study results imply that most of the basement-involved deep thrusts (about 20 km) and deformation. occurred within the upper crustal parts that lie in. Layers-4, 5 and 6, are more likely related to crustal anisotropy.

Two dimensional finite element simulation technique has been applied to analyze the state of stress and fault development within the Subandean foreland region of Southern Bolivian and Northern Argentinean Central Andes, assuming the lithospheric crustal block as a multilayered elastic slab exhuming under plane-strain condition. The computations have been carried out with computer programs, which simulate the elastic models under convergent displacement boundary condition, synonyms to the converging Nazca Plate toward east underneath the South American Plate. The finite element method, the numerical modeling technique computes the stresses with its magnitude, orientation, deformation of rock materials and to predict fracture attributes as a function of physical properties of rock materials along the study area. The failure proximity is defined by adopting the concept of Mohr-Coulomb failure criterion. The failure proximity and stress field in this model postulated a proper assumption of compressive tectonic deformation in the Southern Bolivia and Northern Argentinean Central Andes. The assigned rock rheology and physical properties of each layers in simulation makes the models behave as elastic body in which well defined failure location which gives a hint of fault development. In this study area extensive failure elements developed within the subandean region above the Precambrian basement, is contemporary to the early phase of thrusting within this Tertiary sequence and the failure elements nucleate during the later phase of convergent movement within the comparatively stable rear part of Eastern Cordillera and Altiplano-Puna region as the recent deep thrust. The focal mechanism of major shallow (～33km) recent earthquakes representing compressional event in this area adjacent to the CANP (Main Andean Thrust) and CFP (Main Frontal Thrust) is consistent with thrust development in Subandean frontal Tertiary sequences.

Finite element method (FEM) is a general method of structural analysis in which a continuum or continuous structure is replaced by a finite number of elements interconnected at a finite number of nodal points. The method can be used to determine the displacements of the nodal points and stresses within the elements developed in two or three dimensional elastic or viscous structures of arbitrary geometrical and material properties. From this point of view, a 2D finite element method has been adopted to characterize the stress field and deformation pattern (mainly faults) in the Himalayan orogenic ranges. The Himalayas represent one of the few places on earth where continental crust is attempting to underthrust continental crust. As the Indian plate underthrusts beneath the Himalaya, it warps down in response to an advancing orogenic load and keeps the entire Himalayan mountain arc seismically active with which continuously influencing the stress field and structures in the regime. A series of elastic finite element models are presented to examine the state of stresses and faults on the models as well as within the incipient zones of major thrusts (MCT, MBT and MFT) after collision. Finally, the tectonic implementation of such simulated structures are drawn with combining the previously published geological, seismological and focal mechanism solution of faults data in the Himalayas. The geologic profiles A, B, C and D of the central Himalayas which are used for the purpose under plane strain condition with elastic rheology. The elasticity of such model's layers considered with regards to the rock layer properties that there is a tendency, i.e. the older profiles might composed of the harder rocks and the older rock layers are the larger properties (e.g. density, Young's modulus and cohesion), whereas friction angle is lesser and Poisson's ratio is constant for all stages of models. The convergent rate of Indian plate has been considered 10 cm/a for profile A (40-20 Ma, The distribution of stress trends and faults calculated in the models are compared with the previously published geological data, earthquakes focal mechanism solutions of faults and active faults analysis data in the Himalayas. Comparison shows the close similarities between the simulated results and the aforesaid published data. Finally on the basis of the numerical modeling results and summary of former geological and geophysical researches, a preliminary hypothesis is proposed on the structural and tectonic development of the Himalayan orogeny, where especial attention has been paid for examining the earlier development stage of stress state and fault in the incipient zones of the future thrusts MCT, MBT and MFT as well as the present stage. The compressive stress and thrust faults within the incipient zones might be the responsible for the earlier development stage of MCT (40 Ma), MBT (20 Ma) and MFT (10 Ma) which may paved the way for posing that the MCT is the oldest and MFT is the youngest thrusts in the Himalayas and they are propagating southward from the initiation time. Thus It is very convenient to state that the continuous propagation of such thrusts might have greatly influenced the tectonics and structures in this compressoinal mountain belt. Presently, around the frontal part of the Himalayas might be more active due to development of large numbers simulated faults which is also responsible for the neotectonic activity in the regimes.

Two dimensional finite element simulation technique has been applied to analyze the state of stress and fault development within the Subandean foreland region of Southern Bolivian and Northern Argentinean Central Andes, assuming the lithospheric crustal block as a multilayered elastic slab exhuming under plane-strain condition. The computations have been carried out with computer programs, which simulate the elastic models under convergent displacement boundary condition, synonyms to the converging Nazca Plate toward east underneath the South American Plate. The finite element method, the numerical modeling technique computes the stresses with its magnitude, orientation, deformation of rock materials and to predict fracture attributes as a function of physical properties of rock materials along the study area. The failure proximity is defined by adopting the concept of Mohr-Coulomb failure criterion. The failure proximity and stress field in this model postulated a proper assumption of compressive tectonic deformation in the Southern Bolivia and Northern Argentinean Central Andes. The assigned rock rheology and physical properties of each layers in simulation makes the models behave as elastic body in which well defined failure location which gives a hint of fault development. In this study area extensive failure elements developed within the subandean region above the Precambrian basement, is contemporary to the early phase of thrusting within this Tertiary sequence and the failure elements nucleate during the later phase of convergent movement within the comparatively stable rear part of Eastern Cordillera and Altiplano-Puna region as the recent deep thrust. The focal mechanism of major shallow (～33km) recent earthquakes representing compressional event in this area adjacent to the CANP (Main Andean Thrust) and CFP (Main Frontal Thrust) is consistent with thrust development in Subandean frontal Tertiary sequences.

Grabens of southern Tibet and the Himalaya represent the Cenozoic extensional tectonic phase, which has affected whole Tibet and northernmost part of the Himalaya. Thakkhola half graben, on the crest of the Himalaya, is one of the north trending grabens that define the Neogene structural pattern of the southern margin of the Tibetan Plateau and is seemingly enigmatic feature in a regionally contractional tectonic setting between the colliding plates. Two-dimensional, elastic, plane-strain, finite element models (FEMs) are generated to simulate the effects of geometry and rock properties on growth of graben faults and their configuration for the Thakkhola half graben evolution. The performed models have shown that the extensional graben faults form at the top of the overburden and propagate downward as we increased extensional displacement. Simulated models have clearly defined the graben bounding faults. Further they are able to suggest that natural grabens have multiple faults on each side rather than single fault. During progressive extensional displacement depth of faulting increases and deformation is mainly localized in downthrown block both in basement and syntectonic deposits. Syntectonic deposit is characterized by normal faulting in tensional tectonic stress field, which is a common feature of the small-scale graben at post rift deformation stage. The width and depth of graben are primarily controlled by the rheology of the upper elastic layer and syntectonic deposits. The applied rock layer properties are able to deduce the first order characteristics of the Thakkhola half graben. Therefore, this simulation constrains probable values for the rock layer properties controlling the Thakkhola half graben evolution. Assumption of the weak zone does not make significant difference on stress distribution and faulting. Thus it seems that the Thakkhola fault system only did not contribute to development of the half graben.

Central Hokkaido is one of the interest areas to study geology because this area has an issue about the position of the plate boundary between the Eurasian (Ammurian) and the North American (Okhotsk) plate. There are two hypotheses about this plate boundary. We have used elastic plastic finite element program (Hayashi, 2002) to estimate displacement rate at Umaoi hills, and compared with the result of earlier study (Kato et al, 2004) which based on balanced cross section. The result indicates that the convergence rate in the front of Hidaka thrust system has not decreased during late Quaternary.

In this study, we propose the cause of thrusting in Nankai Trough off cape of Muroto using Finite Element Method (FEM). Our models are based on cross-section produced by Moore et al. (2001). We divide the section into eight layers. During calculation we change rock layer properties of Landward Dipping Reflector Zone (LDRZ), oceanic crust and hypothetical seismogenic zone. Overall results of the modelling suggest that the seismogenic zone controls the thrust development in the region.

Grabens of southern Tibet and the Himalaya represent the Cenozoic extensional tectonic phase, which has affected whole Tibet and northernmost part of the Himalaya. Thakkhola half graben, on the crest of the Himalaya, is one of the north trending grabens that define the Neogene structural pattern of the southern margin of the Tibetan Plateau and is seemingly enigmatic feature in a regionally contractional tectonic setting between the colliding plates. Two-dimensional, elastic, plane-strain, finite element models (FEMs) are generated to simulate the effects of geometry and rock properties on growth of graben faults and their configuration for the Thakkhola half graben evolution. The performed models have shown that the extensional graben faults form at the top of the overburden and propagate downward as we increased extensional displacement. Simulated models have clearly defined the graben bounding faults. Further they are able to suggest that natural grabens have multiple faults on each side rather than single fault. During progressive extensional displacement depth of faulting increases and deformation is mainly localized in downthrown block both in basement and syntectonic deposits. Syntectonic deposit is characterized by normal faulting in tensional tectonic stress field, which is a common feature of the small-scale graben at post rift deformation stage. The width and depth of graben are primarily controlled by the rheology of the upper elastic layer and syntectonic deposits. The applied rock layer properties are able to deduce the first order characteristics of the Thakkhola half graben. Therefore, this simulation constrains probable values for the rock layer properties controlling the Thakkhola half graben evolution. Assumption of the weak zone does not make significant difference on stress distribution and faulting. Thus it seems that the Thakkhola fault system only did not contribute to development of the half graben.

The Ryukyu arc is an active arc-trench system, associate with active intracontinental back-arc basin (Okinawa Trough). Rifting of the Okinawa Trough has occurred since Miocene. The Ryukyu arc is expected to undergo complex geological history. Paleostress fields of the central and northern Ryukyu island are estimated from fault analysis by use of the Multi-inverse method (Yamaji, 2000) and Ginkgo method (Yamaji, 2003a). The stress transition since middle Miocene is found by measurement of the fault-slip data in the northern and central Ryukyu arc. The fault-striation analysis leads the following tectonic evolution in the central and northern Ryukyu arc. In middle Miocene, E-W extension occurred in the northern Ryukyu arc, which may be related to the opening of the Japan Sea. The compressional stress field occurred in late Miocene to early Pliocene (5-8 Ma), which was simultaneous with the formation of the Taiwan-Shinji Fold Belt. Subsequently E-W extension occurred in Pliocene, which was related to the opening the Okinawa Trough. Around at 2 Ma, complex stress regime occurred in the Ryukyu arc. In this time, opening of the Okinawa Trough resumed and the deposition of the Ryukyu group started. In northern Ryukyu arc, strike-slip stress regime has occurred from 2 Ma ago. NE-SW extension, arc-parallel extension is the latest stress regime in the central and northern Ryukyu arc. Arc parallel extension is observed in the whole Ryukyu arc. On the other hand, before the deposition of the Ryukyu group, stress regime of the southern Ryukyu arc is different from the central and northern Ryukyu arc.

We used 2D finite element method (FEM) to investigate the crustal strength of the Izu collision zone. Our model primarily based on the crustal structure given by Aoike et al. (2001). We set up standard rock layer properties for simulation. During simulation, cohesion was varied systematically. Simulation results show that failure zone decreases with increasing the convergent displacement in both cases i.e. before and after increasing cohesion. We used Mohr-Coulomb failure criterion and Mohr's circle to understand the reason. It is reasonable that rock layers do not break but flow below more than 30km depth. Further we compared simulated results with the focal mechanism data. Our models show only normal faulting which is not consistent with focal mechanism data. Thus our simulation models probably reveal the crustal strength of Izu collision zone.

Finite element method has been used in this study to know the fault development in the Red Sea rift system. Two models with fault zone and without fault zone are calculated to determine the effect of detachment fault on the distribution of deformation in the Red Sea area under plane strain condition with linear elastic rheology. Influence of layer properties on fault development are compared with two models. Three types of displacement boundary condition (extensional, spreading, and spreading with upward) are also used to examine results and to search the. best fitted boundary conditions. Results of this study show that (1) detachment fault controls the normal fault system in the Red Sea area, (2) friction angle and cohesion are important parameters, and sensitive to fault development, (3) spreading and spreading with upward displacement boundary conditions show the reasonable results which support the active rifting hypothesis in the development of the Red Sea rift system.

Finite element method has been used in this study to know the fault development in the Red Sea rift system. Two models with fault zone and without fault zone are calculated to determine the effect of detachment fault on the distribution of deformation in the Red Sea area under plane strain condition with linear elastic rheology. Influence of layer properties on fault development are compared with two models. Three types of displacement boundary condition (extensional, spreading, and spreading with upward) are also used to examine results and to search the. best fitted boundary conditions. Results of this study show that (1) detachment fault controls the normal fault system in the Red Sea area, (2) friction angle and cohesion are important parameters, and sensitive to fault development, (3) spreading and spreading with upward displacement boundary conditions show the reasonable results which support the active rifting hypothesis in the development of the Red Sea rift system.

The Ryukyu arc is an active arc-trench system, associate with active intracontinental back-arc basin (Okinawa Trough). Rifting of the Okinawa Trough has occurred since Miocene. The Ryukyu arc is expected to undergo complex geological history. Paleostress fields of the central and northern Ryukyu island are estimated from fault analysis by use of the Multi-inverse method (Yamaji, 2000) and Ginkgo method (Yamaji, 2003a). The stress transition since middle Miocene is found by measurement of the fault-slip data in the northern and central Ryukyu arc. The fault-striation analysis leads the following tectonic evolution in the central and northern Ryukyu arc. In middle Miocene, E-W extension occurred in the northern Ryukyu arc, which may be related to the opening of the Japan Sea. The compressional stress field occurred in late Miocene to early Pliocene (5-8 Ma), which was simultaneous with the formation of the Taiwan-Shinji Fold Belt. Subsequently E-W extension occurred in Pliocene, which was related to the opening the Okinawa Trough. Around at 2 Ma, complex stress regime occurred in the Ryukyu arc. In this time, opening of the Okinawa Trough resumed and the deposition of the Ryukyu group started. In northern Ryukyu arc, strike-slip stress regime has occurred from 2 Ma ago. NE-SW extension, arc-parallel extension is the latest stress regime in the central and northern Ryukyu arc. Arc parallel extension is observed in the whole Ryukyu arc. On the other hand, before the deposition of the Ryukyu group, stress regime of the southern Ryukyu arc is different from the central and northern Ryukyu arc.

We used 2D finite element method (FEM) to investigate the crustal strength of the Izu collision zone. Our model primarily based on the crustal structure given by Aoike et al. (2001). We set up standard rock layer properties for simulation. During simulation, cohesion was varied systematically. Simulation results show that failure zone decreases with increasing the convergent displacement in both cases i.e. before and after increasing cohesion. We used Mohr-Coulomb failure criterion and Mohr's circle to understand the reason. It is reasonable that rock layers do not break but flow below more than 30km depth. Further we compared simulated results with the focal mechanism data. Our models show only normal faulting which is not consistent with focal mechanism data. Thus our simulation models probably reveal the crustal strength of Izu collision zone.

Paleostress estimation is essential in understanding the tectonics of the Southern Ryukyu arc. We have used Multiple inverse method (Yamaji, 2000) and Ginkgo method (Yamaji, 2003) to detect paleostress from the fault-slip data collected in the Southern Ryukyu arc where 23 groups of stress (stresses A-W) are devided. We suggest that stress transition in the Southern Ryukyu arc is more complex than those from the previous studies. As the slickenside of faults in the Ryukyu Group can be measured, detailed stress transition is deduced. Stress field of the after Ryukyu Group in the Southern Ryukyu arc is divided into three zones; Miyako island and Yaeyama islands excluding for Hateruma island (MY zone), Hateruma island (HA zone) and Yonaguni island (YO zone). We have tried to re-create a complex stress transition after the deposition of the Ryukyu Group by the finite element method imposed on the convergent rate of the Philippine Sea Plate (present, 1 Ma ago and 2 Ma ago). We have re-created stress field in the Southern Ryukyu arc by comparison between the stress pattern detected by the fault-slip analysis and that simulated by the finite element method. From the result of the finite element method, we have proposed that the stress field is caused by the arc-parallel stretching which is responsible to the increasing curvature of the whole arc accompanying the still active back arc basin (Okinawa Trough) proposed by Fabbri and Fournier (1999) and Fabbri (2000). Before 1 Ma when the distance between the Yonaguni island and the collision zone at Taiwan is the nearest by the westward migration of the Philippine Sea Plate, strike-slip fault regime of after Ryukyu Group detected in the Yonaguni island is recognized. This is caused by the effect of collision at Taiwan. Progressive rifting of the Okinawa Trough before 1 Ma gave rise the stress transition of the after Ryukyu Group from NW-SE extension to NE-SW extension in the MY zone. Strike-slip fault regime detected in the Hat

The purpose of this study is to estimate the effect of surface slope-angle (α), decollement slope-angle (β) and boundary condition to the stress field within accretionary wedge by using finite element method (FEM). We impose four types of boundary conditions along the base of four types of models. These boundary conditions are expressed by liner, quadratic (convex and concave) and cubic function. Four types of models have different decollement slope and surface slope. Comparing with two models (models B and C) which have different decollement slopeangle (β), the decollement slope-angle (β) affect weakly to stress field. On the other hand, comparing with the other two models (models A and C) which have different surface slopeangle (α), the surface slope-angle (α) affects fairly to stress field. Failure area propagates wider to the tip of model as the surface slope-angle (α) becomes gentler. However the more effective factor to decide the propagation of failure area is the difference of boundary conditions. Each boundary condition has different tendency of propagation of failure area. The most widely spread failure area is calculated under the boundary condition 3 and followed in the order of boundary condition 2, 1 and 4.

The study area consists of three tectonic zones from south to north, they are the Siwalik, Lesser Himalayan Sequence and Higher Himalaya Crystalline thrust sheet. The Lesser Himalayan Sequence is composed of chlorite, biotite and garnet grade metasediments and augen gneiss. The Higher Himalayan Crystalline thrust sheet is composed of kyanite to sillimanite grade paragneiss, orthogneiss and quartzite. The area is affected mainly by two deformational episodes (i) Syn-MCT metamorphic ductile deformation and (ii) Post-MCT metamorphic deformation. Syn-MCT metamorphic ductile deformation is characterized by (a) development of bedding-parallel foliation and syn-metamorphic stretching lineations trending NNE- SSW and (b) development of S-C structure. Three sub phases of the post-MCT metamorphic deformation are observed in the study area viz. (a) development of folds, (b) new generation of foliation and extensional shearing features and (c) small-scale brittle fault, shear bands and cross-cut veins. The Lesser Himalayan Sequence has been metamorphosed to greenschist-amphibolite facies whereas the Higher Himalayan Crystalline thrust sheet has been metamorphosed to amphibolite to granulite facies. At least two metamorphic events could be recorded in the Lesser Himalayan Sequence. During Pre Himalayan metamorphic phase, the Lesser Himalayan Sequence might be metamorphosed up to anchizone grade prograde metamorphism(?). But Eo-Himalayan metamorphic phase could not affect the Lesser Himalayan Sequence. The Neo-Himalayan (Syn-MCT) metamorphic phase has been recorded on S-C fabric and is revealed by inverted metamorphic zonation in the Lesser Himalayan Sequence. The final phase in the Lesser Himalayan Sequence was retrograde phase shown by chloritization of garnet and biotite. At least three metamorphic events are recognized in the Higher Himalayan Crystalline thrust sheet of the study area. In the Eo-Himalayan (pre-MCT) metamorphic event kyanite grade prograde metamorphism has bee

We examined the nature of stress around the Himalaya via numerical simulation using the 2 dimensional plane strain finite element models with elastic rheology. Mohr-Coulomb failure criterion also adopted to analyze the relationship between stress distribution and its influence on forming faults around the Himalaya. From this point of view, we presented three finite element elastic models and considered convergence displacement that is subjected along the SW-NE horizontal direction. Results point out that the convergence displacement boundary conditions and elastic properties of rock control the distribution, orientation, magnitudes and intensity of stress during the experiments. Some interesting features of our models are: (1) principle stresses are mainly compressive; (2)σ_1 directs vertically in deeper part and horizontally in the upper part of all layer; (3)σ_2 exhibits horizontal direction in the deeper part and vertical in the upper part; (4) magnitudes of both stresses are relatively high in the deeper part compared with the shallower part; (5)some tensional stresses are displayed in the upper part of Higher Himalayan region; (6)most of the elements are failed in layer 2 (Sub-Himalaya) and in the upper part of layers, 1 (Pre-Cambrian Basement), 4 (Higher Himalaya) and 5 (Tethys Himalaya). These features allow us to infer that the nature and direction of compressive and tensional principle stresses are responsible for forming thrust and normal faults in these layers, respectively and they are intensely concentrated along Sub-Himalaya and upper part of other layers. The results from our numerical experiments are in agreement with the seismicity and focal mechanism solutions of earthquakes in the study area.

The study area consists of three tectonic zones from south to north, they are the Siwalik, Lesser Himalayan Sequence and Higher Himalaya Crystalline thrust sheet. The Lesser Himalayan Sequence is composed of chlorite, biotite and garnet grade metasediments and augen gneiss. The Higher Himalayan Crystalline thrust sheet is composed of kyanite to sillimanite grade paragneiss, orthogneiss and quartzite. The area is affected mainly by two deformational episodes (i) Syn-MCT metamorphic ductile deformation and (ii) Post-MCT metamorphic deformation. Syn-MCT metamorphic ductile deformation is characterized by (a) development of bedding-parallel foliation and syn-metamorphic stretching lineations trending NNE- SSW and (b) development of S-C structure. Three sub phases of the post-MCT metamorphic deformation are observed in the study area viz. (a) development of folds, (b) new generation of foliation and extensional shearing features and (c) small-scale brittle fault, shear bands and cross-cut veins. The Lesser Himalayan Sequence has been metamorphosed to greenschist-amphibolite facies whereas the Higher Himalayan Crystalline thrust sheet has been metamorphosed to amphibolite to granulite facies. At least two metamorphic events could be recorded in the Lesser Himalayan Sequence. During Pre Himalayan metamorphic phase, the Lesser Himalayan Sequence might be metamorphosed up to anchizone grade prograde metamorphism(?). But Eo-Himalayan metamorphic phase could not affect the Lesser Himalayan Sequence. The Neo-Himalayan (Syn-MCT) metamorphic phase has been recorded on S-C fabric and is revealed by inverted metamorphic zonation in the Lesser Himalayan Sequence. The final phase in the Lesser Himalayan Sequence was retrograde phase shown by chloritization of garnet and biotite. At least three metamorphic events are recognized in the Higher Himalayan Crystalline thrust sheet of the study area. In the Eo-Himalayan (pre-MCT) metamorphic event kyanite grade prograde metamorphism has bee

There are a number of methods to analyse geological structures. Finite element method is one of them. Numerical modeling based on finite element analysis is an effective tool for studying the elastic behaviors of earth's crust due to tectonic movement. This study describes how to use an advanced numerical modeling technique, the finite element method, to compute rock deformation and to predict stress and fault development as a function of material properties, cohesion and friction angle. Stress distribution and fault development of 2 dimensional plane strain FEM models of four Himalayan cross sections are described. SW (south west) to NE (north east) horizontal shortening up to maximum 375 m (equivalent displacement at 7.5 cm/yr of 5000 yr) is applied at the southwestern end of the cross-sections. Proposed models show that the direction of maximum principal stresses (σ_1) are horizontal along the shallower part of all the models. Variation of the velocity boundary condition indicates the changes of direction of principal stresses along the deeper part while that along the shallower part remain unchanged. According to the Mohr - Coulomb criterion, failure is observed along the shallower part of Siwalik, Tethys and Granitic layer, and a very few near the surface of MBT, MCT and STDS. Failure has not occurred in the deeper part of Higher Himalaya and Lesser Himalaya, inspite of changing physical parameters of rock formations, because of the hydrostatic condition that is observed along the models. Previous studies on focal mechanism solutions of earthquakes in the Himalayan region provide the existence of thrust faults along its EW stretching with one plane dipping gently north beneath the Himalaya. Simulation shows the same distribution of thrust faults along the upper part of the models as shown by the focal mechanism solutions.

Meso-scale faults are investigated for reconstruction of paleostress recorded in the Shimajiri Group during the upper Miocene to lower Pleistocene exposed around southern region of Okinawa-jima (island). Stress field was estimated by means of the stress inversion method. Every fault slip data from the three formations, namely Tomigusuku, Yonabaru and Shinzato Formation, were calculated. The datasets were divided into subgroups by means of histogram which drew angular misfits between observed and predicted slip directions. Newly calculated stresses are classified by the axial directions and stress ratios. As a result, four stress states were obtained for the Shimajiri Group. Stress state A is NW-SE extension, B is NE-SW extension, C is E-W extension and D is strike-slip type. The order of stress state is C and D, B, A from older to newer. These stress states can be interpreted as a result of subduction of the Philippine Sea plate beneath the Eurasia plate and the crustal extension in the Okinawa Trough.

The least square strain technique that was developed by Hayashi (1994) and is based on the least square method is explained in detail. The technique uses the strain ellipses on three non-parallel section planes of oriented rock samples to construct strain ellipsoid. Three series of model are prepared to examine what kind of factor controls the precision of strain where the axial ratio of initial strain marker ellipsoids is assumed to scatter as normal distribution. Three series of model are produced to test three factors; (1) The sample size of markers. (2)The mean of axial ratio of makers and (3) The standard deviation of axial ratio of markers. Results of the simulation are; (1) When the sample size of the initial strain marker ellipsoids is large, higher the precision of strain analysis is. (2) When the mean of axial ratio of the initial strain marker ellipsoids is small, higher the precision of strain analysis is. (3) Although the standard deviation of axial ratio of the initial strain marker ellipsoids varies, the precision of strain analysis does not change.

Three paleostress transitions are measured in Miyako-jima Island. NW-SE and NE-SW extensions are recorded in the Simajiri Group. Strike-slip type stress state where σ_1 directs to N-S and σ_3 directs to E-W, is recorded in the Ryukyu Group. Transition of paleostress is considered as follows. Before the sedimentation of the Shimajiri Group, NW-SE extension occurred in the Miyako-jima because the Miyako-jima was extended toward the Ryukyu Trench by the trench retreat during the subduction of the Philippine Sea plate. During the sedimentation of the Shimajiri Group, the arc-parallel stretching occurred in the Miyako-jima because the Okinawa Trough started to push the Miyako-jima toward the Ryukyu Trench. Then NE-SW extension occurred in the area. During the sedimentation of the Ryukyu Group, strike-slip type stress state occurred by the sinistral strike-slip movement along the central rift in the Okinawa Trough.

The least square strain technique that was developed by Hayashi (1994) and is based on the least square method is explained in detail. The technique uses the strain ellipses on three non-parallel section planes of oriented rock samples to construct strain ellipsoid. Three series of model are prepared to examine what kind of factor controls the precision of strain where the axial ratio of initial strain marker ellipsoids is assumed to scatter as normal distribution. Three series of model are produced to test three factors; (1) The sample size of markers. (2)The mean of axial ratio of makers and (3) The standard deviation of axial ratio of markers. Results of the simulation are; (1) When the sample size of the initial strain marker ellipsoids is large, higher the precision of strain analysis is. (2) When the mean of axial ratio of the initial strain marker ellipsoids is small, higher the precision of strain analysis is. (3) Although the standard deviation of axial ratio of the initial strain marker ellipsoids varies, the precision of strain analysis does not change.

The Teniya area in Okinawa-jima situated just north to the Kayo area where Hayashi (1988,1999,1992) accomplished the strain analysis. The Kayo Formation around Teniya-zaki strikes generally northeast to southwest and dips to the west. The result of the strain analysis of the area is similar to that performed by Hayashi (1989); the bedding plane has the tendency parallel to the XY plane of the strain ellipsoid. The intensity of strain ε_s takes weaker value than that expected from the folding structure developed in the area. The reason is considered to be the competency contrast between quartz grains of strain marker and matrix. As the values of Ι_ and χ^2 obtained from R_f / φ diagram are larger than the critical value of them, we conclude that there was no initial fabrics of quartz grains before deformation.

Basic concept of deformation and strain is described. Strain ellipse and reciprocal strain ellipse are then introduced as the interface between deformation and strain. Mohr's circle is explained as the magnificent tool to obtain axial ratio and direction of axis of strain using graph instead of calculation. Practical techniques in field are described to measure axial ratio and direction of axis of strain. Practical techniques are the methods that use the strain grid, glass shard and fossils. The famous R_f／φ technique is explained that the method is invented from applying the theory of superposition of homogeneous strain.

Contoured diagram of crenulation cleavage shows a small circle, axis of which indicates N45°E, E10°. Contoured diagram of minor fold shows also the same small circle as that of crenulation cleavage. Therefore, crenulation cleavage and minor fold were firstly formed in the area and then a flexural slip fold deformed the crenulation cleavage and minor fold. The axis of the flexural slip fold is the same as the axis of the small circle, that is, N45°E, E10°. The important point obtained by the 3D strain analysis is that the poles of XY plane of strain ellipsoid of 13 gneiss pieces sampled around the area show the same small circle, axis of which indicates N45°E, E10°. The reason why their small circles coincide with together is still an open problem for the future.

Rock bodies are assumed to be composed of a two elements Maxwell and a generalized Maxwell model. The finite element formulation of these two viscoelastic materials is that viscous term within constitutive equation of the materials is regarded as initial strain and the unknown displacements are calculated for suitable time interval by the stiffness equation which is derived from the variational principle of elasticity. The resultant computer program of the two elements Maxwell model is adopted to a simple problem of stress response in order to be tested. Numerical and analytical solutions fit well together. It is interesting that tensile stress is observed parallel to contractive strain in this test.

The paper describes a finite element formulation of the incompressible viscous fluid without inertia term in detail that is based on a variational principle of viscous fluid and is formulated by the direct method of variational calculus. Although the variational principles of two and three dimensional viscous flows are explained, the FEM formulation is performed with regard to only the two dimensional flow. One of the examples of the calculation using the method has already been published and given a fruitful result (Hayashi, 1975).

The Khaptar Gneiss Klippe, which occupies the northern part of the present region, constructs a successive western fringe of the Karnali Klippe. The area between this Klippe and the Dandeldhura Crystallines, which distribute in southern area of the region, is occupied by green mica schist (Doti Schist). The Dandeldhura Crystalline Massif are considered as eastern continuation of the Almora Nappe in Kumaun. The Doti Schist is doubtfully considered to be a member of the MCT zone in the region. The boundary between the Khaptar Gneiss and the underlying Doti Schist seems to be concordant.

The finite element method may be defined as the modern "Method of Weighted Residuals" (MWR). This paper describes how to solve the differential equations which are essential in order to explain quantiatively a number of valuable geological and geodynamic problems. The methods to solve linear differential equation, non-linear equation, non-linear non-steady equation, Laplace equation and incompreeible New tonian flow problem are explained by means of the Galerkin finite element method.

The paper describes a finite element formulation of the incompressible viscous fluid without inertia term in detail that is based on a variational principle of viscous fluid and is formulated by the direct method of variational calculus. Although the variational principles of two and three dimensional viscous flows are explained, the FEM formulation is performed with regard to only the two dimensional flow. One of the examples of the calculation using the method has already been published and given a fruitful result (Hayashi, 1975).