Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of W1,∞(L 2 ) is proved. An L∞(H1 )-error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the heat equation but also for its first derivatives (both spatial and temporal). Even the proof presented in this note is in some sense standard but the stated W1,∞(L 2 )- error estimate seems not to be present in the existing literature of the Crank-Nicolson finite element schemes for parabolic equations.
There are many problems of groundwater flow in a disrupted rock massifs that should be modelled using numerical models. It can be done via "standard approaches'' such as increase of the permeability of the porous medium to account the fracture system (or double-porosity models), or discrete stochastic fracture network models. Both of these approaches appear to have their constraints and limitations, which make them unsuitable for the large-scale long-time hydrogeological calculations. In the article, a new approach to the modelling of groudwater flow in fractured porous medium, which combines the above-mentioned models, is described. This article presents the mathematical formulation and demonstration of numerical results obtained by this new approach. The approach considers three substantial types of objects within a structure of modelled massif important for the groudwater flow - small stochastic fractures, large deterministic fractures, and lines of intersection of the large fractures. The systems of stochastic fractures are represented by blocks of porous medium with suitably set hydraulic conductivity. The large fractures are represented as polygons placed in 3D space and their intersections are represented by lines. Thus flow in 3D porous medium, flow in 2D and 1D fracture systems, and communication among these three systems are modelled together.
This case study presents the verification of two surface subsidence prediction models for longwall mining at depths greater than 400 m. The surface subsidence points were surveyed and compared for both models. The first model uses empirical calculations to predict the surface subsidence. This method is reliable for predicting surface subsidence at shallower depths. At present, however, coal mining has progressed to great depths. The second model is the 2-dimensional finite element method to predict surface subsidence. In contrast to the first method, this method is based on the regional parameters and uses the rock mass properties to evaluate surface subsidence for multi-seams at any depth. Results show that the finite element method gives a better approximation of the measured surface subsidence than the Knothe method. The maximum surface subsidence, which was determined by the FEM method, was used to adjust the extraction coefficient in the Knothe's method. The predicted value differs from the measured value by 8 %. The slope of the predicted subsidence trough was within the range of 2‒8 % from the surveyed subsidence. This case study proposes a procedure for using both models to successfully predict the surface subsidence.
This paper deals with the basic dynamical model of electromotor whose creation motivated by problems of electromotor vibrations and its subsequent fatigue. The model is chaacrterized by a flexible shaft with mounted padket of sheet metals that are equipped with parallel copper bars connected by end shortcircuit rings. Finite element analysis is used for the disretization of the shaft, while the sheet metal packet of cylindrical shape is modelled as a set of rigid bodies joined using chosen viscoelastic forces and torques. The shortcircuit rings are supposed to be rigid in this basic dynamical model and the pieces of copper bars are substituted by massless springs of calculated properties. Problematic model parameters are identified by means of performed experimental modal analysis. and Obsahuje seznam literatury
This paper is devoted to a new approach to the dynamic response of Soil-Structure System (SSS), the far field of which is meshed by decay or mapped elastodynamic infinite elements, based on scaling modified Bessel shape functions to be calculated. These elements are appropriate for Soil-Structure Interaction problems, solved in time or frequency domain and can be treated as a new form of the recently proposed elastodynamic infinite elements with united shape functions (EIEUSF) infinite elements. Here the time domain form of the equations of motion is demonstrated and used in the numerical example. In the paper only the formulation of 2D horizontal type infinite elements (HIE) is used, but by similar techniques 2D vertical (VIE) and 2D corner (CIE) infinite elements can also be added. Continuity along the artificial boundary (the line between finite and infinite elements) is discussed as well and the application of the proposed elastodynamical infinite elements in the Finite element method is explained in brief. A numerical example shows the computational efficiency and accuracy of the proposed infinite elements, baeed on scaling Bessel shape functions. and Obsahuje seznam literatury
We introduce a new efficient way of computation of partial differential equations using a hybrid method composed from FEM in space and FDM in time domain. The overall computational scheme is explicit in time. The key idea of the suggested way is based on a transformation of standard basis functions into new basis functions. The results of this matrix transformation are e-invariants (effective invariants) with such suitable properties which save the number of arithmetical operations needed for a problem solution. The application of this procedure and its effectiveness for 2D problem was the first time published in \cite{halabala}. Now we describe the generalization of this procedure for 3D problem. In order to present the main principle of our process and its advantage, we first explain the main idea of our approach on a simple 1D example and then the application of the e-invariants on an elastodynamics equation using hexahedral elements in 3D is described. Finally, the efficiency of the suggested method in both cases from the point of the required number of arithmetical operations is analyzed. The result of this analysis confirms computational efficiency the suggested method and the usefulness of e-invariants which save only the essential information needed for the computation. Moreover, the method can be used for various types of elements and equations.
The effect of tonsillectomy on production of Czech vowels /a/ and /i/ is numerically examined. Similar experimental studies are difficult to realise on living subjects. Finite element (FE) models of the acoustic spaces corresponding to the human vocal tract for the Czech vowels /a/ and /i/ with acoustic space around the human head are used in numerical simulations of phonation. The acoustic resonant characteristics of the FE models are studied by modal and transient analyses. The production of vowels is simulated in time-domain using transient analysis of FE models excited at the position of vocal folds by analytically desctibed Liljencrants-Fant‘s (LF) glottal signal model. The results show that tonsillectomy causes frequency shifts of some formant frequencies up to 150 Hz. The frequency shift of formants significantly depends on position and size of the tonsils. and Obsahuje seznam literatury
The boundary conditions and loading ways of geostress field of oil and gas trap are the difficulties in the numerical simulation and geomechanical analysis. Owing to the limited data of geostress, unclear tectonic movement and complex geological structure, the stress field cannot be solved directly. Boundary load inversion is a very important method to analyze the stress field of rock mass. Based on the measured in-situ stress of S4 member in C41 fault block of Liangjialou oilfield, the boundary loads of the geological body stress field are inversely calculated. Meanwhile, the optimal boundary stress obtained by the inverse modeling is used to study the stress field near the fault. This method can overcome the shortcomings of common back analysis, such as boundary load adjustment method and regression method, and improve the calculation accuracy of stress field. The results show that the inversion method is simple, reliable, accurate and fast. The distribution of stress field can well reflect the in homogeneity of the magnitude and direction of the stress field near the fault. Therefore, this method has a certain application value in boundary load inversion, and the initial stress field distribution of faults provides a precondition for local stability.
This paper presents a computational procedure for the design of an observer of a nonlinear system. Outputs can be delayed, however, this delay must be known and constant. The characteristic feature of the design procedure is computation of a solution of a partial differential equation. This equation is solved using the finite element method. Conditions under which existence of a solution is guaranteed are derived. These are formulated by means of theory of partial differential equations in L2-space. Three examples demonstrate viability of this approach and provide a comparison with the solution method based on expansions into Taylor polynomials.
The paper presents the solution to the geodetic boundary value problem by the finite element method in area of Slovak Republic. Generally, we have made two numerical experiments. In the first one, Neumann BC in the form of gravity disturbances generated from EGM-96 is used and the solution is verified by the quasigeoidal heights generated directly from EGM-96. In the second one, Neumann BC is computed from gravity measurements and the solution is compared to the quasigeoidal heights obtained by GPS/leveling method.