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Fundamentals of the finite element method
Published in Paulo B. Lourenço, Angelo Gaetani, Finite Element Analysis for Building Assessment, 2022
Paulo B. Lourenço, Angelo Gaetani
As it is possible to observe in Table 1.1, the generalized strain κ associated with bending involves the second derivative of the field variable w. Accordingly, C1 continuity for the deflection is required, i.e. the w field and at least its first derivative must be continuous. The shape functions for the deflection must fulfil these requirements and interpolate the displacements within an element starting from its nodal displacements and rotations (first derivative of deflection). In this regard, it is possible to implement the Hermite interpolation (named after Charles Hermite, 1822–1901), which is a method of interpolating data points and derivatives with a polynomial function. In the case only the first derivative is considered (i.e. the slope), the Hermite polynomial is of the third order, resulting in a smooth continuous spline whose first derivative is still smooth (being a polynomial of the second order).
Finite Element Analysis of Beams
Published in J. N. Reddy, Theories and Analyses of Beams and Axisymmetric Circular Plates, 2022
When the weak form (or variational statement) involves second-order derivatives and the primary variable list contains the function and its derivative, as in the case of the Euler–Bernoulli beam theory, we must interpolate the field variable to have the function values and its derivatives as the nodal degrees of freedom. The resulting interpolation functions are known as the Hermite interpolation functions. The lowest order Hermite interpolation functions are necessarily cubic because a cubic polynomial has four parameters to replace four nodal values – two values of the function and two values of its derivative (i.e., the function and its derivative at each of the two nodes).
Analysis of Rectangular Plates
Published in J.N. Reddy, A. Miravete, Practical Analysis of COMPOSITE LAMINATES, 2018
There are two basic types of interpolations that are used in finite element analyses. The Lagrange interpolation is one in which only the function is interpolated, whereas the Hermite interpolation is one in which the function and its derivatives are interpolated. The finite elements developed using the Lagrange type interpolation are called C0 elements, and finite elements developed using Hermite type interpolation are called Cm elements, where m > 0 is the order of the derivatives included in the interpolation.
A real 3D scene rendering optimization method based on region of interest and viewing frustum prediction in virtual reality
Published in International Journal of Digital Earth, 2022
Pei Dang, Jun Zhu, Jianlin Wu, Weilian Li, Jigang You, Lin Fu, Yiqun Shi, Yuhang Gong
The problem of predicting the VR viewing frustum can be simplified to predict the state of the next time according to the past state through extrapolation. Extrapolation methods mainly include Lagrange interpolation, the Hermite interpolation method, and a Kalman filter. The Lagrange method is a polynomial-based method, which has less computation and is suitable for smooth motion trajectory prediction. However, when the interpolation times are high, the Runge phenomenon will appear, resulting in a large deviation of the interpolation results. Hermite interpolation is an optimization of the Lagrange interpolation method, but it requires the same derivative value at the viewpoint and has great restrictions on use, so it is not suitable for viewing frustum prediction. A Kalman filter is a linear optimal filtering algorithm (Meinhold and Singpurwalla 1983) that uses the mean square error and the criterion of the minimum mean square error to predict the rotation of the VR helmet. It has high accuracy and no requirements for the movement of the viewing frustum (Gómez and Maravall 1994). In VR, to prevent the occurrence of dizziness, teleportation movement mode is mostly used. This movement mode is discontinuous and cannot realize position prediction through extrapolation. Therefore, it is more reasonable to extrapolate and predict the rotation of the VR helmet using a Kalman filter (Zhang and Zhang 2010).
Numerical analysis of density flows in adiabatic two-phase fluids through characteristic finite element method
Published in International Journal of Computational Fluid Dynamics, 2019
Mutsuto Kawahara, Kohei Fukuyama
For interpolation, the Hermite interpolation function based on the third-order polynomials is applied. In this section, the equations are not expressed through the indicial notation. The coordinate is denoted by at node i of an element. Figure 2 shows the Hermite interpolation function of an element. The function values, the values of the first derivative at the nodes of the element, and the function value at the centre of gravity are used as the independent degrees of freedom. Therefore, a total of 10 degrees of freedom are employed for one element. The Hermite interpolation function , , , and are expressed as follows: where represent the nodal coordinates for each triangular element and , , and are the area coordinates, which are the functions of the coordinate . Permutation is used for .
The Study of the Onset of Flow Boiling in Minichannels: Time-Dependent Heat Transfer Results
Published in Heat Transfer Engineering, 2021
Beata Maciejewska, Magdalena Piasecka, Artur Piasecki
The paper focused on flow boiling heat transfer in a vertical minichannel of 1.7 mm depth on the basis of time-dependent experiments. One wall of the minichannel was resistive heated. Thermocouples measured the outer heater surface temperature at 18 points and fluid temperature at the inlet and outlet of the test section. The onset of flow boiling was induced by lowering the pressure in the circulating system (Experiment No. 1) or by increasing the heat flux supplied to the heater (Experiment No. 2). Data acquisition stations monitored all main experimental parameters. The measurements were performed at 1/25 s intervals. The main goal of the work was to determine the heat transfer coefficient by means of the FEM with the spacetime basis functions based on the Trefftz functions and the Hermite interpolation. The approximate foil temperature, being a linear combination of the Trefftz functions, fulfills the heat equation exactly, while it satisfies the initial and boundary conditions only approximately. The use of the Hermite interpolation allows to determine not only the values of an unknown function at the grid nodes but also the values of its derivatives at these nodes. Identification of unknown function derivatives without differentiation of this function is useful for solving of differential equations by the approximate methods. Dynamic changes in thermal and flow experimental parameters accompanying the onset of flow boiling in a minichannel, including: pressure, mass flow rate and temperature of the heater, were shown. The main results were illustrated as local heat transfer coefficients versus time or the distance from the minichannel inlet. It was observed that heat transfer coefficient values increased at the boiling incipience region for both experiments. The relationships known as boiling curves, plotted for selected points in the channel based on the data from Experiment No. 2, were presented. The curves had a typical course in which the boiling incipience was accompanied by “nucleation hysteresis”. The error analysis of the results was provided. In particular, the mean relative errors of the heat transfer coefficient were determined and discussed. The maximum of these errors did not exceed 11%. To validate our results two correlations known from the literature were selected for calculations. The convergence of the results, especially in the middle part of the minichannel was found.