Finite Element Formulation

the various terms of the principle of virtual work have been compiled, containing different temporal and spatial derivatives of the mechanical displacements and rotations as well as of the electric potential of the adaptive beam. In the finite element approach, these continuous functions have to be approximated by discrete values at certain nodal points with adequate local interpolations in between. The degrees of freedom at such nodal points associated with a beam finite element may be summarized in the element vector i/j(t). When elements with two nodes are chosen, the degrees of freedom at both element ends are contained  

For a problem of structural mechanics, the geometric boundary conditions are essential and thus have to be fulfilled to obtain an admissible displacement state. In the process of discretization, this has to be taken into account for the continuity requirements to be warranted by the interpolation functions at the element boundaries. Thus, the beam displacements u x,t), v x,t), w x,t) 

Analogously, such hnear Lagrange poljmomials will be utihzed for the approximation of the electric potential distributions. In the case of the beam twist, consideration of the warping torsion is associated with the twist rate, so continuity is required and can be achieved by the use of cubic Hermite polynomials  

By abandoning the warping effect, the beam torsion problem may also be treated with linear Lagrange polynomials. For the interpolation functions of Eqs. (9.15) and (9.16), the element coordinate Xi is introduced with its origin at the center of the element and the element length /j. Thus, the continuous blade coordinate x can be expressed with the aid of the distance L to the element coordinate origin  

The discretization of the mechanical and electric degrees of freedom as well as of mechanical strains and electric field strengths may be 

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Bell, B.C. and Surana, K. S, 1994. p-version least squares finite element formulations for two-dimensional, incompressible, non-Newtonian isothermal and non-isothcmial fluid flow. hit. J. Numer. Methods Fluids 18, 127-162. 

Hughes, T. J. R., Franca, L. P. and Balestra, M., 1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition - a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolations. Cornput. Methods Appl. Meek Eng. 59, 85-99. 

Finite element formulations for linear stress analysis problems are often derived by direct reasoning approaches. Fluid flow and other materials processing problems, however, are often viewed more easily in terms of their governing differential equations, and this is the... 

The Infiated system (10) Is nonsingular even when the Jacobian of the finite element formulation (8) becomes singular (2fi.) at simple turning points. 

Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains (with A.G. Salinger and J.J. Derby). Int. J. Num. Meth. Fluids 18,1185-1209 (1994). 

In the mathematical literature, the Galerkin method is also known as Galerkin-Bubnov, while the case Wj / Petrov-Galerkin [30,68] and is used in special finite element formulations, such as those where the heat transfer is governed by convective effects. The application of Galerkin s method in the finite element method will be covered in detail in Chapter 9 of this textbook. 

Numerical Implementation of a One-Dimenional Finite Element Formulation... 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

In this section, we will proceed to develop a finite element formulation for the two-dimensional Poisson s equation using a linear displacement, constant strain triangle. Poisson s equation has many applications in polymer processing, such as injection and compression mold filling, die flow, potential problems, heat transfer, etc. The general form of Poisson s equation in two-dimensions is... 

At this point, we can proceed to the finite element formulation of the above governing equations. For this, we will use the isoparametric element presented in the last sections. 

When modeling a system, we try to reduce the problem to a two-, and if possible, to a one-dimensional model. However, often it is not possible to reduce the dimensionality of a problem, forcing us to solve a full three-dimensional model. In principle, solving a problem in 3D using a finite element formulation work the same way as in 1D or 2D. However, set-up effort, and therefore engineering time, as well as computational costs go up drastically when a problem is solved using a full three dimensional model. Most developments as described for ID and 2D finite element formulations are also valid for 3D. In this section, we will present several finite elements and formulations. 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

Based on the control volume approach and using the three-dimensional finite element formulations for heat conduction with convection and momentum balance for non-Newtonian fluids presented earlier, Turng and Kim [10] and [17] developed a three-dimensional mold filling simulation using 4-noded tetrahedral elements. The nodal control volumes are defined by surfaces that connect element centroids and sides as schematically depicted in Fig. 9.33. 

In order to solve viscoelastic problems, we must select the most convenient model for the stress and then proceed to develop the finite element formulation. Doue to the excess in non-linearity and coupling of the viscoelastic momentum equations, three distinct Galerkin formulations are used for the governing equations, i.e., we use different shape functions for the viscoelastic stress, the velocity and the pressure... 

Using these definitions, the Galerkin finite element formulation for the stress equations will be... 

Sun, D.N., Gu, W.Y., Guo, X.E., Lai, W.M., and Mow, V.C. (1999) A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues. International Journal for Numerical Methods in Engineering 45, 1375-1402... 

Levenston, M.E., Frank, E.H., and Grodzinsky, A.J. (1998) Variationally derived 3-field finite element formulations for quasistatic poroelastic analysis of hydrated biological tissues. Comput. Methods Appl. Mech. Eng. 156, 231-246... 

The viscoelastic stress-strain equation, Equation (4) can be expressed in finite element formulation which relates the stress tensor a.. at time index n and cell centre (ij) to the corresponding strain tensor arising from the movement of the adjoining cell corners. Using backward differences for the time step, at time index n. 

BOUNDARY FINITE ELEMENT FORMULATION FOR CONTINUA WITH EQUIVALENT CROSS-SECTIONAL PROPERTIES... 

Application of the same decomposition to the static stiffness matrix of the finite element formulation... 

In order to obtain the boundary finite element formulation, on the one hand the force-displacement relation of the discretized element layer between nodal forces P and nodal displacements u is considered, which can be written in the decomposed form ... 

Everett, M. E., 1999, Finite element formulation of electromagnetic induction with coupled potentials Three-dimensional electromagnetics, Published by the Society of Exploration Geophysics, Tulsa, OK, 444-450. 

This paper describes a finite element formulation designed to simulate polymer melt flows in which both conductive and convective heat transfer may be important, and illustrates the numerical model by means of computer experiments using Newtonian extruder drag flow and entry flow as trial problems. Fluid incompressibility is enforced by a penalty treatment of the element pressures, and the thermal convective transport is modeled by conventional Galerkin and optimal upwind treatments. 

As described above, the temperature field is computed using the finite element formulation of the heat conduction equation, with the viscous heat generation being computed from the stress and velocity fields obtained during the first iteration of the problem. The temperature contours, normalized on the maximum centerline temperature Tc = v V2/3K expected for capillary Poiseuille... 

A final method, the so-called finite-element formulation, requiring a variational formulation or the Galerkin method ofweighted residuals, is beyond the scope of the text (see, for example, Arpaci and Larsen 1984, ch. 7). 

The material presented earlier was confined to steady-state flows over simply shaped bodies such as flat plates, with and without pressure gradients in the streamwise direction, or stagnation regions on blunt bodies. The simplicity of these flow configurations allows reduction of the problems to the solution of steady-state ordinary differential equations. The evaluation of convective heat transfer to more complex three-dimensional configurations, characteristic of real aerodynamic vehicles, involves the solution of partial differential equations. Even when the latter are confined to steady-state problems, they require extensive use of computers in the solution of finite difference or finite element formulations Nonsteady flows further complicate the problems by introducing another dimension, namely, time. 

W. A. Fiveland and J. P. Jessee, A Finite Element Formulation of the Discrete-Ordinate Method For Multidimensional Geometries, in Radiative Heat Transfer Current Research, ASME HTD no. 244, New York, 1993. 

The finite element formulation is obtained using the Standard Galerkin method. 

The model is solved numerically by means of the finite element code ASTER. The equations are discretized within a finite element formulation. The time discretization is implicit and the coupling is solved by means of a global inversion of the system, as explained in Chavant (2(X)1). 

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