Finite Element Method Schemes

IDEAS OAYES NASTRAN MARC F. BANG F. ISOPAR 

and Merwin. J.A. Solution of non-linear problems of elastoplasticity by finite element method. J. AIAA, 6,1968. 

Bangash, M.Y.H. The Automated three-dimensional cracking analysis of prestressed concrete vessels, Proc. 6 Int. Conf. Struct. Mech. Reactor Technology. Paper H3/2 Paris 1981 

Bangash, M.Y.H. The structural Integrity of concrete containment vessels under external impact. Proc. 6 Inter. Conf. Struc. Mech. React. Technology Paper J7/6, Paris, 1981. Bangash, M.Y.H. Reactor Pressure Vessel Design and Practice. Prog. Nucl. Energy, 10, 69-124, 1982 

---

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

Hughes, T. J.R. and Brooks, A.N., 1979, A multidimensional upwind scheme with no cross-wind diffusion. In Hughes, I . J. R. (ed.), Finite Element Methods for Convection Dominated Flows, AMD Vol. 34, ASME, New York. 

Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. 

Derivation of the working equations of upwinded schemes for heat transport in a polymeric flow is similar to the previously described weighted residual Petrov-Galerkm finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. 

In the sequel we will show that scheme (12) Is identical with the scheme emerged in variational difference methods (the finite element method). 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

Today, there an established software tool set does exist for the primary task, the calculation of modes and the description of field propagation. Approaches based on the finite element method (FEM) and finite differences (FD) are popular since long and can be applied to complex problems . The wave matching method, Green functions approaches, and many more schemes are used. But, as a matter of fact, the more dominant numerical methods are, the more the user has to scrutinize the results from the physical point of view. Recent mathematical methods, which can control accuracy absolutely - at least if the problem is well posed, help the design engineer with this. ... 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

Continuum wavefunctions can also be generated by solving the partial differential equation (3.3) directly without first transforming it into a set of ordinary differential equations. One possible scheme is the finite elements method (Askar and Rabitz 1984 Jaquet 1987). Another method, which has been applied for the calculation of multi-dimensional scattering wavefunctions, is the 5-matrix version of the Kohn variational principle (Zhang and Miller 1990). 

The governing equations of the model are discretized in space by means of the finite element method [3, 18], and in time through a fully implicit finite difference scheme (backward difference) [18], resulting in the nonlinear equation set of the following form, [4, 7],... 

Due to the constructive peculiarities in the most of widely used high pressure and temperature apparatuses including both lens and belt type ones temperature field in reaction cell has axial symmetry. Thus, one has possibility to analyze temperature conditions in reaction cells of different design by means of finite element method calculation scheme [4], Beneath we discuss the results and application of this kind of calculations of temperature fields for one possible type of lens-shaped high pressure apparatus [1],... 

Peskin [49] used the Galerkin finite-element method to compute current distribution and shape change for electrodeposition into rectangular cavities. A concentration-dependent overpotential expression including both forward and reserve rate terms was used, and a stagnant diffusion layer was assumed. An adaptive finite-element meshing scheme was used to redefine the problem geometry after each time step. 

According to the requirements specified in the ISO standard [1] deviation of the delamination propagation from mid-plane invalidates the test. In that sense, any data analysis of the cross-ply laminates, therefore, will yield invalid results. This point was discussed extensively in [4] where the authors also used the Finite-Element method to supplement their analysis and concluded that the corrected beam theory data reduction scheme seemed to remain applicable and that the non-symmetric cross-ply material yielded apparently valid fracture toughness data, even though these were probably affected by transverse cracking. 

The disadvantage of the EE scheme is its complexity. Typically, the EE scheme requires more coefficients per element than the conventional finite element method. 

The finite element methods are more mathematically involved than the finite difference schemes, and a detailed explanation of these methods is not within the scope of this work. Thus, we shall briefly elaborate on the extension to multi-... 

One method to solve partial differential equations using the numerical schemes developed for solving time dependent ordinary differential methods is the method of lines. In this method, the spatial derivatives at time t are replaced by discrete approximations such as finite differences or finite element methods such as collocation or Galerkin. The reason for this approach is the advanced stage of development of schemes to solve ordinary differential equations. The resulting numerical schemes are frequently similar to those developed directly for partial differential equations. 

The final feature of the dislocation dynamics method that must be introduced so as to give such methods the possibility of examining real boundary value problems in plastic deformation is the treatment of boundary conditions. In particular, if we wish to consider the application of displacement and traction boundary conditions on finite bodies, the fields of the relevant dislocations are no longer the simple infinite body Volterra fields that have been the workhorse of our discussions throughout this book. To confront the situation presented by finite bodies, a useful scheme described in Lubarda et al. (1993) as well as van der Giessen and Needleman (1995) is to use the finite element method to solve for the amendments to the Volterra fields that need to be considered in a finite body. Denote the Volterra fields for an infinite body as In this case the fields of interest are given by... 

For the spatial solution of the nonlinear coupled multi-field problem given in Sect 1.3, the Finite Element Method (FEM) is applied. The equations for the three fields are solved with a Newton-Raphson algorithm, and the time integration is performed with the implicit Euler backwards scheme. 

All the problems encountered during application of the algorithms presented above illustrate the difficulty of solution of the advection problem in atmospheric transport models. A number of techniques have been developed to treat advection accurately, including flux-corrected transport (FCT) algorithms (Boris and Book, 1973), spectral and finite element methods [for reviews, see Oran and Boris (1987), Rood (1987), and Dabdub and Seinfeld (1994). Bott (1989, 1992), Prather (1986), Yamartino (1992), Park and Liggett (1991), and others have developed schemes specifically for atmospheric transport models. 

Mass balance of solid Mass balance of water Mass balance of air Momentum balance for the medium Internal energy balance for the medium The resulting system of Partial Differential Equations is solved numerically. Finite element method is used for the spatial discretization while finite differences are used for the temporal discretization. The discretization in time is linear and the implicit scheme uses two intermediate points, t and t between the initial 1 and final t limes. Finally, since the problems are nonlinear, the Newton-Raphson method has been adopted following an iterative scheme. 

---