# Finite element approach

Finite element methods [20,21] have replaced finite difference methods in many fields, especially in the area of partial differential equations. With the finite element approach, the continuum is divided into a number of finite elements that are assumed to be joined by a discrete number of points along their boundaries. A function is chosen to represent the variation of the quantity over each element in terms of the value of the quantity at the boundary points. Therefore a set of simultaneous equations can be obtained that will produce a large, banded matrix. [c.91]

The three primary advantages of the finite element approach over finite difference methods are [9] [c.92]

The standard technique for improving the accuracy of finite element approximations is to refine the computational grid in order to use a denser mesh consisting of smaller size elements. This also provides a practical method for testing the convergence in the solution of non-linear problems through the comparison of the results obtained on successively refined meshes. In the h-version of the finite element method the element selected for the domain discretization remains unchanged while the number and size of the elements vary with each level of mesh refinement. Alternatively, the accuracy of the finite element discretizations can be enhanced using higher-order elements whilst the basic mesh design is kept constant. For example, after obtaining a solution for a problem on a mesh consisting of bi-Iinear elements another solution is generated via bi-quadratic elements while keeping the number, size and shape of the elements in the mesh unchanged. In this case the number of the nodes and, consequently, the node-to-element ratio in the mesh will increase and a better accuracy will be obtained. This approach is commonly called the p-version of the finite element method. [c.40]

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 [c.64]

In a significant number of polymer processes the influence of fluid elasticity on the flow behaviour is small and hence it is reasonable to use the generalized Newtonian approach to analyse the flow regime. In generalized Newtonian fluids tire extra stress is explicitly expressed in terms of velocity gradients and viscosity and can be eliminated from the equation of motion. This results in the derivation of Navier-Stokes equations with velocity and pressure as tire only prime field unknowns. Solution of Navier-Stokes equations (or Stokes equation for creeping flow) by the finite element schemes is the basis of computer modelling of non-elastic polymer flow regimes. In contrast, in viscoelastic flow models the extra stress can only be given through implicit relationships with the rate of strain, and hence remains as a prime field unknown in the governing equations. In this case therefore, in conjunction with the governing equations of continuity and momentum (generally given as Cauchy s equation of motion) an appropriate constitutive equation must be solved. Numerical solution of viscoelastic constitutive equations has been the subject of a considerable amount of research in the last two decades. This has given rise to a plethora of methods [c.79]

As discussed in the previous chapters, utilization of viscoelastic constitutive equations in the finite element schemes requires a significantly higher computational effort than the generalized Newtonian approach. Therefore an important simplification in the model development is achieved if the elastic effects in a flow system can be ignored. However, almost all types of polymeric fluids exhibit some degree of viscoelastic behaviour during their flow and deformation. Hence the neglect of these effects, without a sound evaluation of the flow regime characteristics, which may not allow such a simplification, can yield inaccurate results. [c.150]

Under certain conditions it may be appropriate to focus on the modelling of a segment of a larger domain in order to obtain detailed results within that section, while maintaining computing economy. To develop such a model, first the entire flow domain is simulated using a relatively coarse finite element mesh. The results generated at the end of this stage are used to define the boundary conditions at the borders of the selected part. This section is then modelled utilizing a refined mesh to obtain detailed predictions about phenomena of interest in the flow field. The procedure can be repeated using a step-by-step approach in which the zooming on a segment witliin a very large domain is achieved through successive reduction of the size of the segments at each step of modelling. [c.156]

Tlie focus of discussions presented so far in this publication has been on the finite element modelling of polymers as liquids. This approach is justified considering that the majority of polymer-fonning operations are associated with temperatures that are above the melting points of these materials. However, solid state processing of polymers is not uncommon, furthermore, after the processing stage most polymeric materials are used as solid products. In particular, fibre- or particulate-reinforced polymers are major new material resources increasingly used by modern industry. Therefore analysis of the mechanical behaviour of solid polymers, which provides quantitative data required for their design and manufacture, is a significant aspect of the modelling of these materials. In this section, a Galerkin finite element scheme based on the continuous penalty method for elasticity analyses of different types of polymer composites is described. To develop this scheme the mathematical similarity between the Stokes flow equations for incompressible fluids and the equations of linear elasticity is utilized. [c.183]

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. [c.100]

There are currently no standards or regulations governing ANG containers. The existing regulations for CNG cylinders are not appropriate, particularly if non-cylindrical designs are being considered. Thus the use of ANG tanks requires a comprehensive design and evaluation program to ensure safety. Applying standard pressure vessel codes, such as ASME VIII, invariably leads to a heavy and unnecessarily bulky container. However, in the absence of specific codes, ANG tanks that are to be used on public roads must be designed as closely as possible to an existing pressure vessel standard. The design must be checked by finite element analysis, and the design validated by carrying out a program of pressure testing on sample tanks to establish burst safety factors and the fatigue life. This approach allows the design engineer to assess the fitness-for-purpose of the tank. However, it may not be possible to obtain full certification since some aspect of the tank design may fall outside the ASME Vlll Code. [c.280]

The division of micromechanics stiffness evaluation efforts into the mechanics of materials approach and the elasticity approach with its many subapproaches is rather arbitrary. Chamis and Sendeckyj [3-5] divide micro mechanics stiffness approaches into many more classes netting analyses, mechanics of materials approaches, self-consistent models, variational techniques using energy-bounding principles, exact solutions, statistical approaches, finite element methods, semiempirical approaches, and microstructure theories. All approaches have the common objective of the prediction of composite materials stiffnesses. All except the first two approaches use some or all of the principles of elasticity theory to varying degrees so are here classed as elasticity approaches. This simplifying and arbitrary division is useful in this book because the objective here is to merely become acquainted with advanced micromechanics theories after the basic concepts have been introduced by use of typical mechanics of materials reasoning. The reader who is interested in micromechanics should supplement this chapter with the excellent critique and extensive bibliography of Chamis and Sendeckyj [3-5]. [c.137]

The division of micromechanics stiffness evaluation efforts into the mechanics of materials approach and the elasticity approach with its many subapproaches is rather arbitrary. Chamis and Sendeckyj [3-5] divide micro mechanics stiffness approaches into many more classes netting analyses, mechanics of materials approaches, self-consistent models, variational techniques using energy-bounding principles, exact solutions, statistical approaches, finite element methods, semiempirical approaches, and microstructure theories. All approaches have the common objective of the prediction of composite materials stiffnesses. All except the first two approaches use some or all of the principles of elasticity theory to varying degrees so are here classed as elasticity approaches. This simplifying and arbitrary division is useful in this book because the objective here is to merely become acquainted with advanced micromechanics theories after the basic concepts have been introduced by use of typical mechanics of materials reasoning. The reader who is interested in micromechanics should supplement this chapter with the excellent critique and extensive bibliography of Chamis and Sendeckyj [3-5]. [c.137]

This paper compares experimental data for aluminium and steel specimens with two methods of solving the forward problem in the thin-skin regime. The first approach is a 3D Finite Element / Boundary Integral Element method (TRIFOU) developed by EDF/RD Division (France). The second approach is specialised for the treatment of surface cracks in the thin-skin regime developed by the University of Surrey (England). In the thin-skin regime, the electromagnetic skin-depth is small compared with the depth of the crack. Such conditions are common in tests on steels and sometimes on aluminium. [c.140]

Finite element modelling of engineering processes can be based on different methodologies. For example, the preferred method in structural analyses is the displacement method which is based on the minimization of a variational statement that represents the state of equilibrium in a structure (Zienkiewicz and Taylor, 1994). Engineering fluid flow processes, on the other hand, cannot be usually expressed in terms of variational principles. Therefore, the mathematical modelling of fluid dynamical problems is mainly dependent on the solution of partial differential equations derived from the laws of conservation of mass, momentum and energy and constitutive equations. Weighted residual methods, such as the Galerkin, least square and collocation techniques provide theoretical basis for the numerical solution of partial differential equations. However, the direct application of these techniques to engineering probletns is usually not practical and they need to be combined with finite element approximation procedures to develop robust practical schemes. Hence the commonly adopted approach in computer modelling of flow processes in polymer engineering operations is the application of weighted residual finite element methods. [c.18]

Flexibility to cope with irregular domain geometry in a straightforward and systematic manner is one of the most important characteristics of the finite element method. Irregular domains that do not include any curved boLindary sections can be accurately discretized using triangular elements. In most engineering processes, however, the elimination of discretization error requires the use of finite elements which themselves have curved sides. It is obvious that randomly shaped curved elements cannot be developed in an ad hoc manner and a general approach that is applicable in all situations must be sought. The required generalization is obtained using a two step procedure as follows [c.34]

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). [c.74]

The streamline upwinding method is usually employed to obtain the discretized form of Equation (3.73). The solution algorithm in the ALE technique is similar to the procedure used for a fixed VOF method. In this technique, however, the solution found at the end of the nth time step, based on mesh number n, is used as the initial condition in a new mesh (i.e. mesh number n + 1). In order to minimize the error introduced by this approximation the difference between the mesh configurations at successive computations should be as small as possible. Therefore the time increment should be small. In general, adaptive or re-raeshing algorithms are employed to construct the required finite element mesh in successive steps of an ALE procedure (Donea, 1992). In some instances it is possible to generate the finite element mesh required in each step of the computation in advance, and store them in a file accessible to the computer program. This can significantly reduce the CPU time required for the simulation (Nassehi and Ghoreishy, 1998). An example in which this approach is used is given in Chapter 5. [c.103]

Simplification achieved by using a constant mesh in the modelling of the flow field in a single-blade mixer is not applicable to twin-blade mixers. Although the model equations in both simulations are identical the solution algorithm for twin-blade mixers cannot be based on the VOF method on a fixed domain and instead the Arbitrary Lagrangian-Eulerian (ALE) approach, described in Chapter 3, Section 5.2, should be used. However, the overall geometry of the plane of the rotors blades cross-section is known and all of the required mesh configurations can be generated in advance and stored in a file to speed up the calculations. Figure 5.4 shows the finite element mesh corresponding to 19 successive time steps from the start of the simulation in a typical twin-blade tangential rotor mixer. The finite element mesh configurations correspond to counter-rotating blades with unequal rotational velocities set to generate an mieven stress field for enhancing dispersive mixing efficiency. Calculation of mesh velocity, required for modification of the free surface equation (see Equation (3.73)) at each time step, is based on the following equations (Ghoreishy, 1997) [c.146]

Finite element solution of engineering problems may be based on a structured or an unstructured mesh. In a structured mesh the form of the elements and local organization of the nodes (i.e, the order of nodal connections) are independent of their position and are defined by a general rule. In an unstructured mesh the connection between neighbouring nodes varies from point to point. Therefore using a structured mesh the nodal connectivity can be implicitly defined and explicit inclusion of the connectivity in the input mesh data is not needed. Obviously this will not be possible in an unstructured mesh and nodal connectivity throughout the computational grid must be specified as part of the input data. It is important, however, to note that stnictured computational grids lack flexibility and hence are not suitable for engineering problems which, in general, involve complex geometries. Discretization of domains with complicated boundaries using structured grids is likely to result in badly distorted elements, thus precluding robust and accurate numerical solutions. Using an unstructured mesh, geometrical complexities can be handled in a more natural manner allowing for local adaptation, variable element concentration and preferential resolution of selected parts of the problem domain. However, because of the inherent complexity of data handling in unstructured mesh generation this approach requires special programs for the organization and recording of nodes, element edges, surfaces, etc. which involve extra memory requirement. In particular, any increase in the number of elements during mesh refinement requires rapidly rising computational efforts. A further drawback for unstructured grids is the difficulty of handling moving boundaries in a purely Lagrangian approach in the simulation of flow problems. [c.192]

Finite-element approaches can be supplemented by the other main methods to get comprehensive models of different aspects of a complex engineering domain. A good example of this approach is the recently established Rolls-Royce University Technology Centre at Cambridge. Here, the major manufacturing processes involved in superalloy engineering are modelled these include welding, forging, heat-treatment, thermal spraying, machining and casting. All these processes need to be optimised for best results and to reduce material wastage. As the Centre s then director, Roger Reed, has expressed it, if the behaviour of materials can be quantified and understood, then processes can be optimised using computer models . The Centre is to all intents and purposes a virtual factory. A recent example of the approach is a paper by Matan el al. (1998), in which the rates of diffusional processes in a superalloy are estimated by simulation, in order to be able to predict what heat-treatment conditions would be needed to achieve an acceptable approach to phase equilibrium at various temperatures. This kind of simulation adds to the databank of such properties as heat-transfer coefficients, friction coefficients, thermal diffusivity, etc., which are assembled by such depositories as the National Physical Laboratory in England. [c.474]

The fundamental differences ia the quantum mechanical character of the two helium isotopes created much interest ia the properties of mixtures. Several reviews are available (70—72). Mixtures of isotopes of a single element usually behave quite ideally, but ia the case of He— He solutions, nonideaUty reaches the point of forming two immiscible Hquid phases. The Hquid-phase diagram for He— He solutions at low pressure is shown as the soHd curves in Figure 4. The solutions undergo the superfluid transition, but the transition temperature is depressed by increases of He concentration. Below a solution critical point at 0.867 K and 67.5 mol % He (71), two immiscible Hquid phases can form. The He-rich phase is superfluid and the He-rich phase remains normal, at least to below 0.003 K. The solubiHty of He in He appears to approach 2ero as the temperature approaches 2ero, but the solubiHty of He in He does not it remains finite (6.4 mol %) as absolute 2ero is approached (70). [c.9]

See pages that mention the term

**Finite element approach**:

**[c.964] [c.27] [c.141] [c.142]**

Mechanics of composite materials (1999) -- [ c.125 , c.145 , c.289 ]

Machanics of composite materials (1998) -- [ c.125 , c.145 , c.289 ]