Finite Element Applications

This subject has been dealt with in much greater detail in Chapter 3 using dynamic finite element technique. The analysis given under Sections 2.3.1 and 2.3.2 are taken as basis for the finite element approach. Various 

Double walled-double dome resting on piles 

5 m inside cylinder Wall thickness = 0.9 m 12 m buttresses Total height = 59.5 m thick above ground level Base slab = 5 m, base slab = 55.2 m with keys, 1.5 m keys depth variable 

Double wall type Double dome type Total height of walls = 545 m Spaces = 3.34 m along the walls Base slab = 50.90 m 

Country Name Type Containment type Design pressure 

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Solutions for MDOF systems arc usually obtained through the use of finite element procedures. Due to nonlinearities associated with plasticity and possibly large displacements, the direct time integration method should be used. Various direct integration methods for time integration are employed but, the Newmark Method is perhaps he most common. Other methods, such as the Houboult Method, Wilson-T Method and the Central Difference Method are commonly used in finite element applications. Refer to Bathe 1995 for further details. 

Yoganadan N, Kumaresan S, Voo L, Pintar FA (1996) Finite element application in human cervical spine modeling. Spine 21 1824-1834... 

Kavaragh, K. Clough, R. 1972. Finite element application in characterization of elastic solid. International Journal of solids and structures 7 11 23. 

Holzapfel, G. A. (1996). "On large strain viscoelasticity Continuum formulation and finite element applications to elastomeric structures." International Journal for Numerical Methods in Engineering 39(22) 3903-3926. 

In brief, the basis of the finite element method is the representation of a body or a structure by an assemblage of subdivisions called finite element. Simple fimctions are chosen approximate the distribution or variation of the actual displacements over each finite element. The unknown magnitudes or amplitudes of the displacement fimctions are the displacements at the nodal points. Hence, the final solution will yield the approximate displacements at the discrete locations in the body, the nodal points. A displacement function can be expressed in various forms, such as polynomials and trigonometric fimctions. Since polynomials offer ease in mathematical manipulations, they have been employed in finite element applications. 

In certain types of finite element computations the application of isoparametric mapping may require transformation of second-order as well as the first-order derivatives. Isoparametric transformation of second (or higher)-order derivatives is not straightforward and requires lengthy algebraic manipulations. Details of a convenient procedure for the isoparametric transformation of second-order derivatives are given by Petera et a . (1993). 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

Despite the simplicity of the outlined weighted residual method, its application to the solution of practical problems is not straightforward. The main difficulty arises from the lack of any systematic procedure that can be used to select appropriate basis and weight functions in a problem. The combination of finite element approximation procedures with weighted residual methods resolves this problem. This is explained briefly in the forthcoming section. 

Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

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. 

An example describing the application of this algorithm to the finite element modelling of free surface flow of a Maxwell fluid is given in Chapter 5. 

Petera, J. and Nassehi, V., 1996. Finite element modelling of free surface viscoelastic flows with particular application to rubber mixing. Int. J. Numer. Methods Fluids 23, 1117-1132. 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

After application of the 6 time-stepping method (see Chapter 2, Section 2.5) and following the procedure outlined in Chapter 2, Section 2.4, a functional representing the sum of the squares of the approximation error generated by the finite element discretization of Equation (4.118) is formulated as... 

Figure 5.2 shows the finite element mesh corresponding to the configuration shown in Figure 5.1. This mesh consists of 225 nine-node bi-quadratic elements and its utihzation in the present model is based on the application of isoparametric mapping, described in Chapter 2.

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

The required working equations are derived by application of the following finite element schemes to the described governing model ... 

It should be emphasized at this point that the basic requirements of compatibility and consistency of finite elements used in the discretization of the domain in a field problem cannot be arbitrarily violated. Therefore, application of the previously described classes of computational grids requires systematic data transfomiation procedures across interfaces involving discontinuity or overlapping. For example, by the use of specially designed mortar elements necessary communication between incompatible sections of a finite element grid can be established (Maday et ah, 1989). 

G. Dziuk. A boundary element method for curvature flow. Application to crystal growth. In J. E. Taylor, ed. Computational Crystal Growers Workshop, AMS Selected Lectures in Mathematics. Providence, Rhode Island American Mathematical Society, 1992, p. 34 A. Schmidt. Computation of three dimensional dendrites with finite elements. J Comput Phys 125 293, 1996. 

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