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NUMERICAL METHODS:

The following numerical methods are generally used for solving engineering problems.- Finite difference method- Finite element method- Control volume or finite volume Method- Boundary volume method

Finite difference:- The Scheme expresses the derivatives of the governing differential equation. In terms of the variables at the

selected grid points using truncated Taylor series expansion.- The computational domain is filled up with number of grids and grid points selected at the centre of the

corresponding grids.

Finite difference scheme has the following disadvantages.

- Specification of boundary conditions for irregular geometries.- Incorporation of complex material properties.

Finite element method:- It is a popular computer aided numerical method based on the discretisation of the domain, structure or

continuum into number of elements and obtaining the solution.Control volume analysis or finite volume method:

- It formulates the discretised equations at each grid point of the domain by integrating the terms of thedifferential equations over the control volume and assuming the variation of the unknown variable betweengrid points. This method is mainly used for fluid flow problems.

Boundary element method:- It is based on the boundary integral equation formulation of the problem. It needs only a boundary

discretisation in contrast to domain methods. This method has not become Popular compared to finitedifference, finite element and control volume formu1ations.

FINITE ELEMENT FORMULATION:

Steps involved for finite element formulation are briefly discussed below:

1. Discretisation:

o It is the process of dividing the domain, structure or continuum into sub regions or Sub divisionscalled finite elements.

o Element having a simple shape with which a domain or body or structure is discretised.o Nodes are defined for each element, and nodes are the locations or discrete points at which the

unknown variables are to be determined. These unknown variables are called field variables as theunknown variable may be displacement, temperature or velocity depending on the type of problemunder consideration.

o Collection of elements is called mesh. Elements are connected at the nodes.

2. Approximation of field variable in an element:

o Polynomial expressions are normally employed to define the variation of the field variable.o It is easy to differentiate and integrate the terms in polynomial expressions.

3. Formulation of element equations:

o Element equations will be derived by minimisation of a function. The function is an integralexpression and minimisation with respect to nodal variables yields the element equations.

o The formulation of element equations can be accomplished by using one of the following methods.

Variation formulation using calculus of variation. Weighted residual methods out of which Galerkins method is extensively used.

o For structural mechanics problems, the potential energy expression written in an integral formprovides the necessary function.

o Minimisation of potential energy expression with respect to nodal displacements will yield elementequations which are known as equations of equilibrium.

4. Principle of minimum potential energy: Among all valid configurations of a system, those that satisfy theequations of equilibrium make the potential energy stationary with respect to small variations of displacement.

5. Assembly of matrices to form global or system equations: The stiffness matrix and load vector (also calledforce vector) for each element are formulated. Assembling these matrices for all elements will give globalstiffness matrix and global load vector to formulate the global or system equations.

6. Solving the global or system equations will provide the solution for field variable i.e., displacementsfor a solid or structural mechanics problem.

7. Secondary quantities like strain and stress can be calculated once the nodal displacements arc known.

8. The description of finite element method can be given based ono Displacement formulation. As seen aboveo Equilibrium method:

In equilibrium method, stress is field variable and displacement is to be derived fromstress.

o Mixed method: In mixed method, part of the domain is solved using displacement formulation and

remaining part with equilibrium method.

Solid Bar under Axial LoadOne-Dimensional Fluid Flow

Applications of FEM:

- As stated earlier, the finite element method was developed originally for the analysis of aircraftstructures. However, the general nature of its theory makes it applicable to a wide variety ofboundary value problems in engineering. A boundary value problem is one in which a solution issought in the domain (or region) of a body subject to the satisfaction of prescribed boundary (edge)conditions on the dependent variables or their derivatives. Table 1.1 gives specific applications of thefinite element method in the three major categories of boundary value problems, namely,(i) equilibrium or steady-state or time-independent problems,(ii) eigenvalue problems, and

(iii) propagation or transient problems.- In an equilibrium problem, we need to find the steady-state displacement or stress distribution if it is

a solid mechanics problem, temperature or heat flux distribution if it is a heat transfer problem, andpressure or velocity distribution if it is a fluid mechanics problem.

- In eigenvalue problems also. time will not appear explicitly. They may be considered as extensionsof equilibrium problems in which critical values of certain parameters are to be determined inaddition to the corresponding steady-state configurations. In these problems, we need to find thenatural frequencies or buckling loads and mode shapes if it is a solid mechanics or structuresproblem, stability of laminar flows if it is a fluid mechanics problem, and resonance characteristics ifit is an electrical circuit problem.

- The propagation or transient problems are time-dependent problems. This type of problem arises, forexample, whenever we are interested in finding the response of a body under time-varying force inthe area of solid mechanics and under sudden heating or cooling in the field of heat transfer.

- Mostly the choice of the type of element is dictated by the geometry of the body and the number ofindependent coordinates necessary to describe the system.

- If the geometry, material properties, and the field variable of the problem can be described in termsof only one spatial coordinate, we can use the one dimensional or line elements

- The temperature distribution in a rod (or fin), the pressure distribution in a pipe flow. and thedeformation of a bar under axial load, for example, can be determined using these elements.Although these elements have cross-sectional area, they are generally shown schematically as a lineelement.

- For a simple analysis, one-dimensional elements are assumed to have two nodes, one at each end,with the corresponding value of the field variable chosen as the unknown (degree of freedom).

- However, for the analysis of beams, the values of the field variable (transverse displacement) and itsderivative (slope) are chosen as the unknowns (degrees of freedom) at each node.

- When the configuration and other details of the problem can be described in terms of twoindependent spatial coordinates, we can use the two-dimensional elements shown in Figure. Thebasic element useful for two-dimensional analysis is the triangular element.

- Although a quadrilateral (or its special forms, rectangle and parallelogram) element can be obtainedby assembling two or four triangular elements. In some cases the use of quadrilateral (or rectangle orparallelogram) elements proves to be advantageous.

- For the bending analysis of plates, multiple degrees of freedom (transverse displacement and itsderivatives) are used at each node.

- If the geometry, material properties, and other parameters of the body can be described by threeindependent spatial coordinates, we can idealize the body by using the three dimensional elements.

- The basic three-dimensional element, analogous to the triangular element in the case of two-dimensional problems, is the tetrahedron element.In some cases the hexahedron element, which canbe obtained by assembling five tetrahedrons as indicated in Figure. can be used advantageously.

- Some problems, which are actually three-dimensional, can be described by only one or twoindependent coordinates. Such problems can be idealized by using an axi-symmetric or ring type ofelement. The problems that possess axial symmetry, such as pistons, storage tanks, valves, rocketnozzles, and reentry vehicle heat shields, fall into this category.

Type of Elements:

- If the problem involves the analysis of a truss structure under a given set of load conditions, the typeof elements to be used for idealization is obviously the "bar or line elements".

- Similarly, in the case of stress analysis of the short beam.- The finite element idealization can be done using three-dimensional solid elements the number of

degrees of freedom needed, the expected accuracy, the ease with which the necessary equations canbe derived, and the degree to which the physical structure can be modelled without approximationwill dictate the choice of the element type to be used for idealization.

At start finite element analysis relies on two important functions.- A continuous piecewise smooth function is needed to prescribe the field variable within the element.- An integral expression called functional is used to generate clement equations.- Generally, three kinds of forces which can be considered to act on a body, namely,

i) body forcesii) surface tractions andiii) point or concentrated loads.

- We require 15 equations to proceed with the design:1. 6 components of stress,2. 6 components of strain and3. 3 components of displacement at a point.

- These equations are provided by1. 3 Equilibrium equations,2. 6 stress-strain relations and3. 6 strain-displacement relations.

f relates to force per unit volume example weight

One Dimensional Problems:

- Elasticity Problems- Heat Conduction Problems are famous 1D problems

One Dimensional linear element is called as a bar element.One Dimensional linear element or a bar element has two nodes for every element.The polynomial below satisfies the variation of field variable for a bar element.

For every node there is one degree of freedom, so an element with two nodes will have 2 DOFs.

q are the nodal displacements.Global Coordinate System is defined for a complete stepped bar.Shape functions are also called as interpolation functions.

Derivation of Shape Functions for one dimensional bar element:

Displacement Equations using Shape Functions:

Formulation of element matrices:

- To achieve formulations we need to minimize the potential energy function with respect to nodaldisplacements.

For a one dimensional problem:The Strain can be derived from the equation:

- It can be inferred that for obtaining strain-displacement matrix, the expression for strain is essential.- Similarly, for obtaining material property matrix [D], the expression for stress is necessary.

LOOK IN TO THE jpeg 1 FOR ENTIRE PROBLEM:

TREATMENT OF BOUNDARY CONDITIONS:

- There are two approaches that can be used for incorporating boundary conditions in global equations.i) Elimination approachii) Penalty approach.

Elimination Approach:

This approach was already explained in the previous section and this method reduces the size of thematrices.

Penalty Approach

This method is based on the concept of considering the support as a spring having large stiffness so that thedeflection of the spring is very small. A spring of large stiffness C is included and nodal displacements ofthe spring are a1 and qi as shown in Fig. 2.7. The spring which is included will contribute to the strainenergy term in the functional.

when differentiated with respect to q. for the process of minimisation

Hence, by considering a large stiffness spring to model a support, one term Cq1 should be added to thestiffness matrix and another term Ca1 be added to the load vector corresponding to degree of freedom 1. Thechoice of the value C is Normally, one of the diagonal terms in the stiffness matrix will bemaximum. If we consider the stepped bar problem discussed in previous sections, the penalty approachapplied to model the left fixed support, the following modifications need to be incorporated. Addition of C isonly incorporated if the ends are fixed if the bar is fixed at both ends then the m11 and m33 are added withC.