Unit-4-computer Aided Design

UNIT – IV FINITE ELEMENT ANALYSIS 4.1. INTRODUCTION •

The finite element method is a numerical method for solving engineering and mathematical physics problems.

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The typical use of this method to solve the problems in the field of stress analysis, heat transfer, fluid flow, mass transfer and electromagnetic.

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This method can able to solve physical problems involving complicated geometrics, loadings and material properties which can not be solved by analytical method.

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In this method, the domain in which the analysis to be carried out is divided into smaller bodies or unit called as finite elements.

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The properties of each type of finite element is obtained and assembled together and solved as whole to get solution.

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Based on application, the problems are classified into structural and nonstructural problems.

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In structural problems, displacement at each nodal point is obtained. Using these displacement solutions, stress and strain in each element are determined.

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Similarly, the non-structural problems, temperature or fluid properties at each nodal point is/are obtained.

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Using these nodal values, properties like heat flux, fluid flow etc., for each element is determined.

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Since large computations are to be carried out, this method requires highspeed computation facility with large memory.

4.2. BRIEF HISTORY •

The finite element method used today was developed to its present state very recently.

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According to Zinckiewicz, the development is occurred along two major paths, one in mathematics and other in engineering; somewhere in between these two paths are variational and weighted residual methods.

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Both of which requires trial functions to effect a solution. The use of these functions data is sent back to almost 250 years.

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These trial functions are assumed based on physical intution and they are applied globally to get the solution for the problem.

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The use of trial functions is neither considered as development in pure mathematical field nor in engineering field.

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In a paper by Gauss in 1795, trial functions were used in what is now called as the method of weighted residuals.

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Later, Rayleigh used these functions in variational method in 1870 and by Ritz in 1909. In a landmark paper in 1915; Galerkin introduced a particular type of

weighted residual method which is called by his name as “Galerkin weighted residual method”. •

In 1943, Courant introduced piecewise trial functions which are now called as shape functions. These shape functions are applied in a smaller region (i.e. at element level) instead of applying it globally which is made him to solve the real world problems.

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It is widely felt that finite element method has its roots in Courant works but he didn’ t use the terminology “Finite element”.

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In the early 1940’ s, aircraft engineers were developing and using analysis method called force matrix method which is recognized as early form of finite element method.

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In this method, the nodal unknowns are forces not the displacements. When the displacements of each node are taken as unknown, the method is called as “St method”.

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In a paper in 1960, Clough first introduced the term “Finite element”.

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In 1965, Zinekiwicz and Cheung applied FEM, to solve the non-structural problems. Szabo & Leo showed how the weighted residual method, particularly the Galerkin method could be used in non-structural problem analysis.

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However, the present day FEM does not have its roots in any discipline.

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Mathematicians are trying to improve the mathematical background of FEM, while the engineers are interesting in applications where FEM can be used. In most branches of engineering, these developments have made the FEM as one of the most powerful numerical solution method.

4.3. BASIC CONCEPT OF FEM •

The basic concept of FEM is that the structure to be analysed is considered as an assemblage of discrete pieces called “Elements”, that are connected together at a finite number of points (or) nodes.

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The finite element is a geometrically simplified representation of a small part of the physical structure.

Concept: 1. Divide the domain in which analysis is to be carried out. 2. Isolating one of the element from each type and get the property of them. 3. Assembling the finite elements to get the property of whole domain. Example: I Determination of the area of a circle using the areas of triangles 1. Finite-element discretization: •

First the continuous region (i.e. the circle) is represented as a collection of a finite number ‘ n’ of subregions say triangles. This is called discretization of the domain by triangles.

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Each subregion is called as an “element”. Consider the two methods of discretizing of circle as shown in fig 4.1.

Fig 4.1. Two methods of discretizing circle with triangular elements 2. Element equation: •

From each method of discretization, Isolate one of the elements and get the property (i.e., areas of triangle.)

3. Assembly of elements and solution: •

The approximate area of the circle is obtained by adding all areas of triangles, this process is called assembly of the element.

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Total area = Sum of the area of individual elements.

4. Convergence and error estimate: •

The error in the approximation is equal to each difference between the area of the sector and that of the triangle.

Fig 4.4. Elemental error

4.4. PROCEDURE OF FEM The following steps summarize the finite element procedure.

Step (1) & (2) =

Preprocessing phase of FEM

Step (3) to (8) =

Analysis phase of FEM

Step (9)

Post processing phase of FEM

=

4.4.1. Discretization of Domain (Nodal Approximation and Element Type): •

Discretization is the process in which the given body or domain is sub-divided into number of elements.

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In ID problems, the given body is divided by line/bar elements. In 2D problems, using triangular and quadrilateral elements body can be descritized. In case of 3D problems, tetrahedral

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The choice of the types of element for discretization is dictated by the geometry of the body and the number of independent co-ordinates necessary to describe them (i.e. ID, 2D or 3D).

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The size of elements and number of elements will have effect on accuracy and convergence of solution.

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Hence atmost care should be taken to select size and number of elements. If the size of elements is small, the solution will be more accurate with increase in cost of computation.

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Therefore some,times it is necessary to have different size of elements in the same body to balance between cost of computation and accuracy.

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In general wherever a steep gradient of field variable is expected, finer mesh can be used in that region. It is important to note h that as increasing the number of elements the accyracy of the solution increases up to a limit.

Fig 4.9. Location nodes

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The location of node is another important aspect of discretization.

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If a body has abrup changes in geometry, material properties, loads etc, at a point it is necessary to introduce a node a that point as shown in fig 4.9.

4.4.2. Numbering of Nodes and Elements: •

After discretization, the nodes and elements are to be numbered.

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If bandwidth solver or skyline solver is used for solving the resulting simultaneous equation after assembly step and node numbering important.

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Because, node number scheme will decide the size of half bandwidth of the Global stiffness matrix (i.e. memory required for Global stiffness is high).

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To reduce the memory, the numbering of nodes should be such that maximum node number of an element minus minimum node number of that element should be minimum. i.e. maximum difference in node numbers of an element.

i.e. Maximum node number —Minimum node Number = Minimum •

To accomplish this, shorter side should be numbered first and then longer side should be numbered later as explained in fig 4.10.

From fig 4.10. (a) Considering element (1) Maximum node Number = 7 Minimum node Number Difference

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=6

From fig 4.10. (b) considering element (1) Maximum node Number = 6 Minimum node Number = Difference

=5

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Hence, node-numbering system followed in fig 4.10 (b) is selected, because it reduces the memory requirements.

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If frontal solver used in element numbering is important not the node numbering.

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For greater efficiency of this method, consecutive element numbering should be done across the structure in a direction that spans the smallest number of nodes.

4.4.3. Selection of a Displacement Function or Interpolation Function: •

In F.E. analysis assumed the solution function used to represent the behaviour of field variables in each element are called “Displacement function” or “Interpolation function”.

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These functions are usually polynomial of linear, quadratic and cubic form. The accuracy of the solution can be improved by increasing the order of the polynomial.

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Various polynomials used in 1D problems are given below fig. 4.11

4.4.4. Defining Material Behaviour Using Stress-strain and Strain-displace Relationship: •

The results from FEA mainly depend on how well the material behaviour is accu modeled.

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The material behaviour model is defined by stress-strain relationship and displacement relationship.

For example, ID problem —stress due to axial load Stress-strain relationship is given by

4.4.5. Derivation of Element Equation (or) Formulation of Characteristic Matrix Vector: The element equation is in the form of

it can be derived using interpolation function from step 3 and material behaviour from stel by using any one of the following methods of minimization.

I) Variational method 2) Weighted residual method [ used in thermal analysis problems]. 3) Virtual displacement method [ used in stress analysis problems] 4) Energy method [ used in stress analysis problems]. 4.4.6. Assembling: •

Assembling all the element equations of all the elements in the descritized domain using method of superposition is called direct stiffness method to get the global equation of the physical domains, which is in the form of

4.4.7. Applying the boundary conditions: •

The known degrees of freedom at the boundary nodes of a physical domain are applied to global equation before solving for global displacement vector.

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Hence the number of unknown fieli variable; to be solved will be reduced.

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Boundary Conditions (BCs) are classified into primary BCs (also known as essential B.Cs and secondary B.Cs (also known as natural B.Cs).

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The secondary B.Cs are automatically include in the formulation stage (derivation) of element characteristic equations itself, where as th primary boundary conditions are applied at this stage in global stiffness matrix and global forc vector.

4.4.8. Solution: •

After applying the primary B.Cs, the resulting simultaneous equations are solved by using Gauss-elimination method or any other solving method

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By solving, the Global displacement vector {a} is determined.

4.4.9. Obtaining the required results from the solution of displacement vector: •

From the solution of displacement vector {a} stress and strain values [ principal stresse and strains, von-misses stresses and strains] can be obtained.

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For example, to find ID axial strain due to axial load.

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Similarly the required results are obtained from the displacement vector and they are represented either tabular form or graphical form.

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Tabulation results can be used to get exact value of displacement, stress, strain, reaction force etc on particular nodes or elements.

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It is very difficult to understand the displacement pattern, stress distribution etc from the tabulated results.

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To overcome these difficulties, the results are usually represented in the graphical form, animated views, contour plots, graphs etc.

4.5. ADVANTAGES AND DISADVANTAGES OF FEM 4.5.1. Advantages of FEM: I. Irregular geometries can be modeled more accurately and easily. 2. Implementation of any type of boundary conditions is very easy. 3. With very little effort, heterogeneous and anisotropic materials can be modeled. 4. Any type of loading can be handled. 5. The element sizes can be varied throughout the model. Wherever it is necessary. we can use fine meshes. 6. Whether the problem is linear or non linear, the basics (i.e. steps followed/implemented) of FEA remain same. 7. Altering the element model with different loads, boundary conditions and other changes on the model can be done easily. 4.5.2. Disadvantages: 1. FEA softwares are costlier. 2. Output result will vary considerably, when the body is modeled with fine mesh when compared to body modeled with course mesh. 3. Before using a element for a problem, we should know about its capabilities and nature, because no single element is available for all applications. 4. Even though the FEA softwares are user friendly but they are not relatively easier for use. 5. Generally, it is compared with finite difference method (FDM), the FEM seems to have longer execution time. 4.6. APPLICATIONS OF FEA The FEA can be used to analyse both the structural and non-structural problems. (a) Types of structural problems. (i) Stress analysis Linear and Nonlinear analysis Example: Linear Analysis:- Plate with hole subjected to inpiane loads. Non-linear analysis:- Material Non-linearity - Machine element members are subjected to stress more than elastic limit.

Geometrical non-linearity:- Thin shell structure is subjected axial or torsional loads. Both material and geometrical non-linearity: - Thin shell structures are subjected to mechanical loading with high temperature creep effect. (ii) Eigen Buckling Analysis: Example: Connecting rod subjected to axial compression. This analysis can be used to determine mode shape of buckling load and its critical loads. (iii) Vibrational analysis: Example: Beams subjected to different types of loading. This analysis can be used to determine the mode shape of vibration with its natural frequencies. (b) Types of non-structural problems (i) Heat Transfer Analysis: Linear:- Steady state thermal analysis on composite wall. Non-linear:- Thermal Analysis with anisotropic materials. The heat transfer problem can be further classified as Steady state (time independent) Transient (time dependent) (ii) Fluid-flow Analysis: Example: Fluid flow through pipes or channel. (iii) Electromagnetic Analysis: Example: Modeling of electromagnetic field of motor. Recently FEM is used in Bio-Mechanical Engineering field. For example, stress analysis on human parts like bones, joints, skull, tooth etc. 4.7. SOURCES OF ERRORS IN FEM •

The error of the finite element solution depends on discretization which is characterized by the finite element mesh and the choice of elements.

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Computed results are rarely exact.

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A structure is divided into elements whose displacement fields may exclude many of the physically possible deformation modes.

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The computer represents numbers by a finite number of bits or digits.

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Numerical difficulties may arise even when the analyst makes no outright blunder in using a computer program.