STUDY OF FINITE ELEMENT ANALYSIS AND SOFTWARE PACKAGES
History of FEA
Basic ideas of the finite element analysis were developed by aircraft engineers in the early 1940’s.These were primarily the matrix methods of analysis.
The modern development of the finite element method began in the year of 1945 in the field of structural engineering with the work by Hrennikoff. The term finite element was first introduced by Clough in 1960 in the plane stress analysis and he used both triangular and rectangular element in that analysis.
Introduction of FEA
The finite element method (FEM) is the dominant discretization technique in structural mechanics. The basic concept in the physical interpretation of the FEM is the sub division of the mathematical model into disjoint (non-overlapping) components of simple geometry called finite elements or elements for short. The response of each element is expressed in terms of a finite number of degrees of freedom characterized as the value of an unknown function or functions, at a set of nodal points.
Objectives of FEAthe analyst need certain requirements while designing and assembling the parts of the prodiuct those requirements are mentioned below.
To calculate,
Displacement of certain points.
Stress distribution.
Natural frequencies.
Critical buckling loads.
Vibrations.
Pressure, velocity and temperature distribution.
Crack growth, residual strength and fatigue life.
FEA Procedure
The following two general methods are associated with the finite element analysis. They are
Force method.
Displacement or Stiffness method.
Step 1
Discretization of Structure.
The art of subdividing a structure into a convenient number of smaller elements is known as discretization.
Some elements are classified as follow:
One dimensional element.
Two dimensional elements.
Three dimensional elements.
Axisymmetric elements.
Step 2
Numbering of nodes and elements.
The nodes and elements should be numbered after discretization process. The numbering process is most important since it decides the size of the stiffness matrix and it leads the reduction memory requirement. While numbering the nodes, the following condition should be satisfied.
Step 3
Selection of a displacement function or interpolation function.
The polynomial type of interpolation functions are mostly used due to the following reasons.
It is easy to formulate and computerize the finite element equations.
It is easy to perform differentiation or integration.
The accuracy of the results can be improved by increasing the order of the polynomial.
Case1
Linear polynomial
One dimensional problem Φ(x)=a0 +a1 x.
Two dimensional problem Φ(x,y)=a0+a1x+a2y.
Three dimensional problem Φ(x,y,z)=a0+a1x+a2y+a3z.
Case2
Quadratic polynomial
One dimensional problem Φ(x)=a0+a1x+a2x2.
Two dimensional problem Φ(x,y)=a0+a1x+a2y+a3x2+a4y2+a5xy.
Three dimensional problem Φ(x,y,z) =a0+a1x+a2y+a3z+a4x2+a5y2+a6z2+a7xy+a8yz+a9xz.
Step 4
Define the material behavior by using strain-displacement and stress-strain relationship.
In caser of our dimensional deformation, the strain-displacement relationship is given by
e=du/dx.
Where, u=displacement field variable along y direction.
e=strain.
Stress-strain relationship is given by
σ =Ee.
Where , σ = stress in x direction.
E = modulus of elasticity or young’s modulus.
Step 5
Derivation of element stiffness matrix and equations.
The finite element equation is in matrix form as,
f1 k11 k12 k13 … k1n u1
f2 = k21 k22 k23 ... k2n u2
f3 k31 k32 k33 … k3n u3
fu k41 k42 k43 … k4n un
Equation
{Fe} = [ke] {ue}
Step 6
Assemble the element equations to obtain the global or total equations.
{ F} =[k] {u}
Where, {F] = global force vector.
[k] = global stiffness matrix.
{u} = global displacement vector.
Step 7
Interpret the results (post processing)
Analysis and evaluation of the solution results is referred to as post-processing.
Post processor computer programs help the user to interpret the results by displaying them in graphical form.
Advantages
*One of the major advantages of the FEM over other approximate methods is the fact that FEM can handle irregular geometry in a convenient manner.
*Non-homogeneous materials can be handled easily.
*All the various types of boundary conditions are handled.
*Dynamic effects are included.
*Higher order elements may be implemented.
*It handles general load conditions without conditions.
*Vary the size of the elements to make it possible for using small elements where necessary.
Disadvantages FEM
*It requires a digital computer and fairly extensive software.
*It requires longer execution time compared with finite difference method.
*Output result will vary considerably, when the body is modeled with fine mesh when compared to body modeled with course mesh.
*In finite difference method, the governing differential equation of the phenomenon must be known whereas finite element method does not require to express fully.
Application
Structural; analysis is probably the most common application of finite element method. The term structural implies not only civil engineering structures such as bride and buildings, but also naval, aeronautical; and mechanical structures such as ship hulls, aircraft bodies, and machine housings, as well as mechanical components such as pistons, machine parts and tools.
Types of structural analysis
Static analysis.
Modal analysis.
Harmonic analysis.
Transient dynamic analysis.
Spectrum analysis.
Buckling analysis.
Explicit dynamic analysis.
Thermal analysis
A thermal analysis calculates the temperature distribution and related thermal quantities in a system component. Typical thermal quantities of interest are
Temperature distributions.
The amount of heat lost or gained.
Thermal gradients.
Thermal fluxes.
Thermal simulations play an important role in the design of many engineering applications including internal combustion engines, turbines, heat exchangers, piping systems and electronic components.
Loads
The term loads in Ansys includes boundary conditions & externally or internally applied forcing functions. The term loads in different disciplines are.
Structural: Displacements, forces, pressures, gravity temperatures[ for thermal strain].
Thermal: Temperatures, hear flow rates, convections, internal heat generation, infinite surface.
Magnetic: Magnetic potentials, magnetic flux, magnetic current segments, source current density, infinite surface.
Electric: Electric potential (voltage), electric current charge densities, infinite surfaces.
Fluid: Velocities, pressures.
Types of loads
Degree of freedom.
Force
Surface load.
Body load.
Inertia load.
Coupled field loads, etc.
Degree of freedom (DOF)
A DOF constraint fixes a degree of freedom to a known value. ex. of constraints are specified displacements and symmetry boundary conditions in a structural analysis, prescribed temperature in thermal analysis.
Force
A force is a concentrated load applied at a node in the model. Ex. Forces and movements in a structural analysis, heat flow rate in a thermal analysis and current segment in magnetic field analysis.
Surface Load
A surface load is a distributed applied load over a surface. Ex. Pressure in structural analysis, convection and heat flux in thermal analysis.
Body load
A body load is a volumetric (or) Field load. Ex. Temperatures & fluencies in structural analysis, heat generation rate in thermal analysis & current densities in a magnetic field analysis
Inertia loads
Inertia loads are those attributes to the inertia[mass matrix] of a body such as gravitational acceleration, angular velocity, angular acceleration.
Coupler field loads
It is a special case of one of the above loads, where results from one analysis are used as loads in another analysis. For Example: Magnetic forces calculated in a magnetic field analysis as force in a structural analysis.
Related Software of FEA
ANSYS
ANSYS has a substantial commitment to continual improvement of its solver technologies. Improvements in the distributed ANSYS product allow us to run solutions efficiently in parallel across multiple CPU’s Enhanced capability of solver performance directly impacts the return on investment of managing turbine field life and ANSYS has demonstrated and commitment to begin first to market with new solver features
CONCLUSION
Finite element analysis has become a solution to the task of predicting failure due to unknown stresses by showing problem areas in a material and allowing designers to see all of the theoretical stresses within the model. This method of product design and testing is for superior to the manufacturing cost which would occur if each sample was actually built and tested.
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