Industrial machinery and equipment always present great hurdles for designing reliable manufacturing solutions. Most industrial machines are expensive hence the cost of equipment failure in machinery can be costly. Therefore, manufacturing industries do a careful evaluation of any prospective design before it can be utilized for delivering valuable products.

Finite element analysis is one of the techniques that has been used extensively by industries for the analysis and design of machinery and equipment. Earlier, it was used only as a tool for structural analysis but now it has become an integral part of Computer-Aided Engineering. It is now invariably used for developing electromagnetic and biomechanics solutions. Finite element analysis delivers exact solutions to the complicated partial differential equations of complicated machine geometry and boundary conditions. It transforms partial differential equations into algebraic equations.

## The procedure of finite element analysis is simple and can be applied to any of the real-life problems.

- Finite element analysis is started with an assumed trial solution. It is done so because both the differential equations and the boundary conditions are unknown. It should be noted that the assumed trial solution must satisfy the boundary
- Generally, the assumed trial solution does not completely satisfy the differential equation hence there is always an error while satisfying the differential equation. This error is also called domain
- The residual domain varies at different points of the structural domain and it is very difficult to bring it down to nil at every point. Hence, it is ensured that the residual domain is brought down to the smallest unit. So that the weighted sum of the domain residual when estimated over all the structural domain is rendered
- The estimation of the assumed trial solution is a very difficult process and requires expertise for determining an accurate assumed trial solution. Each structural domain should be discretized into several independent segments which are also called finite elements. These consist of numerous continuous trial functions within each finite element.
- These trial functions are used to estimate the value of the field variable at any of the points in the interior of the segment. These key points are called nodes. Trial functions that are used in the finite element are known as element level shape functions.
- This is followed by estimating the weighted sum for the entire structural domain. The element level shape functions are utilized to obtain the weighted sum of domain residual for each of the elements which are further added up over all the elements to compute the weighted sum of the entire structural
- The estimated value of the elements is then transformed into elemental level characteristic matrices are programmed into software like IDEAS, NASTRAN, etc. The program uses the characteristics matrices of each element that are then added up to obtain the system-level

The finite element analysis method is increasingly used to obtain the solution for structural mechanics problems. It offers easy visualization of the machinery and equipment and is applicable to real-life problems of varied physical domains. One of them is the aerospace industry. The finite element analysis is used for the structural analysis in the aerospace industry. It is used for analyzing mode shapes, natural frequencies, and aero-servo-elastic studies. It is also used in aerodynamics and for analyzing natural frequencies.

Finite element analysis is also extensively applicable in the complex configuration of dielectric insulating materials and electrodes. It can be used for enhancing the optimization and dependability of insulated design in high voltage equipment. Finite element analysis can provide an equipotential field plot for a high voltage transformer and can assure the minimization of stray losses in electrical machines. One of the preferred processes for installing a thermal wire bonding between a lead frame and a semiconductor chip is the thermosonic wire bonding. It is connected by a metal wire. Wire bonding technology is a very complex task and works within certain boundaries and specifications, these designs have to meet the objective of lower mass, inertia, and higher dynamic stiffness. In such technologies, finite element analysis helps in analyzing the dynamic rigidity of the system.

The method of final element analysis is applicable to various engineering problems because of the common characteristics in the mathematical formulation of the problems. Problems with the same form of partial differential equation generally correspond to varied physical problems. One of the methods that are very powerful in obtaining the solutions of the differential equations is the Weighted Residual (WR) method. Piecewise trial approximation of the weighted residual technique is the basis of the finite element analysis. The finite element analysis of a given differential equation has the following steps:

- The first step is to note down the weighted residual
- Then, the differential equation is integrated by performing integration by parts significant times so that the distribution of the differential between the various variables and the weighting function is even. Further, the weak form of the weighted residual is developed. It must be noted that the weak form of the differential equation is within the natural boundary
- The n elements of the weak form of the differential equation are added
- Further, the nodes and geometry ie. The finite elements are
- The function is further interpolated to derive the shape and is further used as a weighting
- The next step is to structure the element level equations to develop the system equations for a given structure of finite element
- The equations obtained after the previous step is substituted and solved for the unknown

In manufacturing industries, the crankshaft is an important component of industrial machinery. Crankshafts have to experience fluctuations in torques due to simultaneous strokes in the cylinder. It results in torsional vibrations which occur due to twisting reaction in rotatory shafts. The analysis of the torsional vibration is done by discretizing the crankshaft and other drivelines components. These components are considered as a set of lumped masses and springs. Due to the complex geometry of the multicylinder engine crankshafts, the bending modes are coupled with the torsion modes. Hence the three-dimensional vibrational analysis of crankshaft is done to save time while designing real-life products. Generally, the beam element model is used to analyze automotive crankshaft.

The thermal analysis of a pressure vessel is an important process in equipment manufacturing. It is one of the applications of the finite element analysis. It helps in checking the rigidity and stability of the machinery in challenging conditions. The first step in the thermal analysis of the pressure vessel is the geometrical modeling of the required pressure vessel. The modeling is done for the two-dimensional structure along with the other threads. It should be noted that at a steady-state, the temperature of inside wall should be 300 degrees Celsius while the temperature of the outside wall should be 50 degrees celsius. Further, the distribution of the temperature along the vessel wall and threads should be noted down. The important thing in the thermal analysis of a pressure vessel is to obtain the distribution of the temperature when the gas temperature inside the vessel is at 450 degrees Celcius. The next step is to solve the thermal stresses at the stable state of the vessel which can be followed by finding the temperature distribution of the vessel as a function of time.

## Other applications of the finite element analysis are as follows:

- Structural dynamic analysis of a pressure vessel.
- Dynamics of a hard disk drive head assembly.
- Visualizing stress
- Thermal analysis of an IC Engine
- Analysis of a turbine
- Dynamic analysis of a
- Structural analysis of an automotive chassis dynamics.
- Dynamics of a scooter

In the method of finite element analysis, the accuracy of the solution is determined in terms of the refined element mesh. There are generally two methods for mesh refinement. First, h- refinement where an increasing number of elements are used to design a particular structural domain. Second, p-refinement where interpolation functions are increased by using the order of the polynomials. The refinement is done to estimate the sequential solutions that show the exact solution.

Finite element analysis can be considered a mathematical method for analyzing problems of mathematical and engineering physics. It is applicable to problems with complex loading, material properties, and geometries. Though stress analysis of trusses and beams can be analyzed by finding an analytical solution while finite element analysis is utilized in the situation where the designs are very complex. It is highly required for the situation where the accuracy is essential. It is highly crucial to identify the physical behaviors like fluid flow, strength, and transfer capability of complex objects. It is also useful in understanding the optimal design and predicting the behavior and performance of the design.

Some of the industries that use the finite element analysis method in their product delivery are mechanical engineering, civil engineering, automotive engineering, and aerospace engineering. It is useful for analyzing processes like fluid flow, heat transfer, electromagnetic fields, soil mechanics, acoustics, and biomechanics. Industries are increasingly utilizing the capability of this technique by further increasing the durability and design of their equipment and machinery.

Also Read – Applications Of Computational Fluid Dynamics