Concepts of Finite Element Method

I. Introduction

The Finite Element Method (FEM) is a numerical technique used in CAD CAM to solve complex engineering problems. It is widely used in various fields such as structural analysis, heat transfer analysis, and fluid flow analysis. The method involves dividing a complex problem into smaller, simpler elements and solving them individually. This allows for efficient computation and accurate results.

A. Importance of Finite Element Method in CAD CAM

The Finite Element Method plays a crucial role in CAD CAM as it allows engineers to simulate and analyze the behavior of structures and systems before they are physically built. This helps in reducing costs, optimizing designs, and improving overall performance.

B. Fundamentals of Finite Element Method

The fundamentals of the Finite Element Method are based on the principles of variational calculus and the concept of discretization. Variational calculus involves minimizing the total potential energy of a system, while discretization involves dividing the system into smaller elements.

II. Key Concepts and Principles

In order to understand the Finite Element Method, it is important to grasp the key concepts and principles associated with it.

A. Shape functions

Shape functions are mathematical functions used to approximate the behavior of a system within each element. They define the shape and behavior of the system within the element and allow for the interpolation of values at different points within the element.

1. Definition and purpose

Shape functions are mathematical functions that describe the behavior of a system within an element. They are used to approximate the unknowns within the element and allow for the interpolation of values at different points within the element.

1. Types of shape functions

There are various types of shape functions used in the Finite Element Method, such as linear shape functions, quadratic shape functions, and higher-order shape functions. The choice of shape functions depends on the complexity of the problem and the desired accuracy of the solution.

1. Calculation and selection of shape functions

Shape functions are calculated based on the geometry and properties of the element. They are selected in such a way that they satisfy certain properties, such as continuity and completeness. The selection of shape functions is crucial as it affects the accuracy and convergence of the solution.

B. Element matrices

Element matrices are matrices that represent the behavior of an element. They are used to relate the unknowns within the element to the applied loads and boundary conditions.

1. Definition and purpose

Element matrices are matrices that describe the behavior of an element. They relate the unknowns within the element to the applied loads and boundary conditions. They are used to assemble the global matrix and solve the FE equations.

1. Types of element matrices

There are different types of element matrices used in the Finite Element Method, such as stiffness matrices, mass matrices, and damping matrices. The type of element matrix depends on the physical behavior being modeled.

1. Calculation and assembly of element matrices

Element matrices are calculated based on the properties of the element, such as its geometry, material properties, and boundary conditions. They are then assembled into the global matrix, which represents the behavior of the entire system.

C. Global matrix

The global matrix represents the behavior of the entire system. It is obtained by assembling the element matrices and incorporating the boundary conditions.

1. Definition and purpose

The global matrix is a matrix that represents the behavior of the entire system. It relates the unknowns of the system to the applied loads and boundary conditions. It is used to solve the FE equations and obtain the displacements and stresses.

1. Assembly of global matrix using element matrices

The global matrix is obtained by assembling the element matrices. Each element matrix is added to the corresponding location in the global matrix, taking into account the connectivity of the elements.

1. Incorporation of boundary conditions

Boundary conditions are incorporated into the global matrix by modifying the corresponding rows and columns. This ensures that the solution satisfies the prescribed boundary conditions.

D. Solution of FE equations

The FE equations represent the equilibrium of the system and are solved to obtain the unknowns and displacements.

1. Methods for solving FE equations

There are two main methods for solving FE equations: direct methods and iterative methods. Direct methods involve solving the system of equations directly, while iterative methods involve iteratively improving the solution until convergence is achieved.

1. Solution techniques

Various solution techniques can be used to solve the FE equations, such as Gauss elimination and LU decomposition. These techniques involve manipulating the equations to obtain a solution.

1. Calculation of unknowns and displacements

Once the FE equations are solved, the unknowns and displacements can be calculated. These values represent the behavior of the system and can be used for further analysis and post-processing.

E. Post processing

Post processing involves analyzing and interpreting the results obtained from the Finite Element Method.

1. Calculation of stresses and strains

Stresses and strains can be calculated from the displacements obtained from the FE equations. These values represent the internal forces and deformations within the system.

1. Visualization and interpretation of results

The results obtained from the Finite Element Method can be visualized and interpreted using various techniques, such as contour plots, animations, and graphs. This allows for a better understanding of the behavior of the system.

1. Validation and verification of results

The results obtained from the Finite Element Method should be validated and verified to ensure their accuracy. This can be done by comparing the results with analytical solutions, experimental data, or results from other numerical methods.

F. Convergence requirements

Convergence is the process of obtaining a solution that is accurate and reliable. It is important to ensure that the solution converges to the correct solution.

1. Definition and significance

Convergence refers to the process of obtaining a solution that is accurate and reliable. It is important because an inaccurate or unreliable solution can lead to incorrect predictions and designs.

1. Criteria for convergence

There are various criteria for convergence, such as the residual error, the change in the solution, and the number of iterations. These criteria ensure that the solution is accurate and reliable.

1. Techniques for improving convergence

There are various techniques for improving convergence, such as adjusting the mesh, refining the element size, and using different solution techniques. These techniques help in obtaining a solution that converges faster and more accurately.

III. Step-by-step Walkthrough of Typical Problems and Solutions

In order to understand the Finite Element Method better, it is helpful to go through step-by-step walkthroughs of typical problems and their solutions.

A. Application to Longitudinal/Axial bar problem

1. Problem statement

Consider a bar subjected to axial loading. The goal is to determine the displacements and stresses within the bar.

1. Finite element discretization

The bar is divided into smaller elements, such as beam elements or truss elements. Each element is represented by a set of nodes and has its own shape functions and element matrices.

1. Calculation of element matrices and assembly of global matrix

The element matrices are calculated based on the properties of the element, such as its length and material properties. They are then assembled into the global matrix, taking into account the connectivity of the elements.

1. Incorporation of boundary conditions

The boundary conditions, such as fixed displacements or applied loads, are incorporated into the global matrix by modifying the corresponding rows and columns.

1. Solution of FE equations and post processing

The FE equations are solved to obtain the displacements and stresses within the bar. These values can be used to analyze the behavior of the bar and make design decisions.

B. Beam problem

1. Problem statement

Consider a beam subjected to bending. The goal is to determine the deflections and bending moments within the beam.

1. Finite element discretization

The beam is divided into smaller elements, such as beam elements or shell elements. Each element is represented by a set of nodes and has its own shape functions and element matrices.

1. Calculation of element matrices and assembly of global matrix

The element matrices are calculated based on the properties of the element, such as its length, cross-sectional area, and material properties. They are then assembled into the global matrix, taking into account the connectivity of the elements.

1. Incorporation of boundary conditions

The boundary conditions, such as fixed displacements or applied loads, are incorporated into the global matrix by modifying the corresponding rows and columns.

1. Solution of FE equations and post processing

The FE equations are solved to obtain the deflections and bending moments within the beam. These values can be used to analyze the behavior of the beam and make design decisions.

C. Plane stress/strain problem

1. Problem statement

Consider a 2D plate subjected to plane stress or strain. The goal is to determine the displacements and stresses/strains within the plate.

1. Finite element discretization

The plate is divided into smaller elements, such as quadrilateral elements or triangular elements. Each element is represented by a set of nodes and has its own shape functions and element matrices.

1. Calculation of element matrices and assembly of global matrix

The element matrices are calculated based on the properties of the element, such as its area, thickness, and material properties. They are then assembled into the global matrix, taking into account the connectivity of the elements.

1. Incorporation of boundary conditions

The boundary conditions, such as fixed displacements or applied loads, are incorporated into the global matrix by modifying the corresponding rows and columns.

1. Solution of FE equations and post processing

The FE equations are solved to obtain the displacements and stresses/strains within the plate. These values can be used to analyze the behavior of the plate and make design decisions.

D. Iso-parametric formulation

1. Definition and purpose

Iso-parametric formulation is a technique used in the Finite Element Method to improve the accuracy of the solution. It involves using the same shape functions for both the geometry and the unknowns.

1. Calculation of shape functions and element matrices

The shape functions and element matrices are calculated based on the geometry and properties of the element. The same shape functions are used for both the geometry and the unknowns.

1. Advantages and disadvantages of iso-parametric formulation

The iso-parametric formulation has the advantage of improving the accuracy of the solution, especially for curved or irregular geometries. However, it can also increase the computational cost and complexity of the analysis.

E. Axis symmetric problem

1. Problem statement

Consider a problem that exhibits axisymmetric behavior, such as a cylindrical structure subjected to internal or external pressure. The goal is to determine the displacements and stresses within the structure.

1. Finite element discretization

The structure is divided into smaller elements, such as axisymmetric elements or shell elements. Each element is represented by a set of nodes and has its own shape functions and element matrices.

1. Calculation of element matrices and assembly of global matrix

The element matrices are calculated based on the properties of the element, such as its length, radius, and material properties. They are then assembled into the global matrix, taking into account the connectivity of the elements.

1. Incorporation of boundary conditions

The boundary conditions, such as fixed displacements or applied loads, are incorporated into the global matrix by modifying the corresponding rows and columns.

1. Solution of FE equations and post processing

The FE equations are solved to obtain the displacements and stresses within the structure. These values can be used to analyze the behavior of the structure and make design decisions.

F. Bending of plates

1. Problem statement

Consider a plate subjected to bending. The goal is to determine the deflections and bending moments within the plate.

1. Finite element discretization

The plate is divided into smaller elements, such as quadrilateral elements or triangular elements. Each element is represented by a set of nodes and has its own shape functions and element matrices.

1. Calculation of element matrices and assembly of global matrix

The element matrices are calculated based on the properties of the element, such as its area, thickness, and material properties. They are then assembled into the global matrix, taking into account the connectivity of the elements.

1. Incorporation of boundary conditions

The boundary conditions, such as fixed displacements or applied loads, are incorporated into the global matrix by modifying the corresponding rows and columns.

1. Solution of FE equations and post processing

The FE equations are solved to obtain the deflections and bending moments within the plate. These values can be used to analyze the behavior of the plate and make design decisions.

G. Weighted residual approach

1. Definition and purpose

The weighted residual approach is an alternative method for solving the FE equations. It involves minimizing the residual error by adjusting the unknowns.

1. Calculation of weighted residuals

The weighted residuals are calculated by multiplying the residual error by a weight function. The weight function is chosen in such a way that it minimizes the residual error.

1. Advantages and disadvantages of weighted residual approach

The weighted residual approach has the advantage of allowing for more flexibility in the choice of shape functions and solution techniques. However, it can also increase the complexity of the analysis and the computational cost.

IV. Real-world Applications and Examples

The Finite Element Method has numerous real-world applications in various fields of engineering. Some examples include:

A. Application of Finite Element Method in structural analysis

The Finite Element Method is widely used in structural analysis to determine the behavior of structures under different loading conditions. It can be used to analyze the stresses, deformations, and stability of structures.

B. Application of Finite Element Method in heat transfer analysis

The Finite Element Method is used in heat transfer analysis to determine the temperature distribution within a system. It can be used to analyze the conduction, convection, and radiation heat transfer.

C. Application of Finite Element Method in fluid flow analysis

The Finite Element Method is used in fluid flow analysis to determine the velocity and pressure distribution within a system. It can be used to analyze the flow patterns, turbulence, and pressure drop.

D. Examples of real-world problems solved using Finite Element Method

Some examples of real-world problems solved using the Finite Element Method include the analysis of bridges, buildings, aircraft structures, heat exchangers, and hydraulic systems.

V. Advantages and Disadvantages of Finite Element Method

The Finite Element Method has several advantages and disadvantages that should be considered when using it for engineering analysis.

A. Advantages

1. Flexibility in handling complex geometries

The Finite Element Method allows for the analysis of complex geometries, such as irregular shapes and curved surfaces. This flexibility is particularly useful in CAD CAM, where complex geometries are common.

1. Ability to model different types of physical phenomena

The Finite Element Method can be used to model various types of physical phenomena, such as structural behavior, heat transfer, and fluid flow. This makes it a versatile tool for engineering analysis.

1. Efficient use of computational resources

The Finite Element Method allows for the efficient use of computational resources by dividing the problem into smaller elements. This reduces the computational cost and allows for faster analysis.

B. Disadvantages

1. Approximation errors

The Finite Element Method involves approximating the behavior of the system within each element. This approximation can introduce errors, especially if the elements are not properly chosen or if the shape functions are not accurate.

1. Sensitivity to mesh quality

The accuracy of the solution obtained from the Finite Element Method is highly dependent on the quality of the mesh. A poorly meshed system can lead to inaccurate results and convergence problems.

1. Complexity in implementation and interpretation of results

The Finite Element Method requires a good understanding of the underlying principles and concepts. It also requires expertise in meshing, element selection, and interpretation of results. This complexity can make it challenging to implement and interpret the results.

VI. Conclusion

In conclusion, the Finite Element Method is a powerful numerical technique used in CAD CAM to solve complex engineering problems. It involves the division of a problem into smaller elements, the calculation of element matrices, the assembly of a global matrix, and the solution of FE equations. The method has numerous applications in various fields of engineering and offers advantages such as flexibility, versatility, and efficient use of computational resources. However, it also has limitations such as approximation errors, sensitivity to mesh quality, and complexity in implementation and interpretation of results. Overall, the Finite Element Method is an essential tool for engineers and researchers in CAD CAM and has the potential for further research and development in the field.

Summary

The Finite Element Method (FEM) is a numerical technique used in CAD CAM to solve complex engineering problems. It involves dividing a complex problem into smaller, simpler elements and solving them individually. The key concepts and principles of FEM include shape functions, element matrices, global matrix, solution of FE equations, post processing, and convergence requirements. The method is applied to various problems such as longitudinal/axial bar, beam, plane stress/strain, axis symmetric, bending of plates, and weighted residual approach. Real-world applications of FEM include structural analysis, heat transfer analysis, and fluid flow analysis. The method offers advantages such as flexibility, versatility, and efficient use of computational resources, but also has limitations such as approximation errors, sensitivity to mesh quality, and complexity in implementation and interpretation of results.

Analogy

The Finite Element Method can be compared to solving a jigsaw puzzle. The complex problem is like a jigsaw puzzle with many pieces, and the goal is to assemble the pieces to form a complete picture. In the Finite Element Method, the problem is divided into smaller elements, similar to dividing the puzzle into smaller sections. Each element is then solved individually, just like solving each puzzle piece. Finally, the solutions from each element are combined to obtain the complete solution, similar to assembling the puzzle pieces to form the complete picture.