FEM.pptx on enginiiering design and coading
- 1. Presentation on INTRODUCTION TO FINITE ELEMENT METHOD by PRERNA CHAUHAN Roll no.: 2214910010031 Semester: V Bachelor of Technology Mechanical Engineering Institute of Engineering and Technology Deen Dayal Upadhaya Gorakhpur University, Gorakhpur December,2023
- 2. TABLE OF CONTENTS • FINITE ELEMENT METHOD • WORKING OF FINITE ELEMENT METHOD • GENERAL STEPS IN FINITE ELEMENT METHOD • APPLICATIONS OF FINITE ELEMENT METHOD • BENEFITS OF USING FINITE ELEMENT METHOD • LIMITATIONS OF FINITE ELEMENT METHOD
- 3. FINITE ELEMENT METHOD The finite element method (FEM), or finite element analysis (FEA), is based on the idea of building a complicated object with simple blocks, or, dividing a complicated object into small and manageable pieces. Application of this simple idea can be found everywhere in everyday life, as well as in engineering. FEM originated from the need for solving Complex Elasticity and Structural Analysisproblems in Civil and Aeronautical Engineering. Its development can be traced back to 1941. It is anumerical technique for finding approximate solutions to PDEs as well as integral equations,permitting the numerical analysis of complex structures based on their material properties. Fig: example of FEM from google
- 4. • The Finite Element Mehtod (FEM) ia a numerical method of solving systems of partial differential equation. • It reduces a partial differential equation to a system of algebraic equations that can be solved using traditional linear algebra techniques. • In simple terms, FEM is a method for dividing up a very complicated problem into small elemnts that can be solved in relation to each other. • Useful for problems with complicated geometries, loadings, and material properties where analytical solutions cannot be obtained.
- 5. WORKING OF FINITE ELEMENT METHOD • Finite Element Analysis (FEA) uses a complex system of points called nodes which make a grid called mesh. • This mesh is programmed to contain the maaterial and structural properties which define how the structure will react to certain loading. • Nodes are asssigned at a certain density throughout the material depending on the anticipated stress levels of a certain area. • Regions which will receive a large amount of stressusually have a high node density than those which experience little or no stress. Poitns of interest may consist of: fracture point of previously tested materials, fillets, corners,
- 6. • FEM is based on the principle of discretization and piecewise polynomial interpolation. It first decomposes a geometrically complex domain into simple sub domains (each is called an element) and thecollection is called a grid or finite element mesh. The elements are connected to each other at pointscalled nodes. This process of generating the mesh, the elements, the corresponding nodes, and theboundary conditions together is known as discretization.(Fig. 1).Over each finite element, algebraic equations can be determined withthe help of governing equations of the problem. By assembling therelationships from all the finite elements using certain interelement. Fig: element and mesh
- 7. GENERAL STEPS IN FINITE ELEMENT METHOD We will only consider the structural issue to simplify the presentation of the subsequent steps. The engineer typically tries to figure out the displacements and stresses throughout the structure when it is in equilibrium and under applied loads for the structural stress-analysis problem. The finite element method must be used because conventional methods make it difficult to determine the distribution of deformation for many structures The steps for the same are:
- 8. Fig: Dividing structure into elements Step 1: Discretize and Select the Element Types. The process of dividing a structure into smaller parts, called elements, is called the discretization of a structure in the finite element method. A node is a point in space where coordinates are used to define degrees of freedom (DOFs). The DOFs for this point show the potential movement of this point as a result of the structure’s loading. This can be seen in the figure: Step 2: Select U Function (degrees of freedom) Degree of Freedom (DoF) refers to the “ability” to move in a specific direction. DoF in finite element analysis (FEA) also controls boundary conditions, provide information about stresses, and more. In this step, the U function is displacement. For each element, we can define a vector {u} that contains all of the possible displacements for the nodes of the element, including rotations.
- 9. Step 3: Determination of strain/displacement and stress/strain relationships The relationship between strain, stress, and displacement is required to derive the equations for each finite element. For example, for one-dimensional deformation in the x direction, the strain “εx” is associated with the displacement u by εx=du/dx if the strain is small. Furthermore, stress must be related to strain by the stress-strain law, commonly known as the law of materials. The ability to accurately define material behavior is paramount to achieving acceptable results. Hooke’s law is the simplest of the stress/strain laws and is commonly used in stress analysis. F: External force K: stiffness matrix of element U: displacement of each element ε: strain of each element σ: stress of each element E: Young modulus of material
- 10. Step 4: Derive the Element Stiffness Matrix and Equations The development of the element stiffness matrix and element equations is initially based on the concept of factors affecting stiffness, which requires a background in statics. There are alternative mehods Direct equilibrium Method, Work or energy Methods,Weighted residuals Methods etc. Step 5: Assemble the Element Equations to get the Global or Total Equations, and then add Boundary Conditions The individual element nodal equilibrium equations generated in step 4 are assembled into global nodal equilibrium equations in this step. The element stiffness matrix defines how much each node in the element will displace when a set of forces and moments is applied to the nodes. Figure shows just one element, but our overall mesh will be made up of many more elements. Figure: Stiffness matrix for one element
- 11. For example, in a 2D beam, we can assemble the individual stiffness matrices of all the elements (figure below) into a huge global stiffness matrix that defines how the entire structure will displace when loads are applied to it. Figure : Stiffness matrices for each element of the 2D beam
- 12. Like the element stiffness matrix, the global stiffness matrix is a square matrix, and the number of rows and columns is equal to the total number of degrees of freedom in the model. The element stiffness matrices are assembled together to form the global stiffness matrix based on how the elements are connected together. Figure shows that elements 1 and 2 are connected at node 2, and it tells us that since these two elements are connected at the same node, the displacement for both elements must be the same at the common node. Figure: Global stiffness matrix of a 2D beam element
- 13. Step 6: Determine the Unknown Degrees of Freedom The above equation, with the boundary conditions taken into account, is a set of simultaneous algebraic equations that can be written in expanded matrix form n is the structure’s total number of unknown nodal degrees of freedom. Step 7: Solution for the Element Strains and Stresses Because stress and strain can be directly expressed in terms of the displacements determined in step 6, important secondary quantities of strain and stress (or moment and shear force) can be obtained for the structural stress-analysis problem. Step 8: Analyze the Result The final objective is to use the results in the design/analysis process by interpreting and analyzing them. When making decisions for design or analysis, it is usually important to find the places in the structure where there are a lot of stresses and big deformations. Postprocessor computer programs aid in their interpretation by providing the user with a graphical representation of the results.
- 14. APPLICATIONS OF FINITE ELEMENT METHOD Both structural and nonstructural issues can be examined using the finite element method. The most typical structural areas are: • Stress analysis, including truss and frame analysis • Buckling, such as in columns, frames, and vessels • Vibration analysis, such as in vibratory equipment • Impact problems, including crash analysis of vehicles And nonstructural problems include: • Heat transfer, such as in electronic devices emitting heat as in a personal computer microprocessor chip • Fluid flow, including seepage through porous media • Distribution of electric or magnetic potential
- 15. BENEFITS OF USING FINITE ELEMENT METHOD • Designing bodies with irregular shapes is simple. • Handle stress/strain cases without trouble • Because each element equation is assessed separately, model bodies are made of various materials. • Manage an infinite variety of boundary conditions • Changing the size of the components to enable the usage of small components as necessary and implementing higher-order elements in the finite element model is possible. • Simulating many kinds of material qualities from element to element or even within an element is simple. • Is easy to use, small, and focused on results. • Take care of any nonlinear behavior resulting from significant deformations and materials by incorporating a finite element model. • A variety of computer software programs and books are readily available, making FEM a flexible and effective numerical method.
- 16. LIMITATIONS OF FINITE ELEMENT METHOD • The mesh used for nodal analysis requires a significant amount of data as input. • Idealization of real-life objects can’t be exact for complex shapes. • It is extremely involved and requires just a computer. • The time needed for solving the problems increases with the degree of fineness of the mesh. • The output results will vary considerably. FEM yields an approximate solution. It is tried to minimize the error over the whole domain; as a result, we get the exact solution at nodes only.
- 17. CONCLUSION This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications. The introduction of FEM has substantially decreased the time to take products from concept to the production line. It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated. In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.
- 18. REFRENCES ● https://caeassistant.com/blog/finite-element-method/ ● https://www.researchgate.net/publication/ 336987894_APPLICATIONS_OF_FINITE_ELEMENTS_METHOD_FEM_- AN_OVERVIEW ● https://savree.com/en/encyclopedia ● https://www.pexels.com/