General steps of the finite element method

General Steps used to solve FEA/ FEM Problems. Steps Involves involves dividing the body into a finite elements with associated nodes and choosing the most appropriate element type for the model.

1. General Steps of the Finite Element Method

2. Step 1: Discretize and Select Element Types Step 1 involves dividing the body into a finite elements with associated nodes and choosing the most appropriate element type for the model. (In this step the geometry is divided in a no. of parts and parts are known as element. And Elements are interconnected at points called as Nodes.) The Process of dividing the geometry into finite no. of elements is known as discretization. Element Types: 1D : Bar, Beam, Pipe, Spar (web) 2D: Triangular, Quadrilateral 3D: Tetrahedron, Hexahedron (Brick Element) 2 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

3. 3 Step 1: Contd.. Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

4. Additional Knowledge of Elements 4 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

5. Step 2: Identify Primary Unknown Quantity 5 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

6. Step 3: Select a Proper interpolation or displacement model Since the displacement solution of a complex structure under any specified load conditions cannot be predicted exactly, we assume some suitable solution within an element to approximate the unknown solution. The assumed solution must be simple from a computational standpoint, but it should satisfy certain convergence requirements. In general, the solution or the interpolation model is taken in the form of a polynomial. 6 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

7. Step 4: Derive the Element Stiffness Matrix and Equations On FEA it is necessary to find out primary unknown at nodal points (nodes) and this can be done by establishing equation in terms of primary unknown. From the assumed displacement model, the stiffness matrix and load vector of element are to be derived by using following method Direct Equilibrium Method (used for 1D Problems) Work Energy Method (Used for 2D and 3D Problems) Method of Weighted Residual. (Used for 1D Problems) 7 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

8. Step 5: Derive overall Stiffness Equation (Assemble the Equation) The individual element equations generated in step 4 can now be added together using a method of superposition (called the direct stiffness method). This step assembles all individual element equations derived in Step 4 to provide the “Stiffness equations” for the entire medium. Mathematically, this equation has the form: {F} = [K] {U} Where, {F} = Vector of global nodal forces [K] = Stiffness matrix {U} = Primary Unknown 8 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

9. Step 6: Solve Primary Unknowns {F} = [K] {U} After introducing Boundary Conditions and Forces, Primary Unknown (like Displacement) can be solved by using Gaussian Elimination method or inverse matrix method. 9 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

10. Step 7 Solve for secondary unknowns For the structural stress-analysis problem, important secondary quantities of strain and stress (or moment and shear force) can be obtained because they can be directly expressed in terms of the displacements determined in step 6. 10 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

11. Step 8 Display and Interpretation of Results The final goal is to interpret and analyze the results for use in the design/analysis process. Determination of locations in the structure where large deformations and large stresses occur is generally important in making design/analysis decisions. Postprocessor computer programs help the user to interpret the results by displaying them in graphical form. 11 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)

12. Keep in mind that the analyst must make decisions regarding dividing the structure or continuum into finite elements and selecting the element type or types to be used in the analysis (step 1), the kinds of loads to be applied, and the types of boundary conditions or supports to be applied. The other steps, 2 through 7, are carried out automatically by a computer program. 12 Mr. Mahesh Gaikwad (Mechanical Department SGGSIE&T Nanded)