Finite Element Method

Finite Element Method

1. General procedure

   • Discretise the structure or continuum into finite elements.
     • Specify the approximation equation.
       • Determine the shape functions for each element.
         • Formulate the properties of each element.
           • Determine stiffness matrices and equivalent load vectors for all elements.
             • Assemble elements to obtain the finite element model of the structure or continuum.
               • Apply the known loads
                 • Apply boundary conditions.
                   • Solve simultaneous linear algebraic equations
                     • DOF
                       • nodal displacements in stress analysis
                       • nodal temperatures in heat transfer analysis
                     • stress analysis
                       • Compute stress
                     • heat transfer analysis
                       • compute temperature
                         • Interpret results

2. Advantages of FEM

   • Model complex shaped bodies quite easily.
     • Handle several load conditions without difficulty.
       • Handle different kinds of boundary conditions.
         • Include dynamic effects.
           • Vary the size of the elements to make it possible to use small elements where necessary.
             • Handle time-dependent and time-independent heat transfer problems.

3. Engineering Applications

   • Mechanical Desing
     • Stress concentration problems
     • Stress analysis of pressure vessels
     • Composite materials
     • Linkages and gears
   • Electrical machines and electromagnetics
     • Steady state analysis of synchronous and induction machines
     • eddy current and core losses in electric machines
     • magnetostics
   • Biomedical engineering
     • Stress analysis of eyeballs
     • bones and teeth
     • load-bearing capacity in plant and prosthetic systems
     • mechanics of heart values
   • Nuclear engineering
     • Analysis of nuclear pressure vessels and containment structures
     • Steady state temperature distribution in reactor components

4. Degrees of freedom(DOF)

   • Specify the state of the element. They also function as ‘handles’ through which adjacent elements are connected. DOF are defined as the values (and possibly derivatives) of a primary field variable at nodal points

5. What is FEM?

   • FEM is a powerful tool for the numerical solution of a wide range of engineering problems.
     • 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

6. Boundary conditions

   • Avoid the possibility of the structure moving as a rigid body. Two approaches are used for handling specified displacement boundary conditions.
     • Elimination approach
     • Penalty approach

7. Variational Formulation

   • The Total Potential Energy Functional (TPE)
     • Relation between strain energy density and external energy
   • The Minimum Potential Energy Principle (MPEP)
     • states that the actual displacement solution u(x) is that which renders Pi stationary

8. ANSYS

   • It is one of the finite element analysis computer programming software.
     • ANSYS-Mechanical product is designed for analysing
       • static
       • structural and thermal
       • linear and non-linear
       • buckling
       • sub-structure
       • acoustics
       • dynamic/transient
   • It enables users to solve a wide variety of analyses in mechanical engineering applications

9. Finite Element Discretisation

   • Decompose the TPE functional into a sum of contributions of individual elements.
     • The same decomposition applies to the internal and external energies, as well as to the condition of MPEP
       • Using the fundamental lemma of variational calculus, it can be shown that implies that for a generic element (e) we may write:
         • This variational equation is the basis for the derivation of element stiffness equations once the displacement field has been discretised over a bar element.

10. By César Pineda Leonardo

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