Basics of finite element analysis (FEA) - 2

What kind of formulation is used in FEA?

A system of linear equations is created to solve for displacement at each node of each element. A vector {u} which contains all the possible displacements at the node of each element is formulated.

Before formulation, we need to know what is the meaning of displacement at a node. Consider following line element 1-2 which has 2 nodes. Each node has linear displacement in the x and y direction and each node has angular displacement theta. There are a total of 6 displacements u1, v1, u2, v2, θ1, and θ2. Each of these displacements is called a Degree of Freedom. For the following case, there are 6 degrees of freedom.

The displacement vector {u} for the above element is given by,

Displacement Vector

Depending on the type of element the number of linear and angular displacements varies.

What is Hooke's law?

The spring stiffness 'k' determines how far the spring gets displaced when force 'F' is applied to the spring.

F = -k.x

where k = stiffness of the spring

x = displacement of the spring

F = Force applied to the spring

This analogy is applied to each mesh element to formulate a linear system of equations.

{F}=[k]{u}

where F = nodal forces

k = stiffness matrix for mesh element

u = displacement vector

What is the global stiffness matrix?

All the individual stiffness matrices are assembled into one big stiffness matrix. The assembly of all the local (elemental) stiffness matrices into one big stiffness matrix is called the global stiffness matrix.

Why does a global stiffness matrix called a sparse and banded matrix?

When a global stiffness matrix is formed there are a large number of cells with zero value in the matrix (because there is no relation between corresponding nodes of those elements). The non-zero elements in the global stiffness matrix are usually along the diagonal of the stiffness matrix. For example,

Hence, the global stiffness matrix is called sparse and banded stiffness matrix.

The global stiffness matrix is solved by inversing in the matrix or using the conjugate gradient method.

Once the displacements at each node are known, the derived quantities like stress and strain are calculated using the following relations.

What are the methods to derive the stiffness matrix? Explain.

1. Direct Method (Strong Form)

2. Variation Method (Weak Form)

3. Galerkin Method of Weighted Residuals. (Weak Form)

What is a strong form and what is a weak form of differential equations?

The strong form of differential equations is nothing but a PDE with boundary conditions. The strong form requires continuity of field variables which is a very difficult task. The weak form is nothing but a mathematical manipulation to relax the strong form requirements. The weak forms are formulated using lesser degree polynomials which results in loss of continuity. The differential form is converted to integral form and then the linear system of equations is solved.

The weak forms are the reason why more number of mesh elements results in higher accuracy compared to a solution with a lower number of mesh elements.

Which is the widely used approach to derive a global stiffness matrix?

Galerkin Method of Weighted Residuals. (More details about the Galerkin method in a future post)

What is the procedure for doing FEA?

1. Define the problem -

• Define Materials

• Define Boundary Conditions

• Define Loads

2. Discretization/Meshing

3. Define the element stiffness matrix

• Direct Method

• Variational Method

• Galerkin Method

4. Assembly of the global stiffness matrix

5. Solve for displacement

6. Calculate variables of interest like stress, strain, etc

7. Post-processing and Validation

From the above steps, the ones in orange only have to be performed by users. Steps 3,4,5 are performed by FEA software and (usually) the user does not have to worry about what's going on inside the BlackBox.

Which softwares are currently available for FEA?

ANSYS, ABAQUS, LS-DYNA, COMSOL, NASTRAN, Simscale, Autodesk, etc