Test Your Foundation in Finite Elements

DEFINE THE FOLLOWING:

1. What is the purpose of discretizing (meshing) a continuum?

2. What are the major assumptions in the design of a finite element? What are the assumptions on displacements between nodes?

3. How are finite-element equations formulated?

4. What is the primary unknown in FEA using the displacement method? What are the secondary unknowns?

5. What is a nodal degree of freedom or DOF?

6. What is the relation between the total number of degrees of freedom in a model and the total number of unknowns in the FEA model?

7. Is the total number of DOF in the model equal to the total number of elements, or to the total number of nodes?

8. What is special about the state in which the total potential energy of the model is minimized?

9. What are the components of the total potential energy?

10. Write and explain the fundamental equation for FEA.

11. What is the price to pay for replacing a continuum with a set of finite elements?

ANSWERS

1. What is the purpose of discretizing (meshing) a continuum?

A continuous body has an infinite number of points and, consequently, an infinite number of degrees of freedom. To completely describe the deformation of a continuous body under load, we would need to know displacements at all its points.

This problem is unsolvable unless we can find an analytical solution. The finite-element method allows solving problems for which analytical solutions are difficult or unknown. Using the finite element method we make a compromise for the sake of solving the problem. We say we’ll be happy if we can find displacements only for selected points (call them nodes) in the body. Furthermore, we decide that displacements of any other point (one that is not a node) will be found by interpolating these nodal displacements. This way we have reduced the problem to finding a finite number of nodal displacements.

2. What are the major assumptions in the design of a finite element? What are the assumptions on displacements in-between nodes?

The fundamental assumption in the formulation of a finite element is that everything there is to know about the element is determined by displacements of nodes belonging to that element. Once we know nodal displacements (three translations and three rotations in a 3D structural analysis; temperature in thermal analysis) we can calculate displacements or temperatures any point inside the element or along it’s edge. Formulae with instructions on how to perform these calculations are called shape functions.

Shape functions in most cases are polynomials because element designers find them easy to work with. Simplifying a bit we can say that if a first order polynomial is used, displacement anywhere in the element is a linear function of nodal displacement. If a second-order polynomial is used, displacements are quadratic functions of nodal displacements, and so on.

3. What are basic steps in FEA ?

Step 1: Starting point

A supported body subjected to loads has an infinite number of degrees of freedom. Therefore, an infinite amount of information is required to describe the body’s displacements.

Step 2: Discretize a continuum

We decide that we do not need to know displacements at every point and will be satisfied with displacements only at certain nodes. Based on these displacements we’ll figure out behavior of the entire structure. This way the problem has been discretized. A finite amount of information is now sufficient to describe a body’s displacements under the load.

Step 3: Calculate displacements in-between nodes

Now we divide the body into finite regions (we will call them elements) based on the previously selected nodes. Those elements can be of any shape, but triangles and quadrilaterals (or tets and bricks in 3D) are easiest to work with. Having divided the body into finite elements we now formulate expressions to describe displacements of any point in the element based on displacements of nodes belonging to this element. When done, we will have described a discrete field of displacements anywhere in the body consisting of so-many finite elements.

Step 4: Find nodal displacements by first formulating an expression for total potential energy with nodal displacements as unknowns. Using the principle of minimum total potential energy (It says the state of minimum total potential energy is also the state of equilibrium.) we minimize that expression and find a set of nodal displacements that minimizes the total potential energy. It also corresponds to the new state of equilibrium.

Step 5: Find stresses and strains

After finding all nodal displacements, we can find displacements anywhere else. Then, strains can be found (if necessary) as derivatives of displacements. Stresses are found based on strains and material properties.

4. What is the primary unknown in FEA using the displacement method? What are secondary unknowns?

The primary unknown in FEA using the displacement method is nodal displacement. Depending on the type of analysis such as structural 2D, 3D, or thermal, the nodal displacement can be the actual displacement in x ,y, z, rotation rot_x, rot_y, rot_z, or temperature.

Secondary unknowns are strain, stresses, temperature gradient, and heat flux, in short, everything that can be calculated using nodal displacements.

5. What is a nodal degree of freedom or DOF?

The nodal degree of freedom is an unknown assigned to a node. Physical interpretation would be displacement or temperature. In general 3D structural analysis, each unconstrained node has six degrees of freedom, three translations and three rotations. In thermal analysis, each node has just one degree of freedom, temperature. By defining supports at certain nodes, some degrees of freedom are eliminated or assigned a prescribed value.

6. What is the relation between the total number of nodal degrees of freedom in an FEA model and the total number of unknowns?

For an unconstrained (unsupported) model, the total number of degrees of freedom equals the total number of unknowns. By defining supports, some nodes are assigned displacements. For example, assigning x = 0, y = 0, z = 0, rot_x = 0, rot_y = 0, rot_z = 0 defines a rigidly supported node. More constraints on the model mean there are fewer unknowns to find.

7. Is the total number of DOF in a model equal to the total number of elements, or equal to the total number of nodes?

Neither is correct. The total number of DOF in the model is equal to the total number of nodes times the number of DOF for each node. Therefore, a model with 100 nodes and three DOF at each node has 300 DOF. But not all DOF constitute unknowns because some nodes are constrained or have arbitrarily assigned displacements or temperatures.

8. What is significant about the minimum value of the total potential energy?

The state of minimal total potential energy of an elastic body under load is also the state of equilibrium for the same body. Thus, in a finite-element model, equilibrium can be found by looking for the set of nodal displacements that minimize total potential energy.

9. What are components of total potential energy?

Components of total potential energy are strain energy accumulated in the deformed body and potential energy of external loads.

10. What are the fundamental equations of FEA?

[ F ] = [ K ] * [ d ]

where [ F ] = vector of nodal loads, [ K ] = stiffness matrix, and [ d ] = nodal displacements. These matrix equations are formulated by minimizing the total potential energy of the model.

11. What is the cost for replacing a continuum with a set of finite elements ?

The price to pay is accuracy. Replacing a continuum with discreet elements introduces what’s called discretization error.