Comparing the Explicit and Implicit Methods in FEA

In Ansys, the analysis settings play a very important role in converging the solution and obtaining the results. These involve settings about the timestep size, solver type, energy stabilization etc. 

In this blog, we will look at the meaning and significance of these analysis settings. But to understand them, we have to first understand the difference between the explicit method and the implicit method in Finite Element Analysis (FEA).

To understand these two methods, let us consider an example. Suppose a person is traveling in a car from Delhi to Agra. The distance is approximately 200 km. The person takes 4 hours to travel the entire distance. Now if the person calculates the speed at the end of the journey, it will be 50 km/hr.

This value is the average speed during the entire period of 4 hours. However, from the experience, we know that the actual speed at a particular point of time (the instantaneous speed) is different from this average value. 

For example, at the start of the journey, the person is traveling through the Delhi traffic. At this point of time, the speed is very low, suppose 20 km/hr. But after 2 hours, the person is traveling along the expressway, where speed is very high, say 100 km/hr. Again when the person is close to Agra, the speed reduces due to traffic.

This example illustrates that there is a difference or error between the average speed and the instantaneous speed. 

But now suppose the person measures the speed at every interval of 15 minutes instead of at the end of the journey. For this, the distance traveled in those 15 minutes will be divided by the time value i.e. 15 minutes.

Suppose the journey starts at 10 AM. During the 10 AM - 10:15 AM interval, the speed is measured. Let’s say that the average speed in this interval comes out to be 15 km/hr. This is closer to the instantaneous speed, as the person is driving through traffic. 

If we consider the 12 PM - 12:15 PM interval, the average speed measured comes out to be 90 km/hr. This is also closer to the instantaneous speed. 

This shows that as the time interval of measurement reduces, the average speed and instantaneous speed come out to be closer to each other.

This means that as the time interval of measurement reduces, the error also reduces. This is the basic idea of FEA.

Explicit Method

Let’s consider that in an FEA problem, the duration value is 10 seconds. We have seen that as the time interval of measurement reduces, the error also reduces. Hence in the explicit method, the time duration of 10 seconds is divided into miniscule intervals, in the order of milliseconds or even microseconds. So, if the size of the interval is 1 ms, the entire duration of 10 seconds is divided into 10000 steps.

Each such interval is called a “timestep”. The solver calculates the answers at the end of each timestep. The answers at the end of a particular timestep act will be taken as the input to calculate the answers for the next timestep. 

For example, let’s say that each timestep is of 1 ms duration. Suppose the speed of an object in the 1st time step is 5 mm/ms. Hence the distance traveled by the object in the 1st time step will be 5 mm. In the 2nd timestep, if the speed is 7 mm/ms, the distance traveled in the 2nd time step will be 7 mm. Thus the total distance traveled by the object within these two timesteps will be = 

(Distance traveled in the 1st time step + Distance traveled in the 2nd timestep) = 5+7 = 12 mm.

Thus, the result at the end of the 1st time step was used to calculate the result at the end of the 2nd timestep.

By using this technique, the answers at the end of each timestep are calculated and the problem is completed. 

Thus, the explicit method is a step-wise process, much like climbing a staircase. One can climb a step only by climbing the step before it. But here, the size of the step is very small to reduce the error.

Implicit Method

On the other hand, the implicit method is like an elevator, which takes a person directly from the ground floor to the 5th floor.

In the implicit method also, the problem is divided into multiple steps. Here, the steps are referred to as Load steps, which are not very small in size.   

Let us assume that the duration is 10 seconds. In the explicit method, this time duration is divided into the time steps of the order of milliseconds or even microseconds. 

But in the implicit method, the load step is of the order of seconds i.e., the 10 seconds can only be divided into 10 load steps of 1 second each. So there will be only 10 load steps as compared to 10000-time steps in the explicit method.

Since the duration of a load step in the implicit method is larger, the error will also be larger. Hence at this point, the implicit method uses a mathematical formula called Newton-Raphson method to reduce this error. 

But how much should the error be reduced? This is given by a criterion called tolerance. The Newton-Raphson method will try to reduce the error below this tolerance value. The tolerance value can be decided by the user, based on the accuracy level desired. The tolerance can be specified for the values of quantities like force, total energy, displacement etc.

These operations to reduce the error take place in a series of iterations. Hence, the implicit method is also called an iterative method. However, the mathematics of the Newton-Raphson method is not the topic of discussion of this blog. Let’s go back to establishing the difference between the two methods. 

Let’s say that the tolerance criterion of 1% has been specified for total energy value. This means that the error in the values of total energy should be less than 1%. Now suppose, in a particular problem, the total energy value is supposed to be 100, but the solver calculated it to be 110. Now, the Newton-Raphson iterations start to take this total energy value below 101. This might happen in the manner shown below:

Thus after the 4th iteration, the error value has been reduced below the tolerance value. When this happens, it is said that the solution has converged. The solver will record 100.9 as the final value of total energy. The corresponding values of other physical quantities are also recorded as the final answer.

The above process takes place for every load step. When each load step converges, it is said that the entire solution has converged, and the results can be observed.