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Understanding dynamic simulation and explicit-implicit integration methods

By Ananth Sridhar 30 December 2019

We recently presented a webinar on Nonlinear Explicit Structural Analysis with OnScale Cloud Engineering Simulation. During the webinar we discussed the difference between implicit and explicit methods and the theory powering the nonlinear transient explicit core solvers of our software. We thought we would provide an overview of dynamic analysis and answer the most fundamental question here:

What does nonlinear transient explicit means?

First, let’s have a quick look at those three words. Nonlinear is an easy one to understand, it simply means “not linear”. Transient means dynamic time domain, so something which is dynamic and not static. Explicit refers to the usage of an explicit integration scheme.

Nonlinear Transient Explicit

To understand what nonlinear transient explicit is, you have to understand the different types of Finite Element Analysis (FEA) analysis available.

The 4 main FEA simulation categories

To make it easier we’ll divide FEA analysis in two main categories:

  • A simulation can be either linear or nonlinear.
  • And then it can be either static or dynamic.

Then you have combinations of both, you have linear static, linear dynamic, nonlinear static and nonlinear dynamic.


  • Linear static is the simplest type of simulation, it is generally used by designers to assess their new designs.
  • Linear Dynamic is more complex because it considers a dynamic input.
  • A nonlinear static is also more complex because you have the nonlinearities which start to come into play.
  • Now if you think about it, Nonlinear Dynamic is even more complex than that because both the nonlinearity aspects and the dynamic aspect involved.

What causes nonlinearity?

To understand what nonlinear explicit simulation is, you must first underhand nonlinearities.

Linearity is simple, because it states that the output results are directly proportional to the input. If you multiply the load by 2, the result is increase by 2 times. It is easy to understand, but it is in fact a big simplification because it is only valid for very small loads… The world is nonlinear!

Most of the physical phenomena around us are nonlinear and linearity is just a simplification of the world. There are three main types of nonlinearities which are considered when you do mechanical type of analysis:

  • Nonlinear Material
  • Nonlinear Contact
  • Large Deformations

Linear dynamic analysis overview

Let’s talk about linear dynamic analysis.

Let’s take a structure which has some mass and some stiffness, which are the basic characteristics of the structure. (The structure can represent simple mechanical parts, a building or it can be something larger as well.)

Dynamic Analysis

Prior to doing a dynamic analysis, it is important to grasp the natural characteristics of the structure. To do this we need to understand its natural frequencies and its normal mode shapes. For example, we would need to understand which frequency of vibration will cause deformation and how the structure vibrates. We can obtain both the natural frequencies and the mode shapes with a Modal Analysis, which will help us to prepare a more advanced dynamic time or frequency response analysis.

Natural FrequenciesOnce we have the results of this modal analysis, what we want to know next is how a structure behave under a certain chosen input dynamic load: That’s the purpose of the linear dynamic simulation.Dynamic LoadWhen we talk about linear dynamic the structure itself is linear. So, it doesn’t have the three types of nonlinearity that I presented before. The dynamic component is due mainly to the dynamic nature of the loading. There are four types of linear dynamic analysis, which can be generally considered: Transient response, Frequency response, Spectrum response or Random analysis.

AnalysisLooking at the response, we would generally obtain different kinds of results, including acceleration, energy, velocity, reaction force, displacement and stresses.Dynamic Input

How to integrate the dynamic mechanical equations?

Now that we understand the general process of setting up a dynamic analysis and what the important components are (Dynamic loads, Structure, Analysis Type and Outputs), let’s look at the numerical methods which are implemented in the software to “Integrate” the dynamic mechanical equations and provide the results we are interested in.

There are two ways of integration of the mechanical dynamic equations: Direct or Indirect.

You can either go for the Direct Integration Method (1) which means that you look at the equations of dynamic motions from the second law of Newton and you basically integrate them numerically directly at each time step or for each frequency.

The second way is to use the result of the modal analysis and construct a new base of solution from each of the mode shapes. You can think about this kind of solution as a linear combination of the normal modes of vibration. This is called the Modal Superposition Method (2).

We have an on-demand webinar available on this topic. Click here to check it out!

The difference is that the modal superposition method is linear while the direct integration method can be linear or nonlinear (but it’s also more time consuming).

There are also two ways to do go for a direct integration: you can either do a transient time domain response analysis or you can do a frequency response analysis.

The frequency domain solution is some kind of simplification of the time domain results because to get into the frequency domain you have to make the assumption that the time component of your equation is uncoupled from the other spatial variables of the equation.

There is also a third type under the modal superposition method which is used mainly for seismic analysis: Shock and response analysis using a response spectrum.

Dynamic AnalysisThe Transient response analysis can be integrated using either an implicit or an explicit scheme, both are possible. To understand the difference between those 2 integration methods, we must go into the theory… we talk about it in the webinar and in the next articles.

What makes nonlinear transient explicit so special?

Nonlinear explicit dynamic analysis is very efficient for fast-dynamic applications, such as:

  • Impact analysis of a projectile
  • Crash or drop electronic equipment
  • Nonlinear buckling with large deformations

In OnScale, this method is used to solve piezo-acoustic-mechanical coupled dynamic problems for ultrasonic sensors, RF Filters, Non-destructive Testing and many more very specialized applications.

Check out our on-demand webinar to find out more about explicit and implicit integrations!

Ananth Sridhar
Ananth Sridhar

Ananth Sridhar is an Application Engineer at OnScale. As part of our engineering team he assists with developing applications, improving our existing software and providing technical support to our customers.

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