time-integration-techniques-analysis

Time Integration Schemes: Explicit vs. Implicit Dynamics

Time Integration Schemes: Explicit vs. Implicit Dynamics

Understanding the differences between explicit and implicit time integration schemes is crucial for selecting the appropriate approach for simulations.

Time-integration-scheme-implicit-explicit

Explicit Dynamics:

  • Uses current time step data to determine the system’s state in the next step
  • Efficient per step, avoiding complex equation solving
  • Requires small time steps for stability, especially in high-stiffness systems
  • Conditionally stable, ideal for high-speed events like crashes or explosions

Implicit Dynamics:

  • Solves equations incorporating both current and next step data
  • Computationally intensive per step but allows larger time increments
  • Unconditionally stable, regardless of step size
  • Suitable for slow dynamic processes and structural vibrations

The choice between explicit and implicit methods depends on the nature of the problem. High-speed, transient events favor explicit methods, while slow, stable processes benefit from implicit methods.

time-integration-techniques-analysis

Computational Efficiency

Explicit methods require less computational effort per time step due to their direct calculation approach. They avoid the complex system of equations needed in implicit methods, making each step quick and efficient.

However, explicit methods necessitate many tiny time steps, especially for simulations involving stiff systems or high-strain events. This can increase the total computational cost and extend simulation times.

Implicit methods involve substantial computational effort per time step, solving complex, often nonlinear, equation systems. They utilize techniques like the Newton-Raphson method to iteratively converge on a solution. However, their ability to take larger time steps compensates for this complexity, reducing the total number of steps needed to span the same time period.

Implicit methods are also unconditionally stable for linear problems, enhancing their efficiency for slow, gradual processes. They manage to cover significant time intervals without compromising stability, making them ideal for simulations where long-term robustness is paramount.

“In summary, explicit methods excel in scenarios requiring rapid precision and minimal computational effort per step, while implicit methods leverage larger time increments and unconditional stability to handle slow, prolonged processes efficiently.”

Time Step Size and Stability

The time step size significantly influences the stability of dynamic simulations using explicit and implicit methods.

In explicit dynamics, stability is highly sensitive to the time step size. There’s a critical limit on how large each time step can be, known as conditional stability. This “safe” time step size is derived from the Courant-Friedrichs-Lewy (CFL) condition, ensuring that information can travel through the smallest element in the model within one time increment. Exceeding this critical value can lead to unstable or divergent results.

Implicit dynamics offers unconditional stability, maintaining stability regardless of the step size. This allows for much larger time steps compared to explicit methods, making implicit dynamics well-suited for simulations involving slower processes or long-term events.

For example:

  • In a car crash simulation using explicit dynamics, the time step needs to be minuscule to capture high-frequency events accurately.
  • A building structural vibration simulation under wind loads using implicit dynamics can employ larger time steps to cover extended periods without sacrificing stability or accuracy.

Understanding these implications ensures that engineers can choose the appropriate dynamic method, balancing precision, computational effort, and simulation duration to achieve optimal results.

explicit-vs-implicit-time-integration

Simple, no iteration required. Time-step limited, i.e. is conditionally stable. Good for short-duration phenomena (clashing & snatching) Conditional stability  ’guaranteed’ accuracy. Implicit Integration: Xn+1 – Xn = Dt . F(Xn+1) Stable, large time-steps, but requires iteration. Can lose accuracy,at large time steps – this needs to be assessed. Can miss short-duration phenomena (contact, axial waves, etc) An implicit integration scheme for OrcaFlex is under development.

Application Scenarios

Explicit dynamics are well-suited for applications involving high-speed, high-strain events that require capturing rapid changes within systems. Examples include:

  • Crash Simulations: In automotive safety engineering, explicit dynamics model car crashes, capturing extreme deformations and stress wave propagation through vehicle structures.
  • Blast Analysis: Simulations of explosions or blasts, where shock waves travel at high velocities through different media.
  • Ballistic Events: Military and aerospace applications simulating the effects of projectiles or ballistic impacts.

Implicit dynamics are more suitable for applications involving slow dynamic processes or scenarios where long-term integration is required:

  • Structural Vibrations: Civil engineering applications studying the response of buildings and bridges to dynamic loads such as wind, earthquakes, or operational machinery over extended periods.
  • Quasi-static Simulations: Problems where loads are applied slowly, such as progressive loading of structures or gradual application of pressure on surfaces.
  • Long-term Environmental Loading: Assessing the cumulative impact of environmental factors like temperature fluctuations, moisture, or creep in materials over extended periods.

Choosing the appropriate dynamic method based on the specific requirements of a simulation task ensures optimal performance and accuracy in achieving reliable results.

Dynamic vs. Quasi-static Simulations

Dynamic simulations are essential for purely dynamic problems characterized by inertia effects and high-speed phenomena. These time-dependent simulations accurately capture inertia-induced forces and rapid changes within systems, making them ideal for events like car crashes, explosions, or ballistic impacts.

Quasi-static simulations, or time-independent simulations, are used when inertia effects are negligible. They are suitable for scenarios where loads are applied slowly, and dynamic effects like acceleration and wave propagation do not significantly influence the system’s response. Examples include gradual loading of structures or slow deformation processes.

Implications for simulations:

  1. Accuracy and Relevance: Dynamic simulations are crucial for capturing high-frequency stress wave propagations and rapid deformations. Quasi-static methods are more effective for slow, gradual loading processes.
  2. Computational Demand: Dynamic simulations, particularly those using explicit methods, require significant computational resources due to tiny time steps. Quasi-static simulations are generally more resource-efficient for long-term or slow processes.
  3. Modeling Complexity: Dynamic simulations often demand detailed material models that accommodate rate-dependent behaviors and strain-rate effects. Quasi-static models simplify these requirements, focusing on material and geometric properties under near-equilibrium conditions.

Selecting the appropriate method—dynamic for high-speed, inertia-sensitive phenomena, or quasi-static for slow, inertia-negligible scenarios—ensures accurate, reliable results while optimizing computational resources.

A comparison between dynamic and quasi-static simulations, showing the difference in time-dependency and loading conditions

By understanding the distinct advantages of explicit and implicit time integration methods, engineers can adapt their strategies to match the specific demands of their projects, ensuring both precision and reliability in their results. The choice between these methods can significantly impact simulation accuracy, computational efficiency, and overall project success1.

  1. Hughes TJR. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Courier Corporation; 2012.