What Every Engineer Needs to Know: Selection of Implicit/Explicit Algorithm

Bangalore,  January 29, 2025

Read time: 5 – 7 minutes

Target audience: CAE Analysts, Simulation engineers, Researchers, etc.

As a simulation engineer, one of the most important assessments to make are regarding the nature of the problem, whether static or dynamic, whether linear or nonlinear, and which algorithm to choose – implicit or explicit.

In both explicit and implicit time integration procedures, equilibrium is defined in terms of the external applied forces, P, the internal element forces, I, and the nodal accelerations, u ̈:

where M is the mass matrix.

Implicit Algorithm

Implicit solvers in Abaqus are particularly well-suited for static and quasi-static analyses, where the primary goal is to determine the equilibrium state of a system under applied loads. These solvers use a Newton-Raphson iterative approach to solve the system of nonlinear equations that arise from the finite element discretization. This method involves iteratively updating the solution until convergence criteria are met, ensuring that the equilibrium conditions are satisfied. The key advantage of implicit solvers lies in their ability to efficiently handle simulations involving slow dynamic responses or quasi-static problems by allowing larger, stable increments in the time step size. This makes them ideal for problems where high-frequency inertial effects are negligible.

Abaqus/Standard is the implicit solver, and it uses automatic time incrementation based on the full Newton iterative solution method. Newton’s method aims to satisfy dynamic equilibrium at the end of each increment, at time 𝑑+Δ𝑑, and computes the displacements at that point. The time increment, Δ𝑑, is relatively large compared to the explicit method because the implicit scheme is unconditionally stable. For nonlinear problems, each increment typically requires multiple iterations to achieve a solution within the specified tolerances. During each Newton iteration, a correction, 𝑐𝑗, is computed for the incremental displacements, Δ𝑒𝑗. This involves solving a set of simultaneous equations:

Solving these equations can be computationally expensive for large models. The effective stiffness matrix, , is a combination of the tangent stiffness matrix and the mass matrix for each iteration.

The iterative nature of the Newton-Raphson method can lead to convergence issues, especially in highly nonlinear problems or when dealing with complex contact interactions. These solvers require the computation of the global stiffness matrix and its subsequent inversion, which can be computationally expensive for large-scale models. Despite these challenges, implicit solvers remain a powerful tool for engineers, offering robust solutions for a wide range of applications where precision and stability in reaching equilibrium are crucial.

Β Explicit Algorithm

Explicit solvers in Abaqus viz. Abaqus/Explicit are designed for dynamic analyses, where capturing the transient response of a system is essential. These solvers use an explicit time integration scheme, which calculates the state of the system at a future time step directly from the current state, without the need for iterative solutions.

Abaqus/Explicit uses a central difference rule to integrate the equations of motion explicitly through time, using the kinematic conditions at one increment to calculate the kinematic conditions at the next increment. At the beginning of the increment the program solves for dynamic equilibrium, which states that the nodal mass matrix, M, times the nodal accelerations, , equals the net nodal forces (the difference between the external applied forces, P, and internal element forces, I), :

Since the explicit procedure always uses a diagonal, or lumped, mass matrix, solving for the accelerations is trivial; there are no simultaneous equations to solve. The acceleration of any node is determined completely by its mass and the net force acting on it, making the nodal calculations very inexpensive.

The accelerations are integrated through time using the central difference rule, which calculates the change in velocity assuming that the acceleration is constant. This change in velocity is added to the velocity from the middle of the previous increment to determine the velocities at the middle of the current increment. The velocities are integrated through time and added to the displacements at the beginning of the increment to determine the displacements at the end of the increment.

Thus, satisfying dynamic equilibrium at the beginning of the increment provides the accelerations. Knowing the accelerations, the velocities and displacements are advanced explicitly through time. The term explicit refers to the fact that the state at the end of the increment is based solely on the displacements, velocities, and accelerations at the beginning of the increment.

This approach is particularly advantageous in simulations involving high-speed dynamic events such as impacts, explosions, and wave propagation. The ability to handle large deformations, complex contact scenarios, and nonlinear material behaviour without the convergence issues associated with implicit methods makes explicit solvers highly effective for these applications.

Implicit vs Explicit Methods

In finite element analysis (FEA), both implicit and explicit methods offer different advantages depending on the problem’s nature. The implicit method is suited for simulations with slow, steady processes or complex interactions that allow for larger time steps. In contrast, the explicit method is better for highly dynamic, short-duration events, such as impacts or explosions, where small time increments ensure accuracy. Choosing between the two depends on factors like the problem’s time scale, nonlinearity, computational efficiency and type of application. In the implicit procedure a set of linear equations are solved by a direct solution method. The computational cost of solving this set of equations is high when compared to the relatively low cost of the nodal calculations with the explicit method.

Given below are some of the scenarios and which algorithm to choose for them.

  1. Static structural analysis – Implicit – Handles equilibrium conditions well and allows larger time steps.
  2. Quasi-static analysisImplicit / Explicit – Explicit solvers handle large deformations and complex contacts, while implicit solvers efficiently solve smooth, stable equilibrium conditions.
  3. Dynamic analysis with high-speed impacts – Explicit – Captures transient responses accurately with an explicit time integration scheme.
  4. Large deformations and nonlinear material behaviour – Explicit – Avoids convergence issues and effectively handles complex contact scenarios.
  5. Long-term load application – Implicit – Suitable for equilibrium analysis over long periods with computational efficiency.
  6. Crashworthiness simulations – Explicit – Handles large deformations and high-speed dynamic events without convergence issues.
  7. Wave propagation – Explicit – Captures the transient behaviour of waves accurately due to the explicit time integration.
  8. Structural stability analysis – Implicit – Provides robust solutions for evaluating the stability of structures under static loads.
  9. Simulations involving blast loads – Explicit – Effectively captures the dynamic response of materials and structures to explosive forces.
  10. Seismic analysis of structures – Explicit – Accurately captures the transient response of structures under seismic loads.

The explicit method offers notable cost advantages over the implicit method as the model size grows, given that the mesh stays relatively uniform. The figure below shows a cost comparison between the explicit and implicit methods as the model size increases. Here, the number of degrees of freedom scales with the number of elements.

The advantages of Implicit and Explicit methods are listed below:

Implicit method: –

  1. Superior stability for static and quasi-static analyses, allowing larger time steps.
  2. Strong convergence properties for nonlinear problems due to iterative correction of displacements.
  3. Provides high accuracy in steady-state and low-frequency response scenarios.
  4. Capable of accurately modelling complex boundary and initial conditions.
  5. Better handling of damping effects in simulations.
  6. Well-suited for simulations involving complex contact interactions under steady-state loading.

Explicit method: –

  1. Faster computation for dynamic problems with a direct integration approach.
  2. Eliminates the need for iterative solution processes, resulting in faster convergence per time step.
  3. Handles complex material behaviour and large deformations effectively.
  4. Robust in dealing with dynamic contact and impact scenarios.
  5. Well-suited for transient loading conditions and nonlinear dynamic analyses.
  6. Stability constraints are less stringent for certain applications.

The explicit method easily accommodates contact conditions and other highly discontinuous events, allowing them to be enforced on a node-by-node basis without the need for iteration. Nodal accelerations can be modified to ensure equilibrium between external and internal forces during contact.

Abaqus/Explicit requires less memory for large models since it does not assemble global stiffness matrices, benefiting massive simulations. For problems in which the computational cost of the two programs may be comparable, the substantial disk space and memory savings of Abaqus/Explicit make it attractive.

Having said that, there are solver exclusive analyses which can only be performed with either Implicit or Explicit methods. They are:

Abaqus/Standard or Implicit:-

  1. Other analysis types than mechanics and thermal, such as acoustics
  2. Higher order elements
  3. Frequency analysis
  4. X-FEM (Extended finite element method)
  5. Pressure penetration

Abaqus/Explicit:-

  1. CEL (Coupled Eulerian Lagrangian)
  2. SPH (Smoothed Particle Hydrodynamics)
  3. DEM (Discrete Element Method)

Choosing the right method (implicit or explicit) in FEA is crucial for balancing computational efficiency and accuracy, as it directly impacts the stability, convergence, and runtime of the simulation based on the problem’s nature.

Abaqus utilizes a token-based licensing system to efficiently manage computational resources across simulations. Tokens are shared between implicit and explicit solvers, allowing flexible usage tailored to diverse analysis requirements. In HPC environments, token allocation is optimized via job queuing and cluster distribution for scalability and efficiency. Both implicit and explicit solvers, scale effectively across multiple CPUs and GPUs, significantly reducing simulation runtimes for large models.

Get in touch with us to know more about Abaqus and its powerful simulation capabilities.

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