Explicit Dynamic Analysis Fundamentals

What Is It?

Explicit dynamic analysis solves the equations of motion by advancing the state directly from the current time step to the next, without iteration. The acceleration is computed from the dynamic equilibrium using known forces; the velocity and displacement are then updated using simple time integration. The method is called "explicit" because the state at the next step is computed explicitly from the current state — no system of equations needs to be solved. This makes each time step computationally inexpensive but requires very small time steps for numerical stability.

Why It Matters

Explicit dynamics is the primary method for analysing impact, crash, blast, bird strike, metal forming and other problems involving severe contact, large deformation and material failure. These problems are difficult or impossible for implicit solvers because the frequent contact changes and element failure prevent convergence. Explicit solvers handle these conditions naturally — there is no iteration to fail. Understanding how explicit methods work is essential for credible analysis of rapid transient events.

Central Difference Integration

The explicit method typically uses the central difference time integration scheme. The acceleration at the current time step is computed from the dynamic equilibrium. The velocity at the half-step is updated from the previous half-step velocity and the current acceleration. The displacement is updated from the current displacement and the half-step velocity. This simple update requires no matrix factorisation — it is a vector operation using the diagonal (lumped) mass matrix.

Central difference explicit integration:

1. Acceleration (current step n):

   {ü_n}  =  [M]⁻¹ ( {F_ext_n} − {F_int_n} )

   [M] is diagonal (lumped) → inversion is trivial

2. Velocity (half-step):

   {ū_{n+1/2}}  =  {ū_{n-1/2}}  +  Δt · {ü_n}

3. Displacement (next step):

   {u_{n+1}}  =  {u_n}  +  Δt · {ū_{n+1/2}}

4. Advance:  n → n+1,  repeat

Dynamic Equilibrium

The explicit method solves the dynamic equilibrium equation at each time step. The external forces include applied loads and contact forces. The internal forces include element stresses converted to nodal forces. Damping may be included. The key feature is that the acceleration is computed directly from the known forces and the mass — no stiffness matrix factorisation is needed.

Dynamic equilibrium (explicit form):

[M]{ü}  =  {F_ext}  −  {F_int}  +  {F_damping}

where:
[M]         =  diagonal (lumped) mass matrix
{F_ext}     =  external forces (applied loads + contact)
{F_int}     =  internal forces (element stresses → nodal forces)
{F_damping} =  damping forces (if included)

[M]⁻¹ is trivial — diagonal matrix inversion

Why Diagonal (Lumped) Mass Benefits Explicit

The efficiency of the explicit method depends on the mass matrix being diagonal — a lumped mass matrix where each node has mass but no mass coupling between nodes. With a diagonal mass matrix, the inversion [M]⁻¹ is trivial — each acceleration component is simply the force divided by the nodal mass. With a consistent (non-diagonal) mass matrix, [M]⁻¹ would require a matrix solve at every time step, destroying the computational efficiency. The lumped mass approximation introduces a small error but is essential for the explicit method's speed.

Wave Propagation

Explicit dynamics naturally captures wave propagation — stress waves travelling through the structure at the material wave speed. When an impact occurs, a stress wave propagates from the impact point through the structure. The explicit time step must be small enough to resolve this wave propagation — the wave should not travel more than one element per time step. This is the basis of the stability condition. Wave propagation is physically important in impact and blast problems — the structural response is not instantaneous but develops as waves reflect and interact.

Impact → stress wave propagates from impact point → reflects at boundaries → interacts with other waves → global structural response develops over time

Energy Components

In an explicit dynamic analysis, energy is tracked across several components. Understanding these components is essential for verifying that the analysis is physically credible.

Quasi-Static Use of Explicit

Explicit solvers are sometimes used for problems that are physically quasi-static — no inertia, slow loading. This is done when the implicit solver cannot converge due to severe contact changes. In quasi-static explicit, the loading is applied slowly (relative to the structural natural frequencies) and mass scaling may be used to increase the stable time step. The key verification is that the kinetic energy remains small relative to the internal energy throughout the analysis — if kinetic energy is significant, the response is dynamic, not quasi-static, and the results do not represent the intended slow-loading condition.

EXPLICIT CHECK: For quasi-static explicit analysis, the kinetic energy should remain small relative to the internal energy throughout the analysis. If kinetic energy is significant, the response is dynamic — not quasi-static — and the results do not represent the intended loading condition.

Key Takeaways

• Explicit dynamics advances the state directly — no iteration, no matrix factorisation
• The central difference scheme with lumped mass makes each step computationally cheap
• Very small time steps are required for stability — the wave must not cross more than one element per step
• Explicit naturally captures wave propagation, severe contact and element failure
• Quasi-static explicit is possible but must be verified through the kinetic-to-internal energy ratio