SPH, Lagrangian & Coupled Impact Modelling

What Is It?

SPH (Smoothed Particle Hydrodynamics), Lagrangian, Eulerian and ALE (Arbitrary Lagrangian-Eulerian) are different discretisation methods used in impact and extreme-deformation analysis. The Lagrangian mesh moves with the material — standard FEA. SPH represents material as particles that move and interact without a mesh. Eulerian methods fix the mesh and let material flow through it. ALE combines both — the mesh moves but can also relax. The choice of discretisation method determines how well the analysis can handle extreme deformation, particularly for soft projectiles in impact problems.

Why It Matters

In a bird-strike or other soft-projectile impact, the projectile deforms so severely that a Lagrangian mesh becomes unusable — the elements distort beyond recognition, the time step drops to near zero, and the analysis fails. SPH or Eulerian methods can handle this extreme deformation because they do not rely on a connected mesh that must maintain its shape. Choosing the right discretisation method is essential for practical analysis of extreme-deformation problems. The wrong choice leads to analyses that fail to run or that produce meaningless results due to mesh distortion.

Choose the discretisation method around the expected material deformation. Lagrangian for moderate deformation; SPH for severe deformation of soft materials; coupled SPH-Lagrangian for soft projectile on structural target.

The Lagrangian Mesh

The Lagrangian mesh is the standard FEA mesh — nodes are connected by elements, and the mesh moves with the material. Each node tracks the material that was originally at that location. The advantage is that material boundaries are clearly defined by the element edges, and the method is efficient for moderate deformation. The disadvantage is that under severe deformation the elements distort — they can become degenerate, with near-zero volume or extreme aspect ratios. Distorted elements have poor accuracy, a tiny stable time step and can cause the analysis to fail.

The Extreme Distortion Problem

When a Lagrangian element distorts severely, several problems arise. The element Jacobian (the determinant of the deformation gradient) can become zero or negative — the element inverts. The stable time step, which depends on the element characteristic length, becomes very small — the analysis slows dramatically. The element stiffness becomes ill-conditioned — the results are unreliable. The element may need to be deleted, creating a hole in the mesh. For soft projectiles like birds, the deformation is so severe that Lagrangian elements cannot survive the impact — the method fails.

Smoothed Particle Hydrodynamics (SPH)

SPH is a mesh-free method that represents material as a set of particles. Each particle has mass, position, velocity and material state. The particles interact through a kernel function — a mathematical function that defines how each particle influences its neighbours. The material properties at any point are approximated by summing contributions from nearby particles. Because there is no mesh connecting the particles, there is no mesh to distort — the particles can move freely and separate without any connectivity problems. SPH is well suited to severe deformation, fluid-like behaviour and fragmentation.

SPH concept:

Material represented as particles (no mesh)
Each particle: mass, position, velocity, material state
Particles interact through kernel (smoothing) function
Material property at any point ≈ weighted sum of nearby particles

Advantage: no mesh to distort under severe deformation

SPH for Soft Projectiles

SPH is particularly well suited to soft projectiles like birds, gelatine blocks or ice. The projectile material undergoes extreme deformation — spreading, flowing, fragmenting — that would destroy a Lagrangian mesh. SPH particles can spread, flow and separate naturally, representing the soft projectile behaviour without mesh distortion. The SPH bird model can be initialised with the bird mass, velocity and material properties (density, equation of state, strength model) and can impact the target with realistic deformation behaviour.

Coupled SPH–Lagrangian Approach

The standard approach for bird-strike and similar problems is the coupled SPH–Lagrangian method. The projectile (bird) is modelled with SPH particles. The target structure (leading edge, panel) is modelled with Lagrangian finite elements. The contact between the SPH particles and the Lagrangian elements is defined through a contact algorithm that handles particle-to-surface interaction. This combines the strengths of both methods — SPH handles the extreme projectile deformation; Lagrangian handles the structural target with conventional FEA. The coupled approach is the most common configuration for soft-projectile impact analysis.

Eulerian and ALE Methods

Eulerian methods fix the mesh in space and let material flow through it. This is well suited to fluid-like behaviour and severe deformation — the mesh does not distort because it does not move. However, tracking material boundaries is more complex — the interface between materials must be tracked through the fixed mesh. ALE (Arbitrary Lagrangian-Eulerian) combines both — the mesh can move with the material (Lagrangian) but can also relax or rezoned when deformation becomes severe. ALE is used for some impact and fluid-structure interaction problems. For most engineering impact problems, SPH is simpler to set up than Eulerian or ALE and is the more common choice for soft projectiles.

Comparison

Key Takeaways

• Lagrangian meshes distort under severe deformation — they fail for soft projectiles
• SPH represents material as particles — no mesh to distort; handles extreme deformation
• The coupled SPH–Lagrangian approach is standard for soft-projectile impact analysis
• Eulerian and ALE methods are alternatives for fluid-like behaviour and severe deformation
• SPH is a discretisation method, not a material model — the material behaviour is defined separately