When dealing with ‘large deformation’ problems such as impact, collision, explosion, and fluid-structure coupling, we often encounter a problem: mesh distortion. Behind this problem actually lies the selection of different explicit algorithms.
I. Lagrangian Algorithm
The Lagrangian description is the most classic framework in finite element methods. In LS-DYNA, most solid structures use the Lagrangian Formulation by default.
Its mathematical essence is: mesh nodes are fixed to material points and move and deform with the material. The characteristics of this algorithm are as follows:
1. Continuum mechanics equations (mass, momentum, energy conservation) are solved in the material coordinate system.
2. There is no ‘advection term’, making calculations relatively simple and accurate.
3. Free surfaces, interfaces, and boundary conditions of materials are naturally satisfied—because the mesh is the material itself.
Algorithm Advantages:
1. Tracking material history variables is very convenient (plastic strain, damage, etc.).
2. Contact, friction, and other boundary conditions are handled naturally.
3. High computational efficiency, especially for pure structural problems.
Algorithm Disadvantages:
1. Severe mesh distortion under large deformation: time step drops dramatically, even negative volume causing calculation termination.
2. Not suitable for fluids (large deformation is normal for fluids) or explosion product diffusion.
Typical Application Scenarios:
1. Object collision: Solid structure large deformation without flow, Lagrange is most suitable.
2. High-speed penetration/armor piercing: Projectile and target plate initially use Lagrange, material failure is handled through keyword deletion.
3. Drop, impact: Mobile phones, packaging cushioning, etc.
II. ALE Algorithm
The ALE algorithm is a compromise between Eulerian and Lagrangian algorithms, combining the advantages of both:
Lagrangian characteristic: Mesh can move with the material, maintaining the ability to track material boundaries.
Eulerian characteristic: Mesh can be ‘re-meshed’ in space to avoid excessive distortion.
The comparison of the three is as follows:
Under the ALE framework, mesh motion velocity is independent of material velocity. When mesh distortion reaches a threshold, the algorithm automatically performs mesh re-meshing, ‘straightening’ the deformed mesh, while using advection algorithms to map physical quantities (mass, momentum, energy) from the old mesh to the new mesh.
Algorithm Advantages:
1. Effectively avoids severe mesh distortion, suitable for large deformation fluid problems.
2. Can accurately track moving interfaces (structural boundaries).
3. Supports multi-material flow, explosion product diffusion, and strong fluid-structure interaction.
Algorithm Disadvantages:
1. Larger computational cost than pure Lagrange.
2. Requires careful setting of calculation methods and smoothing parameters to avoid numerical diffusion or leakage.
3. Interface tracking accuracy is slightly inferior to pure Lagrange, requiring fine parameter tuning.
Typical Applications:
1. Large deformation metal forming: Extrusion, rolling, superplastic forming
2. Fluid-structure coupling problems: Fuel tank sloshing, liquid filling
3. High-speed impact and penetration: Projectile armor piercing, bird strike analysis
4. Explosion and detonation: Underwater explosion, air shock wave propagation
Comprehensive comparison is as follows:
Practical selection recommendations:
1. Pure structural large deformation without flow: Prioritize Lagrange + material failure.
2. Explosion, detonation, underwater explosion, bird strike, strong fluid-structure coupling, etc.: ALE (fluid part) + Lagrange (structure).
3. Extreme large deformation (such as hypervelocity impact debris clouds): Can be supplemented with SPH method.
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