水中爆発解析
Theory and Physics
What is an Underwater Explosion (UNDEX)?
Professor, what kind of problem is underwater explosion analysis?
UNDEX (Underwater Explosion) is the problem of shock waves from an underwater explosion acting on the hulls of ships and submarines. It is the most important shock resistance evaluation in naval hull design.
Physics of Underwater Explosions
Characteristic phenomena of underwater explosions:
1. Shock Wave (primary shock wave) — Propagates spherically from the source. Travels at the speed of sound in water (1500 m/s)
2. Bubble Pulse — Repeated expansion → contraction → re-expansion of the gas bubble. Low-frequency pressure pulsation
3. Cavitation — Negative pressure near the water surface due to shock wave reflection → water boils (cavitation) → secondary shock upon collapse
It's different from in air, isn't it?
Water is 800 times denser than air. The energy of the shock wave is orders of magnitude greater, causing several times more damage for the same TNT equivalent compared to in air.
Analysis Methods in FEM
Can the DAA method be used in Nastran?
Define FLUIDEX (external fluid field) in Nastran's SOL 109/112, and input the underwater explosion shock wave into the structure using the DAA approximation. This is the standard method of the U.S. Navy.
Summary
Key Points:
- Underwater explosions have orders of magnitude more energy than in air — Water density is 800 times greater
- Shock Wave + Bubble Pulse + Cavitation — Three phenomena
- DAA Method (Nastran), ALE Method (LS-DYNA), BEM-FEM — Analysis methods
- Most important in naval hull design — Shock resistance performance evaluation
Bubble Collapse in Underwater Explosions Destroys Structures
In underwater explosions (UNDEX), after the initial shock wave, the "bubble pulse" of repeated expansion and contraction of the explosion gas bubble delivers a secondary shock to the structure. The period of this bubble pulsation can be estimated by T=K(W^(1/3)/(D+10)^(5/6)) seconds (K is the TNT constant, D is explosion depth in meters). A typical explosion (100kg TNT, 30m depth) shows a period of about 0.5 seconds. It was later discovered that many cases of ships sunk during World War II were due to keel breaking caused by this bubble pulse.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced your body being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which assumes "forces are applied slowly so acceleration can be ignored". It absolutely cannot be omitted in impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That is Hooke's Law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure" — different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibration energy for a smoother ride. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considers only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity and creep requires constitutive law extensions
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
Implementation of DAA Method
DAA (Doubly Asymptotic Approximation) is a method that simultaneously approximates low-frequency added mass effects and high-frequency acoustic radiation. DAA boundary conditions are set on the structure's wetted surface (surface in contact with water).
Which Nastran solution is used?
Add FLUIDEX (external fluid field) to SOL 109/112. The explosion shock wave is calculated using formulas like Cole's equation and input as an incident wave to the structural surface. Scattered waves are calculated using DAA approximation.
ALE Method (LS-DYNA)
Simultaneously solves water and explosive with Eulerian (ALE) mesh, and ship hull with Lagrangian mesh. Directly simulates all phenomena: shock wave propagation, reflection, and cavitation.
Is the ALE method more accurate?
The ALE method is the most accurate, but requires enormous water mesh (millions to tens of millions of elements). For distant explosions, the DAA method is efficient. For close-range explosions, the ALE method is necessary.
Summary
Simultaneously solves water and explosive with Eulerian (ALE) mesh, and ship hull with Lagrangian mesh. Directly simulates all phenomena: shock wave propagation, reflection, and cavitation.
Is the ALE method more accurate?
The ALE method is the most accurate, but requires enormous water mesh (millions to tens of millions of elements). For distant explosions, the DAA method is efficient. For close-range explosions, the ALE method is necessary.
DAA (Delayed Acoustic Approximation) is Key to Fluid-Structure Coupling
For fluid-structure interaction in UNDEX analysis, DAA (Doubly Asymptotic Approximation), which approximates scattered waves in infinite water domains, is widely used. DAA2 (second-order accuracy) developed by DeRuntz and Geers in 1978 continuously switches between asymptotic solutions for both far-field and near-field, and has been adopted in the U.S. Navy's ship UNDEX certification code USA (Underwater Shock Analysis). The USA code is still managed by the Naval Research Laboratory (NRL) as an external fluid solver for Nastran and Abaqus.
Linear Elements (1st-order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Shows quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying full load at once, increases it in small steps. The Arc-length method (Riks method) can trace beyond extremum points in the load-displacement relationship.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper" — reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer" — starts rough but
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