中央差分法（陽解法）
Theory and Physics
What is the Explicit Method?
Professor, what is the difference between the "Explicit Method" and the "Implicit Method"?
The Implicit Method solves a system of equations at each time step (with equilibrium iterations). The Explicit Method directly calculates the displacement at the next time step using only the inverse of the mass matrix (or its reciprocal if diagonal), without solving a system of equations.
Without solving a system of equations? That sounds fast.
The calculation per step is orders of magnitude faster. However, there is a stability condition, requiring a very small time step ($\Delta t \propto L_{min} / c$, where $c$ is the speed of sound). It is optimal for short-duration phenomena like impact.
Central Difference Method Algorithm
For the equation of motion $[M]\{\ddot{u}\} = \{F\} - [K]\{u\} - [C]\{\dot{u}\}$:
1. Acceleration Calculation: $\{\ddot{u}_n\} = [M]^{-1}(\{F_n\} - \{F_{int,n}\})$
2. Velocity Update: $\{\dot{u}_{n+1/2}\} = \{\dot{u}_{n-1/2}\} + \Delta t_n \{\ddot{u}_n\}$
3. Displacement Update: $\{u_{n+1}\} = \{u_n\} + \Delta t_{n+1/2} \{\dot{u}_{n+1/2}\}$
If $[M]^{-1}$ is a diagonal matrix, the inverse is just taking the reciprocal of each diagonal component... super fast computation!
That's why the Lumped Mass Matrix (diagonal) is essential in explicit methods. Using a consistent mass matrix (non-diagonal) makes the inverse calculation costly, negating the advantage of the explicit method.
Stability Condition (CFL Condition)
Stable time increment for explicit methods:
$$ \Delta t \leq \frac{L_{min}}{c} = \frac{L_{min}}{\sqrt{E/\rho}} $$
Stable time increment for explicit methods:
$L_{min}$ is the minimum element size, $c$ is the material's speed of sound (elastic wave speed).
For steel ($c \approx 5000$ m/s) with $L_{min} = 5$ mm, then $\Delta t \leq 1 \mu s$... that's extremely small!
That's why analyzing one second requires over 1 million steps. An impact (a few ms) can be done in a few thousand steps, but a quasi-static analysis (several seconds) becomes enormous. Explicit methods are suitable for short-duration dynamic phenomena.
Explicit Method vs. Implicit Method
| Characteristic | Explicit Method | Implicit Method |
|---|---|---|
| Computation per step | Very light | Heavy (system of equations) |
| Time Step | Very small (order of $\mu s$) | Large ($ms \sim s$) |
| Stability Condition | Yes (CFL Condition) | No (unconditionally stable) |
| Application | Impact, explosion, collision | Quasi-static, vibration |
| Contact | Easy | Difficult |
| Nonlinearity | Handled naturally | Requires iteration |
Ease of handling contact is a big advantage.
In explicit methods, contact detection and force calculation are performed independently at each step, with no convergence issues. For problems with many simultaneous contacts, like car crashes, explicit methods are overwhelmingly advantageous.
Summary
Let me organize the Central Difference Method (Explicit Method).
Key points:
- Does not solve a system of equations — If $[M]^{-1}$ is diagonal, it's just reciprocals.
- $\Delta t \leq L_{min} / c$ — CFL Condition. Determined by the smallest element.
- Optimal for short-duration dynamic phenomena — Impact, explosion, collision.
- Contact is easy — No convergence problems.
- Lumped mass matrix is essential — Not fast if not a diagonal matrix.
So the explicit method is an algorithm for "short-duration, intense phenomena".
All LS-DYNA analyses use the explicit method. Abaqus/Explicit also uses the explicit method. Automotive crash safety wouldn't be possible without this technique.
The Central Difference Method was born from 1695 calculus
The central difference method, which applies the finite difference concept published by Leibniz in 1695 to dynamics, approximates acceleration as a(t)＝[u(t+Δt)−2u(t)＋u(t−Δt)]/Δt². This second-order accurate scheme leads to the critical time step Δtcr＝2/ωmax from von Neumann stability analysis, constrained not to exceed the natural frequency of the smallest element. This algorithm was first implemented on a large scale in FEM in the 1970s in DYNA3D, the predecessor code of LS-DYNA.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced 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, assuming "forces are applied slowly enough that acceleration is negligible". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under 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". Stiffness is "resistance to deformation", strength is "resistance to failure" — different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while 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 applying "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 vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational energy for a smoother ride. What if damping were zero? Buildings would keep swaying 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, with a linear stress-strain relationship.
- Isotropic material (unless specified otherwise): Material properties are direction-independent (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Plasticity, creep, and other nonlinear material behaviors require 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 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
Explicit Method Implementation
How is the FEM code for the explicit method implemented?
The main loop is extremely simple:
```
for each time step:
1. Internal force calculation: F_int = assemble(sigma B V) $ Element loop
2. External force calculation: F_ext = boundary_conditions()
3. Acceleration: a = (F_ext - F_int) / M_diag $ Diagonal division
4. Velocity update: v = v + dt * a
5. Displacement update: u = u + dt * v
6. Contact check: contact_detection_and_force()
7. Time increment update: dt = CFL * L_min / c
```
Step 3, "diagonal division", is the key point.
Yes. That's the source of the explicit method's speed. In implicit methods, step 3 becomes "solving an $n \times n$ system of equations", with orders of magnitude more computational cost.
Stable Time Increment Management
The stable time increment is determined by the smallest value among all elements:
If just one element is small, the overall $\Delta t$ becomes small.
The whole thing slows down because of one bad element?
Mesh quality directly affects computation speed. Elements with poor aspect ratios or collapsed elements have a very small $L_e$, reducing $\Delta t$. In explicit methods, mesh quality is key to computational efficiency.
Mass Scaling
Mass scaling artificially increases the mass of small elements to enlarge $\Delta t$:
Increasing $\rho$ decreases $c$ and increases $\Delta t$. An essential technique for quasi-static problems using explicit methods.
Doesn't increasing the mass change the inertial effects?
It does. Therefore, it's necessary to verify that kinetic energy / internal energy < 5%. If this condition is met, it can be considered "quasi-static".