直接法周波数応答解析
Theory and Physics
What is the Direct Method?
Professor, how does the "Direct Method" frequency response analysis differ from the modal method?
The Direct Method solves the equations of motion directly at each frequency without expanding into natural modes.
This is a complex system of equations for each frequency $\omega$.
So it's computationally heavy because it solves a system of equations at each frequency?
Yes. It solves a complex system of equations of size $n \times n$ ($n$ = number of DOFs) for each frequency point. The modal method only solves $N$ scalar equations for a single-degree-of-freedom system, so it's orders of magnitude faster.
When the Direct Method is Necessary
So why does the Direct Method exist? It's used for cases that cannot be accurately handled by the modal method:
| Case | Reason |
|---|---|
| Non-proportional Damping | The damping matrix cannot be diagonalized by the modes |
| Frequency-Dependent Material Properties | Viscoelastic materials. $E(\omega), \eta(\omega)$ |
| Structural Damping (Hysteresis) | Complex stiffness $K^* = K(1+ig)$ |
| Systems with Large Damping | Rubber mounts, damping materials |
| External Impedance Boundaries | Soil-structure interaction, etc. |
So the modal method can't be used for materials like viscoelastic ones whose properties "change with frequency".
The modal method uses natural modes (frequency-independent) as a basis, so frequency-dependent materials cannot be naturally incorporated into the modal expansion. The Direct Method can update material properties at each frequency.
Handling Structural Damping
Structural Damping (hysteretic damping) is most naturally handled by the Direct Method:
$g$ is the structural damping coefficient. It is a frequency-independent damping and is often physically more accurate than viscous damping.
Structural damping can't be used in the time domain, right?
Exactly. Structural damping is physically meaningful only in the frequency domain (Direct Method). Using structural damping in the time domain violates causality.
Nastran
```
SOL 108 $ Direct Method Frequency Response
CEND
FREQUENCY = 20
BEGIN BULK
FREQ1, 20, 1., 500., 1.
```
Abaqus
```
*STEP
*STEADY STATE DYNAMICS, DIRECT
1., 500., 500, 1.
*END STEP
```
Ansys
```
/SOLU
ANTYPE, HARMONIC
HROPT, FULL ! Direct Method
HARFRQ, 1., 500.
NSUBST, 500
SOLVE
```
Summary
Let me summarize the Direct Method frequency response.
Key Points:
- Solves the system of equations directly at each frequency — No modal expansion
- Computational cost is 10 to 100 times that of the modal method — Number of frequency points × Number of DOFs
- Handles non-proportional damping, frequency-dependent materials, structural damping — Overcomes limitations of the modal method
- SOL 108 (Nastran), *SSD DIRECT (Abaqus), HARMONIC FULL (Ansys)
- The modal method is sufficient for most problems — The Direct Method is only for special cases
Direct Method Matrix Size is 3 Times the Number of Degrees of Freedom
In Direct Method harmonic response analysis, the complex stiffness matrix [K + iωC − ω²M] is factorized sequentially at each frequency step. Separating the real and imaginary parts of the matrix doubles the effective degrees of freedom, and considering the fill-in from LU decomposition, memory capacity of 3 to 5 times the theoretical degrees of freedom is required. For a 1 million DOF model, it takes several minutes per frequency point, so the choice of sparse solver (e.g., PARDISO) changes the order of magnitude of computation time.
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, which assumes "forces are applied slowly enough that acceleration can be ignored". It 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'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" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure" — different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (pressure, contact forces). 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 ending up with "compression" — sounds like a joke, but it actually happens when coordinate systems are rotated 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 to improve ride comfort. 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 direction-independent (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 behaviors like plasticity and creep 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) | In mm system: N, in m system: N (keep consistent) |
Numerical Methods and Implementation
Improving Direct Method Computational Efficiency
Are there ways to reduce the computational cost of the Direct Method?
There are several techniques:
1. Reusing LU Decomposition of Dynamic Stiffness Matrix
LU decomposition of $[D(\omega)] = -\omega^2[M] + i\omega[C] + [K]$ is the most costly. Decompose the frequency-independent part ($[K]$) only once and treat the frequency-dependent part as an increment using an iterative method.
2. Parallel Computing
Calculations for each frequency point are independent, so full parallelism between frequency points is possible. If 100 frequency points are calculated simultaneously on 100 cores, it effectively takes the time of calculating one frequency point.
3. Reduction Method (Reduced Method)
In Nastran SOL 108, it can be combined with Guyan Reduction or CMS reduction to reduce DOFs before applying the Direct Method. Different from the modal method, but improves efficiency by reducing DOFs.
Modeling Viscoelastic Materials
How are viscoelastic materials handled in the Direct Method?
Complex elastic modulus of viscoelastic materials:
$E'$ is the storage modulus (stiffness), $E''$ is the loss modulus (damping), $\eta = E''/E'$ is the loss factor.
In Abaqus, define Prony series parameters with *VISCOELASTIC, FREQUENCY. Automatically calculates $E^*(\omega)$ at each frequency. In Nastran, define frequency-dependent materials with TABLEM1.
This frequency dependence is crucial in designing damping materials (constrained layer damping), right?
The loss factor $\eta$ of rubbers and viscoelastic polymers depends heavily on frequency and temperature. This dependence cannot be accurately handled without the Direct Method.
Summary
Let me summarize the numerical methods for the Direct Method.
Key Points:
- Improve efficiency with parallel computing across frequency points — Full parallelism is possible
- Complex elastic modulus of viscoelastic materials — $E^*(\omega) = E'(1+i\eta)$
- Model viscoelasticity with Prony series — Abaqus *VISCOELASTIC
- Combine with reduction methods — Reduce computational cost by reducing DOFs
Householder Organized Complex Eigenvalues in 1958
The complex matrix tridiagonalization algorithm, which forms the numerical foundation of the Direct Method, was published by A.S. Householder in 1958. Subsequently, EISPACK, implemented for the IBM System/360 in the 1960s, became a common CAE solver library, and MSC Nastran's SOL 108 directly inherits this lineage. Even in modern Direct Method solvers, the core algorithm remains essentially unchanged from Householder's original insight.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with 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 important.
Full Integration vs. Reduced Integration
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