3次元定常熱伝導
Theory and Physics
Fundamentals of 3D Steady-State Heat Conduction
What changes when moving from 2D to 3D heat conduction analysis?
Since the temperature field changes in three directions, it targets problems where shape simplification is not effective. Many real-world problems like engine blocks, molds, and electronic enclosures are inherently three-dimensional.
Governing Equation
The 3D steady-state heat conduction equation is
Expanding it gives
For anisotropic materials, $k_x \neq k_y \neq k_z$.
There are almost no analytical solutions in 3D, right?
For a rectangular prism with simple boundary conditions on each face, a triple series solution exists, but in practice, numerical methods are essential. Discretization using FEM solid elements (tetrahedra, hexahedra) is the standard approach.
Types of Boundary Conditions
In 3D problems, different boundary conditions can be set for each face.
| Face | Condition | Example |
|---|---|---|
| Heat Generation Surface | Heat Flux q [W/m2] | IC Heat Generation Surface |
| Heat Dissipation Surface | Convection h, T∞ | Heat Sink Outer Surface |
| Contact Surface | Contact Conductance | Bolted Joint |
| Symmetry Plane | Adiabatic (q=0) | Utilizing Symmetry |
| Fixed Temperature | T = const | Cooling Water Surface |
Being able to change conditions per face is a strength of 3D, isn't it?
Correct. In 1D, you can only treat the whole system uniformly, but in 3D, you can set individual conditions for each face and region, allowing you to create models faithful to real-world physics.
General Form of the 3D Heat Conduction Equation
The 3D steady-state heat conduction equation ∇·(λ∇T)+q̇=0 reduces to Laplace's equation if λ is isotropic and homogeneous and there is no internal heat generation. The "Green's function" introduced by Green (1828) became the foundation for solving the 3D Poisson equation and was later applied to the Helmholtz equation and electromagnetism. Green is also known for his origins as the self-taught son of a baker.
Physical Meaning of Each Term
- Heat Storage Term $\rho c_p \partial T/\partial t$: Rate of thermal energy storage per unit volume. 【Everyday Example】An iron frying pan is hard to heat up and cool down, while an aluminum pot heats up and cools down easily—this is due to the difference in the product of density $\rho$ and specific heat $c_p$ (Heat Capacity). Objects with large heat capacity have slower temperature changes. Water has a very high specific heat (4,186 J/(kg·K)), so temperatures near the ocean are more stable than inland. In transient analysis, this term determines the rate of temperature change over time.
- Heat Conduction Term $\nabla \cdot (k \nabla T)$: Heat conduction based on Fourier's law. Heat flux proportional to the temperature gradient. 【Everyday Example】When you put a metal spoon in a hot pot, the handle gets hot—because metal has a high thermal conductivity $k$, heat transfers quickly from the high-temperature side to the low-temperature side. A wooden spoon doesn't get hot because its $k$ is small. Insulating materials (like glass wool) have extremely small $k$, so heat hardly transfers even with a temperature gradient. This is a mathematical formulation of the natural tendency: "Heat flows where there is a temperature difference."
- Convection Term $\rho c_p \mathbf{u} \cdot \nabla T$: Heat transport accompanying fluid motion. 【Everyday Example】Feeling cool under a fan is because the wind (fluid flow) carries away the warm air near your body surface and supplies fresh, cold air—this is forced convection. The ceiling area of a room getting warm with heating is due to natural convection where heated air rises due to buoyancy. The fan in a PC's CPU cooler also dissipates heat via forced convection. Convection is an order of magnitude more efficient heat transport method than conduction.
- Heat Source Term $Q$: Internal Heat Generation (Joule heat, chemical reaction heat, radiation absorption, etc.). Unit: W/m³. 【Everyday Example】A microwave oven heats food via microwave absorption inside the food (volumetric heat generation). The heater wire in an electric blanket warms up via Joule heating ($Q = I^2 R / V$). Heat generation during lithium-ion battery charging/discharging and friction heat from brake pads are also considered as heat sources in analysis. Unlike boundary conditions that supply heat from the "surface" externally, the heat source term represents energy generation "inside" the material.
Assumptions and Applicability Limits
- Fourier's Law: Linear relationship where heat flux is proportional to temperature gradient (non-Fourier heat conduction is needed for extremely low temperatures or ultra-short pulse heating)
- Isotropic Thermal Conductivity: Thermal conductivity does not depend on direction (anisotropy must be considered for composite materials or single crystals, etc.)
- Temperature-Independent Material Properties (Linear Analysis): Assumption that material properties do not depend on temperature (temperature dependency is needed for large temperature differences)
- Treatment of Thermal Radiation: View factor method for surface-to-surface radiation; DO method or P1 approximation for participating media
- Non-Applicable Cases: Phase change (melting/solidification) requires consideration of latent heat. Thermal-stress coupling is essential for extreme temperature gradients
Dimensional Analysis and Unit System
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Temperature $T$ | K (Kelvin) or Celsius | Be careful not to confuse absolute temperature and Celsius. Always use absolute temperature for radiation calculations. |
| Thermal Conductivity $k$ | W/(m·K) | Steel: ~50, Aluminum: ~237, Air: ~0.026 |
| Heat Transfer Coefficient $h$ | W/(m²·K) | Natural Convection: 5–25, Forced Convection: 25–250, Boiling: 2,500–25,000 |
| Specific Heat $c_p$ | J/(kg·K) | Distinguish between specific heat at constant pressure and constant volume (important for gases) |
| Heat Flux $q$ | W/m² | Neumann condition as a boundary condition |
Numerical Methods and Implementation
FEM Element Selection
Which elements should I use for 3D thermal analysis?
Comparison of 3D heat conduction elements.
| Element Type | Number of Nodes | Temperature Distribution | Accuracy | Mesh Generation |
|---|---|---|---|---|
| 4-Node Tetrahedron (TET4) | 4 | Linear | Low | Easy automatic meshing |
| 10-Node Tetrahedron (TET10) | 10 | Quadratic | High | Easy automatic meshing |
| 8-Node Hexahedron (HEX8) | 8 | Trilinear | Medium–High | Requires structured mesh |
| 20-Node Hexahedron (HEX20) | 20 | Quadratic | Very High | Difficult to generate |
In Ansys, SOLID70(HEX8), SOLID87(TET10), SOLID90(HEX20) are representative thermal elements. In Abaqus, they correspond to DC3D4, DC3D10, DC3D8, DC3D20.
Is TET10 a safe choice in practice?
Considering compatibility with automatic meshing, TET10 has high versatility. However, if the shape allows hexahedral meshing, HEX8 can reduce the number of elements. Using Ansys Meshing's Multizone or Sweep methods allows priority generation of HEX elements.
Handling Large-Scale Problems
Element count increases rapidly in 3D problems. Countermeasures for cases exceeding 1 million elements:
- Iterative Solver: PCG method + AMG preconditioner uses about 1/10 the memory of direct methods
- Parallel Computing: Utilize multiple cores with Ansys DMP (Distributed Memory Parallel)
- Submodeling: Refine a local model using the temperature field from the global model as boundary conditions
How do you do submodeling specifically?
Obtain a steady-state solution with a coarse global model, transfer the temperature to the boundary faces of the region of interest, and solve a refined model. This can be easily set up with the Submodel command in Ansys Workbench. Even using a mesh 10 times finer for 10% of the total area results in computational cost roughly 1/100th of refining the entire model.
Birth of FEM Tetrahedral Elements
The tetrahedral element, essential for 3D thermal analysis, was proposed in 1960 by Turner et al. in NASA-sponsored research. Initially for structural analysis, its adaptation to thermal analysis was systematized by Wilson and Bathe in the 1970s. The origins of C3D10 (10-node quadratic tetrahedron) used in current Abaqus/Standard and Ansys Mechanical also date back to this era.
Linear Elements vs. Quadratic Elements
In heat conduction analysis, linear elements often provide sufficient accuracy. For regions with steep temperature gradients (e.g., thermal shock), quadratic elements are recommended.
Heat Flux Evaluation
Calculated from the temperature gradient within the element. Smoothing may be required, similar to nodal stresses.
Convection-Diffusion Problems
When the Peclet number is high (convection-dominated), upwind stabilization (e.g., SUPG) is needed. Not required for pure heat conduction problems.
Time Step for Transient Analysis
Set a time step sufficiently smaller than the characteristic thermal diffusion time $\tau = L^2 / \alpha$ ($\alpha$: Thermal Diffusivity). Automatic time step control is effective for rapid temperature changes.
Nonlinear Convergence
Nonlinearity due to temperature-dependent material properties is often mild, and Picard iteration (direct substitution method) is often sufficient. Newton's method is recommended for strong nonlinearity like radiation.
Steady-State Analysis Determination
Convergence is determined when the temperature change at all nodes falls below a threshold (e.g., $|\Delta T| / T_{max} < 10^{-5}$).
Analogy for Explicit and Implicit Methods
Explicit method is like "predicting the next step using only current information, like a weather forecast"—fast calculation but unstable with large time steps (misses storms). Implicit method is like "prediction considering future states"—stable even with large time steps but requires solving equations at each step. For problems without rapid temperature changes, using the implicit method with larger time steps is more efficient.
Practical Guide
Guidelines for Shape Simplification
If you use CAD data as-is, the element count explodes, right?
CAD geometry cleanup determines the success or failure of 3D thermal analysis.
| Simplification Item | Effect | Caution |
|---|---|---|
| Removing Small Fillets | 50% reduction in element count | Guideline: below 0.5mm |
| Shelling Thin-Walled Parts | No elements needed in thickness direction | Not possible if gradients in thickness direction are important |
| Omitting Bolt Holes | Avoids local refinement | Limited to those not affecting heat paths |
| Utilizing Symmetry | 1/2 to 1/8 model | Boundary conditions must also be symmetric |
So you simplify in SpaceClaim (Ansys) or Design Modeler, right?
Correct. Ansys Discovery Live allows you to see the temperature field in real-time while modifying the geometry, enabling quick judgment of which geometric features affect the temperature field.
Mesh Strategy
Mesh guidelines for 3D steady-state heat conduction:
- Near heat sources: Minimum element size = 1/5 or less of the heat source size
- Far field: Can be coarse (temperature gradient is small)
- Material interfaces: Mesh aligned with the interface (shared nodes or Tied Contact)
- Thin-walled parts: At least 3 layers in the thickness direction
What criteria do you use for convergence verification?
If the maximum temperature is the quantity of interest, it's sufficient if the change in maximum temperature is within 1% when the mesh is refined by a factor of 2. However, local heat flux converges slower, so verification with three or more mesh refinement levels is advised.
Thermal Analysis of an Automotive Engine Head
Toyota used 3D steady-state thermal analysis in developing the GR Yaris (2020) to optimize the temperature distribution around the combustion chamber of the aluminum cast head. They reduced the maximum temperature near the fuel injection nozzle by about 25°C compared to previous designs, achieving both knock resistance and thermal efficiency. The analysis model had over 2 million nodes, with a computation time of about 4 hours on an 8-core PC.
Analogy for Analysis Workflow
Think of the thermal analysis workflow as "designing a bath reheating system"
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