Mortar法接触
Theory and Physics
What is the Mortar Method?
Professor, is the Mortar method the latest contact technique?
The Mortar method is a technique that imposes contact conditions in weak form (integral form). It is more robust to mesh non-conformity than the traditional Node-to-Surface (NTS) contact.
Traditional NTS method: Slave nodes are projected onto the master surface. Contact conditions are evaluated as "points" at each node.
Mortar method: Contact conditions are evaluated by integrating over the entire contact surface. Constraints are satisfied in an averaged sense over the entire surface.
So the difference is whether evaluation is done at "points" or over the "surface".
Advantages of the Mortar method:
- Robust to mesh non-conformity — Master and slave meshes do not need to match
- No contact pressure oscillation — Eliminates checkerboarding that occurs with the NTS method
- Path independent — Less dependent on master/slave selection
Implementation in Abaqus
Abaqus's SURFACE TO SURFACE contact is Mortar-based. NODE TO SURFACE is the traditional NTS method.
```
*CONTACT PAIR, INTERACTION=prop, TYPE=SURFACE TO SURFACE
```
Abaqus's default is SURFACE TO SURFACE (Mortar).
Summary
Key points:
- Integrate contact conditions over the entire surface — More robust than point evaluation
- Robust to non-conforming meshes — Master/slave meshes can be different
- Abaqus's SURFACE TO SURFACE is Mortar-based — Default
- No contact pressure oscillation — Overcomes weaknesses of the NTS method
Bernardi-Maday-Patera 1993
The Mortar method was devised in 1993 by C. Bernardi, Y. Maday, and A.T. Patera for domain decomposition in the spectral element method. It ensures continuity in a weak sense via L² projection between subdomains with different meshes. Its application to contact problems was formulated by Ben Belgacem (1999), who proved that integral accuracy of contact pressure is maintained even between non-conforming meshes.
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 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. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? 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"—they are different concepts.
- External force term (load term): Body force $f_b$ (e.g., gravity) 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), 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 typical pitfall 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 away. 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—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue 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 definition)
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considers only balance between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extension
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 to N in mm system, N in m system |
Numerical Methods and Implementation
Mortar Method Calculation
The Mortar method constructs integral segments between contact surfaces and evaluates constraints surface-to-surface.
- Abaqus: SURFACE TO SURFACE (default)
- Ansys: MPC CONTACT or Mortar contact
- Nastran: MORTAR contact (SOL 400)
So the Mortar method is available in all solvers.
Modern commercial solvers are transitioning to Mortar-based contact. The NTS method remains as legacy, but the Mortar method is recommended for new analyses.
Summary
Implementation of Segment Integration
The core of Mortar contact calculation is finding the intersection segments between master and slave surfaces and performing Gaussian integration on each segment. In the algorithm by Fischer & Wriggers (2005), 3D intersecting polygon clipping is implemented using the Sutherland-Hodgman algorithm, ensuring that integration points do not overlap even for complex curved surface contacts. This processing is one of the most geometrically heavy parts of the code.
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 important.
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. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed 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, apply in small increments. The arc-length method (Riks method) can track beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately by hand calculation"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like "flexible curves"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Mortar Method in Practice
Using the Mortar method "by default" is best practice. There are rare reasons to explicitly choose the NTS method.
Practical Checklist
Wind Turbine Main Shaft Contact Analysis
Vestas began adopting the Mortar method around 2015 for non-conforming mesh contact between wind turbine main shafts and housings. The shaft side mesh density is locally fine around bearing grooves, while the housing side is coarse, but Mortar projection ensures load transfer at the interface exceeds 99.5% of theoretical value. With traditional tie contact (node-to-node), artificial stress concentrations occurred at non-conforming areas, leading to underestimation of fatigue life.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Easily Fall Into
Are you checking mesh convergence? Do you think "calculation ran = results are correct"? This is actually the trap beginners fall into most often. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer is far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is the same as "writing the exam question". If the question is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most important step in the entire analysis.
Software Comparison
Mortar Method Tools
Selection Guide
Sierra/Solid and Mortar Implementation
Sandia National Laboratories in the US implemented mortar contact in the Sierra/Solid code in V4.0 (2008) and has utilized it for component contact analysis in nuclear explosion simulations. Commercially, ABAQUS 6.14 (2014) added the mortar formulation option, opening it up for large assembly analyses with non-conforming meshes. In ANSYS Mechanical, mortar contact was promoted from preview to official feature in version 2019R1 (2019).
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types needed for Mortar method contact? For example, presence of LES support for fluids, contact/large deformation capability for structures can differ.
- "Who will use it?": The user's skill level and available time determine the appropriate tool. High-end tools require expertise but offer high accuracy; low-end tools are easier but may have limitations.
- "What resources are available?": Budget, hardware, and software licenses. High-performance computing (HPC) environments enable large-scale analyses, but cloud costs or license fees must be considered.
Open Source vs Commercial Software
Open source (e.g., CalculiX, Code_Aster) is like "buying ingredients and cooking yourself"—requires effort but offers flexibility and low cost. Commercial software (e.g., Abaqus, ANSYS) is like "eating at a restaurant"—convenient and reliable, but expensive and customization is limited. The choice depends on whether you prioritize "cost and freedom" or "time and reliability".
Cloud Analysis Pros and Cons
Cloud analysis is like "renting a professional kitchen when needed"—no need to own expensive hardware, can scale instantly, and pay per use. However, data transfer time and security concerns exist. It's suitable for projects with fluctuating computational demands or when quick results are needed without capital investment.
Related Topics
なった
詳しく
報告