kMC-FEA

Overview

To understand the advantages and disadvantages of kinetic Monte Carlo (kMC) in Finite Element Analysis (FEA) when compared to Johnson-Mehl-Avrami-Kolmogorov (JMAK) model.

Johnson-Mehl-Avrami-Kolmogorov Model

The JMAK model is a classical phenomenological model describing phase transformation kinetics based on nucleation and growth theory. It assumes random nucleation sites and isotropic growth, leading to an analytical expression for the transformed fraction, $X(t)$, over time:

$$X(t)=1-\exp \left(-k{t}^n\right)$$

where

• $k$ is a temperature-dependent rate constant incorporating nucleation and growth rates,
• $n$ is the Avrami exponent related to the mechanism and dimensionality of growth,
• $t$ is time.

Kinetic Monte Carlo Model

The kinetic Monte Carlo (kMC) model is a stochastic simulation approach that explicitly models atomic or microstructural events such as nucleation, growth, and coarsening with probabilistic rules. Instead of an analytical expression, kMC simulates individual events over discrete time steps, capturing detailed microstructural evolution.

For solid-state phase transformations (SSPT) within FEA, kMC can be coupled by embedding the stochastic microstructural evolution model within each finite element or integration point. This allows spatially-resolved simulation of nucleation and growth kinetics informed by local temperature, stress, and composition fields computed by the FEA solver. The kMC model updates phase fractions or microstructure state variables at each time step, providing a detailed microscale evolution that feeds back into macroscopic properties such as stiffness or thermal conductivity.

Calculation of kMC Variables and Probabilities

In kMC, the evolution is governed by rates of possible events (e.g., nucleation, growth, dissolution), which are translated into probabilities used to randomly select events at each time step.

• Event Rate: Each possible event $i$ has an associated rate $r_i$ determined by Arrhenius-type kinetics:

$$r_i={\nu }_i\exp \left(-\frac{E_i}{k_{B}T}\right)$$

where

• ${\nu }_i$ is the attempt frequency (pre-exponential factor),
• $E_i$ is the activation energy for event $i$,
• $k_B$ is the Boltzmann constant,
• $T$ is the absolute temperature.

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• Total Rate and Selection Probability: The total rate is the sum over all possible events:

$$R=\sum _i{r}_i$$

Each event’s selection probability $p_i$ is:

$$p_i=\frac{r_i}{R}$$

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• Time Increment: The kMC time step $\Delta t$ advances stochastically according to

$$\Delta t=-\frac{\ln (u)}{R}$$

where $u$ is a uniform random number in $(0,1)$.

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• Implementation in FEA: Within each finite element, local conditions (e.g., temperature, stress) influence $E_i$ and ${\nu }_i$. The kMC algorithm at each time step:
  1. Calculate event rates $r_i$ based on local conditions.
  2. Compute total rate $R$ and select an event according to $p_i$.
  3. Update microstructural state variables (e.g., phase fraction, grain size) accordingly.
  4. Increment simulation time by $\Delta t$.
  5. Pass updated microstructure information back to the FEA solver to update material properties.

This approach enables multiscale coupling, capturing microscale stochasticity within a macroscale continuum framework.

Advantages

• Captures detailed, spatially-resolved microstructural evolution.
• Can model complex kinetics and heterogeneous nucleation sites.
• Incorporates microstructural effects such as grain boundaries and anisotropy more naturally.
• Allows direct simulation of discrete events rather than relying on averaged assumptions.

Disadvantages

• Computationally intensive, especially for large-scale FEA domains.
• Requires careful parameterization and validation of event rates and probabilities.
• Less straightforward to couple with macroscopic models due to scale differences.
• Longer simulation times can limit applicability for very large problems.

---
config:
  flowchart:
    htmlLabels: false
    diagramPadding: 1
    useMaxWidth: true
---
flowchart TD
    subgraph kMC["kMC-Coupled FEA"]
        fea2[FEA element]
        kmc["kMC Simulation
        (Event-driven kinetics)"]
        phase2[Track evolving microstructure]
        mat2["`Assign material properties locally per event state
        (Based on dynamic event-based states)`"]
        style mat2 fill:#e78284,stroke:#e78284,stroke-width:4px,color:#fff
        fea2 --> kmc --> phase2 --> mat2
    end

    subgraph JMAK["Traditional FEA – JMAK"]
        fea1[FEA element]
        jmak["JMAK Model
        (Empirical kinetics)"]
        phase1[Compute phase fractions]
        mat1["`Select material properties based on predominant phase
        (Based on phase fraction)`"]
        style mat1 fill:#e78284,stroke:#e78284,stroke-width:4px,color:#fff
        fea1 --> jmak --> phase1 --> mat1
    end