KMC FEA Synthesis

Project Overview

The kMC-FEA project investigates the integration of Kinetic Monte Carlo (kMC) methods within Finite Element Analysis for modelling solid-state phase transformations (SSPT) in additive manufacturing. The central objective is to compare the stochastic, event-driven kMC approach against the classical Johnson-Mehl-Avrami-Kolmogorov (JMAK) model to determine when the additional computational cost of kMC yields meaningful improvements in microstructural prediction accuracy.

The project is anchored by a single element transient thermal model built with deal.II, which serves as the foundation for coupling thermal fields with microstructural evolution. All code is compiled using a Docker-based build system to ensure reproducible development environments.

Phase Transformation Modelling Approaches

JMAK Model: Analytical Efficiency

The JMAK model provides a closed-form analytical expression for transformed fraction as a function of time:

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

Key characteristics:

• Computationally efficient, suitable for large-scale FEA domains
• Based on assumptions of random nucleation sites and isotropic growth
• Avrami exponent $n$ encodes mechanism and dimensionality of growth
• Extended volume concept accounts for overlapping nuclei through phantom grains
• Historically derived independently by Kolmogorov, Johnson-Mehl, and Avrami

Limitations:

• Cannot capture heterogeneous nucleation or anisotropic growth naturally
• Assumes uniform conditions within each element
• Less accurate for complex thermal histories with multiple phase transformations

kMC Method: Stochastic Resolution

The Kinetic Monte Carlo method replaces analytical expressions with explicit event-driven simulation:

• Event rates follow Arrhenius kinetics: $r_i={\nu }_i\exp (-E_i/k_{B}T)$
• Time advances stochastically: $\Delta t=-\ln (u)/R$
• Each event (nucleation, growth, dissolution) is selected probabilistically

Advantages over JMAK:

• Captures spatially-resolved microstructural evolution
• Models heterogeneous nucleation sites and grain boundary effects naturally
• Incorporates anisotropy and local stress/composition fields
• Provides direct simulation of discrete events rather than averaged behaviour

Trade-offs:

• Computationally intensive, especially for large FEA meshes
• Requires careful parameterization of event rates and activation energies
• Scale coupling between microscale kMC and macroscale FEA introduces complexity

Computational Implementation

Transient Thermal Foundation

The single element transient thermal model establishes the computational infrastructure:

• Mesh: cube domain $[0,1{]}^3$ with 512 quadratic hexahedral elements
• Time integration: backward Euler ($\theta =1$ implicit)
• Material properties: temperature-dependent $c_p(T)$ and $k(T)$ from external data files
• Boundary conditions: Dirichlet temperature from input file, clamped outside data range
• Output: .vtu files for ParaView visualization and .gpl for Gnuplot

The TransientThermalProblem class assembles consistent mass and conductivity matrices, interpolating material properties at the average temperature per element. This thermal solver will provide the temperature fields that drive kMC event rates in the coupled simulation.

Build System and Development Workflow

The Docker-based build system ensures consistent compilation across development environments:

• Official dealii/dealii:latest image extended with clangd LSP and uv for Python post-processing
• Volume-mounted project directories enable seamless host-container development
• Neovim LSP configured to run clangd inside the container for accurate code intelligence
• ninja build system replaces make for faster compilation within the container

This infrastructure supports both the JMAK baseline implementation and the kMC extension, ensuring that comparisons between methods are not confounded by build environment differences.

Key Connections and Insights

Multiscale Coupling Strategy

The project implements a two-way coupling between macroscale FEA and microscale kMC:

1. FEA solver computes local temperature, stress, and composition fields
2. These fields inform kMC event rates ($E_i$ and ${\nu }_i$) at each integration point
3. kMC simulates microstructural evolution and updates phase fractions
4. Updated microstructure feeds back into FEA material properties (stiffness, thermal conductivity)

This approach bridges length scales that the JMAK model cannot resolve, capturing grain-level heterogeneity within a continuum framework.

Validation Pathway

The JMAK model serves as a benchmark for validating the kMC implementation:

• Both models should converge to similar predictions under conditions where JMAK assumptions hold (uniform temperature, random nucleation)
• Divergence under complex thermal histories indicates regimes where kMC provides superior accuracy
• The thermal model provides the shared thermal boundary conditions for both approaches

Computational Cost Considerations

The trade-off between accuracy and computational cost is central to the project:

• JMAK: analytical evaluation per element per time step (negligible cost)
• kMC: stochastic simulation with potentially millions of events per element per time step (significant cost)
• The containerized build enables performance profiling across different hardware configurations to quantify this trade-off

External Connections

The notes in this directory reference concepts that connect to broader themes:

• Solid-State Phase Transformations (SSPT): The physical phenomenon being modelled, relevant to materials science and metallurgy
• Additive Manufacturing: The application domain where accurate SSPT prediction is critical for part quality
• deal.II Library: The finite element framework providing the computational backbone for both thermal and microstructural solvers