Since first published in 1949, the Monte-Carlo method has been a prominent technique for tackling the simulation of complex physical systems by solving the underlying multi-dimensional integration and boundary-value problem using random number generation. Numerical solutions based on implicit techniques (finite difference and finite element methods) have been widely adopted for solving boundary-value problems in physical domains, although some inherent shortcomings in these methods made them impractical in several important applications, in particular when meshing large domains or when modeling multi-scale systems. Implicit techniques place a high demand in computational hardware requirements (memory size and computing power) when complex geometries are analyzed, making the simulation time prohibitive when modeling certain systems, such as heat conduction problems with abrupt changes in thermal diffusivity, as present at cryogenic systems with composite layered materials.
The computational bottleneck in implicit methods is given by the need to invert a large matrix, whose size grows with the number of elements in the model. The Monte-Carlo method uses the evolution of “particles” within the domain to solve the boundary value problem, using random number generation to solve the inherent multi-dimensional integration within the domain, thereby bypassing the need of inverting a large matrix. For decades, the practical advantages of using Monte Carlo methods as opposed to finite elements have been marginal or nil due to the available computing hardware, and finite element tools have grown to be a multi-billion dollar industry today. In recent years, however, a hardware and software technology breakthrough has taken place with the advent of GPU computing: powerful and relatively inexpensive massively parallel computing resources are now available, leading to a reassessment of the advantages of Monte Carlo methods over finite elements in problems where the shortcomings of implicit methods make simulation times impractical.
Reza Bahadori (PhD candidate) and Professor Hector Gutierrez at the Florida Institute of Technology, in collaboration with Dr. Shashikant Manikonda and Dr. Rainer Meinke from AML Superconductivity and Magnetics, Florida, have developed a novel approach for the simulation of three-dimensional transient conductive heat transfer from a homogeneous media to a non-homogeneous multi-layered composite material with temperature dependent thermal properties using a mesh-free Monte-Carlo method, published in the International Journal of Heat and Mass Transfer.
The proposed approach accounts for the impact of thermal diffusivities from source to sink in the calculation of the particles’ step length, and a derivation of the three-dimensional peripheral integration to account for the influence of material properties around the sink on its temperature. Then, a transient Bessel function solution is combined with a steady-state peripheral integral method to simulate the transient heat conduction in composite media with temperature dependent material properties. Simulations developed by the proposed approach were compared against both experimental measurements and results from a finite element simulation.
The proposed mesh-free method is well suited for modelling intricate geometries and multi-scale systems where solution by conventional finite element tools would require very large number of elements, leading to prohibitively long simulation times. The results were validated by both experimental measurements and finite element simulations, showing that accurate results using the Monte Carlo approach can be achieved with a relatively small number of “particles”.
The proposed approach holds great promise for simulation of multi-scale problems such as the multi-physics analysis of quench in superconducting magnets, where proper representation of superconducting tapes and insulation remains a significant challenge using conventional finite element tools. Solution to the first and second types of thermal boundary conditions have been developed and verified – the extension of the proposed approach to the third kind of boundary condition is currently under development.
Reza Bahadori, Hector Gutierrez, Shashikant Manikonda, Rainer Meinke. A mesh-free Monte-Carlo method for simulation of three-dimensional transient heat conduction in a composite layered material with temperature dependent thermal properties.. International Journal of Heat and Mass Transfer, volume 119 (2018) page 533–541.Go To International Journal of Heat and Mass Transfer