# A 2D Moving Mesh Finite Element Analysis of Heat Transfer in Arctic Soils

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Phase Change Modeling—Issues and Approaches to Latent Heat

**v**is the velocity of the interface, and L is the volumetric latent heat.

## 2. Method

#### 2.1. Meshing

#### 2.2. Phase-Change Method

_{i}is the temperature at elemental node i, {θ} is the column vector composed of the nodal temperatures of the element, S

_{i}is the linear shape function corresponding to node i and is dependent on the position coordinates, and [S] is an array of the shape functions for an element, then temperature anywhere in the element, T, is approximated by [8]:

_{i}is the coordinate of node i, we can expand the derivative of [S] with respect to t [17]:

^{T}, and integrating over the element area. To evaluate the term involving the second spatial derivative (the Laplacian in higher dimensions), the second terms were converted into first-order terms using the chain rule, and rearranging terms (see Moaveni Ch. 9 [8]):

_{i}with respect to time also comes out of the integral and can be calculated from the difference between nodal coordinates from the current and last time step. Elemental matrices are assembled into subdomain matrices separately for each of the two subdomains, leading to an ordinary differential equation of this form for each subdomain. The remaining integrations involve only S

_{i}or S

_{i}S

_{j}terms. These are easily solved as well, either using the form of the shape function in one dimension or relationships for the area integration of triangular shape functions (S

_{1}, S

_{2}and S

_{3}) [18]:

_{n}/dt, where dS

_{n}is the change in position of the boundary, or thaw-front):

_{s}is the thermal conductivity in the solid/frozen region, and k

_{l}is the thermal conductivity in the liquid/thawed region.

#### 2.3. Boundary Conditions

#### 2.4. Initial Conditions

## 3. Results

#### 3.1. Analytical Check

#### 3.2. Check against 15 Days of Barrow Data and Finite Difference Model Results

#### 3.3. Check against 49 Days of Barter Island Data

_{b}= −5.5 C to match the Barter Island borehole data, and that the surface temperature boundary condition function was modified slightly to better match the initial and final surface temperatures in the model:

T_{s} = −6.33 − 13.5(2π((i Δt/86.4 × 10^{3}) + 170 − 26)/365), for model start at Julian day 170 |

#### 3.4. ANSYS Comparison

## 4. Conclusions

- One-dimensional formulation of the models were checked and compared well with those presented in the reference used for the algorithm [16].
- One-dimensional and the two-dimensional version of the model were verified, with constant temperature boundary conditions, against the Neumann solution to the Stefan problem (a classic analytical solution to a simple two-phase phase change problem).
- Temperatures with depth agreed reasonably well, and thaw depths agreed to within a few percent at all times in simulations up to 60 days in length, or within 0.02 m, often an order of magnitude better.
- Thaw predictions of the moving mesh model agreed well with two borehole thermistor data sets collected for 15 days of from Barrow, and for 49 days from Barter Island.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- IPCC. Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change; Stocker, T.F., Qin, D., Plattner, G.-K., Tignor, M., Allen, S., Boschung, J., Nauels, A., Xia, Y., Bex, V., Midgley, P., Eds.; Cambridge University Press: Cambridge, UK, 2013; 1535p. [Google Scholar] [CrossRef] [Green Version]
- Barnhart, K.R.; Anderson, R.S.; Overeem, I.; Wobus, C.; Clow, G.D.; Urban, F.E. Modeling erosion of ice-rich permafrost bluffs along the Alaskan Beaufort Seacoast. J. Geophys. Res. Earth Surf.
**2014**, 119, 1155–1179. [Google Scholar] [CrossRef] - Zottola, J. Feasibility of Thermosyphons to Impede the Progress of Coastal Permafrost Erosion Along the Northern Coastline of Alaska. Master’s Thesis, University of Alaska, Anchorage, AK, USA, 2016. [Google Scholar]
- Reimnitz, E.; Kempema, E.; Barnes, P.W. Anchor ice, seabed freezing, and sediment dynamics in shallow Arctic Seas. J. Geophys. Res. Ocean.
**1987**, 92, 14671–14678. [Google Scholar] [CrossRef] - Ravens, T.; Hailu, G.; Peng, J.; Ulmgren, M.; Wilber, M. Arctic-capable coastal geomorphic change modeling with application to Barter Island, Alaska. In Proceedings of the Oceans 17 Conference MTS/IEEE, Anchorage, AK, USA, 18–21 September 2017; Available online: http://www.oceans17mtsieeeanchorage.org (accessed on 28 September 2022).
- Reimnitz, E.; Graves, M.; Barnes, P.V. Beaufort Sea Coastal Erosion, Sediment Flux, Shoreline Evolution, and the Erosional Shelf Profile; Text to Accompany U.S. Geological Survey Series Map I-1182-G; Department of Tile Inferior, U.S. Geological Survey: Reston, VA, USA, 1988; 22p. Available online: https://pubs.usgs.gov/imap/1182g/report.pdf (accessed on 25 September 2022).
- Çengel, Y.A.; Ghajar, A.J. Heat and Mass Transfer, 5th ed.; McGraw Hill Education: New York, NY, USA, 2016. [Google Scholar]
- Moaveni, S. Finite Element Analysis: Theory and Application with ANSYS, 4th ed.; Pearson: London, UK, 2015. [Google Scholar]
- Voller, V.R. An Overview of Numerical Methods for Solving Phase Change Problems. In Advances in Numerical Heat Transfer; Minkowycz, W.J., Sparrow, E., Eds.; CRC Press: Boca Raton, FL, USA, 1996; Chapter 9; Volume 3. [Google Scholar]
- Huy, H.; Argyropoulos, S.A. Mathematical modeling of solidification and melting: A Review. Model. Simul. Mater. Sci. Eng.
**1996**, 4, 371–396. [Google Scholar] [CrossRef] [Green Version] - Ling, F.; Zhang, T. Numerical simulation of permafrost thermal regime and talik development under shallow thaw lakes on the Alaskan Arctic Coastal Plain. J. Geophys. Res.
**2003**, 108, 4511. [Google Scholar] [CrossRef] - Borisov, V. Mathematical Modeling of Underground Construction Temperature Influence on Permafrost Soils. Procedia Comput. Sci.
**2015**, 66, 112–121. [Google Scholar] [CrossRef] [Green Version] - Liandi, F. Applications of the Finite-Element Method to the Problem of Heat Transfer in a Freezing Shaft Wall; CRREL Report 86-8; Cold Regions Research and Engineering Laboratory: Hanover, NH, USA, 1986; Available online: http://www.dtic.mil/dtic/tr/fulltext/u2/a172552.pdf (accessed on 25 September 2022).
- Darrow, M.M. Measurement of Temperature and Soil Properties for Finite Element Model Verification; Alaska Department of Transportation and Public Facilities and Alaska University Transportation Center: Juneau, AK, USA, 2010.
- Persson, P.; Strang, G. A simple mesh generator in MATLAB. SIAM Rev.
**2004**, 46, 329–345. [Google Scholar] [CrossRef] [Green Version] - Mori, M. A finite element method for solving the two-phase Stefan problem. Publ. Res. Inst. Math. Sci. Ser. A
**1977**, 13, 723–753. [Google Scholar] [CrossRef] [Green Version] - Beckett, G.; Mackenzie, J.; Robertson, M. A moving mesh finite element method for the solution of two-dimensional Stefan Problems. J. Comput. Phys.
**2001**, 168, 500–518. [Google Scholar] [CrossRef] - Zienkiewicz, O.C.; Taylor, R.; Zhu, J. Chapter 6—Shape Functions, derivatives and integration. In The Finite Element Method: Its Basis and Fundamentals; Butterworth-Heinemann: Oxford, UK, 2013. [Google Scholar]

**Figure 1.**A meshed domain. The phase change boundary (in blue) separates the thawed and frozen subdomains, which are meshed and modeled separately, subject to the Stefan condition at the boundary.

**Figure 3.**Comparison of Stefan analytical solution (‘exact’) with 2-D numerical solution at end of one-day simulation.

**Figure 4.**Deviation of analytical solution (blue) from 2-D numerical solution (red) for thaw depth with time for one-day simulation.

**Figure 5.**Percent difference between analytical and 2-D numerical solution of thaw depth with time for one-day simulation.

**Figure 6.**Comparison of Stefan analytical solution (‘exact’) with 2-D numerical solution at end of one-day simulation neglecting the moving-mesh correction.

**Figure 7.**Barrow borehole thermistor data compared to model results. Initial thermistor data is in blue and used as initial conditions for the models. Thermistor temperature data after 15 days is in orange and was used to calibrate soil thermal parameters for the 1D Finite difference model, in red. Results of this 2D finite element model, using these soil parameters, are in black.

**Figure 8.**Map of data location for Barter Island borehole temperature data, borehole location in yellow.

**Figure 11.**Initial (

**left**) and final (

**right**) meshes for 49-day Barter Island Simulation. The thawed region is shown separately from, and above the frozen region in each case. The gap between the regions is not physical, the top is spatially continuous with the bottom, the discontinuity represents the phase-change boundary accounted for in the model.

**Figure 12.**Temperature with depth results for one and ten day runs of the moving mesh and ANSYS models compared with the Neumann Analytical Solutions.

Total length, | l = 2 m |

Thermal conductivity of thawed soil, | k_{1} = 1.6 J/(s m C) |

Thermal conductivity of frozen soil, | k_{2} = 1.2 J/(s m C) |

Volumetric heat capacity of thawed soil, | c_{1} = 2.55 × 10^{6} J/(m^{3}C) |

Volumetric heat capacity of frozen soil, | c_{2} = 2.35 × 10^{6} J/(m^{3}C) |

Volumetric latent heat of water, | L_{w} = 3.34 × 10^{8} J/m^{3} |

Volumetric moisture content of soil, | W = 40% |

Volumetric latent heat of soil, | L = W∗L_{w} |

Phase change temperature, | T_{m} = 0 C |

Boundary temperature at soil surface, | T_{s} = 4 C |

Boundary temperature at bottom, | T_{b} = −4 C |

Number of divisions in thawed and frozen regions, | n_{1} = n_{2} = 5 |

Total time of numerical simulation, | t = 86,400 s |

time step, | Δt = 0.1 s |

**Table 2.**Parameters used in numerical comparison to Barrow data (also used for comparison to Barter Island data with slightly modified d, t, T

_{s}and T

_{b}).

Total horizontal dimension of model, | l = 4 m |

Total depth of model, | d = 9 m |

Node spacing at phase change boundary, | 0.05 m |

Thermal conductivity of thawed soil, | k_{1} = 1.6 J/(s m C) |

Thermal conductivity of frozen soil, | k_{2} = 1.2 J/(s m C) |

Volumetric heat capacity of thawed soil, | c_{1} = 2.55 × 10^{6} J/(m^{3}C) |

Volumetric heat capacity of frozen soil, | c_{2} = 2.35 × 10^{6} J/(m^{3}C) |

Volumetric latent heat of water, | L_{w} = 3.34 × 10^{8} J/m^{3} |

Volumetric moisture content of soil, | W = 40% |

Volumetric latent heat of soil, | L = W∗L_{w} |

Phase change temperature, | T_{m} = 0 C |

Boundary temperature at soil surface, | T_{s} = −6.8 − 13.5cos(2π((i Δt/86.4 × 10^{3}) + 165 − 26)/365) ^{1} |

Boundary temperature at bottom, | T_{b} = −7.787 C |

Total time of numerical simulation, | t = 15 days |

time step, | Δt = 10 s |

^{1}T

_{s}= −6.8 − 13.5cos(2π((i Δt/86.4 × 10

^{3}) + 165 − 26)/365)—temperature at soil surface for each time step, i, is functionally the same as the surface temperature condition given by Zottola [3], in that I = 0 corresponds with Julian day 165.

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**MDPI and ACS Style**

Wilber, M.; Hailu, G.
A 2D Moving Mesh Finite Element Analysis of Heat Transfer in Arctic Soils. *Thermo* **2023**, *3*, 76-93.
https://doi.org/10.3390/thermo3010005

**AMA Style**

Wilber M, Hailu G.
A 2D Moving Mesh Finite Element Analysis of Heat Transfer in Arctic Soils. *Thermo*. 2023; 3(1):76-93.
https://doi.org/10.3390/thermo3010005

**Chicago/Turabian Style**

Wilber, Michelle, and Getu Hailu.
2023. "A 2D Moving Mesh Finite Element Analysis of Heat Transfer in Arctic Soils" *Thermo* 3, no. 1: 76-93.
https://doi.org/10.3390/thermo3010005