TWO-AND-THREE DIMENSIONL FINITE ELEMENT
of the roadway embankments and pos- sible roadway failure. Analysis of such embankments to estimate the maximum penetration of the freezing front provides information for use in designing mitiga- tion methods to protect from freeze/thaw degradation impacts.
The typical roadway embankment under study has a two-dimensional geometry depicted in Figure 1. In Figure 1, the problem domain is discretized into triangular shaped finite elements. (Note: For three-dimensional problems, tetrahedrons are the basis elements used in the FEM model. The tetrahedron ele- ments are subsequently further resolved from modeling “bricks” used in model definition in order to simplify model buildup.) Figure 2 shows the partition- ing of a triangular finite element into “nodal domains” or, similarly termed “finite volumes (areas)” or “control vol- umes”. The shaded area indicates that portion of the partitioned finite element associated by the phase change algo- rithm to model node “j”.
The assembly of all associated FEM partitioned areas to node j results in the total nodal domain or control volume associated in the algorithm to node j (see Figure 3). Figure 4 depicts the variation of the state variable over the nodal domain for node j. For modeling the soil water phase change within the node j nodal domain, the basis function used is the step function as depicted in Figure 5. Figure 6 shows phase change occurring at both the upper and lower boundary regions of the node j nodal domain. Heat flux estimates are comput- ed using the flux directions as depicted in Figure 7. Figure 7 applies to both the heat transport as well as soil water flow regimes with respect to model node j. Figure 8 defines the computer model outcome estimates for frozen soil water content associated with node k. Sharp interface displacement geometry used in the isothermal phase change algorithm follows the geometry depicted in Figure 9. The computer model calculates volu- metric quantities, such as volumetric ice content. The end-user interprets these quantities to develop locations of the evolving freezing front.
Computer Program Source Code
The Documentation includes the FORTRAN source code for the computer models. Also provided in this report is the documentation for the program PROTO0
Figure 4. Node J basis function depiction of state varialbe value.
Figure 9. Phase change moving boundary sharp interface depiction of algoithm.
which can be obtained from the U.S. Army Corps of Engineers, Cold Regions and Research Laboratory (CRREL). PROTO0 provides a streamlined input setup for typical roadway and high- way cross sections and embankments. Problem geometry and mesh generation, as well as initial and boundary condi- tions are guided input.
References
Hromadka II, T.V., McInvale, D., Gatzke, B., Phillips, M., Espinosa, B. 2014, A Cumulative Departure Model of the Cryosphere during the Pleistocene. ASCE Journal of Cold Regions Engineering, published online April 2014.
Figure 5. Node J nodal domain ice volume tracking.
Figure 6. Node J definition of phase change accumulation of frozen soil water.
Figure 1. Discretization of roadway embank- ment domin into finite elements.
Figure 2. Parition of finite element into nodal domains or finite volumes (areas).
Figure 7. Node J heat flux transport across nodal domain control surface (bondary).
Figure 3. Nodal domain associated to node J.
Figure 8. Phase change accounting algorithm depiction.
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