Three-Dimensional Stress Analysis of Mountain Ranges: A Novel Approach Using Marching Volume Polytopes Algorithm and Finite Cell Method

The negative feedback between relief formation due to valley incision, increasing topographic stress towards a critical stress state dependent on rock strength, and consequently relief-destroying (and stress-reducing) landslides determines the geometry of alpine landscapes. Hence, the computation of topographic stresses for entire mountain massifs is crucial to identify potential landslide hotspots at steep landforms close to rock failure, determining the maximum strength of rocks and rock sequences at the mountain scale, and explaining contrasting geometries of alpine landscapes in dependence on the prevailing rock types. Traditional 2D stress and displacement calculations on valley cross-sections tend to oversimplify the complicated stress pattern, particularly where valleys converge or around ridges and peaks. 3D stress calculations based on standard finite element methods are computationally expensive and not feasible for entire mountain massifs at a reasonable expense.

Our study addresses this limitation by employing a novel three-dimensional approach, utilizing the Marching Volume Polytopes Algorithm for mesh generation and the Finite Cell Method as an alternative to the widely used finite element method. Incorporating an octree-like structure and advancing-front meshing techniques, the Marching Volume Polytopes Algorithm accurately represents given surface data through a tetrahedral mesh. In the Finite Cell Method representing a fictitious domain approach, the difficulty of generating adequate grids for physical domains with complicated geometry is transformed into the problem of specifying an adequate integration scheme for the finite cells and thus saving degrees of freedom. The computational efficiency of our approach is particularly advantageous when dealing with equidistant grids such as digital elevation models for mesh generation.

In a first study, we use our model to compute the 3D topographic stress distribution for the three Austrian UNESCO Global Geoparks known for over-steepened valley flanks and high landslide activity. Initial results show high shear stress maxima occurring predominantly at over-deepened glacial valleys bordered by rock faces, with stress maxima at valley flanks but also at or slightly below the valley floors. Unexpected stress patterns occur in areas with a complicated landscape geometry, where valleys converge, or intersecting ridge lines form pyramid peaks. Lithological contrasts of the investigated mountain massifs are reflected in very different stress patterns, with shear stress maxima showing the highest values in carbonate-dominated units.

In addition to local topographic metrics, the spatial distribution of observed landslides and the rock types that occur, modelled topographic stresses provide a new data set for assessing landslide potential. Beyond that, modeling topographic stresses of entire mountain massifs offers new insights into the evolution of alpine landscapes in the competition between relief-forming and relief-destroying processes.