The identification of complex geometry and 3-D flow tracking are the key steps in the process-response simulation between buildings and debris flows. However, the current numerical codes used for debris flow simulations consider the flow depth to be small relative to the tangential length scale and simplify this factor to represent shallow water flow. They have only second-order accuracy in space, as the effects of complex 3-D topography and the vertical mobility of debris flows are not considered (Y. Zhang et al., 2021). To avoid the abovementioned limitations, it is necessary to use an efficient 3-D numerical approach to accurately capture the debris flow behaviors considering the influences of building geometry.

FLOW-3D, a 3-D finite-volume-based CFD model, is considered one of the most efficient tools for predicting hydraulic phenomena with strong turbulent components and inconsistent free water surfaces. FLOW-3D was designed to address the Reynolds-averaged Navier–Stokes equations (RANS) (Jones and Launder, 1972), implementing the volume of fluid (VOF) methods (Hirt and Nichols, 1981) and fractional area–volume obstacle representation (FAVOR) (Hirt and Sicilian, 1985). The advanced TruVOF method can be used to precisely track the 3-D transient free fluid surface. Its unique FAVOR mesh processing technology can define independent and complex geometry within the structured mesh and avoid the shortcomings of the traditional finite difference method in complex boundary fitting (Y. Zhang et al., 2021). The FAVOR processor can generate area fractions for each cell face in the grid by determining which corners of the face are inside of a defined geometry and incorporate geometry effects into the governing equations. As a result of a robust capacity to deal with the data in both the fluid and solid phases, the FLOW-3D code is considered to be appropriate for analyzing the interactions between debris flows and exposed buildings.

In this study, the renormalized group (RNG) model-based k-ε turbulence model and the general moving objects (GMO) model are applied to build a fluid–solid-coupled model of the debris flow impact. The RNG k-ε model is a modification of the standard k-ε model, which takes the turbulent vortex into account and provides an analytic formula for the Prandtl number, as well as an analytic formula for low Reynolds number flow viscosity (Franco et al., 2021). These features make the RNG model more reliable and accurate for a broader flow than the standard k-ε model (Yin et al., 2015). In recent years, the RNG model has been applied in simulations of landslide surges (Yin et al., 2015; Hu et al., 2020), the entrainment effects of debris avalanches (Hu et al., 2019), dam-break floods (Zhuang et al., 2020) and the runout characteristics of debris flows (Y. Zhang et al., 2021), with good results.

The turbulent kinetic energy and the turbulence dissipation balance equations of the RNG k-ε model in FLOW-3D are as following: 1∂kT∂t+1VFuAx∂kT∂x+vAy∂kT∂y+wAz∂kT∂z=PT+GT+DiffkT-εT,2∂εT∂t+1VFuAx∂εT∂x+vAyR∂εT∂y+wAz∂εT∂z=CDIS1⋅εTkTPT+CDIS3⋅GT+Diffε-CDIS2εT2kT, where kT is the turbulent kinetic energy; VF is the fractional volume open to flow; and Ax, Ay and Az are the fractional area open to flow in the x, y and z directions, respectively. PT is the turbulent kinetic energy production term, GT is the buoyancy production term, Diff is the diffusion term, and εT is the turbulence dissipation term. In the RNG model of FLOW-3D, CDIS1 and CDIS3 are dimensionless user-adjustable parameters that have defaults of 1.42 and 0.2, respectively, and CDIS2 is determined from kT and PT (Flow Science, Inc., 2014).

In this study, a series of debris flow simulations based on the RNG model with various building characteristics were executed. From the characteristics of the RNG k-ε model, the type of debris flow involved in this study was determined as mudflow or viscous debris flow, in which a single-phase fluid was assumed and solid particles were treated as suspension and mixed with the fluid phase well. The division between solid and fluid was assumed to be difficult; therefore, the granular deposition was not considered in this study.

With the help of the GMO model in FLOW-3D, the combined hydraulic force due to normal pressure and shear stress can be calculated in the space system. The normal pressure and shear force of an impacted object in x, y and z directions can be gained at each time step. Due to the complex geometry and variable built environments, the impacted elements of target building were changing in the different scenarios. In this study, therefore, the target building was treated as a whole bearing structure to keep consistency of analysis. All over the grids covering building surface would be calculated when contacting with the flow. The GMO model can simulate the rigid body motion, which is either user prescribed or dynamically coupled with fluid flow. In this study, the target building and surrounding buildings were all prescribed to be the fixed and nondeformable rigid models. The raw impact force data were collected at an interval of 0.001 s from the numerical code and averaged manually over 10 points (0.01 s) to reduce uncertainty (Song et al., 2021).

The interaction between a dam break and the structure has become a classic benchmark for the validation of fluid–structure interaction (Liu et al., 2021). The accuracy of the model will be validated by means of the experimental setup previously used in Gomez-Gesteira and Dalrymple (2004). This experiment has been referred to as a “bore in a box”, where it was a dam-break and structure-impact problem confined within a rectangular box. The geometric dimensions of the experimental model are shown in Fig. 1. The rectangular tank is 1.60 m long, 0.61 m wide and 0.75 m high. The tank is considered to be a smooth surface, and its surface roughness (k) is set to 0 m (Liu et al., 2021). The volume of water initially contained behind a thin gate at one end of the box is 0.4 m long, 0.61 m wide and 0.3 m high. An initial layer of water (approximately 1 cm deep) existed on the bottom of the tank. The obstacle, which is 0.12 m × 0.12 m × 0.75 m in size, is placed 0.5 m downstream of the gate and 0.24 m from the nearest sidewall of the tank. The surface roughness of the obstacle is not determined. Therefore, ko (0, 0.001, 0.002 or 0.003 m) is selected for sensitivity analysis to determine if this parameter could reasonably reflect the macromechanical behaviors of the dam-break test. The time history of the impact force on the structure was measured with a load cell.

In the numerical simulation, the analysis domain was discretized into a grid with a cell size of 0.01 m, which was equal to a cube with a 0.01 m side in a 3-D model. The fluid properties were set to be the density of 1000 kg m-3 and viscosity of 0.001 Pa s. The motion of fluid was computed by means of the RNG k-ε model in FLOW-3D. The obstacle and gate were controlled by the GMO module; specifically this obstacle was set as a fixed and nondeformable rigid body, and the gate was prescribed to be lifted 0.3 m along the z+ direction. The time history of impact forces and the corresponding dynamic processes were selected to validate the accuracy of the numerical simulation. The direction of the force was considered positive when exerted in the y+ direction.

Figure 2a shows the general agreement of numerical forces obtained by means of the RNG and GMO coupled model with experimental data, particularly the positions of both peaks, which correspond to the wave hitting the front and the back of the structure and were reasonably reproduced by the numerical model. This indicates that the impact force has a very low sensitivity to ko; this study does not pay too much attention to the value of this parameter, and the surface roughness (k) of all the impacted objects in the numerical simulation is set to 0 m. Figure 2b shows the evolution of the wave generated by the dam break and the initial layer of water on the bottom. At t = 0.32 s, the wave is colliding with the front of the obstacle. At t = 0.58 s, the wave is wrapping around the structure, colliding together and continues moving toward the tank wall. At t = 1.44 s, the reflected wave is hitting the back of the obstacle.

As shown in Fig. 3, a depositional fan model with a length of 120 m, width of 120 m and monogradient of 5∘ was numerically constructed. This deposition fan was treated as a rigid bed model; therefore, the bed material scour would not happen in this study. To sum up, the bed variation induced by sediment transport processes, including sediment deposition and bed scour, was not considered. The surface roughness (k) of 0.05 m was set, meaning that the deposition fan surface was roughened with 5 cm diameter particles, for the representation of the natural environment surrounding mountainous buildings.

The realistic debris flow inflow discharge and duration are essential for a truthful analysis of impact forces. A 3-D numerical simulation with a realistic hydrograph, however, needs a large amount of computer memory and processing time. It is difficult to accept in the sensitivity analysis, in which the sufficient experimental groups are required. In this study, a synthetic and simplified inflow hydrograph was set at the top center of the deposition fan. Specifically, the debris flow discharge was fixed as 500 m3 s-1, which can be of some interest for a large magnitude of debris flow, and the inflow duration was limited to 5 s. The inflow cross section was rectangular, with a channel base width of 20 m, flow depth of 2.5 m, inclination angle of 5∘ and an initial velocity of 10 m s-1. In consideration of the distance between the inflow cross section and target building, a computation time of 15 s, triple the inflow duration, was set for the FLOW-3D modeling, to ensure all the debris flow can flow through the target building. The peak impact force was treated as the maximum value in a relatively complete impact process. In order to compare with some realistic building damage cases, for example the Qipan gully and Zhouqu debris flows in the west of China, the rheological properties of the numerical model was set as the viscous debris flow. Therefore, the debris flow density was set as 2000 kg m-3 and viscosity was empirically 1.0 Pa s (Takahashi, 2007).

The target and surrounding building models, with a length of 15 m, width of 10 m, height of 6 m (equal to two floors) and wall thickness of 0.35 m, were designed in accordance with representative buildings in the mountainous areas of western China. To maintain a balance between the computational accuracy and time cost, the whole computation domain was discretized at intervals of 0.5 m. The planar center of the target building was set at 80 m downstream of the debris flow inflow, as shown in Fig. 3. The embedded domain of the target building should be refined further following two principles: (1) its cell size must be less than the wall thickness of 0.35 m. This is because it is possible that the wall geometry may intersect a cell face more than once in the case of the building rotating, and the corresponding cell edge is assumed to be either fully inside the object or fully outside; some lacks of wall surface will be produced. (2) The boundaries of the embedded and external meshes must be overlapped for the computation stability, that is, the external cell size is a multiple of embedded cell size. To sum up, the cell size of the target building domain was determined as 0.25 m. The total number of computational cells was 1 645 600, and the computer memory of a single simulation was about 20 GB.

Because the sensitivity analysis requires extensive simulation results to consider the combinations of all the factors, the representative factors that could be adjusted in the FLOW-3D simulation were chosen (Kim et al., 2021). The orientation (Or) and opening scale (Op) of the target building and the azimuthal angle (A) and distance (D) of the surrounding buildings with respect to the target building were considered to be the key built environment parameters for debris flow impact loads.

In this study, the orientation (Or) of the target building was defined as the angle between the building's long axis and the debris flow's main flow path, as shown in red and green in Fig. 4a, and ranged from 0 to 90∘. An orientation of 0∘ meant that the long axis of the building was parallel to the debris flow path, and the perpendicular case was represented by an orientation of 90∘. As far as building openings are concerned, it is well known that several features of openings are of great importance in terms of building damage, such as which wall they are located on and their size, height and structure (Gems et al., 2016; Faisal et al., 2018; Papathoma-Köhle et al., 2019). However, it remains challenging to analyze these parameters with a single model. In this study, the size of the opening was only selected to be analyzed to reduce computing costs. As shown in Fig. 4b, two symmetrical openings with a constant height of 3 m were placed in wall A. Therefore, the opening scale (Op) was defined as the proportion of the total opening width (double w) to the length of wall A.

The azimuthal angle (A) of the surrounding buildings was defined as the angle between the line of the geometric center of buildings (e.g., line B–B' in Fig. 4c) and the main flow path (green line in Fig. 4c). There were no surrounding buildings downstream of the target building. Except for the scenario of an azimuthal angle of 0∘, two surrounding buildings were placed symmetrically on both sides of the target building. The distance (D) between the surrounding building and target building was defined as the straight-line distance between two points located on the overlap between the line of the buildings' geometric centers and building envelope, as shown by the orange line C–C' in Fig. 4c.

Abundant simulation results are required for assessing the effect of built environment parameters on the debris flow impact force. Due to the considerable time consumed and computational cost, a mathematical metamodel was constructed using a small fraction of the simulation results. A metamodel, referred to as a surrogate model, is a “model of a model”, a simplified model of an actual model using mathematical construction. Numerous accurate simulation results can be generated based on the metamodel relation or algorithm between input and output (Booker et al., 1999; Hoffman et al., 2003). Of the various metamodel modeling methods, the Kriging model – or Gaussian process (GP) – is considered to be the most suitable for the unbiased prediction of a deterministic model and is also fitted to simulation I/O data obtained for global experimental areas (Kleijnen, 2016). In this study, the GP was selected and executed through JMP^® Pro 16.0.0, a commercial statistical software designed by the SAS Institute Inc.

In this study, the metamodels representing the objective function were created using the 160 samples shown in Table 1. In the metamodel modeling, the variation range of the orientation (Or) was from 0 to 90∘, the opening scale (Op) was from 0 to 0.8∘, the azimuthal angle (A) was from 0 to 90∘ and the distance (D) was from 5 to 30 m. The coefficient of determination (R2) of regression analysis in the GP model was 0.88. At this accuracy, a total of 10 000 new simulation results were obtained from random values in the specified ranges of the four input variables for use in the subsequent sensitivity analysis.

Sensitivity analysis aims to understand the relative importance of uncertain input variables to the model response (J. Zhang et al., 2021). As opposed to local sensitivity analysis, GSA can be performed by an all-at-a-time method, where output variations are induced by varying all input factors simultaneously, and thus, the sensitivity of each factor considers the direct influence of the factor, as well as the joint influence caused by the factor interactions (Kim et al., 2019). GSA allows a ranking among the input parameters to be established according to their influence on the variability of the output. In this study, GSA was conducted to simultaneously consider both the main and interaction effects of input parameters on debris flow impact. A variance-based GSA (VBSA), also referred to as Sobol's indices, is usually recommended. This method is applicable over the whole space of random input data and can also deal with nonlinear responses and measure the effect of interactions in nonadditive systems (Saltelli et al., 2010). The basic principle of Sobol's indices is that the variance of model output is decomposed into fractions within a probabilistic framework that can be attributed to inputs and sets of inputs (Sobol, 1993): 3V=Varfx=∑i=1dVi+∑1≤i<j≤ddVij+…+V1,2,…,d, where the partial variances are calculated as follows: 4Vi1,…,is=∫fi1,…,is2xi1,…,xispxi1,…,xisdxi1,…,xis,s=1,…,d. Sobol's indices are defined as the relative contribution of the partial variances to the total variance following the decomposition in Eq. (1): 5Si1,…,is=Vi1,…,isV=Vi1,…,is∑i=1dVi+∑1≤i<j≤ddVij+…+V1,2,…,d, such that 6∑i=1dSi+∑1≤i<j≤dSij+…+S1,2,…,d=1, where the index Si measures the separate contribution of each variable xi to the output variance without interaction with any other inputs; hence, Si is commonly referred to as the first-order effect index or the main effect index. The higher-order indices in Eq. (4) measure the interactive contribution to the total variance. Using Si, Sij and higher-order indices, we can therefore infer the impacts of each input variable and the interaction of variables on the output variance (J. Zhang et al., 2021). In this study, only the second-order effect index, reflecting the interaction between every two factors, was considered. The total contribution of variable i is as follows: 7SiT=∑i⊂i1,…,isVi1,…,isV, which measures the contributions of variable xi and its interactions to the output variance. If the input xi has a SiT=0.5, then it contributes 50 % of the overall variance of output. In Sobol's sensitivity analysis, the input factors with a sensitivity index below 0.01 are usually considered noninfluential to the output (Sarrazin et al., 2016). Unlike the first-order indices, 8∑i=1dSiT≥1, because the interaction effect between, for example, xi and xj is included in both SiT and SjT. The sum of SiT is equal to 1 if and only if the model is purely additive without any interaction effects.

In this study, Sobol's global sensitivity indices were calculated using the SobolGSA model, a general purpose GUI-driven GSA software developed by Kucherenko and Zaccheus (2023; https://www.imperial.ac.uk/process-systems-engineering/research/free-software/sobolgsa-software/, last access: 23 February 2023). SobolGSA evaluates the effect of a factor while all other factors are varied as well, and thus, it accounts for interactions between variables and does not depend on the choice of a nominal point like local sensitivity analysis methods. The set of available GSA techniques includes screening method- (the Morris measure), variance- (Sobol's indices, FAST) and derivative-based sensitivity measures. All techniques implemented in SobolGSA make use of either quasi-Monte Carlo sampling based on Sobol's sequences or standard Monte Carlo sampling (Sobol et al., 2011; Kucherenko et al., 2015).

Global sensitivity indices and total sensitivity indices for debris flow impacts are listed in Table 2, and the main and total effects of each parameter are expressed in Fig. 5. From the main effect indices, the peak impact force is most sensitive to the azimuth angle, with a maximum value of 0.6303, which represents 63.03 % of the overall variance in the debris flow peak impact loads. The most influential second-order effect index, with a value of 0.1842, was obtained for the interaction between the azimuth angle and the distance of the surrounding buildings. This result also highlights the importance of the surrounding buildings' azimuth angle to the debris flow impact over the distance's main effect index of 0.0095. The sums of all main effect and second-order effect indices are 0.7648 and 0.2352, respectively, indicating that single built environment parameters have a more significant effect on the debris flow impact on a building.

From the total effect indices, on the other hand, the importance of the built environment parameters to the debris flow impact responses is ranked as follows: azimuth angle (A) > distance (D) > orientation (Or) > opening scale (Op), and their total effect indices are 0.8531, 0.2035, 0.1649 and 0.0137, respectively. The sum of the main effects of the target building's properties (Or + Op) is 0.1786 and that of the surrounding buildings' properties (A + D) is 1.0566, indicating that the surrounding buildings' properties are more significant than the target building's properties on the peak impact forces.

Finally, it is concluded that the azimuth angle and distance of the surrounding buildings and the target building's orientation are the key factors and must be carefully considered in the assessment of building vulnerability to debris flow impacts. It is highly recommended to further study the effect of the surrounding buildings' azimuth angles. Although the scale of building openings indeed has a nonsignificant effect on debris flow impacts in this study, the other features of building openings, such as their location, height and structure, should be discussed in more depth.

As shown in Fig. 6, the peak impact forces of the debris flow change with the increasing azimuth angles in the simulation scenarios at an orientation of 90∘, an opening scale of 0.4 and a distance of 5 m (Or90-Op0.4-Ax-D5, where x represents a variable). The variations in the peak impact forces are calculated with the background value of 3717 kN from the scenario with an orientation of 90∘, opening scale of 0.4 and no surrounding buildings (Or90-Op0.4-Anull-Dnull, where null means the value is not relevant). The azimuth angles have different kinds of effects on debris flow impacts, as shown in the different colored zones in Fig. 6:

Shielding effect. In the cases of azimuth angles of 0∘ (Or90-Op0.4-A0-D5) and 30∘ (Or90-Op0.4-A30-D5), the peak impact forces are only 796 and 1193 kN, and the corresponding variations (ΔF) are -78.58 % and -67.90 %, respectively, which are located in the blue shielding effect region (ΔF < -10 %) in Fig. 6. It is indicated that the target building is protected effectively in a shielding area produced by the surrounding buildings. This result is consistent with previous case studies wherein some representative catastrophic debris flow events were observed. In the Zhouqu debris flow event shown in Fig. 7, which occurred in the province of Gansu in northwestern China, on 7 August 2010, building B, next to the extensively damaged building A, suffered no damage apart from its first story being buried (Hu et al., 2012). Similarly, in the Qipan gully debris flow case (Zeng et al., 2015), which occurred in the city of Wenchuan in southwestern China during rainstorms on 11 July 2013, building B, shielding protected from the completely damaged building A, was exposed to only slight damage, as shown in region I of Fig. 8. As shown in Fig. 9, a deflection wall was designed following the principles of the shielding effect and can be repeatedly used to protect an entire building ensemble from gravitational mass movements, such as the snow avalanches, in the mountain areas of Austria (Holub et al., 2012). This design of local protection can provide an effective reference for debris flow mitigation.

As shown in Fig. 10, the variations in peak impact forces change with the surrounding buildings' distances in conjunction with the influences of azimuth angles. The scenario of the orientation of 90∘ and opening of 0.4 (Or90-Op0.4-Ax-Dx) is taken as an example, and the background value (Or90-Op0.4-Anull-Dnull) is still 3717 kN. According to the results of the sensitivity analysis, the most significant second-order effect comes from the interaction between the azimuth angle and distance, which can be divided into an amplification of the shielding effect or a reduction in the canalization effect, as shown in Fig. 10.

Amplification of the shielding effect. The peak impact forces are found to be reduced gradually with increasing surrounding building distance in the case of an azimuth angle of 0∘ (Or90-Op0.4-A0-Dx), with a 5 m distance yielding 796 kN and a 30 m distance yielding 467 kN, and the corresponding variations in peak impact forces are approximately -78.58 % and -87.44 %. The shielding effects at an azimuth angle of 0∘ are amplified with increasing surrounding distance. This process occurs because there is a broader shielding area further downstream of the surrounding building when the debris flow path is separated by an obstacle at a fixed angle of spread, as shown in Fig. 11. Furthermore, there is likely a small-scale debris flow conflux zone close to the surrounding building, as shown in the red zone of Fig. 11. Buildings upstream, therefore, may suffer from a higher impact load in the same debris flow shielding area. As shown in region III of Fig. 8, the lower a building's damage degree, the further away the completely damaged building D is in the back shielding area. Building E, next to building D, is exposed to moderate damage, and other shielding-protected buildings, such as buildings F, G and H, experience only slight damage from lateral abrasion and accumulation. However, the amplified shielding effect inevitably disappears further downstream due to the confluence of debris flow runoff. The range of the shielding area mainly depends on the debris flow properties, especially the friction coefficient and dynamic viscosity (Liang et al., 2021). Further investigation of the effective shielding-protection area is needed.

​​​​​​​A smaller surrounding building distance, however, does not necessarily indicate a larger impact force. With surrounding building distances of 5 and 10 m, impact force shielding effects occur under the condition of an azimuth angle of 30∘ (Or90-Op0.4-A30-D5 and Or90-Op0.4-A30-D10), as shown in Fig. 13, and the corresponding variations in peak impact loads are -67.90 % and -22.03 %, respectively. The width of the narrowed flow path, which is determined by the factors of the azimuth angle and surrounding building distance, could account for this fact. The flow paths are so narrow, with widths of 1.6 and 6.6 m, respectively, like bottlenecks, that not much flow passes through them, as shown in Fig. 13a and b. Thereafter, the canalization effect occurs at distances beyond 15 m and reduces and even disappears gradually with increasing surrounding building distance and the corresponding wider flow paths. It is found that the ratio of the width of the narrowed flow path to the length of the target building has a significant effect on the increase in impact force. The largest peak impact force is most likely to be found under a ratio of approximately one, for example, the ratios of 0.8 in the case of Or90-Op0.4-A45-D5 (Fig. 12a) and 1.1 of Or90-Op0.4-A30-D20 (Fig. 13d).

According to the sensitivity analysis results, the opening scale is the least important factor for debris flow impacts, as it yields the minimum first-order effect index, 0.0050, and the minimum total effect index, 0.0137. From the results with an orientation of 90∘ and no surrounding buildings (Or90-Opx-Anull-Dnull), shown in Fig. 20, the peak impact forces of the target building change very slightly with increasing opening scale. There is a maximum peak impact force of 3945 kN in the case of openings with a scale of 0.8 (Or90-Op0.8-Anull-Dnull), which is approximately 15.38 % larger than the 3419 kN of openings with a scale of 0.2 (Or90-Op0.2-Anull-Dnull). Interestingly, the impact responses of the target building are different with different opening scales. Specifically, there are smaller impact forces in the cases of larger opening scales when only wall A of the target building is impacted, that is, in the early stage of debris flow impact from 5.3 to 6.2 s, as shown in Fig. 21. In this stage, the maximum impact pressure of 865 kN of an opening scale of 0.8 is approximately half of the 1618 kN of an opening scale of 0.2; this is due to the difference in the effective impacted areas. From this perspective, the mitigation performance of single wall elements with more openings is proven (Mazzorana et al., 2014; Gems et al., 2016). Thereafter, the impact load of the overall building increases rapidly following the abundant intrusion of materials through openings after 6.2 s, as shown in Fig. 21. Due to the greater accessibility and higher flow velocity, there is faster growth in impact pressure in the scenarios with the larger opening scales. Finally, after the two above-described impact stages are combined, there are only slight differences between multiple scales of openings in terms of peak impact forces. This indicates that the mitigation function of openings for the whole building is very limited if the time for material intrusion is sufficient.