Drainage channels with step-pool system are widely used to control debris ﬂow. The blocking of debris ﬂow often gives rise to local damage at the steps and baes. Hence, the estimation of impact force of debris ﬂow is crucial for designing step-pools channel. Existing empirical models for impact pressure prediction cannot consider the inﬂuence of bae shape. In this work, a three-dimensional smoothed particle hydrodynamics (SPH) study on the impact behaviour of debris ﬂows in step-pool systems is presented, where debris material is modelled using the regularized Bingham model. The SPH method is ﬁrst checked using the results from two laboratory tests. Then it is used to investigate the inﬂuence of bae shape and ﬂow density. Numerical results show that the impact pressure at the ﬁrst bae highly depends on the bae shape; however, the largest impact pressure usually occurs at subsequent baes due to the violent impact induced by jet ﬂows. The peak impact pressure at the ﬁrst bae initially grows with increasing ﬂow density; however, it starts to drop as density is beyond a threshold. Based on the numerical results, an empirical relation considering the inﬂuence of bae shape is proposed for better prediction of debris impact pressure. Keywords: Debris ﬂow, Impact pressure, Step-pool systems, Bae conﬁguration, Smoothed particle hydrodynamics 1. Introduction The step-pool bed conﬁguration, a common geomorphologic phenomenon in high-gradient mountainous terrain (3-30%), consists of a sequence of steps and pools forming a staircase-like structure in longitudinal proﬁles (Chin, 1999; Waters and Curran, 2012). This self-organized system can enhance channel bed resistance, slow down mass ﬂows, and e ectively dissipate ﬂow energy Maxwell et al. (2001); Wilcox et al. (2011). Inspired by this interesting phenomenon in natural stream ﬂow, similar protection systems with step-pools were introduced to mitigate debris ﬂow Yu et al. (2007); Wang et al. (2012). For instance, Wang et al. Wang et al. (2012) reported an experiment on debris ﬂow on artiﬁcial step-pool system. It was found that the step-pool system can result in increased energy dissipation; thus, it reduces the erosion of gully bed and the risk of debris ﬂow outbreak. Debris-ﬂow drainage channels are important hazard mitigation infrastructure because they are usually constructed in the headwaters of deposit fan, where the life lines are susceptible to debris ﬂow hazards. On the other hand, the analogy between step-pool sequences and energy dissipation structures stimulates the investigation of artiﬁcial step- pool in drainage channels. A number of studies were carried out to understand the transport processes of debris ﬂow in drainage channel with artiﬁcial step-pools Chen et al. (2009, 2015, 2018); Huang et al. (2009); Liu et al. (2016). These experimental studies indicate that the step-pool systems are capable of maximizing ﬂow resistance, reducing velocity, and dissipating the energy of debris ﬂow. For debris ﬂow drainage channels with step-pools, the stability of steps and baes is the key factor for the safe operation. Generally, the foundations of steps and baes in the step-pool system are prone to fail (see Fig. 1) due to the large impact force between debris materials and structures, Li (1997); Fei and Shu (2004). Therefore, the stability Corresponding author. E-mail: wei.wu@boku.ac.at Preprint submitted to Landslides December 23, 2020This is a preprint of the paper published in Landslides Accepted citation: Li, S., Peng, C., Wu, W., Wang, S., Chen, X., Chen, J., Zhou, G.G. and Chitneedi, B.K., 2020. Role of baffle shape on debris flow impact in step-pool channel: an SPH study. Landslides, 17, pp.2099-2111. DOI: 10.1007/s10346-020-01410-w and durability of step-pools under debris ﬂow impact forces should be carefully considered when designing drainage channels. Figure 1: Site examples of broken step-pool in debris-ﬂow drainage channels of (a) Daduo (b) Daqiaohe. Dynamic impact of debris ﬂow has long been a research subject of considerable interest. Many ﬂume experiments H¨ubl and Holzinger (2003); Zanuttigh and Lamberti (2006); Armanini et al. (2011); Cui et al. (2015); Vagnon and Segalini (2016); Choi et al. (2017); Song et al. (2018), and some ﬁeld tests Zhang (1993); Hu et al. (2011); Wendeler et al. (2007) were conducted to study the impact pressure of debris ﬂows. Based on these results, several models were proposed by integrating either hydrostatic or dynamic pressures to estimate the impact force. In these models, the impact pressure of debris ﬂows is assumed to be proportional to the ﬂow depth and the square of ﬂow velocity. Apart from the ﬂow depth and velocity, the impact pressure is also highly dependent on the geometry of structures Shieh et al. (2008). However, little attention has been paid to the impact pressure in step-pool systems with di erent bae shapes. This may be attributed to the fact that conventional approaches like physical modelling and ﬁeld testing are either time-consuming or costly. Furthermore, in physical modelling it is usually dicult to get all the desired ﬂow properties. On the other hand, ﬁeld testing involves high degree of uncertainty, and is sometimes dangerous. Alternatively, numerical simulation can be applied to investigate debris ﬂow impact. Traditionally, depth-integration methods are widely used in debris ﬂow modelling owing to their relative simplicity and eciency Hutter et al. (1994); Iverson and Denlinger (2001). However, it is dicult to obtain accurate impact force using depth integration methods, especially for problems with complex ﬂow dynamics and violent free surfaces involving jumps and fragmentations. On the other hand, three-dimensional simulations, which consider the complete mass and momentum conservations, can provide detailed results on impact forces and ﬂow dynamics. Nevertheless, not all numerical methods are suitable for this task. Traditional grid-based methods such as ﬁnite element method (FEM) or ﬁnite volume method (FVM) need to track the violent free-surface using additional techniques Larese et al. (2008), which usually involve complex formulation and implementation, increased computational cost, and numerical error Lv et al. (2011). Recently, particle-based methods are developed for problems with free-surface ﬂows and large deformation, e.g. Discrete Element Method (DEM) Leonardi et al. (2019); Calvetti et al. (2019), Material Point Method (MPM) Fern and Soga (2016), Smoothed Particle Hydrodynamics (SPH) Neto and Borja (2018), and Particle Finite Element Method Li et al. (2018). Among them, smoothed particle hydrodynamics (SPH) is well established and has a long application history in computational ﬂuid dynamics Monaghan (1994) and solid mechanics Monaghan (2000). Owing to its meshless and Lagrangian nature, SPH is suitable for problems with violent free-surface and large deformation. Consequently, in the past years SPH was used in landslide and debris ﬂow modelling Chambon et al. (2011); Huang et al. (2012); Dai et al. (2017); Peng et al. (2016); Wang et al. (2016); Pastor et al. (2018). The results indicate that the SPH is robust and reliable in modelling the fundamental dynamic behaviour of mass ﬂows. Therefore, SPH is adapted in this work to investigate the inﬂuence of bae shape on the impact pressure in step-pool systems. To accelerate the SPH computations, acceleration based on Graphic Processing Unit (GPU) is utilized. The remaining part of the paper is organized as follows. First, the SPH method with a regularized Bingham model is introduced. Then, two ﬂume tests on triangular and square baes with ﬁxed densities are carried out and the results are used as benchmarks to check the performance of the SPH model. With the calibrated SPH model, the impact force on three bae shapes, i.e. square, triangular, and trapezoidal, are investigated under varying ﬂow densities. Finally, the numerical results are compared with empirical models from literature. A new relation considering the inﬂuence 2 of bae shape is proposed to estimate the impact pressure. 2. SPH with a regularized Bingham model 2.1. Governing equations and constitutive model The governing equations for debris ﬂows consisting of the classical (Navier-Stokes) mass and momentum conser- vation equations are as follows d dt=r v (1) dv dt=1 r+g (2) where d()=dt represents the material time derivative, ris the gradient operator and denotes the inner product.  andvare the ﬂow density and velocity. andgdenote Cauchy stress tensor and body force, respectively. The stress tensorcan be decomposed into a pressure and a shear stress tensor: =pI+ (3) where Iis an identity matrix. In this work, debris materials are simulated as a weakly-incompressible non-Newtonian ﬂuid. This assumption allows us to compute the pressure directly from density using an equation of state as: p=c2(0) (4) where cis the speed of sound taken as a numerical constant, 0is the reference density when there is no pressure. More complex equations of state are also found in SPH computations, but the results are essentially identical Chambon et al. (2011); Marrone et al. (2011). In this study, debris materials are modeled using the Bingham model, which is widely used in debris ﬂow mod- elling Dai et al. (2014); Wang et al. (2016); Lin et al. (2019). The Bingham model is suitable for debris materials with considerable ﬁne particles, where the ﬂow exhibit strong viscous behaviors. The debris materials employed in the experimental tests are such type of material; thus, the Bingham model is adopted. The model is characterized by the existence of a threshold termed yield stress 0. Only when the stress state reaches the yield condition the material starts to ﬂow. Beyond the yield stress, the material behaves like a Newtonian ﬂow with a constant viscosity 0. In general, the shear stress tensor is computed as: =2app˙" (5) whereappis the apparent viscosity, and ˙"is the strain rate tensor deﬁned as ˙"=[rv+(rv)T]=2. In Bingham model, the apparent viscosity has the following form: app=0+ 0=˙ (6) where ˙ is deﬁned as ˙ =p 2˙":˙". It is found that the above model is singular in the static state where ˙ =0. Therefore, we employ the regularized model proposed by Zhu et al. Zhu et al. (2005), which has the following form: app=0+ 0 ˙ (1em˙ ) (7) where mis a regularization parameter which has the dimension of time. With this regularization, when the strain rate tends to zero, the viscosity has a limit 0+ 0m. It is found that the regularized Bingham model is closer to the original Bingham model if a larger mis used. However, a very large mleads to extremely high apparent viscosity in the quasi-static ﬂow region, which results in very small time step in the SPH method and prohibitive computational cost. In numerical experiments, we found that simulations using m=100 s and m=1000 s give almost identical ﬂow dynamics. Therefore, m=100 s is used throughout the work. 3 2.2. SPH formulations In SPH, the computational domain is partitioned using particles which can be viewed as material elements carrying ﬁeld variables and moving with the material. The carried variables usually include mass, density, stresses, velocity components, and other needed ﬁeld variables. The governing equations can be solved by tracking the motions of the particles and changes of the carried variables. The essential part of SPH is to approximate a function f(x) and its gradient rf(x) through the discrete values f in the vicinity of xusing a weighting function fi=nX j=1mj jfjWij (8) rfi=nX j=1mj jfjriWi j (9) where fiandrfiare the approximated function and gradient at the position xi, whereas fj,mj, andjare the function value, mass, and density at the position xj, respectively. Wi j=W(xixj;h) is the weighting function called smoothing kernel. The weighting function depends on the scalar distance between the two positions xiandxjand the constant h, called smoothing length. The soothing length hdetermines the size of the weighting function, and nis the number of particles in the inﬂuence area of the kernel function. Various smoothing kernels are employed in SPH computation. Here the C2 Wendland kernel function Wendland (1995) is used. By using the SPH kernel approximation, the governing equations can be rewritten as the following discretized form di dt=nX j=1mj(vivj)riWi j+hcnX j=1mj j i jriWi j (10) dvi dt=nX j=1mj(pi 2 i+pj 2 j+  i j)+nX j=12m j(i+j)ri jriWi j ij(r2 i j+0:01h2)vi j+gi (11) Eq. (10) is the SPH discretized form of the mass conservation equation. It is well-known that the classical SPH method su ers from pressure oscillations. To increase the accuracy of impact force computation, a numerical di usion term called -SPH is employed with =0:1 Marrone et al. (2011). The numerical di usion term smooths out the oscillations and results in better pressure results. The ﬁrst and second terms in the discretized momentum conservation equation (11) denote the contribution from pressure gradient and shear stress gradient, respectively. The discretization of the shear stress gradient follows the form employed in Szewc Szewc (2017). For details of the derivation, interested readers can refer to Szewc Szewc (2017) and the