Model description paper 22 Dec 2021
Model description paper  22 Dec 2021
An explicit GPUbased material point method solver for elastoplastic problems (ep23De v1.0)
 ^{1}Institute of Earth Sciences, University of Lausanne, 1015 Lausanne, Switzerland
 ^{2}Swiss Geocomputing Center, University of Lausanne, 1015 Lausanne, Switzerland
 ^{3}Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991, Russia
 ^{1}Institute of Earth Sciences, University of Lausanne, 1015 Lausanne, Switzerland
 ^{2}Swiss Geocomputing Center, University of Lausanne, 1015 Lausanne, Switzerland
 ^{3}Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991, Russia
Correspondence: Emmanuel Wyser (manuwyser@gmail.com)
Hide author detailsCorrespondence: Emmanuel Wyser (manuwyser@gmail.com)
We propose an explicit GPUbased solver within the material point method (MPM) framework using graphics processing units (GPUs) to resolve elastoplastic problems under two and threedimensional configurations (i.e. granular collapses and slumping mechanics). Modern GPU architectures, including Ampere, Turing and Volta, provide a computational framework that is well suited to the locality of the material point method in view of highperformance computing. For intense and nonlocal computational aspects (i.e. the backandforth mapping between the nodes of the background mesh and the material points), we use straightforward atomic operations (the scattering paradigm). We select the generalized interpolation material point method (GIMPM) to resolve the cellcrossing error, which typically arises in the original MPM, because of the C_{0} continuity of the linear basis function. We validate our GPUbased inhouse solver by comparing numerical results for granular collapses with the available experimental data sets. Good agreement is found between the numerical results and experimental results for the free surface and failure surface. We further evaluate the performance of our GPUbased implementation for the threedimensional elastoplastic slumping mechanics problem. We report (i) a maximum 200fold performance gain between a CPU and a singleGPUbased implementation, provided that (ii) the hardware limit (i.e. the peak memory bandwidth) of the device is reached. Furthermore, our multiGPU implementation can resolve models with nearly a billion material points. We finally showcase an application to slumping mechanics and demonstrate the importance of a threedimensional configuration coupled with heterogeneous properties to resolve complex material behaviour.
Graphics processing units, or GPUs, have revolutionized the entire field of highperformance computing (HPC) in the last decade. GPUs are manycore processors that were originally developed by the gaming industry in the mid1990s to accelerate graphics and video rendering. Currently, GPUs are widely employed hardware accelerators used in various applications, including artificial intelligence (AI) and machine learning. GPUs are also increasingly used for highperformance scientific computing (see Dong et al., 2015b; Omlin et al., 2018; Räss et al., 2018; Zhang et al., 2021; Alkhimenkov et al., 2021). The majority of the scientific algorithms on manycore (e.g. GPU) hardware accelerators are memorybounded, meaning that data transferring (reading and writing) limits the performance of a solver. This is in contrast to the recent computebounded algorithms, where arithmetic floating point calculations are the main limiting factor in solver performance. This GPU supercomputing breakthrough requires reengineering existing scientific codes or developing new algorithmic structures to efficiently take advantage of the intrinsic lowlevel parallelism of GPUs.
The material point method (MPM) was first proposed by Sulsky et al. (1994) and was further advanced by the generalized interpolation material point method (GIMPM) by Bardenhagen and Kober (2004). It can be thought of as a finiteelement method (FEM) in which (a) integration points (i.e. material points) move and (b) convey state variables, e.g. stress and strain components. The continuum is discretized by material points. The nodal momentum equations are solved on a background mesh, and nodal basis functions provide a mapping framework between the mesh and the material points to transfer either the updated nodal solution or material point properties. The background mesh is reset and actually never deforms. It has been widely used for large deformation geomechanical problems such as retrogressive failure, coupled hydromechanical landslides or granular collapses (Tran and Sołowski, 2019; Bandara and Soga, 2015; Dunatunga and Kamrin, 2015).
From a computational point a view, it is critical for MPM to be able to simulate largescale problems in both two and threedimensional configurations. From this perspective, a few researchers have exploited parallel computing using a single or multiple GPU strategy (Dong et al., 2015a; Dong and Grabe, 2018) to efficiently implement an explicit GIMPM for twodimensional configurations. More recently, some researchers in the graphics community presented a similar implementation (Gao et al., 2018; Hu et al., 2019; Wang et al., 2020) for threedimensional configurations. One of the most computationally expensive operations in MPM is mapping between material points and their associated nodes, which is supported by basis functions. When implementing a GPU, the two most common approaches are gathering and scattering. The former gathers the material point's state variables (i.e. mass, velocity component or stresses) to the nodes, whereas the latter scatters (i.e. distributes) the material point's state variables to their associated nodes. This leads to write conflicts, as several threads are writing into the same memory location at the same time. Gao et al. (2018) demonstrated the superiority of scattering over gathering, provided that the write conflicts are handled without atomic operations. Gao et al. (2018) proposed parallel scattering that results in a performance of an order of magnitude higher than that of a naive atomic implementation. Recently, Wang et al. (2020) proposed an Array of Structures of Arrays (AoSoA) as an efficient layout. It is largely responsible for CPU or GPU performances, as it dictates the memory access pattern by ensuring coalesced memory accesses (Wang et al., 2020).
We propose an explicit GIMPM implementation in a threedimensional configuration on a single GPU and multiple GPUs (ep23De v1.0), taking advantage of the efficient vectorized algorithmic structure of the MPM solver proposed by Wyser et al. (2020a). Our GPUbased solver relies on builtin functions of atomic operations for the mapping between material points and their associated nodes (i.e. scattering). For largescale simulations, the main hardware limit is the GPU onchip memory, which was well documented by Dong and Grabe (2018). To resolve the GPU onchip memory limitation, we rely on a distributed memory parallelization using the message passing interface (MPI) standard. The multiGPU implementation can resolve models with nearly a billion material points. The GPU solver ep23De v1.0^{1} combines MATLAB for pre and postprocessing activities with the massive power of the most recent GPU architectures available (Ampere, Turing and Tesla architectures). This approach allows the user to easily set the problem's geometry and initialize the material points as well as their state variables. Everything needed is then passed to the GPU, which further performs the computations. We propose a formal framework to evaluate the performance of our GPUbased implementation based on the metric for memorybounded codes, i.e. the effective memory throughput (Omlin, 2017). Since the memory wall has been reached, the memory bandwidth becomes the limiting factor for performance. In addition, it is an easily comparable metric. Similarly, we also report the average number of iterations per second for the same reason: it indicates a relative performance, and it does not depend on material properties (e.g. bulk or shear moduli). We also implement the solver ep23De v1.0 under a singleCPU architecture to provide a reference baseline for the performance evaluation of the GPUbased implementation. For the validation of our solver, we simulate the granular collapse problem in a threedimensional configuration and compare the result against the wellknown experimental results of Bui et al. (2008).
In this section, we briefly describe the governing equations implemented in the MPM solver. We use a linear elastoplastic rheology. Large deformations are carried out via a ratedependent formulation with the Jaumann stress rate.
2.1 Governing equations
The conservation of linear momentum is given by (using the Einstein summation convention)
where σ_{kl} is the Cauchy stress tensor, ${v}_{k}=\partial {u}_{k}/\partial t$ is the velocity, u_{k} is the displacement, g_{k} is the body force, and $k,l=\stackrel{\mathrm{\u203e}}{\mathrm{1}\mathrm{\dots}\mathrm{3}}$. The conservation of angular momentum is given by σ_{kl}=σ_{lk}. Dirichlet and Neumann boundary conditions are
where ${\stackrel{\mathrm{\u203e}}{u}}_{k}$ and ${\stackrel{\mathrm{\u203e}}{\mathit{\tau}}}_{k}$ are prescribed displacements, and n_{k} is a unit normal vector pointing outward from the boundary ∂Ω of the domain Ω. Following the standard FEM procedure, we use the updated Lagrangian framework; thus, the weak form of Eq. (1) is written in the current spatial configuration. The weak form of Eq. (1) can be obtained by multiplying it with a test function ϕ and then applying integration by parts and divergence theorem, leading to
where $\partial {v}_{k}/\partial t={a}_{k}$ is the acceleration, ϕ is any test function that vanishes on ∂Ω_{u}, and ${\stackrel{\mathrm{\u203e}}{\mathit{\tau}}}_{k}$ is the external traction applied on the boundary ∂Ω, $k=\stackrel{\mathrm{\u203e}}{\mathrm{1}\mathrm{\dots}\mathrm{3}}$. However, in our MPM implementation, tractions on the boundary are not used. Equation (4) can be solved using a finiteelement approach leading to the following compact form:
where ${M}_{ij}={\sum}_{p=\mathrm{1}}^{{n}_{p}}{m}_{p}{\mathit{\varphi}}_{i}\left({\mathit{x}}_{p}\right){\mathit{\varphi}}_{j}\left({\mathit{x}}_{p}\right)$ is the consistent mass matrix with ϕ_{i}(x_{p}) being the basis function between node i and material point p. This work adopts a lumped mass matrix, i.e. ${m}_{i}\equiv {M}_{ii}={\sum}_{p=\mathrm{1}}^{{n}_{p}}{m}_{p}{\mathit{\varphi}}_{i}\left({\mathit{x}}_{p}\right)$, to avoid an expensive matrix inversion (Sulsky et al., 1994; Bardenhagen and Kober, 2004; González Acosta et al., 2020). The external ${f}_{k,n}^{\text{ext}}$ and internal ${f}_{k,n}^{\text{int}}$ forces at node n are then defined by
where m_{p} is the material point's mass, v_{p} is the material point's volume and σ_{kl,p} is the material point's Cauchy stress tensor. Solving Eq. (5) for the acceleration a_{k,n}, the updated velocity is obtained via a forwardEuler scheme,
where the velocity is given by ${v}_{k,n}^{t}={m}_{n}^{\mathrm{1}}{\sum}_{p=\mathrm{1}}^{{n}_{p}}{\mathit{\varphi}}_{n}\left({\mathit{x}}_{p}\right){m}_{p}{v}_{k,p}$ and v_{k,p} is the material point's velocity. Boundary conditions are enforced on the boundary nodes. The material point velocity v_{k,p} and coordinates x_{k,p} are defined by mapping (i.e. an interpolation) between the updated solution on the mesh and the material points, i.e.
where n_{n} is the number of associated nodes n to a material point p. The remaining tasks are (i) to update the material point volume and (ii) to solve for the constitutive stress–strain relationship.
2.2 Rate formulation
The large deformation framework necessitates a suitable stress–strain formulation. Some studies prefer the finite deformation framework and employ a linear relationship between Kirchhoff stresses and logarithmic strains (Charlton et al., 2017; Gaume et al., 2018; Coombs et al., 2020). In the present work, we adopt a ratedependent framework by applying the Jaumann rate (e.g. Huang et al., 2015; Wang et al., 2016c, b; Bandara et al., 2016), which yields an objective stress rate measure.
The Jaumann rate of the Cauchy stress is given by
where C_{ijkl} is the fourthrank tangent stiffness tensor. Thus, the Jaumann stress derivative may be written as
where ${\mathit{\omega}}_{ij}=({\partial}_{i}{v}_{j}{\partial}_{j}{v}_{i})/\mathrm{2}$ is the vorticity tensor, and $\mathrm{D}{\mathit{\sigma}}_{ij}/\mathrm{D}t$ corresponds to the material derivative
By rearranging the Jaumann stress derivative in Eq. (12), we obtain
where ${\mathit{\sigma}}_{ij}^{\mathcal{R}}$ represents the rotation of the Cauchy stress tensor, which satisfies the stress objectivity for the ratedependent formulation.
Let us expand ${\mathit{\sigma}}_{ij}^{\mathcal{R}}$ in Eq. (14) using identities σ_{ij}=σ_{ji}, ${\dot{\mathit{\omega}}}_{ij}={\dot{\mathit{\omega}}}_{ji}$ and ${\dot{\mathit{\omega}}}_{kk}=\mathrm{0}$. The Cauchy stress tensor is written using the socalled Voigt notation (as a vector $\mathit{\sigma}=\mathit{\{}{\mathit{\sigma}}_{xx},{\mathit{\sigma}}_{yy},{\mathit{\sigma}}_{zz},{\mathit{\sigma}}_{xy},{\mathit{\sigma}}_{yz},{\mathit{\sigma}}_{xz}\mathit{\}}$). After expanding, collecting and rearranging terms, the objective stress terms ${\mathit{\sigma}}_{ij}^{\mathcal{R}}$ for a threedimensional configuration are
and, for a twodimensional configuration assuming plane strain conditions, Eqs. (15), (16) and (18) reduce to
2.3 Elastoplastic deformation
A nonassociated Drucker–Prager (DP) model with a tension cutoff is used in this study, similar to Huang et al. (2015), Liu et al. (2020), Nguyen et al. (2020) and Zuo et al. (2020), because of its straightforward implementation within explicit numerical solvers. The DP model has been established as an approximation of the Mohr–Couloumb (MC) model (Krabbenhoft et al., 2012; Alejano and Bobet, 2012), i.e. a conical yield surface that approximates the MC yield surface in the principal stress space. The former can be adjusted by parameters, so it passes either through the outer or inner edges of the MC yield surface (Jiang and Xie, 2011; De Borst et al., 2012).
The DP yield function f (see Fig. 1) is typically defined in terms of invariants: the first invariant of the Cauchy stress tensor I_{1}=σ_{kk} and the second invariant ${J}_{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}{\mathit{\tau}}_{ij}{\mathit{\tau}}_{ji}$ of its deviatoric part τ_{ij}, where the deviatoric part of the Cauchy stress is ${\mathit{\tau}}_{ij}={\mathit{\sigma}}_{ij}+{\mathit{\delta}}_{ij}p$ with the pressure $p=\frac{\mathrm{1}}{\mathrm{3}}{\mathit{\sigma}}_{kk}$. The DP yield surface is made of two surfaces (i.e. representing shear and tensile yield criteria), delimited by
where $\mathit{\tau}=\sqrt{{J}_{\mathrm{2}}}$ is the effective shear stress, ${\mathit{\sigma}}_{\mathrm{m}}=p$ is the mean stress, q_{ϕ} and k_{ϕ} are the material parameters defined by ϕ as the internal friction angle, σ^{t} is the tensile strength, and c is the cohesion. Cohesion varies with the accumulated plastic strain ${\stackrel{\mathrm{\u203e}}{\mathit{\u03f5}}}_{\mathrm{p}}$ when considering a strainsoftening material, i.e. $c=f\left({\stackrel{\mathrm{\u203e}}{\mathit{\u03f5}}}_{\mathrm{p}}\right)$. These two surfaces define two plastic regions (see Fig. 1) corresponding to either the shear or tensile failure mode. We use a nonassociated plastic flow law for shear and tensile failures; thus, the plastic potential function g is written as
where q_{ψ} is a material parameter estimated with the dilation angle ψ.
The line segment $h({\mathit{\sigma}}_{\mathrm{m}},\mathit{\tau})=\mathrm{0}$ represents the diagonal line between ${f}^{\mathrm{s}}({\mathit{\sigma}}_{\mathrm{m}},\mathit{\tau})=\mathrm{0}$ and ${f}^{\mathrm{t}}({\mathit{\sigma}}_{\mathrm{m}},\mathit{\tau})=\mathrm{0}$ in the (σ_{m},τ) plane; i.e. h is the boundary between shear and tensile failure modes. The function h(σ_{m},τ) is given by
with the constants ${\mathit{\tau}}^{P}={k}_{\mathit{\varphi}}{q}_{\mathit{\varphi}}{\mathit{\sigma}}^{\mathrm{t}}$ and ${\mathit{\alpha}}^{P}=(\mathrm{1}{q}_{\mathit{\varphi}}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}{q}_{\mathit{\varphi}}^{\mathrm{2}}$. We consider an inner adjustment of the DP yield surface with respect to the MC yield surface (de Souza Neto et al., 2011), and the model parameter used in Eqs. (24) and (26) are given by
In the following, we briefly detail the return mapping strategy used to return the trial Cauchy stress ${\mathit{\sigma}}_{ij}^{\text{tr}}$ (i.e. assuming pure elastic deformation only) onto the yield surfaces considering ψ=0. A complete description of such return mapping can be found in Huang et al. (2015). Shear failure is declared when (i) ${f}^{\mathrm{s}}({\mathit{\sigma}}_{\mathrm{m}}^{\text{tr}},{\mathit{\tau}}^{\text{tr}})>\mathrm{0}$ and ${\mathit{\sigma}}_{\mathrm{m}}^{\text{tr}}<{\mathit{\sigma}}^{\mathrm{t}}$ or if (ii) $h({\mathit{\sigma}}^{\text{tr}},{\mathit{\tau}}^{\text{tr}})>\mathrm{0}$ and ${\mathit{\sigma}}_{\mathrm{m}}^{\text{tr}}\ge {\mathit{\sigma}}^{\mathrm{t}}$. The corrected Cauchy stress tensor now reads
with δ the Kronecker tensor. Tensile failure is declared when $h({\mathit{\sigma}}^{\text{tr}},{\mathit{\tau}}^{\text{tr}})\le \mathrm{0}$ and ${\mathit{\sigma}}_{\mathrm{m}}^{\text{tr}}\ge {\mathit{\sigma}}^{\mathrm{t}}$. The corrected Cauchy stress tensor reads as
We propose an explicit generalized interpolation material point method (GIMPM) implementation (Dong and Grabe, 2018; Wang et al., 2020) in a threedimensional configuration on a GPU, taking advantage of the efficient vectorized algorithmic structure (Wyser et al., 2020a, b). We select an explicit GIMPM implementation, which is valid for a variety of problems compared to other recent variants (Wang et al., 2019; Coombs et al., 2020), i.e. CPDI or CPDI2q. Additionally, we use a doublemapping approach (MUSL; see Nairn, 2003; Buzzi et al., 2008), which consists of updating the stress at the end of the time step. We implement the following domainupdate methods: (a) no update of the material point domain, further labelled uGIMPM, and (b) a domain update controlled by the determinant of the deformation gradient, i.e. det(F_{ij}), further labelled cpGIMPM. These domainupdate methods are commonly used in the literature (Baumgarten and Kamrin, 2019; Tran and Sołowski, 2019). The limitation of the two methods is that they are not ideally suited for specific tests: simple stretching and compression modes (Coombs et al., 2020).
3.1 Implementation on a graphical processing unit (GPU)
Graphical processing units (GPUs) are manycore processors originally designed to refresh screen pixels (e.g. for computer games) independently. A schematic representation of the main architecture differences between a CPU and a GPU is depicted in Fig. 2. On the GPU chip, most of the physical space is dedicated to arithmetic logical units, whereas on a CPU, most of the physical space is dedicated to chip host scheduling and control microsystems. GPUs feature many more cores, a lower threadscheduling cost and a higher memory bandwidth than CPUs. The programming model is based on a parallel principle called single instruction and multiple data (or SIMD); i.e. every single instruction is executed on different data. GPUs feature a hierarchical structure. The lowest computational unit is the thread. Threads are organized into blocks of threads, the whole constituting a hierarchical grid of blocks of threads. A GPU typically launches thousands of threads, which execute the same instruction in parallel, thus achieving massive parallelism. Additionally, the most recent GPUs offer a high throughput (close to a TB per second peak memory throughput).
Currently, most of the algorithms are memorybounded, meaning that memory transfers limit the performance, in contrast to computerbounded algorithms, where floating point (arithmetic) operations limit the performance. Thus, for an efficient implementation of an algorithm, one must consider (a) limiting the memory transfers to the bare minimum and (b) avoiding complex data structures (Räss et al., 2019a) to benefit from the high throughput capabilities of GPUs. The ability of a GPU is particularly well suited to efficiently execute a large number of local operations in parallel, i.e. SIMD programming. In the case of a GIMPM implementation, this includes the calculation of shape functions and the update of various quantities at the material point level (i.e. stresses, domain lengths, material point volumes, etc.). Below, we present key aspects of our GPUbased implementation using the Compute Unified Device Architecture (CUDA C) language of the Nvidia Corporation, which is a syntax extension of the C programming language.
3.2 The multiGPU code implementation
One of the major limitations of a singleGPU implementation is the onchip memory. It is then essential to overcome this limit in order to resolve larger computational domains with a greater amount of material points. We address this concern by implementing a distributed memory parallelization using the message passing interface (MPI) standard. However, we limit our implementation efforts by considering (1) a onedimensional GPU topology, (2) no computation–communication overlaps, and (3) only meshrelated quantities are shared amongst GPUs; i.e. the material points are not transferred between GPUs during a simulation. We also selected a nonadaptive time step to avoid the collection of the material point's velocities located in different GPUs at the beginning of each calculation cycle.
3.2.1 Algorithm workflow
In our implementation, MATLAB acts as an architect (see Fig. 3). It (1) defines the problem geometry (i.e. the background mesh, material
point locations and related quantities, etc.), which can be tedious to initialize in a CUDA C environment. It also calls an external MATLAB script,
which compiles the necessary source codes, i.e. gpu.cu
or cpu.cu
. It further (2) calls either a CUDA C or plain C executable, i.e.
gpu.exe
or cpu.exe
, within a Windows operating system (OS) to solve for the numerical problem and finally (3) imports the results of calculations for
further postprocessing tasks.
This is a powerful combination between a highlevel language such as MATLAB and a performant lowlevel language such as CUDA C or plain C. It is also
easy to invoke system commands directly via MATLAB, i.e. to compile source codes and/or run executables using the builtin command
system('…')
. We focus on OSfree scripting in MATLAB using a builtin command (i.e. isunix
or ispc
) to ensure that
it performs well under all OS architectures. In addition, such a workflow can be easily extended to other highlevel languages such
as Python.
3.2.2 Kernels and launch configuration
We briefly describe our GPUbased implementation (gpu_main.cu
) while focusing mainly on the computational aspects of the
implementation. Implementation of an explicit GIMPM solver into the CUDA C language requires dispatching computational activities into several
kernels, i.e. similar to classic functions for a serial implementation in the C language. Each kernel is operated by the GPU only, and kernel launch
configuration parameters must be defined for its proper execution. Among them, one must define the number of active threads per block (i.e. the block
size) and the number of blocks (i.e. the grid size). A typical kernel is executed N times in parallel by N distinct threads organized into blocks
of threads, i.e. a grid of blocks of threads. The principal hardware limitation is the total number of threads within a block: it cannot exceed 1024
threads per block. One must ensure that the maximal size of a block is lower than or equal to this limit.
The computational activities are handled by multiple GPU kernels; 11 kernels are successively launched over a computational cycle. An overall
description is given in Fig. 4. A while
loop is used to perform the computational cycles, and an MPM step is solved
at every cycle. n_{IO} (i.e. the number of accesses to the GPU global memory) is reported in Fig. 4 for each kernel
and is estimated by a careful examination of relevant operations within the kernels. Note that all calculations are performed on the GPU, except the
calculation of the adaptive time step, which is serially executed by the CPU.
In our GPUbased implementation, we define two distinct types of kernel launch parameters: (1) those used for mapping between material points and background nodes (i.e. accumulations and projections between material points with their associated nodes and back and forth) and (2) those used for local calculation at the material point or node level (i.e. update of material point stresses or the solution to the momentum balance equations on the Eulerian background mesh). We use regular background mesh because it is straightforward to find the material point's location. However, computing a material point's location using an irregular background mesh is more complicated.
3.2.3 Adaptive time step
An adaptive time step is implemented. For threedimensional configurations, the maximum elastic wave speed of the material (Anderson, 1987; Zhang et al., 2017) reads as
where ${c}_{\text{el}}=\left(\right(K+\mathrm{4}G/\mathrm{3})/\mathit{\rho}{)}^{\frac{\mathrm{1}}{\mathrm{2}}}$ is the elastic wave speed of the material; K and G are the bulk and shear moduli, respectively; ρ is the material density; and v_{x,p}, v_{y,p} and v_{z,p} are the material point velocity components. The time step Δt is then restricted by the CFL condition,
where $\mathit{\alpha}\in [\mathrm{0};\mathrm{1}]$ is the time step multiplier, and $\mathrm{\Delta}x,\mathrm{\Delta}y,$ and Δz are the background mesh resolutions.
This requires evaluation of the maximum velocity of all material points at the beginning of each calculation cycle. We choose to sequentially find the maximum velocity using the CPU instead of a parallel implementation on the GPU. This results in systematic memory transfers between the GPU global memory and the random access memory (RAM) of the CPU. However, we report a low performance loss due to these transfers, i.e. a maximal loss of 2 %–5 % in performance, which is acceptable.
3.2.4 Backandforth mapping between material points and their associated nodes
The GPUbased algorithm relies heavily on the use of arrays p2e
and e2n
(Wyser et al., 2020a). Elements are numbered with an increasing
index. Associated nodes are also numbered in a similar manner. The array e2n
of dimension n_{el}×n_{n}, where
n_{el} is the total number of nodes and n_{n} is the number of nodes associated with an element e, describes the topological
relation between the elements and the nodes of the mesh. Similarly, the array p2e
describes the topological relation between the material
points and the element in which they are located. These two arrays provide an intuitive definition of the relations between (i) the material points
and the nodes they are associated with (i.e. p2n
) and (ii) the element and their nodes (i.e. e2n
). Then, it is a computationally
straightforward process to identify which nodes n are associated with a material point p, which is occupying an element e.
The GPUbased implementation relies on the builtin function atomicAdd()
in CUDA C. It performs atomic operations, which avoid the data race
of multiple threads, from the same or different blocks to update the same memory location. Atomic operations are extensively used to calculate
internal and external force contributions (Eqs. 6 and 7), as well as the lumped mass matrix, and to update the
material point's properties such as velocities and coordinates (Eqs. 9 and 10). Dong et al. (2015a) and Wang et al. (2020)
reported (for older GPU architectures such as Pascal or Kepler) that atomic scattering can be significantly slower compared to an optimized parallel
implementation. However, atomic operations are (a) intuitive to both understand and implement, and (b) they avoid a complex data layout, such as
recently proposed in Wang et al. (2020). The use of builtin atomic operations considerably reduces programming efforts.
3.2.5 Treatment of volumetric locking for loworder elements
When loworder elements are used in a GIMP formulation, volumetric locking arises and results in spurious oscillations of the stress field (Jassim et al., 2013; Coombs et al., 2018; González Acosta et al., 2019, 2021). We implement a simple procedure to mitigate volumetric locking when considering nearincompressible behaviour for isochoric plastic flows. Cuomo et al. (2019) and Lei et al. (2020) introduced an elementbased averaging method, following Mast et al. (2012). Selected material point properties are reconstructed based on an average value calculated at the element's centre at the end of a time step. However, we propose averaging only the volumetric part of the stress tensor, i.e. the pressure $p=\frac{\mathrm{1}}{\mathrm{3}}{\mathit{\sigma}}_{kk}$, while its deviatoric part ${\mathit{\tau}}_{ij}={\mathit{\sigma}}_{ij}p{\mathit{\delta}}_{ij}$ remains unchanged. We believe our approach is conceptually similar to the Bbar technique (Hughes, 1980; Bisht et al., 2021). This results in the following:
where v_{p} is the material point's volume. This gives a constant distribution of the pressure field over an element because of its zeroorder reconstruction (Lei et al., 2020). The Cauchy stress tensor σ_{ij,p} of a material point p occupying an element e is corrected as
where δ_{ij} is the Kronecker delta and (p_{e})_{p} is the averaged pressure within an element e and assigned to a material point p.
3.3 Available computational resources
The CPU and GPUbased simulations are performed on a modern workstation running on a Windows 10 operating system with the latest CUDA version v11.2. The CPU is an Intel Core i910900K with 10 physical cores of base clock speed (or frequency) of 3.70 GHz, which can rise up to a maximum clock speed of 5.30 GHz, supported with 64 GB DDR4 RAM. It hosts a consumer electronics Nvidia RTX 3090 GPU (the latest Ampere architecture) with 82 streaming multiprocessors (SM units) with a base frequency of 1.40 GHz. This results in 10490 CUDA cores that are supported with an onchip memory of 24 GB GDDR6 (i.e. the GPU global memory). Other GPUs installed on older desktops are also used to compare their respective GPU performances, i.e. an RTX 2080 Ti (workstation) and a GTX 1650 (laptop), both running on a Windows 10 operating system. Additional simulations were also run on a workstation equipped with the latest Nvidia A100 GPU at the Lomonosov Moscow State University.
Furthermore, GPUbased simulations are also performed on the Octopus GPU supercomputer at the Swiss Geocomputing Centre, University of Lausanne, Switzerland. In particular, the GPUbased simulations are run on the Volta node, hosting a 16 GB Nvidia Tesla V100 (Volta architecture), supported by an Intel(R) Xeon(R) E52620 v2 (Haswell) with 2.1 GHz CPU. The latest CUDA version installed is v11.0, and the supercomputer Octopus is operated under a CentOS 6.9. environment. To summarize the computational resources in use, Table 1 presents the main characteristics of the GPUs used in this study.
The multiGPU simulations are run on the two different systems. The first one is an Nvidia DGX1 – like node hosting eight Nvidia Tesla V100 Nvlink (32 GB) GPUs and two Intel Xeon Silver 4112 (2.6 GHz) CPUs. The second one is composed of 32 nodes, each featuring four Nvidia GeForce GTX Titan X Maxwell (12 GB) GPUs and two Intel XEON E52620V3 4112 (2.4 GHz) CPUs. To summarize the computational resources in use, Table 2 presents the main characteristics of the GPUs used in this study.
3.4 Measuring computational performance on a GPU
Omlin (2017), Räss et al. (2019a, b) and Alkhimenkov et al. (2021) demonstrated that a pertinent metric to quantify the performance of memorybounded algorithms is the effective memory throughput, i.e. MTP_{eff} in GB s^{−1}. It quantifies the efficiency of data transfers between the global memory (i.e. the onchip memory of the GPU) and the arithmetic logical units (ALUs) of the GPU. To determine the effective memory throughput, one must estimate (or quantify) the overall set of memory operations (readandwrite or readonly), i.e. n_{IO}, which are needed to resolve a given problem. Consequently, we carefully estimate the minimum number of memory operations while considering a GIMPMbased implementation. This results in the following effective memory throughput:
where n_{p} is the arithmetic precision (i.e. singleprecision floatingpoint format FP32 or doubleprecision floatingpoint format FP64) and t_{GPU} is the wallclock time in seconds to complete the n_{iter} iterations to solve for the numerical problem. For threedimensional problems, we estimate the minimal number of memory operations for an explicit GIMP implementation as
where n_{mp} is the number of material points, n_{n} is the number of associated nodes for an element (i.e. n_{n}=16 in 2D and n_{n}=64 in 3D), n_{no} is the number of nodes, and n_{el} is the number of elements. Additionally, we also report the count of calculation cycles per second of the GPU, i.e. iterations s^{−1} as well as the wallclock time. These two metrics give an intuitive sense of the timetosolution, which is convenient for potential application purposes.
In this section, we present two numerical models using the solver ep23De v1.0, namely,

Model 1, the granular collapse, which serves as
 a.
a validation benchmark against the results of the widely accepted experiment of Bui et al. (2008) under a threedimensional configuration and
 b.
a demonstration of the influence of the mesh resolution on plastic strain localization under a plane strain configuration;
 a.

Model 2, the threedimensional earth slump (Varnes, 1958, 1978), which serves as
 a.
an evaluation of the relative performances of a single and multipleGPUbased and CPUbased implementations of the solver ep23De v1.0 considering a variety of recent GPU architectures and
 b.
a showcase of a potential application of the solver ep23De v1.0 for an elastoplastic problem considering different isotropic peak cohesion fields (homogeneous and heterogeneous).
 a.
4.1 Model 1
4.1.1 Settings for Models 1a and 1b
We investigate the granular collapse of an aluminiumbar assemblage (Bui et al., 2008) under threedimensional or plane strain configurations. The geometry of the problem is shown in Fig. 5, and its variables are summarized in Table 3 for both threedimensional and plane strain configurations. Note that for Model 1a, we use the same number of elements in the x direction ${n}_{\text{el},x}=\mathrm{80}$ as in Huang et al. (2015). As a direct comparison for Model 1b under a plane strain configuration, Huang et al. (2015) used n_{el}=15 360, $\mathrm{\Delta}x=\mathrm{\Delta}z=\mathrm{2.5}$ mm and n_{mp}=25 600.
We consider a noncohesive granular material of density ρ = 2650 kg m^{−3}, with a bulk modulus K = 0.7 MPa and a Poisson's ratio ν=0.3, as in Huang et al. (2015). The cohesion is c = 0 Pa, and the internal friction angle is ϕ = 19.8^{∘} with a dilatancy angle ψ=0 according to Bui et al. (2008). However, the density and stiffness properties have negligible effects on the granular flow dynamics, as reported by Nguyen et al. (2020). We introduce local damping D (see Wang et al., 2016b) to resolve numerical results that are compatible with the experimental results of Bui et al. (2008). We find that D=0.025 results in the most compatible dynamics. The reasons for the introduction of local damping can be found in Appendix C. Fully fixed boundary conditions (i.e. no slip) are enforced at the bottom and rollers on the sidewalls. The total simulation time is 1.0 s, considering a the time step multiplier α=0.5.
4.1.2 Model 1a: the threedimensional granular collapse
To validate the numerical implementation under a GPU architecture, we first compare it against the wellknown granular collapse experiments initially performed by Bui et al. (2008). Here, we present and compare numerical results without focusing on the performance of the GPUbased implementation. All the simulations are performed on a consumer electronics RTX 3090 GPU with doublearithmetic precision (i.e. n_{p} = 8 bytes).
The results from the numerical simulation under a threedimensional configuration are shown in Fig. 6. A direct and visual comparison demonstrates excellent agreement between the numerical solver and the experiments of Bui et al. (2008). We observe a slightly lower runout distance, but the overall geometry of both the failure surface and the free surface is very close to the experimental data. We also report an angle of repose of ≈ 13^{∘}. This value is also consistent with the value reported by Bui et al. (2008), i.e. 14^{∘}. The good agreement between the numerical results and the experimental work of Bui et al. (2008) demonstrates that the solver ep23De v1.0 is suitable to simulate large deformation elastoplastic problems such as granular collapses.
The equivalent accumulated plastic strain ${\mathit{\u03f5}}_{\text{eqv}}^{\mathrm{p}}$ is shown in Fig. 7. We observe a coherent deformation of the granular material with a large shear zone that propagates backward from the base of the material to the top of the granular material. The mobilized granular material flows along a principal failure surface. However, the overall deformation pattern is rather coarse, i.e. fine structures or local shear bands are not yet observed, even though slight deformation heterogeneities can be observed. This coarse behaviour of shear banding is also consistent with previous studies (see Huang et al., 2015; Chalk et al., 2020; Zhang et al., 2021). This is mainly due to the background mesh resolution used in the numerical simulation. We further investigate shear banding using a higher background mesh resolution under a plane strain configuration in Model 1b.
4.1.3 Model 1b: the plane strain granular collapse
We investigate granular collapse under a plane strain configuration, as this allows an increase in the number of elements, resulting in an even finer background mesh (see Table 3). For Model 1a, the numerical solution is in agreement with the experimental work of Bui et al. (2008) regarding either the free surface or the failure surface (see Fig. 8). This demonstrates that both the threedimensional and plane strain configurations are in agreement with each other. However, we observe a lower runout for the granular collapse under a plane strain configuration.
An interesting feature of granular collapse is the equivalent accumulated plastic strain (see Figs. 9 and 10a and b). The GPUbased implementation allows both the background mesh resolution and the total number of material points to be increased. This results in finer plastic strain localizations, as demonstrated in Fig. 10a by the various shear bands and their complex interactions during collapse. Such detailed shear bands are almost impossible to obtain at lower resolutions, which demonstrates the importance of a GPUbased implementation to overcome the hardware limitation of a CPUbased implementation, i.e. mainly longer wallclock times.
Furthermore, Fig. 10a and b demonstrate the influence of the mesh resolution over shear banding: the finer the background mesh, the thinner the shear bands. This is significant since it shows that the dynamics of shallower granular avalanches appears to be more complex for higher resolutions.
4.2 Model 2
4.2.1 Settings for Models 2a and 2b
Here, we select a cohesive elastoplastic isotropic material (i.e. a homogeneous or heterogeneous peak cohesion field) with no dilatancy behaviour. It is modelled with a pressuresensitive Drucker–Prager model with linear strainsoftening behaviour. It is well known that the numerical solutions (as in FEM) are meshdependent when considering the strainsoftening behaviour of the material. We did not implement techniques to address this issue, but the use of nonlocal plasticity (Galavi and Schweiger, 2010; Burghardt et al., 2012) or viscoplastic formulations (Duretz et al., 2019) are possible ways to address this specific task.
We have chosen an arbitrary geometry (see Fig. 11 and Table 5), which represents an idealized threedimensional setting, to observe elastoplastic slumps (i.e. earth slumps according to the original classification proposed by Varnes, 1958, 1978), which are now classified as rotational slides in the recent update of the Varnes classification proposed by Hungr et al. (2014). The geometrical setting differs from the one typically used in the literature, as in Zhang et al. (2021). However, it promotes the compression of the toe, which is an expected feature we want to reproduce. The size of the physical domain ${l}_{z}\times {l}_{x}\times {l}_{y}$ is, at most, 12 m × 64 m × 1024 m for Model 2a, whereas it is 12 m × 64 m × 16 m for Model 2b.
We assume this setting features the principal firstorder characteristics of a typical rotational earth slump (Varnes, 1958, 1978), i.e. a complex zone of scarps (minor and major) delimiting a crownlike structure, followed by a transition (or depletion) zone in which the material flows homogeneously along internal shear zones due to severe plastic strain localizations and, finally, a compression (or accumulation) zone resulting in complex thrusting at the toe of the slump. Because of the nature of the boundary condition at the bottom of the material (i.e. freeslip), an additional horizontal sliding component is introduced within the rotational part of the displacement. This results in stronger deformations, which we want to highlight. However, the bottom boundary condition influences the shear band propagation and the overall behaviour by introducing a stronger horizontal component in the motion.
We select material properties (i.e. bulk and shear moduli K and G, friction angle ϕ, and peak and residual cohesion c_{peak} and c_{res}) that result in severe deformation processes and strain localizations. The material properties are presented in Table 4. They are close to the values commonly used in the literature (Wang et al., 2016b, a; Bandara et al., 2016; Zhang et al., 2021). To increase deformations even more, we also introduce a weak layer of thickness 0.3 × l_{z} m at the base of the material with a lower friction angle ϕ_{weak}. A time step multiplier α=0.5 is selected; i.e. Δt_{min} = 1.56 × 10^{−2} s is obtained over the whole simulation according to the CFL condition for both Models 2a and b. As in Zhang et al. (2021), elastic loading dynamic relaxation is applied for a period of t = 8 s (i.e. Models 2a and b), and the elastoplastic behaviour is activated for an additional 7 s, resulting in a total simulation time t = 15 s (i.e. Model 2b only).
Gaussian random fields (see Appendix B) are used to initialize the peak cohesion field c_{peak}, which is parametrized by an average cohesion ${\stackrel{\mathrm{\u203e}}{c}}_{\text{peak}}$ and its standard deviation σ (see Table 5) along with the residual cohesion ${c}_{\text{res}}={c}_{\text{peak}}/\mathrm{4}$. This allows us to account for heterogeneities within the material, which lead to complex and heterogeneous displacement fields. We first perform preliminary simulations with a constant cohesion field and notice a homogenous solution of the displacement field in the y direction. Using Gaussian fields allows us to mitigate this homogeneity.
Freeslip boundary conditions are applied on the sides and the bottom of the computational domain; only the normal component to the boundary is constrained, while the two others are free. Finally, and as suggested in Wang et al. (2016b) for landslide applications, we introduce local damping, i.e. D=0.1.
4.2.2 Model 2a: single GPU performances
Here, we investigate the computational performances of the solver ep23De v1.0 under a threedimensional configuration on a variety of GPUs with recent architectures: Ampere, Turing and Volta. Furthermore, we restrict our performance analysis only for the elastic loading phase (i.e. 8 s of simulation) because it is more complex to determine the exact number of material points that are yielding during each computational cycle (see Fig. 4) and to infer the exact effective memory throughput.
All the numerical simulations are performed on the computational resources and GPU hardware presented in Table 1 under doublearithmetic precision (i.e. n_{p} = 8 bytes in Eq. 38). As a reference baseline, we use the performance obtained for a CPUbased singlethreaded implementation of ep23De v1.0 on an i910900K CPU (e.g. latest Intel CPU chip). However, this is not representative of a highly optimized multithreaded implementation under a CPU architecture.
We report the effective memory throughput MTP_{eff} of the solver ep23De v1.0 on various GPUs and CPUs (see Fig. 12). An increase in the effective memory throughput is observed as the number of material points increases. All GPUs reach a maximum effective throughput, but the Tesla V100 scores a maximum effective throughput of ≈ 650 GB s^{−1}. This corresponds to 88 % of its peak throughput (for the GPU's hardware limit, see Table 1). We report a similar observation for the RTX 2080 Ti, MTP_{eff}≈ 320 GB s^{−1} corresponding to 62 % of its hardware limit. RTX 3090 and GTX 1650 reach MTP_{eff}≈ 405 GB s^{−1} and MTP_{eff}≈ 75 GB s^{−1}, respectively, which correspond to 52 % and 44 % of their respective hardware limits. Finally, we report a memory throughput of at least MTP_{eff}≈ 5 GB s^{−1} for the i910900K CPU (10 % of its hardware limit).
The overall results suggest, as in Räss et al. (2019b), that most recent GPUs, such as the datacentre Tesla V100 (Volta), offer significant performances compared to entrylevel consumer electronics GPUs, such as the GTX 1650. In terms of absolute performance, the more recent the GPU is, the higher its performance. A demonstration is given by the absolute effective throughput between the RTX 2080 Ti and the RTX 3090: the latter achieves an additional 20 % throughput compared to the former. We highly suspect the hardware itself to be the main reason for this. We further investigate the performances of the most recent datacentre GPU, i.e. the A100 (Ampere architecture), with its predecessor the V100 (Tesla architecture). The A100 reaches ≈ 1100 GB s^{−1}, which yields a 1.6fold performance gain with respect to the Tesla V100. When compared to the maximum effective memory throughput in Table 1, this corresponds to 97 % of the hardware limit.
Finally, we report the wallclock time for various computing architectures (see Fig. 13a). As expected by the maximum effective memory throughput, A100 delivers the fastest solution, regardless of the number of material points n_{mp}. The A100 GPU resolves a geometry of n_{mp}≈ 3.2 × 10^{6} in less than a minute (29 s), whereas the i910900K CPU resolves the same problem in more than an hour (5949 s). This corresponds to a 200fold performance gain (123fold performance gain for the V100; see Fig. 13b compared to the CPUbased implementation of ep23De v1.0. The RTX 2080 Ti and the RTX 3090 reach a 60fold and 77fold performance gain, respectively. However, the entrylevel GTX 1650 is only 10 times faster than i910900 K. As already shown in Fig. 12a, these performance gains are only expected when the different GPUs reach their maximum effective memory throughput. In terms of runtime, the performance gain (Fig. 13b) is in agreement with the memory throughputs reported in Fig. 12a.
4.2.3 Model 2a: multiGPU performances
To avoid transfers of frequent material points transfers amongst the GPUs, we consider an overlap of eight elements between neighbouring meshes, i.e. nine nodes. This results in a onedimensional GPU topology, for which both material points and meshes are distributed in the y direction of the global computational domain (see Figs. 11 and 14). Arranging GPUs in this direction allows the need to transfer material points amongst GPUs to be overcome, provided that the material point's displacement is not greater than the buffer zone, i.e. the element overlap. The evaluation of the multiGPU implementation is based on the Model 2a, with slight modifications; i.e. the number of elements in the y direction is largely increased. The size of the physical domain ${l}_{z}\times {l}_{x}\times {l}_{y}$ is, at most, 12 m × 64 m × (64 × 2048) m.
We consider two distributed computing systems for parallel GPU computation, using up to 8 Tesla V100 (Volta architecture) or 128 Geforce GTX Titan X
(Maxwell architecture) GPUs. All numerical simulations are performed using a singlearithmetic precision (i.e.
n_{p} = 4 bytes). This allows the maximum number of material points and mesh dimensions to be increased. In addition, our GPU
implementation relies on the usage of the builtin function atomicAdd()
. It does not support the doubleprecision floatingpoint format FP64
for GPUs with compute capabilities lower than 6.0, i.e. the Maxwell architecture amongst others. Note that, unlike the Tesla V100, the Geforce GTX
Titan X only delivers an effective memory throughput of MTP_{eff}≈ 100 GB s^{−1}. This corresponds to 38 % of
its hardware limit. This was already reported by Räss et al. (2019a) and Alkhimenkov et al. (2021), and it could be attributed to its older Maxwell
architecture (Gao et al., 2018). This performance drop is even more severe, mainly due to the use of builtin functions like atomicAdd()
.
We first performed parallel simulations with a moderate number of GPUs, up to 8 Tesla V100 NVlink (32 GB). The respective wallclock times are reported in Fig. 15. We report a wallclock time of ≈ 110 s for n_{mp}≈ 10^{8}. If n_{mp} is increased by a factor 2, 4 or 8, the wallclock time is roughly similar to the baseline, i.e. n_{GPU}=1. The effective memory throughput MTP_{eff} is shown in Fig. 16 (the total sum of MTP_{eff} across all the GPUs). Based on the memory throughput of one GPU, an estimation of a perfect weak scaling is possible. For eight GPUs, an ideal weak scaling corresponds to MTP_{eff} = 4824 GB s^{−1}, whereas we report MTP_{eff} = 4538 GB s^{−1}. This gives a parallel efficiency of ≈ 94 % and, an effective 7.5fold speedup. Similar observations are made for n_{GPU}=2 and n_{GPU}=4.
We investigate a parallel GPU computation using up to 128 Geforce GTX Titan X GPUs. This allows even larger geometries to be addressed, as shown in Fig. 17 where the geometry of nearly n_{mp}≈ 9.75 × 10^{8} is resolved in less than 8 min. For parallel computations of up to 64 GPUs, the wallclock time evolution is smooth. For 128 GPUs, the wallclock time is chaotic for fewer material points, whereas it stabilizes as the number of material points increases. We suspect the absence of computation–communication overlaps to be the main reason for this erratic behaviour. The communication between many GPUs requires careful synchronization between GPUs which can be hidden under computation–communication overlap (Räss et al., 2019a; Alkhimenkov et al., 2021). The total size of the overlap is constant, regardless of the y dimension. As the number of material points increases, the time spent on computation becomes larger compared to the time spent on exchanges between GPUs and the wallclock time stabilizes. The effective memory throughput MTP_{eff} is shown in Fig. 18. An ideal weak scaling corresponds to the effective memory throughput MTP_{eff} = 12 800 GB s^{−1} for 128 GPUs, whereas we report only MTP_{eff} = 11 326 GB s^{−1}. This gives a parallel efficiency of ≈ 90 % and an effective ≈ 113fold speedup.
4.3 Model 2b: homogeneous and heterogeneous slumps
As a final experiment, we show the results of the ep23De v1.0 solver for a slump with homogeneous or heterogeneous cohesion fields. In this numerical model, we only show the displacement field at the end of the numerical simulation at t = 15 s. The interested reader is referred to Appendix D for an overview of the temporal evolution of the equivalent plastic strain ϵ_{eqv} for the slump under the three settings of the peak cohesion field. All the numerical simulations are run on a laptop equipped with GTX 1650; t_{GPU}≈ 30 s with the settings presented in Table 5. In the following, we present the main results for the three peak cohesion fields, and we discuss the main characteristics obtained for typical slumping mechanics.
4.3.1 Homogeneous peak cohesion field
The homogeneous solution gives preliminarily interesting results (see Fig. 19). The firstorder characteristics of a slump can be observed, even though their magnitude is relatively fair compared to the real slump. The most striking feature is the development of one major shear zone, along which the material flows (i.e. depletion) towards the toe of the slump, resulting in a compression zone (i.e. thrusting and folding deformations). The crownlike structure develops linearly in the y direction and is highly localized at the surface of the slump (at x≈ 20 m in Fig. 19). However, the material flows homogeneously in the x direction (see the vertical profile in Fig. 19), as shown by the displacement field. The lateral variation of the displacement field (along the y direction) is almost nonexistent, which is mainly due to the spatial homogeneity of the peak cohesion field.
4.3.2 Isotropic Gaussian covariance
Considering heterogeneities with a Gaussian covariance function for the cohesion field, the displacement field starts to resolve a differential behaviour (see Fig. 20). Higher and/or weaker values of the peak cohesion field yield lower and/or greater displacements. This is obvious, especially in the transition zone where this differential is observable. In addition, the compression zone also starts to resolve spatial variations due to weaker and stronger cohesion values.
A striking difference is the shear zone itself (see Fig. D2): the shear zone exhibits a more complex spatial pattern, whereas only one major shear zone is observed in Fig. D2. Retrogressive shear banding appears during the time evolution of the slump, which suggests the development of a secondary shear zone within the slump. Moreover, the crownlike structure is now curved and not linear in the y direction. Its spatial extent is more important and is not as localized as in the homogeneous case. Nevertheless, a more complex arrangement of major and minor scarps within the crownlike structure has not yet been observed. Such a structure is more evident if one observes the accumulated equivalent plastic strain ${\mathit{\u03f5}}_{\text{eqv}}^{\mathrm{p}}$ in Fig. D2 in Appendix D.
The high magnitude of the displacement field in the areas $x\in [\mathrm{20};\mathrm{40}]$ and y≥ 8 m is due to a weaker zone in the peak cohesion field (see Fig. D2). This shows a strong influence of the heterogeneous peak cohesion field on the final displacement field. A lower shear strength of the material yields faster strainsoftening behaviour, promoting a faster response of shear banding.
4.3.3 Isotropic exponential covariance
Shear banding activities become even more complex when an exponential covariance function is used, relative to Fig. 19 and even to Fig. 20 to some extent. The spatial distribution of the peak cohesion (see Fig. D3) resolves finer heterogeneities with a smaller length scale compared to when Gaussian covariance is used. Principal differences are observed at the top and toe of the slump, where the crownlike structure turns into a complex zone made of minor and major scarps (see Fig. 21). The displacement field becomes highly heterogeneous, particularly at the toe and the top of the slump. However, it is also more homogeneous when compared with Fig. 20, particularly in $x\in [\mathrm{25};\mathrm{35}]$. The difference is evident between Figs. 22 and 20 at this particular location.
The difference between the Gaussian and exponential covariance of the peak cohesion suggests the following. Heterogeneous displacement fields could be influenced by larger and/or coarser fluctuations of the shear strength within the material. By extrapolation, this could imply that the magnitude of the heterogeneity might be related to the fluctuation scales of the peak cohesion field. Locally rather homogeneous fluctuations of the peak cohesion (i.e. Gaussian covariance) seem to promote an increasingly heterogeneous displacement field at the surface. The characteristic length scale of spatial fluctuations could have important implications for highly heterogeneous displacements within landslides. The same assumption could hold for understanding the more complex crownlike structure of slumps (see Fig. D3)
5.1 GIMPM suitability
We investigated granular collapses in both threedimensional and plane strain configurations. Our numerical results demonstrated the suitability of GIMPM to correctly reproduce experimental granular collapses. They also demonstrated that the results did not significantly differ between these two spatial configurations and that both approaches give similar numerical solutions.
5.2 Collapse limitation
For Model 1a, the principal hardware limit is the onchip memory of the GPU. Even though RTX3090 is supported by 24 GB DDR4, it is physically impossible to achieve the resolution used for plane strain granular collapse. This would require more than 24 GB of onchip memory. Model 1b demonstrated the importance of the background mesh resolution over strain localization. Using a higher numerical resolution (i.e. finer background mesh) allows fullresolution plastic strain localization. Similarly, future additional development efforts towards MPI implementation could resolve highly detailed threedimensional granular collapse simulations in the future. This will definitely benefit future studies on complex strain localization.
The wallclock time for Model 1b is t_{GPU} = 1470.5 s (25 min), and the number of iterations per second is 85.5 iterations s^{−1} for n_{mp} = 819 200. As a preliminary example, the same numerical model was performed by Wyser et al. (2020a), who reported 19.98 iterations s^{−1} for n_{mp} = 12 800. Proportionally, this corresponds to a performance gain factor of 275 for the GPUbased implementation (ep23De v1.0) over the MATLABbased implementation (fMPMMsolver v1.1) (Wyser et al., 2020a).
5.3 Performance
The performance analysis we carried out in Model 2a demonstrated that even though the algorithm heavily relies on atomic operations to accumulate material point quantities on the nodes, the effective memory throughput reaches 88 % at most (for Tesla V100). We expected a much lower throughput due to the use of these atomic operations, since they are likely known to undermine the computational performances of an implementation under previous GPU architectures (e.g. Kepler) (Dong et al., 2015a; Dong and Grabe, 2018; Gao et al., 2018). Our actual understanding (at least for a GPUbased implementation of GIMPM) is that the latest GPU architecture (Ampere and Turing) is now efficient when dealing with atomic operations and that the need to use a complex data layout for scattering is not as important as before. Furthermore, we identify the memory throughput as the main bottleneck: an additional 12 % performance improvement on the V100 before reaching the hardware limit of the memory bandwidth. The A100 shows that the solver reaches the hardware limit with an effective memory throughput which is very close (i.e. 97 %) to the actual maximum memory throughput. Similarly, the true limiting factor of the single GPU implementation is the hardware limit of the GPU onchip memory.
The multiGPU implementation resolves the onchip memory limitation problem. Our multiGPU implementation is particularly wellsuited to resolve highly detailed threedimensional shearbanding. We also reported decent wallclock times (less than 8 min) for simulations with nearly a billion material points. However, investigating highresolution threedimensional granular collapses is not possible under the assumptions made, because of small displacement required in the y direction. This is incompatible with threedimensional granular collapses. Hence, this motivates future deeper investigations toward a more versatile multiGPU implementation. In addition, we report a slight drop of the parallel efficiency, as the number of GPUs increases. Future works should be directed toward a parallel strategy that hides communication latency, as proposed in Räss et al. (2019a), Räss et al. (2020) and Alkhimenkov et al. (2021). This will allow us to achieve an optimal parallel efficiency of 95 %–98 % of the weak scaling tests involving up to 128 GPUs.
5.4 Slumping mechanics
We show the application of the GIMPM solver ep23De v1.0 for slumping mechanics. We have presented various slump results and demonstrated the significant influence of heterogeneities within the peak cohesion field over the displacement field or the equivalent plastic strain. However, we have arbitrarily selected values that resulted in severe deformations of the material, which we wanted to highlight to demonstrate the potential of the solver. Further efforts should now be oriented towards numerical models that are closer to real and welldocumented cases, such as in Tran and Sołowski (2019) and Ying et al. (2021). Despite the simplifications we made, we have reported threedimensional simulations that resolve all the firstorder characteristics of slumps, including complex major and minor scarps, different shear zones of various activities, and a complex arrangement within the compression zone. The use of a threedimensional GIMPM implementation under a GPU architecture will highly benefit future studies in the field, allowing faster and more detailed numerical simulations of heterogeneous and complex strain localization problems.
5.5 Local damping coefficient
Due to our explicit formulation, a damping relaxation term should be introduced to mitigate dynamic wave propagations (Wang et al., 2016c). In this work, we selected damping values that were either commonly accepted (e.g. D=0.1 for slumps) or that were better at resolving experimental results (e.g. D=0.025 for granular collapses). Future investigations should specifically address the influence of damping terms on the material's behaviour.
5.6 Code portability
Our numerical models showed the efficient computing capabilities of modern GPUs under the latest Nvidia GPU architectures. An important concern is the code portability. CUDA C is only applicable for Nvidia's GPUs and is not yet compatible with other corporations' GPUs, such as AMD (ATI Technologies). As such, an extension of the ep23De v1.0 solver towards an OpenCLbased implementation would ensure better code portability in the future.
We developed ep23De v1.0, an explicit GPUbased implementation of the generalized interpolation material point method that exploits the capabilities of the most recent GPU architectures (Ampere, Turing and Volta). We achieved fast execution times on a single GPU with a scattering approach that relies on extensive usage of atomic operations. We report, at most, an effective memory bandwidth of 88 % relative to the maximal hardware capabilities of the GPUs. We achieve, at most, a 200fold performance gain on a single GPU compared to a singlethreaded CPUbased implementation of the solver. On entrylevel customer electronics GPUs, we report a ≈ 10fold performance gain. Our multiGPU implementation permits geometries with almost a billion material points to be resolved and demonstrates fast execution times. We achieve a parallel efficiency of ≈ 94 % on weak scaling tests for 8 GPUs and ≈ 90 % for 128 GPUs. We also report that the memory bandwidth is the main limiting performance factor. We validated our solver against the wellknown experimental results of the granular collapse problem in a threedimensional configuration. We show applications of the solver to model slumping mechanics in threedimensional configurations considering different material heterogeneities.
One of the most important problems of any sMPM formulation is the cellcrossing instability (or error; see Steffen et al., 2008; Wilson et al., 2021). As material points move through the mesh, they cross element boundaries. The discontinuous gradient due to the C_{0} continuity of the basis functions results in spurious oscillations of the stress field and internal forces (González Acosta et al., 2020, 2019; Bardenhagen and Kober, 2004) when material points cross element boundaries.
To solve for this instability, Bardenhagen and Kober (2004) introduced the generalized interpolation material point method (GIMPM). Whereas the material point is treated as a point in sMPM, Bardenhagen and Kober (2004) assigned a spatial extent or a domain to the material point. Alternative basis functions are constructed, i.e. to consider the material point domain, as follows:
where v_{p} is the material point volume, Ω_{p} denotes the material point domain, χ_{p}(x) is the particle characteristic function, N_{n}(x) is the basis function (or shape function) for the mapping between the material point p and its associated nodes n, and $\mathit{x}={\mathit{x}}_{p}{\mathit{x}}_{n}$ are the local coordinates between node n and material point p.
The particle characteristic function must satisfy the partition of unity property, i.e. ${\sum}_{p}{\mathit{\chi}}_{p}\left(\mathit{x}\right)=\mathrm{1}$ (Bardenhagen and Kober, 2004). The simplest particle characteristic function is given by the hat function, i.e.
The GIMPM basis functions and derivatives are constructed analytically (Coombs et al., 2020; Charlton et al., 2017) in one dimension from a convolution of the standard finiteelement basis functions and the material point characteristic function (Steffen et al., 2008), i.e.
where l_{p} is the length of the material point domain, h is the mesh resolution, and $x={x}_{p}{x}_{n}$, where x_{p} is the coordinate of a material point and x_{n} is the coordinate of its associated node n. The twodimensional basis function of a node n with its material point p is constructed as
for which the gradient is defined as
In earth sciences, random fields (Christakos, 1992) are numerically generated predictions of a geophysical property (i.e. rock or soilrelated properties) with probabilistic spatial variability. These predictions are based on (i) an assumed probability density function, i.e. characterized by a mean value μ with a standard deviation σ, and (ii) an assumed spatial correlation function, characterized by fluctuation scales in a vector format, i.e. $\mathit{\lambda}=({\mathit{\lambda}}_{x},{\mathit{\lambda}}_{y},{\mathit{\lambda}}_{z})$. In regard to numerical modelling, the principal requirement is that both small and large scales are simultaneously resolved over the computational mesh to ensure physically meaningful solutions.
Recently, Räss et al. (2019b) presented an efficient implementation based on a spectral representation of Gaussian random fields for geophysical applications using either Gaussian or exponential covariance functions. The numerical codes, named GRFS, were made available by Räss et al. (2019b) in both native MATLAB and CUDA C languages^{2}. However, a sufficiently large number of harmonics should be used to obtain convergent Gaussian random fields, as stated in Räss et al. (2019b).
Similar to the random material point method (RMPM; see Wang et al., 2016a; Liu et al., 2019; Remmerswaal et al., 2021) initially proposed by Fenton and Vanmarcke (1990) to generate RFs for a finiteelement mesh (RFEM), we combined this approach with the codes proposed by Räss et al. (2019b) to generate an isotropic peak cohesion field to demonstrate its influence on the mechanical behaviour.
In Huang et al. (2015), no volumetric locking mitigation strategy was introduced, even though loworder elements were used. This should promote volumetric locking and an overall stiffer response of the granular material. In addition, Huang et al. (2015) used the standard (or original) material point method (instead of the generalized interpolation material point method), which is well known to introduce spurious oscillations of internal forces (González Acosta et al., 2020).
When implementing the proposed volumetric locking mitigation strategy, we observed (a) larger deformations of the granular material with a stronger vertical compaction (i.e. stronger vertical displacement) and (b) slightly longer runout distances when compared to the experimental data. The softer mechanical response of the granular material had to be compensated for somehow, which can be achieved by the introduction of a small local damping parameter.
We reproduced the numerical setting used in Huang et al. (2015) with the same mesh resolution, i.e. Δx=Δy = 2.5 mm, and a similar number of material points n_{mp} = 28 800 with an initial number of material points per initially filled element n_{pe}=9. The material parameters used for this preliminary investigation are presented in Sect. 4.1.1.
Figure C1a and b show the major differences between either a lockingfree or a lockingprone solution and the experimental results. As mentioned before, a slightly longer runout distance is obtained for the lockingfree solution. As a result, the numerical prediction given by the lockingfree solution of the free surface is underestimated. However, the most noticeable difference is the failure surface. Whereas the failure surface predicted by the lockingprone solution fits with the experiment of Bui et al. (2008), it diverges for a lockingfree solution. In particular, the onset of the failure surface at the top of the material is underestimated by the lockingfree solution compared to the experimental results. This is due to the softer response of the granular material when volumetric locking is mitigated, which promotes greater vertical compaction and a stronger runout distance at the same time.
Even though the introduction of local damping better resolves the experimental results, one can argue that the lockingfree solution without the introduction of local damping still agrees with the experiment of Bui et al. (2008). The overall response of the numerical granular collapse is still very close to the actual physical experiment, and the differences between the numerical and experimental results can still be considered acceptable.
We further present additional threedimensional results for Model 2b for a homogeneous cohesion field (see Figs. C2 and C3). Threedimensional simulations of cohesive material better illustrate the influence of volumetric locking. Figure C3 demonstrates that a significantly smoother pressure field is resolved with the proposed method.
In addition, the pressure field is certainly smoothed, but it does not significantly differ from the original pressure field (in locations where locking is minimum). Volumetric locking is particularly highlighted within shear bands due to isochoric plastic flows, resulting in significant stress oscillations.
The solver ep23De v1.0 developed in this study is licensed under the GPLv3 free software licence. The solver ep23De v1.0 archive (v1.0) is available from a permanent DOI repository (Zenodo) at https://doi.org/10.5281/zenodo.5600373 (Wyser et al., 2021) (the latest version of the code is available for download from GitHub at https://github.com/ewyser/ep23De, last access: 26 October 2021).
The data used can be found in Bui et al. (2008).
EW and YA wrote the original manuscript and developed the first version the ep23De v1.0 solver. MJ and YYP supervised the early stages of the study and provided guidance. All authors have reviewed and approved the final version of the paper.
The contact author has declared that neither they nor their coauthors have any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Yury Alkhimenkov gratefully acknowledges support from the Swiss National Science Foundation (grant no. 172691). Yury Alkhimenkov and Yury Y. Podladchikov gratefully acknowledge support from the Russian Ministry of Science and Higher Education (project no. 0751520191890). We gratefully acknowledge Thomas Poulet, Quoc Anh Tran and José León González Acosta for constructive suggestions that helped us to improve the quality of the paper.
This research has been supported by the Swiss National Science Foundation (grant no. 172691) and the Russian Ministry of Science and Higher Education (project no. 0751520191890).
This paper was edited by Thomas Poulet and reviewed by José León González Acosta and Quoc Anh Tran.
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The routines of the ep23De v1.0 solver are available for download from GitHub at https://github.com/ewyser/ep23De (last access: 26 October 2021). The routines archive (v1.0) (Wyser et al., 2021) is available from a permanent DOI repository (Zenodo) at https://doi.org/10.5281/zenodo.5600373.
The GRFS routines are available at https://bitbucket.org/lraess/grfs/src/master/ (last access: 25 February 2021).
 Abstract
 Introduction
 Numerical implementation
 GIMPM implementation under a GPU architecture
 Results
 Discussion
 Conclusions
 Appendix A: GIMPM basis functions and derivatives
 Appendix B: Gaussian random cohesion fields
 Appendix C: Volumetric locking and damping corrections
 Appendix D: Heterogeneities for the peak cohesion field
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Numerical implementation
 GIMPM implementation under a GPU architecture
 Results
 Discussion
 Conclusions
 Appendix A: GIMPM basis functions and derivatives
 Appendix B: Gaussian random cohesion fields
 Appendix C: Volumetric locking and damping corrections
 Appendix D: Heterogeneities for the peak cohesion field
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References