Material Point Method Code Research Articles

Easily porting material point methods codes to GPU

AbstractThe material point method (MPM) is computationally costly and highly parallelisable. With the plateauing of Moore’s law and recent advances in parallel computing, scientists without formal programming training might face challenges in developing fast scientific codes for their research. Parallel programming is intrinsically different to serial programming and may seem daunting to certain scientists, in particular for GPUs. However, recent developments in GPU application programming interfaces (APIs) have made it easier than ever to port codes to GPU. This paper explains how we ported our modular C++ MPM code to GPU without using low-level hardware APIs like CUDA or OpenCL. We aimed to develop a code that has abstracted parallelism and is therefore hardware agnostic. We first present an investigation of a variety of GPU APIs, comparing ease of use, hardware support and performance in an MPM context. Then, the porting process of to the Kokkos ecosystem is detailed, discussing key design patterns and challenges. Finally, our parallel C++ code running on GPU is shown to be up to 85 times faster than on CPU. Since Kokkos also supports Python and Fortran, the principles presented therein can also be applied to codes written in these languages.

MPM vs. CEL: Numerical modelling of penetration processes

MPM vs. CEL: Numerical modelling of penetration processes

Two different phase field models of the explicit material point method for brittle dynamic fracture

This paper proposes two different phase-field models for brittle dynamic fracture using the explicit material point method (MPM). Based on the consistent stiffness matrix, Model I directly solves the linear equations of the activated Euler grid to obtain the nodal phase field; Model II presents a method for calculating the lumped stiffness matrix and solves the phase field by using a residual iterative method. In addition, we developed the phase field MPM (PH-MPM) code based on GPU parallelization. Based on two one-dimensional models, we study the homogeneous and nonhomogeneous performances of the two models and verify their accuracy by comparing them with the analytical solution. We explored the computational accuracy and efficiency of the two models by performing classic two-dimensional numerical experiments. The results show that both models can analyse fractures with high precision. Meanwhile, compared with Model Ⅰ, the computational efficiency of Model Ⅱ has apparent advantages.

GPU material point method (MPM) and its application on slope stability analysis

Slope stability analysis is a popular area of interest in geotechnical engineering and is significant to landslide prevention and other geohazards. Owing to the success of the combined numerical methods of Lagrange and Euler, the material point method (MPM) has been developed. The benefit of MPM for the study of the evolution process for the slope is its application to efficiently simulate small- to large-scale deformation processes. The high-performance computing GPU-based MPM code named as CoSim-MPM is developed and the strength reduction method (SRM) is introduced to perform the quantitative analysis of slope failure risk assessment. Based on the developed code, a method to define the factor of stability (Fos) is provided and validated for two typical slopes: a homogenous slope and a slope with a weak interlayer. By comparing the MPM results with that of the limit equilibrium method (LEM), it is shown that both of the potential slide surfaces and Fos correspond well with each other. The accuracy of the MPM-SRM results surpasses that of the LEM for the failure mechanical and deformation characteristics of the landslide. As a practical application, the stability and failure mechanisms of Dongmiaojia slope, downstream of the Xiaolangdi Dam, China, is analyzed. All the results show that the GPU-based MPM can have powerful advantages for the stability analysis of slope with high efficiency. In addition, CoSim-MPM has strong scalability and can be applied to large-scale simulation research in the field of geotechnical engineering.

Numerical Simulation of Mesodamage Behavior of Concrete Based on Material Point Method

Concrete consists of coarse aggregates, mortar matrix, and interfacial transition zone (ITZ) between them at the mesoscale. Considering these three phases, many numerical tests have been conducted to study the mesodamage behavior of concrete, in which a variety of numerical methods have also been adopted. These methods are mainly based on the finite element method (FEM); however, some other methods have been proven to be helpful as well. For example, the material point method (MPM) has the advantage of building a numerical model based on pixel or voxel of the image and is capable of solving large deformation problems. In view of this, MPM is introduced in this paper. Firstly, a method for establishing the numerical specimen is put forward, considering the original sample of its mesoscopic geometric character. Then, a stochastic damage constitutive model considering the heterogeneity of the concrete is proposed. Next, the numerical model and the constitutive model are incorporated into an MPM code to conduct numerical tests. The uniaxial tension and compression tests of a random‐aggregate model and a double‐aggregate specimen under uniaxial tension are then simulated numerically to validate the proposed method. Results show that the proposed method can well capture the main macroscopic mechanical behavior of concrete and the mesoscopic damage initiation and propagation. It is also found that MPM can save the time of model establishing and improve calculation efficiency. The influences of different parameters of the proposed constitutive model are also clarified through a parametric study. The proposed method can provide a useful tool for concrete numerical testing and for studying the mechanical behavior of concrete at mesoscale.

A fully coupled hydro-mechanical material point method for saturated dense granular materials

The stability of a dense granular assembly can be greatly reduced by a pore pressure of the interstitial fluid, and the body may fail and transit from a solid-like state to a fluid-like state. This process involves two major problems: large deformation and hydro-mechanical coupling. In this work, a three-dimensional fully coupled hydro-mechanical model using material point method (MPM) is developed. Darcy's law, considering the inertial effect, is adopted to govern the motion of interstitial water, and the conservation of momentum of the mixture is used to govern the motion of the solid, i.e., granular materials. The spatial discretization schemes for these equations are derived using the generalized integration material point method (GIMP), and the proposed coupled MPM formulation is implemented in a three-dimensional numerical code. The developed model is first quantitatively validated by comparing the simulation results of temporal evolution of spatial distribution of hydraulic pressure in a one-dimensional oedometer test with the analytical results. An experiment is designed to observe the failure of a saturated sand pile, in which the partial-saturated region is avoided by increasing the hydraulic head at the input boundary, and the kinetic energy of water is dissipated by a filtering cloth. The failure process is simulated with the MPM code. It is found that the location of the shear band in the simulation agrees with the location of the sliding surface in the experiment. The temporal evolutions of the spatial distributions of hydraulic pressure and the solid velocity distribution at a specific time are given to provide insight into the mechanism of the failure process. This work would be helpful in understanding the initiation mechanism of debris flows induced by rainfall, and sand production in gas hydrate-bearing sediments due to increasing fluid content associated with hydrate dissociation.

Programming the material point method in Julia

This article presents the implementation of the material point method (MPM) using Julia. Julia is an open source, multi-platform, high-level, high-performance dynamic programming language for technical computing, with syntax that is familiar to Matlab and Python programmers. MPM is a hybrid particle-grid approach that combines the advantages of Eulerian and Lagrangian methods and is suitable for complex solid mechanics problems involving contact, impact and large deformations. We will show that a Julia based MPM code, which is short, compact and readable and uses only Julia built in features, performs much better (with speed up of up to 8) than a similar Matlab based MPM code for large strain solid mechanics simulations. We share our experiences of implementing MPM in Julia and demonstrate that Julia is a very interesting platform for rapid development in the field of scientific computing.

An Implicit Material Point Method Applied to Granular Flows

The main objective of this work lies in the development of a variational implicit Material Point Method (MPM), implemented in the open source Kratos Multiphysics framework. The ability of the MPM technique to solve large displacement and large deformation problems is widely recognised and its use ranges over many problems in industrial and civil engineering. In the current work the continuum based implicit MPM is applied to engineering applications, where granular material flow is involved.For the resolution of the length and time scale of these particular problems, both continuum and discrete models are typically used. Even if discrete techniques predict more feasible results, nowadays, their use is limited to the investigation of element tests of particles, or to the simulation of reduced systems, not allowing to make important decisions in the analysis and design of granular processes. Some advantages of MPM over discrete methods are tested, such as, the ability to simulate granular flow at the large scale with acceptable computational cost and the capability to get information of stress and strain state in a more straightforward way.The focus of this paper is a comparative study between an irreducible and a mixed formulation, both implemented in the MPM code, to assess the improvement in accuracy and reliability of the numerical results when the latter formulation is adopted.

Development of an implicit material point method for geotechnical applications

An implicit material point method (MPM), a variant of the finite element method (FEM), is presented in this paper. The key feature of MPM is that the spatial discretisation uses a set of material points, which are allowed to move freely through the background mesh. All history-dependent variables are tracked on the material points and these material points are used as integration points similar to the Gaussian points. A mapping and re-mapping algorithm is employed, to allow the state variables and other information to be mapped back and forth between the material points and background mesh nodes during an analysis. In contrast to an explicit time integration scheme utilised by most researchers, an implicit time integration scheme has been utilised here. The advantages of such an approach are twofold: firstly, it addresses the limitation of the time step size inherent in explicit integration schemes, thereby potentially saving significant computational costs for certain types of problems; secondly, it enables an improved algorithm accuracy, which is important for some constitutive behaviours, such as elasto-plasticity. The main purpose of this paper is to provide a unified MPM framework, in which both quasi-static and dynamic analyses can be solved, and to demonstrate the model behaviour. The implementation closely follows standard FEM approaches, where possible, to allow easy conversion of other FEM codes. Newton’s method is used to solve the equation of motion for both cases, while the formation of the mass matrix and the required updating of the kinematic variables are unique to the dynamic analysis. Comparisons with an Updated Lagrangian FEM and an explicit MPM code are made with respect to the algorithmic accuracy and time step size in a couple of representative examples, which helps to illustrate the relative performance and advantages of the implicit MPM. A geotechnical application is then considered, illustrating the capabilities of the proposed method when applied in the geotechnical field.

Validación numérica, teórica y experimental del método del punto material para resolver problemas geotécnicos

This paper investigates the possibility of using the material point method (MPM) to solve small strains quasi-static problems and dynamic problems related to large distortions. Traditional methods such as the finite element method (FEM) face difficulties when large strains are involved. Therefore, tools such as the MPM have become more important in recent years. As a new tool, the MPM needs to prove its functionality for geotechnical engineering problems. In first place the MPM mathematical formulation is shortly described. Next, numerical simulations of a shallow foundation, an unconfined compression test and a slope problem are performed in an open source MPM code. The results are compared with FEM simulations, analytical solutions and real laboratory tests. The study shows qualitative and quantitative agreement when compared; a better performance of MPM for solving stresses better than strains is detected. The set of simulations validates the MPM to solve geotechnical engineering problems when dealing with small and large strains. However, the traditional FEM showed a better performance for quasi-static cases.

Material point method of modelling and simulation of reacting flow of oxygen

Aerospace vehicles are continually being designed to sustain flight at higher speeds and higher altitudes than previously attainable. At hypersonic speeds, gases within a flow begin to chemically react and the fluid's physical properties are modified. It is desirable to model these effects within the Material Point Method (MPM). The MPM is a combined Eulerian–Lagrangian particle-based solver that calculates the physical properties of individual particles and uses a background grid for information storage and exchange. This study introduces chemically reacting flow modelling within the MPM numerical algorithm and illustrates a simple application using the AeroElastic Material Point Method (AEMPM) code. The governing equations of reacting flows are introduced and their direct application within an MPM code is discussed. A flow of 100% oxygen is illustrated and the results are compared with independently developed computational non-equilibrium algorithms. Observed trends agree well with results from an independently developed source.

Coupling of finite element method with material point method by local multi-mesh contact method

As a Lagrangian particle method, the material point method (MPM) has the potential to model extreme deformation of materials, where the traditional finite element method (FEM) often encounters mesh distortion and element entanglement which lead to numerical difficulties. However, FEM is more accurate and efficient than MPM for problems with small deformation. It is therefore desirable to model the body with extreme deformation by MPM and the body with small deformation by FEM, respectively. In this paper, a method to handle the contact interaction between the MPM body and the FEM body is proposed, which is implemented on the background grid of MPM. By this method, FEM is coupled with MPM and a hexahedral element is incorporated into our 3D explicit MPM code MPM3D®. Several numerical examples, including plate impact, sphere rolling, perforation of thick plate, and fluid–structure interaction problems, are studied and the numerical results are in good agreement with analytical solution and results available in the literature. The coupling of FEM and MPM offers advantages of both FEM and MPM.

Contact algorithms for the material point method in impact and penetration simulation

AbstractThe inherent no‐slip contact constraint in the standard material point method (MPM) creates a greater penetration resistance. Therefore, the standard MPM was not able to treat the problems involving impact and penetration very well. To overcome these deficiencies, two contact methods for MPM are presented and implemented in our 3D explicit MPM code, MPM3D. In MPM, the impenetrability condition may not satisfied on the redefined regular grid at the beginning of each time step, even if it has been imposed on the deformed grid at the end of last time step. The impenetrability condition between bodies is only imposed on the deformed grid in the first contact method, while it is imposed both on the deformed grid and redefined regular grid in the second contact method. Furthermore, three methods are proposed for impact and penetration simulation to determine the surface normal vectors that satisfy the collinearity conditions at the contact surface. The contact algorithms are verified by modeling the collision of two elastic rings and sphere rolling problems, and then applied to the simulation of penetration of steel ball and perforation of thick plate with a particle failure model. In the simulation of elastic ring collision, the first contact algorithm introduces significant disturbance into the total energy, but the second contact algorithm can obtain the stable solution by using much larger time step. It seems that both contact algorithms give good results for other problems, such as the sphere rolling and the projectile penetration. Copyright © 2010 John Wiley & Sons, Ltd.

Shared Memory OpenMP Parallelization of Explicit MPM and Its Application to Hypervelocity Impact

The material point method (MPM) is an extension of particle-in-cell method to solid mechanics. A parallel MPM code is developed using FORTRAN 95 and OpenMP in this study, which is designed primarily for solving impact dynamic problems. Two parallel methods, the array expansion method and the domain decomposition method, are presented to avoid data races in the nodal update stage. In the array expansion method, two-dimensional auxiliary arrays are created for nodal variables. After updating grid nodes in all threads, the auxiliary arrays are assembled to establish the global nodal array. In the domain decomposition method, the background grid is decomposed into some uniform patches, and each thread deals with a patch. The information of neighbor patches is exchanged through shared variables. After updating nodes in all patches, their nodal variables are assembled to establish the global nodal variables. The numerical tests show that the domain decomposition method has much better parallel scalability and higher parallel efficiency than the array expansion method. Therefore, a parallel computer code, MPM3DMP, is developed based on the domain decomposition method. Finally, MPM3DMP is applied to a large-scale simulation with 13,542,030 particles for obtaining the high-resolution results of debris cloud in hypervelocity impact.

Implicit time integration for the material point method: Quantitative and algorithmic comparisons with the finite element method

AbstractAn implicit integration strategy was developed and implemented for use with the material point method (MPM). An incremental‐iterative solution strategy was developed around Newton's method to solve the equations of motion with Newmark integration to update the kinematic variables. Test problems directly compared the implicit MPM solutions with those obtained using an explicit MPM code and implicit finite element (FE) code. Results demonstrated very good agreement with FE predictions and also illustrated several advantages in comparison to calculations using the explicit MPM code. In particular, the accuracy of the implicit solution was superior to the explicit MPM when compared to validated FE solutions, and by definition the implicit time integration is unconditionally stable. Similarities between the assembly of the implicit MPM equations and those of the FE method were identified and should allow further algorithmic improvement. Copyright © 2003 John Wiley & Sons, Ltd.