Abstract
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.
Highlights
The material point method (MPM) has been shown to be a robust spatial discretisation method for simulating multi-phase interactions involving large deformations and failure evolution
The computational mesh may be maintained in its original position, or it can be adjusted in an appropriate way to avoid mesh distortion after each time/loading step, thereby removing the disadvantage of the finite element method (FEM) for which extreme mesh distortion may occur due to large deformations
Implicit dynamic MPM formulations [16,17] have been reported, this paper aims to provide a clear and straightforward description of all the necessary techniques for adapting an existing FEM implementation into one based on the implicit MPM
Summary
The material point method (MPM) has been shown to be a robust spatial discretisation method for simulating multi-phase interactions involving large deformations and failure evolution. During 1994–96, Sulsky et al [1,2,3] first developed and applied the method for modelling solid materials This led to researchers, from different fields, recognising the potential of the method and adapting it to various applications, e.g. silo discharge and plastic forming [4,5], explosion problems, exploiting its ability to represent an arbitrary geometry [6,7], large-scale response of cellular constructs in biomechanics [8], and, more recently, for. The term implicit MPM refers to a framework where both dynamic and quasi-static problems (with inertial terms neglected) can be solved effectively. The subsequent section focuses on a series of representative examples to investigate and validate the presented framework for quasi-static and dynamic analyses, respectively, with the results being compared with those obtained from an explicit code, in order to gain a thorough understanding of how the implicit algorithm behaves
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