Discrete Element Method Modelling Of Complex Gr...

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A simple and fast original method to create irregular particle shapes for the discrete element method using overlapping spheres is described. The effects of its parameters on the resolution of the particle shape are discussed. Overlapping spheres induce a non-uniform density inside the particle leading to incorrect moments of inertia and therefore rotational behaviour. A simple method to reduce the error in the principal moments of inertia which acts on the individual densities of the spheres is also described. The pertinence of the density correction is illustrated by the case of free falling ballast particles forming a heap on a flat surface. In addition to improve behaviour, the correction reduces also computational time. The model is then used to analyse the interaction between ballast and geogrid by simulating pull-out tests. The pulling force results show that the model apprehends better the ballast geogrid interlocking than models with simple representation of the shape of the particles. It points out the importance of modelling accurately the shape of particles in discrete element simulations.

Modelling is an important tool in understanding the behaviour of biological tissues. In this paper we advocate a new modelling framework in which cells and tissues are represented by a collection of particles with associated properties. The particles interact with each other and can change their behaviour in response to changes in their environment. We demonstrate how the propose framework can be used to represent the mechanical behaviour of different tissues with much greater flexibility as compared to traditional continuum based methods.

In this paper we present a discrete-element agent-based modelling environment to model cell-cell and cell-ECM mechanical interactions. In contrast to the overwhelming majority of agent based models of biological tissues which use a single discrete agent to represent each cell, we propose to represent cells and ECM using multiple agents. This has the immediate advantage of increasing the simulation resolution, allowing large deformation of individual cells (e.g. to model cell spreading or epithelial-to-mesenchymal transitions), realistic cell-cell and cell-ECM physical interactions and a more nuanced control of cell behaviour, which is difficult in a continuum mechanics approach. The main disadvantage is an increase in computational expense; nevertheless, the proposed solution method is completely explicit and therefore perfectly suitable for parallel implementation on Graphics Processing Units, using techniques similar to those presented in [29]. We will demonstrate the flexibility of the proposed agent-based model on a number of biological problems of intense ongoing interest and compare it to competing simulation methods in order to demonstrate its benefits.

Different properties of a cell can be created via suitable adjustment to the various properties of the component agents. External properties, such as basement adhesion and stromal stiffness, can be similarly recreated. Finer scale structures and processes can be represented by having more and smaller agent-based particles in a simulation. While there have been extraordinary advances in digital computers, which now permit the representation of tissues by many millions of particles, it remains important to do the computations as efficiently as possible. For this reason, multiscale agent-based discrete models may be developed. In this approach, a region of tissue may be represented generally using large particles, but particular regions of the tissue where more information is required may be represented using smaller particles, which provide higher local resolution. As importantly, for such particle-based methods, the position, material and biochemical properties are only stored at the particles rather than at every node in a spatial grid, avoiding unnecessary computations required for re-generating the grid every time the geometry of the cell structure changes. This allows the optimization of storage and reduction of processing bandwidth, enabling efficient parallelization of computations.

The total drag force acting on a particle I due to all its neighbouring particles becomes, taking into account eq (8), as(11)and is therefore proportional to the difference between the velocity of particle I and the average velocity of the surrounding particles which exercise an influence on particle I. The chosen form for the drag force, which originates from multiple dissipative processes taking part at cellular level, is different from the form used in other discrete element simulations, where it only acts in the normal direction between particles [26]. This drag force resembles more a viscous drag, as it is proportional to the velocity of the reference particle relative to its surrounding particles. More complex constitutive equations can be employed to represent particular dissipative processes as required.

Experimental cell diameters were in the range of 10 to 15 μm, with a nucleus diameter range of 3 to 7 μm. In our model we created a cell having a 15 μm diameter with a 6 μm nucleus. The computational domain for a single cell is represented by 981 circular discrete elements (726 cytoplasm, 155 membrane and 100 nucleus discrete elements).

In this paper we present a framework for modelling biological tissues based on discrete particles. The cell and tissue models consist of a collection of particle types and a set of simple interactions between particles. The particles interactions are controlled by parameters define at particle level, which simplifies the implementation of the computational framework. An explicit solution method leads to an algorithm easy to implement on parallel hardware, such as GPU.

In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field.

In the 1990s a new class of meshfree methods emerged based on the Galerkin method. This first method called the diffuse element method[4] (DEM), pioneered by Nayroles et al., utilized the MLS approximation in the Galerkin solution of partial differential equations, with approximate derivatives of the MLS function. Thereafter Belytschko pioneered the Element Free Galerkin (EFG) method,[5] which employed MLS with Lagrange multipliers to enforce boundary conditions, higher order numerical quadrature in the weak form, and full derivatives of the MLS approximation which gave better accuracy. Around the same time, the reproducing kernel particle method[6] (RKPM) emerged, the approximation motivated in part to correct the kernel estimate in SPH: to give accuracy near boundaries, in non-uniform discretizations, and higher-order accuracy in general. Notably, in a parallel development, the Material point methods were developed around the same time[7] which offer similar capabilities. Material point methods are widely used in the movie industry to simulate large deformation solid mechanics, such as snow in the movie Frozen.[8] RKPM and other meshfree methods were extensively developed by Chen, Liu, and Li in the late 1990s for a variety of applications and various classes of problems.[9] During the 1990s and thereafter several other varieties were developed including those listed below.

The primary areas of advancement in meshfree methods are to address issues with essential boundary enforcement, numerical quadrature, and contact and large deformations.[24] The common weak form requires strong enforcement of the essential boundary conditions, yet meshfree methods in general lack the Kronecker delta property. This make essential boundary condition enforcement non-trivial, at least more difficult than the Finite element method, where they can be imposed directly. Techniques have been developed to overcome this difficulty and impose conditions strongly. Several methods have been developed to impose the essential boundary conditions weakly, including Lagrange multipliers, Nitche's method, and the penalty method.

One recent advance in meshfree methods aims at the development of computational tools for automation in modeling and simulations. This is enabled by the so-called weakened weak (W2) formulation based on the G space theory.[28][29] The W2 formulation offers possibilities to formulate various (uniformly) "soft" models that work well with triangular meshes. Because a triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence enables automation in modeling and simulation. In addition, W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the Smoothed Point Interpolation Methods (or S-PIM).[16] The S-PIM can be node-based (known as NS-PIM or LC-PIM),[30] edge-based (ES-PIM),[31] and cell-based (CS-PIM).[32] The NS-PIM was developed using the so-called SCNI technique.[26] It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.[33] The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments. The W2 formulation has also led to the development of combination of meshfree techniques with the well-developed FEM techniques, and one can now use triangular mesh with excellent accuracy and desired softness. A typical such a formulation is the so-called smoothed finite element method (or S-FEM).[34] The S-FEM is the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler. 781b155fdc