Material-point-method simulation

Due to the sheer number, it is not possible to include in this review all of the proposed hybrid methodologies that deal with the coupling of continuum and atomistic regions in dynamical simulations. However, before leaving this section, we want to mention a few more types of methodologies. First, the multigrid methods,of which the recent work of Waisman and Fish is a good example. Also, the MPM/MD method, where the material point method (MPM) is used instead of the finite-element method (FEM) to couple continuum mechanics to conventional molecular dynamics. 

Two-Dimensional Mixed Mode Crack Simulation Using the Material Point Method. 

Multiscale Simulation from Atomistic to Continuum-Coupling Molecular Dynamics (MD) with the Material Point Method (MPM). 

Figure 1.4 Computer simulated fragmentation of a ceramic coating on a solid substrate using the material point method, from Li et al. (2012). The method can be adapted to model device fragmentation during later stage of degradation.

Computer analysis of crack propagation through finite element grids was developed by several authors. Cracks are represented by discontinuities of the finite element mesh, and smeared crack models were also applied. Cement-based matrices were considered as linear elastic bodies up to the point where cracks open and later their behaviour becomes highly non-linear. Various methods are applied to represent non-linear and heterogeneous materials and to simulate their behaviour under load (cf. Petersson 1981). In discrete models, cracks are represented as discontinuities in the finite element mesh. This is also where smeared crack models are introduced. 

Simulation of Dynamic Crack Growth Using the Generalized Interpolation Material Point (GIMP) Method. 

Numerical simulations offer several potential advantages over experimental methods for studying dynamic material behavior. For example, simulations allow nonintrusive investigation of material response at interior points of the sample. No gauges, wires, or other instrumentation are required to extract the information on the state of the material. The response at any of the discrete points in a numerical simulation can be monitored throughout the calculation simply by recording the material state at each time step of the calculation. Arbitrarily fine resolution in space and time is possible, limited only by the availability of computer memory and time. 

The work described in this paper is an illustration of the potential to be derived from the availability of supercomputers for research in chemistry. The domain of application is the area of new materials which are expected to play a critical role in the future development of molecular electronic and optical devices for information storage and communication. Theoretical simulations of the type presented here lead to detailed understanding of the electronic structure and properties of these systems, information which at times is hard to extract from experimental data or from more approximate theoretical methods. It is clear that the methods of quantum chemistry have reached a point where they constitute tools of semi-quantitative accuracy and have predictive value. Further developments for quantitative accuracy are needed. They involve the application of methods describing electron correlation effects to large molecular systems. The need for supercomputer power to achieve this goal is even more acute. 

Alternatively, fundamental parameter methods (FPM) may be used to simulate analytical calibrations for homogeneous materials. From a theoretical point of view, there is a wide choice of equivalent fundamental algorithms for converting intensities to concentrations in quantitative XRF analysis. The fundamental parameters approach was originally proposed by Criss and Birks [239]. A number of assumptions underlie the application of theoretical methods, namely that the specimens be thick, flat and homogeneous, and that, for calibration purposes, the concentrations of all the elements in the reference material be known (having been determined by alternative methods). The classical formalism proposed by Criss and Birks [239] is equivalent to the fundamental influence coefficient formalisms (see ref. [232]). In contrast to empirical influence coefficient methods, in which the experimental intensities from reference materials are used to compute the values of the coefficients, the fundamental influence coefficient approach calculates... 

In this type of apparatus, the two phases do not come to equilibrium, at any point in the contactor and the simulation method is based, therefore, not on a number of equilibrium stages, but rather on a consideration of the relative rates of transport of material through the contactor by flow and the rate of interfacial mass transfer between the phases. For this, a consideration of mass transfer rate theory becomes necessary. 

For the purposes of considering diffusion at microelectrodes, it is convenient to introduce two categories of electrodes those to which diffusion occurs in a linear fashion and those to which diffusion occurs in a nonlinear fashion. The former category consists of cylindrical and spherical electrodes. As shown schematically in Figure 12.2A, the lines of flux (i.e., the pathway followed by material diffusing to the electrode) are straight, and the current density is the same at all points on the electrode. Thus, the diffusion problem is one-dimensional (i.e., distance from the electrode surface) and involves solution of the appropriate form of Fick s second law, Equation 12.7 or 12.8, either by Laplace transform methods or by digital simulation (Chap. 20). 

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