*Vasily Bulatov and Wei Cai*

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198526148
- eISBN:
- 9780191916618
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198526148.003.0016
- Subject:
- Computer Science, Software Engineering

The phase field method (PFM) can be used as an approach to dislocation dynamics simulations alternative to the line DD method discussed in Chapter 10. The degrees of freedom in PFM are continuous ...
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The phase field method (PFM) can be used as an approach to dislocation dynamics simulations alternative to the line DD method discussed in Chapter 10. The degrees of freedom in PFM are continuous smooth fields occupying the entire simulation volume, and dislocations are identified with locations where the field values change rapidly. As we will see later, as an approach to dislocation dynamics simulations PFM holds several advantages. First, it is easier to implement into a computer code than a line DD model. In particular, the complex procedures for making topological changes (Section 10.4) are no longer necessary. Second, the implementation of PFM can take advantage of well-developed and efficient numerical methods for solving partial differential equations (PDEs). Another important merit of PFM is its applicability in a wide range of seemingly different situations. For example, it is possible to simulate the interaction and co-evolution of several types of material microstructures, such as dislocations and alloying impurities, within a unified model. PFM has become popular among physicists and materials scientists over the last 20 years, but as a numerical method it is not new. After all, it is all about solving PDEs on a grid. Numerical integration of PDEs is a vast and mature area of computational mathematics. A number of efficient methods have already been developed, such as the finite difference method [121], the finite element method [122], and spectral methods [123], all of which have been used in PFM simulations. The relatively new aspects of PFM are associated with the method’s formulation and applications, which are partly driven by the growing interest in understanding material microstructures. In Section 11.1, we begin with the general aspects of PFM demonstrated by two simple applications of the method not related to dislocations. Section 11.2 describes the elements required to adapt PFM to dislocation simulations. There we will briefly venture into the field of micromechanics and consider the concept of eigenstrain. The elastic energy of an arbitrary eigenstrain field is derived in Section 11.3. Section 11.4 discusses an example in which the PFM equations for dislocations are solved using the fast Fourier transform method.
Less

The phase field method (PFM) can be used as an approach to dislocation dynamics simulations alternative to the line DD method discussed in Chapter 10. The degrees of freedom in PFM are continuous smooth fields occupying the entire simulation volume, and dislocations are identified with locations where the field values change rapidly. As we will see later, as an approach to dislocation dynamics simulations PFM holds several advantages. First, it is easier to implement into a computer code than a line DD model. In particular, the complex procedures for making topological changes (Section 10.4) are no longer necessary. Second, the implementation of PFM can take advantage of well-developed and efficient numerical methods for solving partial differential equations (PDEs). Another important merit of PFM is its applicability in a wide range of seemingly different situations. For example, it is possible to simulate the interaction and co-evolution of several types of material microstructures, such as dislocations and alloying impurities, within a unified model. PFM has become popular among physicists and materials scientists over the last 20 years, but as a numerical method it is not new. After all, it is all about solving PDEs on a grid. Numerical integration of PDEs is a vast and mature area of computational mathematics. A number of efficient methods have already been developed, such as the finite difference method [121], the finite element method [122], and spectral methods [123], all of which have been used in PFM simulations. The relatively new aspects of PFM are associated with the method’s formulation and applications, which are partly driven by the growing interest in understanding material microstructures. In Section 11.1, we begin with the general aspects of PFM demonstrated by two simple applications of the method not related to dislocations. Section 11.2 describes the elements required to adapt PFM to dislocation simulations. There we will briefly venture into the field of micromechanics and consider the concept of eigenstrain. The elastic energy of an arbitrary eigenstrain field is derived in Section 11.3. Section 11.4 discusses an example in which the PFM equations for dislocations are solved using the fast Fourier transform method.

*Vasily Bulatov and Wei Cai*

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198526148
- eISBN:
- 9780191916618
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198526148.003.0011
- Subject:
- Computer Science, Software Engineering

As was discussed in Chapter 2, stable and accurate numerical integration of the MD equations of motion demands a small time step. In MD simulations of solids, the integration step is usually of the ...
More

As was discussed in Chapter 2, stable and accurate numerical integration of the MD equations of motion demands a small time step. In MD simulations of solids, the integration step is usually of the order of one femtosecond (10−15 s). For this reason, the time horizon ofMDsimulations of solids rarely exceeds one nanosecond (10−9 s). On the other hand, dislocation behaviors of interest typically occur on time scales of milliseconds (10−3 s) or longer. Such behaviors remain out of reach for direct MD simulations. Time-scale limits of a similar nature also exist in MC simulations. For instance, the magnitude of the atomic displacements in the Metropolis algorithm has to be sufficiently small to ensure a reasonable acceptance ratio, which results in a slow exploration of the configurational space. This disparity of time scales can be traced to certain topographical features of the potential-energy function of the many-body system, typically consisting of deep energy basins separated by high energy barriers. The system spends most of its time wandering around within the energy basins (metastable states) only rarely interrupted by transitions from one basin to another. Whereas the long-term evolution of a solid results from transitions between the metastable states, direct MDand MC simulations spend most of the time faithfully tracing the unimportant fluctuations within the energy basins. In this sense, most of the computing cycles are wasted, leading to very low simulation efficiency. Because the transition rates decrease exponentially with the increasing barrier heights and decreasing temperature, this problem of time-scale disparity can be severe.
Less

As was discussed in Chapter 2, stable and accurate numerical integration of the MD equations of motion demands a small time step. In MD simulations of solids, the integration step is usually of the order of one femtosecond (10−15 s). For this reason, the time horizon ofMDsimulations of solids rarely exceeds one nanosecond (10−9 s). On the other hand, dislocation behaviors of interest typically occur on time scales of milliseconds (10−3 s) or longer. Such behaviors remain out of reach for direct MD simulations. Time-scale limits of a similar nature also exist in MC simulations. For instance, the magnitude of the atomic displacements in the Metropolis algorithm has to be sufficiently small to ensure a reasonable acceptance ratio, which results in a slow exploration of the configurational space. This disparity of time scales can be traced to certain topographical features of the potential-energy function of the many-body system, typically consisting of deep energy basins separated by high energy barriers. The system spends most of its time wandering around within the energy basins (metastable states) only rarely interrupted by transitions from one basin to another. Whereas the long-term evolution of a solid results from transitions between the metastable states, direct MDand MC simulations spend most of the time faithfully tracing the unimportant fluctuations within the energy basins. In this sense, most of the computing cycles are wasted, leading to very low simulation efficiency. Because the transition rates decrease exponentially with the increasing barrier heights and decreasing temperature, this problem of time-scale disparity can be severe.