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Preferred contact method Email Text message. When will my order be ready to collect? In prior work [ 73 , 82 , 83 ], it was found that spectral representations on functions on the local state space offered many advantages both in compact representation and in reducing the computational cost. In such cases, h indexes the spectral basis functions employed.
The selection of these functions depends on the nature of local state descriptors. Examples include i the primitive basis or indicator functions used to represent simple tessellation schemes [ 71 , 72 , 74 , 75 , 76 , 77 , 78 , 79 , 84 ], ii generalized spherical harmonics used to represent functions over the orientation space [ 73 , 82 ], and iii Legendre polynomials used to represent functions over the concentration space [ 83 ]. The discretization scheme for both the microstructure function and the vector space needed to define the spatial correlations, illustrated on a simple two-phase composite material.
The discretized vectors r describe the relative positions between different spatial locations. More detailed explanations on the MKS homogenization and localization linkages can be found in prior literature [ 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 83 , 84 ].
The MKS homogenization workflow left consists of four steps. Discretize the raw microstructure with the microstructure function. Compute 2-point statistics using local states Eq.
Create low dimensional microstructure descriptors using dimensionality reduction techniques Eq. Establish a linkage with low dimensional microstucture descriptors using machine learning. The MKS localization workflow right consists of 2 steps. Calibrate physics-based kernels using regression methods Eq. It provides a high-level, computationally efficient framework to implement data pipelines for classification, cataloging, and quantifying materials structures for PSP relationships.
PyMKS is written in Python, a natural choice for scientific computing due to its ubiquitous use among the data science community as well as many other favorable attributes [ 86 ]. PyMKS is licensed under the permissive MIT license [ 87 ] which allows for unrestricted distribution in commercial and non-commercial systems.
PyMKS consists of four main components including a set of tools to compute 2-point statistics, tools for both homogenization and localization linkages, and tools for discretizing the microstructure. In addition, PyMKS has modules for generating data sets using conventional numerical simulations and a module for custom visualization of microstructures. This is a high-level system for combining multiple data and machine learning transformations into a single customizable pipeline with only minimal required code.
This approach makes cross-validation and parameter searches simple to implement and avoids the complicated book keeping issues associated with training, testing, and validating data pipelines in machine learning. The starting point for an MKS homogenization analysis is to use 2-point statisics as outlined in Eq. The calibration of the influence kernels is achieved using a variety of linear regression techniques described in numerous previous studies [ 71 , 72 , 73 , 83 ]. The MKSLocalizationModel object uses fit and predict methods to follow the standard interface for a Scikit-learn estimator object.
To use either the homogenization or the localization models in PyMKS, the microstructure first needs to be represented by a microstructure function, m j [ h , s ]. The bases module in PyMKS contains four transformer objects for generating the m j [ h , s ] using a varietly of discretization methods [ 71 , 72 , 73 , 74 , 75 , 76 , 77 , 83 ].
These four objects can be thought of as materials-specific extension to the feature extraction module in Scikit-learn. A PrimitiveBasis object uses indicator or hat functions and is well suited for microstructures that have discrete local states e. The LegendreBasis and FourierBasis objects create spectral representations of microstructure functions defined on nonperiodic and periodic continuous local state spaces, respectively.
For example, functions over a range of chemical compositions can be described using LegendreBasis , while functions over orientations in two-dimensional space can be described using FourierBasis. Furthermore, GSHBasis creates compact spectral representations for functions over lattice orientation space such as those needed to describe polycrystalline microstructures [ 88 , 89 , 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 , 98 , 99 ].
The MicrostructureGenerator object creates stochastic microstructures using digital filters. This assists users in creating PyMKS workflows even when data is unavailable.
PyMKS has objects for generating sample data from both a spinodal decomposition simulation using the CahnHilliardSimulation object and a linear elasticity simulation using the ElasticFESimulation object. PyMKS comes with custom functions for visualizing microstructures in elegant ways in the tools module.
These are used extensively in the PyMKS example notebooks to minimize incidental code associated with visualization. NumPy arrays are the primary data structure used throughout PyMKS and provide the basic vector and matrix manipulation operations. SfePy is used to simulate linear elasticity to create sample response field data. Matplotlib is used to generate custom microstructure visualizations.
https://alsismangspam.tk PyMKS leverages from existing tools, standards, and web resources wherever possible. Additionally, a Google group is used as a public forum to discuss the project development, support, and announcements see pymks-general googlegroups. The Travis CI continuous integration tool is used to automate running the test suite for branches of the code stored on GitHub. Code standards are maintained by following the Python PEP8 standards and by reviewing code using pull requests on GitHub.
A demonstration of the MKS homogenization and localization workflows as shown in Fig.
The method is demonstrated by the evolution of an approved acetylcholinesterase inhibitor drug donepezil I into brain-penetrable ligands with either specific polypharmacol. The individual pointed out that that approach does not provide any information about grain boundary composition. Frontiers in Materials , 4 DOI: The MKS homogenization workflow left consists of four steps. In previous sections, we have addressed the acceptance of materials data infrastructures from a technological viewpoint.
In this example, the MKSHomogenizationModel is used to create a structure-property linkage between a 2-phase composite material and effective stiffness C x x. This function has several arguments. Lastly, seed is used as the seed for the random number generator.
One sample from each of the 16 different microstructure classes used for calibration of the homogenization model. Before an instance of the MKSHomogenizationModel can be made, an instance of a basis class is needed to specify the discretization method for the microstructure functions see Fig. For this particular example, there are only two discrete phases numerated by 0 and 1.
It has been shown that the primitive basis provides the most compact representation of discrete phases [ 71 , 71 , 74 , 77 , 78 , 79 , 81 , 84 ]. Mean R -squared values and standard deviation as a function of the order of the polynomial and the number of principal components. The mean R -squared values indicated by the points and the standard deviation indication by the shared regions as a function of the number of principal components for the first three orders of a polynomial function Color figure online. Goodness-of-fit plot for effective stiffness C x x for the homogenization model Color figure online.
The model is calibrated using delta microstructures. Delta microstructure right and its associated strain field left. The delta microstructures and their local response fields are used to calibrate the localization model Color figure online. Random microstructure and its local strain field found using finite element analysis Color figure online. A comparison between the local strain field computed using finite element left and the prediction from the localization model right Color figure online.
The MKS framework offers a practical and computationally efficient approach for distilling and disseminating the core knowledge gained from physics-based simulations and experiments using emerging concepts in modern data science.