Percentage porosity computation of three-dimensional non-convex porous geometries using the direct Monte Carlo simulation

Abstract

The pursuit of more representative numerical models for open-cell metallic foams requires the computation of volume and percentage porosity of geometries containing randomly distributed interconnected pores, which is one of the main characteristics that determines its mechanical properties. From a mathematical standpoint, the analytical definition of foam geometries forms a three-dimensional non-convex set. It is known that the volume computation of n-dimensional polytopes and sets is a P-hard problem. A common way to approach this problem is using the Monte Carlo techniques; however, efforts are oriented toward the treatment of convex polytopes and polyhedrons. In this article, the Direct Monte Carlo Simulation (DMCS) is used to compute the percentage porosity of three-dimensional non-convex sets. A single-thread Python code was implemented, and tests were run to estimate the percentage porosity of three-dimensional open-cell porous geometries. Measurements of percentage porosity and runtime requirements over cubical and cylindrical geometries containing from 100 to 4000 overlapping spherical pores showed high accuracy and consistency in non-convex three-dimensional sets, while the proposed algorithm achieved a significant reduction in computing time with respect to the currently available method. In the same manner, results from the proposed algorithm were compared with a similar software available, showing a gain in both performance time and accuracy.

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Correspondence to Mauricio Campillo.

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Appendices

Appendix A

In this appendix, the complete Python code script is presented. This code runs in single core configuration allowing to estimate the porosity of cylindrical- or cubic-shaped foam geometries containing randomly distributed spherical pores when a complete analytical description of it (i.e. overall dimensions and pores location and dimension) is given in a JavaScript script file. The code uses Eq. 15 to average the results obtained by a series of Monte Carlo simulations, based on an LHS strategy, according Eqs. 5, 6, 7, and 8. The code will iterate until two established criteria are met, which are a minimum number of iteration and a maximum standard error, according to Eq. 18. This code requires the user to provide the filename with its extension as system argument (e.g., Cube_100_1.js). The path to the file is assumed to be the current work directory

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Appendix B

In this appendix, more detailed information regarding the computed percentage porosity for the data set obtained by means of the proposed algorithm and McVol after 20 independent runs are presented in Table 2.

Table 2 Minimum, maximum, and average percentage porosity and absolute error respect to reference value obtained from ANSYS Design Modeler Module obtained after 20 independent runs using the proposed implementation and McVol

Appendix C

In this appendix, more detailed information regarding the computed runtimes for the data set for the proposed algorithm and McVol after 20 independent runs are presented in Table 3.

Table 3 Minimum, maximum, and average runtime after 20 independent runs using the proposed implementation and McVol

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Campillo, M., Pérez, P., Daher, J. et al. Percentage porosity computation of three-dimensional non-convex porous geometries using the direct Monte Carlo simulation. Engineering with Computers (2019). https://doi.org/10.1007/s00366-019-00866-2

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Keywords

  • Direct Monte Carlo simulation
  • Percentage porosity
  • Non-convex sets
  • Metallic foams
  • Python