Cellular Materials - CellMat 2014 October 22 – 24
International Congress Center Dresden Dresden, Germany
Digital Representation of Complex Cellular Structures for
Nu-merical Simulations
J. Ettrich1,2, A. August1,2, M. Roelle1, B. Nestler1,2
1
Karlsruhe University of Applied Sciences, Institute of Materials and Processes, Moltkestrasse 30, 76133 Karlsruhe, Germany
2
Karlsruhe Institute of Technology, Institute of Applied Materials, Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germany
Abstract
Cellular Solids are a rather young group of materials which gain increasing attention of aca-demics and engineers during the recent decades. Utilising metals as base material results in excellent mechanical, electrical, thermal and acoustic properties, whereas specific properties like thermal expansion coefficient, electric conductivity or melting temperature remain un-changed compared to the raw material [1]. Characterisation of the governing physical proper-ties of these materials in terms of experimental investigations is a time consuming, costly and sometimes cumbersome task. Thus, the application of appropriate numerical methods to ana-lyse structure property correlations is of increasing interest in each of the relevant disciplines. A challenge for all numerical methods is the realistic representation of the microstructure. The majority of the latest publications employ either expensive or time consuming investigations using industrial computed tomography (CT) scanning, or numerical modelling using repre-sentative elementary volumes (REV) in terms of simple unit cells. The latter method contra-dicts the stochastic nature of the real or natural structures, and lacks from capturing natural defects and structure specific features. In our contribution, we present the results of the nu-merical modelling of open cell metal foam structures using a heuristic algorithm, embedded in a modern phase-field method in comparison with measurements of real samples. Furthermore, an outlook on applications of these cellular structures in a fluid flow and heat transfer simula-tion is given.
Keywords: Phase-Field, lattice-Boltzmann, Heat-Transfer, Cellular Solids, Open Cell Metal Foam
1 Introduction
With regards to the difficulties associated with the measurement at the pore scale level of cel-lular solids it seems obvious to employ advanced numerical methods. However, modelling and simulation does not come for free.
Making the complex geometric structures of cellular solids available to a numerical simula-tion requires the representasimula-tion in a digitally, machine readable format. Today, processing power is falling in cost, which permits for the numerical simulation of even bigger computa-tional domains and more complex physics. However, the efforts spent on the tasks of mesh generation, model specification and evaluation of the results are still the most time
consum-structure under consideration, but also to provide an elegant and smart numerical treatment of the complex domain. Thus, the success of a computational approach heavily depends on the representation of the underlying geometrical and topological entities.
The majority of the latest publications employs either expensive and time consuming inves-tigations using industrial computed tomography (CT) scanning, e.g. [2, 3], or numerical mod-elling using representative elementary volumes (REV) in terms of unit cells ranging from simple cubic cells to regular dodecahedra and even more complex tetracaidecahedra, e.g. [4–8]. Modelling methods based on stochastic approaches, sophisticated tessellation methods and growth algorithms are presented in e.g. [9–12].
In the present work we present an alternative approach for the numerical modelling of cel-lular solid using a manufacturing inspired heuristic algorithm [13], embedded in a modern phase-field method and simulation framework PACE3D[14]. We provide exemplary applica-tions for realistic open cell foam structures, as well as comparison to experimental data and an outlook on numerical simulations of heat transfer and fluid flow in cellular structures provid-ed in an accompanying publication in the same conference proceprovid-edings.
2 Open cell foam modelling
The geometrical and topological elements of synthetic structures require a qualitative repre-sentation in order to ensure the success of numerical simulation which is strongly dependent on them. It is a challenge from the mid-19th century, the beginning of numerical methods, until today, where broad compendium of tools and numerical methods are available to be used with many applications.
In order to be able to use the complex geometric structures of cellular solids in a numerical simulation, one requires the representation to be available in a digital format, so as to be read-able by a machine. Foam structures can be modelled by a number of methods such as repre-sentative elementary volumes (REV), which include characteristics from a simple cubic cell up to a regular dodecahedron [5] and an even more complex tetrakaidecahedron [4, 7].
(a) (b) (c) (d)
Figure 1. (a) Compact packing of spheres (red) with uniform pore diameter. (b) 3D Voronoi diagram, where the different colours represent different Voronoi regions. (c) The struts computed from the Voronoi diagram, (d) the final structure [15].
For the description of the pore structure, a representative unit cell is constructed by the models which mark the structure and configuration of the foam. Nodes that are connected by edges make up the unit cell. In this way open-cell metal foam can be defined as a combination of nodes and edges. The distribution of mass in these models is performed at the struts and the knots of the regular structures. Typical foam characteristics such as porosity or structure per-formance data like thermal conductivity, are then derived analytically by means of these mod-els.
The foam structure, discussed in [5], is modelled by way of a simple cubic elementary cell, which provides a rather simple representative version of the structure. Based on the elemen-tary cell, a theoretical model for pressure loss and heat transfer can be derived. The method presented in [16] is even simpler, because it assumes a one-dimensional heat transfer and models a foam structure with the help of a batch of cylinders. In [5], the shape of the inter-connected edges is investigated, which is cross-sectional of the dodecahedron with twelve flat pentagonal faces. Modelling and subsequent comparison with regard to porosity and pressure loss is done with elementary cells surrounded by prismshaped and round edges. The tetrakaidecahedron can be described as a polyhedron which has six quadratic and eight hex-agonal faces [17], better reflecting an imbalance in the shape of the face and the edge in a real foam structure. Apart from other scientists, this model is employed by [4, 7, 18–20] to model porosity, pressure loss and heat transfer in open-cell metal foams.
Because of the simple representation by means of the above-mentioned unit cells, the tailed numerical investigations either show structural deficiencies at a pore scale level, or de-pend on a geometry, which requires an expensive tomography in order to be captured, cf. [2, 8, 21].
A new algorithm which is able to create three-dimensional cellular solids, foam or fabric-like structures with open or closed cells, in a synthetic fashion, is introduced in [13]. The workflow includes three single steps:
Step 1: Gradual filling of the computational domain with a compact packing of spheres, which controls the scattering of the pores, cf.fig. 1(a).
Step 2: Derivation of the basic topology of the structure from a Voronoi decomposition of the spatial domain, see fig. 1(b).
Step 3: Creation of the cellular structure from the boundaries of the Voronoi diagram, see fig. 1(c).
There is often a dissimilarity in the cross-sectional shape of the struts of the cellular struc-tures. It can often be observed, that aluminium foams have a triangular cross-sectional shape and show a thickening in the vicinity of the knots. The thickness of the struts can be related to the distance of their knots, which results in a re-sampling of the last mentioned property. Tak-ing the connections of two Voronoi regions into account also leads to a re-samplTak-ing of the triangular shape. In case material is set or not, the distance to the nearest boundary region provides another criterion, fig. 1(d).
Phase field models are very useful for solving free boundary problems mathematically, in an easy and efficient way. The advantage of the phase field method, as compared to the classical front tracking method, is the fact, that phase boundaries are not treated explicitly and that the application of the boundary conditions at the interface is also done implicitly.
The phase field approach is favourable and unique, because of the naturally embedded in-terface capturing and the possibility of setting up multidisciplinary couplings to various phys-ical models, which distinguishes the phase field method from other methods. Interfacial mo-tion and complex geometrical evolumo-tion such as catastrophic phase terminamo-tions are problems of phase transformations, for which the phase field method has successfully been applied to.
The phase field method used in this work builds on the original formulation of [14]. Mass diffusion, temperature and front tracking is solved simultaneously in multiphase and multi-component materials. In the following application, we assume two different phases, a solid and a fluid phase. The order parameter in the bulk solid and the bulk fluid phase is solid = 1
and fluid = 0, respectively, and vice versa. The condition solid + fluid = 1, is valid all the
time for the entire numerical domain. With fluid = and solid = 1, the representation of
the order parameters can be reduced to a single field variable .
The distribution of the order parameter specifies the cellular structures of interest and shows areas of pure solid and pure fluid. The interfaces between the two phase states are de-scribed by a diffuse interface of finite width , where 0 < solid , fluid < 1. For simplicity, no
phase dynamics such as an evolution of the phase states is considered in this work. As a con-sequence, the phase field equations are not solved. However, the profiles of the diffuse inter-face serve as a means to describe complex microstructure geometries by generating a smooth phase boundary. The diffuse interface representation serves as an efficient and straightforward concept to numerically interpolate physical parameters such as fluid flow and heat transfer, which are examples of other physical quantities, across the boundary of adjacent phase states.
Table 1. Quantitative comparison of the porosity of experimental and artificial foam structures
sample experiment modelling error
Nr.1 88.441 % 88.440 % 0.001 % Nr.2 92.777 % 92.780 % 0.003 % Nr.3 87.707 % 87.700 % 0.007 % Nr.4 83.846 % 83.850 % 0.004 % Nr.5 87.889 % 87.890 % 0.001 %
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Exemplary Applications
The above mentioned algorithm developed by [13] is used for the reconstruction of open cell metal foam structures, e.g. in [15]. The method is controlled by statistical measures (mean and standard deviation) of pore and edge diameter, whereas the porosity and specific gravity are compared to experimental data. Thanks to the heuristic nature of the underlying algorithm, we
are able to reflect the stochastic nature of the foam structures. As a consequence, realistic ef-fects in fluid flow and heat transfer are recovered in respective simulations, cf. [15]. Exempla-rily, fig. 2 depicts the qualitative comparison of topological features observed in both, real foam samples and computational model structures. For a quantitative comparison, foams are generated using experimentally derived pore scale measures. Table 1 shows the quantitative comparison of experimentally measured and algorithmically generated foam structures and illustrates high agreement by small deviations.
Figure 2. Qualitative comparison of typical topological features found in real foam samples as well as in compu-tational model structures [15].
The qualitative impact of the statistical control parameters – namely pore-diameter, edge-thickness and the related standard deviations – are depicted in figs. 3. A specified level of statistical variation takes adequate account of the intrinsic stochastic scattering of the pore scale measures as a result of manufacturing. Furthermore, it allows for the modelling of dif-ferent levels of homogeneity. Compared to classical sharp interface methods, the numerical
(a) (b) (c)
Figure 3. Alteration of the statistical parameters of the pore scale control parameters (pore-diameter and edge-thickness) shows an impact on the homogeneity of the pore distribution and the variation of the foam ligaments. While the mean value of the pore-diameter and edge-thickness are kept constant, the deviation is reduced from (a) to (c). As a consequence, (a) shows a vast variety of pore- and edge diameters, (b) gives a more homogeneous distribution, whereas (c) shows the result of almost constant pore diameters and small deviations in
representation of a solid phase within the phase-field simulation package PACE3D allows for the superposition of different structures seamlessly and with great ease, and therefore permits to create even more complex arrangements. Exemplarily, fig. 4 displays a structure with two completely different basic scales. Since foam structures of different basie materials (e.g. cop-per and aluminium) have different cross sectional shapes of the ligaments [1], the modelling algorithm provides an option to create structures with both, round as well as triangular cross sectional edges, cf. figs. 5.
(a) (b)
Figure 4. The modelling algorithm allows to take into account the cross sectional shape of the ligaments. Whereas the (a) round cross sectional shape is suitable e.g. for copper foams, the (b) triangular cross sectional shape very well matches the real shape of aluminium type foam structures.
5 Outlook and Conclusion
In this publication, we present the application of an algorithm [13], capable to artificially gen-erate realistic geometries of cellular solids. It permits a realistic representation of foam struc-tures, which qualitatively comprise the same topological and geometrical features as their real counterparts. Furthermore, foam structures generated from experimental pore scale measures reveal extremely good quantitative agreement with real samples in terms of porosity. Compar-ison exhibits deviations of less than a fraction of one percentage. With computational simula-tion of fluid flow or heat transfer in view, porosity is a key property for the correct physical representation and capturing of fundamental effects. Most of the classic sharp interface meth-ods require obvious user interaction, adaptation and time. Here, the most striking fact in uti-lizing the above mentioned diffuse interface method is, that it provides a seamless, straight-forward numerical approach with minimum user interaction and minimal turnaround times.
6 Acknowledgments
The authors thank the scientific staff at the Institute of Materials and Processes at the Karls-ruhe University of Applied Sciences for their support and work on the PACE3D package. Support provided by the Federal Ministry of Education and Research (BMBF) through the FHProfUnt founding project SimFoam (FKZ 17029X10) is also greatly acknowledged. We are grateful for financial support through the Helmholtz POF III (progam oriented funding) within the program EMR (energy efficiency, resources and materials).
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