Tech Briefs

A two-stage approach is used for element-by-element analysis of a micro-structure.

Synthetic and natural micro-architectures occur frequently, and multiphase functionally graded composites are becoming increasingly popular for applications requiring optimized/tailored material properties. When dealing with such materials computationally, one issue that immediately arises is the analysis of the mechanical properties of macroscopically inhomogeneous multi-scale structures. The bulk response of these structures can be determined by performing full finite element analysis (FEA); that is, with the entire geometry discretized at a resolution high enough to accurately model the smallest length scale of interest. However, these full models may easily exceed hundreds of millions, or potentially billions, of degrees of freedom, and solving problems of this magnitude may only be possible with the use of supercomputing facilities.

The process of Generating the Macro Mesh: Femoral head modeling (a) input dataset, (b) coarse tetrahedralization, and (c) micro-element.

Additionally, in an iterative optimization process where the performance of the structure may be evaluated thousands of times, the use of full FEA simulations becomes highly impractical. When the performance of a structure is evaluated in an optimization process, typically only some aspect of the bulk response, such as deflection, is considered. For such properties, the use of full FEA simulations to model the problem may be excessive. In the present project, a novel, two-stage approach to solving such a large problem by performing element-by-element homogenization of the micro-structure, followed by solving the global problem with a coarser mesh, was explored.

The approach taken is based on creating a coarse tetrahedral/hexahedral discretization of the domain using traditional volume meshing techniques, and assigning appropriate material properties based on a finite element homogenization based on high-resolution mesh at the microstructural level of the macro tetrahedra or hexahedra. In effect, two length scales are decoupled by computing effective properties using the finite element approach for each macro-element. The novelty here lies in effectively discretizing the full 3D mesh into larger tetrahedra and hexahedra, and computing homogenized properties for each macro-element based on exact meshed domains representing the full microstructural complexity within the macro-elements.

For the more general case, this work exploited the use of robust, all-tetrahedral volume meshing as a method for dividing an irregular domain into smaller subvolumes for homogenization. As each sub-volume conformed to its parent macro-element, a method for calculating their effective properties was developed.

At the highest level, the developed homogenization process involves treating the sub-volume as if it were the actual macro-element. Appropriate boundary conditions are applied based on the shape functions of the macro-element such that the sub-volume is constrained to the same modes of deformation. A series of finite element simulations is then performed in order to determine the subvolume’s effective properties.

As part of the validation of the developed homogenization technique for tetrahedral sub-volumes, a homogeneous sub-volume with fully anisotropic material properties was used as a “sanity test.” The input material properties were accurately recovered. More interestingly, the technique was also used to recover the effective properties of a real-world structure and compared to the results obtained using classical methods. The Schoen Gyroid was chosen for this purpose as its periodic geometry allows periodic boundary conditions to be used. These boundary conditions are often considered to be the exact solution.

Following the development of a technique to determine an effective constitutive matrix for an arbitrary tetrahedral subvolume, the issue of multi-scale problems was addressed. Of particular interest is the set of problems having an irregular (i.e. non-cuboidal) domain. While problems of a more regular nature may be addressed with more conventional methods of determining effective constitutive matrices, they are nevertheless addressable using the methods developed in this work.

For many problems, it is highly impractical to attempt to include all length scales in a finite element model; consequently, it is often desirable to only capture the coarser details. Rather than excluding the smaller length scales, a coarse mesh with appropriate homogeneous material properties was produced. The process of generating the macro mesh is outlined in the figure. The homogeneous domain should conform to the bounds of the original domain as closely as possible, representing the result of a “shrink-wrap” operation.

Each of the macro-elements in the generated mesh is subsequently homogenized using the developed technique. As each macro-element is considered as an independent sub-volume, the processing may occur in either series or in parallel, depending on the available computational resources. The final result is a macroscopic homogenous model with varying material properties that can be exported to a traditional finite element package.

This work was done by Philippe G. Young of the University of Exeter; and David Raymont, Joonshik Kim, and Viet Bui Xuan of Simpleware Ltd. for the Air Force European Office of Aerospace Research & Development (EOARD). AFRL-0225

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