Homogenization based on Optimization (subproject of DK Numerical Simulations in Technical Siences)

Research Areas

Every material possesses a specific micro structure which governs the macroscopic material behavior. Even steel, usually considered as a homogeneous material, has a distinct micro structure which is not homogeneous. Fortunately, this micro structure is several length scales smaller than the macro scale so that the homogeneous behavior is a very good approximation, i.e., within the linear theory the isotropic Hooke's law can be used. This is different, e.g., in case of reinforced materials where the reinforcement is sometimes even observable on the macro scale. In such cases anisotropic material laws are used. Even then a homogenous continuum is assumed using effective material data in the constitutive equations. In this case the problem of determining the effective material constants arises. The approach of determining them from the microscopic behavior by a transfer to the macroscopic level suggests itself. This can be done by homogenization.
Using homogenization, different approaches can be followed. There is the mathematically more involved multiscale approach which does not rely on effective material data (see, e.g.[10], [12]). This is a general purpose method which, however, is computationally intensive for concrete applications. On the other hand, there is the determination of effective material properties as discussed above. This approach tries to map the microscopic properties of a material to the material law on the macroscopic level. The constants determined for the material law are then used as the effective material data. This approach is cheaper than the first one but not so general, i.e., for each different micro-structure the process has to be reapplied.
Here, the latter approach will be used where the mapping between the micro and macro scale will be performed with an optimization procedure. The optimization is necessary because contrary to other approaches, here, the inertia on the micro scale will be taken into account. This results in a time dependent behavior on the micro scale which influences also the macroscopic behavior.

State of the Art

The effective material properties of materials with micro-structure are mostly determined by homogenization procedures. A typical example is given by composite materials (see, e.g.,[2],[ 4]). The techniques developed for such materials are often also applicable to other materials with micro-structure. For the special case of cellular foams with special microscopic shape even analytical methods are available (see,e.g. [3]). Framelike structures in particular can be treated by analytical methods (see, e.g. [5]). For granular materials mostly computational approaches are chosen (see, e.g. [9]). Bounds of these approaches to determine effective material data on the macroscopic level may be found in [11] and the principles for a computational approach to tackle micro-mechanical problems can be found, e.g., in [12].
All these approaches have in common the assumption that the micro structural behavior is static, i.e., inertia effects are neglected on the micro scale. This is mostly motivated by the scale separation which is assumed to be large as stated in [8].
Only a few approaches are found in literature which take inertia effects on the micro level into account [7]. In this case, the microscopic behavior is time dependent or if a time harmonic excitation is assumed it is frequency dependent due to the inertia terms in the governing equations on the micro-scale. Therefore, the effective macroscopic material properties are also functions of time or frequency, respectively. Material models with a time dependent characteristic are, e.g., visco- or poroelastic constitutive equations.
On the contrary, to the static micro structure an analytical homogenization for a general purpose configuration seems not be possible or is highly complicated. Instead, an optimization problem can be defined to determine the macroscopic material data from the microscopic behavior [1].

Project Topics

A genetic Algorithm is under study to solve this optimization process and to avoid local minima. At present, only a simple cost function without constraints is used for the time/frequency dependent material data. Since the macroscopic material law must be thermodynamically consistent, i.e., the second law of thermodynamics has to be fulfilled, it is necessary to find an optimization procedure which includes the thermodynamic restrictions. Moreover, the performance of the genetic algorithm is not satisfactory. Hence, a strategy will be developed which combines the global features of the genetic algorithm with efficient tools for a local optimization processes.
In a first step, the micro structure is modelled by a simple beam model in the frequency domain as input for the optimization procedure. On the macro scale a three-dimensional viscoelastic material law is chosen for which parameters have to be found.
In a next step, more expensive calculations on the micro scale with even three-dimensional continuum models will be tackled. In this context, additionally, micro-structures yielding anisotropic macroscopic material laws will be treated. Based on the experience thus obtained poroelastic materials will be considered to model micro structures with two phases.

References

[1] S. Alvermann, M. Schanz: Influence of Inertia on the Microscale in Homogenization. Proc. Appl. Math. Mech. 4 (2004) 179-180.
[2] H. J. Böhm: Modeling the Mechanical Behavior of Short Fiber Reinforced Composites. In Mechanics of Microstructured Materials (H. J. Böhm, ed.), CISM International Centre for Mechanical Sciences. Courses and Lectures, Nr. 464, Springer-Verlag, Wien New York, 2004.
[3] L. J. Gibson, M. F. Ashby: Cellular Solids. Pergamon Press, Oxford, 1988.
[4] Z. Hashin: Analysis of Composite Materials - A Survey. J. Appl. Mech. ASME 50 (1983) 481-504.
[5] J. Hohe, W. Becker: Effective Stress-Strain Relations for Two-Dimensional Cellular Sandwich Cores: Homogenization, Material Models, and Properties. Appl. Mech. Rev. 55 (2002) 61-87.
[6] M. Hintermüller, K. Ito, K. Kunisch: The Primal-Dual Active Set Strategy as a Semi-Smooth Newton Method, SIAM J. Optim. 13 (2002) 865-888.
[7] R. S. Lakes: Micromechanical Analysis of Dynamic Behavior of Conventional and Negative Poisson's Ratio Foams. J. Engrg. Mat. Tech. 118 (1996) 285-288.
[8] C. Miehe: Computational Micro-to-Macro Transitions for Discretized Micro-Structures of Heterogeneous Materials at Finite Strains based on the Minimization of Averaged Incremental Energy. Comp. Meth. Appl. Mech. Eng. 192 (2003) 559-591.
[9] C. Miehe, J. Dettmar: A Framework for Micro-Macro Transitions in Periodic Particle Aggregates of Granular Materials. Comp. Meth. Appl. Mech. Eng. 193 (2004) 225-256.
[10] E. Sanchez-Palenzia: Non-Homogeneous Media and Vibration Theory, Vol. 127, Lecture Notes in Physics, Springer-Verlag, Berlin Heidelberg, 1980.
[11] S. Torquato: Random Heterogenous Media: Microstructure and Improved Bounds on Effective Properties. Appl. Mech. Rev. 44 (1991) 37-76.
[12] T. I. Zohdi, P. Wriggers: Introduction to Computational Micromechanics, Vol. 20, Lecture Notes in Applied and Computational Mechanics. Springer Verlag, Berlin, Heidelberg, New York, 2005.