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

Project: Research project

## Project Details

### 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.
Status Finished 1/10/07 → 1/10/11

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Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.
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• ### Homogenization of microstructures in dynamics using periodic boundary conditions

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Research output: Contribution to journalArticle