Abstract

Given a heterogeneous material, the mechanical behavior of its microstructure can be investigated by an algorithm that uses the Fourier representation of the Lippmann-Schwinger equation. Incorporating a model order reduction technique based on calculations with a reduced set of Fourier modes, the computational cost of this algorithm can be decreased. It was shown that the accuracy of this model order reduction technique strongly depends on the choice of Fourier modes by considering a geometrically adapted rather than a fixed sampling pattern to define the reduced set of Fourier modes. Since it is difficult to define a geometrically adapted sampling pattern for complex microstructures, additionally a strain-based sampling pattern was introduced. The accuracy and adaptability of this strain-based reduced set of Fourier modes is shown by incorporating a polycrystalline microstructure.

Abstract

The properties of shape memory alloys (SMA) are mainly influenced by phase transformations between austenite and martensite. The complex material behavior is described by a variational method which describes the evolution of the phase fractions. We combined the method with a microstructural analysis based on fast Fourier transformations. Such a highly resolved microstructural analysis comes along with a high computational effort.To reduce the later one, we propose a model order reduction technique that uses just a reduced set of Fourier modes, which is adapted to the underlying microstructure. The presentation of the theoretical background as well as of the implemented algorithm is followed by numerical results that underline the performance of our method.