Computational homogenization of non-linear and inelastic material laws in the tensor train format (TT-Hom)

• Applicant: Professor Dr. Matti Schneider
• Subject Area: Mechanics
• Term: from 2019 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 418247895

Modern 3D imaging techniques like micro-computed tomography provide extremely detailed microstructure images of materials. To compute directly on these images, computational techniques specialized for regular grids have been established. For example, FFT-based computational homogenization, which avoids storing the stiffness matrix and still treats non-linear and inelastic problems robustly.State-of-the-art micro-computed tomographs produce volume images consisting of several billion voxels. To compute on these images, on the one hand the solution vector needs to be stored, and, on the other hand, material laws have to be evaluated. The first point concerns memory demands, whereas the second point involves computational time.Conventional computational homogenization methods are stretched to their limits if a high number of load steps, a high resolution or parameter studies are of concern. Examples can be found in creeping behavior, fatigue, fine shear bands in plasticity or in damage/fracture mechanics. The mathematical difficulty is rooted in the complexity of the problems which is bounded from below by (degrees of freedom)x(number of load steps)x(number of computations).The proposal targets applying tensor compression methods (like the tensor trainformat ) cleverly to computational homogenization on digital images. Exploiting low rank representations the computational complexity can be reduced below the apparent barrier.From a methodical point of view the efficient compression of real microstructures and the efficient representation of the differential and integral operators of elasticity in the tensor train format are central. Furthermore, the application to non-linear elasticity, viscoelasticity and elastoplasticity is addressed. In particular, space-time preconditioners need to be used, and some functional tensor train representations need to be generalized to non-smooth operations.The proposed project targets:- computing effective material laws for - complex image-based microstructures - material laws with internal variables - multiaxial loading incorporating load reversal - high cycle count - using tensor data compression techniques - incorporating modern non-linear solversThe proposed project shall generate knowledge, which can be classified as follows. Firstly, consistently using tensor compression methods enables archiving microstructure and simulation results in a long-term manner. Secondly, we target handling problems that appear, due to a large number of degrees of freedom or many load steps, untractable with conventional methods, like multi-scale fatigue computations for composites.

• DFG Programme: Research Grants