Dynamic multiscale topology optimization for multi-regional micro-structured cellular composites

© 2018 Elsevier Ltd In this paper, a new dynamic multiscale topology optimization method for cellular composites with multi-regional material microstructures is proposed to improve the structural performance. Firstly, a free-material distribution optimization method (FMDO) is developed to generate the overall configuration for the discrete element densities distributed within a multi-regional pattern. The macrostructure is divided into several sub regions, and each of them consists of a number of elements but with the same densities. Secondly, a dynamic topology optimization formulation is developed to perform the concurrent design of the macrostructure and material microstructures, subject to the multi-regional distributed element densities. A parametric level set method is employed to optimize the topologies of the macrostructure and material microstructures, with the effective macroscopic properties evaluated by the homogenization. In the numerical implementation, the quasi-static Ritz vector (QSRV) method is incorporated into the finite element analysis so as to reduce the computational cost in numerical analysis, and some kinematical connectors are introduced to make sure the connectivity between adjacent material microstructures. Finally, 2D and 3D numerical examples are tested to demonstrate the effectiveness of the proposed dynamic multiscale topology optimization method for the material-structural composites

Numerical Methods for Multilattices

Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method has attracted growing interest in recent years. The QC method was first developed for crystalline materials with Bravais lattice and was later extended to multilattices (Tadmor et al, 1999). Another existing numerical approach to modeling multilattices is homogenization. In the present paper we review the existing numerical methods for multilattices and propose another concurrent macro-to-micro method in the numerical homogenization framework. We give a unified mathematical formulation of the new and the existing methods and show their equivalence. We then consider extensions of the proposed method to time-dependent problems and to random materials.Comment: 31 page

Topology optimization for multiscale design of porous composites with multi-domain microstructures

© 2018 Elsevier B.V. This paper proposes a new multiscale topology optimization method for the design of porous composites composed of the multi-domain material microstructures considering three design elements: the topology of the macrostructure, the topologies of multiple material microstructures and their overall distribution in the macrostructure. The multiscale design involves two optimization stages: the free material distribution optimization and the concurrent topology optimization. Firstly, the variable thickness sheet (VTS) method with the regularization mechanism is used to generate multiple element density distributions in the macro design domain. Hence, different groups of elements with the identical densities can be uniformly arranged in their corresponding domains, and each domain in the space will be periodically configured by a unique representative microstructure. Secondly, with the discrete material distributions achieved in the macro domain, the topology of the macrostructure and topologies of multiple representative microstructures are concurrently optimized by a parametric level set method combined with the numerical homogenization method. Finally. Several 2D and 3D numerical examples are provided to demonstrate the effectiveness of the proposed multiscale topology optimization method

Model Reduction of Synchronized Lur'e Networks

In this talk, we investigate a model order reduction schemethat reduces the complexity of uncertain dynamical networks consisting of diffusively interconnected nonlinearLure subsystems. We aim to reduce the dimension ofeach subsystem and meanwhile preserve the synchronization property of the overall network. Using the upperbound of the Laplacian spectral radius, we first characterize the robust synchronization of the Lure network bya linear matrix equation (LMI), whose solutions can betreated as generalized Gramians of each subsystem, andthus the balanced truncation can be performed on the linear component of each Lure subsystem. As a result, thedimension of the each subsystem is reduced, and the dynamics of the network is simplified. It is verified that, withthe same communication topology, the resulting reducednetwork system is still robustly synchronized, and the apriori bound on the approximation error is guaranteed tocompare the behaviors of the full-order and reduced-orderLure subsyste

Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to PDE-based continuum methods, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for non-crystalline materials, it may still be used as a first order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to 2-D cellular-scale models by assessing the mechanical behaviour of model biological tissues, including crystalline (honeycomb) and non-crystalline reference states. The numerical procedure consists in precribing an affine deformation on the boundary cells and computing the position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For centre-based models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete/continuous modelling, adaptation of atom-to-continuum (AtC) techniques to biological tissues and model classification

DAG-based software frameworks for PDEs

pre-printThe task-based approach to software and parallelism is well-known and has been proposed as a potential candidate, named the silver model, for exas-cale software. This approach is not yet widely used in the large-scale multi-core parallel computing of complex systems of partial differential equations. After surveying task-based approaches we investigate how well the Uintah software and an extension named Wasatch fit in the task-based paradigm and how well they perform on large scale parallel computers. The conclusion is that these approaches show great promise for petascale but that considerable algorithmic challenges remain