Objective multiscale analysis of random heterogeneous materials

The multiscale framework presented in [1, 2] is assessed in this contribution for a study of random heterogeneous materials. Results are compared to direct numerical simulations (DNS) and the sensitivity to user-defined parameters such as the domain decomposition type and initial coarse scale resolution is reported. The parallel performance of the implementation is studied for different domain decompositions

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM

Multiscale Methods for Random Composite Materials

Simulation of material behaviour is not only a vital tool in accelerating product development and increasing design efficiency but also in advancing our fundamental understanding of materials. While homogeneous, isotropic materials are often simple to simulate, advanced, anisotropic materials pose a more sizeable challenge. In simulating entire composite components such as a 25m aircraft wing made by stacking several 0.25mm thick plies, finite element models typically exceed millions or even a billion unknowns. This problem is exacerbated by the inclusion of sub-millimeter manufacturing defects for two reasons. Firstly, a finer resolution is required which makes the problem larger. Secondly, defects introduce randomness. Traditionally, this randomness or uncertainty has been quantified heuristically since commercial codes are largely unsuccessful in solving problems of this size. This thesis develops a rigorous uncertainty quantification (UQ) framework permitted by a state of the art finite element package \texttt{dune-composites}, also developed here, designed for but not limited to composite applications. A key feature of this open-source package is a robust, parallel and scalable preconditioner \texttt{GenEO}, that guarantees constant iteration counts independent of problem size. It boasts near perfect scaling properties in both, a strong and a weak sense on over $15,000$ cores. It is numerically verified by solving industrially motivated problems containing upwards of 200 million unknowns. Equipped with the capability of solving expensive models, a novel stochastic framework is developed to quantify variability in part performance arising from localized out-of-plane defects. Theoretical part strength is determined for independent samples drawn from a distribution inferred from B-scans of wrinkles. Supported by literature, the results indicate a strong dependence between maximum misalignment angle and strength knockdown based on which an engineering model is presented to allow rapid estimation of residual strength bypassing expensive simulations. The engineering model itself is built from a large set of simulations of residual strength, each of which is computed using the following two step approach. First, a novel parametric representation of wrinkles is developed where the spread of parameters defines the wrinkle distribution. Second, expensive forward models are only solved for independent wrinkles using \texttt{dune-composites}. Besides scalability the other key feature of \texttt{dune-composites}, the \texttt{GenEO} coarse space, doubles as an excellent multiscale basis which is exploited to build high quality reduced order models that are orders of magnitude smaller. This is important because it enables multiple coarse solves for the cost of one fine solve. In an MCMC framework, where many solves are wasted in arriving at the next independent sample, this is a sought after quality because it greatly increases effective sample size for a fixed computational budget thus providing a route to high-fidelity UQ. This thesis exploits both, new solvers and multiscale methods developed here to design an efficient Bayesian framework to carry out previously intractable (large scale) simulations calibrated by experimental data. These new capabilities provide the basis for future work on modelling random heterogeneous materials while also offering the scope for building virtual test programs including nonlinear analyses, all of which can be implemented within a probabilistic setting

An extreme-scale implicit solver for complex PDEs: highly heterogeneous flow in earth's mantle

Mantle convection is the fundamental physical process within earth's interior responsible for the thermal and geological evolution of the planet, including plate tectonics. The mantle is modeled as a viscous, incompressible, non-Newtonian fluid. The wide range of spatial scales, extreme variability and anisotropy in material properties, and severely nonlinear rheology have made global mantle convection modeling with realistic parameters prohibitive. Here we present a new implicit solver that exhibits optimal algorithmic performance and is capable of extreme scaling for hard PDE problems, such as mantle convection. To maximize accuracy and minimize runtime, the solver incorporates a number of advances, including aggressive multi-octree adaptivity, mixed continuous-discontinuous discretization, arbitrarily-high-order accuracy, hybrid spectral/geometric/algebraic multigrid, and novel Schur-complement preconditioning. These features present enormous challenges for extreme scalability. We demonstrate that---contrary to conventional wisdom---algorithmically optimal implicit solvers can be designed that scale out to 1.5 million cores for severely nonlinear, ill-conditioned, heterogeneous, and anisotropic PDEs