Stochastic and deterministic algorithms for set-based data-driven computational mechanics

RWTH Publications (RWTH Aachen)

Abstract

Data-driven methods have transformed the field of computational mechanics in recent years. Inspired by remarkable advances in data science and artificial intelligence across various disciplines, researchers in computational mechanics have rethought the way to tackle existing challenges. In the field of solid mechanics, a central challenge is the realistic description of the material behavior due to the enormous complexity and variety of both natural and engineered materials. While the required field theories of equilibrium and kinematic compatibility are widely valid and mathematically very well characterized, the classical method of constitutive, phenomenological material modeling is a laborious task with many modeling uncertainties. The models have to be specifically designed, calibrated, verified and validated based on the current knowledge and data about the material. This process has to be repeated as new information and data are obtained, turning it into an open-ended process. A significant portion of recently developed data-driven methods target the automation of this process by implementing universal function approximators such as artificial neural networks. Such methods are extremely flexible and are designed to handle large amounts of data efficiently. Furthermore, basic principles from materials theory can be implemented in order to regularize the functions and reduce the required amount of data. In 2016, Kirchdoerfer and Ortiz [82] proposed an alternative approach, herein called the \textit{set-based} data-driven method. Instead of working with functional approximations for the material behavior, the boundary value problem is formulated explicitly in the material data set. The principal goals of the method are 1) reduction of modeling uncertainties 2) no loss of information for noisy data 3) correct description of materials where no single-valued functional relationship exists and 4) simple and automated process-cycle of data acquisition and inference. This cumulative dissertation aims to contribute to the set-based data-driven method with a set of novel algorithms addressing several open research questions. A central theme throughout the dissertation is the identification of whether either a stochastic or a deterministic algorithm is the best choice for a given problem. The findings are often surprising; for instance, computing the global minimum of the discrete optimization problem with combinatorial complexity is best achieved using a stochastic algorithm, whereas discovering all local minima from a non-convex energy landscape in high dimensions was only tractable via a deterministic algorithm. In the first article, a stochastic algorithm to compute the global minimum of the set-based minimization problem is developed, introducing a randomized data search which effectively helps to escape local minima emerging from the discrete nature of the problem. This solver further enables a new goal-oriented data acquisition scheme in the context of multiscale mechanics driven by a weighted k-means clustering. It was shown that the error can be consistently reduced by adding a minimal amount of optimally selected data, saving a substantial amount of costly microstructure simulations. The second article introduces the foundation for a stochastic description of the set-based method by considering the whole posterior distribution of possible system outcomes as compared to a single, deterministic solution. A stochastic, simulated annealing algorithm explores the distribution by randomly propagating a Markov Chain. While the algorithm is capable of sampling multi-modal distributions by maintaining a set of solutions, it is limited to rather small system sizes. This limitation is overcome by the third article. The proposed algorithm is based on a basis for the space of equilibrium stress fields, where sparsity of the basis is crucial for numerical efficiency. Existing methods to compute such a sparse null basis for general systems often rely heavily on expensive numerical factorizations or sophisticated graph algorithms, occasionally leading to conditioning problems. To overcome these problems, an efficient algorithm to compute the sparse null basis for the important special case of linear tetrahedral elements is proposed. Equipped with the basis for equilibrium stress fields, a deterministic annealing algorithm based on [144] is developed. While slowly resolving details of the multi-modal distribution, a partial eigenvalue decomposition in the space of equilibrium stress and compatible strain fields detects the formation of local minima and provides the directions where to initialize two separate deterministic minimization runs. By this process, a hierarchical solution set is constructed which provides the full posterior distribution, serving the purpose of uncertainty quantification. The method is demonstrated with a brittle fracture problem featuring a local material with stochastic tensile strength, where every solution represents one possible crack pattern. As the data set cannot be represented by a single-valued functional relationship, this result may be seen as a prime example for the set-based data-driven method.

Authors 1

1. E. Prume corresponding Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen