Emerging Discretization Methods in Solid Mechanics

This course provides an advanced exploration of numerical methods utilized in the analysis of solid mechanics problems. It includes sophisticated emerging techniques essential for addressing complex engineering challenges, focusing on both theoretical understanding and practical implementation. The techniques range from non-classical methods to machine learning. Lectures will provide an in-depth exploration of lattice Boltzmann methods (LBMs) and their applications in solid mechanics. LBMs are a class of computational fluid dynamics techniques based on kinetic theory, but they have been extended to simulate a wide range of phenomena beyond fluid flow. Students will learn the fundamentals of LBMs and how they can be adapted to address various solid mechanics scenarios. Another set of lectures deals with the least squares finite element method (LSFEM), which provides a robust framework for solving a wide range of boundary value problems. The theoretical and algorithmic formulations are derived for applications in solid mechanics. In addition, mixed finite element methods considering linear and nonlinear formulations are discussed and applied to engineering analysis problems. Virtual element methods (VEMs) have a broad range of applications. VEM represents a new modern numerical discretization technique capable of handling complex geometries and heterogeneous material properties. The lectures provide students with theoretical foundations and hands-on experience in utilizing VEMs for solving solid mechanics problems efficiently and accurately. A comprehensive description of the field of electro-magneto-mechanics will be discussed, starting from Maxwell and Cauchy equations, moving into sophisticated computational formulations and finishing with practical Finite Element technologies. With a primary focus on large strains, the course will cover fundamental aspects such as well-posedness and specific spatial discretization techniques. For applications into soft robotics (i.e. Electro-Active Polymers), development of structure preserving time integrators and data-driven Machine Learning type of constitutive models will demonstrate a broad range of realistic applications. This course explores and critically questions the use of machine learning (ML) in solid mechanics applications. ML has proven to be a helpful tool for reducing computational costs and enabling accurate representations of data ranging from experiments to high-fidelity simulations. The lectures cover the principles of ML techniques and discuss their possibilities to solve partial differential equations, facilitate multiscale problems, and describe constitutive behavior. Further lectures explore automation of discretization techniques in the context of the numerical methods discussed in the course. It will be shown that automation helps to develop more complex emerging discretization techniques like the virtual element method. The course will appeal to doctoral students and postdoctoral researchers from academia and industry. Participants will gain expertise in employing different numerical methods to accurately model and simulate a wide range of solid mechanics phenomena.