Conference Agenda

We live in a world of exploding complexity with enormous challenges. Digital twins, tightly integrating the real and the digital world, are a key enabler for decision making in the context of complex systems. While the digital twin has become an intrinsic part of the product creation process, its true power lies in the connectivity of the digital representation with its physical counterpart.

To be able to use a digital twin scalable in this context, the concept of an executable digital twin has been proposed. An executable digital twin is a stand-alone and self-contained executable model for a specific set of behaviors in a specific context. It can be leveraged by anyone at any point in lifecycle. To achieve this, a broad toolset of mathematical technologies is required - ranging from model order reduction, calibration to hybrid physics- and data-based models.

In this presentation, we review the concept of executable digital twins, address mathematical key building blocks such as model order reduction, real-time models, state estimation, and co-simulation and detail its power along a few selected use cases.

The numerical simulation of nonlinear problems in contact mechanics and mixed-dimensional coupling poses significant challenges for high-performance computing (HPC). The considerable computational effort introduced by the evaluation of discrete contact or mixed-dimensional coupling operators requires an efficient framework that scales well on parallel hardware architectures and is, thus, suitable for the solution of high-fidelity models with potentially many million degrees of freedom. Another unsolved task is the efficient solution of the arising linear systems of equations on parallel computing clusters. In this study, we investigate the scalability of computational strategies and algebraic multigrid (AMG) solvers to address these challenges effectively and ultimately target a faster time-to-solution. Two problem classes of utmost practical relevance serve as prototypes for this endeavor.

First, we investigate the finite element analysis of nonlinear contact problems based on non-matching mortar interface discretizations. Mortar methods enable a variationally consistent imposition of almost arbitrarily complex coupling conditions but come with considerable computational effort for the evaluation of the discrete coupling operators (especially in 3D). We identify bottlenecks in parallel data layout and domain decomposition that hinder a truly efficient evaluation of the mortar operators on modern-day HPC systems and then propose computational strategies to restore optimal parallel communication and scalability. In particular, we suggest a dynamic load balancing strategy in combination with a geometrically motivated reduction of ghosting data. Using increasingly complex 3D examples, we demonstrate strong and weak scalability of the proposed algorithms up to 480 parallel processes. In addition to the computational strategies, we investigate the integration of tailored AMG preconditioners for the resulting saddle point-type linear systems of equations into a then fully scalable simulation pipeline.

Second, we present the mixed-dimensional interaction of slender fiber- or rod-like structures with surrounding solid volumes, thus leading to a 1D-3D approach that we refer to as beam-to-solid coupling. In terms of practical applications, natural and artificial fiber-reinforced materials (e.g., biological tissue, composites) come to mind. Again, not so different from contact problems, the main challenges on the way to a scalable computational framework lie in the efficient and parallelizable evaluation of the underlying mixed-dimensional coupling operators and the design of tailored AMG preconditioners. Here, we will focus on a novel physics-based block preconditioning approach based on AMG that uses multilevel ideas to approximate the block inverses appearing in the system. Eventually, we will assess the performance and the weak scalability of the proposed block preconditioner using large-scale numerical examples.