Diffusion and ecological problems : mathematical models

Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.

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Digital Twin: Values, Challenges and Enablers From a Modeling Perspective

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

/pdf/finite-element-methods-for-surface-pdes-1w1k25wxnx.pdf

Finite element methods for surface PDEs

Color science is a multidisciplinary field with broad applications in industries such as digital imaging, coatings and textiles, food, lighting, archiving, art, and fashion. Accurate definition and measurement of color appearance is a challenging task that directly affects color reproduction in such applications. Color Appearance Models addresses those challenges and offers insight into the preferred solutions. Extensive research on the human visual system (HVS) and color vision has been performed in the last century, and this book contains a good overview of the most important and relevant literature regarding color appearance models.

https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging/volume-23/issue-2/029901/Color-Appearance-Models/10.1117/1.JEI.23.2.029901.pdf

Color Appearance Models

Towards a semantic Construction Digital Twin: directions for future research

Digital Twins are one of the hottest digital trends. In this contribution we will shortly review the concept of Digital Twins and the chances for novel industrial applications. Mathematics are a key enabler and the impact will be highlighted along four specific examples addressing Digital Product Twins democratizing Design, Digital Production Twins enabling robots to mill, Digital Production Twins driving industrialization of additive manufacturing, and Digital Performance Twins boosting operations. We conclude the article with an outlook on the next wave of Digital Twins, Executable Digital Twins, and will review the associated challenges and opportunities for mathematics.

Digital Twins.

Here, we report on a new approach for adaptive path finding in microscopic simulations of pedestrian dynamics. The approach extends a widely used concept based on scalar navigation fields—the so-called floor field method. Adopting a continuum perspective, navigation fields used in our approach correspond to the shortest distances to the pedestrian's targets with respect to arbitrary metrics, e.g. metrics depending on the local terrain. If the metric correlates inversely with the expected speed, these distances could be interpreted as expected travel times. Following this approach, it is guaranteed that virtual pedestrians navigate along the steepest descent of the navigation field and thus follow geodesics. Using the Eikonal equation, i.e. a continuum model, navigation fields can be determined with respect to arbitrary metrics in an efficient manner. The fast marching method used in this work offers a fast method to solve the Eikonal equation (complexity N log N, where N is degree of freedom). Increasing computational efforts with respect to classical approaches only mildly, the consideration of locally varying metrics allows a realistic adaptive movement behavior like the avoidance of certain terrains. The method is outlined using a simple cellular automaton approach. Extensions to other microscopic models, e.g. cellular automata approaches or social force models, are possible.

https://iopscience.iop.org/article/10.1088/1367-2630/12/4/043032/pdf

Adaptive pedestrian dynamics based on geodesics

Inspired by doubts from social scientists on the validity of computer models that see a crowd as a pure aggregation of individuals, we develop a mathematical model for group formation within crowds. It is based on a few simple characteristics. Most importantly, small groups stick together as they thread their way through a crowd. Additionally, groups have a tendency to walk abreast to ease communication. Through simulation, we establish that the occurrence of groups significantly impacts crowd movement, namely evacuation times. Further, we complement and validate the simulations by a small experiment: a classroom egress. The simulation results match the measurements qualitatively. We get a good quantitative match after calibrating the supposed desire to communicate while walking—and hence to walk abreast. We conclude that it is one of the crucial parameters to calibrate the group model against reality. While working on a mathematically complete model, a new gap between the mathematical modelling and the s...

On modelling the influence of group formations in a crowd

An increasing number of disruptive innovations with high economic and social impact shape our digitalizing world. Speed and extending scope of these developments are limited by available tools and paradigms to master exploding complexities. Simulation technologies are key enablers of digitalization. They enable digital twins mirroring products and systems into the digital world. Digital twins require a paradigm shift. Instead of expert centric tools, engineering and operation require autonomous assist systems continuously interacting with its physical and digital environment through background simulations. Model order reduction (MOR) is a key technology to transfer highly detailed and complex simulation models to other domains and life cycle phases. Reducing the degree of freedom, i.e., increasing the speed of model execution while maintaining required accuracies and predictability, opens up new applications. Within this contribution, we address the advantages of model order reduction for model-based system engineering and real-time thermal control of electric motors.

Model Order Reduction a Key Technology for Digital Twins

One of the most important aspects for a realistic prediction of pedestrian flows is the modelling of human navigation in normal situations such as early design phases of buildings or transportation systems and hubs as well as in evacuation studies to enhance safety in existing infrastructures. To overcome the limitations of current navigation models, this paper proposes a new hybrid multi-scale model, which closely links information between the small-scale and large-scale navigation layer to improve the navigational behaviour. In the presented hybrid navigation model, graph-based methods using visibility graphs are used to model large-scale wayfinding decisions. The pedestrians' movements between two nodes of the navigation graph are modelled by means of a dynamic navigation field. The navigation field is updated dynamically during the runtime of the simulation, explicitly considering other pedestrians for determining the fastest path. The proposed hybrid approach provides a realistic modelling of human navigational behaviour and thus a realistic prediction of flows since it reflects the human cognitive processes triggered by wayfinding tasks. This includes taking into account other pedestrians for routing decisions who are visible from the current position of the considered pedestrian. The paper discusses the concept and the technical details of the proposed hybrid multi-scale approach in detail and presents an extensive case study demonstrating its advantages.

Dirk Hartmann

Papers

Digital Twins.

Adaptive pedestrian dynamics based on geodesics

On modelling the influence of group formations in a crowd

Model Order Reduction a Key Technology for Digital Twins

A hybrid multi-scale approach for simulation of pedestrian dynamics