Modeling is one of the main challenges in engineering science for the predictive description and simulation of the responses of systems and processes under complex actions in solid mechanics. This includes mechanical and mathematical modeling of single- and multi-phase materials as well as coupled problems, scale-bridging of micro-heterogenous materials, multiscale design of tailored materials, characterization of soft biological tissues and the tremendous development of data driven science in engineering using concepts of artificial intelligence, machine and manifold learning. All systems related to engineering need predictions of the behavior, durability and efficiency and hence models have to be developed that can be solved by analytical or numerical methods.

Therefore we focus on mechanical and mathematical modeling in single-phase solid mechanics, discussing theoretical models as well as simulation models associated with engineering applications in solid mechanics. Within the lectures examples will be considered in which different models and discretization schemes are compared.

Another challenging problem is the modeling and characterization of soft biological tissues. Some properties of a biological system, might be directly measurable, others, and especially in-vivo properties, can only be explored with proper models at hand. In addition, prospective events, i.e. questions such as ‘What would be the outcome from a certain clinical intervention?’ can only been explored through modeling. Soft biological tissues are non-man-made highly complex systems, and the first main task of the modeler is to decide what properties are important for the Intended Model Application.

A variety of engineering applications exhibit a coupling between individual thermodynamical fields, (e.g. thermoelasticity), electro-mechanical coupling, multiphase-systems. Hence in this course we discuss the general formulation of thermoelasticity and their solution using finite elements, the coordinate-invariant modeling of piezo-electricity within the framework of the invariant-theory and a macroscopic model of a saturated porous solids consisting of ice and water, presented within the theory of mixtures.

Computational homogenization schemes and multi-scale modeling can be seen as one of the driving forces in virtual material design. This field requires modeling and simulation at the scale of heterogeneous microstructures with an implicit or explicit connection to other scales. Particular emphasis is given to nonlinear and multiphysics phenomena at a micro-scale to incorporate related simulation challenges. We discuss numerical or virtual material testing across the scales to realize two-scale analyses equivalent to FE2 methods.

Modern advances in small-scale fabrication techniques have enabled the creation of a wide range of engineered meta-materials with tailored mechanical properties, whose design calls for theoretical and computational techniques that link microscale and nanoscale architectures to macroscale properties. We will discuss a number of state-of-the-art computational approaches and associated architected materials systems.

A tremendous development of Artificial Intelligence techniques took place in the last decades. Machine learning and manifold learning, and, notably, deep learning techniques, have assisted to an unprecedented growth in the wide range of applications they can be envisaged for. With the irruption of data-enabled science and engineering, applied science is today a symbiosis of theory, experiments and simulation. This lecture covers some of the most outstanding applications of AI and data-science in engineering sciences.

The course is intended for doctoral and postdoctoral researchers in civil and mechanical engineering, applied mathematics and physics as well as industrial researchers, who are interested in conducting research in the topic. A problem for young scientists trying to do high level research in this area is the number of topics one has to be familiar with: Advanced constitutive modeling, homogenization, data driven science in engineering as well as numerical discretization schemes and applications of these methods to coupled problems, biomechanics and tailored materials. Furthermore, there are neither adequate textbooks nor advanced courses at research/university level available. The aim of this CISM course is to fill this gap.