Simulations are used in many areas of mechanical engineering, such as the strength and crash calculation of components or the calculation and design of flow processes. They offer the possibility of predicting real physical effects with the aid of computers, thus providing a deeper understanding of the observed effects and their causes. The aim of the course is to provide students with the fundamentals for the successful generation and execution of simulations in mechanical engineering.

Conservation equations are formulated for modelling. To close the system of equations, constitutive equations must also be formulated. The resulting partial differential equations (PDEs) are usually space and time dependent and form the fundamental components of modelling. The PDEs in question usually cannot be solved analytically, so that numerical methods and, in most cases, discretization methods are required.

The creation of suitable simulation models, the execution and evaluation of simulation studies and the avoidance of typical errors can be learnt, but also require some practice. Therefore, the course consists of three elements: Lectures, classroom exercises and computer exercises. As part of the computer exercises, students learn to implement (in Python), apply and evaluate appropriate simulation methods for given problems.

Content (with lectures (L), exercises (Ex) and/or computer exercises (PC))

• Introduction: model definition, overview of numerical simulation methods (L)
• Tensor calculus, Python and Git (L, Ex, PC)
• Equilibrium equations, constitutive laws, model reduction (L, Ex)
• Method of Weighted Residuals (MWR) (L, Ex)
• Finite Difference Method (FDM) (L, Ex, PC)
• Finite Element Method (FEM) (L, Ex, PC)
• Finite Volume Method (FVM) (L, Ex, PC)
• Selection of methods (L)

Learning objectives:

The students will be able to

• name and explain the balance equations and give examples of constitutive laws,
• construct models for continuum mechanical problems,
• explain model reduction approaches,
• explain the basic assumptions and techniques of the numerical methods covered (MWR, FDM, FEM, FVM), implement them in software and evaluate their suitability for given problems.

• Bathe, K.-J.: Finite Element Procedures, Prentice Hall, Pearson Education (1st ed.) / Watertown, MA (2nd ed.), 2014. ISBN: 978-0-9790049-5-7
• Belytschko, T., Liu, W. K., Moran, B., & Elkhodary, K.. Nonlinear finite elements for continua and structures. John Wiley & Sons, 2014.
• Ferziger, J. H., Peric, M.: Numerische Strömungsmechanik, Springer-Verlag, 2008.
  https://doi.org/10.1007/978-3-662-46544-8 
• Ferziger, J. H., Peric, M.: Computational Methods for Fluid Dynamics. Springer-Verlag, 2020. https://doi.org/10.1007/978-3-319-99693-6 
• Gurtin, M.E.; Fried E.; Anand, L.: The mechanics and thermodynamics of continua, Cambridge University Press, 2010. ISBN: 978-0-521-40598-0
• Hirsch, C.: Numerical Computation of Internal and External Flows, Vol. I, II, Wiley, 2007.
  ISBN: 978-0-7506-6594-0
• Schäfer, M.: Computational Engineering – Introduction to Numerical Methods, Springer-Verlag, 2006. https://doi.org/10.1007/978-3-030-76027-4