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The book examines innovative numerical methods for computational solid and fluid mechanics that can be used to model complex problems in engineering. It also presents innovative and promising simulation methods, including the fundamentals of these methods, as well as advanced topics and complex applications. Further, the book explores how numerical simulations can significantly reduce the number of time-consuming and expensive experiments required, and can support engineering decisions by providing data that would be very difficult, if not impossible, to obtain experimentally. It also includes chapters covering topics such as particle methods addressing particle-based materials and numerical methods that are based on discrete element formulations; fictitious domain methods; phase field models; computational fluid dynamics based on modern finite volume schemes; hybridizable discontinuous Galerkin methods; and non-intrusive coupling methods for structural models.

### Discrete Element Methods: Basics and Applications in Engineering

Abstract
A computational approach is presented in this contribution that allows a direct numerical simulation of 3D particulate movements. The given approach is based on the Discrete Element Method (DEM) The particle properties are constitutively described by specific models that act at contact points. The equations of motion will be solved by appropriate time marching algorithms. Additionally coupling schemes with the Finite Element Method (FEM) are discussed for the numerical treatment of particle-solid and particle-fluid interaction. The presented approach will be verified by computational results and compared with those of the literature. Finally, the method is applied for the simulation of different engineering applications using computers with parallel architecture.
Peter Wriggers, B. Avci

### Adaptive Integration of Cut Finite Elements and Cells for Nonlinear Structural Analysis

Abstract
Fictitious domain methods facilitate the discretization of boundary value problems by applying simple meshes containing finite elements or cells that do not conform to the geometry of the domain of interest. In this way, the effort of meshing complex domains is shifted to the numerical integration of those elements/cells that are cut by the boundary of the domain. In this chapter, we will first introduce a high-order fictitious domain method and then present adaptive methods that are suited for the numerical integration of broken elements and cells. Since the quadrature schemes presented in this chapter are quite general, they can be applied to the different versions of fictitious domain methods.
Alexander Düster, Simeon Hubrich

### Numerical Implementation of Phase-Field Models of Brittle Fracture

Abstract
These lecture notes address the main challenging computational aspects of phase-field modeling of brittle fracture. We focus, in particular, on the irreversibility constraint and the iterative solution strategy for non-convex minimization problems. In the former case, we present multiple options of incorporating the constraint and discuss the equivalence of the resulting formulations. For the latter, we also consider various available options and critically assess the efficiency and robustness of some of them. Numerical examples on well-known benchmark problems illustrate the theoretical findings.
Laura De Lorenzis, Tymofiy Gerasimov

### Practical Computational Fluid Dynamics with the Finite Volume Method

Abstract
This chapter covers the fundamental aspects of Computational Fluid Dynamics simulation tools and introduces the terminology and principles of the second order accurate Finite Volume Method with polyhedral mesh support, as implemented in OpenFOAM (Weller et al. 1998). The first part is dedicated to types and properties of computational meshes, followed by a description of the Finite Volume discretisation for a generic scalar transport equation. Algorithms for pressure-velocity coupling for single–phase incompressible Newtonian fluid flow, as described by the Navier-Stokes equations are presented, supported by an overview of algorithms for the solution of the resulting linear system of algebraic equations. Application of the polyhedral FVM solver for real engineering problems is illustrated on some practical examples.
Hrvoje Jasak, Tessa Uroić

### Tutorial on Hybridizable Discontinuous Galerkin (HDG) Formulation for Incompressible Flow Problems

Abstract
A hybridizable discontinuous Galerkin (HDG) formulation of the linearized incompressible Navier-Stokes equations, known as Oseen equations, is presented. The Cauchy stress formulation is considered and the symmetry of the stress tensor and the mixed variable, namely the scaled strain-rate tensor, is enforced pointwise via Voigt notation. Using equal-order polynomial approximations of degree k for all variables, HDG provides a stable discretization. Moreover, owing to Voigt notation, optimal convergence of order $$k+1$$ is obtained for velocity, pressure and strain-rate tensor and a local postprocessing strategy is devised to construct an approximation of the velocity superconverging with order $$k+2$$, even for low-order polynomial approximations. A tutorial for the numerical solution of incompressible flow problems using HDG is presented, with special emphasis on the technical details required for its implementation.
Matteo Giacomini, Ruben Sevilla, Antonio Huerta

### Non Intrusive Global/Local Coupling Techniques in Solid Mechanics: An Introduction to Different Coupling Strategies and Acceleration Techniques

Abstract
The aim of this paper is to provide, for a reader not familiar with the non-intrusive coupling method, the simplest possible example on which all the different iterative coupling strategies can be solved by hands. Among them, the basic algorithm, Aitken’s method, mixed interface conditions...A drawback of this example is that, for some acceleration techniques, the convergence can be achieved in one iteration after the initialization. Nevertheless it allows to easily become acquainted with the different techniques. Some examples from previous papers are then used to illustrate the same properties on more complex examples involving nonlinearity.
Olivier Allix, Pierre Gosselet