Meshless Finite Element Methods -- Meshless discretisation of nonlocal damage theories -- Three-dimensional non-linear fracture mechanics by enriched meshfree methods without asymptotic enrichment -- Accounting for weak discontinuities and moving boundaries in the context of the Natural Element Method and model reduction techniques -- Discontinuous Galerkin Methods -- Modeling Evolving Discontinuities with Spacetime Discontinuous Galerkin Methods -- Analysis of a finite element formulation for modelling phase separation -- Finite Element Methods with Embedded Discontinuities -- Recent Developments in the Formulation of Finite Elements with Embedded Strong Discontinuities -- Evolving Material Discontinuities: Numerical Modeling by the Continuum Strong Discontinuity Approach (CSDA) -- A 3D Cohesive Investigation on Branching for Brittle Materials -- Partition-of-Unity Based Finite Element Methods -- On Applications of XFEM to Dynamic Fracture and Dislocations -- Some improvements of Xfem for cracked domains -- 2D X-FEM Simulation of Dynamic Brittle Crack Propagation -- A numerical framework to model 3-D fracture in bone tissue with application to failure of the proximal femur -- Application of X-FEM to 3D Real Cracks and Elastic-Plastic Fatigue Crack Growth -- Accurate Simulation of Frictionless and Frictional Cohesive Crack Growth in Quasi-Brittle Materials Using XFEM -- On the Application of Hansbo’s Method for Interface Problems -- An optimal explicit time stepping scheme for cracks modeled with X-FEM -- Variational Extended Finite Element Model for Cohesive Cracks: Influence of Integration and Interface Law -- An Evaluation of the Accuracy of Discontinuous Finite Elements in Explicit Dynamic Calculations -- A discrete model for the propagation of discontinuities in a fluid-saturated medium -- Single Domain Quadrature Techniques for Discontinuous and Non-Linear Enrichments in Local Partion of Unity FEM -- Other Discretization Methods -- Numerical determination of crack stress and deformation fields in gradient elastic solids -- The variational formulation of brittle fracture: numerical implementation and extensions -- Measurement and Identification Techniques for Evolving Discontinuities -- Conservation under Incompatibility for Fluid-Solid-Interaction Problems: the NPCL Method.

With mechanics focusing on smaller and smaller length scales, the need to properly model discontinuities increases. Technically important interface problems appear in solid mechanics, at fluid-solid boundaries, e.g. in welding and casting processes, and in aeroelasticity. Discretization methods have traditionally been developed for continuous media and are less well suited for treating discontinuities. Indeed, they are approximation methods for the solution of the partial differential equations, which are valid on a domain. Discontinuities divide this domain into two or more parts and at the interface special solution methods must be employed. Also, fluid-solid interfaces cannot be solved accurately except at the expense of complicated and time-consuming remeshing procedures. In recent years, discretization methods have been proposed, which are more flexible and which have the potential of capturing (moving) discontinuities in a robust and efficient manner. This volume assembles contributions of leading experts with the most recent developments in this rapidly evolving field.