Preface GAMM Mitteilungen

Abstract

The aim of this Priority Programme is to pool expertise of mathematics and mechanics in Germany and to create new collaborations and strengthen existing networks. Over the last years the main objective of the Priority Programme has been the development of modern nonconventional discretization's, based on for example, mixed (Galerkin or least-squares) finite elements, discontinuous Galerkin formulations, finite cell methods, collocation techniques, or isogeometric analysis. These developments include the mathematical analysis for geometrically as well as physically nonlinear problems in the fields of, for instance, incompressibility, anisotropies, and discontinuities (cracks or contact). Numerical simulation techniques are an essential component for the construction, design and optimization of cutting-edge technologies as for example innovative products, new materials as well as medical-technical applications and production processes. These important developments pose great demands on the quality, reliability, and efficiency of numerical methods, which are used for the simulation of the aforementioned complex problems. Existing computer-based solution methods often provide approximations, which cannot guarantee or fulfill substantial, absolutely necessary stability criteria. Specifically in the field of geometrical and material nonlinearities such uncertainties appear. Consequently, the Priority Programme 1748 focuses on novel approaches for reliable simulation techniques in solid mechanics, especially in the development of nonstandard discretization methods accompanied with mechanical and mathematical analysis. The topics addressed in this special issue will deal with mathematical and mechanical aspects of nonconventional discretization methods. The investigation of the sensitivity of phase-field approaches with respect to model specific parameters, that is, the critical length of regularization, the degradation function and the mobility is discussed in “A detailed investigation of the model influencing parameters of the phase-field fracture approach” by C. Bilgen, A. Kopaničáková, R. Krause, and K. Weinberg. The insights of diffusive models for fracture formulations are presented by a phase field model for ductile fracture with linear isotropic hardening in “3D phase field simulations of ductile fracture” by T. Noll, C. Kuhn, D. Olesch, and R. Müller. A stress equilibration procedure for hyperelastic material models based on a displacement-pressure approximation is investigated in the paper “Weakly symmetric stress equilibrium for hyperelastic material models” by F. Bertrand, M. Moldenhauer, and G. Starke. A mixed least-squares formulation with explicit consideration of the balance of angular momentum is discussed from an engineering point of view for the fulfillment of support reactions in the contribution “A mixed least-squares finite element formulation with explicit consideration of the balance of moment of momentum, a numerical study” by M. Igelbüscher, J. Schröder, and A. Schwarz. All contributions represent the vital ongoing research in the framework of the development of modern nonconventional discretization methods and thus provide an extensive overview of the Priority Programme 1748.