Preface GAMM Mitteilungen

Abstract

The aim of this Priority Programme is to pool the 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 discretizations 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 and 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. Such uncertainties appear specifically in the field of geometrical and material nonlinearities. 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 by mechanical and mathematical analysis. The topics addressed in this special issue will deal with mathematical and mechanical aspects of nonconventional discretization methods. An application of a diffuse modeling approach for embedded material interfaces to nonconforming meshes is presented for linear elasticity in the paper “A diffuse modeling approach for embedded interfaces in linear elasticity” by P. Hennig, R. Maier, D. Peterseim, D. Schillinger, B. Verfürth, and M. Kästner. Mixed finite-element formulations for gradient elasticity are presented in a finite strain hyperelastic setting in the contribution “Three-field mixed finite element formulations for gradient elasticity at finite strains” by the authors J. Riesselmann, J. Ketteler, M. Schedensack, and D. Balzani. The investigation of mesh adaptivity for monolithic phase-field fractures in brittle materials by a reliable and efficient residual-type error estimator is discussed in the contribution “Mesh adaptivity for quasi-static phase-field fractures based on a residual-type a posteriori error estimator” by K. Mang, M. Walloth, T. Wick, and W. Wollner. The application of hp-basis functions with higher differentiability properties in the context of the finite cell method and numerical simulations on complex geometries is presented in “hp-basis functions of higher differentiability in the Finite Cell Method” by S. Kollmannsberger, D. D'Angella, E. Rank, W. Garhuom, S. Hubrich, A. Düster, P. Di Stolfo, and A. Schröder. All contributions represent 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.