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This book presents new ideas in the framework of novel, finite element discretization schemes for solids and structure, focusing on the mechanical as well as the mathematical background. It also explores the implementation and automation aspects of these technologies. Furthermore, the authors highlight recent developments in mixed finite element formulations in solid mechanics as well as novel techniques for flexible structures at finite deformations. The book also describes automation processes and the application of automatic differentiation technique, including characteristic problems, automatic code generation and code optimization. The combination of these approaches leads to highly efficient numerical codes, which are fundamental for reliable simulations of complicated engineering problems. These techniques are used in a wide range of applications from elasticity, viscoelasticity, plasticity, and viscoplasticity in classical engineering disciplines, such as civil and mechanical engineering, as well as in modern branches like biomechanics and multiphysics.



Engineering Notes on Concepts of the Finite Element Method for Elliptic Problems

In this contribution, we discuss some basic mechanical and mathematical features of the finite element technology for elliptic boundary value problems. Originating from an engineering perspective, we will introduce step by step of some basic mathematical concepts in order to set a basis for a deeper discussion of the rigorous mathematical approaches. In this context, we consider the boundedness of functions, the classification of the smoothness of functions, classical and mixed variational formulations as well as the \(H^{-1}\)-FEM in linear elasticity. Another focus is on the analysis of saddle point problems occurring in several mixed finite element formulations, especially on the solvability and stability of the associated discretized versions.
Jörg Schröder

Sensitivity Analysis Based Automation of Computational Problems

The paper describes automation of primal and sensitivity analysis of computational models formulated and solved by the finite element method. Based on the symbolic system AceGen (http://​symech.​fgg.​uni-lj.​si/​), fast and reliable code can be created with minimum effort and immediately tested and verified by using the associated finite element program AceFEM . Automation of first- and second-order sensitivity analysis with respect to an arbitrary parameter is presented. In an example, it is shown how sensitivity analysis has become an indispensable part of modern computational algorithms.
Jože Korelc, Teja Melink

Equilibrated Stress Reconstruction and a Posteriori Error Estimation for Linear Elasticity

Based on the displacement–pressure approximation computed with a stable finite element pair, a stress equilibration procedure for linear elasticity is proposed. Our focus is on the Taylor–Hood finite element space, with emphasis on the behavior for (nearly) incompressible materials. From a combination of displacement in the standard continuous finite element spaces of polynomial degrees k+1 and pressure in the standard continuous finite element spaces of polynomial degrees k, we construct an H(div)-conforming, weakly symmetric stress reconstruction. Explicit formulas are first given for a flux reconstruction and then for the stress reconstruction.
Fleurianne Bertrand, Bernhard Kober, Marcel Moldenhauer, Gerhard Starke

A Concept for the Extension of the Assumed Stress Finite Element Method to Hyperelasticity

The proposed work extends the well-known assumed stress elements to the framework of hyperelasticity. In order to obtain the constitutive relationship, a nonlinear set of equations is solved implicitly on element level. A numerical verification, where two assumed stress elements are compared to classical enhanced assumed strain elements, depicts the reliability and efficiency of the proposed concept. This work is closely related to the publication of Viebahn et al. (2019)
Nils Viebahn, Jörg Schröder, Peter Wriggers

A Fully Nonlinear Beam Model of Bernoulli–Euler Type

This work presents a geometrically exact Bernoulli–Euler rod model. In contrast to Pimenta (1993b), Pimenta and Yojo (1993), Pimenta (1996), Pimenta and Campello (2001), where the hypothesis considered was Timoshenko’s, this approach is based on the Bernoulli–Euler theory for rods, so that transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. The fact that both the first Piola–Kirchhoff stress tensor and the deformation gradient appear again as primary variables is also appealing. A straight reference configuration is assumed for the rod, but, in the same way, as in Pimenta (1996), Pimenta and Campello (2009), initially curved rods can be accomplished, if one regards the initial configuration as a stress-free deformed state from the straight position. Consequently, the use of convective non-Cartesian coordinate systems is not necessary, and only components on orthogonal frames are employed. A cross section is considered to undergo a rigid body motion and parameterization of the rotation field is done by the rotation tensor with the Rodrigues formula that makes the updating of the rotational variables very simple. This parametrization can be seen in Pimenta et al. (2008), Campello et al. (2011). A simple formula for the incremental Rodrigues parameters in function of the displacements derivative and the torsion angle is also settled down. A 2-node finite element with Cubic Hermitian interpolation for the displacements, together with a linear approximation for the torsion angle, is displayed within the usual Finite Element Method, leading to adequate \(C_{1}\) continuity.
Paulo de Mattos Pimenta, Sascha Maassen, Cátia da Costa e Silva, Jörg Schröder

Isogeometric Analysis of Solids in Boundary Representation

In this chapter, we present boundary-oriented numerical methods to analyze three-dimensional solid structures. For the analysis, the original geometry of the solid is employed according to the isogeometric paradigm. For the parametrization of the domain, the idea of the scaled boundary finite element method is adopted. Hence, the boundary of the solid is sufficient to describe the entire domain. The presented approaches employ analytical and numerical solution methods such as the Galerkin and collocation methods. To illustrate the applicability in the analysis procedure, three formulations are elaborated and demonstrated by means of numerical examples. The advantages compared to standard numerical methods are discussed thoroughly.
Sven Klinkel, Margarita Chasapi
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