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The existence of a complete, embedded minimal surface of genus one, with three ends and whose total Gaussian curvature satisfies equality in the estimate of Jorge and Meeks, was a sensation in the middle eighties. From this moment on, the surface of Costa, Hoffman and Meeks has become famous all around the world, not only in the community of mathematicians. With this article, we want to fill a gap in the injectivity proof of Hoffman and Meeks, where there is a lack of a strict mathematical justification. We exclusively argue topologically and do not use additional properties like differentiability or even holomorphy.

A method to correct the elastic stress tensor at a fixed point of an elastoplastic body, which is subject to exterior loads, is presented and analysed. In contrast to uniaxial corrections (Neuber or ESED), our method takes multiaxial phenomena like ratchetting or cyclic hardening/softening into account by use of Jiang's model. Our numerical algorithm is designed for the case that the scalar load functions are piecewise linear and can be used in connection with critical plane/multiaxial rainflow methods in high cycle fatigue analysis. In addition, a local existence and uniqueness result of Jiang's equations is given.

A gradient based algorithm for parameter identification (least-squares) is applied to a multiaxial correction method for elastic stresses and strains at notches. The correction scheme, which is numerically cheap, is based on Jiang's model of elastoplasticity. Both mathematical stress-strain computations (nonlinear PDE with Jiang's constitutive material law) and physical strain measurements have been approximized. The gradient evaluation with respect to the parameters, which is large-scale, is realized by the automatic forward differentiation technique.

In this article, we give some generalisations of existing Lipschitz estimates for the stop and the play operator with respect to an arbitrary convex and closed characteristic a separable Hilbert space. We are especially concerned with the dependency of their outputs with respect to different scalar products.

Error estimates for quasistatic global elastic correction and linear kinematic hardening material
(2006)

We consider in this paper the quasistatic boundary value problems of linear elasticity and nonlinear elastoplasticity with linear kinematic hardening material. We derive expressions and estimates for the difference of solutions (i.e. stress, strain and displacement) of both models. Further, we study the error between the elastoplastic solution and the solution of a postprocessing method, that corrects the solution of the linear elastic problem in order to approximate the elastoplastic model.

In this article, we give an explicit homotopy between the solutions (i.e. stress, strain, displacement) of the quasistatic linear elastic and nonlinear elastoplastic boundary value problem, where we assume a linear kinematic hardening material law. We give error estimates with respect to the homotopy parameter.

In this article, we present an analytic solution for Jiang's constitutive model of elastoplasticity. It is considered in its stress controlled form for proportional stress loading under the assumptions that the one-to-one coupling of the yield surface radius and the memory surface radius is switched off, that the transient hardening is neglected and that the ratchetting exponents are constant.

In this article we give a sufficient condition that a simply connected flexible body does not penetrate itself, if it is subjected to a continuous deformation. It is shown that the deformation map is automatically injective, if it is just locally injective and injective on the boundary of the body. Thereby, it is very remarkable that no higher regularity assumption than continuity for the deformation map is required. The proof exclusively relies on homotopy methods and the Jordan-Brouwer separation theorem.

In this paper, we present a viscoelastic rod model that is suitable for fast and accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (‘stiff’ dof), bending and torsion (‘soft’ dof). For inner dissipation, a consistent damping potential proposed by Antman is chosen. We parametrise the rotational dof by unit quaternions and directly use the quaternionic evolution differential equation for the discretisation of the Cosserat rod curvature. The discrete version of our rod model is obtained via a finite difference discretisation on a staggered grid. After an index reduction from three to zero, the right-hand side function f and the Jacobian \(\partial f/\partial(q, v, t)\) of the dynamical system \(\dot{q} = v, \dot{v} = f(q, v, t)\) is free of higher algebraic (e. g. root) or transcendental (e. g. trigonometric or exponential) functions and therefore cheap to evaluate. A comparison with Abaqus finite element results demonstrates the correct mechanical behavior of our discrete rod model. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computational times within milliseconds, it is suitable for interactive applications in ‘virtual reality’ as well as for multibody dynamics simulation.

A simple transformation of the Equation of Motion (EoM) allows us to directly integrate nonlinear structural models into the recursive Multibody System (MBS) formalism of SIMPACK. This contribution describes how the integration is performed for a discrete Cosserat rod model which has been developed at the ITWM. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.