## Fachbereich Mathematik

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#### Keywords

- isogeometric analysis (2)
- FEM-FCT stabilization (1)
- NURBS (1)
- Nonsmooth contact dynamics (1)
- adjoint approach (1)
- harmonic balance (1)
- modal derivatives (1)
- model reduction (1)
- monlinear vibration (1)
- partial differential-algebraic equations (1)
- positivity preserving time integration (1)
- shape optimization (1)
- weight optimization (1)

For the prediction of digging forces from a granular material simulation, the
Nonsmooth Contact Dynamics Method is examined. First, the equations of motion
for nonsmooth mechanical systems are laid out. They are a differential
variational inequality that has the same structure as classical discrete algebraic equations. Using a Galerkin projection in time, it becomes possible to derive
nonsmooth versions of the classical SHAK and RATTLE integrators.
A matrix-free Interior Point Method is used for the complementarity
problems that need to be solved in every time step. It is shown that this method
outperforms the Projected Gauss-Jacobi method by several orders of magnitude
and produces the same digging force result as the Discrete Element Method in comparable computing time.

To write about the history of a subject is a challenge that grows with the number of pages as the original goal of completeness is turning more and more into an impossibility. With this in mind, the present article takes a very narrow approach and uses personal side trips and memories on conferences,
workshops, and summer schools as the stage for some of the most important protagonists and their contributions to the field of Differential-Algebraic Equations (DAEs).

We develop a framework for shape optimization problems under state equation con-
straints where both state and control are discretized by B-splines or NURBS. In other
words, we use isogeometric analysis (IGA) for solving the partial differential equation and a nodal approach to change domains where control points take the place of nodes and where thus a quite general class of functions for representing optimal shapes and their boundaries becomes available. The minimization problem is solved by a gradient descent method where the shape gradient will be defined in isogeometric terms. This
gradient is obtained following two schemes, optimize first–discretize then and, reversely,
discretize first–optimize then. We show that for isogeometric analysis, the two schemes yield the same discrete system. Moreover, we also formulate shape optimization with respect to NURBS in the optimize first ansatz which amounts to finding optimal control points and weights simultaneously. Numerical tests illustrate the theory.

In this paper we present a method for nonlinear frequency response analysis of mechanical vibrations of 3-dimensional solid structures.
For computing nonlinear frequency response to periodic excitations, we employ the well-established harmonic balance method.
A fundamental aspect for allowing a large-scale application of the method is model order reduction of the discretized equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information.
For an efficient spatial discretization of continuum mechanics nonlinear partial differential equations, including large deformations and hyperelastic material laws, we use the isogeometric finite element method, which has already been shown to possess advantages over classical finite element discretizations in terms of higher accuracy of numerical approximations in the fields of linear vibration and static large deformation analysis.
With several computational examples, we demonstrate the applicability and accuracy of the modal derivative reduction method for nonlinear static computations and vibration analysis.
Thus, the presented method opens a promising perspective on application of nonlinear frequency analysis to large-scale industrial problems.

Non-smooth contact dynamics provides an increasingly popular simulation framework for granular material. In contrast to classical discrete element methods, this approach is stable for arbitrary time steps and produces visually acceptable results in very short computing time. Yet when it comes to the prediction of draft forces, non-smooth contact dynamics is typically not accurate enough. We therefore propose to combine the method class with an interior point algorithm for higher accuracy. Our specific algorithm is based on so-called Jordan algebras and exploits the relation to symmetric cones in order to tackle the conical constraints that are intrinsic to frictional contact problems. In every interior point iteration a linear system has to be solved. We analyze how the interior point method behaves when it is combined with Krylov subspace solvers and incomplete factorizations. We show that efficient preconditioners and efficient linear solvers are essential for the method to be applicable to large-scale problems. Using BiCGstab as a linear solver and incomplete Cholesky factorizations, we substantially improve the accuracy in comparison to the projected Gauss-Jacobi solver.

In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle.
As application, the straight nonlinear Euler-Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.

An extension of the finite element method–flux corrected transport stabilization (FEM-FCT) for hyperbolic problems in the context of partial differential-
algebraic equations (PDAEs) is proposed. Given a local extremum diminishing
property of the spatial discretization, the positivity preservation of the one-step
θ−scheme when applied to the time integration of the resulting differential-
algebraic equation (DAE) is shown, under a mild restriction on the time step-
size. As crucial tool in the analysis, the Drazin inverse and the corresponding
Drazin ODE are explicitly derived. Numerical results are presented for non-
constant and time-dependent boundary conditions in one space dimension and
for a two-dimensional advection problem where the advection proceeds skew to
the mesh.