## 65-XX NUMERICAL ANALYSIS

- Pedestrian Flow Models (2014)
- There have been many crowd disasters because of poor planning of the events. Pedestrian models are useful in analysing the behavior of pedestrians in advance to the events so that no pedestrians will be harmed during the event. This thesis deals with pedestrian flow models on microscopic, hydrodynamic and scalar scales. By following the Hughes' approach, who describes the crowd as a thinking fluid, we use the solution of the Eikonal equation to compute the optimal path for pedestrians. We start with the microscopic model for pedestrian flow and then derive the hydrodynamic and scalar models from it. We use particle methods to solve the governing equations. Moreover, we have coupled a mesh free particle method to the fixed grid for solving the Eikonal equation. We consider an example with a large number of pedestrians to investigate our models for different settings of obstacles and for different parameters. We also consider the pedestrian flow in a straight corridor and through T-junction and compare our numerical results with the experiments. A part of this work is devoted for finding a mesh free method to solve the Eikonal equation. Most of the available methods to solve the Eikonal equation are restricted to either cartesian grid or triangulated grid. In this context, we propose a mesh free method to solve the Eikonal equation, which can be applicable to any arbitrary grid and useful for the complex geometries.

- Nonlinear frequency response analysis of structural vibrations (2014)
- 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.

- Isogeometric analysis of nonlinear Euler-Bernoulli beam vibrations (2013)
- 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 Inexact Interior Point Method for the Large-Scale Simulation of Granular Material (2013)
- 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.

- On Finite Element Method–Flux Corrected Transport Stabilization for Advection-Diffusion Problems in a Partial Differential-Algebraic Framework (2013)
- 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.