Schnelle Löser für partielle Differentialgleichungen

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Combining Parameterizations, Sobolev Methods and Shape Hessian Approximations for Aerodynamic Design Optimization

Aerodynamic design optimization, considered in this thesis, is a large and complex area spanning different disciplines from mathematics to engineering. To perform optimizations on industrially relevant test cases, various algorithms and techniques have been proposed throughout the literature, including the Sobolev smoothing of gradients. This thesis combines the Sobolev methodology for PDE constrained flow problems with the parameterization of the computational grid and interprets the resulting matrix as an approximation of the reduced shape Hessian. Traditionally, Sobolev gradient methods help prevent a loss of regularity and reduce high-frequency noise in the derivative calculation. Such a reinterpretation of the gradient in a different Hilbert space can be seen as a shape Hessian approximation. In the past, such approaches have been formulated in a non-parametric setting, while industrially relevant applications usually have a parameterized setting. In this thesis, the presence of a design parameterization for the shape description is explicitly considered. This research aims to demonstrate how a combination of Sobolev methods and parameterization can be done successfully, using a novel mathematical result based on the generalized Faà di Bruno formula. Such a formulation can yield benefits even if a smooth parameterization is already used. The results obtained allow for the formulation of an efficient and flexible optimization strategy, which can incorporate the Sobolev smoothing procedure for test cases where a parameterization describes the shape, e.g., a CAD model, and where additional constraints on the geometry and the flow are to be considered. Furthermore, the algorithm is also extended to One Shot optimization methods. One Shot algorithms are a tool for simultaneous analysis and design when dealing with inexact flow and adjoint solutions in a PDE constrained optimization. The proposed parameterized Sobolev smoothing approach is especially beneficial in such a setting to ensure a fast and robust convergence towards an optimal design. Key features of the implementation of the algorithms developed herein are pointed out, including the construction of the Laplace-Beltrami operator via finite elements and an efficient evaluation of the parameterization Jacobian using algorithmic differentiation. The newly derived algorithms are applied to relevant test cases featuring drag minimization problems, particularly for three-dimensional flows with turbulent RANS equations. These problems include additional constraints on the flow, e.g., constant lift, and the geometry, e.g., minimal thickness. The Sobolev smoothing combined with the parameterization is applied in classical and One Shot optimization settings and is compared to other traditional optimization algorithms. The numerical results show a performance improvement in runtime for the new combined algorithm over a classical Quasi-Newton scheme

On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Amp\`{e}re Operator

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step $h\rightarrow 0$. We considered also test problems with no known exact solutions

A three-dimensional macroscopic fundamental diagram for mixed bi-modal urban networks

Recent research has studied the existence and the properties of a macroscopic fundamental diagram (MFD) for large urban networks. The MFD should not be universally expected as high scatter or hysteresis might appear for some type of networks, like heterogeneous networks or freeways. In this paper, we investigate if aggregated relationships can describe the performance of urban bi-modal networks with buses and cars sharing the same road infrastructure and identify how this performance is influenced by the interactions between modes and the effect of bus stops. Based on simulation data, we develop a three-dimensional vehicle MFD (3D-vMFD) relating the accumulation of cars and buses, and the total circulating vehicle flow in the network. This relation experiences low scatter and can be approximated by an exponential-family function. We also propose a parsimonious model to estimate a three-dimensional passenger MFD (3D-pMFD), which provides a different perspective of the flow characteristics in bi-modal networks, by considering that buses carry more passengers. We also show that a constant Bus-Car Unit (BCU) equivalent value cannot describe the influence of buses in the system as congestion develops. We then integrate a partitioning algorithm to cluster the network into a small number of regions with similar mode composition and level of congestion. Our results show that partitioning unveils important traffic properties of flow heterogeneity in the studied network. Interactions between buses and cars are different in the partitioned regions due to higher density of buses. Building on these results, various traffic management strategies in bi-modal multi-region urban networks can then be integrated, such as redistribution of urban space among different modes, perimeter signal control with preferential treatment of buses and bus priority

Optimal control of multiphase steel production

An optimal control problem for the production of multiphase steel is investigated, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the ferrite phase fraction. The optimal control problem is analyzed and the first-order necessary and second-order sufficient optimality conditions are derived. For the numerical solution of the control problem reduced sequential quadratic programming (rSQP) method with a primal-dual active set strategy (PDAS) was applied. The numerical results were presented for the optimal control of a cooling line for production of hot rolled Mo-Mn dual phase steel

Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications

International audienceParameterization of computational domain is a key step in isogeometric analysis just as mesh generation is in finite element analysis. In this paper, we study the volume parameterization problem of multi-block computational domain in isogeometric version, i.e, how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Firstly, we show how to find good volume parameterization of single-block computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parametrization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. By using this method, the resulted volume parameterization has no self-intersections, and the isoparametric structure has good uniformity and orthogonality. Then we extend this method to the multi-block case, in which the continuity condition between the neighbor B-spline volume should be added to the constraint term. The effectiveness of the proposed method is illustrated by several examples based on three-dimensional heat conduction problem