Meinhard Paffrath

Adapted polynomial chaos expansion for failure detection

Abstract In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator–prey model and a chemical reaction problem.

Intrusive versus Non-Intrusive Methods for Stochastic Finite Elements

In this paper we compare an intrusive with an non-intrusive method for computing Polynomial Chaos expansions. The main disadvantage of the nonintrusive method, the high number of function evaluations, is eliminated by a special Adaptive Gauss-Quadrature method. A detailed efficiency and accuracy analysis is performed for the new algorithm. The Polynomial Chaos expansion is applied to a practical problem in the field of stochastic Finite Elements.

Development of a Three-Dimensional Geometry Optimization Method for Turbomachinery Applications

This article describes the development of a method for optimization of the geometry of three-dimensional turbine blades within a stage configuration. The method is based on flow simulations and gradient-based optimization techniques. This approach uses the fully parameterized blade geometry as variables for the optimization problem. Physical parameters such as stagger angle, stacking line, and chord length are part of the model. Constraints guarantee the requirements for cooling, casting, and machining of the blades. The fluid physics of the turbomachine and hence the objective function of the optimization problem are calculated by means of a three-dimensional Navier-Stokes solver especially designed for turbomachinery applications. The gradients required for the optimization algorithm are computed by numerically solving the sensitivity equations. Therefore, the explicitly differentiated Navier-Stokes equations are incorporated into the numerical method of the flow solver, enabling the computation of the sensitivity equations with the same numerical scheme as used for the flow field solution. This article introduces the components of the fully automated optimization loop and their interactions. Furthermore, the sensitivity equation method is discussed and several aspects of the implementation into a flow solver are

Method of temporary coordinate domains for moving boundary value problems (semiconductor processing simulation)

An approach to the numerical solution of moving boundary value problems that is based on the idea of temporary coordinate domains is presented. The method generalizes the concept of coordinate transformations in two respects. First, the physical structure itself at some moment is chosen as the coordinate domain. Thus, the coordinate domain is no longer determined a priori and without regard to the actual physical structure. Second, the coordinate domain is adapted to the geometrical evolution of the structure by redefining it at appropriate time points. In this way the geometrical restrictions of conventional coordinate transformation methods are eliminated. The method is used to simulate thermal processing. It is assumed that the geometry and the material flow are handled by a separate module for simulations of oxide growth. The redistribution of dopants, including segregation at material interfaces, is modeled by diffusion equations and interface conditions. >

Three-Dimensional Optimization of Turbomachinery Bladings Using Sensitivity Analysis

This paper presents an approach to optimize the geometry of turbomachinery blades utilizing an automated optimization loop. Optimization examples of turbomachinery blade geometries with selected objective functions and a set of design variables are introduced. The presented optimization examples are performed for a 1.5-stage turbine test case. The blade geometries are fully parameterized, enabling three-dimensional changes to the blade shape during the optimization. Therefore various non-physical and physical parameters such as stacking line or stagger angle can be selected as design variables. Three-dimensional steady-state numerical flow simulations and a sensitivity equation method are part of the optimization process. The design sensitivities used within the optimization are obtained by numerically solving the analytical sensitivity equations. This optimization scheme uses the same numerical method for the flow simulation and for the computation of the sensitivity equation. A Navier-Stokes flow solver, which has especially been designed for turbomachinery applications was used for the implementation of the sensitivity equation method. The focus of this paper is on the application of the described optimization strategy to turbomachinery flows. The presented optimization examples are used to demonstrate and to discuss the capabilities of this approach.Copyright

An accuracy comparison of polynomial chaos type methods for the propagation of uncertainties

In (Augustin et al. in European J. Appl. Math. 19:149-190, 2008) we considered the Polynomial Chaos Expansion for the treatment of uncertainties in industrial applications. For many applications the method has been proven to be a computationally superior alternative to Monte Carlo evaluations. In the current overview we compare the accuracy of Polynomial Chaos type methods for the propagation of uncertainties in nonlinear problems and verify them on two examples relevant for industry. For weakly nonlinear time-dependent models, the generalized Kalman filter equations define an efficient method, yielding good approximations if the quantities of interest are restricted to the first two moments of the solution. Secondly, stochastic collocation is discussed. The method is applied to delay differential equations and random ordinary differential equations. Finally, a generalized PC method is discussed which is based on a subdivision of the random space. This approach is even suitable for highly nonlinear models.