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11 Layout of a paper machine. The process starts from the headbox, then continues from right to left and ends-up as a finished paper on the roll. Courtesy of Metso Paper, Inc.
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There exist several computational strategies of different efficiency for the solution of model-based optimization problems — particularly, in the case of models based on challenging CFD problems. Applied mathematics provides means for their analysis and for advice on their proper usage.
In this chapter, methods are mainly analyzed based on the expl...
Citations
... Temporal discretization involves the Euler method [18][19][20], explicit Runge-Kutta method [24], backward differentiation formula [15,16,21,23], orthogonal collocation on finite elements (OCFE) [17,22,25], and Cayley-Tustin framework [26,27]. Regarding to dynamic optimization strategies, single shooting [15,16,20,21,24], multiple shooting [18,23,28], and simultaneous collocation [17,22,25] all have applications. However, many studies pay more attention to the accuracy of temporal discretization so that high-order methods are usually used in time while low-order methods are employed in space, which may affect the convergence of optimization and even lead to unreliable results. ...
... whereṗ andż are tangent vectors,z andp are adjoint vectors, and the adjoint mode (27) indicates the solution X of M z linear Eqs. (28). ...
... whereẊ is the solution of the linear Eqs. (31) that are obtained by applying the forward differentiation to the linear system (28). The item (∂ 2 z/∂p 2ṗ ) Tz has the form of vector-Hessian-vector product, wherez is received from the gradient evaluation of the Lagrangian. ...
Dynamic optimization constrained by partial differential equations (PDE) involves the state distribution on space, and can generate improved operational policies compared to ODE-constrained dynamic optimization. With the help of space-time orthogonal collocation on finite elements (ST-OCFE) which can guarantee relatively high accuracy in both space and time, PDE models are transformed into a set of algebraic equations. For solving PDE-constrained optimization problems, three direct approaches, namely ST-OCFE based single shooting, ST-OCFE based multiple shooting, and ST-OCFE based simultaneous collocation are proposed. The similarities and differences among these approaches are analyzed, where the discretized nonlinear programming problems to be solved in each algorithm are described. Furthermore, the first-order and second-order sensitivity computation for ST-OCFE based shooting approaches is introduced with algorithmic differentiation to efficiently evaluate the exact Hessian of the Lagrangian. Three cases including different dynamics and constraints are investigated. The optimal solution given by these ST-OCFE based direct approaches is consistent, demonstrating the effectiveness of these algorithms.
The paper describes general methodologies for the solution of design optimization problems. In particular we outline the close relations between a fixed point solver based piggy back approach and a Reduced SQP method in Jacobi and Seidel variants. The convergence rate and general efficacy is shown to be strongly dependent on the characteristics of the state equation and the objective function. In the QP scenario where the state equation is linear and the objective quadratic, finite termination in two steps is obtained by the Seidel variant with Newton state solver and perfect design space preconditioning. More generally, it is shown that the retardation factor between simulation and optimization is bounded below by 2 with the difference depending on a cross-term representing the total sensitivity of the adjoint equation with respect to the design.
Recently, optimization has become an integral part of the aerodynamic design process chain. However, because of uncertainties with respect to the flight conditions and geometry uncertainties, a design optimized by a traditional design optimization method seeking only optimality may not achieve its expected performance. Robust optimization deals with optimal designs, which are robust with respect to small (or even large) perturbations of the optimization setpoint conditions. That means, the optimal designs computed should still be good designs, even if the input parameters for the optimization problem formulation are changed by a non-negligible amount. Thus even more experimental or numerical effort can be saved. In this paper, we aim at an improvement of existing simulation and optimization technology, developed in the German collaborative effort MEGADESIGN1, so that numerical uncertainties are identified, quantized and included in the overall optimization procedure, thus making robust design in this sense possible. We introduce two robust formulations of the aerodynamic optimization problem which we numerically compare in a 2d testcase under uncertain flight conditions. Beside the scalar valued uncertainties we consider the shape itself as an uncertainty source and apply a Karhunen-Loève expansion to approximate the infinite-dimensional probability space. To overcome the curse of dimensionality an adaptively refined sparse grid is used in order to compute statistics of the solution.
