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10 Comparison of CD velocity for the optimal control (Step 2) between HOCS Fiber and the 3D CFD model
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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...
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... paper machine [1,24,35]. The headbox is the first part of the paper machine from where the fiber-water mixture starts the papermaking process by going through the header as the first flow passage ( Fig. 9.2). The fiber- water mixture coming from a pump is turned 90 • into the machine direction and distributed inside the header as illustrated in Fig. 9.3. An uneven flow rate distribution may cause an uneven basis weight profile in the paper web produced. Inside the headbox, a non-optimal header may also be a potential source of cross-directional flows which in turn causes misaligned fibers in the paper since fibers approximately follow the velocity vectors at the outlet jet of the ...
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... vectors at the outlet jet of the headbox. Optimization is utilized to minimize these disturbances. The design of the tapered header has been identified as an optimal shape design problem. A cost function describes how even the flow rate distribution after the header should be and design variables define the location of the back Initial Optimized Fig. 9.4 Initial and optimized back wall of the tapered header wall of the header. The first numerical experiments in header optimization were reported in the early 1990's ...
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... of the headbox flows includes certain special features. First, the tube bundles (see Figs. 9.2 and 9.3) consist of hundreds of small tubes. They cannot be included in detail in CFD but are taken into account as an ef- fective porous medium. A three-dimensional (3D) CFD model would also be too time-consuming for optimization. Hence, specific two-dimensional (2D) models have been developed for the headbox applications [12]. The ...
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... at the paper mill. Despite the model reductions, the accu- racy of the HOCS Fiber has been confirmed: typically, for the first paper machine start-up, only one slice opening tuning proposed by the software is needed. HOCS Fiber has been successfully used at dozens of paper mills and one example of a real trouble shooting case is given in Fig. 9.6. As seen in the figure, the fiber orientation angle is much better after optimization than the original one measured at the mill. The original fiber orientation profile may cause problems in printing machines, but the optimized profile fulfills all the market ...
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... reduced model, that is, the depth-averaged Navier-Stokes equations, has been validated by comparing the results to a complete CFD model which includes the headbox slice channel and its free jet in 3D. The coordinate frame of the model is shown in the sketch in Fig. 9.7. The x-axis follows the main flow direction, while the z-axis lies in the span-wise direction. The 3D CFD model utilized has been validated using wind tunnel experiments [31]. The fiber orientation angle profile is mostly determined by the CD velocity component, because variations in the MD velocity are only of the order of one per ...
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... using wind tunnel experiments [31]. The fiber orientation angle profile is mostly determined by the CD velocity component, because variations in the MD velocity are only of the order of one per mille. Thus, the CD velocity is the most important component in the validation. The CD velocity profiles at the slice opening for Step 1 are presented in Fig. 9.8 (Step 1 is the diagnostics step determining flow rate profiles inside the headbox which creates the measured fiber orientation pro- file). The whole CD velocity field is presented in Fig. 9.9 to illustrate how the cross-directional flows develop in the slice channel. As can be seen in Fig. 9.8, the prediction given by HOCS Fiber ...
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... mille. Thus, the CD velocity is the most important component in the validation. The CD velocity profiles at the slice opening for Step 1 are presented in Fig. 9.8 (Step 1 is the diagnostics step determining flow rate profiles inside the headbox which creates the measured fiber orientation pro- file). The whole CD velocity field is presented in Fig. 9.9 to illustrate how the cross-directional flows develop in the slice channel. As can be seen in Fig. 9.8, the prediction given by HOCS Fiber agrees well with the result calculated using the whole 3D model, even though the reduced model does not take into account the effect of the ...
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... at the slice opening for Step 1 are presented in Fig. 9.8 (Step 1 is the diagnostics step determining flow rate profiles inside the headbox which creates the measured fiber orientation pro- file). The whole CD velocity field is presented in Fig. 9.9 to illustrate how the cross-directional flows develop in the slice channel. As can be seen in Fig. 9.8, the prediction given by HOCS Fiber agrees well with the result calculated using the whole 3D model, even though the reduced model does not take into account the effect of the ...
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... the diagnostics step, the fiber orientation is optimized. The CD pro- files obtained from Step 2 are presented in Fig. 9.10. As can be seen, the difference between the reduced and the full model is now slightly bigger than in Step 1, but both models predict very small CD velocities resulting in op- timal fiber orientation angle profile. The validation shows that the depth-averaged equations are sufficiently accurate for use in solving industrial fiber ...
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... Fiber can perform both optimization steps, that is, both the diagnostics and op- timal control steps, in a couple of minutes. If the full 3D model had been coupled with optimization by using finite difference approximation for the cost function gradient, this would lead to computing times of days or weeks, which is unreasonable. As illustrated in Fig. 9.1, compromises need to be made and CFD models need to be reduced when developing tools for control ...
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... in the previous sections, the unit-processes of the paper machine have been individually modeled and optimized for more than ten years. Nowa- days, focus has extended to handle larger ensembles including the whole pa- permaking process. This poses a real challenge since a paper machine consists of a number of consecutive sub-ensembles as shown in Fig. 9.11. There are four main parts essential in the papermaking process: the headbox, wire section, press section and the drying section. Thus, the model of the whole process represents the combination of different unit-process models as a chain where the output of one unit-process is an input for the following unit-processes. In order to ...
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... are numerous optimization approaches that are able to utilize uncertainty information such as optimization under uncertainty [6,19,25], stochastic programming [2] and robust optimization [8,30] among others. Figure 9.12 illustrates on a simple example how model uncertainty may affect reliability of the optimization results. ...
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.
