C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationSpace mapping and defect correction D. Echeverría, P.W. Hemker Space mapping and defect correction ABSTRACT In this paper we show that space-mapping optimization can be understood in the framework of defect correction. Then, space-mapping algorithms can be seen as special cases of defect correction iteration. In order to analyze properties of space mapping and the space-mapping function we introduce the new concept of flexibility of the underlying models. The best spacemapping results are obtained for so-called equally flexible models. By introducing an affine operator as a left preconditioner, two models can be made equally flexible, at least in the neighborhood of a solution. This motivates an improved space-mapping (or manifold-mapping) algorithm. The left preconditioner complements traditional space mapping where only a right preconditioner is used. In the last section a few simple examples illustrate some of the phenomena analyzed in this paper. REPORT MAS-E0506 MARCH 2005 ABSTRACTIn this paper we show that space-mapping optimization can be understood in the framework of defect correction. Then, space-mapping algorithms can be seen as special cases of defect correction iteration. In order to analyze properties of space mapping and the space-mapping function we introduce the new concept of flexibility of the underlying models. The best space-mapping results are obtained for so-called equally flexible models.By introducing an affine operator as a left preconditioner, two models can be made equally flexible, at least in the neighborhood of a solution. This motivates an improved space-mapping (or manifold-mapping) algorithm. The left preconditioner complements traditional space mapping where only a right preconditioner is used. In the last section a few simple examples illustrate some of the phenomena analyzed in this paper. Mathematics Subject Classification: 65K05, 65B05Keywords and Phrases: Space mapping, defect correction, two-level optimization IntroductionThe space-mapping idea was conceived by Bandler [4] in the field of microwave filter design. It aims at reducing the cost of accurate optimization computations by iteratively correcting a sequence of rougher approximations. In technological applications this allows us to couple simple rules that represent expert knowledge accumulated over the years with the accuracy of expensive simulation techniques based on the numerical solution of partial differential equations. This combination may yield an efficient method with good accuracy of the final solution. As an example, in Section 5.2 we significantly accelerate the solution process for an optimization problem from magnetostatics by combining finite elements for the precise computations of the magnetic field with rather simple magnetic circuit calculations. The space-mapping technique has been mainly applied in electromagnetics [6,8] but, since the underlying principles...

SUMMARYDeflating the shifted Laplacian with geometric multigrid vectors yields speedup. To verify this claim, we investigate a simplified variant of Erlangga and Nabben presented in [Erlangga and Nabben, ETNA, 2008;31:403-424]. We derive expressions for the eigenvalues of the two-level preconditioner for the onedimensional problem. These expressions show that the algorithm analyzed is not scalable. They also show that the imaginary shift can be increased without delaying the convergence of the outer Krylov acceleration. An increase of the number of grid points per wavelength results in convergence acceleration. This contrasts to the use of the shifted Laplace preconditioner. Our analysis also shows that the use of deflation results in a spectrum more favorable to the convergence of the outer Krylov acceleration. The near-null space components are still insufficiently well resolved, and the number of iterations increases with the wavenumber. In the two-dimensional case, the number of near-zero eigenvalues is larger than in the one-dimensional case. We perform numerical computations with the two-level and multilevel versions of the algorithm on constant and nonconstant wavenumber problems. Our numerical results confirm our spectral analysis.

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationManifold-mapping optimization applied to linear actuator design ABSTRACT Optimization procedures in practice are based on highly accurate models that typically have an excessive computational cost. By exploiting auxiliary models that are less accurate but much cheaper to compute, space-mapping has been reported to accelerate such procedures. However, the space-mapping solution does not always coincide with the accurate model optimum. We introduce manifold mapping, an improved version of space mapping that finds this precise solution with the same computational efficiency. By an example in linear actuator design we show that our technique delivers a significant speed-up compared to other optimization schemes.2000 Mathematics Subject Classification: 65K10, 65M60, 65N55, 65Y20, 90C31

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Avoiding the formation of rings in rotary kilns is an issue of primary concern to the cement production industry. We developed a numerical combustion model that revealed that in our case study rings are typically formed in zones of maximal radiative heat transfer. This local overheating causes an overproduction of the liquid phase of the granular material, which tends to stick to the oven's wall and to form rings. To counteract for this phenomenon, we propose to increase the amount of secondary air injected to cool the oven. Experimental validation at the plant has confirmed that our solution is indeed effective. For the first time in years, the kiln has been operating without unscheduled shut-downs, resulting in hugely important cost savings.

This work studies how non-premixed turbulent combustion in a rotary kiln depends on the geometry of the secondary air inlet channel. We target a kiln in which temperatures can reach values above 1800 degrees Kelvin. Monitoring and possible mitigation of the thermal nitric-oxide (NOx) formation is of utmost importance. The performed reactive flow simulations result in detailed maps of the spatial distribution of the flow, thermodynamics and chemical conditions of the kiln. These maps provide valuable information to the operator of the kiln. The simulations show the difference between the existing and the newly proposed geometry of the secondary air inlet. In the existing configuration, the secondary air inlet is rectangular and located above the base of the burner pipe. The secondary air flows into the furnace from the top of the flame. The heat release by combustion is unevenly distributed throughout the flame. In the new geometry, the secondary air inlet is an annular ring placed around the burner pipe. The secondary air flows circumferentially around the burner pipe. The new secondary air inlet geometry is shown to result in a more homogeneous spatial distribution of the heat release throughout the flame. The peak temperatures of the flame and the production of thermal NOx are significantly reduced. Further research is required to resolve limitations of various choices in our modeling approach.

This series contains volumes on scientific computing for a wide range of electromagnetics problems. The electronics industry, in a very broad sense, is at the forefront of innovation, and our daily life is very much dependent on achievements in this area. These are mainly enabled by rapid developments in sophisticated virtual design environments, numerical methods being at the core of these. Volumes in the series provide details on the modeling, analysis and simulation of problems, as well as on the design process of robust solution methods. Applications range from simple benchmark problems to industrial problems at the forefront of current research and development.For more information on this series and our other book series, please visit our website: www.atlantis-press.com Atlantis Press