Ales Janka

Nested and self-adaptive Bézier parameterizations for shape optimization

This article is a sequel of [J.-A. Desideri, Hierarchical optimum-shape algorithms using embedded Bezier parameterizations, in: Y. Kuznetsov et al., (Ed.), Numerical Methods for Scientific Computing, Variational Problems and Applications, CIMNE, Barcelona, 2003], in which we defined formally a hierarchical shape optimization method based on a multi-level shape representation by nested Bezier parameterizations (FAMOSA), and [J.-A. Desideri, A. Janka, Multi-level shape parameterization for aerodynamic optimization - application to drag and noise reduction of transonic/supersonic business jet, in: E. Heikkola et al., (Ed.), European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyvaskyla, 24-28 July 2004] where we conducted some preliminary numerical experiments of shape optimization in aerodynamics. Here, we are testing the full multi-level optimum-shape algorithm (analogous in logical structure to the classical full multigrid method). Second, we propose a technique for parameterization self-adaptivity. Both methodological enhancements are assessed by novel numerical experiments on an inverse shape model problem, confirming both are very effective.

Numerical Methods for Ferromagnetic Plates

We present two numerical methods for the simulation of ferromagnetic phenomenons in a metallic plate, with or without holes. First we briefly recall the physical model we use for describing the ferromagnetic phenomenon. This model is based on the use of a scalar potential while other models rather use a vector potential as in [1] or [2]. Next we present the discretization methods we use. We then apply these methods on the simple test-case of a thin ferromagnetic plate placed in front of a rectilineal electric conductor. We show the various obtained results: magnetic field on a line perpendicular to the plate and relative permeability on a given line in the plate. Finally we illustrate our results with an industrial device.

Mathematical Modeling of the Dynamics of Shoot-Root Interactions and Resource Partitioning in Plant Growth

Plants are highly plastic in their potential to adapt to changing environmental conditions. For example, they can selectively promote the relative growth of the root and the shoot in response to limiting supply of mineral nutrients and light, respectively, a phenomenon that is referred to as balanced growth or functional equilibrium. To gain insight into the regulatory network that controls this phenomenon, we took a systems biology approach that combines experimental work with mathematical modeling. We developed a mathematical model representing the activities of the root (nutrient and water uptake) and the shoot (photosynthesis), and their interactions through the exchange of the substrates sugar and phosphate (Pi). The model has been calibrated and validated with two independent experimental data sets obtained with Petunia hybrida. It involves a realistic environment with a day-and-night cycle, which necessitated the introduction of a transitory carbohydrate storage pool and an endogenous clock for coordination of metabolism with the environment. Our main goal was to grasp the dynamic adaptation of shoot:root ratio as a result of changes in light and Pi supply. The results of our study are in agreement with balanced growth hypothesis, suggesting that plants maintain a functional equilibrium between shoot and root activity based on differential growth of these two compartments. Furthermore, our results indicate that resource partitioning can be understood as the emergent property of many local physiological processes in the shoot and the root without explicit partitioning functions. Based on its encouraging predictive power, the model will be further developed as a tool to analyze resource partitioning in shoot and root crops.

ALGEBRAIC DOMAIN DECOMPOSITION SOLVER FOR LINEAR ELASTICITY

We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korns constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.

Some Mathematical and Numerical Aspects in Aluminum Production

In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in an aluminum production cell as well as its numerical approximation by a finite element method. We put the emphasis on the magnetic effects which live in the whole three dimensional space and which are solved numerically with a domain decomposition method.

A domain decomposition solver for ferromagnetism

We model ferromagnetic effects by using reduced scalar potential. An overlapping domain-decomposition technique is proposed to solve the underlying problem in unbounded domain. In the near-field, adjacent to the ferromagnetic object, we employ finite element discretization, while the far-field, going to infinity, is modelled through Green representation formulas. In this way, we circumvent the need to store and invert full matrices, which usually occur when using techniques based on boundary element method on fictitious boundary. We estimate the CPU and memory costs for our approach and compare its performance to other known techniques on a simple testcase.

A Flux Correction Multigrid for Compressible Flow

A finite-volume based linear multigrid algorithm is proposed and used within an implicit linearized scheme to solve Navier–Stokes equations for compressible laminar flows. Coarse level problems are constructed algebraically based on convective and diffusive fluxes, without the knowledge of coarse geometry. Numerical results for complex 2D geometries such as airfoils, including stretched meshes, show mesh size independent convergence and efficiency of the method compared to other finite-volume-based multigrid method.