On the optimization of microstructurally motivated On the optimization of microstructurally motivated calculations in engineering mechanicscalculations in engineering mechanics

O optimalizaci mikrostrukturálně motivovaných O optimalizaci mikrostrukturálně motivovaných výpočtů v inženýrské mechanicevýpočtů v inženýrské mechanice

4. matematický 4. matematický workshopworkshop 20. října 200520. října 2005

Jiří Vala (Jiří Vala (vala.j@fce.vutbr.czvala.j@fce.vutbr.cz))Ústav matematiky a deskriptivní geometrie Fakulty stavební VUT v BrněÚstav matematiky a deskriptivní geometrie Fakulty stavební VUT v Brně

Topics

1. Macro- and microanalysis in computational mechanicsMakro- a mikroanalýza ve výpočetní mechanice

2. Iterative algorithm for a model problemIterační algoritmus pro modelový problém

3. Generalizations and examples of technical applicationsZobecnění a příklady technických aplikací

Keywords computations “from nano-scale particles to terrestrial

bodies” homogenization techniquestwo-scale grids two-scale convergence

classical FEM mesh-free methods

Structure of some building materials

gas concrete

straw pannel Stramit

fire-clay brick foam polyethylen

2 types of polyurethan-based insulation

Laboratory of Building Physics, Faculty of Civil Engineering, Brno University of Technology

Example: temperature field in the rubber-based insulation in certain part of the window construction

local thermal fluxes

ANSYS-supported calculations

0.1 mm

Ni superalloy CMSX4

Ni-Al-Cr-Ta alloy superaustenitic iron NICROFER

Bi-Sn-Zn alloy

Microstructure of some advanced alloys

Institute of Physics of Materials,Academy of Sciences of the Czech Republic in Brno

Two basic approaches to two-scale problems:

1. some multiple levels of not necessarily nested grids considered (and some successive corrections needed) without deeper analysis of microstructural phenomena (Rech et al. 2003, Glowinski et al. 2005, …)

2. mathematical two-scale convergence (homogenization) theory (as generalization of G-, H-, Γ-, … convergence) applied (Nguetseng 1989, Allaire 1992, Holmbom 1997, ...)

Notation

Model elliptic problem

Two-scale convergence

Construction of homogenized characteristics

Iterative algorithm

Projection onto micro- and macroscopic scales

Convergence analysis

Finite element interpolation theory (cf. Zlámal 1968)

Several concluding references to recent author’s papers

• heat propagation in buildings: thermal insulation and accumulation J. V. & S. Šťastník, Modelling, July 2005 in Pilsen

• diffusive phase transformation in substitutional alloys J. V. & J. Svoboda, Algoritmy, March 2005 in PodbanskéJ. Svoboda & J. V., Defect & Diffusion Forum 240 (2005), 647-653J. V., Equadiff, July 2005 in Bratislava

• convergence analysis for an iterative algorithm in case of classical FEM J. V., Numerical Methods in Computational Mechanics, August 2005 in ŽilinaJ. V., Journal of Mechanical Engineering, to appear

Thank you for your attention. Questions and remarks are welcome.

Supported by GA ČR, Reg. No. 103/05/0292Better results are being prepared for the 5-th workshop

(probably) in Brno in October 2006….