Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...

<p>A common technique to evaluate load paths in complex structures is to perform FE-calculations with relative large elements. This procedure gives no information regarding stress concentrations at e.g. holes or radius but this phenomenon can later on be investigated in details with local individual submodels. Displacements is taken from the global model and used to analyse stress concentrations and crack driving parameters in the submodel.</p><p>Today, the crack controlling stress intensity factors are in general cases obtained from handbook solutions of elementary cases. This method requires engineering judgements in a conservative manner and one way to improve the solution is to model the crack in its correct surroundings in a local three-dimensional submodel.</p><p>This master thesis is focused on the development of an automated support for analysing three-dimensional cracks in submodels. The results from a global Nastran model can be imported to Trinitas and used for a more accurate stress and fatigue life analysis in a local model. Here a three-dimensional crack tip subdomain can be generated inside an eight point brick volume. The crack tip subdomain is specially designed and adjusted for accurate determination of stress intensity factors along the crack front. For example, all points are adjusted with respect to the brick volume and the crack size, triangular wedge elements are applied around the crack tip, the midpoints for these elements are moved to quarter points and the crack front is curved. The crack tip subdomain is validated against several reference cases and shows sufficiently good results with respect to the stress intensity factor.</p><p>Finally, the automated crack tip subdomain generation is applied to a geometrically complex part of a main wing carry-through bulkhead of a fighter aircraft in order to show the applicability of the procedure in an industrial environment.</p>

The design process includes many different activities inwhich various computational mechanics tools are used forbehaviour modelling of mechanical systems and their buildingblocks, e.g. machine elements. These tools usually supportlarge and complex models and they produce large quantities ofdata with a high degree of complexity. In these situations,efficient data management and the ability to search and sharedata are important issues to achieve an efficient designprocess. Today, this ability is usually not supported by theindividual applications even though this probably would improveand facilitate the ability to search for data on a higher levelin the engineering information system. This work investigates the ability of searching andcomparing analysis data within behaviour models of technicalsystems as well as over the analysis results. This is done byinvestigating the potential benefits of integrating moderndatabase technology with a multibody system (MBS) analysissoftware in the same manner that has been successfully done forbusiness and administrative applications. This has resulted inan implemented pilot system, named MECHAMOS, that integratesthe main-memory resident object-relational database managementsystem (DBMS) AMOSwith the symbolic multibody system (MBS)software SOPHIA operating in MapleV. This provides MECHAMOSwith both symbolic and numeric mathematical capabilities forMBS analysis and data management capabilities to search andcompare engineering data in the database. The approach, making data managing tools available in acomputer aided engineering software, considerably improves theanalysis of technical systems. The analysis is brought to ahigher level through the available query language and thedesired data is specified, fairly intuitively, in a query. Whenthe query is processed, the DBMS knows how to retrieve andautomatically derive the required data. As shown in someexamples, the ability to search over stored and derived data inthe database is not restricted to a single MBS-model inMECHAMOS. Because of the implemented materialisation handling,it is also possible to search, combine, and compare data fromseveral simulation results which are based on several differentmodels in the database. This extends the ability to performoptimisation from a traditional parameter study to thepossibility to analyse and compare different technical conceptsthrough the query language and hereby retrieve those conceptsthat fulfil certain requirements. If submodel techniques aresupported, queries over a set of components in the databasewould automatically create, analyse and compare the possibleconcepts. This would assist the designer in choosing the bestcomponents for a design.

A common technique to evaluate load paths in complex structures is to perform FE-calculations with relative large elements. This procedure gives no information regarding stress concentrations at e.g. holes or radius but this phenomenon can later on be investigated in details with local individual submodels. Displacements is taken from the global model and used to analyse stress concentrations and crack driving parameters in the submodel. Today, the crack controlling stress intensity factors are in general cases obtained from handbook solutions of elementary cases. This method requires engineering judgements in a conservative manner and one way to improve the solution is to model the crack in its correct surroundings in a local three-dimensional submodel. This master thesis is focused on the development of an automated support for analysing three-dimensional cracks in submodels. The results from a global Nastran model can be imported to Trinitas and used for a more accurate stress and fatigue life analysis in a local model. Here a three-dimensional crack tip subdomain can be generated inside an eight point brick volume. The crack tip subdomain is specially designed and adjusted for accurate determination of stress intensity factors along the crack front. For example, all points are adjusted with respect to the brick volume and the crack size, triangular wedge elements are applied around the crack tip, the midpoints for these elements are moved to quarter points and the crack front is curved. The crack tip subdomain is validated against several reference cases and shows sufficiently good results with respect to the stress intensity factor. Finally, the automated crack tip subdomain generation is applied to a geometrically complex part of a main wing carry-through bulkhead of a fighter aircraft in order to show the applicability of the procedure in an industrial environment.

<p>The design process includes many different activities inwhich various computational mechanics tools are used forbehaviour modelling of mechanical systems and their buildingblocks, e.g. machine elements. These tools usually supportlarge and complex models and they produce large quantities ofdata with a high degree of complexity. In these situations,efficient data management and the ability to search and sharedata are important issues to achieve an efficient designprocess. Today, this ability is usually not supported by theindividual applications even though this probably would improveand facilitate the ability to search for data on a higher levelin the engineering information system.</p><p>This work investigates the ability of searching andcomparing analysis data within behaviour models of technicalsystems as well as over the analysis results. This is done byinvestigating the potential benefits of integrating moderndatabase technology with a multibody system (MBS) analysissoftware in the same manner that has been successfully done forbusiness and administrative applications. This has resulted inan implemented pilot system, named MECHAMOS, that integratesthe main-memory resident object-relational database managementsystem (DBMS) AMOSwith the symbolic multibody system (MBS)software SOPHIA operating in MapleV. This provides MECHAMOSwith both symbolic and numeric mathematical capabilities forMBS analysis and data management capabilities to search andcompare engineering data in the database.</p><p>The approach, making data managing tools available in acomputer aided engineering software, considerably improves theanalysis of technical systems. The analysis is brought to ahigher level through the available query language and thedesired data is specified, fairly intuitively, in a query. Whenthe query is processed, the DBMS knows how to retrieve andautomatically derive the required data. As shown in someexamples, the ability to search over stored and derived data inthe database is not restricted to a single MBS-model inMECHAMOS. Because of the implemented materialisation handling,it is also possible to search, combine, and compare data fromseveral simulation results which are based on several differentmodels in the database. This extends the ability to performoptimisation from a traditional parameter study to thepossibility to analyse and compare different technical conceptsthrough the query language and hereby retrieve those conceptsthat fulfil certain requirements. If submodel techniques aresupported, queries over a set of components in the databasewould automatically create, analyse and compare the possibleconcepts. This would assist the designer in choosing the bestcomponents for a design.</p>

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