3D Topology Optimization with Fatigue Constraints
Author
Hersbøll, Joachim Emil Kokholm
Term
4. term
Education
Publication year
2018
Submitted on
2018-06-01
Pages
74
Abstract
Dette speciale udvikler metoder til topologioptimering i 2D og 3D for store problemer med mange designvariable. Målet er at reducere massen, samtidig med at konstruktionen overholder krav til flydning (permanent deformation) og træthed (skade ved gentagne belastninger). Først gennemgås velkendte udfordringer i topologioptimering: skakbrætsmønstre (checkerboarding), net-/meshafhængige løsninger, minimum længdeskala og rene solid/void-strukturer. Disse illustreres med den klassiske formulering, hvor man minimerer compliance (eftergivelighed, dvs. omvendt stivhed) under en volumebegrænsning. Dernæst behandles spændingsbaserede begrænsninger. Formuleringen af spændingskrav inden for densitetsbaseret topologioptimering gennemgås, herunder problemet med singulære optima, som håndteres med en qp-tilgang (kvadratisk programmering). Fordi spændingsgrænser er lokale og talrige, anvendes en aggregeringsfunktion, der samler mange lokale begrænsninger i et fåtal glatte, beregningseffektive mål. På dette grundlag formuleres en flydebegrænsning med von Mises-flydekriteriet. Derudover opstilles et træthedskrav under antagelse af proportional belastning med variabel amplitude og lineært elastisk adfærd i højcyklusområdet. Belastningshistorien reduceres til reverseringer via rainflow-optælling. Hver reversering giver en multiaxial spændingstilstand, som omsættes til en ækvivalent uniaxial amplitudespænding med enten Sines-kriteriet eller signeret von Mises; middelspændingseffekter indregnes med den modificerede Goodman-relation. Antallet af cyklusser til brud estimeres med Basquin-kurven, og Palmgren-Miners lineære skadehypotese bruges til at formulere en lokal skadebegrænsning, som ligeledes håndteres ved hjælp af aggregering. Metoderne afprøves på standard 2D- og 3D-benchmark-eksempler, der viser, hvordan træthedsbegrænsninger påvirker de optimale materialefordelinger.
This thesis develops methods for 2D and 3D topology optimization aimed at large problems with many design variables. The goal is to reduce mass while ensuring the structure meets safety limits for yielding (permanent deformation) and fatigue (damage from repeated loading). First, the work reviews common issues in topology optimization: checkerboarding, mesh-dependent solutions, minimum length scale, and solid/void designs. These are illustrated using the classical formulation that minimizes compliance (flexibility, i.e., inverse stiffness) under a volume constraint. Next, the thesis addresses stress-based constraints. It examines how to formulate stress constraints within density-based topology optimization and discusses the well-known issue of singular optima, handled here with a qp approach (quadratic programming). Because stress limits are local and numerous, an aggregate function is used to combine many local constraints into a small number of smooth, computationally efficient measures. On this basis, a yield constraint is formulated using the von Mises criterion. A fatigue constraint is also formulated under assumptions of proportional loading with variable amplitude and linear elastic behavior in the high-cycle regime. Load histories are reduced to reversals by rainflow counting. Each reversal produces a multiaxial stress state that is converted to an equivalent uniaxial amplitude using either the Sines criterion or signed von Mises; mean stress effects are included via the modified Goodman relation. From the equivalent stress, the Basquin curve estimates cycles to failure, and Palmgren-Miner’s linear damage rule defines a local damage constraint, which is likewise handled by aggregation. The methods are demonstrated on standard 2D and 3D benchmark problems, highlighting how fatigue constraints influence optimal material layouts.
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