# topology - Topology Optimization Mathematics for Design...

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1 Topology Optimization Mathematics for Design Homogenization Design Method (HMD) Why topology ? Change in shape & size may not lead our design criterion for reduction of structural weight.

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2 Structural Design 3 Sets of Problems • Sizing Optimization • thickness of a plate or membrane • height, width, radius of the cross section of a beam • Shape Optimization • outer/inner shape • Topology Optimization • number of holes • configuration Shape of the Outer Boundary Location of the Control Point of a Spline thickness distribution hole 2 hole 1 Sizing Optimization Starting of Design Optimization 1950s : Fully Stressed Design 1960s : Mathematical Programming ( L. Schmit at UCLA ) σσ = allowable in a structure min max allowable u Total Weight u Design Sensitivity Analysis
3 Ku = f N 1 DesignVelocity Sensitivity Dg g g D Dg g g D ∂∂ =+ =⇒ + =  +   u dd d u Ku f Ku f u K d Kf d d u Equilibrium : State Equation Design Sensitivity Performance Functions g Typical Performance Functions Strain Energy Density For Structural Design (This must be constant !) Mises Equivalent Stress For Strength Design and Failure Analysis Mean Compliance & Maximum Displacement For Stiffness Design Maximum Strain For Formability Study of Sheet Metals

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4 Hemp in 1950s Size to Topology Eliminate unnecessary bars by designing the cross sectional area. An Optimization Algorithm P1 P2 E, A min max max Ku f u = = σσ ρ e allowable i u eee e N AL 1 K uK u f DBu DB u u u u u =− + ∂σ = = =• AAA AA A e e e ee e e i e i i i e ± ±±± ±± ± bg Design Sensitivity
5 Prager in 1960s Design Optimization Theory Maximizing the minimum total potential energy 11 1 2 ee NN TT e e e e == Π= Π = ∑∑ dKd df max min e e design A Leads Equilibrium Π d ±²³ Why Total Potential ?

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## This document was uploaded on 12/08/2011.

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topology - Topology Optimization Mathematics for Design...

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