Solving extremely ill-posed structures

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Solving extremely ill-posed structures

Lars Beex Marc Duflotb_{Laurent Adam}b

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Introduction

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Problem: no out-of-plane stiffness

x y

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Energy minimisation without Hessian

x y

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Energy minimisation: steepest descent

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START: u₀ E(u₀)

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START: u₀ E(u₀) -f(u₀)

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UPDATE: u_i+1 = u_i– α f(u_i)

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CONDITION: small stepsize α

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Steepest descent with backtracking linesearch (BTLS)

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IDEA: once you know the search direction,

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Cross-section along search direction

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BACKTRACKING: α_i = α₀

while E(α_i+1) < E(α_i) α_i+1 = 0.5 α_i

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Steepest descent vs Steepest descent with BTLS

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Problem BACKTRACKING: New point is higher than starting point

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Steepest descent with BTLS and Armijo Rule (AR)

Solution: apply Armijo Rule, before backtracking E(u_i – α_i+1f(u_i)) ≤ E(u_i) – c₁ α₀f(u_i)T _f(u

i)

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Steepest descent with BTLS vs Steepest descent with BTLS & AR

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Steepest descent with BTLS and Armijo Rule

Consider the following case:

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Steepest descent with BTLS and Armijo rule

Include Curvature Rule to ensure that the slope changes enough

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Steepest descent with BTLS incl. AR and CR

Curvature Rule:

f(u_i)T _f(u

i+1) ≤ c2 f(ui)T f(ui)

Terminology:

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Steepest descent with BTLS & AR vs Steepest descent with BTLS & WC

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Conjugate gradient instead of Steepest descent

trond.hjorteland.com

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Conjugate gradient vs Steepest descent

Steepest descent update:

u_i+1 = u_i– α f(u_i)

Conjugate gradient update for 2nd_step:

u₂ = u₁– α ( f(u₁) - β f(u₀) ) where β: β(f(u₁), f(u₀))

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Steepest descent with BTLS & AR vs Steepest descent with BTLS & WC

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Truss network: Conjugate gradient with BTLS & WC

x y

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Truss network: Conjugate gradient with BTLS & WC

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Final remarks

Contact cannot be included via standard penalty methods or Lagrange multipliers

Energy landscape must be smooth

à no discontinuous potential

à no dissipation, BUT varational dissipation!! à no friction in contact