Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard...

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Solving extremely ill-posed structures Lars Beex Marc Duflot b Laurent Adam b b e-Xstream Engineering

Transcript of Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard...

Page 1: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Solving extremely ill-posed structures

Lars Beex Marc Duflotb Laurent Adamb

be-Xstream Engineering

Page 2: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Introduction

Page 3: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Introduction

480 µm

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Problem: no out-of-plane stiffness

x

y z

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Creating out-of-plane stiffness: Approach 1

x

y z

Lwarp

Lweft

h

h << Lwarp h << Lweft

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Creating out-of-plane stiffness: Approach 2

x

y z

bending

torsion

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Energy minimisation without Hessian

x

y z

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Energy minimisation: steepest descent

View 1 Top view

START: u0 E(u0)

Page 9: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Energy minimisation: steepest descent

View 1 Top view

START: u0 E(u0) -f(u0)

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Energy minimisation: steepest descent

View 1 Top view

UPDATE: ui+1 = ui – α f(ui) E(ui+1)

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Energy minimisation: steepest descent

View 1 Top view

CONTINUE until E(ui+1) > E(ui)

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Energy minimisation: steepest descent

View 1 Top view

CONDITION: small stepsize α à this makes the procedure inefficient

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Steepest descent with backtracking linesearch (BTLS)

View 1 Top view

IDEA: once you know the search direction, optimise stepsize α such that E(ui+1) is minimum

Page 14: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with backtracking linesearch (BTLS)

Cross-section along search direction

IDEA: once you know the direction, optimise stepsize α such that E(ui+1) is minimum

Page 15: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with backtracking linesearch (BTLS)

BACKTRACKING: αi = α0 while E(αi+1) < E(αi) αi+1 = 0.5 αi

Cross-section along search direction

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Steepest descent vs Steepest descent with BTLS

Steepest descent Steepest descent with BTLS

Page 17: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with backtracking linesearch (BTLS)

Problem BACKTRACKING: New point is higher than starting point

Cross-section along search direction

Page 18: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with BTLS and Armijo Rule (AR)

Solution: apply Armijo Rule, before backtracking E(ui – αi+1 f(ui)) ≤ E(ui) – c1 α0 f(ui)T f(ui)

Cross-section along search direction

Page 19: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with BTLS vs Steepest descent with BTLS & AR

Steepest descent with BTLS Steepest descent with BTLS & AR

Page 20: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with BTLS and Armijo Rule

Consider the following case:

Cross-section along search direction

AR

Page 21: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with BTLS and Armijo rule

Include Curvature Rule to ensure that the slope changes enough

Cross-section along search direction

AR

Page 22: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Steepest descent with BTLS incl. AR and CR

Curvature Rule:

f(ui)T f(ui+1) ≤ c2 f(ui)T f(ui) Terminology: Armijo Rule + Curvature Rule = Wolfe conditions

Page 23: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

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

Steepest descent AR Steepest descent AR & CR

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Conjugate gradient instead of Steepest descent

trond.hjorteland.com

Combine current search direction with previous search direction

Enough about backtracking linesearch, consider this:

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Conjugate gradient vs Steepest descent

Steepest descent update:

ui+1 = ui – α f(ui) Conjugate gradient update for 2nd step:

u2 = u1 – α ( f(u1) - β f(u0) )

where β: β(f(u1), f(u0))

(e.g. Polak-Ribiere)

Page 26: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

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

Steepest descent AR & CR Conjugate gradient AR & CR

Page 27: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Truss network: Conjugate gradient with BTLS & WC

x

y z

Page 28: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

Truss network: Conjugate gradient with BTLS & WC

no scaling

Page 29: Solving extremely ill-posed structures · Final remarks Contact cannot be included via standard penalty methods or Lagrange multipliers Energy landscape must be smooth ! no discontinuous

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

Alternative to avoid ill-posedness for truss structures:

à beams


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