Local control in topology optimization: from additive manufacturing to stress constraints
Author:
Jose Antonio Torres Lerma
PhD Research Plan
Supervisors:
Dr. Alex Ferrer Ferre
Dr. Fermin Otero Gruer
Presentation Outline
Introduction
Additive manufacturing constraints
A new topology optimization algorithm with limited parameters
Stress constraints
Summary
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Introduction
Motivation
4
Need to reduce fuel consumption and environmental impact in the transport industry
Aerospace industry
Automotive industry
Topology
Optimization
Topology Optimization
5
Problem formulation
Design variables
The original design variable is discontinuous
Common functionals:
Density
Level set
Additive manufacturing
6
Use of additive manufacturing (3D CAD data printing)
3D printing process failed!
Not enough sensitivity of the machine
50%
75%
85%
01
02
03
Topology optimized design
Minimum length scale constraint
Additive manufacturing
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Use of additive manufacturing (3D CAD data printing)
3D printing process failed!
Material fell during deposition process
50%
75%
85%
01
02
03
Topology optimized design
Overhang constraint
Objectives overview
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Objective 1
Local AM constraints
Objective 3
Stress constraints
Objective 2
SQP + TO multiple constraints
Manufacturability and integrity in TO
Additive manufacturing constraints
Additive manufacturing constraints
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Length scales
constraints
Overhang
constraints
Bound formulation and three density fields representation
Implicit penalization with larger nº inequality/PDE constraints
Isotropic Perimeter as penalty term in the cost function
Efficient fulfillment of a global length
Overhang density filtering through threshold projection
Layer-by-layer computation of the gradient
Mechanical constraints in shape optimization
Complex shape derivatives
?
Objective
To define a simple and efficient local control of additive manufacturing constraints in topology optimization, working for density and level set
Global perimeter
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Overhang
constraints
The Perimeter is a functional that computes the length of Ω boundaries.
Relative perimeter
INTERNAL BOUNDARIES
Total perimeter
INTERNAL + EXTERNAL BOUNDARIES
Shape derivative (level set)
Gradient methods (density)
1. Domain filtering
2. Perimeter computation
Global perimeter
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Overhang
constraints
1. Domain filtering
Global isotropic relative perimeter: H1(D)
projection with Neumann boundary conditions.
Similar
Finite gradient
(S. Amstutz, C. Dapogny & A. Ferrer, 2022)
2. Perimeter computation
Global perimeter convergence
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Overhang
constraints
Global perimeter
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Other filtering strategies
1. Isotropic perimeter
2. Anisotropic perimeter
3. Non-linear anisotropic perimeter
Implementation
Already seen
Pending to implement
Current findings of global perimeter
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Cantilever beam
Current findings of global perimeter
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Gripper compliant mechanism
Extension to local perimeter
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New motivation: control the length of boundaries and overhang more locally.
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Global
constraint
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Multiple
local constraints
One Lagrange multiplier per subdomain.
A new topology optimization algorithm with limited parameters
Optimization schemes
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Goal: to solve a non-linear optimization problem with multiple constraints (PDE and others) in a versatile way
Augmented
lagrangian
Interior point
methods
MMA
Null space
algorithm
Gradient methods
Shape derivatives
Topological derivatives
Parameter dependent
Gradient methods
Shape derivatives
Topological derivatives
Gradient methods
Shape derivatives
Topological derivatives
Parameter dependent
Parameter dependent
Parameter free
Gradient methods
Shape derivatives
Topological derivatives
Topological derivatives
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Null space algorithm (multiple constraints and parameter free)
Well suited for gradient methods, but not enough versatile
Spherical linear interpolation scheme
(Slerp)
Topological derivative
How the cost changes with an inclusion
Dr. Ferrer dissertation
Objective
To couple null space algorithm with topological derivatives and multiple local constraints
Null space optimizer
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Null space algorithm (multiple constraints and parameter free)
Null space flow
Range space flow
Complete gradient flow
Current findings of null space/Slerp
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Analytic problem with η=1
Topology optimization with density and η=1
Topology optimization using topological derivatives and η=1
Stress constraints
Additive manufacturing
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Use of additive manufacturing (3D CAD data printing)
3D printing process completed!
50%
75%
100%
01
02
03
Topology optimized design
Three-point flexural test
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Now, we should test if the printed design fulfills its function, according to the application.
Plasticity/damage generated!
The maximum stress has been exceeded
Objective
To include stress constraints in the entire framework
Stress computation
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Elastic problem weak form
Stress tensor
But… what do we understand for homogenized stresses?
Intermediate material (micro). Which one?
Stress constraint
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We choose a metric for the maximum stress and then… we obtain more constraints to consider!
The maximum function is not differentiable, and we can not use bound formulation since the Von Mises stresses are defined at each gauss point.
Thus, we may relax the problem:
Summary
Main goal and its objectives
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We will ensure the manufacturability and integrity of topology optimized designs by addressing:
The methodology will be based in constraining locally the anisotropic perimeter and the stresses, and then solving the optimization problem with the versatile null space method.
Applications
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Structures with maximum stiffness
Compliant mechanisms
Microstructures with optimized homogenized properties
Finite elements for linear elasticity. Using yield stress as maximum value
(2D and 3D)
Finite elements for linear periodic cell problem
(2D and 3D)
Next steps
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Publications and participation in conferences
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Publications:
Participation in conferences:
Thank you for your attention
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Jose Antonio Torres Lerma