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STUDY OF OPTIMAL DESIGN OF 3D MECHANICAL METAMATERIALS

Author: Ariadna Sorribas Bono

Director: Alex Ferrer Ferre

Codirector: Ton Creus Costa

Examination session: Spring, 2023

Introduction

Theoretical Background

Methodology

Conclusions

Results

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Introduction

Theoretical Background

Methodology

Conclusions

Results

01

02

03

Development of 3D Numerical Methods for Elastic Problem Solving

Application of Topology Optimization techniques in 3D metamaterials design

Investigation of elastic direct homogenization problem in 3D material design

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Introduction

Theoretical Background

  • Elasticity Theory
  • Material Characterization
  • Topology Optimization
  • Material Design
  • Metamaterials

Methodology

Conclusions

Results

KEY PRINCIPLES

  • Equilibrium Equations
  • Compatibility
  • Constitutive Relationships

BOUNDARY CONDITIONS

  • Dirichlet BC
  • Neumann BC

2D Boundary Conditions

3D Boundary Conditions

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Introduction

2D Boundary Conditions

3D Boundary Conditions

Macro scale

Micro scale

MULTI-SCALE ANALYSIS

Homogenization theory

PERIODIC BOUNDARY CONDITIONS

Theoretical Background

  • Elasticity Theory
  • Material Characterization
  • Topology Optimization
  • Material Design
  • Metamaterials

Methodology

Conclusions

Results

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Introduction

DENSITY - BASED METHODS

LEVEL SET - BASED METHODS

Theoretical Background

  • Elasticity Theory
  • Material Characterization
  • Topology Optimization
  • Material Design
  • Metamaterials

Methodology

Conclusions

Results

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Introduction

Weighted Inverse Homogenized Elasticity Matrix Function

Rational Weighted Inverse Homogenized Elasticity Matrix Function

Inverse Homogenization Matrix Function

Theoretical Background

  • Elasticity Theory
  • Material Characterization
  • Topology Optimization
  • Material Design
  • Metamaterials

Methodology

Conclusions

Results

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Introduction

Theoretical Background

  • Elasticity Theory
  • Material Characterization
  • Topology Optimization
  • Material Design
  • Metamaterials

Methodology

Conclusions

Results

Optical metamaterial

Positive Poisson’s ratio:

Negative Poisson’s ratio:

Acoustic metamaterial

Mechanical metamaterial

Acoustic metamaterial

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UnfittedMesh Class - UML Diagram

Finite Element Method Class – UML Diagram

Introduction

Theoretical Background

Methodology

  • Clean Code Introduction
  • 3D Optimization

​

Conclusions

Results

GitHub Introduction

Clean Code Practices

Object-Oriented Programming

SWAN Code Debugging

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Introduction

Theoretical Background

Methodology

  • Clean Code Introduction
  • 3D Optimization

​

Conclusions

Results

Boundary Conditions

Simulations

3D Mesh Creation

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3D MESH DATA

Element type

Hexahedra

Element size

0,0866025

Mesh

Structured

Division / line

20

Problem type data

SWAN

PROBLEM DATA

Unit System

SI

Dimensions

3D

Type of Problem

Plane Stress

Physical Type

Elastic

Macro / Micro

Micro

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Introduction

Theoretical Background

Methodology

  • Clean Code Introduction
  • 3D Optimization

​

Conclusions

Results

Boundary Conditions

Simulations

3D Mesh Creation

3D MESH DATA

Element type

Hexahedra

Element size

0,0866025

Mesh

Structured

Division / line

20

Problem type data

SWAN

PROBLEM DATA

Unit System

SI

Dimensions

3D

Type of Problem

Plane Stress

Physical Type

Elastic

Macro / Micro

Micro

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Introduction

Theoretical Background

Methodology

  • Clean Code Introduction
  • 3D Optimization

​

Conclusions

Results

Boundary Conditions

Master Slave - front view (1)

Simulations

3D Mesh Creation

Master Slave - back view (2)

1

2

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Introduction

Theoretical Background

Methodology

  • Clean Code Introduction
  • 3D Optimization

​

Conclusions

Results

3D SIMULATIONS

​

METAMATERIALS MICROSTRUCTURES

Cases

Case 1

​

Case 2

​

Case 3

​

​

Final volume fraction

​

​

3D SIMULATIONS

​

NORMAL MICROSTRUCTURES

Optimizer

MMA (Density – based method)

Null Space (Level set – based method)

​

Cases

Cilinder

​

Vertical plate

​

Sphere

​

Diagonal

​

Shear

​

​

Final volume fraction

​

​

2D SIMULATIONS

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

​

​

​

​

​

​

​

Boundary Conditions

Simulations

3D Mesh Creation

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Introduction

Theoretical Background

Methodology

Conclusions

Results

  • 2D Simulations
  • 3D Simulations
  • Challenges

 

 

 

 

 

 

 

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Introduction

Theoretical Background

Methodology

Conclusions

Results

  • 2D Simulations
  • 3D Simulations
  • Challenges

 

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Introduction

Theoretical Background

Methodology

Conclusions

Results

  • 2D Simulations
  • 3D Simulations
  • Challenges

NORMAL MATERIALS

METAMATERIALS

 

 

 

 

 

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Introduction

Theoretical Background

Methodology

Conclusions

Results

  • 2D Simulations
  • 3D Simulations
  • Challenges

NORMAL MATERIALS

METAMATERIALS

 

MMA (density – based)

Null Space (level set – based)

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Introduction

Theoretical Background

Methodology

Conclusions

Results

  • 2D Simulations
  • 3D Simulations
  • Challenges

NORMAL MATERIALS

METAMATERIALS

 

 

 

 

 

 

 

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Introduction

Theoretical Background

Methodology

Conclusions

Results

  • 2D Simulations
  • 3D Simulations
  • Challenges

CHALLENGES

SOLUTIONS

Mesh Refinement

  • GID software license
  • Matlab Java heap memory
  • Matlab array size

Mesh Element Type

  • Tetrahedral elements
  • Non-symmetric simulation results

Simulation Performance

  • Incorrect material addition
  • Volume constraint

non-compliance

  • Study to determine the maximum line divisions and minimum element size.
  • Hexahedral elements
  • Corresponding code adaptation
  • Precise parameter adjustment for each simulation

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Introduction

Theoretical Background

Methodology

Conclusions

  • Discussion
  • Future development

​

Results

Dimensionality - Convergence Speed Relationship

  • Expected behaviour
  • Faster convergence for 2D simulations

Cost Parameter

  • Constant tendency
  • Effective convergence and progress

MMA – Null Space Comparison

  • Null Space optimizer needed more iterations
  • Identical simulation outcomes

Metamaterials Singularities

  • Longer simulation times
  • Final volume fraction influence

Analogy with Composite Materials’ Structure

  • Link between achieved results and composite material design
  • Code accuracy validated

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Introduction

Theoretical Background

Methodology

Conclusions

  • Discussion
  • Future development

​

Results

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Introduction

Theoretical Background

Methodology

Conclusions

  • Discussion
  • Future development

​

Results

Exploration of 3D Meshes

Mesh-Independence Study

Comparative Analysis of Additional Optimizers

Development of J3 Function for 3D Cases

Metamaterial simulations for smaller final volume fractions

Iterative Solver

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STUDY OF OPTIMAL DESIGN OF 3D MECHANICAL METAMATERIALS

Author: Ariadna Sorribas Bono

Director: Alex Ferrer Ferre

Codirector: Ton Creus Costa

Examination session: Spring, 2023