Complex Fluids and Soft Materials: A Numerical Perspective - - PowerPoint PPT Presentation

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Complex Fluids and Soft Materials: A Numerical Perspective - - PowerPoint PPT Presentation

Apr 05, 2023 691 likes •1.62k views

Complex Fluids and Soft Materials: A Numerical Perspective Bo Zhu MIT CSAIL boolzhu@csail.mit.edu Complex Physical Systems Geometry, Topology, Dynamics Material, Structure, Codimension,

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Complex Fluids and Soft Materials: A Numerical Perspective

复杂流体和柔性材料的计算方法

Bo Zhu

朱 博

MIT CSAIL

boolzhu@csail.mit.edu

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Complex Physical Systems

Geometry, Topology, Dynamics Material, Structure, Codimension, Transition

Computer Graphics, Computational Fluid Dynamics, Computational Fabrication, 3D Printing, Biomedical Engineering, Robotics

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Flame

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𝜍ℎ 𝑣ℎ − 𝐸 = 𝜍𝑔(𝑣ℎ − 𝐸) 𝜍ℎ 𝑣ℎ − 𝐸 2 + 𝑞ℎ = 𝜍𝑔 𝑣𝑔 − 𝐸

2 + 𝑞𝑔

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Bubbles

[ Fabian Oefner ]

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[Bremond N and Villermaux E 2006]

u u

Two liquid jets collide with each other

[John Bush Lab, MIT, 2004]

Impinging Jets

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Water Bell

[F. Savart 1833] [R. Buckingham and J. Bush 2001]

http://www.phikwadraat.nl/

[John Bush Lab, MIT, 2004]

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Waterbell

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Non-Newtonian Flow

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Viscosity matters!

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Math behind a pizza piece

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Paint Orchid

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Functional Soft Bodies

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What are they?

Why do they happen?

How to make it?

What are inside these flames? …

Why are splash crown-shaped?

…

How to make a glider fly?

…

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Geometric Data Structures

Meshing Numerical PDE Solvers Real-time Simulators User Interface Large-Scale Optimization Fabrication Adaptive/Reduced Discretizations

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Meshing Numerical PDE Solvers Real-time Simulators User Interface Large-Scale Optimization Fabrication Adaptive/Reduced Discretizations

Geometric Data Structures

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Large-scale Simulation for Film Visual Effects

Bo Zhu, Wenlong Lu, Matthew Cong, Byungmoon Kim, and Ron Fedkiw. A New Grid Structure for Domain Extension. ACM Trans. Graph. (SIGGRAPH 2013), 32, 63.1-63.8.

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Sim time on a far-field grid:

1x 12x 160x

Domain Extension

3.1x 6.1x

_ _

Sim time on a uniform grid:

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New Grid Structure

δx δx δx δx 2δx 2δx 2δx 2δx 2δx 4δx 4δx δx 2δx 2δx 2δx 2δx 2δx δx δx δx δx δx 4δx 4δx

X-Axis: Layer 1: 4 Layer 2: (2, 3) Layer 3: (1, 1) Y-Axis: Layer 1: 6 Layer 2: (1, 4) Layer 3: (0, 2)

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Two Grid Boxes

  • The interior box with the finest

resolution to resolve fine details

  • The exterior box with gradually

coarsened resolutions to enclose the entire fluid

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Fast Index Access

1

( ) 2

i i i

x x I x I x 

         2 2

m m

m m m m i x x m m

 

   

1D Array for Layer Information

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Solving Incompressible Flow on Stretched Grid Cells

  • Use the volume weighted divergence to solve the Poisson

equation for pressure on stretched cells in order to obtain a SPD system

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Meshing Real-time Simulators User Interface Large-Scale Optimization Fabrication Adaptive/Reduced Discretizations Numerical PDE Solvers

Geometric Data Structures

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Computational Tools for Exploring Fundamental Sciences

Bo Zhu, Ed Quigley, Matthew Cong, Justin Solomon, and Ron Fedkiw. Codimensional Surface Tension Flow on Simplicial Complexes. ACM Trans. Graph. (SIGGRAPH 2014). Wen Zheng, Bo Zhu, Byungmoon Kim, and Ron Fedkiw. A New Incompressibility Discretization for a Hybrid Particle MAC Grid Representation with Surface Tension. J. Comp. Phys., 280, 94-142, 2015. Bo Zhu, Minjae Lee, Ed Quigley, and Ron Fedkiw. Codimensional Non-Newtonian Fluids. ACM Trans. Graph. (SIGGRAPH 2015).

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Anisotropic Thin Features

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Embed a Lagrangian mesh in a grid

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What will happen if the features get even thinner? Vanishingly thin?

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These phenomena are not rare…

Membrane:

Oefner’s photography

Jets and sheets:

Bush’s experiments, MIT Applied Math Lab fabianoefner.com

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Simplicial Complex

A geometric structure that consists of points, segments, triangles, and tetrahedra

surface tension, adhesion, gravity, etc.

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Codimension-0 Codimension-1 Codimension-2 Codimension-3 Tetrahedra Triangles Segments Points

Discrete Geometric Analogues

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Droplet Filament Film (and Rim)

Film interior Film boundary (Rim) Filament boundary Filament interior

Reduced Geometry

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Codimensional Volume-Weighted Gradient

  • For all the simplexes incident to a particle:
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Discretized Poisson Equation

  • Poisson equation:
  • Volume weighted formula:

Discretizing

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lf,n

→

Surface Tension

  • Discretization:

lf,n

→

dr,n

→

Film interior

de,n

→

Film boundary (Rim) Filament

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Meshing Algorithm

For each timestep

//Volumetric meshing Tetrahedron edge/face flip Tetrahedron edge split Skinny tetrahedron collapse Tetrahedron edge/face flip //Thin film meshing Triangle edge split Triangle edge collapse Triangle edge flip Triangle crumple merge //Filament meshing Segment edge split Segment edge collapse //Topological merging/breaking Boundary vertex snap Thin triangle break Thin segment break

Edge to Face Flip Face to Edge Flip Edge Split Face Split Triangle Edge Split Triangle Edge Flip Triangle Edge Collapse Segment Edge Split Segment Edge Collapse α Edge Collapse Crumple Merge Vertex Snap

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Example: Blowing Bubbles

http://www.soapbubble.dk/

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[J Eggers and E Villermaux 2008]

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Example: Film Catenoid

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Example: Waterbell

[F. Savart 1833] [R. Buckingham and J. Bush 2001] http://www.phikwadraat.nl/

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Bo Zhu, Minjae Lee, Ed Quigley, and Ron Fedkiw. Codimensional Non-Newtonian Fluids. ACM Trans. Graph. (SIGGRAPH 2015).

Numerical Simulation of Non-Newtonian Fluids

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Different Material Models

Paint: Shear Thinning Quicksand: Shear Thickening Mud: Bingham Plastic

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Variable Viscosity

  • Semi-Implicit viscosity force:
  • Volume weighted formula for the implicit part:
  • Non-Newtonian flow:

Explicit part Implicit part

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Meshing Numerical PDE Solvers Real-time Simulators User Interface Large-Scale Optimization Fabrication Adaptive/Reduced Discretizations

Geometric Data Structures

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An interactive system for cardiovascular surgeons

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Anisotropic thin pipes Reduced Graph

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  • Hydraulics
  • Hydrodynamics

Qn=-MQe MDeMTPn = Qn

N-S equations with Dirichlet Boundary

Reduced Geometry

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Meshing Numerical PDE Solvers Real-time Simulators User Interface Large-Scale Optimization Fabrication Adaptive/Reduced Discretizations

Geometric Data Structures

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Motivation: Direct Design v.s. Generative Design

Direct Design Generative Design

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Topology Optimization

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Challenges

Software: SIMP Topology Optimization

  • Up to millions of elements
  • Difficult to handle multiple materials

Hardware: Object-1000 Plus

  • Up to 39.3 x 31.4 x 19.6 in.
  • 600dpi (~40 microns)
  • 5 trillion voxels
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Previous Work: Fabrication-Oriented Optimization

[Musialski et.al. 2016] [Lu et.al. 2014] [Schumacher et.al. 2015] [Panetta et.al. 2015] [Xu et.al. 2015] [Matinez et.al. 2016]

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Topology Optimization

[Wu et.al. 2016] [Langlois et.al. 2016] [Matinez et.al. 2015] [Liang et.al. 2015]

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Two-scale Topology Optimization

𝑸𝒑𝒋𝒕𝒕𝒑𝒐′𝒕 𝑺𝒃𝒖𝒋𝒑 Young’s Modulus 𝑻𝒊𝒇𝒃𝒔 𝑵𝒑𝒆𝒗𝒎𝒗𝒕

Base materials Force Grip Design Goal Continuous Optimization Material Property Space Continuous Representation Fabrication

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Two-scale Topology Optimization

𝑸𝒑𝒋𝒕𝒕𝒑𝒐′𝒕 𝑺𝒃𝒖𝒋𝒑 Young’s Modulus 𝑻𝒊𝒇𝒃𝒔 𝑵𝒑𝒆𝒗𝒎𝒗𝒕

Base materials Force Grip Design Goal Continuous Optimization Material Property Space Continuous Representation Fabrication

Material Property Space

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Young’s 𝐐𝐩𝐣𝐭𝐭𝐩𝐨 𝐓𝐢𝐟𝐛𝐬

Base materials

Microstructure

𝝉 = 𝑫𝝑

= 1 − 𝜉 𝐹 𝜉 𝐹 𝜉 𝐹 1 − 𝜉 𝐹 𝜉 𝐹 1 − 𝜉 𝐹 𝜈 𝜈 𝜈

Cubic Material

𝑞 = 𝐹, 𝜉, 𝜈 𝑈

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Continuous Representation: Levelset

𝜚 𝑞 = 0

Young’s 𝐐𝐩𝐣𝐭𝐭𝐩𝐨 𝐓𝐢𝐟𝐛𝐬

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Expanding the Achievable Property Domain

𝜚 𝑞 = 0

Stochastically-Ordered Sequential Monte Carlo

Young’s 𝐐𝐩𝐣𝐭𝐭𝐩𝐨 𝐓𝐢𝐟𝐛𝐬

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Expanding the Achievable Property Domain

𝜚 𝑞 = 0

Continuous Microstructure Optimization

Young’s 𝐐𝐩𝐣𝐭𝐭𝐩𝐨 𝐓𝐢𝐟𝐛𝐬

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𝜚 𝑞 = 0

Young’s 𝐐𝐩𝐣𝐭𝐭𝐩𝐨 𝐓𝐢𝐟𝐛𝐬

Lame parameters space 4D space

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Two-scale Topology Optimization

𝑸𝒑𝒋𝒕𝒕𝒑𝒐′𝒕 𝑺𝒃𝒖𝒋𝒑 Young’s Modulus 𝑻𝒊𝒇𝒃𝒔 𝑵𝒑𝒆𝒗𝒎𝒗𝒕

Base materials Force Grip Design Goal Continuous Optimization Material Property Space Continuous Representation Fabrication

Continuous Optimization

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Topology Optimization

min

p : 𝑇(𝒒, 𝒗)

𝑡. 𝑢. : 𝐺 𝒒, 𝒗 = 0 𝜚 𝒒 ≤ 0

Linear Elastic FEM: 𝐺 𝒒, 𝒗 = 𝐿 𝒒 𝒗 − 𝒈 = 0 (Adjoint Method) Levelset Constraints (Interior Point Method, Finite difference for 𝜚𝑦) Minimum Compliance/Target Deformation

𝒒 = [𝝇𝟐, 𝑭1, 𝝃1, 𝝂𝟐, 𝝇𝟑, 𝑭2, 𝝃2, 𝝂𝟑, … ] Material property for each cell

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Minimum Compliance

𝑇𝑑 𝒒, 𝒗 = 𝒗𝑈𝑳𝒗

Density -> Density, Young’s modulus, Poisson’s Ratio, … (0,1] -> Levelset boundary

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Target Deformation

𝑇𝑒 𝒒, 𝒗 = 𝒗 − 𝒗 𝑈𝑬(𝒗 − 𝒗)

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Two-scale Topology Optimization

𝑸𝒑𝒋𝒕𝒕𝒑𝒐′𝒕 𝑺𝒃𝒖𝒋𝒑 Young’s Modulus 𝑻𝒊𝒇𝒃𝒔 𝑵𝒑𝒆𝒗𝒎𝒗𝒕

Base materials Force Grip Design Goal Continuous Optimization Material Property Space Continuous Representation Fabrication

Fabrication

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Microstructure Mapping

Map points in continuous space to discrete microstructures

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Example: Soft Gripper

Deform Push

Optimization Fabrication

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Example: Different Gripping Mechanisms

Convergence rate Different gripper structures optimized for the same target deformation

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Example : Bridge

8.5 inches 5k pixels 17 inches 10k pixels 34 inches 20k pixels

Push A Trillion Voxels

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Example : Flexure

Multiple objectives

Stiff Soft

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Example: Soft Ray

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Thank you!