Mathmatics Behind CFD - AIMCAL · The Finite Volume Method in Computational Fluid Dynamics: An...

Mathematics Behind CFDapplied to modeling slot die coating hoppers

©2018 Carestream

Key Fields for CFD• Mathematics• Physics• Software Engineering

The goal of the training- turn the ferocious 3 headed dog into a cute

puppy

Credits : 1. https://en.wikipedia.org/wiki/Cerberus2. https://www.deviantart.com/lorenzolivrieri/art/12-Cerberus-12-Labours-of-Hercules-583392360

©2018 Carestream

3 Key Fields

•Each of these three fields deserve equal respect when setting out to develop Simulations to drive product and process development in industry

Mathematics

Physics /Engineering

Computational /Software Eng

©2018 Carestream

Key Fields: Mathematics• Tensor Calculus

• Partial Differential Equations in Tensor form• Coupled & non-coupled• Linear & non linear

• Finite Methods• Finite Differences Methods• Finite Element Methods• Finite Volume Methods

• Fourier Transforms• Specific Algorithms for Finite Methods Solvers and Matrix Solver• Linear Algebra• Solving Systems of Linear and Nonlinear Matrices

•Different notation then most engineers are used to

•Discretization must be evaluated for each problem

•Balancing under-convergance versus over convergance

• Stability analysis

Mathematics


Computational /Software Eng©2018 Carestream

Key Fields: Physics• Fluid Dynamics

• Navier Stokes Equations• Simplifications for specific sections of modeling:

• Ie:1d lubrication model for die/web fluid interface (the coating bead)

• Solids Dynamics• Fluid solids interactions due to bar deflection and deformation during

operation• Heat and Mass Transfer

• Fluid properties and composition may be temperature dependent• Heat transfer from coating die to fluid• Heat transfer due to reaction of fluid

•All of these can be combined into Multiphysics Simulations

Mathematics



Key Fields: Computational/Software Engineering

• Software Engineering• Designing code that meets needs while being sustainable and maintainable

• HPC Engineering• Building hardware tailored to fit the computational demand• Standard equipment not up to the task

• due to high IO requirements & big data needs

• Special High Performance Requirements• Static and Dynamic Memory Management• Parallel computing• Gpu computing• Asynchronous Simulation computing

•“low Level programming languages”

•Special hardware and networking requirements

•Isolated equipment

Mathematics



Tensor Calculus Applied to Slot Die Modeling…time to shake off the rust….

Mathematics



Key Mathematics• Tensor Calculus

• Partial differential equations• Differential Equations in Tensor form

• Finite Methods• Finite Differences Methods (FDM)• Finite Element Method (FEM)• Finite Volume Method (FVM)

• Fourier Transforms• Specific Algorithms for Finite Methods Solvers and Matrix Solvers

Mathematics



Fundamental Theorem of Tensor Calculus

1. Key Concepts to Refresh on 1. Line integrals 2. Green’s Theorem3. Stokes Theorem4. Divergence Theorem5. Leibniz Integral Rule

Mathematics



Line Integral & Gradient of a Curve in Tensor Calculus

𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝐿𝐿𝐿𝐿𝐺𝐺𝐺𝐺 𝐼𝐼𝐺𝐺𝐼𝐼𝐺𝐺𝐼𝐼𝐺𝐺𝐺𝐺𝐺𝐺 �𝑡𝑡0

𝑡𝑡𝑡𝑡

𝑓𝑓 𝑥𝑥(𝐼𝐼),𝑦𝑦(𝐼𝐼) 𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑥𝑥 2 + 𝑑𝑑𝑦𝑦 2

𝑑𝑑𝑑𝑑 =𝑑𝑑𝐼𝐼𝑑𝑑𝐼𝐼∗ 𝑑𝑑𝑥𝑥 2 + 𝑑𝑑𝑦𝑦 2

�𝑡𝑡𝑖𝑖

𝑡𝑡𝑡𝑡

𝑓𝑓 𝑥𝑥 𝐼𝐼 ,𝑦𝑦 𝐼𝐼𝑑𝑑𝑥𝑥𝑑𝑑𝐼𝐼

2

+𝑑𝑑𝑦𝑦𝑑𝑑𝐼𝐼

2

𝑑𝑑𝐼𝐼

Think of the z vertical coordinate as the value of the function F(x,y). For these system to work both x and y must be functions of a shared parameter (t in this case).

Mathematics



Gradient of a Line• 𝐼𝐼𝐺𝐺𝐺𝐺𝑑𝑑𝐿𝐿𝐺𝐺𝐺𝐺𝐼𝐼 𝑜𝑜𝑓𝑓 𝐼𝐼𝑡𝐺𝐺 𝑐𝑐𝑐𝑐𝐺𝐺𝑐𝑐𝐺𝐺 ⇒∫𝑐𝑐 𝛻𝛻 � 𝑑𝑑𝐺𝐺 = 𝑑𝑑 𝐺𝐺 𝑏𝑏 − 𝑑𝑑(𝐺𝐺 𝐺𝐺 )

Applications in coating modeling include- Particle tracing in Lagrangian Models- Basic fluid flow mathematics- Differential equation development

Mathematics



Mathematics



Computational Engineering and Software Design

©2018 Carestream

Key Fields: Software Engineering• memory management:

• How the computer stores data• When it stored that data• Volume of data generation• Velocity of data IO to different devices

• Ie Disk, Ram, CPU

• Parallel processing:• Breaking up the computing job across multiple

• Servers, CPUs, Cores on CPUs

• GPU Computing• What we will all be talking about in 12 years

• HPC job scheduling• Management of large data sets• Commanding servers to compute jobs and analyze data sets

• MPI communication• Communication language upon which both, parallel processes and HPC management operates

Mathematics



http://www.advancedclustering.com/products/hpc-clusters/#

©2018 Carestream

Is this Overkill?Maybe for academic research, but when industry needs to accelerate the development process a high priority project needs to generate results faster.

100s of jobs

Lots of simulations to organize to meet customers needs

1 day turn around

Need to deliver information to

engineering as fast as industry moves

Hundreds of gb’s of

data

One simulation weighs over [3-6] gb…This is “Big Data”

Mathematics


Computational /Software Eng ©2018 Carestream

Big Data• Highly trending in the tech sector• Multiple Definitions

• Simplest definition: • any data management where each file >2gb

• Real Definition: • Data which is not able to be handled by standard methods based on

regular database management• 3 V’s of Big Data

• Volume, Variety Velocity

Mathematics



High Performance Computing Engineering

Big Data Management

High Performance

Code

High PerformanceComputing

High Performance

Hardware

Mathematics



Memory Management

In essence how the computer stores information which is not being written to disk

Is is being written to the HEAP or the Stack

These are critical pieces of information to successfully understand how to write high performance programs.

Mathematics



Physical Phenomena: Applied to Slot Die Modeling…time to shake of the rust….

©2018 Carestream

Key Fields: Physics

• Fluid Dynamics• Naiver Stokes Equations

• Simplifications for specific sections of modeling: • Ie:1d lubrication model for die/web fluid interface (the coating bead)

• Solids Dynamics• Fluid solids interactions due to bar deflection and deformation during operation

• Heat and Mass Transfer• Fluid properties and composition may be temperature dependent• Heat transfer from coating die to fluid• Heat transfer due to reaction of fluid

Mathematics



Fundamental Physics For FVM

• Reynold’s Transport Theorem 𝑑𝑑𝑑𝑑𝑑𝑑𝐼𝐼 𝑚𝑚𝑚𝑚

�𝑉𝑉

[𝜕𝜕𝜕𝜕𝐼𝐼

𝜌𝜌𝑏𝑏 + 𝛻𝛻 � (𝜌𝜌𝒗𝒗𝑏𝑏)]𝑑𝑑𝑑𝑑

• Here b is any intensive value (dB/dm)=b• Conservation equation can then be reapplied for each of the following

• Conservation of Mass • Conservation of Linear Momentum• Non-Conservative Form • Stress Tensor and Momentum Equation for Newtonian fluids• Conservation of Energies

• Moukalled, F.; Mangani, L.; Darwish, M.. The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM® and Matlab (Fluid Mechanics and Its Applications) (Kindle Location 20). Springer International Publishing. Kindle Edition.

Mathematics



Coordinate Transformation

• Continuum Equations• Lagrangian coordinates to Eularian coordinate

𝑑𝑑𝑑𝑑𝑑𝑑𝐼𝐼 𝑀𝑀𝑉𝑉

= 0 ⇒ �𝑚𝑚

𝐷𝐷𝜌𝜌𝐷𝐷𝐼𝐼

+ 𝜌𝜌𝛻𝛻 � �⃑�𝑐 𝑑𝑑𝑑𝑑 = 0

Mathematics



How to Numerically Solve PDE’s

Finite Methods

Finite Differences Method (FDM)

Finite Element Method(FEM)

Finite Volume Method(FVM)

Mathematics



Finite Volume Methods

• Basic concept• Partial differential equation (PDE) is evaluated into a modified differential

equation• Navier Stokes equation for Slot die modeling internal to the cavity and slot geometry

• (if only accounting for fluid flows)

Mathematics



Bringing it All together

©2018 Carestream

Geometry

Mesh

Solve

Analyze

Validate

Redesign

Steps in CFD Process

p.26©2018 Carestream

Mathematics



There is a lot going on Between each of these 3 fields the researcher must remain patient and have a clear identification of what they plan to perform with FVM or FEM

take small steps

… develop mathematics

… develop physics

… develop code

Repeat and Add Capabilities to the modeling with Each iteration

Mathematics



Ref: https://www.ossila.com/pages/slot-die-coating-theory

©2018 Carestream

Carestream R+DDeveloping Custom Modeling and Process/Product Development

©2018 Carestream

Additional Modeling Capabilities from CFD (1/2)• Ability to test Modifications to existing dies

• Approximate shim dimensions, optimize control on wet thickness (downweb and crossweb)• Model different inlets, different manifold designs, non newtonian fluids (this may be future?)

• Waste and cost reduction, better quality, faster time to market • Improved models will reduce the number of trials required to establish coating configuration

• Error reduction, root cause analysis• Computational Models have more control over error. • Simulate Different formulations and compare results directly against each other, knowing

that there was not some unknown variable that could have caused poor coating conditions• Improved communication with Customer

• Use 3d simulations to identify and share opportunities for equipment improvement• Deliver results faster through in-house resources

• Research in Coating Field has made clear agreement on viability of coating design through Multiphysics Simulations

p.29 ©2018 Carestream

Additional Modeling Capabilities from CFD

• Ability to Visualize the Coating Problem:• See Flow inside of the die

• where recirculation's are forming• where are particles settling or agglomerating• where stagnation may result in unwanted changes in coating fluid over time

• Parametric Studies: to determine how variables impact overall performance

• Parametric studies to evaluate:• Perturbations in delivery rate• Perturbations in fluid homogeneity• Errors in die construction, pressure driven deformation, warping (temperature gradients)

©2018 Carestream

Evaluating Value to Business

Cons Pros

©2018 Carestream

Additional Modeling Capabilities from CFD

• Ability to Visualize the Coating Problem:• See Flow inside of the die

• where recirculation's are forming• where are particles settling or agglomerating

• Parametric Studies: to determine how variables impact overall performance

• Parametric studies to evaluate:• Perturbations in delivery rate• Perturbations in fluid homogeneity• Errors in die construction

p.32 ©2018 Carestream

Using Simulation to break down the problem

Ex: simplify and isolate sections of the geometry for analysis

Whether or not the ability to simulate just the cavity is critical, the flexibility is what is important to understand.

example:• If you wanted to test a

• new shim or new inlet • Determine if the flow is caused by:

• Bad inlet geometry triggering turbulence that causes shear thinning

• Rough interface between slot and cavity• Optimize Individual sections then

determine the impact on the overall geometry

https://www.ossila.com/pages/slot-die-coating-theory©2018 Carestream

Applications for Shape Optimization Case:Vw Audi Group

•These same modeling methods can be re-applied to product and process development in the coating industry

p.34©2018 Carestream

Bad simulations happen, so make sure to learn from them!

When modeling for products we must understand physics in the real system and how it is being transformed and modified through the model mathematically. Then understand how that is being executed through code. This enables the engineer ensure that the correct solution can be scaled to fit the business needs.

…results are unclear fall back on the math, and re-trace the code

©2018 Carestream

Case StudyComparison of different non-newtonain polymeric solutions and their performance in an end fed die.

©2018 Carestream

Comparison of Two Non Newtonian Solutions

©2018 Carestream

Comparison of Two Non Newtonian Solutions

©2018 Carestream

Testing a Series of Different Boundary Conditions

©2018 Carestream

Insight into developing better geometriesIf our customers have a solution that cannot be modified the Modeling software can be re-developed to design a new die which does generate a distribution that is acceptable. Maybe Modify an Inlet?

©2018 Carestream

Case Study 2Delivering Best Knowledge to Product Development -> as Fast as possible!-> as Cheap as Possible!

©2018 Carestream

Use your Software Skills to speed up your work and others. Develop Automate

TestAutomate

Deploy Automate

The development cycles can be automated by writing programs that generate multiple case simulations, varying geometries and flow conditions

The test cycles can also be automated to generate clear flags for simulations that are well outside of acceptable

Validation process can be automated by writing programs which generate reports and analyze those against structured data.

Good Developer look for all opportunities. We all know automation of our work is coming. It is better to be on the side performing the automation and save ourselves some stress.

©2018 Carestream

scale up to iteratively run hundreds of cases

©2018 Carestream

Reapplying Programming knowledge from writing CFD code, to Programming Intelligent Software

©2018 Carestream

Summary• There is enough information to fill multiple lifetimes

• Must strive to balance the three fields equally• Do not underestimate the Fields most Coating Engineers are

unfamiliar with (Software Eng)

Special thanks to Carestream to Funding the development of this research

• And thanks to my family for supporting me while I write code with my headphones on in a large portion of my free time

Visit us at our booth to learn more about how Carestream can accelerate coating product development through Contract Manufacturing and Growth Development

©2018 Carestream

©2018 Carestream

Date post:	01-Jun-2020
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

Mathmatics Behind CFD - AIMCAL · The Finite Volume Method in Computational Fluid Dynamics: An...

Documents