Mathematics Behind CFD

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Stanley Johns
5 years ago
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1 Mathematics Behind CFD applied to modeling slot die coating hoppers

Credits : 1. https://en.wikipedia.org/wiki/cerberus 2. https://www.

2 Key Fields for CFD Mathematics Physics Software Engineering The goal of the training- turn the ferocious 3 headed dog into a cute puppy Credits :

3 3 Key Fields Mathematics Physics /Engineering Computational /Software Eng Each of these three fields deserve equal respect when setting out to develop Simulations to drive product and process development in industry

4 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 underconvergance versus over convergance Stability analysis Mathematics Physics /Engineering Computational /Software Eng

5 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 Physics /Engineering Computational /Software Eng

6 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 Physics /Engineering Computational /Software Eng

7 Tensor Calculus Applied to Slot Die Modeling time to shake off the rust. Mathematics Physics /Engineering Computational /Software Eng

8 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 Physics /Engineering Computational /Software Eng

9 Fundamental Theorem of Tensor Calculus 1. Key Concepts to Refresh on 1. Line integrals 2. Green s Theorem 3. Stokes Theorem 4. Divergence Theorem 5. Leibniz Integral Rule Mathematics Physics /Engineering Computational /Software Eng

10 Line Integral & Gradient of a Curve in Tensor Calculus tttt GGGGGGGGGGGGGG LLLLLLLL IIIIIIIIIIIIIIII ff xx(tt), yy(tt) dddd tt0 ttii tttt ff xx tt, yy tt dddd = dddd 2 + dddd 2 dddd = dddd dddd dddd 2 + dddd 2 dddd dddd 2 + dddd dddd 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 2 dddd Physics /Engineering Computational /Software Eng

11 Gradient of a Line gggggggggggggggg oooo ttttt cccccccccc cc dddd = ss rr bb ss(rr aa ) Applications in coating modeling include - Particle tracing in Lagrangian Models - Basic fluid flow mathematics - Differential equation development Mathematics Physics /Engineering Computational /Software Eng

12 Mathematics Physics /Engineering Computational /Software Eng

13 Computational Engineering and Software Design

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

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

14 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 Physics /Engineering Computational /Software Eng

15 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. Need to deliver information to engineering as fast as industry moves 1 day turn around 100s of jobs Lots of simulations to organize to meet customers needs Mathematics Hundreds of gb s of data One simulation weighs over [3-6] gb This is Big Data Physics /Engineering Computational /Software Eng

be handled by standard methods based on regular database management 3 V s of Big Data

16 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 Physics /Engineering Computational /Software Eng

17 High Performance Computing Engineering Big Data Management Mathematics High Performance Computing Physics /Engineering Computational /Software Eng High Performance Code High Performance Hardware

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

18 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 Physics /Engineering Computational /Software Eng

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

20 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 Physics /Engineering Computational /Software Eng

21 Fundamental Physics For FVM Reynold s Transport Theorem dddd dddd mmmm VV [ 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 Physics /Engineering Computational /Software Eng

22 Coordinate Transformation Continuum Equations Lagrangian coordinates to Eularian coordinate dddd dddd MMMM = 0 vv DDρρ DDDD + ρρ vv dddd = 0 Mathematics Physics /Engineering Computational /Software Eng

23 How to Numerically Solve PDE s Finite Methods Finite Differences Method (FDM) Finite Element Method (FEM) Finite Volume Method (FVM) Mathematics Physics /Engineering Computational /Software Eng

24 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 Physics /Engineering Computational /Software Eng

25 Bringing it All together

26 Steps in CFD Process Geometry Redesign Mesh Validate Solve Mathematics Analyze Physics /Engineering Computational /Software Eng p.26

develop physics develop code Ref: https://www.ossila.

27 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 Ref: om/pages/slot-diecoating-theory Repeat and Add Capabilities to the modeling with Each iteration Mathematics Physics /Engineering Computational /Software Eng

28 Carestream R+D Developing Custom Modeling and Process/Product Development

29 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

30 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)

31 Evaluating Value to Business Cons Pros

32 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

Using Simulation to break down the problem Whether or not the ability

the flow is caused by: Bad inlet geometry triggering turbulence that

Optimize Individual sections then determine the impact on the overall

33 Using Simulation to break down the problem 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 Ex: simplify and isolate sections of the geometry for analysis

34 Applications for Shape Optimization Case: Vw Audi Group These same modeling methods can be reapplied to product and process development in the coating industry p.34

35 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

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

37 Comparison of Two Non Newtonian Solutions

38 Comparison of Two Non Newtonian Solutions

39 Testing a Series of Different Boundary Conditions

40 Insight into developing better geometries If 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?

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

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

43 scale up to iteratively run hundreds of cases

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

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

45 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

1.2 Numerical Solutions of Flow Problems

1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian