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Scientific Computing Chad Sockwell Florida State University kcs12j@my.fsu.edu October 27, 2015 Chad Sockwell (FSU) Scientific Computing October 27, 2015 1 / 73 Main Points What is Scientific Computing (SC)? Agreement Between SC and

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Scientific Computing

Chad Sockwell

Florida State University kcs12j@my.fsu.edu

October 27, 2015

Chad Sockwell (FSU) Scientific Computing October 27, 2015 1 / 73

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Main Points

What is Scientific Computing (SC)? Agreement Between SC and Experiment SC complementing Experiments SC complementing Theory The SC department and program

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What is Scientific Computing?

Scientific Computing (SC) or Computational Science is an Interdisciplinary Science. Combining Mathematics, Computer Science, Engineering, and Natural Sciences to solve problems. Scientific Computing = Computer Science

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How is it used?

Typically Computational Scientist fall into two groups. Using algorithms and Improving / Implementing algorithms. The second group shares the true spirit of Scientific Computing. Scientific Computing is aimed at finding ways to improve problem solving and solving new problems

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How is it used? (Cont.)

Physical phenomena can be modeled by numerical algorithms to take advantage of what computers do best. Some of these advantages are seen in:

Complicated domains and non-linearities in PDE’s. Large statical analysis or Monte Carlo methods. Large matrix equations or eigenvalue problems

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Why Should You Care?

Computers are becoming more powerful. FSU’s RCC HPC has 403 nodes, 6464 CPU cores and 109.5 Tflops (1012 operations per second). Symbiotic relation with Science. Numerical simulations can complement experiments. Complementing complicated theories.

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SC Agreeing With Experiment

Verification and Validation are critical Some may say nothing new was done. Some also are suspect of the numerical error associated with the algorithm, but this is were verification comes in. Experimental uncertainty and numerical error

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Some examples

Qiang Du, shows that his algorithm for the Gross-Pitaevskii

  • equations. This a BEC of an alkali-metal gas. The voctices are

nucleated by laser stirring and rotating magnetic traps. His results for the vortices in the substance are shown in (a). Experimental results from MIT are shown in (b).

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Gravitational Lensing

Bin Chen of FSU’s RCC uses his Linearized backward gravitational lensing code to reproduce a scene from a movie. All though the movie is pretty graphic art, Bin’s algorithm reproduces a similar realistic simulation.

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Gravitational Lensing

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Omega Laser and Super Nova

The experiment (left) Simulates shock in a Super Nova. The simulation (right) tries to replicate it but is slightly off.

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Omega Laser and Super Nova

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Experimental setup

Channel has dimensions 5.2mm × 45mm × 0.1mm Water injected at 0.01mm3/s Re = 0.005 Particle Image Velocimetry (PIV) applied to 8 million neutrally buoyant spherical particles (diameter = 1µm)

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Velocity Field

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PDF of v

  • 0.05

0.05 0.1 0.15 0.2 10−2 10−1 100 101 102 Longitudinal Velocity PDF Numerical Experimental

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PDF of v

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 10−2 10−1 100 101 102 Transverse Velocity PDF Numerical Experimental

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Mixing

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SC Complementing Experiment

After verification and validation, we can simulate experiments SC be can used where experiments are too expensive or not feasible. Simulations can used to filter through experiential situations

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Projectile Impact

FSU’s Dr. Shanbhag and Steve Henke, developed a simulation of a projectile crashing into a brittle material. The aim of the study was to show how the setup of the numerical grid affects the results.

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Projectile Impact

This Simulation could used to test materials ballistic strength without destroying precious materials

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Projectile impact

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Jet crash

Another example of precious materials is a billion dollar fighter jet.

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SC Complementing Theory.

Solving theories analytically can call for crafty approximations Simple domains, dimensional reduction, and asymptotic behavior Numerical methods on computers can crank out complicated calculations SC aims at improving algorithms that can handle complicated models These algorithms can used to make through theoretical predictions.

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Ginzburg Landau Equations for Superconductivity

Γ

∂ψ

∂t + i(Jy σ )κψ

  • +
  • |ψ|2 − 1
  • ψ + ( i

κ∇ − A)2ψ = 0 ∇ × H + J = σ∂A ∂t + ∇ × ∇ × A + i 2κ (ψ∗∇ψ − ψ∇ψ∗) + A|ψ|2 where σ∇φ = J A · n = 0 , i κµ ∇ψµ · n = 0 , (∇ × A) × n = (He − HJ) × n on ∂Ω

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GL

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Faraday Rotation

Bin also reproduced the results of another research group They are looking for Gravitational Faraday rotation from a Galaxy This can seen in the X-ray polarization. This simulation can tell astronomers what to look for.

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The Scientific Computing Department

Located on the 4th floor of Dirac (Secret Elevator) We specialize in implementing and improving algorithms www.sc.fsu.edu Advisor: Mark Howard, 403, mlhoward@fsu.edu Double Major (47 Credits) or Minor? Skills are valuable for research and grad school. TEA AND COOKIES, Wednesdays at 3:00

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Courses

ISC 3313(0) Intro to SC ISC 3222 (3) Symbolic and Numerical Computations ISC 4304 (4) Programming for Scientific Applications ISC 4220 (4) Algorithms for Science Applications I ISC 4221 (4) Algorithms for Science Applications II ISC 4223 (4) Computational Methods for Discrete Problems ISC 4232 (4) Computational Methods for Continuous Problems ISC 4943 (3) Practicum in Computational Science

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Second Major

All core courses 3 seminars 6 hours of SC electives 12 hours of other electives

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Prerequisites

Calc I and II Basic programming: COP 3014 or ISC 3313 Science with lab Collateral: Linear algebra and Stats (3000 +) Double count electives

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Minor

Both

ISC 3222 (3) Symbolic and Numerical Computations ISC 4304 (4) Programming for Scientific Applications

and one of

ISC 4220 (4) Algorithms for Science Applications I ISC 4221 (4) Algorithms for Science Applications II

and 1 more elective (14 hours total)

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Skills

Interpolation Approximation (Least Squares) Numerical Linear Algebra Numerical Differentiation and Integration (Quadrature) Non-Linearities and Optimization Game theory applications Statistics and Probabilities ODE’s and PDE’s

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Programming Languages

MATLAB Mathemitca Python Fortran C / C++ Java

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Why SC is Useful to a Physicist

Simulating experiments. SC can help explore problems deeper. Computing skills are valuable for grad school and industry. SC teaches how to implement math problems on computers.

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Quadrature Example

A Riemann Sum is descried as

b

a

f (x) dx ≈

N

  • k=1

f (x∗

k )∆x

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Quadrature Example

Consider the integral that yields the area of the unit circle. A =

R 2π

r dθ dr We can throw darts instead, known as the Monte Carlo method.

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Monte Carlo

For a given Nin and Ntotal. A ≈ Nin Ntotal Atotal 4*0.8 = 3.2 ≈ π. Why should you care? Error ∝ √Ntotal In the Riemann Sum (or any analytic quadrature), Error ∝ (∆x)n What happens as n grows? Monte Carlo wins.

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Monte Carlo

Now consider a complicated domain.

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My story

I do physics, math and SC Started with Superconductivity (physics) I wanted to understand vortex dynamics

Figure : A magnet floating on top of Superconductor.

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Vortex Dynamics (Physics)

Figure : Vortex Dynamics with applied current and field (left), the set up (right).

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Ginzburg Landau Model (Physics and Math)

I had to understand the GL model (math) Gs =

  • fn + α(T)|ψ|2 + 1

2β(T)|ψ|4 + 1 2ms |(−i∇ − es c A)ψ|2 +1 2 B · (B − He) 4π dΩ I learned Calculus of variations and how to rescale equations. −ψ + |ψ|2ψ + ( i κ∇ − A)2ψ = 0 = δG δψ in Ω Js = ∇ × (∇ × A − He) = − i 2κ(ψ∗∇ψ − ψ∇ψ2) − |ψ|2A in Ω ( i κ∇ − A)ψ · n = 0 on Γ ∇ × A × n = H × n on Γ

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Gauge invariance (Physics Math)

Including time demanded an understanding of gauge invariance. φ → φ − 1 c ∂f ∂t , A → A + ∇f , ψ → ψeif E = (−1 c ∂A ∂t − ∇φ), B = ∇ × A Now we disturb the system and account for the total current. Γ(∂ψ ∂t + iκφψ) = δG δψ J = Jn + Js = σE + Js

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GL equations ( Physic and Math)

Γ

∂ψ

∂t + i(Jy σ )κψ

  • +
  • |ψ|2 − 1
  • ψ + ( i

κ∇ − A)2ψ = 0 ∇ × H + J = σ∂A ∂t + ∇ × ∇ × A + i 2κ (ψ∗∇ψ − ψ∇ψ∗) + A|ψ|2 where σ∇φ = J A · n = 0 , i κµ ∇ψµ · n = 0 , (∇ × A) × n = (He − HJ) × n on ∂Ω

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Numerical Analysis (NA)

I have a system of non-linear, time dependent, PDE’s Now I had to learn how to solve PDE’s on a computer (NA) I used my skills to convert math to code The Finite element Method was used to solve the PDE.

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Discretization of The Domain (NA)

Discretization was needed

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Numerical Analysis (NA)

Implementing derivatives, integrals, non-linearities, and the time

  • domain. (NA)

xn+1 = xn − f ′(xn) f (xn) → J(xn)(xn+1 − xn) = −F(xn)

Figure : Newton’s method (top), Euler’s method (left), Riemann sum (right)

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Numerical Analysis (NA)

Now I’m ready solve my matrix equation! I run the code and....

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Compression (CS)

I ran out of space, so i quit storing 0’s. (CS)

Figure : The black dots are non zero values in the matrix

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Compression (CS)

Compressed Spare Row storage did the trick

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Parallel Computing (CS)

The I ran out out time... I need Parallel computing (HPC)

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Parallel Computing Problems (CS)

I need a linear algebra solver, SuperLU I learned how to use make files and link libraries (CS) Lots of debugging.

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Finally

I got my Data!!!! Then I had use plotting software (CS) to make my plots and poster

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Pinning Effects in Two-Band Superconductors

Chad Sockwell, Max Gunzburger, Janet Peterson

Scientific Computing, Florida State University, Email: kcs12j@my.fsu.edu Objectives

  • Numerically simulate the vortex dynamics in

the superconducting material MgB2.

  • Devise a way to raise the maximum amount of

current carried in the superconductor.

  • Simulate how impurities raise the maximal

current in MgB2.

Introduction

Most are familiar with the waste heat produced by resistance in an electrical wire. This wasted energy can be avoided by using resistance free superconduc-

  • tors. However superconductors posses critical (max-

iumum) values for their temperature, magnetic field, and current, only below which they operate as re- sistance free. Introducing impurities can raise the critical current by preventing flux flow. Practical superconducting devices could revolutionize technol-

  • gy but numerical simulations are needed to give

insight.

MgB2

  • Magnesium Diboride (MgB2) is a two band

superconductor that can carry resistance free current under a temperature of 39K (-389.47 ◦ F).

  • The bands act as pathways for electrons, each

possessing their own properties seen in Table 1 .

  • The bands interact to give composite direction

and temperature dependent magnetic properties. λ1=47.8 nm λ2= 36.6 nm ξ1=13 nm ξ2=51 nm κ1=3.61 κ2=0.658 ν = 2.757 η = −0.1701

Table 1: The material parameters for MgB2.

Flux Flow

  • In the presence of a external magnetic field,

normal materials are completely penetrated by the field. However superconductors such as MgB2 are only penetrated by small magnetic flux vortices.

  • The flux vortices interact with the applied

current, J to produce a Lorentz force, F, perpendicular to J.

  • The movement of the vortices, known as flux flow,

induces an electric field, E, parallel to the applied current, creating an effective resistance.

  • Figure 3 shows how the force moves the vortices

( red ) as time increases.

Figure 1: The vortices can be seen where ψ1 is at is smallest (red). They are pushed to the right by the Lorentz force. At later times (t_3) the vortices rearrange themselves.

Methods

The finite element, Euler, and Newton methods were used together to solve the model equations. Super- computers were used at F.S.U.’s R.C.C. for calcula- tions.

Figure 2: The set up for the numerical simulations. The mag- netic field penetrates the sample as flux vortices and an applied current is transported across the sample.

Mathematical Model

The Modified 2B-TDGL model describes supercon- ductivity and contains ψ1 and ψ2, the density func- tions for the current carriers, the magnetic vector potential, A, and takes the parameters from Table 1 as input. The vortices can be seen where ψ1 is at its

  • smallest. Numerical simulations of vortex dynamics

from the model are seen in Figures 1, 4

Results

Figure 3: The critical current for different numbers of impurity

  • sites. This under a temperature of 30 K and magnetic field of

0.106 Tesla.

Impurities were successfully modeled in the mate-

  • rial. The pinning effects are shown in Figure 4. The

impurity sites are outlined by the open black cir-

  • cles. A raise in the critical current Jc was found

by increasing the normals N (Figure 3). However too many impurities degraded the superconducting material and lower Jc, as seen where N=25.

Conclusion

MgB2 was successfully modeled using the simula-

  • tion. Figure 5 shows directional dependence on the

critical magnetic fields, comparable to experiments. An algorithm to model impurities in the sample was successful in raising the critical current.

Figure 4: From top left to bottom right, vortices (red) are gener- ated from a magnetic field. They become pinned to the normal site (black circles). When a current J is applied, the vortices remained pinned, unlike Figure 2. Figure 5: Magnetic properties comparable to experiment, γH = Hab

c2/Hc c2.

Acknowledgements: I would like to acknowledge and graciously thank The Center for Undergraduate Research and Academic Engagement at F.S.U. and their private donors for the M.R.C.E. award that supported this research.

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Put It All Together

Physics + Math+ NA+ CS =SC I didn’t know I could do it. The SC and Physics departments prepared me well!

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Why You Should SC

Now we see how SC is useful You can expand your research Computer’s play a huge role today and you can do it too! www.sc.fsu.edu Advisor: Mark Howard, 403, mlhoward@fsu.edu Tea and cookies Wednesday 3:00 (colloquium after)!

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References

Max Gunzburger for his insightful comments on the subject. Qiang Du, Penn State Sachin Shanbahg, FSU, Department of Scientific Computing Bin Chen, FSU Research Computing Center https://www.youtube.com/watch?v=bPvQ48gIa6U

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References

Bryan Quaife for his Fluid dynamics examples. Tomasz Plewa, Tim Handy and Dr. Plewa’s post docs for their Omega laser examples.

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Finite Elements

FEM is a method used to solve PDE’s Discretization and Basis Functions Consider a surface plot from a function u(x,y)

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Finite Elements

Now the Domain is discretized.

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Finite Elements

Basis Functions

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Finite Elements

Basis Functions

ϕ1 ϕ2 ϕ3 ϕ4 ϕ5 ϕ6

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Finite Elements

Consider Poisson’s equation −∆u = f (x, y) in Ω; u=0 on Γx and ∇u · n = g(x) on Γy Multiply by a smooth bound test function, φ, and IBP −

  • (∆u)φ dΩ = −
  • ∇u · nφ dΓy +
  • ∇u · ∇φ dΩ =
  • f (x, y)φ dΩ

Derivative BC

  • ∇u · ∇φ dΩ =
  • f (x, y)φ dΩ +
  • g(x)φ dΓy

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Finite Elements

Now test for ALL basis function in our set, φi

  • ∇u · ∇φi dΩ =
  • f (x, y)φi dΩ +
  • g(x)φi dΓy

Expand u in terms of φi since it is a complete basis. u(x, y) =

N

  • j

cjφj(x, y) Now we have a matrix equation cj

  • ∇φj · ∇φi dΩ =
  • f (x, y)φi dΩ +
  • g(x)φi dΓy

if φi is Γx node, set ij term to 1 and RHS i term to 0

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Non-dimensionalization ND

We can rescale with familiar quantities. We can close the magnitude gap. Let’s introduce ˜ x = λx, ˜ ψ = ψ

  • −α

β , ˜ A = √ 2Hcλ, ˜ H = √ 2HcH λ(T) =

  • − msβ(T)c2

4πα(T)e2∗ , ξ(T) =

  • −

2 2msα(T), κ = λ ξ

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ND

Inserting ˜ x, ˜ ψ, ˜ A, ˜ H and forming λ and ξ. We divide through by the common term. The ND gives us a much cleaner form (with the tilde’s dropped). gs = fn+|ψ|2+ 1 2|ψ|4+ 1 2ms |(−i∇− es c A)ψ|2+ 1 2 (∇ × A) · (∇ × A − He) 4π (Several lines of algebra) gs = fn − |ψ|2 + 1 2|ψ|4 + |( i κ∇ − A)ψ|2 + 1 2(∇ × A) · (∇ × A − He)

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What We Do with the Free Energy?

We want the Free energy minimized Functions ψ and A must minimize it. Calculus of Variations!!!

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Calculus of Variations

We want the path that minimizes

g(ψ, A) dΩ for ψ and A.

We vary these two variables instead of x and ˙ x. Starting with ψ∗ (the complex conjugate), lets take a functional derivative (not ψ). δgs δψ = lim

η→0

1 η

  • Ω

gs(ψ + ηφ) − gs(ψ) dΩ = 0

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Calculus of Variations

Now we’re only left with terms with a ψ factor δgs(ψ, ψ∗, A) δψ = lim

η→0

1 η

  • Ω

gs(ψ, ψ∗ + ηφ, A) − gs(ψ, ψ∗, A) dΩ = 0 = lim

η→0

1 η

  • −(ψ∗+ηφ)ψ+1

2((ψ∗+ηφ)ψ)2+(−i κ ∇−A)(ψ∗+ηφ)·( i κ∇−A)ψ −(|ψ|2 + 1 2|ψ|4 + |( i κ∇ − A)ψ|2) dΩ

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The Variational form

After some simplification, we have the Variational form. This coincides with the weak from of the Finite element method. To get the EL equations, we need Integration by parts. Gs =

  • −ψφ + |ψ|2ψφ + (−i

κ ∇ − A)φ · ( i κ∇ − A)ψ dΩ = 0 Gs =

  • −ψφ + |ψ|2ψφ + D∗φ · Dψ dΩ = 0

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IBP

We need to use IBP on the derivative term to get rid of φ The i’s change a few things

  • D∗φ · Dψ dΩ =
  • Dψφ · n ds +
  • (D2ψ)φ dΩ

A Boundary Condition can eliminate the surface integral Dψ · n = ( i κ∇ − A)ψ · n = 0

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The EL equation for ψ

Gs =

  • (−ψ + |ψ|2ψ + ( i

κ∇ − A)ψ)2φ dΩ = 0 The Integrand must be zero, The EL equation for ψ is, with the boundary condition −ψ + |ψ|2ψ + ( i κ∇ − A)2ψ = 0 ( i κ∇ − A)ψ · n = 0

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The EL equation for A

Using the same process and a lot few vector calculus identities. δgs δA = lim

η→0

1 η

  • Ω

gs(A + ηφ) − gs(A) dΩ = 0 The EL equation of A is found to be, with a boundary condition to cancel the surface integrals. Js = ∇ × (∇ × A − H) = − i 2κ(ψ∗∇ψ − ψ∇ψ2) − |ψ|2A ∇ × A × n = H × n

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The ND GL equations

Putting everything together, we have a nonlinear system of PDE’s −ψ + |ψ|2ψ + ( i κ∇ − A)2ψ = 0 in Ω ∇ × ∇ × A + i 2κ(ψ∗∇ψ − ψ∇ψ2) + |ψ|2A = ∇ × H in Ω ( i κ∇ − A)ψ · n = 0 on Γ ∇ × A × n = H × n on Γ

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