[PPT] - Space-Time Discontinuous Petrov-Galerkin Finite Elements for PowerPoint Presentation

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Space-Time Discontinuous Petrov-Galerkin Finite Elements for Transient Fluid Mechanics

Truman Ellisa, Leszek Demkowiczb, Jesse Chanc, Nate Robertsa and Robert Moserb

a Sandia Laboratories b ICES, University of Texas at Austin c Dept. of Math., Rice University

Workshop on Space-Time Methods for PDEs RICAM, Linz, Nov. 7-11, 2016

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Navier-Stokes Equations

Numerical Challenges

Robust simulation of unsteady fluid dynamics remains a challenging issue. Resolving solution features (sharp, localized viscous-scale phenomena)

Shocks Boundary layers - resolution needed for drag/load Turbulence (non-localized)

Stability of numerical schemes

Nonlinearity Nature of PDE changes for different flow regimes Coarse/adaptive grids Higher order

Shock Boundary layer

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Motivation

Initial Mesh Design is Expensive and Time-Consuming

Surface mesh must accurately represent geometry Volume mesh needs sufficient resolution for asymptotic regime Engineers often forced to work by trial and error Bad in the context of HPC

Formula 1 Mesh by Numeca

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Motivation

Initial Mesh Design is Expensive and Time-Consuming

Surface mesh must accurately represent geometry Volume mesh needs sufficient resolution for asymptotic regime Engineers often forced to work by trial and error Bad in the context of HPC We desire an automated computational technology

Formula 1 Mesh by Numeca

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DPG on Coarse Meshes

Adaptive Solve of the Carter Plate Problem1 Re = 1000

Temperature on Initial Mesh Temperature after 8 Refinements Temperature after 4 Refinements Temperature after 11 Refinements

1J.L. Chan. ‘‘A DPG Method for Convection-Diffusion Problems’’. PhD thesis. University of Texas at Austin, 2013.

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SLIDE 8

Lessons from Other Methods

Streamline Upwind Petrov-Galerkin: Adaptively changing the test space can produce a method with better stability. Discontinuous Galerkin: Discontinuous basis functions are a legitimate option for finite element methods. Hybridized DG: Mesh interface unknowns can facilitate static condensation -- reducing the number of DOFs in the global solve. Least-Squares FEM: The finite element method is most powerful in a minimum residual context (i.e. as a Ritz method). Space-Time FEM: Highly adaptive methods should have adaptive time

integration. Superior framework for problems with moving
boundaries. Requires a method that is both temporally and

spatially stable.

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Overview of DPG

DPG is a Minimum Residual Method

Find u ∈ U such that b(u, v) = l(v)

∀v ∈ V

with operator B : U → V ′ defined by b(u, v) = Bu, vV ′×V. This gives the operator equation Bu = l

∈ V ′ .

We wish to minimize the residual Bu − l ∈ V ′: uh = arg min

wh∈Uh

1 2 Bwh − l2

V ′ .

Dual norms are not computationally tractable. Inverse Riesz map moves the residual to a more accessible space: uh = arg min

wh∈Uh

1 2

R−1

V (Bwh − l)

2

V .

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Overview of DPG

Petrov-Galerkin with Optimal Test Functions

Taking the Gâteaux derivative to be zero in all directions δu ∈ Uh gives,

R−1

V (Buh − l), R−1 V Bδu

V = 0,

∀δu ∈ U,

which by definition of the Riesz map is equivalent to

Buh − l, R−1

V Bδuh

= 0

∀δuh ∈ Uh ,

with optimal test functions vδuh := R−1

V Bδuh for each trial function δuh.

Resulting Petrov-Galerkin System

This gives a simple bilinear form b(uh, vδuh) = l(vδuh), with vδuh ∈ V that solves the auxiliary problem

(vδuh, δv)V = RVvδuh, δv = Bδuh, δv = b(δuh, δv) ∀δv ∈ V.

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Overview of DPG

Mixed Formulation

Identifying the error representation function:

ψ := R−1

V (Buh − l)

allows us to develop an alternative interpretation of DPG.

DPG as a Mixed Problem

Find ψ ∈ V, uh ∈ Uh such that

(ψ, δv)V − b(uh, δv) = −l(δv) ∀δv ∈ V

b(δuh, ψ) = 0

∀δuh ∈ Uh

In this unconventional saddle-point problem, the approximate solution uh comes from a finite-dimensional trial space and plays the role of the Lagrange multiplier for the error representation function

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SLIDE 13

Overview of DPG

DPG is the Most Stable Petrov-Galerkin Method

Babuška’s theorem guarantees that discrete stability and approximability imply

convergence. If bilinear form b(u, v), with M := b satisfies the discrete inf-sup

condition with constant γh, sup

vh∈Vh

|b(u, v)| vhV ≥ γh uhU ,

then the Galerkin error satisfies the bound

uh − uU ≤ M γh

inf

wh∈Uh wh − uU .

Optimal test function realize the supremum guaranteeing that γh ≥ γ.

Energy Norm

If we use the energy norm, uE := BuV ′ in the error estimate, then M = γ = 1. Babuška’s theorem implies that the minimum residual method is the most stable Petrov-Galerkin method (assuming exact optimal test functions).

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Overview of DPG

Other Features

Discontinuous Petrov-Galerkin

Continuous test space produces global solve for optimal test functions Discontinuous test space results in an embarrassingly parallel solve

Hermitian Positive Definite Stiffness Matrix

Property of all minimum residual methods b(uh, vδuh) = (vuh, vδuh)V = (vδuh, vuh)V = b(δuh, vuh)

Error Representation Function

Energy norm of Galerkin error (residual) can be computed without exact solution

uh − uE = B(uh − u)V ′ = Buh − lV ′ =

R−1

V (Buh − l)

V
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Overview of DPG

High Performance Computing

Eliminates human intervention Stability Robustness Adaptivity Automaticity Compute intensive Embarrassingly parallel local solves Factorization recyclable Low communication SPD stiffness matrix Multiphysics

Stampede Supercomputer at TACC Mira Supercomputer at Argonne

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Locally Conservative DPG

DPG for Convection-Diffusion

Start with the strong-form PDE.

∇ · (βu) − ǫ∆u = g

Rewrite as a system of first-order equations. 1

ǫ σ − ∇u = 0 ∇ · (βu − σ) = g

Multiply by test functions and integrate by parts over each element, K. 1

ǫ (σ, τ)K + (u, ∇ · τ)K − u, τn∂K = 0 −(βu − σ, ∇v)K + (βu − σ) · n, v∂K = (g, v)K

Use the ultraweak (DPG) formulation to obtain bilinear form b(u, v) = l(v). 1

ǫ (σ, τ)K + (u, ∇ · τ)K − ˆ

u, τn∂K

− (βu − σ, ∇v)K + ˆ

t, v∂K = (g, v)K

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Locally Conservative DPG

Local Conservation for Convection-Diffusion

The local conservation law in convection diffusion is

∂K

ˆ

t =

K

g , which is equivalent to having vK := {v, τ} = {1K, 0} in the test space. In general, this is not satisfied by the optimal test functions. Following Moro et al2 (also Chang and Nelson3), we can enforce this condition with Lagrange multipliers: L(uh, λ) = 1 2

R−1

V (Buh − l)

2

V −

K

λK Buh − l, vK

ˆ

t,1K∂K−g,1KK

,

where λ = {λ1, · · · , λN}.

2D. Moro, N.C. Nguyen, and J. Peraire. ‘‘A Hybridized Discontinuous Petrov-Galerkin Scheme for Scalar Conservation Laws’’. In: Int. J. Num.
Meth. Eng. (2011).

3C.L. Chang and J.J. Nelson. ‘‘Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation’’. In: SIAM

J. Num. Anal. 34 (1997), pp. 480–489.
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Locally Conservative DPG

Locally Conservative Saddle Point System

Finding the critical points of L(u, λ), we get the following equations.

Locally Conservative Saddle Point System ∂L(uh, λ) ∂uh = b(uh, R−1

V Bδuh) − l(R−1 V Bδuh)

−

K

λKb(δuh, vK) = 0 ∀δuh ∈ Uh ∂L(uh, λ) ∂λK = −b(uh, vK) + l(vK) = 0 ∀K

A few consequences: Minimization problem turns into a constrained minimization problem. Optimal test function are in the orthogonal complement of constants.

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Locally Conservative DPG

Optimal Test Functions

For each u = {u, σ, ˆ u,ˆ t} ∈ Uh, find vu = {vu, τ u} ∈ V such that

(vu, w)V = b(u, w) ∀w ∈ V

where V becomes Vp+∆p in order to make this computationally tractable. We recently developed this modification to the robust test norm4 which behaves better in the presence of singularities.

Convection-Diffusion Test Norm (v, τ)2

V,Ωh =

min
1

√ǫ,

1

|K|
τ
2

+ ∇ · τ − β · ∇v2 + β · ∇v2 + ǫ ∇v2 + v2

No longer necessary
4J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test

norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795.

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Locally Conservative DPG

Optimal Test Functions

For each u = {u, σ, ˆ u,ˆ t} ∈ Uh, find vu = {vu, τ u} ∈ V such that

(vu, w)V = b(u, w) ∀w ∈ V

where V becomes Vp+∆p in order to make this computationally tractable. We recently developed this modification to the robust test norm4 which behaves better in the presence of singularities.

Convection-Diffusion Test Norm (v, τ)2

V,Ωh =

min
1

√ǫ,

1

|K|
τ
2

+ ∇ · τ − β · ∇v2 + β · ∇v2 + ǫ ∇v2 +

1

|K|

K v

2

Scaling term
4J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test

norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795.

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Locally Conservative DPG

Stability and Robustness Analysis5

We follow Brezzi’s theory for an abstract mixed problem:

    

u ∈ U, p ∈ Q a(u, w) + c(p, w)

= l(w) ∀w ∈ U

c(q, u)

= g(q) ∀q ∈ Q

where a, c, l, g denote the appropriate bilinear and linear forms. a(u, w) = b(u, R−1

V Bw) = (R−1 V Bu, R−1 V Bw)V

c(p, w) =

K λK ˆ

t, 1K∂K Locally conservative DPG satisfies inf-sup and inf-sup in kernel conditions. Robustness is proved by switching to energy norm in Brezzi analysis.

5T.E. Ellis, L.F. Demkowicz, and J.L. Chan. ‘‘Locally Conservative Discontinuous Petrov-Galerkin Finite Elements For Fluid Problems’’. In: Comp.

Math. Appl. 68.11 (2014), pp. 1530 –1549.
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Numerical Experiments

Stokes Flow Around a Cylinder

Horizontal Velocity

1 Refinement 6 Refinements

Nonconservative

1 Refinement 6 Refinements

Conservative

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Numerical Experiments

Stokes Flow Around a Cylinder

Percent Mass Loss at x = [−1, −0.95, 0, 0.95, 3]

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x location 20 40 60 80 100 percent mass loss

1844 DOFs 4526 DOFs 17552 DOFs 66056 DOFs 92084 DOFs 195674 DOFs 339545 DOFs

Nonconservative

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x location 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 percent mass loss 1e 8

1844 DOFs 3980 DOFs 12380 DOFs 33272 DOFs 75326 DOFs 183326 DOFs 358739 DOFs

Conservative

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SLIDE 25

Numerical Experiments

Stokes Flow Over a Backward Facing Step

Horizontal Velocity

Initial Mesh 8 Refinements

Nonconservative

Initial Mesh 8 Refinements

Conservative

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SLIDE 26

Numerical Experiments

Stokes Flow Over a Backward Facing Step

Percent Mass Loss at x = [0, 0.5, · · · , 9.5, 10]

2 4 6 8 10 x location 5 10 15 20 25 30 35 40 percent mass loss

8110 DOFs 12084 DOFs 14760 DOFs 16757 DOFs 18754 DOFs 20751 DOFs 22748 DOFs 24745 DOFs 26742 DOFs

Nonconservative

2 4 6 8 10 x location 0.0 0.5 1.0 1.5 2.0 percent mass loss 1e 12

8110 DOFs 10107 DOFs 12104 DOFs 14101 DOFs 16098 DOFs 18095 DOFs 20092 DOFs 22089 DOFs 24086 DOFs

Conservative

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Space-Time DPG

Extending DPG to Transient Problems

Time stepping techniques are not ideally suited to highly adaptive grids Space-time FEM proposed as a solution ✦ Unified treatment of space and time ✦ Local space-time adaptivity (local time stepping) ✦ Parallel-in-time integration (space-time multigrid) ✪ Spatially stable FEM methods may not be stable in space-time ✪ Need to support higher dimensional problems DPG provides necessary stability and adaptivity Courtesy of XBraid by LLNL

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Space-Time DPG for Convection-Diffusion

Space-Time Divergence Form

Equation is parabolic in space-time.

∂u ∂t + β · ∇u − ǫ∆u = f

This is just a composition of a constitutive law and conservation of mass.

σ − ǫ∇u = 0 ∂u ∂t + ∇ · (βu − σ) = f

We can rewrite this in terms of a space-time divergence. 1

ǫ σ − ∇u = 0 ∇xt ·

βu − σ

u

= f
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SLIDE 30

Space-Time DPG for Convection-Diffusion

Ultra-Weak Formulation with Discontinuous Test Functions

Multiply by test function and integrate by parts over space-time element K.

1

ǫ σ, τ

K

+ (u, ∇ · τ)K − ˆ

u, τ · nx∂K = 0

−

βu − σ

u

, ∇xtv
K

+ ˆ

t, v∂K = f where

ˆ

u := tr(u)

ˆ

t := tr(βu − σ) · nx

+ tr(u) · nt

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SLIDE 31

Space-Time Convection-Diffusion

L2 Equivalent Norms

Bilinear form with group variables: b ((u, ˆ u) , v) = (u, A∗

hv)L2(Ωh)+

u, [

[v] ]Γh

For conforming v∗ satisfying A∗v∗ = u

u2

L2(Ωh) = b(u, v∗) = b(u, v∗)

v∗V v∗V ≤ sup

v∗=0

|b(u, v∗)| v∗ v∗ = uE v∗V

Necessary robustness condition:

v∗V uL2(Ωh) ⇒ uL2(Ωh) uE

Analytical Solution

e−lt(eλ1(x−1) − eλ2(x−1))

+

1 − e

1 ǫ x

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SLIDE 32

Space-Time Convection-Diffusion

L2 Equivalent Norms

A norm should be: bounded by uL2(Ωh), have good conditioning, not produce boundary layers in the optimal test function.

102 103 104 105 106 107 DOFs 10-7 10-6 10-5 10-4 10-3 10-2 10-1 Error ǫ =10−2 L2 Error ǫ =10−2 V ∗ Error ǫ =10−4 L2 Error ǫ =10−4 V ∗ Error ǫ =10−6 L2 Error ǫ =10−6 V ∗ Error ǫ =10−8 L2 Error ǫ =10−8 V ∗ Error

(v, τ)2 =

∇ · τ − ˜

β · ∇xtv

2

+

1

ǫ τ + ∇v

2

+ v2 + τ2

102 103 104 105 106 107 DOFs 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Error ǫ =10−2 L2 Error ǫ =10−2 V ∗ Error ǫ =10−4 L2 Error ǫ =10−4 V ∗ Error ǫ =10−6 L2 Error ǫ =10−6 V ∗ Error ǫ =10−8 L2 Error ǫ =10−8 V ∗ Error

(v, τ)2 =

∇ · τ − ˜

β · ∇xtv

2

+ min

1

h2 , 1

ǫ

τ2

+ ǫ ∇v2 + β · ∇v2 + v2

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SLIDE 33

Steady Convection-Diffusion

Ideal Optimal Shape Functions

Graph Norm

0.0 0.2 0.4 0.6 0.8 1.0 0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06

v

0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

τ

Coupled Robust Norm

0.0 0.2 0.4 0.6 0.8 1.0 0.0015 0.0010 0.0005 0.0000 0.0005 0.0010 0.0015

v

0.0 0.2 0.4 0.6 0.8 1.0 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10

τ

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SLIDE 34

Steady Convection-Diffusion

Approximated (p = 3) Optimal Shape Functions

Graph Norm

0.0 0.2 0.4 0.6 0.8 1.0 0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06

v

0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

τ

Coupled Robust Norm

0.0 0.2 0.4 0.6 0.8 1.0 0.0015 0.0010 0.0005 0.0000 0.0005 0.0010 0.0015

v

0.0 0.2 0.4 0.6 0.8 1.0 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10

τ

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SLIDE 35

Steady Convection-Diffusion

Two Robust Norms for Steady Convection-Diffusion

The following norms are robust for steady convection-diffusion. The robust norm was derived in6:

(v, τ)2 = β · ∇v2 + ǫ ∇v2 + min ǫ

h2 , 1

v2

+ ∇ · τ2 + min

1

h2 , 1

ǫ

τ2 .

The case for the coupled robust norm was made in7:

(v, τ)2 = β · ∇v2 + ǫ ∇v2 + min ǫ

h2 , 1

v2

+ ∇ · τ − β · ∇v2 + min

1

h2 , 1

ǫ

τ2 .
6J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test

norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795.

7J.L. Chan. ‘‘A DPG Method for Convection-Diffusion Problems’’. PhD thesis. University of Texas at Austin, 2013.

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SLIDE 36

Space-Time Convection-Diffusion

Two Robust Norms for Transient Convection-Diffusion

Let ˜

β :=

β

1

and ∇xtv :=
∇v

∂v ∂t

.

The following norms are robust for space-time convection-diffusion. Robust Norm:

(v, τ)2 =

˜

β · ∇xtv

2

+ ǫ ∇v2 + min ǫ

h2 , 1

v2

+ ∇ · τ2 + min

1

h2 , 1

ǫ

τ2 .

Coupled Robust Norm

(v, τ)2 =

˜

β · ∇xtv

2

+ ǫ ∇v2 + min ǫ

h2 , 1

v2

+

∇ · τ − ˜

β · ∇xtv

2

+ min

1

h2 , 1

ǫ

τ2 .
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SLIDE 37

Robust Norms for Transient Convection-Diffusion

Adjoint Operator

Consider the problem with homogeneous boundary conditions 1

ǫ σ − ∇u = 0 ˜ β · ∇xtu − ∇ · σ = f βnu − ǫ∂u ∂n = 0 on Γ−

u = 0 on Γ+ u = u0 on Γ0. The adjoint operator A∗ is given by A∗(v, τ) =

1

ǫ τ + ∇v, −˜ β · ∇xtv + ∇ · τ

.
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SLIDE 38

Robust Norms for Transient Convection-Diffusion

Controlling Different Field Variables

We decompose the continuous adjoint problem A∗(τ, v) = (σ, u) into

Continuous part with forcing u

1

ǫ τ 1 + ∇v1 = 0 −˜ β · ∇xtv1 + ∇ · τ 1 = u τ 1 · nx = 0 on Γ−

v1 = 0 on Γ+ v1 = 0 on ΓT

Continuous part with forcing σ

1

ǫ τ 2 + ∇v2 = σ −˜ β · ∇xtv2 + ∇ · τ 2 = 0 τ 2 · nx = 0 on Γ−

v2 = 0 on Γ+ v2 = 0 on ΓT

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SLIDE 39

Robust Norms for Transient Convection-Diffusion

Proved Bounds at Our Disposal

Proofs of these lemmas can be found in8.

Lemma (1)

If ∇ · β = 0, we can bound

v2 + ǫ ∇v2 ≤ u2 + ǫ σ2

where v = v1 + v2.

Lemma (2)

If

∇β − 1

2∇ · βI

L∞ ≤ Cβ, we can bound
˜

β · ∇xtv1

u .
8T. Ellis, J. Chan, and L. Demkowicz. ‘‘Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential

Equations,x Eds. G.R. Barrenechea et al.’’ In: vol. 114. Lecture Notes in Computational Science and Engineering. in print, see also ICES Report 2015/21. Springer, 2016. Chap. Robust DPG Methods for Transient Convection-Diffusion.

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SLIDE 40

Robust Norms for Transient Convection-Diffusion

Control of u

Bound on (v1, τ 1)

Lemma (2) ⇒

˜

β · ∇xtv1

u

Lemma (2) ⇒

∇ · τ 1 ≤ u +

˜

β · ∇xtv1

2 u

Lemma (2) ⇒

∇ · τ 1 − ˜

β · ∇xtv1

= u

Lemma (1) ⇒

v12 + ǫ ∇v12 ≤ u2

Lemma (1) ⇒ 1

ǫ τ 1 = ǫ ∇v1 ≤ u

We can guarantee robust control

(u, 0)L2(Ωh) (u, σ)E .

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SLIDE 41

Robust Norms for Transient Convection-Diffusion

Control of σ

Bound on (v2, τ 2)

Definition ⇒

∇ · τ 2 − ˜

β · ∇xtv2

= 0 ≤ σ

Lemma (1) ⇒

v22 + ǫ ∇v22 ≤ ǫ σ2

Lemma (1) ⇒ 1

ǫ τ 2 = σ + ǫ ∇v2 = (1 + ǫ) σ

We have not been able to prove bounds on

˜

β · ∇xtv2

r ∇ · τ 2.

We can not guarantee robust control

(0, σ)L2(Ωh) ✓

✓

(u, σ)E .

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SLIDE 42

Robust Norms for Transient Convection-Diffusion

Transient Analytical Solution

Transient impulse decays to Eriksson-Johnson steady state solution. u = exp(−lt) [exp(λ1x) − exp(λ2x)] + cos(πy) exp(s1x) − exp(r1x) exp(−s1) − exp(−r1) t = 0.0 t = 0.5 t = 1.0

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SLIDE 43

Robust Norms for Transient Convection-Diffusion

Robust Convergence to Analytical Solution

102 103 104 105 106 107 DOFs 10-5 10-4 10-3 10-2 10-1 100 Error L2 p =1 Robust V ∗ p =1 Robust L2 p =2 Robust V ∗ p =2 Robust L2 p =4 Robust V ∗ p =4 Robust L2 p =1 CoupledRobust V ∗ p =1 CoupledRobust L2 p =2 CoupledRobust V ∗ p =2 CoupledRobust L2 p =4 CoupledRobust V ∗ p =4 CoupledRobust

ǫ = 10−2

102 103 104 105 106 107 DOFs 10-3 10-2 10-1 100 Error L2 p =1 Robust V ∗ p =1 Robust L2 p =2 Robust V ∗ p =2 Robust L2 p =4 Robust V ∗ p =4 Robust L2 p =1 CoupledRobust V ∗ p =1 CoupledRobust L2 p =2 CoupledRobust V ∗ p =2 CoupledRobust L2 p =4 CoupledRobust V ∗ p =4 CoupledRobust

ǫ = 10−6

102 103 104 105 106 107 DOFs 10-2 10-1 100 Error L2 p =1 Robust V ∗ p =1 Robust L2 p =2 Robust V ∗ p =2 Robust L2 p =4 Robust V ∗ p =4 Robust L2 p =1 CoupledRobust V ∗ p =1 CoupledRobust L2 p =2 CoupledRobust V ∗ p =2 CoupledRobust L2 p =4 CoupledRobust V ∗ p =4 CoupledRobust

ǫ = 10−4

102 103 104 105 106 107 DOFs 10-3 10-2 10-1 100 Error L2 p =1 Robust V ∗ p =1 Robust L2 p =2 Robust V ∗ p =2 Robust L2 p =4 Robust V ∗ p =4 Robust L2 p =1 CoupledRobust V ∗ p =1 CoupledRobust L2 p =2 CoupledRobust V ∗ p =2 CoupledRobust L2 p =4 CoupledRobust V ∗ p =4 CoupledRobust

ǫ = 10−8

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SLIDE 44

Space-Time Incompressible Navier-Stokes

Nonlinear Form

Space-time divergence form: 1

ν D − ∇u = 0 ∇xt ·

u ⊗ u − D + pI

u

= f

∇ · u = 0

Multiply by S ∈ H(div, Q), v ∈ H 1

xt(Q), q ∈ H1(Q), and integrate by parts:

1

ν D, S

+ (u, ∇ · S) − ˆ

u, S · nx = 0

−

u ⊗ u − D + pI

u

, ∇xtv
+ ˆ

t, v = (f , v)

− (u, ∇q) + ˆ

u · n, q = 0

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SLIDE 46

Space-Time Incompressible Navier-Stokes

Robust Norms

Recall the adjoint and robust norm for convection-diffusion:

(σ, 1 ǫ τ + ∇v) + (u, ∇ · τ − β · ∇v − ∂v ∂t ) (v, τ)2

V,K :=

β · ∇v + ∂v

∂t

2

K

+ ǫ ∇v2

K + min

ǫ

h2 , 1

v2

K

+ ∇ · τ2

K + min

1

ǫ , 1

h2

τ2

K

For incompressible Navier-Stokes the adjoint comes from:

∆D, 1

ν S + ∇v

+
∆u, ∇ · S − ∇q −
˜

u · ∇v + ˜ u · (∇v)T + ∂v

∂t

+ (p, −∇ · v)
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SLIDE 47

Space-Time Incompressible Navier-Stokes

Norms for Navier-Stokes come from analogy

Convection-Diffusion Navier-Stokes

ǫ → ν τ → S ∇v → ∇v ∇ · τ → ∇ · S − ∇q β · ∇v + ∂v

∂t

→ ˜

u · ∇v + ˜ u · (∇v)T + ∂v

∂t

Robust norm:

(v, D, q)2

V,K :=

˜

u · ∇v + ˜ u · (∇v)T + ∂v

∂t

2

K

+ ν ∇v2

K

+ min ν

h2 , 1

v2

K + ∇ · S − ∇q2 K

+ min

1

ν , 1

h2

S2

K + ∇ · v2 K + q2 K

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SLIDE 48

Space-Time Incompressible Navier-Stokes

Taylor-Green Vortex

u = e− 2

Re t

sin x cos y

− cos x sin y

103

104 105 106 107 Degrees of Freedom 10-6 10-5 10-4 10-3 10-2 10-1 100 Error

Coupled Robust Norm

L2 Re =101 p =1 L2 Re =101 p =2 L2 Re =101 p =4 L2 Re =103 p =1 L2 Re =103 p =2 L2 Re =103 p =4 L2 Re =105 p =1 L2 Re =105 p =2 L2 Re =105 p =4 V ∗ Re =101 p =1 V ∗ Re =101 p =2 V ∗ Re =101 p =4 V ∗ Re =103 p =1 V ∗ Re =103 p =2 V ∗ Re =103 p =4 V ∗ Re =105 p =1 V ∗ Re =105 p =2 V ∗ Re =105 p =4

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SLIDE 49

Space-Time Incompressible Navier-Stokes

Flow Over a Cylinder, Initial Mesh

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SLIDE 50

Space-Time Incompressible Navier-Stokes

Flow Over a Cylinder, 4 Refinements

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SLIDE 51

Space-Time Incompressible Navier-Stokes

Solve Times and Strong Scaling

Transient Flow Over a Cylinder

1 Node 4 Nodes 32 Nodes Ref Elems DOFs Time Time Scaling vs 1 Time Scaling vs 4 80 31,304 1,772 453 3.91 451 1.01 1 605 225,908 8,190 3,574 2.29 717 4.98 2 3,013 1,081,598 32,008 12,076 2.65 2,648 4.56 3 9,726 3,429,384 28,744 6,319 4.54 4 11,742 4,144,674 8,510

Computations on Lonestar, 1 node = 24 processors 32,008 seconds = 8.8 hours 28,744 seconds = 8.0 hours 8,510 seconds = 2.4 hours

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SLIDE 52

Space-Time Compressible Navier-Stokes

First Order System with Primitive Variables

Assuming Stokes hypothesis, ideal gas law, and constant viscosity: 1

µD − ∇u = 0

Pr Cpµq + ∇T = 0

∇xt ·

ρu

ρ

= fc

∇xt ·

ρu ⊗ u + ρRTI −
D + DT − 2

3 tr(D)I

ρu
= f m

∇xt ·

ρu
CvT + 1

2u · u

+ ρRTu + q − u ·
D + DT − 2

3 tr(D)I

ρ
CvT + 1

2u · u

= fe
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SLIDE 54

Space-Time Compressible Navier-Stokes

Compact Notation

Conserved quantities Cc := ρ Cm := ρu Ce := ρ(CvT + 1 2u · u) Euler fluxes Fc := ρu

Fm := ρu ⊗ u + ρRTI

Fe := ρu

CvT + 1

2u · u

+ ρRTu

Viscous fluxes Kc := 0

Km := D + DT − 2

3 tr(D)I Ke := −q + u ·

D + DT − 2

3 tr(D)I

Viscous variables

MD := D

M q := Pr Cp q GD := 2u Gq := −T Use change of variables to get conservation or entropy variables.

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SLIDE 55

Space-Time Compressible Navier-Stokes

Conservation Variables (Popular for Time-Stepping)

Change of variables:

ρ = ρ

m = ρu E = ρ

CvT + 1

2u · u

Euler fluxes:

Fc

c = m

Fc

m = m ⊗ m

ρ + (γ − 1)

E − m · m

2ρ

I

Fc

e = γE m

ρ − (γ − 1)m · m

2ρ2 m

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SLIDE 56

Space-Time Compressible Navier-Stokes

Entropy Variables (Symmetrize the Bubnov-Galerkin Stiffness Matrix)

Change of variables:

Vc =

−E + (E −

1 2ρm · m)

γ + 1 − ln
(γ−1)(E− 1

2ρ m·m)

ργ

E −

1 2ρm · m

V m = m E −

1 2ρm · m

Ve =

−ρ

E −

1 2ρm · m

Euler fluxes:

Fe

c =

γ − 1 (−Ve)γ

1

γ−1

exp

−γ + Vc −

1 2Ve V m · V m

γ − 1

V m

Fe

m =

γ − 1 (−Ve)γ

1

γ−1

exp

−γ + Vc −

1 2Ve V m · V m

γ − 1 −V m ⊗ V m

Ve

+ (γ − 1)I

Fe

e =

γ − 1 (−Ve)γ

1

γ−1

exp

−γ + Vc −

1 2Ve V m · V m

γ − 1

V m

Ve

1

2Ve V m · V m − γ

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SLIDE 57

Space-Time Compressible Navier-Stokes

Define Group Variables

Group terms C := {Cc , Cm , Ce} F := {Fc , Fm , Fe} K := {Kc , Km , Ke} M := {MD , M q} G := {GD , Gq} f := {fc , f m , fe} Group variables W := {ρ , u , T}

ˆ

W :=

2ˆ

u , −ˆ T

Σ := {D, q}

ˆ

t := {ˆ te , ˆ tm, , ˆ te}

Ψ := {S , τ}

V := {vc , vm, , ve} Navier-Stokes variational formulation is

1

µM, Ψ

+ (G, ∇ · Ψ) −

ˆ

W, Ψ · nx

= 0

−

F − K

C

, ∇xtV
+ ˆ

t, V = (f , V)

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SLIDE 58

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 1 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 59

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 2 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 60

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 3 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 61

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 4 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 62

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 5 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 63

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 6 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 64

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 7 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 65

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 8 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 66

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 9 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 67

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 10 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 68

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 11 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 69

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 12 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 70

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 13 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 71

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 14 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 72

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

Mesh 15 Primitive Variables Conservation Variables Entropy Variables

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SLIDE 73

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 0 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 0 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 0 Refinements Exact Primal Conservation Entropy

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SLIDE 74

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 1 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 1 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 1 Refinements Exact Primal Conservation Entropy

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SLIDE 75

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 2 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 2 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 2 Refinements Exact Primal Conservation Entropy

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SLIDE 76

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 3 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 3 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 3 Refinements Exact Primal Conservation Entropy

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SLIDE 77

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 4 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 4 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 4 Refinements Exact Primal Conservation Entropy

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SLIDE 78

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 5 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 5 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 5 Refinements Exact Primal Conservation Entropy

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SLIDE 79

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 6 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 6 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 6 Refinements Exact Primal Conservation Entropy

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SLIDE 80

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 7 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 7 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 7 Refinements Exact Primal Conservation Entropy

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SLIDE 81

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 8 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 8 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 8 Refinements Exact Primal Conservation Entropy

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SLIDE 82

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 9 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 9 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 9 Refinements Exact Primal Conservation Entropy

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SLIDE 83

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 10 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 10 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 10 Refinements Exact Primal Conservation Entropy

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SLIDE 84

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 11 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 11 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 11 Refinements Exact Primal Conservation Entropy

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SLIDE 85

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 12 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 12 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 12 Refinements Exact Primal Conservation Entropy

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SLIDE 86

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 13 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 13 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 13 Refinements Exact Primal Conservation Entropy

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SLIDE 87

Space-Time Compressible Navier-Stokes

Sod Shock Tube with µ = 10−5

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 density

Density 14 Refinements Exact Primal Conservation Entropy

0.0 0.2 0.4 0.6 0.8 1.0 x 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 velocity Velocity 14 Refinements Exact Primal Conservation Entropy 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 pressure Pressure 14 Refinements Exact Primal Conservation Entropy

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SLIDE 88

Space-Time Compressible Navier-Stokes

Entropy Scaled Test Norms

Let W, U, and V be the set of primitive, conservation, and entropy variables. The entropy function H = −ρ log(pρ−γ) provides a natural residual for the Navier-Stokes equations. A0 = H,UU = V,U is known as the symmetrizer and (U, A0U) provides a natural metric for the linearized Euler equations. In primitive variables:

(U, A0U) = (U,WW, V,UU,WW) = (W, U T

,WV,UU,WW) = (W, A0(W)W)

where A0(W) = U T

,WV,UU,W =

  

γ−1 ρ ρ

CvT

ρ

T 2

   (W, A0W) has units of density.

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SLIDE 89

Space-Time Compressible Navier-Stokes

Entropy Scaled Test Norms

Bilinear form with group variables: b

W, ˆ

W

, v
= (W, A∗

hv)L2(Ωh) +

ˆ

W, [

[v] ]

Γh

For conforming v∗ satisfying A∗v∗ = A0W

A

1 2

0 W

2

= b(W, v∗) = b(W, v∗) v∗V v∗V ≤ sup

v∗=0

|b(W, v∗)| v∗ v∗ = WE v∗V .

Necessary robustness condition:

v∗V

A

1 2

0 W

L2(Ωh)

⇒

A

1 2

0 W

L2(Ωh)

WE

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SLIDE 90

Space-Time Compressible Navier-Stokes

Entropy Scaled Test Norms

We load our adjoint equations with A0W: 1

µM∗(Ψ) + K∗(∇V) = 0 −

F ∗

C∗

(∇xtV) + G∗(∇Ψ) = A0W

This leads to the entropy scaled robust norm:

(V, Ψ)2

V,K :=

A

− 1

2

0 (F ∗ + C∗)

2

K

+ µ

A

− 1

2

K∗

2

K

+ min µ

h2 , 1

A

− 1

2

V

2

K

+

A

− 1

2

G∗

2

K

+ min

1

µ, 1

h2

A

− 1

2

M∗

2

K

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SLIDE 91

Space-Time Compressible Navier-Stokes

Entropy Scaled Test Norms

We load our adjoint equations with A0W: 1

µM∗(Ψ) + K∗(∇V) = 0 −

F ∗

C∗

(∇xtV) + G∗(∇Ψ) = A0W

This leads to the entropy scaled robust norm:

(V, Ψ)2

V,K :=

A

− 1

2

0 (F ∗ + C∗)

2

K

+ µ

A

− 1

2

K∗

2

K

+ min µ

h2 , 1

A

− 1

2

V

2

K

+

A

− 1

2

G∗

2

K

+ min

1

µ, 1

h2

A

− 1

2

M∗

2

K

Numerical results were disappointing.

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SLIDE 92

Space-Time Compressible Navier-Stokes

Noh Implosion with Primitive Variables, Robust Norm, µ = 10−3

1.0 0.8 0.6 0.4 0.2 0.0 x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 density

Exact Initial 4 Elements 5 Refinements 10 Refinements

After 5 refinements After 10 refinements

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SLIDE 93

Space-Time Compressible Navier-Stokes

Piston with µ = 10−2

ˆ

tc =

√

2(−ρu + ρ)

ˆ

tm =

√

2(−ρu2 − ρRT + ρu)

ˆ

te =

√

2(−ρu(CvT + 1 2u2) − uρRT + ρ(CvT + 1 2u2))

ˆ

u = 1

ˆ

tc = 0

ˆ

tm − ˆ te = 0

Density Velocity

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SLIDE 94

Space-Time Compressible Navier-Stokes

Piston with µ = 10−2

Mesh after 8 adaptive refinements

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SLIDE 95

Future Directions

Improve performance: line smoothing for multigrid Shock capturing: DPG makes no promises when it comes to Gibbs phenomenon Non-Hilbert DPG: L1 is known to limit oscillations Anisotropic refinements: necessary for time slabs More extensive 2D results: shedding vortex problems, 2D shock problems 3D results: will not be cheap Incompressible Flow Over a Cylinder

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SLIDE 96

Thank You!

Recommended References

◮

J.L. Chan. ‘‘A DPG Method for Convection-Diffusion Problems’’. PhD thesis. University of Texas at Austin, 2013.

◮

D. Moro, N.C. Nguyen, and J. Peraire. ‘‘A Hybridized Discontinuous Petrov-Galerkin Scheme

for Scalar Conservation Laws’’. In: Int. J. Num. Meth. Eng. (2011).

◮

C.L. Chang and J.J. Nelson. ‘‘Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation’’. In: SIAM J. Num. Anal. 34 (1997), pp. 480–489.

◮

J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II:

Adjoint boundary conditions and mesh-dependent test norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795.

◮

T.E. Ellis, L.F. Demkowicz, and J.L. Chan. ‘‘Locally Conservative Discontinuous Petrov-Galerkin Finite Elements For Fluid Problems’’. In: Comp. Math. Appl. 68.11 (2014), pp. 1530 –1549.

◮

T. Ellis, J. Chan, and L. Demkowicz. ‘‘Building Bridges: Connections and Challenges in

Modern Approaches to Numerical Partial Differential Equations,x Eds. G.R. Barrenechea et al.’’ In: vol. 114. Lecture Notes in Computational Science and Engineering. in print, see also ICES Report 2015/21. Springer, 2016. Chap. Robust DPG Methods for Transient Convection-Diffusion.

◮

L.F. Demkowicz and J. Gopalakrishnan. ‘‘Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations (eds. X. Feng, O. Karakashian, Y. Xing)’’. In: vol. 157. IMA Volumes in Mathematics and its Applications, 2014. Chap. An Overview of the DPG Method, pp. 149–180.

◮

N.V. Roberts. ‘‘Camellia: A Software Framework for Discontinuous Petrov-Galerkin Methods’’. In: Comp. Math. Appl. 68.11 (2014), pp. 1581 –1604.

◮

L.F. Demkowicz and N. Heuer. ‘‘Robust DPG Method for Convection-Dominated Diffusion Problems’’. In: SIAM J. Numer. Anal. 51.5 (2013), pp. 1514–2537.

◮

N. Roberts, T. Bui-Thanh, and L. Demkowicz. ‘‘The DPG method for the Stokes problem’’. In:
Comp. Math. Appl. 67.4 (2014), pp. 966 –995.
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SLIDE 97

Scaling Issues

Multigrid and Convection-Diffusion

Convection-diffusion,

ǫ = 10−2, 64 × 64 mesh

Conjugate gradient Geometric multigrid preconditioner Multiplicative V-cycle Overlapping additive Schwarz smoother Hierarchy of p−coarsening followed by h−coarsening

500 1000 1500 2000 iteration 10

5

10

4

10

3

10

2

10

1

10 residual

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SLIDE 98

Scaling Issues

Incompressible Flow Over a Cylinder, Initial Mesh

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SLIDE 99

Scaling Issues

Incompressible Flow Over a Cylinder, 4 Refinements

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SLIDE 100

Scaling Issues

Solve Times and Strong Scaling

Transient Flow Over a Cylinder

1 Node 4 Nodes 32 Nodes Ref Elems DOFs Time Time Scaling vs 1 Time Scaling vs 4 80 31,304 1,772 453 3.91 451 1.01 1 605 225,908 8,190 3,574 2.29 717 4.98 2 3,013 1,081,598 32,008 12,076 2.65 2,648 4.56 3 9,726 3,429,384 28,744 6,319 4.54 4 11,742 4,144,674 8,510

Computations on Lonestar, 1 node = 24 processors 32,008 seconds = 8.8 hours 28,744 seconds = 8.0 hours 8,510 seconds = 2.4 hours

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SLIDE 101

Scaling Issues

Solve Times and Strong Scaling

Taylor-Green Vortex

1 Node 4 Nodes Ref Elems DOFs Time Time Scaling vs 1 60 21,302 331 140 2.35 1 312 108,410 945 290 3.25 2 2,020 691,834 4,880 1,363 3.58 3 9,244 3,043,024 6,171 Computations on Lonestar, 1 node = 24 processors 4,880 seconds = 1.4 hours 6,171 seconds = 1.7 hours

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SLIDE 102

Scaling Issues

Space-Time Slabs

Assumptions: The maximum required spatial resolution is much finer than the required temporal resolution. Regions requiring high spatial resolution are concentrated in relatively compact parts of the domain. Only isotropic refinements are permitted. The number of time slabs is a power of 2. Test problem: convection diffusion with exact solution u = 1 − e

x

ǫ

n space-time domain [−1, 0] × [0, 1].
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SLIDE 103

Scaling Issues

Space-Time Slabs

t x

Single Slab Strategy

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SLIDE 104

Scaling Issues

Space-Time Slabs

t x

Naive Time Slab Strategy

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SLIDE 105

Scaling Issues

Space-Time Slabs

t x

Smarter Time Slab Strategy

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SLIDE 106

Scaling Issues

Space-Time Slabs

Number of time slabs = 2k.

2 4 6 8 10 k 10 10

1

10

2

10

3

Element Count Ratio

Ratio of total element counts

10 10

1

10

2

10

3

10

4

Number of Space-Time Slabs 10

2

10

3

10

4

10

5

Total Solve Time

Total solve time using smart time slabs Without anisotropic refinements, time slabs don’t significantly speed up computations. Time slabs could be useful for memory constrained problems.

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SLIDE 107

Camellia: DPG for the Masses

Overview

Design Goal

Make DPG research and experimentation as simple as possible, while maintaining computational efficiency and scalability. Built on Trilinos (Teuchos, Intrepid, Shards, Epetra, etc). Mature support for: Rapid specification of DPG variational forms, inner products, etc. Distributed computation of stiffness matrix 1D - 3D geometries Curvilinear elements h- and p-refinements (anisotropic in h) Experimental support for: Space-time computations Iterative solvers (tested up to 32,768 cores)

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SLIDE 108

Convection-Diffusion in Three Slides

Building the Bilinear Form

VarFactory vf; //fields: VarPtr u = vf.fieldVar("u", L2); VarPtr sigma = vf.fieldVar("sigma", VECTOR_L2); // traces: VarPtr u_hat = vf.traceVar("u_hat"); VarPtr t_n = vf.fluxVar("t_n"); // test: VarPtr v = vf.testVar("v", HGRAD); VarPtr tau = vf.testVar("tau", HDIV); double eps = .01; FunctionPtr beta_x = Function::constant(1); FunctionPtr beta_y = Function::constant(2); FunctionPtr beta = Function::vectorize(beta_x, beta_y); BFPtr bf = Teuchos::rcp( new BF(vf) ); bf->addTerm((1/eps) * sigma, tau); bf->addTerm(u, tau->div()); bf->addTerm(-u_hat, tau->dot_normal()); bf->addTerm(sigma - beta * u, v->grad()); bf->addTerm(t_n, v); RHSPtr rhs = RHS::rhs();

Find u ∈ L2(Ωh), σ ∈ L2(Ωh),

ˆ

u ∈ H

1 2 (Γh), ˆ

tn ∈ H− 1

2 (Γh)

such that 1

ǫ (σ, τ) + (u, ∇ · τ) − ˆ

u, τ · n

− (βu − σ, ∇v) + ˆ

tn, v = (f , v) for all v ∈ H1(K), τ ∈ H(div, K), where ǫ = 10−2, β = (1, 2)T and f = 0.

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SLIDE 109

Convection-Diffusion in Three Slides

Boundary Conditions and Mesh

int k = 2; int delta_k = 2; MeshPtr mesh = MeshFactory::quadMesh(bf, k+1, delta_k); BCPtr bc = BC::bc(); SpatialFilterPtr y_equals_one = SpatialFilter::matchingY(1.0); SpatialFilterPtr y_equals_zero = SpatialFilter::matchingY(0); SpatialFilterPtr x_equals_one = SpatialFilter::matchingX(1.0); SpatialFilterPtr x_equals_zero = SpatialFilter::matchingX(0.0); FunctionPtr zero = Function::zero(); FunctionPtr x = Function::xn(1); FunctionPtr y = Function::yn(1); bc->addDirichlet(t_n, y_equals_zero, -2 * (1-x)); bc->addDirichlet(t_n, x_equals_zero, -1 * (1-y)); bc->addDirichlet(u_hat, y_equals_one, zero); bc->addDirichlet(u_hat, x_equals_one, zero);

Create a square mesh

[0, 1] × [0, 1] with boundary

conditions

ˆ

tn = 2x − 2 on y = 0

ˆ

tn = x − 1 on x = 0

ˆ

u = 0 on y = 1

ˆ

u = 0 on x = 1 Note Can subclass SpatialFilter to match any geometry Adding new mesh readers is straightforward

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SLIDE 110

Convection-Diffusion in Three Slides

Test Norm, Solving, and Adaptivity

IPPtr ip = bf->graphNorm(); SolutionPtr soln = Solution::solution(mesh, bc, rhs, ip); double threshold = 0.20; RefinementStrategy refStrategy(soln, threshold); int numRefs = 10;

stringstream refName;

refName << "ConvectionDiffusion"; HDF5Exporter exporter(mesh,refName.str()); for (int refIndex=0; refIndex < numRefs; refIndex++) { soln->solve(); double energyError = soln->energyErrorTotal(); cout << "After " << refIndex << " refinements, energy error is " << energyError << endl; exporter.exportSolution(soln, vf, refIndex); if (refIndex != numRefs) refStrategy.refine(); }

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SLIDE 111

Convection-Diffusion in Three Slides

Computed Solution

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Space-Time Discontinuous Petrov-Galerkin Finite Elements for Transient Fluid Mechanics

Truman Ellisa, Leszek Demkowiczb, Jesse Chanc, Nate Robertsa and Robert Moserb

a Sandia Laboratories b ICES, University of Texas at Austin c Dept. of Math., Rice University

Workshop on Space-Time Methods for PDEs RICAM, Linz, Nov. 7-11, 2016

Table of Contents

1

Motivation: Automating Scientific Computing

2

DPG: A Framework for Computational Mechanics

3

Locally Conservative DPG

4

Space-Time Convection-Diffusion

5

Space-Time Incompressible Navier-Stokes

6

Space-Time Compressible Navier-Stokes

Table of Contents

1

Motivation: Automating Scientific Computing

2

DPG: A Framework for Computational Mechanics

3

Locally Conservative DPG

4

Space-Time Convection-Diffusion

5

Space-Time Incompressible Navier-Stokes

6

Space-Time Compressible Navier-Stokes

Navier-Stokes Equations

Numerical Challenges

Robust simulation of unsteady fluid dynamics remains a challenging issue. Resolving solution features (sharp, localized viscous-scale phenomena)

Shocks Boundary layers - resolution needed for drag/load Turbulence (non-localized)

Stability of numerical schemes

Nonlinearity Nature of PDE changes for different flow regimes Coarse/adaptive grids Higher order

Shock Boundary layer

Motivation

Initial Mesh Design is Expensive and Time-Consuming

Surface mesh must accurately represent geometry Volume mesh needs sufficient resolution for asymptotic regime Engineers often forced to work by trial and error Bad in the context of HPC

Formula 1 Mesh by Numeca

Motivation

Initial Mesh Design is Expensive and Time-Consuming

Surface mesh must accurately represent geometry Volume mesh needs sufficient resolution for asymptotic regime Engineers often forced to work by trial and error Bad in the context of HPC We desire an automated computational technology

Formula 1 Mesh by Numeca

DPG on Coarse Meshes

Adaptive Solve of the Carter Plate Problem1 Re = 1000

Temperature on Initial Mesh Temperature after 8 Refinements Temperature after 4 Refinements Temperature after 11 Refinements

Lessons from Other Methods

spatially stable.

Table of Contents

1

Motivation: Automating Scientific Computing

2

DPG: A Framework for Computational Mechanics

3

Locally Conservative DPG

4

Space-Time Convection-Diffusion

5

Space-Time Incompressible Navier-Stokes

6

Space-Time Compressible Navier-Stokes

Overview of DPG

DPG is a Minimum Residual Method

Find u ∈ U such that b(u, v) = l(v)

∀v ∈ V

with operator B : U → V ′ defined by b(u, v) = Bu, vV ′×V. This gives the operator equation Bu = l

∈ V ′ .

We wish to minimize the residual Bu − l ∈ V ′: uh = arg min

wh∈Uh

1 2 Bwh − l2

V ′ .

Dual norms are not computationally tractable. Inverse Riesz map moves the residual to a more accessible space: uh = arg min

wh∈Uh

1 2

V (Bwh − l)

V .

Overview of DPG

Petrov-Galerkin with Optimal Test Functions