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The future of computational steering:
Interactive Design-through-Analysis
Matthias Möller1
, Casper van Leeuwen2
1Department of Applied Mathematics, TU Delft
2Scientific Visualisation, HPCV, SURF
SURF Research Day 2023, Amersfoort
Joint work with Deepesh Toshniwal, Frank van Ruiten (TU Delft),
Paul Melis (SURF), and Jaewook Lee (TU Vienna)
1 / 32
Computational steering
What do you know about it?
2 / 32
Computational steering – Dutch roots
execute simulations :and
speeds. High
are needed to interactive-
ofdata produced by HPC
e speeds cannot be ob-
merits of computational
is an attractive concept,
s cumbersome and time
r must cooperate with
aces and visualization to
lysis of the output of the
is ready, aftersome weeks
gh that the interests ofthe
so, further analysis of the
esearch questions, which
e tool. The close cooper-
r and the visualization
period is required. More
rovide an environment in
lves can build interfaces
simulation. This would
nd efficient model- simu-
nvironment for computa-
SE provides a collection of
d tools that enable re-
tational steering. The for-
llows: First, a number of
elieve are fundamental for
re given. We then present
CSE's architecture and the
ualization of and interac-
Researoher
l
graphicel cHrect visuaJlza.
input manipulation
Computatiooal Steering Environment
state
perameters
, variables
results
Simulation
must be provided that allow for two-waycommuni-
cation: both input and output. Itmust be possibleto
select and drag visualization objects, thereby direc-
tly controlling parameters and statevariables ofthe
simulation.
The simulation receives from the CSE new par-
ameter values, and returns newly calculated results
to the CSE. We assume that the simulation can
handle changes of parameters on the fly, and that it
can provide meaningful intermediate results within
a time-interval that is acceptable to the researcher.
The process ofachieving insight via simulation is
an incremental one. The researcher must be able to
create and refine the interlace to the simulation
COMPUTATIONAL STEERING AND MEASURING IN VIRTUAL REALITY 85
Figure 9. Diode laser simulation.
computes some fixed points in the phase space for a given
set of parameters. The user can interactively set the values
of selected parameters using sliders. The fixed points serve
as starting point of the simulation. These points are visual-
ized, and the scientist can directly select one of these points
Figure 10. Interactive molecular dynamics.
integration methods, force field parameters, and restart ca-
pabilities. VMD [9] is a visualization environment for struc-
tural biology which has been widely used for plain visual-
ization, docking studies, structure refinement or trajectory
analysis. An important feature is that it is possible to cou-
ple a NAMD running simulation to VMD to study trajecto-
ries of molecules. Moreover, some local forces can be ap-
plied using either a 2D graphical interface or a 3D haptic
feedback tool [9]. In [13], the authors describe how they
Left: R. v. Liere, J.D. Mulder, and J.J. v. Wijk, CWI 1997. Right: L.Renambot, H.E. Bal, D. Germans, and H.J.W. Spoelder, VU 2001.
3 / 32
Computational steering – the concept
The EPSN project by LaBRI (https://www.labri.fr/projet/epsn/) – 2007
4 / 32
Computational steering – the concept
The EPSN project by LaBRI (https://www.labri.fr/projet/epsn/) – 2007
4 / 32
Computational steering – the early 2010s
Opening of the CAVE at HLRS High Performance Computing Center Stuttgart in 2012 (https://www.stuttgarter-zeitung.de/inhalt.forschung-in-stuttgart-ein-
wuerfel-fuer-die-virtuelle-zukunft.1272d909-f2f8-4a53-8ad4-2c1712ab8842.html)
5 / 32
Computational steering – today
CAVE at HLRS High Performance Computing Center Stuttgart today (https://www.hlrs.de/solutions/systems/cave)
6 / 32
Computational steering – today
EU-funded project VITV – 2018-2021 (https://www.b-tu.de/fg-medientechnik/forschung/virtuelles-triebwerk-v)
7 / 32
Computational steering – the future
Siemens blog: Virtual Reality in Engineering - Are You Ready? – 7 July 2021 (https://blogs.sw.siemens.com/teamcenter/virtual-reality-in-engineering-are-you-
ready/)
8 / 32
Computational steering – the future
Microsoft blog: The future of mobility is now: Five themes to watch at CES 2023 – 4 January 2023 (https://blogs.microsoft.com/blog/2023/01/04/the-
future-of-mobility-is-now-five-themes-to-watch-at-ces-2023/)
9 / 32
Computational steering – the future
That is, combining Computer-Aided Design and
Computer-Aided Engineering Analysis to a
unified Design-through-Analysis workflow.
10 / 32
Design-through-Analysis
What do you know about it?
11 / 32
Design-through-Analysis – the inception
J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs, Wiley 2009
12 / 32
Design-through-Analysis – further back in time to the 1970s
EXPERIMENTAL AUTOMOBILE STRUCTURE 2 1 93
* GRID POINT
• ASET GRID POINT
Fig. 14 - Beam elements on the CVS analytical
model
points in Fig. 14, while usually only one
normal (to plate) displacement was kept on the
plate grid points. This resulted in a total
of 193 degrees of freedom for the symmetric
case and 174 for the antisymmetric case (the
difference is due to the grid points in the
centerline which has one degree of freedom less
major modes, and one would generally require a
much more detailed model in order to obtain
reasonable values for local modes. The fre-
quencies given for all the local modes should
only be considered as an indication that local
flexibilities are low, and that panel vibration
or local modes might occur in the location
J.A. Augustitus, M.M. Kamal, and L.J. Howell. Design through analysis of an experimental automobile structure. SAE Transactions, 86:2186-2198, 1977
13 / 32
Design-through-Analysis – further back in time to the 1970s
“The project described herein (which was completed early in 1975) is thought
to have been the first coordinated design through analysis of an entire
automobile, followed by construction and experimental verification.”
EXPERIMENTAL AUTOMOBILE STRUCTURE 2 1 93
* GRID POINT
• ASET GRID POINT
Fig. 14 - Beam elements on the CVS analytical
model
points in Fig. 14, while usually only one
normal (to plate) displacement was kept on the
plate grid points. This resulted in a total
of 193 degrees of freedom for the symmetric
case and 174 for the antisymmetric case (the
difference is due to the grid points in the
centerline which has one degree of freedom less
major modes, and one would generally require a
much more detailed model in order to obtain
reasonable values for local modes. The fre-
quencies given for all the local modes should
only be considered as an indication that local
flexibilities are low, and that panel vibration
or local modes might occur in the location
J.A. Augustitus, M.M. Kamal, and L.J. Howell. Design through analysis of an experimental automobile structure. SAE Transactions, 86:2186-2198, 1977
13 / 32
Design-through-Analysis – further back in time to the 1970s
“The project described herein (which was completed early in 1975) is thought
to have been the first coordinated design through analysis of an entire
automobile, followed by construction and experimental verification.”
EXPERIMENTAL AUTOMOBILE STRUCTURE 2 1 93
* GRID POINT
• ASET GRID POINT
Fig. 14 - Beam elements on the CVS analytical
model
points in Fig. 14, while usually only one
normal (to plate) displacement was kept on the
plate grid points. This resulted in a total
of 193 degrees of freedom for the symmetric
case and 174 for the antisymmetric case (the
difference is due to the grid points in the
centerline which has one degree of freedom less
major modes, and one would generally require a
much more detailed model in order to obtain
reasonable values for local modes. The fre-
quencies given for all the local modes should
only be considered as an indication that local
flexibilities are low, and that panel vibration
or local modes might occur in the location
“[...] the potential value of design through analysis was demonstrated
by a significant reduction in structural weight of the project vehicle.”
J.A. Augustitus, M.M. Kamal, and L.J. Howell. Design through analysis of an experimental automobile structure. SAE Transactions, 86:2186-2198, 1977
13 / 32
Computational steering: Interactive Design-through-Analysis
Vision: unified computational framework for rapid prototyping (design exploration phase)
and thorough analysis (design optimization phase) of engineering designs
Ingredients
• physics-informed machine learning for rapid prototyping
• isogeometric analysis for accurate analysis
14 / 32
Computational steering: Interactive Design-through-Analysis
Vision: unified computational framework for rapid prototyping (design exploration phase)
and thorough analysis (design optimization phase) of engineering designs
Ingredients
• physics-informed machine learning for rapid prototyping
• isogeometric analysis for accurate analysis
Let’s see a live demo
14 / 32
Let’s have a look under the hood
15 / 32
The big picture
Front-ends
IgANet-frontend
by SURF by TU Vienna
WebSockets protocol for interactive Design-through-Analysis
Back-ends
16 / 32
B-spline basis functions
Cox de Boor recursion formula
knot vector Ξ = [0, 1, 2, 3, 4]
b0
i (ξ) =
(
1 if ξi ≤ ξ < ξi+1
0 otherwise
bp
i (ξ) =
ξ − ξi
ξi+p − ξi
bp−1
i (ξ)
+
ξi+p+1 − ξ
ξi+p+1 − ξi+1
bp−1
i+1 (ξ)
17 / 32
B-spline basis functions
Cox de Boor recursion formula
knot vector Ξ = [0, 1, 2, 3, 4]
b0
i (ξ) =
(
1 if ξi ≤ ξ < ξi+1
0 otherwise
bp
i (ξ) =
ξ − ξi
ξi+p − ξi
bp−1
i (ξ)
+
ξi+p+1 − ξ
ξi+p+1 − ξi+1
bp−1
i+1 (ξ)
Many good properties: compact support [ξi, ξi+p+1), positive function values over
support interval, derivatives of B-splines are combinations of lower-order B-splines, ...
17 / 32
Isogeometric Analysis
Paradigm: represent ‘everything’ in terms of tensor products of B-spline basis functions
Bi(ξ, η) := bp
i (ξ) · bq
k(η), i := (k − 1) · ni + i, 1 ≤ i ≤ ni, 1 ≤ k ≤ nk,
Fig. 1. 1D and 2D B-spline basis functions.
bp
i (ξ) bq
k(η)
Bi(ξ, η)
18 / 32
Isogeometric Analysis
Paradigm: represent ‘everything’ in terms of tensor products of B-spline basis functions
Bi(ξ, η) := bp
i (ξ) · bq
k(η), i := (k − 1) · ni + i, 1 ≤ i ≤ ni, 1 ≤ k ≤ nk,
Fig. 1. 1D and 2D B-spline basis functions.
bp
i (ξ) bq
k(η)
Bi(ξ, η)
Many more good properties: partition of unity
n
P
i=1
Bi(ξ, η) ≡ 1, Cp−1 continuity, ...
18 / 32
Isogeometric Analysis
Geometry: bijective mapping from the unit square to the physical domain Ωh ⊂ Rd
xh(ξ, η) =
n
X
i=1
Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2
=: Ω̂
• the shape of Ωh is fully specified by the
set of control points xi ∈ Rd
19 / 32
Isogeometric Analysis
Geometry: bijective mapping from the unit square to the physical domain Ωh ⊂ Rd
xh(ξ, η) =
n
X
i=1
Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2
=: Ω̂
• the shape of Ωh is fully specified by the
set of control points xi ∈ Rd
• interior control points must be chosen
such that ‘grid lines’ do not fold as this
violates the bijectivity of xh : Ω̂ → Ωh
19 / 32
Isogeometric Analysis
Geometry: bijective mapping from the unit square to the physical domain Ωh ⊂ Rd
xh(ξ, η) =
n
X
i=1
Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2
=: Ω̂
• the shape of Ωh is fully specified by the
set of control points xi ∈ Rd
• interior control points must be chosen
such that ‘grid lines’ do not fold as this
violates the bijectivity of xh : Ω̂ → Ωh
• refinement in h (knot insertion) and p
(order elevation) preserves the shape of
Ωh and can be used to generate finer
computational ‘grids’ for the analysis
19 / 32
Isogeometric Analysis
Model problem: Poisson’s equation
−∆uh = fh in Ωh, uh = gh on ∂Ωh
with
(geometry) xh(ξ, η) =
n
X
i=1
Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2
(solution) uh ◦ xh(ξ, η) =
n
X
i=1
Bi(ξ, η) · ui ∀(ξ, η) ∈ [0, 1]2
(r.h.s vector) fh ◦ xh(ξ, η) =
n
X
i=1
Bi(ξ, η) · fi ∀(ξ, η) ∈ [0, 1]2
(boundary conditions) gh ◦ xh(ξ, η) =
n
X
i=1
Bi(ξ, η) · gi ∀(ξ, η) ∈ ∂[0, 1]2
20 / 32
Isogeometric Analysis
Abstract representation
Given xi (geometry), fi (r.h.s. vector), and gi (boundary conditions), compute



u1
.
.
.
un


 = A−1






x1
.
.
.
xn


 ,



g1
.
.
.
gn





 · b






x1
.
.
.
xn


 ,



f1
.
.
.
fn


 ,



g1
.
.
.
gn






Any point of the solution can afterwards be obtained by a simple function evaluation
(ξ, η) ∈ [0, 1]2
7→ uh ◦ xh(ξ, η) = [B1(ξ, η), . . . , Bn(ξ, η)] ·



u1
.
.
.
un



21 / 32
Isogeometric Analysis
Abstract representation
Given xi (geometry), fi (r.h.s. vector), and gi (boundary conditions), compute



u1
.
.
.
un


 = A−1






x1
.
.
.
xn


 ,



g1
.
.
.
gn





 · b






x1
.
.
.
xn


 ,



f1
.
.
.
fn


 ,



g1
.
.
.
gn






Any point of the solution can afterwards be obtained by a simple function evaluation
(ξ, η) ∈ [0, 1]2
7→ uh ◦ xh(ξ, η) = [B1(ξ, η), . . . , Bn(ξ, η)] ·



u1
.
.
.
un



Let us interpret the sets of B-spline coefficients {xi}, {fi}, and {gi} as an efficient
encoding of our PDE problem that is fed into our IgA machinery as input.
The output of our IgA machinery are the B-spline coefficients {ui} of the solution.
21 / 32
Isogeometric Analysis + Physics-Informed Machine Learning
IgANet: replace computation



u1
.
.
.
un


 = A−1






x1
.
.
.
xn


 ,



g1
.
.
.
gn





 · b






x1
.
.
.
xn


 ,



f1
.
.
.
fn


 ,



g1
.
.
.
gn






22 / 32
Isogeometric Analysis + Physics-Informed Machine Learning
IgANet: replace computation by physics-informed machine learning



u1
.
.
.
un


 = IgANet






x1
.
.
.
xn


 ,



f1
.
.
.
fn


 ,



g1
.
.
.
gn


 ; (ξ(k)
, η(k)
)
Nsamples
k=1



22 / 32
Isogeometric Analysis + Physics-Informed Machine Learning
IgANet: replace computation by physics-informed machine learning



u1
.
.
.
un


 = IgANet






x1
.
.
.
xn


 ,



f1
.
.
.
fn


 ,



g1
.
.
.
gn


 ; (ξ(k)
, η(k)
)
Nsamples
k=1



Compute the solution from the trained neural network as follows
uh(ξ, η) = [B1(ξ, η), . . . , Bn(ξ, η)] ·



u1
.
.
.
un


 ,



u1
.
.
.
un


 = IgANet






x1
.
.
.
xn


 ,



f1
.
.
.
fn


 ,



g1
.
.
.
gn






22 / 32
IgANet architecture
x1
xn
f1
fn
g1
gn
σ
σ
σ
σ
σ
σ
σ
σ
σ
u1
un
loss = lossPDE + lossBDR
loss < ε end training
∂loss
∂(w, b)
→ update w, b
and continue training
geometry
r.h.s.
vector
bdr.
cond.
coords (ξ(k)
, η(k)
)N
k=1
23 / 32
Loss function
Model problem: Poisson’s equation with Dirichlet boundary conditions
lossPDE =
α
NΩ
NΩ
X
k=1
∆
h
uh ◦ xh

ξ(k)
, η(k)
i
− fh ◦ xh

ξ(k)
, η(k)
 2
lossBDR =
β
NΓ
NΓ
X
k=1
uh ◦ xh

ξ(k)
, η(k)

− gh ◦ xh

ξ(k)
, η(k)
 2
Express derivatives with respect to physical space variables using the Jacobian J, the
Hessian H and the matrix of squared first derivatives Q (Schillinger et al. 2013):





∂2B
∂x2
∂2B
∂x∂y
∂2B
∂y2





= Q−⊤










∂2B
∂ξ2
∂2B
∂ξ∂η
∂2B
∂η2





− H⊤
J−⊤


∂B
∂ξ
∂B
∂η







24 / 32
Two-level training strategy
For [x1, . . . , xn] ∈ Sgeo, [f1, . . . , fn] ∈ Srhs, [g1, . . . , gn] ∈ Sbcond do
For a batch of randomly sampled (ξk, ηk) ∈ [0, 1]2
(or the Greville abscissae) do
Train IgANet






x1
.
.
.
xn


 ,



f1
.
.
.
fn


 ,



g1
.
.
.
gn


 ; (ξk, ηk)
Nsamples
k=1


 7→



u1
.
.
.
un



EndFor
EndFor
25 / 32
Computational costs
Working principle of PINNs
x 7→ u(x) := NN(x; f, g, G) = σL(WLσ(. . . (σ1(W1x + b1))) + bL)
• use AD engine (automated chain rule) to compute derivatives, e.g., ux = NNx
• use AD engine on top of AD tree (!!!) to compute gradients w.r.t. weights for training
26 / 32
Computational costs
Working principle of PINNs
x 7→ u(x) := NN(x; f, g, G) = σL(WLσ(. . . (σ1(W1x + b1))) + bL)
• use AD engine (automated chain rule) to compute derivatives, e.g., ux = NNx
• use AD engine on top of AD tree (!!!) to compute gradients w.r.t. weights for training
Working principle of IgANets
[xi, fi, gi]i=1,...,n 7→ [ui]i=1,...,n := NN(xi, fi, gi, i = 1, . . . , n)
• use mathematics to compute derivatives, e.g., ∇xu = (
Pn
i=1 ∇ξBi(ξ)ui) J−t
G
• use AD to compute gradients w.r.t. weights for training, i.e. (illustrated in 1D)
∂(dr
ξu(ξ))
∂wk
=
n
X
i=1
∂(dr
ξbp
i ui)
∂wk
=


XXXXXXXX
X
n
X
i=1
dr+1
ξ bp
i
∂ξ
∂wk
ui +
n
X
i=1
dr
ξbp
i
∂ui
∂wk
26 / 32
Towards an ML-friendly B-spline evaluation
Major computational task (illustrated in 1D)
Given sampling point ξ ∈ [ξi, ξi+1) compute for r ≥ 0
dr
ξu(ξ) =
h
dr
ξbp
i−p(ξ), . . . , dr
ξbp
i (ξ)
i
· [ui−p, . . . , ui]
| {z }
network’s output
Textbook derivatives
dr
ξbp
i (ξ) = (p − 1)


−dr−1
ξ bp−1
i+1 (ξ)
ξi+p − ξi+1
+
dr−1
ξ bp−1
i (ξ)
ξi+p−1 − ξi


with
bp
i (ξ) =
ξ − ξi
ξi+p − ξi
bp−1
i (ξ) +
ξi+p+1 − ξ
ξi+p+1 − ξi+1
bp−1
i+1 (ξ), b0
i (ξ) =
(
1 if ξi ≤ ξ  ξi+1
0 otherwise
27 / 32
An ML-friendly B-spline evaluation
Algorithm 2.22 from (Lyche and Morken 2011) with slight modifications
1 b = 1
2 For k = 1, . . . , p − r
1 t1 = (ξi−k+1, . . . , ξi)
2 t21 = (ξi+1, . . . , ξi+k) − t1
3 mask = (t21  tol)
4 w = (ξ − t1−mask) ÷ (t21−mask)
5 b = [(1 − w) ⊙ b, 0] + [0, w ⊙ b]
3 For k = p − r + 1, . . . , p
1 t1 = (ξi−k+1, . . . , ξi)
2 t21 = (ξi+1, . . . , ξi+k) − t1
3 mask = (t21  tol)
4 w = (1−mask) ÷ (t21−mask)
5 b = [−w ⊙ b, 0] + [0, w ⊙ b]
where ÷ and ⊙ denote the element-wise division and multiplication of vectors, respectively.
28 / 32
Performance evaluation - bivariate B-splines
re
f
1
e
0
1
e
1
1
e
2
1
e
3
1
e
4
2
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2
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2
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4
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4
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2
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1
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6
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f
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5
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1
e
6
10−1
100
101
102
103
104
105
106
Wallclock
time
in
ns/entry
Tesla V100S PCIe 32G AMD EPYC 7402 24-Core Processor reference
p = 1 p = 2 p = 3 p = 4 p = 5
29 / 32
Performance evaluation - bivariate B-splines
re
f
1
e
0
1
e
1
1
e
2
1
e
3
1
e
4
2
.5
e
4
5
e
4
1
e
5
2
.5
e
5
5
e
5
1
e
6
re
f
1
e
0
1
e
1
1
e
2
1
e
3
1
e
4
2
.5
e
4
5
e
4
1
e
5
2
.5
e
5
5
e
5
1
e
6
re
f
1
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0
1
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1
1
e
2
1
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3
1
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4
2
.5
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4
5
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4
1
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5
2
.5
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5
5
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5
1
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6
re
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1
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1
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2
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3
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5
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5
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re
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1
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2
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.5
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5
5
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5
1
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6
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100
101
102
103
104
105
106
Wallclock
time
in
ns/entry
Fujitsu A64FX 48-Core Processor AMD EPYC 7402 24-Core Processor reference
p = 1 p = 2 p = 3 p = 4 p = 5
30 / 32
Let’s move on to the front-end
31 / 32
The future of computational steering:
Interactive Design-through-Analysis
Matthias Möller1
, Casper van Leeuwen2
1Department of Applied Mathematics, TU Delft
2Scientific Visualisation, HPCV, SURF
SURF Research Day 2023, Amersfoort
Joint work with Deepesh Toshniwal, Frank van Ruiten (TU Delft),
Paul Melis (SURF), and Jaewook Lee (TU Vienna)
Thank you very much!
32 / 32

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Computational steering Interactive Design-through-Analysis for Simulation Sciences - Matthijs Möller & Casper van Leeuwen - SRD23

  • 1. The future of computational steering: Interactive Design-through-Analysis Matthias Möller1 , Casper van Leeuwen2 1Department of Applied Mathematics, TU Delft 2Scientific Visualisation, HPCV, SURF SURF Research Day 2023, Amersfoort Joint work with Deepesh Toshniwal, Frank van Ruiten (TU Delft), Paul Melis (SURF), and Jaewook Lee (TU Vienna) 1 / 32
  • 2. Computational steering What do you know about it? 2 / 32
  • 3. Computational steering – Dutch roots execute simulations :and speeds. High are needed to interactive- ofdata produced by HPC e speeds cannot be ob- merits of computational is an attractive concept, s cumbersome and time r must cooperate with aces and visualization to lysis of the output of the is ready, aftersome weeks gh that the interests ofthe so, further analysis of the esearch questions, which e tool. The close cooper- r and the visualization period is required. More rovide an environment in lves can build interfaces simulation. This would nd efficient model- simu- nvironment for computa- SE provides a collection of d tools that enable re- tational steering. The for- llows: First, a number of elieve are fundamental for re given. We then present CSE's architecture and the ualization of and interac- Researoher l graphicel cHrect visuaJlza. input manipulation Computatiooal Steering Environment state perameters , variables results Simulation must be provided that allow for two-waycommuni- cation: both input and output. Itmust be possibleto select and drag visualization objects, thereby direc- tly controlling parameters and statevariables ofthe simulation. The simulation receives from the CSE new par- ameter values, and returns newly calculated results to the CSE. We assume that the simulation can handle changes of parameters on the fly, and that it can provide meaningful intermediate results within a time-interval that is acceptable to the researcher. The process ofachieving insight via simulation is an incremental one. The researcher must be able to create and refine the interlace to the simulation COMPUTATIONAL STEERING AND MEASURING IN VIRTUAL REALITY 85 Figure 9. Diode laser simulation. computes some fixed points in the phase space for a given set of parameters. The user can interactively set the values of selected parameters using sliders. The fixed points serve as starting point of the simulation. These points are visual- ized, and the scientist can directly select one of these points Figure 10. Interactive molecular dynamics. integration methods, force field parameters, and restart ca- pabilities. VMD [9] is a visualization environment for struc- tural biology which has been widely used for plain visual- ization, docking studies, structure refinement or trajectory analysis. An important feature is that it is possible to cou- ple a NAMD running simulation to VMD to study trajecto- ries of molecules. Moreover, some local forces can be ap- plied using either a 2D graphical interface or a 3D haptic feedback tool [9]. In [13], the authors describe how they Left: R. v. Liere, J.D. Mulder, and J.J. v. Wijk, CWI 1997. Right: L.Renambot, H.E. Bal, D. Germans, and H.J.W. Spoelder, VU 2001. 3 / 32
  • 4. Computational steering – the concept The EPSN project by LaBRI (https://www.labri.fr/projet/epsn/) – 2007 4 / 32
  • 5. Computational steering – the concept The EPSN project by LaBRI (https://www.labri.fr/projet/epsn/) – 2007 4 / 32
  • 6. Computational steering – the early 2010s Opening of the CAVE at HLRS High Performance Computing Center Stuttgart in 2012 (https://www.stuttgarter-zeitung.de/inhalt.forschung-in-stuttgart-ein- wuerfel-fuer-die-virtuelle-zukunft.1272d909-f2f8-4a53-8ad4-2c1712ab8842.html) 5 / 32
  • 7. Computational steering – today CAVE at HLRS High Performance Computing Center Stuttgart today (https://www.hlrs.de/solutions/systems/cave) 6 / 32
  • 8. Computational steering – today EU-funded project VITV – 2018-2021 (https://www.b-tu.de/fg-medientechnik/forschung/virtuelles-triebwerk-v) 7 / 32
  • 9. Computational steering – the future Siemens blog: Virtual Reality in Engineering - Are You Ready? – 7 July 2021 (https://blogs.sw.siemens.com/teamcenter/virtual-reality-in-engineering-are-you- ready/) 8 / 32
  • 10. Computational steering – the future Microsoft blog: The future of mobility is now: Five themes to watch at CES 2023 – 4 January 2023 (https://blogs.microsoft.com/blog/2023/01/04/the- future-of-mobility-is-now-five-themes-to-watch-at-ces-2023/) 9 / 32
  • 11. Computational steering – the future That is, combining Computer-Aided Design and Computer-Aided Engineering Analysis to a unified Design-through-Analysis workflow. 10 / 32
  • 12. Design-through-Analysis What do you know about it? 11 / 32
  • 13. Design-through-Analysis – the inception J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs, Wiley 2009 12 / 32
  • 14. Design-through-Analysis – further back in time to the 1970s EXPERIMENTAL AUTOMOBILE STRUCTURE 2 1 93 * GRID POINT • ASET GRID POINT Fig. 14 - Beam elements on the CVS analytical model points in Fig. 14, while usually only one normal (to plate) displacement was kept on the plate grid points. This resulted in a total of 193 degrees of freedom for the symmetric case and 174 for the antisymmetric case (the difference is due to the grid points in the centerline which has one degree of freedom less major modes, and one would generally require a much more detailed model in order to obtain reasonable values for local modes. The fre- quencies given for all the local modes should only be considered as an indication that local flexibilities are low, and that panel vibration or local modes might occur in the location J.A. Augustitus, M.M. Kamal, and L.J. Howell. Design through analysis of an experimental automobile structure. SAE Transactions, 86:2186-2198, 1977 13 / 32
  • 15. Design-through-Analysis – further back in time to the 1970s “The project described herein (which was completed early in 1975) is thought to have been the first coordinated design through analysis of an entire automobile, followed by construction and experimental verification.” EXPERIMENTAL AUTOMOBILE STRUCTURE 2 1 93 * GRID POINT • ASET GRID POINT Fig. 14 - Beam elements on the CVS analytical model points in Fig. 14, while usually only one normal (to plate) displacement was kept on the plate grid points. This resulted in a total of 193 degrees of freedom for the symmetric case and 174 for the antisymmetric case (the difference is due to the grid points in the centerline which has one degree of freedom less major modes, and one would generally require a much more detailed model in order to obtain reasonable values for local modes. The fre- quencies given for all the local modes should only be considered as an indication that local flexibilities are low, and that panel vibration or local modes might occur in the location J.A. Augustitus, M.M. Kamal, and L.J. Howell. Design through analysis of an experimental automobile structure. SAE Transactions, 86:2186-2198, 1977 13 / 32
  • 16. Design-through-Analysis – further back in time to the 1970s “The project described herein (which was completed early in 1975) is thought to have been the first coordinated design through analysis of an entire automobile, followed by construction and experimental verification.” EXPERIMENTAL AUTOMOBILE STRUCTURE 2 1 93 * GRID POINT • ASET GRID POINT Fig. 14 - Beam elements on the CVS analytical model points in Fig. 14, while usually only one normal (to plate) displacement was kept on the plate grid points. This resulted in a total of 193 degrees of freedom for the symmetric case and 174 for the antisymmetric case (the difference is due to the grid points in the centerline which has one degree of freedom less major modes, and one would generally require a much more detailed model in order to obtain reasonable values for local modes. The fre- quencies given for all the local modes should only be considered as an indication that local flexibilities are low, and that panel vibration or local modes might occur in the location “[...] the potential value of design through analysis was demonstrated by a significant reduction in structural weight of the project vehicle.” J.A. Augustitus, M.M. Kamal, and L.J. Howell. Design through analysis of an experimental automobile structure. SAE Transactions, 86:2186-2198, 1977 13 / 32
  • 17. Computational steering: Interactive Design-through-Analysis Vision: unified computational framework for rapid prototyping (design exploration phase) and thorough analysis (design optimization phase) of engineering designs Ingredients • physics-informed machine learning for rapid prototyping • isogeometric analysis for accurate analysis 14 / 32
  • 18. Computational steering: Interactive Design-through-Analysis Vision: unified computational framework for rapid prototyping (design exploration phase) and thorough analysis (design optimization phase) of engineering designs Ingredients • physics-informed machine learning for rapid prototyping • isogeometric analysis for accurate analysis Let’s see a live demo 14 / 32
  • 19. Let’s have a look under the hood 15 / 32
  • 20. The big picture Front-ends IgANet-frontend by SURF by TU Vienna WebSockets protocol for interactive Design-through-Analysis Back-ends 16 / 32
  • 21. B-spline basis functions Cox de Boor recursion formula knot vector Ξ = [0, 1, 2, 3, 4] b0 i (ξ) = ( 1 if ξi ≤ ξ < ξi+1 0 otherwise bp i (ξ) = ξ − ξi ξi+p − ξi bp−1 i (ξ) + ξi+p+1 − ξ ξi+p+1 − ξi+1 bp−1 i+1 (ξ) 17 / 32
  • 22. B-spline basis functions Cox de Boor recursion formula knot vector Ξ = [0, 1, 2, 3, 4] b0 i (ξ) = ( 1 if ξi ≤ ξ < ξi+1 0 otherwise bp i (ξ) = ξ − ξi ξi+p − ξi bp−1 i (ξ) + ξi+p+1 − ξ ξi+p+1 − ξi+1 bp−1 i+1 (ξ) Many good properties: compact support [ξi, ξi+p+1), positive function values over support interval, derivatives of B-splines are combinations of lower-order B-splines, ... 17 / 32
  • 23. Isogeometric Analysis Paradigm: represent ‘everything’ in terms of tensor products of B-spline basis functions Bi(ξ, η) := bp i (ξ) · bq k(η), i := (k − 1) · ni + i, 1 ≤ i ≤ ni, 1 ≤ k ≤ nk, Fig. 1. 1D and 2D B-spline basis functions. bp i (ξ) bq k(η) Bi(ξ, η) 18 / 32
  • 24. Isogeometric Analysis Paradigm: represent ‘everything’ in terms of tensor products of B-spline basis functions Bi(ξ, η) := bp i (ξ) · bq k(η), i := (k − 1) · ni + i, 1 ≤ i ≤ ni, 1 ≤ k ≤ nk, Fig. 1. 1D and 2D B-spline basis functions. bp i (ξ) bq k(η) Bi(ξ, η) Many more good properties: partition of unity n P i=1 Bi(ξ, η) ≡ 1, Cp−1 continuity, ... 18 / 32
  • 25. Isogeometric Analysis Geometry: bijective mapping from the unit square to the physical domain Ωh ⊂ Rd xh(ξ, η) = n X i=1 Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2 =: Ω̂ • the shape of Ωh is fully specified by the set of control points xi ∈ Rd 19 / 32
  • 26. Isogeometric Analysis Geometry: bijective mapping from the unit square to the physical domain Ωh ⊂ Rd xh(ξ, η) = n X i=1 Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2 =: Ω̂ • the shape of Ωh is fully specified by the set of control points xi ∈ Rd • interior control points must be chosen such that ‘grid lines’ do not fold as this violates the bijectivity of xh : Ω̂ → Ωh 19 / 32
  • 27. Isogeometric Analysis Geometry: bijective mapping from the unit square to the physical domain Ωh ⊂ Rd xh(ξ, η) = n X i=1 Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2 =: Ω̂ • the shape of Ωh is fully specified by the set of control points xi ∈ Rd • interior control points must be chosen such that ‘grid lines’ do not fold as this violates the bijectivity of xh : Ω̂ → Ωh • refinement in h (knot insertion) and p (order elevation) preserves the shape of Ωh and can be used to generate finer computational ‘grids’ for the analysis 19 / 32
  • 28. Isogeometric Analysis Model problem: Poisson’s equation −∆uh = fh in Ωh, uh = gh on ∂Ωh with (geometry) xh(ξ, η) = n X i=1 Bi(ξ, η) · xi ∀(ξ, η) ∈ [0, 1]2 (solution) uh ◦ xh(ξ, η) = n X i=1 Bi(ξ, η) · ui ∀(ξ, η) ∈ [0, 1]2 (r.h.s vector) fh ◦ xh(ξ, η) = n X i=1 Bi(ξ, η) · fi ∀(ξ, η) ∈ [0, 1]2 (boundary conditions) gh ◦ xh(ξ, η) = n X i=1 Bi(ξ, η) · gi ∀(ξ, η) ∈ ∂[0, 1]2 20 / 32
  • 29. Isogeometric Analysis Abstract representation Given xi (geometry), fi (r.h.s. vector), and gi (boundary conditions), compute    u1 . . . un    = A−1       x1 . . . xn    ,    g1 . . . gn       · b       x1 . . . xn    ,    f1 . . . fn    ,    g1 . . . gn       Any point of the solution can afterwards be obtained by a simple function evaluation (ξ, η) ∈ [0, 1]2 7→ uh ◦ xh(ξ, η) = [B1(ξ, η), . . . , Bn(ξ, η)] ·    u1 . . . un    21 / 32
  • 30. Isogeometric Analysis Abstract representation Given xi (geometry), fi (r.h.s. vector), and gi (boundary conditions), compute    u1 . . . un    = A−1       x1 . . . xn    ,    g1 . . . gn       · b       x1 . . . xn    ,    f1 . . . fn    ,    g1 . . . gn       Any point of the solution can afterwards be obtained by a simple function evaluation (ξ, η) ∈ [0, 1]2 7→ uh ◦ xh(ξ, η) = [B1(ξ, η), . . . , Bn(ξ, η)] ·    u1 . . . un    Let us interpret the sets of B-spline coefficients {xi}, {fi}, and {gi} as an efficient encoding of our PDE problem that is fed into our IgA machinery as input. The output of our IgA machinery are the B-spline coefficients {ui} of the solution. 21 / 32
  • 31. Isogeometric Analysis + Physics-Informed Machine Learning IgANet: replace computation    u1 . . . un    = A−1       x1 . . . xn    ,    g1 . . . gn       · b       x1 . . . xn    ,    f1 . . . fn    ,    g1 . . . gn       22 / 32
  • 32. Isogeometric Analysis + Physics-Informed Machine Learning IgANet: replace computation by physics-informed machine learning    u1 . . . un    = IgANet       x1 . . . xn    ,    f1 . . . fn    ,    g1 . . . gn    ; (ξ(k) , η(k) ) Nsamples k=1    22 / 32
  • 33. Isogeometric Analysis + Physics-Informed Machine Learning IgANet: replace computation by physics-informed machine learning    u1 . . . un    = IgANet       x1 . . . xn    ,    f1 . . . fn    ,    g1 . . . gn    ; (ξ(k) , η(k) ) Nsamples k=1    Compute the solution from the trained neural network as follows uh(ξ, η) = [B1(ξ, η), . . . , Bn(ξ, η)] ·    u1 . . . un    ,    u1 . . . un    = IgANet       x1 . . . xn    ,    f1 . . . fn    ,    g1 . . . gn       22 / 32
  • 34. IgANet architecture x1 xn f1 fn g1 gn σ σ σ σ σ σ σ σ σ u1 un loss = lossPDE + lossBDR loss < ε end training ∂loss ∂(w, b) → update w, b and continue training geometry r.h.s. vector bdr. cond. coords (ξ(k) , η(k) )N k=1 23 / 32
  • 35. Loss function Model problem: Poisson’s equation with Dirichlet boundary conditions lossPDE = α NΩ NΩ X k=1 ∆ h uh ◦ xh ξ(k) , η(k) i − fh ◦ xh ξ(k) , η(k) 2 lossBDR = β NΓ NΓ X k=1 uh ◦ xh ξ(k) , η(k) − gh ◦ xh ξ(k) , η(k) 2 Express derivatives with respect to physical space variables using the Jacobian J, the Hessian H and the matrix of squared first derivatives Q (Schillinger et al. 2013):      ∂2B ∂x2 ∂2B ∂x∂y ∂2B ∂y2      = Q−⊤           ∂2B ∂ξ2 ∂2B ∂ξ∂η ∂2B ∂η2      − H⊤ J−⊤   ∂B ∂ξ ∂B ∂η        24 / 32
  • 36. Two-level training strategy For [x1, . . . , xn] ∈ Sgeo, [f1, . . . , fn] ∈ Srhs, [g1, . . . , gn] ∈ Sbcond do For a batch of randomly sampled (ξk, ηk) ∈ [0, 1]2 (or the Greville abscissae) do Train IgANet       x1 . . . xn    ,    f1 . . . fn    ,    g1 . . . gn    ; (ξk, ηk) Nsamples k=1    7→    u1 . . . un    EndFor EndFor 25 / 32
  • 37. Computational costs Working principle of PINNs x 7→ u(x) := NN(x; f, g, G) = σL(WLσ(. . . (σ1(W1x + b1))) + bL) • use AD engine (automated chain rule) to compute derivatives, e.g., ux = NNx • use AD engine on top of AD tree (!!!) to compute gradients w.r.t. weights for training 26 / 32
  • 38. Computational costs Working principle of PINNs x 7→ u(x) := NN(x; f, g, G) = σL(WLσ(. . . (σ1(W1x + b1))) + bL) • use AD engine (automated chain rule) to compute derivatives, e.g., ux = NNx • use AD engine on top of AD tree (!!!) to compute gradients w.r.t. weights for training Working principle of IgANets [xi, fi, gi]i=1,...,n 7→ [ui]i=1,...,n := NN(xi, fi, gi, i = 1, . . . , n) • use mathematics to compute derivatives, e.g., ∇xu = ( Pn i=1 ∇ξBi(ξ)ui) J−t G • use AD to compute gradients w.r.t. weights for training, i.e. (illustrated in 1D) ∂(dr ξu(ξ)) ∂wk = n X i=1 ∂(dr ξbp i ui) ∂wk = XXXXXXXX X n X i=1 dr+1 ξ bp i ∂ξ ∂wk ui + n X i=1 dr ξbp i ∂ui ∂wk 26 / 32
  • 39. Towards an ML-friendly B-spline evaluation Major computational task (illustrated in 1D) Given sampling point ξ ∈ [ξi, ξi+1) compute for r ≥ 0 dr ξu(ξ) = h dr ξbp i−p(ξ), . . . , dr ξbp i (ξ) i · [ui−p, . . . , ui] | {z } network’s output Textbook derivatives dr ξbp i (ξ) = (p − 1)   −dr−1 ξ bp−1 i+1 (ξ) ξi+p − ξi+1 + dr−1 ξ bp−1 i (ξ) ξi+p−1 − ξi   with bp i (ξ) = ξ − ξi ξi+p − ξi bp−1 i (ξ) + ξi+p+1 − ξ ξi+p+1 − ξi+1 bp−1 i+1 (ξ), b0 i (ξ) = ( 1 if ξi ≤ ξ ξi+1 0 otherwise 27 / 32
  • 40. An ML-friendly B-spline evaluation Algorithm 2.22 from (Lyche and Morken 2011) with slight modifications 1 b = 1 2 For k = 1, . . . , p − r 1 t1 = (ξi−k+1, . . . , ξi) 2 t21 = (ξi+1, . . . , ξi+k) − t1 3 mask = (t21 tol) 4 w = (ξ − t1−mask) ÷ (t21−mask) 5 b = [(1 − w) ⊙ b, 0] + [0, w ⊙ b] 3 For k = p − r + 1, . . . , p 1 t1 = (ξi−k+1, . . . , ξi) 2 t21 = (ξi+1, . . . , ξi+k) − t1 3 mask = (t21 tol) 4 w = (1−mask) ÷ (t21−mask) 5 b = [−w ⊙ b, 0] + [0, w ⊙ b] where ÷ and ⊙ denote the element-wise division and multiplication of vectors, respectively. 28 / 32
  • 41. Performance evaluation - bivariate B-splines re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 10−1 100 101 102 103 104 105 106 Wallclock time in ns/entry Tesla V100S PCIe 32G AMD EPYC 7402 24-Core Processor reference p = 1 p = 2 p = 3 p = 4 p = 5 29 / 32
  • 42. Performance evaluation - bivariate B-splines re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 re f 1 e 0 1 e 1 1 e 2 1 e 3 1 e 4 2 .5 e 4 5 e 4 1 e 5 2 .5 e 5 5 e 5 1 e 6 10−1 100 101 102 103 104 105 106 Wallclock time in ns/entry Fujitsu A64FX 48-Core Processor AMD EPYC 7402 24-Core Processor reference p = 1 p = 2 p = 3 p = 4 p = 5 30 / 32
  • 43. Let’s move on to the front-end 31 / 32
  • 44. The future of computational steering: Interactive Design-through-Analysis Matthias Möller1 , Casper van Leeuwen2 1Department of Applied Mathematics, TU Delft 2Scientific Visualisation, HPCV, SURF SURF Research Day 2023, Amersfoort Joint work with Deepesh Toshniwal, Frank van Ruiten (TU Delft), Paul Melis (SURF), and Jaewook Lee (TU Vienna) Thank you very much! 32 / 32