Stochastic Wave Dynamicsand Uncertainty Quantification
Allan P. Engsig-Karup, Daniele Bigoni and Stefan L.
GlimbergTechnical University of Denmark (DTU)
MotivationTo address challenges in reliable prediction of
extremeevents for the design and safe operation of marinesystems,
simulation-based engineering tools can beused. Such tools are
increasingly cost-efficient and canbe used to quantify
uncertainties and evaluate impacthereof in critical engineering
design problems wheremeasurements are infeasible, impractical or
too costly.To characterise uncertainties in wave dynamics
andwave-structure responses our objective is to considermodern
spectral techniques for uncertainty quantifica-tion to describe
stochastic properties as accurately aspossible in practical times.
The spectral techniques pro-vides the basis for meeting the
accuracy requirementsince these techniques may achieve much faster
conver-gence rates than conventional techniques. To make
ourapproaches practical we seek to combine knowledge inmodern
algorithms and many-core hardware technolo-gies in a framework to
enable efficient stochastic hydro-dynamics calculations. Our scope
is relevant for com-putationally intensive (fx. large scale
problems) wheremany simulations are intractable by conventional
tech-niques and approaches.
Case studiesIn a first preliminary step, we revisit some
classicalbenchmarks for applications and investigate feasibilityof
using spectral techniques for quantification of uncer-tainty in
wave dynamics. Possible uncertainty sourcesare assumed to beI
Boundary conditions (wave generation signal, e.g.
wave period and wave amplitude).I Bathymetry function (sea bed),
h(x, ω).
ContributionsI Stochastic formulation of fully nonlinear and
dispersive wave equations.I Investigation of spectral
uncertainty quantification
techniques.I Integration of our research in fast
hydrodynamics
simulations with spectral uncertainty
quantificationtechniques.
Figure 1 : Snap shot of deterministic wave fieldproduced by
Berkhoff experiment.
Deterministic formulationTo describe nonbreaking irrotational
ocean waves, a fullynonlinear and dispersive water wave is used.
Dynamicand kinematic free surface boundary conditions are
∂tζ = −∇ζ · ∇φ̃+ w̃(1 +∇ζ · ∇ζ),
∂tφ̃ = −gζ −1
2
(∇φ̃ · ∇φ̃− w̃2(1 +∇ζ · ∇ζ)
),
where φ̃ = φ(x, ζ, t), ζ(x, t) and w̃ = ∂zφ|z=ζ arefree surface
quantities and g gravitational acceleration.A Laplace problem needs
to be solved
φ = φ̃, z = ζ(x, t),
∇2φ+ ∂zzφ = 0, −h ≤ z < ζ(x, t),∂zφ+∇h · ∇φ = 0, z = −h.
from which closure is obtained by (u, w) = (∇, ∂z)φ.
Stochastic formulationTo enable quantification of uncertainties,
a stochasticformulation is obtained by introducing ω ∈ Ω as ran-dom
input of the system defined in the probability space(Ω,F ,P), where
Ω is the sample space, F is a σ-fieldand P is a probability
measure. This makes the solutiona random quantity ζ(x, t, ω) : D̄FS
× [0, T ] × Ω → Rand φ(x, t, ω) : D̄ × [0, T ] × Ω → R. D̄ is the
closedspatial domain volume with FS indicating the restrictionto
the free surface, D̄ = {x|x ∈ ξ}. A parametrizationof the
stochastic model is required in order to solve it nu-merically. A
random vector Z : Ω→ Rd, is introduced tocharacterise random
inputs, where d ≥ 1 the stochasticdimension. The stochastic
formulation is then
∂tζ(x, t,Z) = −∇ζ · ∇φ̃+ w̃(1 +∇ζ · ∇ζ),
∂tφ̃(x, t,Z) = −gζ −1
2
(∇φ̃ · ∇φ̃− w̃2(1 +∇ζ · ∇ζ)
),
where for any (random) sea state, the Laplace problemis
fulfilled to obtain closure. This is a stochastic systemwhere
unknown variables are random processes.
Generalized Polynomial ChaosWe use generalized Polynomial Chaos
(gPC) to createsurrogate functions of stochastic variables of the
form
f(z) ≈ f̃(z) = PNf(z) =N∑i=0
f̂iΦi(z), f̂i =(f,Φi)ρz‖Φi‖ρz
.
From these, cheap and exponentially accurate statisticsfor
uncertainty quantification can be obtained
E[f(z)] ≈ E[f̃(z)] = f̂0,
Var[f(z)] ≈ Var[f̃(z)] =N∑i=1
f̂2i ‖Φi‖2ρz.
The unknown gPC expansion coefficients are deter-mined from a
solution ensemble by forward propagationof uncertainties via a
stochastic collocation method.
Massively Parallel ComputingTo enable fast analysis and
resolution of large mar-itime areas, we take advantage of
distributed massivelyparallel high-performance and heterogenous
computingon modern many-core hardware. A massively paral-lel solver
has been prototyped in our in-house GPU-Lab library and can be used
for stochastic wave dy-namics calculations. The fast model enable
accelerationof sampling-based (non-intrusive) UQ algorithms in
thefield of study.
Figure 2 : Absolute run timings in single precision
forheterogenous multi-GPU configurations as a function ofnumber of
grid points for iterative PDC method.I Efficient multigrid
Preconditioned Defect Correction
(PDC) method for arbitrary-order discretizations.I Minimal
memory requirements via short recurrence
iterative PDC method, matrix-free stencilsimplementations of
sparse operators and single ormixed-precision calculations.
I Fast massively parallel execution on hardwaresystems of
arbitrary size ranging from desktops tosuper clusters via hybrid
MPI-CUDA.
I Fault tolerance and resilience via robust multileveliterative
methods.
I Predictable and scalable performance.
Discretization MethodsTo develop a numerical model we useI
Tuneable numerics provides tradeoffs between
accuracy and efficiency.I A flexible-order boundary-fitted
Finite Difference
Method in space.I Multigrid Preconditioned Defect Correction
(PDC)
Method for efficient and scalable iterative solution ofLaplace
problem every Runge-Kutta stage.
I Data-parallel domain decomposition methodimplemented for
distributed computations.
For the time discretization we useI An explicit fourth-order
Runge-Kutta method in time.
Due to bounded operator eigenspectra conditionalCFL stability
without strict step size penalisation dueto high-order numerics
and/or refined grids.
I A parallel in time (Parareal) discretization to
introducealgorithmic concurrency.
For the stochastic discretization we useI A spectral stochastic
collocation method for forward
propagation of parametric uncertainty in input data.
Numerical results
Figure 3 : Uncertainty quantification of harmonicscontributions
to the steady one-wave period solution inWhalin experiment (T = 2s)
with respect to waveheight and wave period. The shaded areas show
onestandard deviation from the mean (full lines).
PerspectivesI Promising practical aspects for spectral
uncertainty
quantification techniques in maritime engineering
forlow-dimensional stochastic problem.
I Speedup solutions via parallel computations tosignificantly
improve analysis in practical times (butdoes not resolve curse of
dimensionality).
I Stochastic simulation and uncertainty quantificationbecoming
increasingly important for reliable analysisof impact of
uncertainties on engineering designs.
I Next steps: better predictions of wave statistics inlarge
(near-costal) finite depths areas and reliableestimations of
extreme events.
ReferencesI Engsig-Karup, A. P., Glimberg, L. S., Nielsen, A.
S.
and Lindberg, O. 2013. Fast hydrodynamics onheterogenous
many-core hardware. Part of: RaphäelCouturier (Ed). Designing
Scientific Applications onGPUs, 2013, CRC Press / Taylor &
Francis Group.
I Engsig-Karup, A. P., Madsen, M. G. and Glimberg, S.L. A
massively parallel GPU-accelerated model foranalysis of fully
nonlinear free surface waves.E-published in International Journal
for NumericalMethods in Fluids, July, 2011.
I Engsig-Karup, A.P., Bingham, H.B. and Lindberg, O.2009 An
efficient flexible-order model for 3D nonlinearwater waves. Journal
of Computational Physics, 288,pp. 2100–2118.
Contacts: Allan: [email protected], Daniele: [email protected], Stefan:
[email protected]
