Colour-Field Based Particle Categorization for Residual Stress
Detection and Reduction in Solid SPH Simulations
Gizem Kayar
a
Computer Science Department, New York University, 251 Mercer Street, New York, U.S.A.
Keywords: Smoothed Particle Hydrodynamics, Residual Stress, Von-MissesYield, Physically-Based Simulations,
Solid Simulations.
Abstract: Residual stress remains in an object even in the absence of external forces or thermal pressure, which, in turn,
may cause significant plastic deformations. In case the residual stress creates unwanted effects on the material
and so is undesirable, an efficient solution is necessary to track and eliminate this stress. Smoothed Particle
Hydrodynamics has been extensively used in solid mechanics simulations and the inherent colour-field
generation approach is a promising tracker for the residual stress. In this paper, we propose a way to use the
colour-field approach for eliminating the residual stress and prevent the undesirable premature failure of solid
objects.
1 INTRODUCTION
Residual stresses describe any stress remain in the
material even after the external forces and thermal
effects are excluded from the environment. These
stresses may originate from many causes, e.g. cooling
rates, volume changes, etc. and they may be created
by welding, rolling, forging, casting, machining, heat
treatments or surface treatments. After that point,
residual stress is generated when an object is stressed
beyond its elastic limit, resulting in plastic
deformation.
Residual stress should be managed properly for
the design, manufacturing, and maintenance phases.
Although it may be desirable in some certain
conditions, residual stress generally causes the
material to fail prematurely. It may affect the fatigue
life, stability, resistance and also brittle fractures of
objects. Elimination process of residual stresses are
extensively researched in material science.
Simulations of such conditions help the researchers to
understand the problem further and therefore, to
create a solution.
Since its introduction (Gingold & Monaghan,
1977; Lucy, 1977), Smoothed Particle
Hydrodynamics (SPH) has been accepted as one of
the major development mediums for the mechanics of
a
https://orcid.org/0000-0002-7811-9357
continuum media. Although it was introduced for
astrophysical problems, later it has been used in
various research fields including but not limited to
fluid simulations (e.g. Solenthaler & Pajarola, 2009;
Shadloo et al., 2016; Gissler et al., 2019) and solid
mechanics (e.g. Libersky & Petschek, 1990; Libersky
et al., 1993). Although SPH is still in the process of
development, the method has been drastically
improved over the years to overcome some major,
inherent problems. SPH is still one of the easy-to-use
numerical methods to model complex systems.
Residual stress has been generated and/or
determined by researchers in some SPH simulations
for different purposes, such as arc welding (Das &
Cleary, 2010), friction stir welding (Eivani et. al,
2021), etc. In our experiments, we observed that the
undesired residual stress is the major cause of
instability in SPH solid simulations.
Our contribution in this ongoing work is to
eliminate the undesired residual stress behaviour
from solid mechanics simulations by applying a
colour-field technique which is extensively used in
SPH fluid simulations. Colour-field approach helps
us to identify the particles with potential residual
stress which, in turn, are handled differently to reduce
the overall stress.
Kayar, G.
Colour-Field Based Particle Categorization for Residual Stress Detection and Reduction in Solid SPH Simulations.
DOI: 10.5220/0011716100003417
In Proceedings of the 18th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2023) - Volume 1: GRAPP, pages
237-241
ISBN: 978-989-758-634-7; ISSN: 2184-4321
Copyright
c
 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
237

Figure 1: Compressed ductile material implemented with our SPH model and visualized with pressure-based colouring.
2 RELATED WORK
SPH has started to be researched in computer
graphics for the last couple of years and still
extensively researched in material science. For
instance, Ihmsen et al. discusses the force
computation problems and visual inefficiency in
granular SPH in (Ihmsen et al, 2013). Here, they
propose to use a coarsely sampled simulation for
force computations, and couple high resolution set of
particles to base particles via up-sampling which
improves both the computation time and visual
quality. In (Nguyen et al, 2017), Nguyen et al. discuss
how to use viscous damping and stress regularisation
for granular flows. Later, Ghaitanellis et al. proposed
a new elastic-viscoplastic model for granular flows in
(Ghaitanellis et al, 2018).
In 2016, Pan et al. proposed an application where
SPH is used create solitary wave impact on offshore
platforms to understand the interaction between large
waves and offshore structures which is a good
example of SPH’s usage in civil engineering and geo-
numerics field (Pan et al., 2016). In the same year,
Holmes et al. proposed a model for grain-scale fluid
flows in porous rocks (Holmes et al., 2016).
Regarding our focus, the elimination process of
residual stresses is vital especially when the stress is
unwanted. Simulations of such conditions help the
researchers to understand the problem further and
therefore, to create better solutions.
Residual stress has been generated and/or
determined by researchers in some SPH simulations
for different purposes, such as arc welding (Das &
Cleary, 2010), friction stir welding (Eivani et. al,
2021), etc. Eivani shows a specific example on AZ91
Mg alloy where they combine SPH with neuro-fuzzy
computations and ultrasonic testing.
Another specific example comes from Saleh,
Luzin and Spencer where they analyse the residual
stress in cold spray technique using some numerical
and empirical methods (Saleh et al, 2014). Liu and
his fellow teammates also propose another technique
for the numerical simulation and elimination of
residual stress by using shot peening in (Liu et al.,
2019)
Discussing other fields, such as fluids, is beyond
the scope of this paper. However, state of the art
reports like (Koschier et al, 2022) are some of the
good resources for those who are interested in the
topic.
3 SPH
SPH is a method of discretizing spatial quantities
using a set of particles equipped with a kernel
function. Each particle is defined by a position, mass
and a support radius where mass can be computed as:
𝑚=𝜌∙𝑉 (1)
with 𝜌:fluid density and 𝑉:particle volume. The
word “smoothed” in SPH comes from the smoothing
operation which actually means calculating any
physical quantity of the particle using the weighted
sum of the same quantity of the neighboring particles
that lie in the range of a kernel function. So, after the
continuous approximation is discretized, he
smoothing operation looks as:
𝐴

=
∑
𝑚





𝑊
(
𝐱

−𝐱

,ℎ
)

(2)
where A is an arbitrary scalar quantity, x is the
position, b is the iterator over all contributing
particles and h is the smoothing length.
GRAPP 2023 - 18th International Conference on Computer Graphics Theory and Applications
238

As we stated before, contributions of the
neighbouring particles are governed by the kernel
function and can be calculated as, e.g.:
𝑊
(
𝐱

−𝐱

,ℎ
)
=
𝜎
ℎ

⎩
⎪
⎨
⎪
⎧
6𝑞

−6𝑞

+1 0≤𝑞≤
1
2
2(1−𝑞)

1
2
<𝑞≤1
0𝑞>1
(3)
𝑞=
|
𝐱

−𝐱

|
/ℎ
𝜎 varies depending on the dimensionality of the
system.
In such a system, for both fluids and solids,
equations of motion in Lagrangian perspective can be
given as:
𝑚

𝐯


=−𝑉

𝛻𝑝

+𝑉

𝜇𝛻

𝐯

+𝑉

𝐟

(4)
which, in turn, can be written as the sum of forces as:
𝑚

𝐯


=𝐅
𝐢
(5)
𝐅
𝐢
=𝐅


+𝐅


+𝐅


(6)
Here, forces can be computed using SPH
interpolation and later be integrated using one of the
explicit or implicit numerical integration schemes,
e.g. Euler-Cromer, Verlet, etc, With this approach,
thousands to billions of particles can be simulated
efficiently (see Figure 2).
4 SPH FOR SOLIDS AND
RESIDUAL STRESS
DETECTION
It is important to mention that all solid SPH
simulations in our work rely on the methodology
discussed in the previous section.
However, the SPH method should be extended so
that it may reflect the solid behaviour and yielding
criterion (see Figure 1 and Figure 3). Therefore, we
firstly implemented elastoplastic solid behaviour in
our system by integrating the momentum equation:



=






+𝑔

(7)
where
𝜎

=−𝑃𝛿

+𝑆

(8)
with pressure P, deviatoric stress tensor S. Stress can
be computed using Hooke’s Law.
At this stage, maximum distortion energy
criterion can be introduced to the system to estimate
the yield of ductile materials. We can even compare
Figure 2: Our SPH fluid simulation with 1.7 million
particles.
the material’s yield stress to Von Misses stress to
observe the its resistance and yielding thresholds.
Our main focus in this ongoing work is to
determine particles which are potentially carrying
residual stress. We therefore propose to use the colour
field approach. This approach was first proposed by
Müller et al. in (Müller et. al, 2003) for determining
surface particles in fluid flows. The same approach
helps us to categorize the solid particles. We can
identify a particle as a surface particle in two
conditions combined: 1) if it has less than a certain
number of neighbouring particles and 2) if its surface
normal n is showing more than a meaningful
threshold. Surface normal in this situation can be
computed as:
𝐧=∇𝑐

(9)
and
𝑐

(𝐩

)=
∑




𝑊(𝐩

−𝐩

,ℎ)

(10)
After identifying surface particles, we generated
one more additional layer right behind the surface
Colour-Field Based Particle Categorization for Residual Stress Detection and Reduction in Solid SPH Simulations
239

particles and we constrain those two sets to apply only
hydrostatic stress to other particles.
Figure 3: Compressed iron with pressure based particle
coloring and marching cubes based surface reconstruction.
5 RESULTS
During this approach, we observed that the stability
of the system improved significantly and we could
use larger time steps for our simulations. To be
specific, our simulation time steps has been increased
almost twice for all presented scenes. Additionally,
we could prevent all undesired crashes and undesired
forces from our system.
As an unexpected effect, we also noticed that we
could simulate brittle fractures without using an ad-
hoc damage model (see Figure 4).
6 CONCLUSIONS
In this paper, we presented a technique to improve the
stability of a solid system simulation by using
smoothed particle hydrodynamics. Based on the
conventional SPH model, we integrated elastoplastic
solid physics and used the idea of surface extraction
for isolating the force computation on certain fields.
We observed that the technique works well for both
ductile and brittle materials.
This is obviously an ongoing work and needs
further improvement. We plan to utilize more test
scenes and various comparisons for different material
properties in the future. Furthermore, we only
discussed the general stability but we did not discuss
the performance in detail. In the future, we would like
to investigate parallelization techniques for the
model.
Figure 4: Surface particle generation (green), inner layer
additional surface set particles (red) and completely fully
particles (blue) during a brittle fracture.
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240

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