Science & Technology Development Journal – Engineering and Technology, 4(1):721-729
Open Access Full Text Article Research Article
Impact force analysis using the B-spline material point method
Vay Siu Lo1,2,*, Nha Thanh Nguyen1,2, Minh Ngoc Nguyen1,2, Thien Tich Truong1,2,*
ABSTRACT
In the MPM algorithm, all the particles are formulated in a single-valued velocity field hence the
non-slip contact can be satisfied without any contact treatment. However, in some impact and
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artphone to scan this penetration problems, the non-slip contact condition is not appropriate and may even yield unrea-
QR code and download this article sonable results, so it is important to overcome this drawback by using a contact algorithm in the
MPM. In this paper, the variation of contact force with respect to time caused by the impact is in-
vestigated. The MPM using the Lagrange basis function, so causing the cell-crossing phenomenon
when a particle moves from one cell to another. The essence of this phenomenon is due to the
discontinuity of the gradient of the linear basis function. The accuracy of the results is therefore also
affected. The high order B-spline MPM is used in this study to overcome the cell-crossing error.The
BSMPM uses higher-order B-spline functions to make sure the derivatives of the shape functions are
1Department of Engineering Mechanics, continuous, so that alleviate the error. The algorithm of MPM and BSMPM has some differences in
Faculty of Applied Sciences, Ho Chi defining the computational grid. Hence, the original contact algorithm in MPM needs tobemodi-
Minh City University of Technology, fied to be suitable in order to use in the BSMPM. The purpose of this study is to construct asuitable
Vietnam. contact algorithm for BSMPM and then use it to investigate the contact force caused by impact.
2 Some numerical examples are presented in this paper, the impact of two circular elastic disks and
Vietnam National University Ho Chi the impact of a soft circular disk into a stiffer rectangular block. All the results of contact force ob-
Minh City, Vietnam.
tained from this study are compared with finite element results and perform a good agreement,
Correspondence the energy conservation is also considered.
Key words: BSMPM, contact algorithm, contact force, impact, MPM
Vay Siu Lo, Department of Engineering
Mechanics, Faculty of Applied Sciences,
Ho Chi Minh City University of
Technology, Vietnam.
the Convected Particle Domain Interpolation (CPDI)
Vietnam National University Ho Chi Minh INTRODUCTION
12
City, Vietnam. The material point method (MPM) was first devel- was introduced by Sadeghirad et al. . Zhang et
Email: losiuvay@hcmut.edu.vn oped in 1994 by Sulsky and his colleagues 1. Over al. modified the gradient of shape functions to en-
hance the MPM 13. Steffen et al. introduced the B-
Correspondence 25 years of development, the number of researchers
spline MPM (BSMPM) 14 by applying the high order
Thien Tich Truong, Department of working on it is increasing more and more. Many uni-
Engineering Mechanics, Faculty of versities and institutes around the world have investi- B-spline function into MPM algorithm. The BSMPM
Applied Sciences, Ho Chi Minh City gated this method, such as Delft University of Tech- is then further improved by Tielen et al. 2, Gan et al. 15,
University of Technology, Vietnam.
nology 2, Stuttgart University 3, Cardiff University 4. Wobbes et al. 16.
Vietnam National University Ho Chi Minh
City, Vietnam. The MPM uses both Lagrangian description and Eu- In the MPM algorithm, a single-valued velocity field is
lerian description 1 so it has the advantages of both
Email: tttruong@hcmut.edu.vn used for all particles so the non-slip contact condition
descriptions. MPM has been widely used to simu- between two bodies is satisfied automatically 1. How-
History late high-velocity problems such as impact 5 and ex-
• Received: 18-11-2020 ever, in some impact and penetration problems, the
plosion 6, large deformation problems 7, fracture 8 and
• Accepted: 11-3-2021 non-slip contact condition is not appropriate, so it is
also Fluid-Structure Interaction 9.
• Published: 30-3-2021 important to develop a contact algorithm for MPM.
However, the original MPM has a major shortcom-
DOI : 10.32508/stdjet.v4i1.794 York et al. proposed a simple contact algorithm for
ing that affects the simulation results. When a par-
MPM 17, Bardenhagen et al. proposed an algorithm
ticle moves across a cell boundary, it will lead to nu-
for multi-velocity field 18, and many other improve-
merical errors due to the discontinuity of the gradi-
ments can be mentioned as Hu and Chen 19, Huang et
ent of the basis functions 1. This is called the “cell-
Copyright 20 21 22
2 al. , Nairn , Ma et al. .
© VNU-HCM Press. This is an open- crossing error” . In order to alleviate the effect of
access article distributed under the this phenomenon, different methods were proposed. This study using the BSMPM to mitigate the cell-
terms of the Creative Commons Bardenhagen et al. proposed the Generalized In- crossing error. The BSMPM and MPM have differ-
Attribution 4.0 International license. terpolation Material Point method (GIMP) 10. Vari- ences in computational grid definition. Therefore, the
ants in the GIMP branch were also introduced, Stef- contact algorithm for MPM cannot be directly applied
fen et al. proposed the Uniform GIMP (uGIMP) 11, to BSMPM. In this paper, the contact algorithm is
Cite this article : Lo V S, Nguyen N T, Nguyen M N, Truong T T. Impact force analysis using the B-spline
material point method. Sci. Tech. Dev. J. – Engineering and Technology; 4(1):721-729.
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Science & Technology Development Journal – Engineering and Technology, 4(1):721-729
modified to a suitable form to the BSMPM. The im-
plementation steps are mentioned in Section 2.3. The
contact force obtained from impact of two elastic ob-
jects are compared with the result from FEM, a slight
difference between FEM and MPM (and BSMPM) re-
sults is observed and explained in Section 3.
METHODOLOGY
B-spline basis functions
Considering a vector containing non-decrease val-
ues X=fx1;x2;:::;xn+d;xn+d+1g, where n is the num-
ber of basis functions, d is the polynomial order.
Each value in this vector is called knot and satis-
fies the relation x1 ≤ x2 ≤ ::: ≤ xn+d ≤ xn+d+1.
Vector X contains a sequence of knots is called the
knot vector 2. The B-spline basis functions are con-
structed by a knot vector. A uniform knot vector is a
knot vector containing equally distributed knots, e.g
X=f0;1;2;3;4;5g is a uniform knot vector. From
the relation of the knots sequence, one notices that
the value of adjacent knots can be repeated, if x1 and
xn+d+1 are repeated d+1 times, it is an open knot vec-
tor 2, e.g X=f0;0;0;1;2;3;4;5;5;5g is an open knot
vector with n = 7 and d = 2.
The i-th B-spline basis function of order d (Ni;d) is
defined by using Cox-de Boor recursion formula 15.
Firsly, the zeroth order (d=0) basis function must be Figure 1: (a) Quadratic B-spline basis functions (d
defined X =
( = 2) built from an open uniform knot vector
f0;0;0;0:5;1;1;1g and (b) Cubic B-spline basis func-
1 i f xi ≤ x ≤ x
N = i+1 (1) tions (d = 3) defined by X = f0;0;0;0:5;1;1;1g
i;0 0 otherwise
the non-zero intervals [xi; xi+1) are called knot
2 ≥
spans .After obtaining Ni;0, higher order (d i ) ba- where p and q are the order of the univariate basis
sis functions are defined as the formula below function.
x − xi Two important properties of B-spline basis functions
Ni;d (x) = Ni;d−1 (x)
xi+d − xi are: they are non-negative for all x and the functions
x − x (2) n
i+d+1 x have the partition of unity property, i.e. ∑ Ni;d =
+ x − x Ni+1;d−1 ( ) i=1
i+d+1 i+1 1 15.
in which the fraction 0/0 is assumed to be zero. Fig-
ure 1 shows the high order B-spline basis functions B-spline Material Point Method
(d=2, d=3). In 2D BSMPM, the computational domain is dis-
15
The derivatives of basis function Ni;d fxg are calcu- cretized by a parametric grid . This grid is de-
lated as following fined by two open{ knot vectors on two} orthogo-
X = x ;x ;:::;x ;x I =
dN (x) {nal directions 1 2 } n+p n+p+1 and
i;d d x
= Ni;d−1 ( ) h1;h2;:::;hm+q;hm+q+1 as shown in Figure 2. The
dx xi+d − xi
d (3) numbers of basis functions in x and h direction are n
− Ni+1;d−1 (x)
xi+d+1 − xi+1 and m, respectively, so the total number of basis func-
tions is n × m. A tensor product grid with the total
In two dimensions, the bivariate B-spline functions
of n × m nodes is constructed as shown in Figure 3,
can be built from the tensor product of the univari-
each node of this grid corresponds to one B-spline ba-
ate ones 8
sis function as defined in Eq. (4). For example, the
Ni; j (x;h) = Ni;p (x)Nj;q (h) (4) node with the position (1, 3) on the grid corresponds
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Science & Technology Development Journal – Engineering and Technology, 4(1):721-729
to the basis function N1;3 (x;h) = N1;p (x)N3;q (h). The figure also shows a particle located in the upper-
All the nodes on this tensor product grid are defined middle cell, so this particle is mapped to [7, 8, 9, 12,
as control points in BSMPM (the same role for grid 13, 14, 17, 18, 19].
node in MPM), and in practice these control points
are arbitrary distribution 15.
Figure 4: A quadratic (d=2) BSMPM grid
Unlike the original MPM, the particles in BSMPM are
considered in the whole discretized domain, instead
Figure 2: A 2D parametric grid constructed from of a specific cell, as shows in the equation below 23
two open knot vectors X and I
x − x
x = min ;
xmax − xmin
y − y (5)
h = min
ymax − ymin
where (xmin, ymin) is the lower-left control point and
(xmax, ymax ) is the upper-right control point. This is
the formula for mapping between the parameter space
to the physical space.
The derivatives of the B-spline basis functions are
given as below 23
2 3
[ ] [ ] ¶x ¶x [ ]
¶N ¶N 6 ¶x ¶y 7 ¶N −
= 4 5 = J 1 (6)
¶x ¶x ¶h ¶h ¶x x
¶x ¶y
where J is the Jacobian matrix and defined by
2 3
¶x ¶x
6 ¶x ¶h 7
= 6 7
Figure 3: A tensor product grid containing n × m J 4 ¶y ¶y 5 (7)
nodes (control points) ¶x ¶h
and the components are computed as
As shown in Figure 4 a second order (quadratic) ¶x ¶N (x)
BSMPM grid. The cell is made from 3 knot spans in x ¶x = ∑ PA ¶x (8)
A=1
direction and 2 knot spans in y direction, so the num-
ber of knots in knot vectors are 4 and 3, respectively. where P denotes the coordinates of the control points
The number of control points in x direction is 3 (knot and A is the global index of control point 23.
spans) + 2 (order) = 5 and in y direction is equal 4. In the BSMPM, for convenient the knot
These control points play the role of grid nodes in the vector for an interval [0;L] is defined by
original MPM, the knots from knot vectors are only X = f0;:::;0;4x;24x;:::;L − 4x;L;:::;Lg,
used for creating a computational grid. At can be seen where 4x denotes the length of knot span 15.
in Figure 4, each cell has 9 control points, for example, And note that the knot vector must be normal-
the lower-left cell related to [1, 2, 3, 6, 7, 8, 11, 12, 13]. ized before a parametric grid is created, so the
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Science & Technology Development Journal – Engineering and Technology, 4(1):721-729
knotn vector is rewritten as the followingo form where GI is the derivatives of the B-spline basis func-
X0 4x 24x L−4x
= 0;:::; L ; L ;:::; L ;1;:::;1 , this is also tions.
a difference in the parameter space between B-spline Before applying into Eq. (9) for checking contact, the
basis functions and other functions. The control normal vector in Eq. (13) must be normalized 23
points are arbitrary distributed and they are in the i
i nI
physical coordinate (x, y). nI = i (14)
nI
Contact algorithm The implementation of contact algorithm into the
This section presents the algorithm proposed by Bar- BSMPM algorithm can be summarized as following
denhagen et al. 18 and makes appropriate modifica- steps:
tion to apply into the BSMPM. Step 1: Mapping data from particles to control points
When two bodies are approaching each other, there 1. Compute the mass of( I-th) control point from the
is a region where they have some of the same control i;t ∑ i;t i
i-th body: mI = p NI xp Mp
points. These control points are viewed as the con- 2. Compute the momentum( of I-th) control point from
tact points, the contact algorithm is applied on these i;t ∑ i;t i
the i-th body: pI = p NI xp (Mv)p
points only.
18 3. Compute external force at control point I from i-th
In the contact region, the following equation is used ext;i;t
body: f
as a condition to check if two bodies are in contact or I
4. Compute internal force at control( point) I from i-th
release
int;i;t −∑ i;t s i;t ∇ i;t
( body: fI = p Vp p NI xp
( ) ≥
i;t − cm i 0 contact i;t
vI vI :nI = (9) 5. Compute the total force at control point I: fI =
< 0 release ext;i;t int;i;t
fI + fI
where i denotes the i-th body in the computational Step 2: Update the control point momentums:
cm 23 i;t+4t i;t i;t 4
domain, vI is the center-of-mass velocity of the pI = pI + fI t
control point I-th for each pair in contact Step 3: Imposed boundary conditions at specific con-
1;t+4t 2;t+4t trol points (if needed)
cm pI + pI
vI = (10) Step 4: Contact force calculating (for contact points
m1;t + m2;t
I I only).
i
In Eq. (9), nI is the normal vector of control point I-th 1. Calculate the normal vector from Eq. (14)
of body i-th and computed as following steps. 2. Calculate the center-of-mass velocity using Eq.
Firstly, the density rc for each cell in contact state is (10)
computed as below 23 3. Check the contact condition in Eq. (9)
np ( ) If two body are in contact, continue sub-step 4 and 5.
ri 1 i 2 i − i
c = ∑ mpS xp xc (11) If not, move to Step 5.
Ve p=1
4. Compute contact( force at contact) control points I:
contract;i;t mi;t cm;t i;t
where Ve is volume of cell e-th, xc is the center of cell I −
fI = 4ti vI vI
e-th. Remember that in the BSMPM each cell is made 5. Correct the control point momentums:
of knot spans (see Figure 4). correct;i;t i;t+4t contract;i;t
fI = fI + fI
x
In 1D, the function S (x) is given by the following def- Step 5: Mapping data from control points to particles
inition 23 i;t+4t i;t
1. Update particle( velocities:) vp = vp +
4 ( )
Sx (x) t ∑ N xt f i;t + f contact;i;t
8 mi;t I I p I I
> 1 3 9 3 1 I
> x2 + x + ; − ≤ x ≤ − i;t+4t i;t
> 2h2 2h 8 2h 2h 2. Update( ) particle positions: xp = xp +
1 3 1 1 4t ∑ t correct;i;t+4t
− x2 + ; − ≤ x ≤ (12) i;t I NI xp pI
2 mI
= > h 4 2h 2h
> 1 2 3 9 1 3 3. For MUSL only, get control point velocities:
> − ≤ ≤ i;t+4t correct;i;t+4t i;t
> 2 x x + ; x
:> 2h 2h 8 2h 2h vp = pI =mI
; i;t+4t
0 otherwise 4. Compute( ) particle gradient velocity: Lp =
∑ ∇ i;t i;t+4t
The function S2 (x;y) in Eq. (11) is obtained by mul- I NI xp vI
tiplying two 1D functions S2 (x;y) = Sx (x)Sy (y): 5. Update( particle gradient) deformation tensor:
i;t+4t i;t+4t i;t
Finally, the normal vector of control point I-th is ob- Fp = I + Lp 4t Fp
tained 23 i;t+4t
6.( Update) particle volume: Vp =
( ) 4
i ∑ i ri det Fi;t+ t V i;0
nI = c GI xc c (13) p p
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7. ( Compute) strain increment: 4ep =
i;t+4t
sym Lp 4t, then compute the stress in-
crement 4sp.
i;t+4t i;t
8. Update particle stresses: sp = sp + 4sp
Then, reset the computational grid and move to the
next time step.
RESULTS
Two numerical examples are presented in this section,
particularly:
• Collision of two circular disks.
• Collision of a circular disk onto a rectangular
block. Figure 5: Impact of two circular disks.
The first example investigates the contact of two cir-
cular surface with the same material. The second ex-
ample studies the contact of a soft circular surface and
hard flat surface.
To validate the results from these two examples, cor-
responding FEM models are created from ABAQUS
software. FEM model is prepared with very fine mesh
and set up with the same parameters and initial con-
ditions as MPM (and BSMPM) model.
Collision of two circular disks
The problem is shown in Figure 5, two elastic disks
with the same radius R = 0:2 m and the thickness is
one unit. The material properties used in this prob- Figure 6: Kinetic and strain energy.
lem are: Young’s modulus E = 1000 Pa, Poisson ra-
3
tio v = 0:3, and the mass density r0 = 1000 kg=m .
The coordinate of the center of the lower-left disk is loss from friction, the strain energy loss in BSMPM is
(0:2; 0:2), the upper-right disk is (0:7; 0:7), two disks caused by other error factors.
× 2
are in a square domain of size 0:9 0:9 m . The initial The variation of contact force during the impact pro-
velocities of the particles v = (0:1; 0:1) m=s, for the cess is shown in Figure 7. The FEM model used to
upper-right disk, the velocities of the particles are set simulate this problem has 3288 nodes. The results
−
to vp = v and for the lower-left vp = v. from MPM and BSMPM show that the impact of two
The computational domain is discretized into 40×40 bodies occurs earlier than the result in FEM as men-
knot spans. Each computational cell has 9 particles. tioned before. This is because the contact force in
The original MPM with Lagrange basis and quadratic MPM is computed in the node of the computational
BSMPM (d=2) are concerned in this example. grid (or control point in BSMPM), not in the particle
The time step for this simulation is chosen as 4t = of the body, so when two bodies approach the contact
0:001 s, the total simulation time is 3 s. So, there is region and have the same control points, the contact
3000 steps in this simulation. is detected immediately although two bodies have not
The kinetic and strain energy obtained from BSMPM touched each other yet. In FEM, the contact is only
and FEM is shown in Figure 6. Kinetic energy in detected when two bodies touch each other, so the
BSMPM decreases earlier than the result from FEM contact force obtained in FEM is later than MPM.
and strain energy in BSMPM increases earlier. This is The contact force obtained from BSMPM using higher
reasonable for the contact in BSMPM algorithm and order B-spline functions also shows the smooth curve
will be explained in the comment of Figure 7. The compared to the MPM and FEM.
value of kinetic energy in both case are the same, while Figure 8 shows the von-Mises stress field during the
the strain energy in BSMPM is lower than FEM. Both impact process of two disks using the BSMPM. In de-
case are in frictionless contact, so there is no energy tail, two disks approaching each other in Figure 8 (a),
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Science & Technology Development Journal – Engineering and Technology, 4(1):721-729
Figure 7: Impact force obtained from FEM, MPM
and BSMPM (d=2) Figure 9: A circular disk collides with a rectangular
block.
then two disks touch each other as shown in Figure 8
(b), two disks deform during the impact as shown in
Figure 8 (c) and then bounce back in Figure 8 (d). Af-
ter impact, two disks move far away as shown inFig-
ure 8 (e).
Collision of a circular disk onto a rectangu-
lar block
In this example, a circular disk collides onto a stiffer
rectangular block, as shown in Figure 9.
The radius of the circular disk is R = 0:2 m, and the
thickness is one unit. The material properties used
for circular disk are: Young’s modulus E1 = 1000 Pa, Figure 10: Impact force of example 5.2 obtained
Poisson ratio v1 = 0:3, and the mass density r1 = from FEM, MPM and BSMPM (d=2)
1000 kg=m3. The rectangular block is made from
6
stiffer material with Young’s modulus E2 = 10 Pa,
r
Poisson ratio v2 = 0:3, and the mass density 1 = FEM and MPM.
3 × 2
5000 kg=m , the rectangular size is 1 0:2m . Dis- Figure 11 shows the collision of two objects, the von-
tance between the center of the circular disk to the top Mises stress field and maximum stress field are pre-
of rectangular block is 0:3 m. The computational do- sented.
× 2
main is a square with dimension of 1:2 1:2 m . The To investigate the convergence of BSMPM and MPM,
−
initial velocity of the disk is v = (0; 0:2) m=s. In this the computational domain with a set of 60 × 60 knot
simulation, the gravitational acceleration is ignored. spans is retained. Different numbers of particles per
The computational domain is discretized into a set of cell (PPC) 4, 9 and 16 are analyzed. Figure 12 shows
×
60 60 knot spans. Each cell has 9 particles. The the total energy of the system respect to time. From
nodes (or control points) on the bottom line of the the initial conditions, the total energy can be com-
rectangular is fixed in two direction x and y. puted as rpR2tv2=2 = 2:512 J and plotted by the
The time step size is chosen as 4t = 0:001 s, and the black line in the figure. As shown in Figure 12, the
total simulation time is 2 s. So, there is 2000 steps in case of MPM with PPC = 4 gives a very large devi-
this simulation. ation, and when PPC = 9, the result is significantly
The contact force obtained in this example also shows improved. In the case of BSMPM, there are no signif-
the similarity to the conclusions from the previous ex- icant deviations and the results are slightly improved
ample. Figure 10 also shows that the impact occurs when increasing PPC.
earlier in BSMPM, because BSMPM has more control
points (nodes) than MPM so the contact is detected DISCUSSIONS
earlier. Similarly to the previous example, the con- As present in Section 3, there is a slight difference
tact force in BSMPM is smoother than the curve from in the results of MPM, BSMPM and FEM. The mag-
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Science & Technology Development Journal – Engineering and Technology, 4(1):721-729
Figure 8: Impact of two circular disks.
Figure 12: Energy of the system respect to time with
different number of particles per cell.
two objects have not touched each other yet, there is
still a gap. And because BSMPM uses higher-order
shape functions, so the BSMPM has more control
points (grid node) than MPM, the contact is therefore
detected earlier. Moreover, if the contact algorithm
is not used in MPM, the non-slip contact can still be
Figure 11: Circular disk deforms during the impact determined automatically when two objects have the
process onto a stiffer surface. (a) von-Mises stress same grid node. In FEM, the contact is only detected
and (b) Maximum stress.
when two objects touch each other (or even penetrate
into each other), so the contact force obtained in FEM
is later than MPM and BSMPM.
nitude of the contact force in the three methods is
the same, but the impact occurs earlier in MPM and CONCLUSIONS
BSMPM. This is because the contact force in MPM is The contact algorithm has been successfully modi-
computed in the node of the computational grid (or fied and applied into the BSMPM. The contact force
control point in BSMPM), not in the discrete parti- obtained from this research is compared to FEM. A
cle of the object, so when two objects approach the slight difference in the result is observed, this is be-
contact region and have same the grid node (control cause the contact force is calculated at the control
points), the contact is detected immediately although point in the computational grid instead of the discrete
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Science & Technology Development Journal – Engineering and Technology, 4(1):721-729
particles. This is an inherent property of the MPM al- 8. Nairn JA. Material point method calculations with ex-
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Tạp chí Phát triển Khoa học và Công nghệ – Kĩ thuật và Công nghệ, 4(1):721-729
Open Access Full Text Article Bài Nghiên cứu
Phân tích lực va chạm bằng phương pháp Điểm vật liệu sử dụng
hàm dạng B-spline
Lồ Sìu Vẫy1,2,*, Nguyễn Thanh Nhã1,2, Nguyễn Ngọc Minh1,2, Trương Tích Thiện1,2,*
TÓM TẮT
Trong giải thuật MPM, các điểm vật liệu được xây dựng trong một trường vận tốc đơn trị nên sự
tương tác/tiếp xúc không trượt giữa

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